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Lf 2020 stochastic_calculus_ito-i

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luc faucheux
luc faucheux

part I of an introduction to stochastic calculus part II is n PDE/SDE mapping part III on the Maxwell demon and the Ito-Stratanovitch controversy part IV is on the Langevin equation

Economy & Finance
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Luc_Faucheux_2020 Stochastic Calculus - Part I Integrals - Ito lemma – SDEs and SIEs 1
Luc_Faucheux_2020 What is this class, and what it is not Β¨ Not a formal, so please interrupt if any question Β¨ More of a pragmatic approach on how to approach stochastic calculus Β¨ I have tried as much as possible to alert when there is something to be careful about Β¨ I have also tried as much as possible to be as rigorous as possible, without getting lost in the notations, or being too formal just for the sake of being formal Β¨ Those notes are more β€œnotes of a practitioner”, and by no means I would dare to hope to substitute a robust course in stochastic calculus Β¨ Those slides originated from a class I taught in 2018 at the hedge fund DRW to their first year associate class, 40 students or so from various backgrounds. The class was a general Fixed-Income class over 8 full days. It was intense, exhilarating, and kept me on my toes the whole time Β¨ Starting from the usual Ito lemma as in most textbooks (Hull), play with it for a while, then start again from the proper formulation of the stochastic integrals 2
Luc_Faucheux_2020 How those slides came about Β¨ Those slides originated from a class I taught in 2018 at the hedge fund DRW in the great city of Chicago. I have modified them and added to them over the past couple of years. Β¨ The class was composed of 40 students or so with various backgrounds, ranging from computer science students with almost no background in Finance, to recent graduates of Masters in Math, to some graduates of the prestigious MSCF (Master of Science in Computational Finance) at Carnegie Melon University under the guidance of Steven Shreve, to some who had almost no mathematical background in options, bond math, random processes but had advanced degrees in Economy, so it was rather a tricky bunch Β¨ The class was a general Fixed-Income class over 8 full days. It was intense, exhilarating, and kept me on my toes the whole time. It pushed me to realize what I had not understood about stochastic processes for 20 years or so, because I never bothered to ask the β€œwhat if” questions and took a lot for granted out of sheer intellectual laziness Β¨ The textbook we used was Hull so you will see pages reference to this book, as we used it in class as a starting point to further explorations of the derivative pricing theory. 3
Luc_Faucheux_2020 How those slides came about - II Β¨ It is often said that the only person who learns anything out of a class is the teacher. Β¨ I hope that my DRW students got something out of it. Β¨ But I know for a fact that without those two weeks in Chicago teaching, I would not have gotten those slides off the ground, and most of the materials would still be in disparate pieces of papers flying around my desk. Β¨ Hopefully the end result is not complete garbage, at least I know that I greatly enjoyed putting those together. Β¨ So, even if the end result is not up to your standards, I am extremely grateful to DRW for having given me the opportunity to teach the associate class that summer of 2018, and even more so grateful to the students who during those two weeks, took me out of my comfort zone, and forced me to confront what I knew and what I realized I was rather ignorant of. Β¨ When you are teaching in front of a bunch of super smart people with different backgrounds for 8 days straight, there is no hiding behind the curtain Β¨ So thanks again to DRW and the 2018 associate class ! I miss you guys. 4
Luc_Faucheux_2020 How those slides came about - III Β¨ Also for those of you who have spent more than 5 minutes with me, you will have noticed that I bring the Ito-Stratonovitch controversy a lot. Β¨ First of all that is sort of a hobby of mine, ever since my PhD thesis (Appendix B) Β¨ Second, I have found it to be quite enlightening, because everyone takes Ito for granted, and then you ask the question β€œwhat if”, and that forces you to make sure that your understanding of Ito was solid. So I am using the example of Stratonovitch to really test the fact that my understanding of ITO is solid Β¨ Apologies on the Powerpoint format, there are battles with Microsoft that you cannot win (including the fact that the Solver is a Macro and not a function, as opposed to some of the earlier spreadsheets like Wingz) Β¨ At times it might feel like going down the rabbit hole, or going up the river to meet Kurtz, but (at least in my experience), I have found those discursions to be useful. Β¨ So this is not really something great about Stratonovitch, it is using Stratonovitch to convince ourselves that we understand ITO 5
Luc_Faucheux_2020 How those slides came about - IV Β¨ Again quite frankly those notes are more for me (sorry) Β¨ I had noticed that I had a number of handwritten notes flying around my desk, and when needed at times I would just rederive from scratch rather then finding the right one. Β¨ So this is trying to put in one place, in a format that I can copy/past easily, most of what I had to go through and still use. I have tried (and failed at many places) to keep the notation consistent Β¨ Again, this is from a β€œpractitioner” point of view, I have tried to be rigorous when I felt it was needed, and be less so when I felt that this was a detail that was not needed and at times would obscure the intuition. Β¨ Notations are hard to keep consistent. I have also find at times that the right notation can be illuminating, whereas the full explicit one can be cumbersome. So I have tried to adapt the notation to what was needed to grasp the concept, and at times be more careful about it to make sure that we do not fall into a trap. Β¨ I guess Godel knew that, the right notation can change completely the proof…. 6
Luc_Faucheux_2020 If you really want to master stochastic calculus. Β¨ The uncontested bible in the field of stochastic calculus for Finance. Quite dense and concise. It sometimes take me 40 pages to understand what Steven Shreve does in 2 lines. 7
Luc_Faucheux_2020 If you really want to master stochastic calculus - II Β¨ An absolute wonderful short book. The notations at times are infuriating, but an absolute must read 8
Luc_Faucheux_2020 If you really want to master stochastic calculus - III Β¨ If you are like me coming from a Physics background, this book is still relevant today, a testament to the genius of Van Kampen 9
Luc_Faucheux_2020 If you really want to master stochastic calculus - IV Β¨ You cannot ignore this book. Every sentence carries meaning, and is worth reading time and time again. 10
Luc_Faucheux_2020 If you really want to master stochastic calculus - V Β¨ Another wonderful short book. The logic is clear, concise and beautiful. The exercises are worth going through. From one of the most respected options traders in the field. He also has quite a ferocious appetite for chocolate cakes. 11
Luc_Faucheux_2020 If you really want to master stochastic calculus - VI Β¨ Quite applied. The appendix on SDEs is rather beautiful, and follows Mikosch. Great for practical applications in the field of Fixed-Income 12
Luc_Faucheux_2020 If you really want to master stochastic calculus - VII Β¨ What not to say about this book? An absolute gem. The 1900 Ph.D. thesis of Louis Bachelier (in French!) with an amazing translation by Davis and Etheridge, and some great chapters about the history of modern finance. Bachelier did it all, 80 years or so before everyone else 13
Luc_Faucheux_2020 If you really want to master stochastic calculus - VIII Β¨ I could not but not add this one here. It is a movie about the life of Vincent (Wolfgang) Doblin (Doebling) who essentially discovered Ito calculus at least 5 to 10 years before Ito in circumstances so incredible that they made a movie out of his life. Ito lemma and calculus is now usually referred to as Ito-Doblin lemma and calculus in recognition of Vincent’s incredible accomplishments 14
Luc_Faucheux_2020 If you really want to master stochastic calculus - IX Β¨ Amazing book if you are coming from a Physics background 15
Luc_Faucheux_2020 If you really want to master stochastic calculus - X Β¨ Found this book as I was almost finished with those slides, and thought about throwing them away, because this book has pretty much anything you want. Pretty heavy on the operators formalism, which is quite elegant once you get used to it, but that it a step that you need to go through 16
Luc_Faucheux_2020 What is so hard about Stochastic Calculus? Β¨ It is quite recent Β¨ It is quite cumbersome Β¨ It is not intuitive Β¨ It is incomplete Β¨ No one really knows how to do it. Β¨ This presentation is trying to strike a balance between being practical and being rigorous, so apologies for the many terms in β€œβ€, whereas a rigorous math class will define what terms like β€œstable” or β€œgot to 0” or β€œgo to infinity” or β€œconverge” much more exactly. What we will say is β€œalmost” rigorous and works in practice 99% of the time Β¨ This is more of a practitioner’s point of view on how to use (or not) stochastic calculus, as it relates to finance and Physics 17
Luc_Faucheux_2020 Stochastic Calculus is quite recent Β¨ Geometry ~ -6,000 BC (navigating looking at the stars, buildings, ships,…) Β¨ Fractions (music with scale, buildings,,) Β¨ Probabilities (~1,600 AD), Pascal triangle, combinatory analysis Β¨ Calculus (Newton, Leibniz, finite differences) (~1,600 AD) (we started shooting canons long range, Electricity, magnets, how do they work? The ICP is still asking) Β¨ Taylor Expansion (1720) Β¨ Brown (1890), Wiener (1940), Bachelier (1900), Einstein (1905),Langevin (1908), Doblin(1940), Ito (1950), Feynman-Kac (1950), Stratonovitch (1966), Black-Sholes (1972) Β¨ So let’s go a little easy on ourselves, shall we? Β¨ Quantum Stochastic Calculus (1980), random processes (diffusion) in fractal geometries 18
Luc_Faucheux_2020 Stochastic calculus is a French thing (except Gauss) Β¨ Black-Sholes might have gotten a Nobel prize in 1972, but Louis Bachelier did it all in his Ph.D. thesis in 1900 (almost) Β¨ Ito might have been known until recently for the Ito calculus and Ito lemma, but Vincent Doelin wrote it all while on the Ardennes front in World War I. This is now being recognized and some textbooks use the term β€œIto-Doeblin” instead of β€œIto” Β¨ Paul Langevin also essentially wrote the textbook on SDEs Β¨ And for the math, all you need is Laplace, Fourier, Cauchy Β¨ Taylor expansion is the only non-French, but it is essentially L’Hospital rule, so again…French… Β¨ Note: L’Hopital rule is very powerful and often overlooked. Β¨ If two functions 𝑓(π‘₯) and 𝑔(π‘₯) and are differentiable, and lim !β†’# [ $%(!) (%(!) ] exists, then Β¨ lim !β†’# [ $(!) ((!) ] = lim !β†’# [ $%(!) (%(!) ] 19
Luc_Faucheux_2020 The Ph.D. thesis of Louis Bachelier Β¨ Reading the original thesis (both in French if you can and the excellent translation by Mark Davis and Alison Etheridge) is humbling. Β¨ Without a strong well-developed theory of stochastic calculus (Ito lemma) that only came about in the 1960s or so Β¨ Without a strong theoretical footing of what is a numeraire and how to price a derivative in the risk-neutral probability associated to that numeraire (Pliska 1980 or so) Β¨ Without yet the strong connection between PDE (Partial Differential Equations) and SDE (Stochastic Differential Equations) that really came about from the Feynman-Kac formula (1950 roughly) Β¨ Louis Bachelier managed to not only built a theory of option pricing that is nowadays coming back in fashion with a vengeance, but perusing through the rather short thesis, one cannot but be amazed at the breadth of his genius, but also at his attention to details. Bachelier at times go through numerical examples with the same precision and clarity of thoughts that he displays in the other more theoretical parts of his thesis. 20
Luc_Faucheux_2020 Stochastic Calculus is not intuitive Β¨ Regular calculus has usually to do with ”things that are around some other things” Β¨ Taylor expansion and derivatives (expansion around a value, local derivatives at or around a point) Β¨ Finite difference for integrating functions on an axis or a path (keep following the path in a continuous fashion) Β¨ Stochastic Calculus tries to address the problem of dealing with β€œthings around things that are not there”. What do I mean ? Β¨ Take the coin flipping problem (Head is +1, Tail is -1). The coin is either head or tail (+1 or - 1), never anything else. And yet we will try to calculate expansions, derivations, integrations of functions around the mean or average (0), which is NOT a possible state of the coin. Β¨ Without stating the obvious or oversimplifying, this is the crux of the problem with stochastic calculus, and also that we are so used to usual calculus that we take a lot of things for granted 21
Luc_Faucheux_2020 Stochastic processes are β€œnon-differentiable” Β¨ A stochastic process essentially β€œflips a coin” at each point in time. Β¨ A β€œregular” process, meaning it is differentiable, would have a unique tangent for every point in time. If 𝑋 𝑑 is differentiable, there is a unique )*(+) )+ Β¨ The stochastic process does NOT have a unique tangent, because, using the coin flip idea, and being somewhat liberal with scaling, at each point in time, 𝑋 𝑑 goes to either {𝑋 𝑑 + 𝛿𝑋} or {𝑋 𝑑 βˆ’ 𝛿𝑋} with some probability (50% in the simplest case, or driftless case). Β¨ The tangent is NOT the average (0) of the two possible tangents. Β¨ So in essence for a stochastic process, writing something like )*(+) )+ is meaningless Β¨ In stochastic processes, the only real thing that we can do is integrals, almost never differential calculus (not surprising as it is not differentiable) Β¨ NOTE that non-differentiable does NOT mean not smooth or not continuous 22
Luc_Faucheux_2020 Stochastic calculus is usually self-similar Β¨ We will go over this in more details, but essentially the simplest stochastic process is the Wiener process or also called the standard Brownian motion. We will start with it Β¨ π‘Š 𝑑 has a normal 𝑁(0, 𝑑) distribution function Β¨ π‘Š 𝑑 βˆ’ 𝑠 has obviously the same normal 𝑁(0, 𝑑 βˆ’ 𝑠) distribution function Β¨ {π‘Š 𝑑 βˆ’ π‘Š(𝑠)} has ALSO the same 𝑁(0, 𝑑 βˆ’ 𝑠) distribution Β¨ Note that they are NOT the same, writing π‘Š 𝑑 βˆ’ π‘Š 𝑠 = π‘Š(𝑑 βˆ’ 𝑠) is obviously wrong but you will find sometimes the notation: Β¨ π‘Š 𝑑 βˆ’ π‘Š 𝑠 ≝ π‘Š(𝑑 βˆ’ 𝑠), meaning distributional identity, NOT pathwise identity Β¨ The simple Brownian motion is self-similar with coefficient (1/2) Β¨ 𝑇, π‘Š 𝑑- , 𝑇, π‘Š 𝑑. , . . , 𝑇, π‘Š 𝑑/ ≝ π‘Š 𝑇𝑑- , π‘Š 𝑇𝑑. , . . , π‘Š 𝑇𝑑/ , with 𝐻 = 1/2 Β¨ This is sometimes used to numerically construct Brownian motion. Β¨ If you β€œscale” up the time scale by a factor 𝑇, the space scale gets only magnified by a factor 𝑇. 23
Luc_Faucheux_2020 Stochastic Calculus is cumbersome Β¨ Usual knowledge and tricks of calculus do not apply anymore Β¨ Chain rule does not work Β¨ Functional derivation does not work : 𝑑 𝑙𝑛𝑆 β‰  ( ⁄𝑑𝑆 𝑆) ! Β¨ Integration and especially derivations are not well defined Β¨ Usual calculus π‘Š(𝑑), when (𝛿𝑑) β†’0, (π›Ώπ‘Š)~𝛿𝑑, (π›Ώπ‘Š).~(𝛿𝑑). Β¨ Stochastic calculus we still have (π›Ώπ‘Š) β†’ 0, BUT WE ALSO HAVE (π›Ώπ‘Š).~𝛿𝑑 so higher orders are mixed together Β¨ Coin toss (+1, -1). Average is 0, variance scales linearly with the number of flips Β¨ Can you think of processes where variance goes to 0 and we need to go to the next order? Β¨ Can you think of a process where (π›Ώπ‘Š). scales maybe as (𝛿𝑑)0, where π‘ˆ < 1 ? Β¨ Also, if you think about it, when π›Ώπ‘Šis stochastic, somehow (π›Ώπ‘Š). is now deterministic, or at least has a deterministic component, the inverse is NOT true 24
Luc_Faucheux_2020 Always better to integrate than to differentiate Β¨ If we have a stochastic process π‘Š(𝑑), we would like to work with functions 𝑓(π‘Š) (please note that those functions are usually well behaved, meaning differentiable and such, without going into too much math) Β¨ 𝑓(π‘Š) is differentiable in π‘Š Β¨ π‘Š(𝑑) is NOT differentiable in 𝑑 Β¨ But we are really dealing with 𝑓(π‘Š 𝑑 ) so we would like to write things like Β¨ 𝛿𝑓 = )$ )1 . )1 )+ . 𝛿𝑑 which is one way to write the traditional β€œchain rule” Β¨ In integral form, the chain rule would read something like: Β¨ 𝑓 π‘Š 𝑑 βˆ’ 𝑓 π‘Š 0 = ∫234 23+ )$ )1 . )1 )2 . 𝑑𝑠 Β¨ The crux of the issue in stochastic calculus is that we do not know what )1 )2 means, and so we should NOT expect to be able to rely on the usual chain rule 25
Luc_Faucheux_2020 The whole Ito-Stratanovitch thing Β¨ Essentially, ITO breakthrough was to find a way to define: Β¨ 𝑓 π‘Š 𝑑 βˆ’ 𝑓 π‘Š 0 = ∫234 23+ )$ )1 . )1 )2 . 𝑑𝑠 = ∫234 23+ )$ )1 . π‘‘π‘Š Β¨ ITO invented the field of stochastic integrals Β¨ The essence of it is that π‘‘π‘Š(𝑑) is a β€œjump” Β¨ ITO (1950) defines the ITO integral as a limit of a sum where the value of )$ )1 is taken β€œbefore the jump”. A consequence of this convention is that the usual rules of calculus (chain rule, Leibniz rule,..) do not apply. For example 𝑑 𝑙𝑛𝑆 β‰  ( ⁄𝑑𝑆 𝑆) ! On the other hand the ITO integral has the nice property to be a martingale (zero expected value) Β¨ Stratanovitch (STRATO) defines the STRATO integral as a limit of a sum where the value of )$ )1 is taken β€œin the middle of the jump”. A rather intriguing consequence is that in STRATO calculus the usual rules of calculus are (formally!) respected. 𝑑 𝑙𝑛𝑆 = ( ⁄𝑑𝑆 𝑆). HOWEVER the STRATO integral is NOT a martingale. 26
Luc_Faucheux_2020 The whole Ito-Stratanovitch thing - II Β¨ So we will see how to not get confused between the two equally valid interpretations of the integrals (from a really theoretical point of view mathematicians prefer ITO because it is defined over a wider range of functions than STRATO, but really nothing we should concern ourselves at this point). Β¨ This will take us some time, first going over the Riemann integral in regular calculus Β¨ Then defining the ITO Β¨ Then looking at the relationship between ITO and STRATO integral Β¨ Then explicitly proving the ITO lemma, then the equivalent STRATO lemma Β¨ From then we look at the SDE (SIE), and we try to map the relations between a specific SDE and its associated PDE for the PDF Β¨ SDE: Stochastic Differential Equation Β¨ SIE: Stochastic Integral Equation Β¨ PDE: Partial Differential Equation (that is usual regular calculus, but not easy by any means) Β¨ PDF: Probability Density Function 27
Luc_Faucheux_2020 The whole Ito-Stratanovitch thing - III Β¨ So apologies in advance if those slides feel pedestrian at time, but unfortunately in order to be somewhat rigorous without losing the intuition, and in order to convince ourselves that we are on somewhat firm ground to justify what we write (without having a 100% rigorous mathematical proof), we have to walk before we run. Β¨ Alternatively, you could be a genius like Vincent Doelin and bypass the entire theory of stochastic integrals and express everything as a Brownian time change, 40 years or so before everyone else, while fighting WWII in the Ardennes as a radio operator. Β¨ If I have time, part V of those decks will be on that Β¨ Also the ITO-STRATO controversy in the 1990s was linked to the concept of thermal ratchets, Brownian motors, biological motors, hence quite a few articles on the subject. Β¨ This is also linked to an individual that physicists refer to as the Maxwell’s demon Β¨ In deck III we will revisit this unsavory character, who prompted me spending a lot of time on ITO-STRATO over my life and as a PhD student. 28
Luc_Faucheux_2020 The whole Ito-Stratanovitch thing - IV Β¨ The Maxwell demon putting the moves on Mr. Tompkins fiancΓ©e and trying to impress her with his tennis skills 29
Luc_Faucheux_2020 The whole Ito-Stratanovitch thing - V Β¨ A more common representation of the demon 30
Luc_Faucheux_2020 The whole Ito-Stratanovitch thing - VI Β¨ There are actually applications in Finance of the Maxell demon, known as the Parrondo paradox. Β¨ Two trading strategies (PM at a hedge fund) on average lose money (B and C, blue and green line) Β¨ However you can alternate between the two strategies to create one (A-red line) that on average will be profitable, like the thermal ratchet who is extracting work out of thermal noise, this Parrondo construct extract positive return out of random switches between two losing strategies 31
Luc_Faucheux_2020 Just a taste of how powerful Vincent Doelin was Β¨ SDE have usually the form: Β¨ 𝑑𝑋 𝑑 = π‘Ž 𝑋 𝑑 , 𝑑 . 𝑑𝑑 + 𝑏 𝑋 𝑑 , 𝑑 . π‘‘π‘Š, which really should always be written as SIE: Β¨ 𝑋 𝑑5 βˆ’ 𝑋 𝑑6 = ∫+3+6 +3+5 𝑑𝑋 𝑑 = ∫+3+6 +3+5 π‘Ž 𝑋 𝑑 , 𝑑 . 𝑑𝑑) + ∫+3+6 +3+5 𝑏 𝑋 𝑑 , 𝑑 . π‘‘π‘Š(𝑑) Β¨ The whole theory of ITO is trying to define what exactly is : ∫+3+6 +3+5 𝑏 𝑋 𝑑 , 𝑑 . π‘‘π‘Š(𝑑) Β¨ What Doelin did essentially was to say: hey I do not need to define this integral, which is subject to the exact convention of β€œwhere to take the value of 𝑏 𝑋 𝑑 , 𝑑 before, during or after the jump π‘‘π‘Š(𝑑)”, and run into all sort of Ito-Stratanovitch confusion, because I am a genius, whatever that integral is, it is equal to : Β¨ ∫+3+6 +3+5 𝑏 𝑋 𝑑 , 𝑑 . π‘‘π‘Š(𝑑) ≝ π‘Š(∫+3+6 +3+5 𝑏 𝑋 𝑑 , 𝑑 .. 𝑑𝑑) which is perfectly well defined. Β¨ Boom. Microphone drop. Β¨ And by the way he burnt most of his research before killing himself to avoid capture by the Germans, so who knows what else he had discovered…. 32
Luc_Faucheux_2020 ITO and DOELIN comparison in 2 pictures 33
Luc_Faucheux_2020 Ito-Doblin and stochastic calculus is.. Β¨ β€œFirst and foremost defined as an integral calculus” Β¨ Really the only thing that works is integrating stochastic processes. Β¨ There are theories of differentiations for stochastic variables Β¨ Malliavin calculus (1980) Β¨ One day I will try to understand what that actually means. I still have no idea. But I should try after all Malliavin is also French Β¨ So even the Ito lemma as we know it is really better expressed in integral form Β¨ However most textbooks do present it in β€œdifferential” form like a Taylor expansion, or even deal with SDE quite liberally. This is sometimes for ease of notations and we will fall into that pattern also. Note that this is a FORMAL equivalence to regular calculus equations like Taylor expansion. Taking those literally leads to mistakes, as we will demonstrate 34
Luc_Faucheux_2020 Why are PDEs so important, and why SDEs are terrifying Β¨ PDEs are Partial Differential Equations )!7(!,+) )!! = )7(!,+) )+ Β¨ Usually in the context of stochastic calculus they will appear for the PDF (Probability Density Function) of the stochastic variable 𝑋 (like a Gaussian), 𝑃(π‘₯, 𝑑) or for functions of that variable (call payoff or 𝐢(𝑋, 𝜎, 𝐾, 𝑇) for example) Β¨ PDEs are well defined and well known (since 1700), tons of knowledge on how to deal with those (Navier-Stokes in fluid mechanic, Maxwell equations in electro-magnetism, Fokker- Planck, Feynman-Kac,..) Β¨ Once you know the PDE, you know ALL the moments of the distribution in the case of a PDE Β¨ PDEs are complete, SDEs are incomplete Β¨ SDEs are Stochastic Differential equations 𝑑𝑆 = πœ‡. 𝑆. 𝑑𝑑 + 𝜎. 𝑆. π‘‘π‘Š Β¨ No one really knows how to deal with them, especially if the volatility is a function of the stochastic variable (and it always is, or when you look at functions of X) Β¨ But sometimes you can get rid of a SDE and use a related PDE (that was the trick that Black Sholes discovered and got a Nobel prize for), or Dupire equation Β¨ Also stay away from SDE, always look for the PDE 35
Luc_Faucheux_2020 The structure of those slides Β¨ We will first start with the usual textbook (Hull) that presents essentially what are Ito SDEs and Ito lemma being a regular Taylor expansion just making sure that we go high enough to keep all the terms linear in time and linear in the stochastic driver Β¨ So in some ways, in the usual calculus π‘Š(𝑑), when (𝛿𝑑) β†’0, (π›Ώπ‘Š)~𝛿𝑑, (π›Ώπ‘Š).~(π›Ώπ‘Š). Β¨ Stochastic calculus we still have (π›Ώπ‘Š) β†’ 0, BUT WE ALSO HAVE (π›Ώπ‘Š).~𝛿𝑑 so higher orders are mixed together, but if we keep the right terms we should be fine Β¨ This is usually where most textbooks in finance stops Β¨ We will show by using something called Stratonovitch convention, that treating the SDEs and the Taylor expansion the way we would do in regular calculus is wrong. Β¨ In fact the Ito lemma and the Ito SDEs are just a formal manner to write Integrals and SIE, which is really the only thing that you can do in stochastic calculus Β¨ Remember, you always read that a stochastic process is NOT differentiable, yet somehow we all proceed happy to write Stochastic DIFFERENTIAL Equations, and writing Ito lemma as a Taylor expansion, without really thinking twice about it 36
Luc_Faucheux_2020 The structure of those slides - II - Integrals Β¨ We then take a couple of steps back and go over a review of the regular integrals (Riemann) in the regular calculus Β¨ We extend this to the realm of stochastic calculus Β¨ We show that unlike the regular case, the point taken in the partition buckets (the mesh) actually matters. One convention is to take the starting point (ITO). Another one is to take the middle point (Stratonovitch) Β¨ We then derive the relationships between the Ito and the Strato integral Β¨ At this point there is no finance theory but it will be worth keeping in mind that the Ito integral will have the property of being a martingale (zero expected value) Β¨ Note that things like β€œpartition” or ”mesh” will not be super rigorously defined, but we leave that to mathematicians, we use those concepts assuming that they are well defined, and we can check that indeed they are over a certain range of β€œnon-pathological” functions or geometries 37
Luc_Faucheux_2020 The structure of those slides - III - Lemma Β¨ From what we learned from looking at those integrals, we then look at the issue around making some operations on those integrals. This is where we look at Ito lemma Β¨ Ito lemma is crucial in finance Β¨ We will see that formally we can write the lemma, and perform operations on functions and integral, in a formal manner that looks like regular calculus, with the exceptions that you have to go β€œone more up” in any kind of derivations or Taylor expansion, in order to capture ALL the terms linear in time and linear in the stochastic driver Β¨ In doing so the β€œusual” rules of calculus (derivations, integrations, Leibniz,..) are no longer true in stochastic calculus using the Ito interpretation of the integral Β¨ We will compare this with the Strato integral Β¨ We will show that the ”usual” rules of calculus are still formally present in Strato calculus (remember this is a formal analogy). This can be useful when explicitly solving equations Β¨ We then show the relationship between the Ito and the Strato lemma 38
Luc_Faucheux_2020 The structure of those slides - IV - SDE Β¨ From what we learned looking at the integrals and how to manipulate them, we can try to look at what would be SDE, Stochastic Differential Equations Β¨ Remember though, that just the same way we can formally write Ito and Strato lemma in β€œdifferential” form for ease of notation, those are ALWAYS simpler way to write down what would really be equations dealing with integrals Β¨ In stochastic calculus, I do not know what a differentiation would actually mean Β¨ All I can do really is to integrate Β¨ Like the integral and the lemma, we will show the relationship between the Ito SDE and the Strato SDE (or really more exactly the Ito SIE and the Strato SIE), and introduce the so-called β€œdrift” between the two representations Β¨ SIE: Stochastic Integral Equations 39
Luc_Faucheux_2020 The structure of those slides - V - PDE Β¨ From the SIE, we then explore the correspondence between the SIE and the PDE (Partial Differential Equation) for the PDF (Probability Distribution Function) of the stochastic variable. Β¨ PDE are just part of the regular calculus, there is no issue there Β¨ SIE and SDE depends on the interpretation (Ito or Strato). Β¨ We will then by extension look at PDE that would correspond to a specific SDE under Ito, and similarly under Strato. Β¨ This is where it can get a little confusing (or even more confusing that it already is) Β¨ We will revisit our old friend the Maxwell demon and see why it was so confusing in the 1990s when applied to biological concepts of thermal ratchets or Brownian motors Β¨ Because a PDE is β€œexact” (once you know the PDE, and if you can solve it you have the PDF, so you have exactly all the moments of the stochastic variable) whereas an SDE only has the first two moments in most cases, and is also subject to interpretation, it is worth keeping in mind that fact: a PDE is not subject to interpretation. An SDE is subject to interpretation, and needs to be treated with great caution 40
Luc_Faucheux_2020 The structure of those slides - VI - PDE Β¨ We will look at some cases of PDEs from the world of Physics in order to gain some more intuition on what is a firm ground to stand on (PDE), and a somewhat more recent and still a little shaky one (SDE) Β¨ This will be a somewhat indirect introduction to a fundamental theorem that links the world of β€œregular” well known calculus of PDE (400 years in the making) with the more recent one having to do with probability and stochastic calculus (only 100 years in the making), and still very new. Β¨ This year Abel prize was given to pioneers of the ergodic theory, that essentially to crudely oversimplify, try to find the solutions of PDEs by using properties of the associated SDEs Β¨ We can then go back to Black-Sholes with a renewed belief in how justified we are in our derivation, in particular we tried to β€œget away” using simple Taylor expansions and just keeping some higher orders, we showed however how wrong it can get very quickly. 41
Luc_Faucheux_2020 Β¨ Since we will first fall into the trap of treating an SDE the way we would do usual calculus, a number of slides in this deck are WRONG. It is sometimes more useful to learn from mistakes than to follow a magistral correct demonstration. Β¨ So in order to not make the whole deck a complete garbage of my random wrong ramblings on stochastic calculus while going up the river to meet colonel Kurtz, I have put a stamp β€œWRONG” across those slides. Β¨ So do not read those slides at face value, but know that those are WRONG. Β¨ It might be an interesting exercise for you to convince yourself why those slides are wrong. Β¨ Again, I left them there because I think it is quite instructive to identify the fault in a derivation. Β¨ We could have kept all the slides in ITO calculus, but it is quite enlightening to also look at STRATANOVITCH calculus, if only to convince ourselves that we really understand ITO (If we understand the difference between two things, then most likely that increases our knowledge about each of those things) A note of caution 42
Luc_Faucheux_2020 Why this whole thing about Ito calculus? Hull textbook Β¨ Hull – White chapters 13, 14 and 15 Β¨ People got excited about stock prices trading as a percentage (people expect a β€œreturn”), p.306, and so what mattered was the return of the stock 𝑆, or β„βˆ†π‘† 𝑆 Β¨ So then they started writing things like : 𝑑𝑆 = πœ‡. 𝑆. 𝑑𝑑 + 𝜎. 𝑆. π‘‘π‘Š, (p.307) Β¨ And then they got stuck, because 𝑑π‘₯ = π‘Ž π‘₯, 𝑑 . 𝑑𝑑 + 𝑏 π‘₯, 𝑑 . π‘‘π‘Š, where 𝑏 π‘₯, 𝑑 is a function of the stochastic variable π‘₯, is not something we know how to deal with (p.306, and no, it is NOT a β€œsmall approximation” as they claim) Β¨ So you need to use a ”guess” on how to deal with 𝑏 π‘₯, 𝑑 , which is why it is called a β€œlemma” Β¨ Ito (1951) guessed that you can write 𝑑π‘₯ = π‘Ž π‘₯, 𝑑 . 𝑑𝑑 + 𝑏 π‘₯, 𝑑 . π‘‘π‘Š as βˆ†π‘₯ = π‘Ž π‘₯, 𝑑 . βˆ†π‘‘ + 𝑏 π‘₯, 𝑑 . βˆ†π‘Š or Ξ΄π‘₯ = π‘Ž π‘₯, 𝑑 . 𝛿𝑑 + 𝑏 π‘₯, 𝑑 . π›Ώπ‘Š Β¨ That seems like a good guess but then the rules of calculus are no longer applicable, you can barely derive without making a mistake, and forget about trying to integrate (p. 311) Β¨ Now you get this weird thing where 𝑑 𝑙𝑛𝑆 = ( ⁄𝑑𝑆 𝑆) βˆ’ ( β„πœŽ. 2). 𝑑𝑑, (p.312) 43
Luc_Faucheux_2020 Why chose Ito then? 44 Β¨ Someone else made a β€œbetter” guess, Stratonovitch (1966) Β¨ Strato guessed that you can write 𝑑π‘₯ = π‘Ž π‘₯, 𝑑 . 𝑑𝑑 + 𝑏 π‘₯, 𝑑 . π‘‘π‘Š as βˆ†π‘₯ = π‘Ž π‘₯, 𝑑 . βˆ†π‘‘ + 𝑏 π‘₯ + Ξ”π‘₯/2, 𝑑 . βˆ†π‘Š or Ξ΄π‘₯ = π‘Ž π‘₯, 𝑑 . 𝛿𝑑 + 𝑏 π‘₯ + 𝛿π‘₯/2, 𝑑 . π›Ώπ‘Š Β¨ Within Strato’s convention, the usual rules of calculus FORMALLY apply 𝑑 𝑙𝑛𝑆 = ( ⁄𝑑𝑆 𝑆), and the chain rule is verified in the usual manner Β¨ So this is super confusing Β¨ We will look at both Ito and Strato, and understand how to go from the SDE (Stochastic Differential Equation) to the PDE (Partial Differential Equation) and the correct PDF (Probability Density Function). Β¨ We will go over the binomial trees and binomial distribution, and its limit the Gaussian distribution (HW chapter 13, p.296-299) Β¨ We will show why the Gaussian distribution is so common and so central to everything (central limit theorem)
Luc_Faucheux_2020 Actually it is not called Ito, but Ito-Doeblin Β¨ It was discovered quite recently (2000) that Vincent Doeblin, born Wolfgang Doeblin, Ph.D. at 23, and drafted in the French army in 1938, while posted in the Ardennes as a phone operator, essentially worked out Ito calculus on his own and sent his results to the Academie des Sciences β€œsous plis scelle”. Β¨ He shot himself after burning the rest of his notes when the German army was about to advance and take over his positions Β¨ So wo knows what else was in his notes? Malliavin calculus maybe ? 45
Luc_Faucheux_2020 Ito and Stratonovitch β€œTaylor” expansion Β¨ Ito (1951) guessed that you can write 𝑑π‘₯ = π‘Ž π‘₯, 𝑑 . 𝑑𝑑 + 𝑏 π‘₯, 𝑑 . π‘‘π‘Š as Ξ΄π‘₯ = π‘Ž π‘₯, 𝑑 . 𝛿𝑑 + 𝑏 π‘₯, 𝑑 . π›Ώπ‘Š Β¨ If 𝐹(π‘₯) a function of x, the corresponding SDE as a result of Taylor expansion in ITO is: Β¨ 𝛿𝐹 = )9 )! . 𝛿π‘₯ + - . . )!9 )!! . 𝑏.. 𝛿𝑑 Β¨ That is the celebrated Ito lemma, or how the chain rule gets modified in Ito stochastic calculus Β¨ Strato guessed that you can write 𝑑π‘₯ = π‘Ž π‘₯, 𝑑 . 𝑑𝑑 + 𝑏 π‘₯, 𝑑 . π‘‘π‘Š as Ξ΄π‘₯ = π‘Ž π‘₯, 𝑑 . 𝛿𝑑 + 𝑏 π‘₯ + 𝛿π‘₯/2, 𝑑 . π›Ώπ‘Š Β¨ So you think that you could write something like this: the SDE for 𝐹(π‘₯) in Stratonovitch convention is: Β¨ 𝛿𝐹 = )9 )! . 𝛿π‘₯ + - . . )!9 )!! . 𝑏.. 𝛿𝑑 + - . . )9 )! . )5 )! . 𝑏. 𝛿𝑑 46
Luc_Faucheux_2020 Ito and Stratonovitch β€œTaylor” expansion - II Β¨ We would like to write something like this for Stratonovitch, following the rule of expanding to the second order and keeping the terms linear in time and linear in the stochastic driver (so essentially in a way use Ito calculus in the Stratonovitch convention) Β¨ Bear in mind that this is absolutely wrong Β¨ It is quite insightful to go through it though and see where it is wrong Β¨ So we have a function 𝐹 of the stochastic variable π‘₯, the function 𝐹 in itself is nothing weird, it is a regular function that we assume to be differentiable 𝐹(π‘₯) Β¨ Ito calculus tells us that if we want to look at the variations of that function in terms of the stochastic variable π‘₯, assuming a process 𝑑π‘₯ = π‘Ž π‘₯, 𝑑 . 𝑑𝑑 + 𝑏 π‘₯, 𝑑 . π‘‘π‘Š Β¨ You can formally write within Ito calculus: Ξ΄π‘₯ = π‘Ž π‘₯, 𝑑 . 𝛿𝑑 + 𝑏 π‘₯, 𝑑 . π›Ώπ‘Š Β¨ 𝛿𝐹 = )9 )! . 𝛿π‘₯ + - . . )!9 )!! . 𝑏.. 𝛿𝑑 47
Luc_Faucheux_2020 Ito and Stratonovitch β€œTaylor” expansion - III Β¨ More exactly Β¨ 𝛿𝐹 = )9 )! . 𝛿π‘₯ + - . . )!9 )!! . 𝛿π‘₯. Β¨ And : Ξ΄π‘₯ = π‘Ž π‘₯, 𝑑 . 𝛿𝑑 + 𝑏 π‘₯, 𝑑 . π›Ώπ‘Š Β¨ So keeping the terms linear in time and in the stochastic driver Β¨ 𝛿π‘₯. = 𝑏.. 𝛿𝑑 Β¨ And we get the usual Ito formula: 𝛿𝐹 = )9 )! . 𝛿π‘₯ + - . . )!9 )!! . 𝑏.. 𝛿𝑑 Β¨ The β€œcanonical” example is usually 𝐹 π‘₯ = 𝐿𝑛(π‘₯), with )9 )! = - ! , and )!9 )!! = :- !!, and with the stochastic process: Ξ΄π‘₯ = π‘₯. π›Ώπ‘Š, so 𝑏 = π‘₯, and 𝑏. = π‘₯. Β¨ And so we get: 𝛿 𝐿𝑁 π‘₯ = - ! . 𝛿π‘₯ + - . . :- !! . 𝑏.. 𝛿𝑑 = - ! . π‘₯. π›Ώπ‘Š + - . . :- !! . π‘₯.. 𝛿𝑑 48
Luc_Faucheux_2020 Ito and Stratonovitch β€œTaylor” expansion - IV Β¨ Or again: Β¨ 𝛿 𝐿𝑁 π‘₯ = 𝛿𝑧 βˆ’ - . . 𝛿𝑑 = ;! ! βˆ’ - . . 𝛿𝑑 Β¨ Whereas the β€œusual rule of calculus would read : 𝛿 𝐿𝑁 π‘₯ = ;! ! Β¨ That is the usual example in most textbooks, especially in Finance Β¨ NOW comes the weird little Stratanovitch trick, where we assume that we can write something like : Β¨ Strato guessed that you can write 𝑑π‘₯ = π‘Ž π‘₯, 𝑑 . 𝑑𝑑 + 𝑏 π‘₯, 𝑑 . π‘‘π‘Š as Ξ΄π‘₯ = π‘Ž π‘₯, 𝑑 . 𝛿𝑑 + 𝑏 π‘₯ + 𝛿π‘₯/2, 𝑑 . π›Ώπ‘Š Β¨ So again we follow the β€œIto” rule of calculus by expanding in linear terms in time and in the stochastic driver 49
Luc_Faucheux_2020 Ito and Stratonovitch β€œTaylor” expansion - V Β¨ 𝛿𝐹 = )9 )! . 𝛿π‘₯ + - . . )!9 )!! . 𝛿π‘₯. Β¨ And : Ξ΄π‘₯ = π‘Ž π‘₯, 𝑑 . 𝛿𝑑 + 𝑏 π‘₯ + 𝛿π‘₯/2, 𝑑 . π›Ώπ‘Š Β¨ Or: Ξ΄π‘₯ = π‘Ž π‘₯, 𝑑 . 𝛿𝑑 + 𝑏 π‘₯, 𝑑 . π›Ώπ‘Š + ;! . . )5 )! . π›Ώπ‘Š, which keeping only the terms linear in time and in the stochastic driver π›Ώπ‘Š, leads to: Β¨ Ξ΄π‘₯ = π‘Ž π‘₯, 𝑑 . 𝛿𝑑 + 𝑏 π‘₯, 𝑑 . π›Ώπ‘Š + - . . )5 )! . 𝑏. 𝛿𝑑 Β¨ And so: 𝛿𝐹 = )9 )! . 𝛿π‘₯ + - . . )!9 )!! . 𝛿π‘₯. Β¨ 𝛿𝐹 = )9 )! . 𝛿π‘₯ + - . . )!9 )!! . 𝑏.. 𝛿𝑑 + - . . )9 )! . )5 )! . 𝑏. 𝛿𝑑 50
Luc_Faucheux_2020 Ito and Stratonovitch β€œTaylor” expansion - VI Β¨ So we think that within the Ito rules of calculus (doing a Taylor expansion and keeping only the terms linear in time and the stochastic driver) but following the Stratonovitch convention we can write something like: Β¨ 𝛿𝐹 = )9 )! . 𝛿π‘₯ + - . . )!9 )!! . 𝑏.. 𝛿𝑑 + - . . )9 )! . )5 )! . 𝑏. 𝛿𝑑 Β¨ The β€œcanonical” example again is 𝐹 π‘₯ = 𝐿𝑛(π‘₯), with )9 )! = - ! , and )!9 )!! = :- !!, and with the stochastic process: Ξ΄π‘₯ = π‘₯. π›Ώπ‘Š, so 𝑏 = π‘₯, and 𝑏. = π‘₯., and )5 )! = 1 Β¨ So we get: Β¨ 𝛿 𝐿𝑁 π‘₯ = - ! . 𝛿π‘₯ βˆ’ - . . - !! . 𝛿𝑑 + - . . - ! . 1. π‘₯. 𝛿𝑑 Β¨ The last two terms cancel out and we get: 𝛿 𝐿𝑁 π‘₯ = - ! . 𝛿π‘₯ Β¨ Which is the usual result in ”regular” (non stochastic) calculus, and we think that we now understood stochastic calculus because textbooks are telling us that the usual rules of calculus are preserved in Strato 51
Luc_Faucheux_2020 Ito and Stratonovitch β€œTaylor” expansion - VII Β¨ So far we seem to be pretty happy because we found that for the canonical example, the β€œusual rules of calculus” are preserved when using the Stratonovitch convention within the Ito calculus (meaning that we are using formally a Taylor expansion, making sure to keep all the terms linear in time and in the stochastic driver, i.e. the Ito rule of calculus, but assuming that we are taking the β€œmid-point” of the jump for functions multiplying this jump, i.e. what we think to be what the Stratonovitch convention is) Β¨ We will show that nothing could be more wrong Β¨ Both Ito and Stratonovitch are calculus on their own on the same footing Β¨ We just cannot do the Taylor expansion within the Stratonovitch framework Β¨ Not super obvious, took me a long time to be confused about it, the point is to always go back to the fact that in stochastic calculus the only thing that you can write is an integral, and sometimes for ease of notation we write something that looks like a Taylor expansion or and SDE. But this is only a formal way of writing, and the fact that it looks like regular calculus does NOT allow us to use those equations β€œas is” 52
Luc_Faucheux_2020 Ito and Stratonovitch β€œTaylor” expansion – VII - a Β¨ So for example, when in Ito calculus we write the chain rule as: Β¨ 𝛿𝐹 = )9 )1 . π›Ώπ‘Š + - . . )!9 )1! . π›Ώπ‘Š. Β¨ What we are really writing, (and what we should always write from time to remind us, since we do not have enough paper and ink to write it all the time at every step) is: Β¨ 𝑓 π‘Š 𝑑5 βˆ’ 𝑓 π‘Š 𝑑6 = ∫+3+6 +3+5 )$ )* . ([). π‘‘π‘Š(𝑑) + - . ∫+3+6 +3+5 )!9 )1! (π‘Š 𝑑 ). 𝑑𝑑 Β¨ In the ”limit” of small time increments, this can be written formally as the Ito lemma: Β¨ 𝛿𝑓 = )$ )* . π›Ώπ‘Š + - . . )!9 )1! . 𝛿𝑑 Β¨ We will go over it once we rebuild our knowledge of stochastic calculus around the integral Β¨ Here we are following mostly Thomas Mikosch β€œElementary Stochastic Calculus with Finance in View”, a wonderful little book that is at times frustrating for some of the weird notations that he uses. 53
Luc_Faucheux_2020 Ito and Stratonovitch β€œTaylor” expansion - VIII Β¨ A very quick manner to realize how wrong we are is to use another example: Β¨ 𝑑π‘₯ = π‘Ž π‘₯, 𝑑 . 𝑑𝑑 + 𝑏 π‘₯, 𝑑 . π‘‘π‘Š Β¨ with 𝑏 π‘₯, 𝑑 = Οƒ. π‘₯ , )5 )! = 𝜎 , 𝑏. = 𝜎.. π‘₯. and π‘Ž π‘₯, 𝑑 = 0, Β¨ 𝐹 π‘₯ = π‘₯., )9 )! = 2π‘₯ , )!9 )!! = 2 Β¨ Using ITO, 𝛿𝐹 = )9 )! . 𝛿π‘₯ + - . . )!9 )!! . 𝑏.. 𝛿𝑑 = 2π‘₯. 𝛿π‘₯ + 𝜎.. 𝛿𝑑 Β¨ Using Strato, 𝛿𝐹 = )9 )! . 𝛿π‘₯ + - . . )!9 )!! . 𝑏.. 𝛿𝑑 + - . . )9 )! . )5 )! . 𝑏. 𝛿𝑑 Β¨ We then get: 𝛿𝐹 = 2π‘₯. 𝛿π‘₯ + 𝜎.. 𝛿𝑑 + - . . 2π‘₯. Οƒ. Οƒ. π‘₯. 𝛿𝑑 = 2π‘₯. 𝛿π‘₯ + 2.𝜎.. 𝛿𝑑 Β¨ So neither what we call Ito and what we call Strato do follow the usual rule of calculus. That should be a sign that we did something very wrong when we loosely used the Taylor expansion and applied it to the Stratonovitch case, because textbooks are telling us that the usual rules of calculus (chain rule) are ”preserved” (similar in form) in Strato 54
Luc_Faucheux_2020 Ito and Stratonovitch β€œTaylor” expansion - IX Β¨ The answer is not completely obvious in identifying what we did wrong Β¨ Conversely, someone could say it is absolutely obvious, because stochastic process are NOT differentiable, and so any kind of Taylor expansion is wrong Β¨ As a note, even though in most textbooks in Finance the Ito lemma is expressed as a Taylor expansion using the formal rules of calculus, the rule is that with stochastic processes you can NEVER differentiate (at least in a manner that makes sense and is safe), you can ONLY integrate Β¨ And so a more formally correct formulation of the Ito Lemma is: Β¨ 𝐹 π‘Š 𝑑 = 𝑇 βˆ’ 𝐹 π‘Š 𝑑 = 𝑆 = ∫+3< +3= )9 )1 . π‘‘π‘Š + - . ∫+3< +3= )!9 )1! . 𝑑𝑑 Β¨ Or also: Β¨ ∫+3< +3= 𝑑𝐹(π‘Š 𝑑 ) = ∫+3< +3= )9 )1 . π‘‘π‘Š(𝑑) + - . ∫+3< +3= )!9 )1! . 𝑑𝑑 55
Luc_Faucheux_2020 Ito and Stratonovitch β€œTaylor” expansion - X Β¨ What is then the correct formulation of Stratonovitch lemma? Β¨ Is that? Β¨ 𝐹 π‘₯ 𝑑 = 𝑇 βˆ’ 𝐹 π‘₯ 𝑑 = 𝑆 = ∫+3< +3= )9 )! . 𝑑π‘₯ + ∫+3< +3= [ - . )!9 )!! . 𝑏. + - . . )9 )! . )5 )! . 𝑏]. 𝑑𝑑 Β¨ Or: Β¨ ∫+3< +3= 𝑑𝐹(π‘₯ 𝑑 ) = ∫+3< +3= )9 )! . 𝑑π‘₯ + ∫+3< +3= [ - . )!9 )!! . 𝑏. + - . . )9 )! . )5 )! . 𝑏]. 𝑑𝑑 Β¨ Or to identify the distinctions between the two: Β¨ STRATO(∫+3< +3= 𝑑𝐹(π‘₯ 𝑑 ))=ITO(∫+3< +3= 𝑑𝐹(π‘₯ 𝑑 ))+ - . ∫+3< +3= )9 )! . )5 )! . 𝑏. 𝑑𝑑 Β¨ This is still wrong, as we mixed Taylor expansion using the usual rules of calculus (Ito rules) with something completely different. We will now show what is the correct way to look at it using integrals. 56
Luc_Faucheux_2020 Simple rules of stochastic calculus (cheat sheet) Β¨ NEVER EVER EVER work with processes where the volatility is a function of the stochastic variable Β¨ If you do or have no choice, transform it into a constant volatility equation (Hull p. 320), or find a way to set the volatility term to 0 (p.330), or give up and come up with something different (SABR model) Β¨ If you are working in Finance (and ”discrete processes) -> use ITO Β¨ If you are working with a DIGITAL computer -> use ITO Β¨ If you are working with an ANALOG computer -> use STRATO Β¨ If you are working in physics, and you do not like to break the time invariance, and you are also not that smart, so you want the usual rules of calculus -> use STRATO Β¨ In ALL cases, especially when working with the SDE, and discrete computer simulations, ALWAYS check that your drift has not been β€œpolluted” by the variable diffusion (β€œspurious” drift, Ryter 1980) 57
Luc_Faucheux_2020 58 Building our knowledge of stochastic calculus around the integral
Luc_Faucheux_2020 We have to start from the basics Β¨ Because clearly using ITO lemma as a Taylor expansion where we keep certain terms and using still the usual rules of calculus is wrong. Β¨ As we hinted at it, we should never write something that looks like a differential but always as an integral. Β¨ Let’s now go through that derivation, and start with the usual Riemann integral in the usual calculus Β¨ We then show that extending that concept to a stochastic variable is not well defined, and needs a convention, or interpretation of the integral. ITO is one interpretation, STRATO is another one. Β¨ We show the correspondence between the two interpretations. Β¨ Extending the concept of the integral being a limit of sums subject to an interpretation, we then derive the ITO lemma as well as the STRATO lemma 59
Luc_Faucheux_2020 β€œRegular” Riemann integrals (definite integrals) Β¨ Riemann integrals are the regular integrals Β¨ Interval [a,b] on regular ”continuous” variable t Β¨ N sections of width (𝑏 βˆ’ π‘Ž)/𝑁, left side 𝐿>, right side 𝑅>, and middle 𝑀> Β¨ The Riemann integral ∫+36 +35 𝑓 𝑑 . 𝑑𝑑 is the limit when (𝑁 β†’ ∞) of the Riemann sums Β¨ LEFT Riemann Sum: 𝐿𝑅𝑆 = 5:6 / βˆ‘>3- >3/ 𝑓(𝐿>) Β¨ RIGHT Riemann Sum: 𝑅𝑅𝑆 = 5:6 / βˆ‘>3- >3/ 𝑓(𝑅>) Β¨ MIDDLE Riemann Sum: 𝑀𝑅𝑆 = 5:6 / βˆ‘>3- >3/ 𝑓(𝑀>) Β¨ SOMETHING Riemann Sum: 𝑆𝑅𝑆 = 5:6 / βˆ‘>3- >3/ 𝑓(𝑆>) where 𝑆> is somewhere in the section indexed by k, we could also define irregular partitions or β€œmesh” if we like Β¨ All those different sums converge to the same integral 60
Luc_Faucheux_2020 61 ITO and STRATO integrals in the simple case W(t)
Luc_Faucheux_2020 Stochastic Integrals Β¨ When not integrating over a β€œregular” continuous variable t but over a stochastic variable X, those sums do NOT converge to the same value. What does that even look like? Β¨ So first of all we would like to write something like this ∫*3*6 *3*5 𝑓 𝑋 . 𝑑𝑋 Β¨ The problem is that this is not well defined, what is the path over 𝑋 that we will integrate over? Remember 𝑋 is really 𝑋(𝑑) and is stochastic (jumps all over the place) Β¨ So really the only integration we can do is over 𝑑 Β¨ So we are looking for something like this: ∫+3+6 +3+5 𝑓 𝑋 𝑑 . 𝑑𝑋(𝑑) Β¨ When 𝑑 increases by a small amount 𝛿𝑑, 𝑋(𝑑) jumps by a small amount δ𝑋(𝑑) Β¨ So, breaking the time interval into 𝑁 sections of small time increment 𝛿𝑑 = (+5:+6) / Β¨ We looking at something like : SOMETHING = βˆ‘>3- >3/ 𝑓(𝑋(𝑑>)). [𝑋(𝑑>?-) βˆ’ 𝑋(𝑑>)] 62
Luc_Faucheux_2020 Ito Integral Β¨ That something is the ITO sum, which converges to the ITO integral. Note that at this point it is just how we define it Β¨ ∫+3+6 +3+5 𝑓 𝑋 𝑑 . ([). 𝑑𝑋(𝑑) = lim /β†’@ {βˆ‘>3- >3/ 𝑓(𝑋(𝑑>)). [𝑋(𝑑>?-) βˆ’ 𝑋(𝑑>)]} Β¨ Note in the integral the usual (.) is replaced by ([) to explicitly indicate ITO convention Β¨ This is to indicate that we take for 𝑓(𝑋(𝑑>) the value of 𝑓 𝑋 BEFORE the jump Β¨ This is to be compared to the ITO lemma where 𝑑π‘₯ = π‘Ž π‘₯, 𝑑 . 𝑑𝑑 + 𝑏 π‘₯, 𝑑 . π‘‘π‘Š gets written in discrete manner as βˆ†π‘₯ = π‘Ž π‘₯, 𝑑 . βˆ†π‘‘ + 𝑏 π‘₯, 𝑑 . βˆ†π‘Š or Ξ΄π‘₯ = π‘Ž π‘₯, 𝑑 . 𝛿𝑑 + 𝑏 π‘₯, 𝑑 . π›Ώπ‘Š Β¨ Note also that to be fairly rigorous there are a lot of conditions that need to be verified for that β€œSOMETHING” to converge in a well defined manner to a well defined function 63
Luc_Faucheux_2020 Stratonovitch Integral Β¨ Similar to the Stratonovitch lemma, where 𝑑π‘₯ = π‘Ž π‘₯, 𝑑 . 𝑑𝑑 + 𝑏 π‘₯, 𝑑 . π‘‘π‘Š gets written in the discrete manner as βˆ†π‘₯ = π‘Ž π‘₯, 𝑑 . βˆ†π‘‘ + 𝑏 π‘₯ + Ξ”π‘₯/2, 𝑑 . βˆ†π‘Š or Ξ΄π‘₯ = π‘Ž π‘₯, 𝑑 . 𝛿𝑑 + 𝑏 π‘₯ + 𝛿π‘₯/2, 𝑑 . π›Ώπ‘Š Β¨ The Stratonovitch integral is defined as: Β¨ ∫+3+6 +3+5 𝑓 𝑋 𝑑 . (∘). 𝑑𝑋(𝑑) = lim /β†’@ {βˆ‘>3- >3/ 𝑓 [𝑋(𝑑> + 𝑋(𝑑>?-)]/2). [𝑋(𝑑>?-) βˆ’ 𝑋(𝑑>)]} Β¨ If ITO was the β€œleft side Riemann”, Stratonovitch is the β€œmiddle Riemann” Β¨ Surprise surprise, they do NOT converge to the same value Β¨ Notice in the integral the usual (.) is replaced by (∘) to indicate that we take the middle point 64
Luc_Faucheux_2020 Reverse ITO integral Β¨ Similarly we can also define a Reverse ITO (OTI?) integral as Β¨ ∫+3+6 +3+5 𝑓 𝑋 𝑑 . (]). 𝑑𝑋(𝑑) = lim /β†’@ {βˆ‘>3- >3/ 𝑓(𝑋(𝑑>?-)). [𝑋(𝑑>?-) βˆ’ 𝑋(𝑑>)]} Β¨ Notice in the integral the usual (.) is replaced by (]) to indicate that we take the right side Β¨ The Reverse Ito lemma would then expand 𝑑π‘₯ = π‘Ž π‘₯, 𝑑 . 𝑑𝑑 + 𝑏 π‘₯, 𝑑 . π‘‘π‘Š in the discrete manner as βˆ†π‘₯ = π‘Ž π‘₯, 𝑑 . βˆ†π‘‘ + 𝑏 π‘₯ + Ξ”π‘₯, 𝑑 . βˆ†π‘Š or Ξ΄π‘₯ = π‘Ž π‘₯, 𝑑 . 𝛿𝑑 + 𝑏 π‘₯ + 𝛿π‘₯, 𝑑 . π›Ώπ‘Š Β¨ This would be equivalent as β€œreading into the future” because we would take the value after the jump, but we do not know yet what the jump will be Β¨ Stratonovitch is also sometimes said to be β€œsimilar”, as in you need to know the jump ahead of time to evaluate the function, even though you do not know the jump ahead of time Β¨ ITO is well adapted to finance and to β€œMartingales” (you do not know the future, expected value is the current value). The technical term is that the ITO integral is β€œnon-anticipating” 65
Luc_Faucheux_2020 Couple of notes here Β¨ It would seem that ITO would be well suited for processes that are β€œdiscontinuous” or β€œdiscrete” in nature (a very poor choice of words on my part), like: Β¨ Discrete computer simulation (on a digital computer) Β¨ Finance, gambling, games of chance Β¨ Radioactive decay (number of particles is discrete and the rate only depends on the previous state) Β¨ On the other hand STRATO seems to be better suited for β€œcontinuous” processes like most processes in Physics and Biology (diffusion, advection, chemotaxis, Brownian motors,..) Β¨ Part of the confusion will arise when trying to model in a discrete fashion a β€œcontinuous” process 66
Luc_Faucheux_2020 Conversion between ITO and Stratonovitch Β¨ ∫+3+6 +3+5 𝑓 𝑋 𝑑 . (∘). 𝑑𝑋(𝑑) = lim /β†’@ {βˆ‘>3- >3/ 𝑓 [𝑋(𝑑> + 𝑋(𝑑>?-)]/2). [𝑋(𝑑>?-) βˆ’ 𝑋(𝑑>)]} Β¨ ∫+3+6 +3+5 𝑓 𝑋 𝑑 . (∘). 𝑑𝑋(𝑑) = lim /β†’@ {βˆ‘>3- >3/ 𝑓 𝑋(𝑑> + [*(+"#$):*(+")] . ). [𝑋(𝑑>?-) βˆ’ 𝑋(𝑑>)]} Β¨ ∫+3+6 +3+5 𝑓 𝑋 𝑑 . (∘). 𝑑𝑋(𝑑) = lim /β†’@ {βˆ‘>3- >3/ 𝑓 𝑋(𝑑> ). [𝑋(𝑑>?-) βˆ’ 𝑋(𝑑>)]} + lim /β†’@ {βˆ‘>3- >3/ 𝑓′ 𝑋(𝑑> . [*(+"#$):*(+")] . ). [𝑋(𝑑>?-) βˆ’ 𝑋(𝑑>)]} Β¨ ∫+3+6 +3+5 𝑓 𝑋 𝑑 . (∘). 𝑑𝑋(𝑑) = ∫+3+6 +3+5 𝑓 𝑋 𝑑 . ([). 𝑑𝑋(𝑑) + lim /β†’@ {βˆ‘>3- >3/ 𝑓′ 𝑋(𝑑> . ( [*(+"#$):*(+")] . ). [𝑋(𝑑>?-) βˆ’ 𝑋(𝑑>)]} Β¨ 𝑆𝑇𝑅𝐴𝑇𝑂(𝑓) = 𝐼𝑇𝑂(𝑓) + lim /β†’@ {βˆ‘>3- >3/ 𝑓′ 𝑋(𝑑> . ( [*(+"#$):*(+")] . ). [𝑋(𝑑>?-) βˆ’ 𝑋(𝑑>)]} 67
Luc_Faucheux_2020 Conversion between ITO and Stratonovitch 2 Β¨ 𝑆𝑇𝑅𝐴𝑇𝑂(𝑓) = 𝐼𝑇𝑂(𝑓) + lim /β†’@ {βˆ‘>3- >3/ 𝑓′ 𝑋(𝑑> . ( [*(+"#$):*(+")] . ). [𝑋(𝑑>?-) βˆ’ 𝑋(𝑑>)]} Β¨ NOW this is a limit of a Riemann sum, because (𝛿𝑋).~𝛿𝑑 and is now deterministic (in the case of a simple Wiener process for 𝑋 = π‘Š) Β¨ In the more general case of 𝑑𝑋 𝑑 = π‘Ž 𝑋 𝑑 , 𝑑 . 𝑑𝑑 + 𝑏 𝑋 𝑑 , 𝑑 . π‘‘π‘Š(𝑑) we will show that Β¨ (𝛿𝑋).~𝑏 𝑋 𝑑 , 𝑑 . 𝛿𝑑 but for now let’s keep 𝑋 𝑑 = π‘Š(𝑑) the simple Brownian motion Β¨ That Riemann sum converges to the definite Riemann integral - . ∫+3+6 +3+5 𝑓′ π‘Š 𝑑 . 𝑑𝑑 Β¨ ∫+3+6 +3+5 𝑓 π‘Š 𝑑 . (∘). π‘‘π‘Š(𝑑) = ∫+3+6 +3+5 𝑓 π‘Š 𝑑 . ([). π‘‘π‘Š(𝑑) + - . ∫+3+6 +3+5 𝑓′ π‘Š 𝑑 . 𝑑𝑑 Β¨ 𝑆𝑇𝑅𝐴𝑇𝑂(𝑓) = 𝐼𝑇𝑂 𝑓 + - . ∫+3+6 +3+5 𝑓′ π‘Š 𝑑 . 𝑑𝑑 Β¨ 𝑆𝑇𝑅𝐴𝑇𝑂(𝑓) = 𝐼𝑇𝑂 𝑓 + - . . 𝑅𝐼𝐸𝑀𝐴𝑁𝑁(𝑓%) 68
Luc_Faucheux_2020 69 ITO lemma and Stratonovitch lemma
Luc_Faucheux_2020 Chain Rule (Ito lemma)- I Β¨ ITO integral: Β¨ ∫+3+6 +3+5 𝑓 𝑋 𝑑 . ([). 𝑑𝑋(𝑑) = lim /β†’@ {βˆ‘>3- >3/ 𝑓(𝑋(𝑑>)). [𝑋(𝑑>?-) βˆ’ 𝑋(𝑑>)]} Β¨ Another way to express it is the following: Β¨ 𝑓 𝑋 𝑑5 βˆ’ 𝑓 𝑋 𝑑6 = βˆ‘>3- >3/ {𝑓(𝑋(𝑑>)) βˆ’ 𝑓(𝑋(𝑑>:-))} Β¨ 𝑓 𝑋 𝑑5 βˆ’ 𝑓 𝑋 𝑑6 = lim /β†’@ βˆ‘>3- >3/ {𝑓(𝑋(𝑑>)) βˆ’ 𝑓(𝑋(𝑑>:-))} Β¨ As long as we have defined a reasonable β€œmesh” for the sequence of 𝑋(𝑑>) Β¨ 𝑓(𝑋(𝑑>)) = 𝑓(𝑋(𝑑>:-)) + )$ )* . ([). 𝛿𝑋 + - . . )!$ )!! . ([). (𝛿𝑋). Β¨ Where we use the ([) notation to reflect the β€œleft-hand” Riemann. Β¨ )$ )* . ([). 𝛿𝑋 = )$ )* 𝑋 = 𝑋(𝑑>:- ]. [𝑋(𝑑>) βˆ’ 𝑋(𝑑>:-)] 70
Luc_Faucheux_2020 Chain Rule (Ito lemma)- II Β¨ - . . )!9 )!! . ([). 𝛿𝑋 . = - . . )!$ )!! [𝑋 = 𝑋(𝑑>:-)]. [𝑋(𝑑>) βˆ’ 𝑋(𝑑>:-)]. Β¨ )$ )* . ([). 𝛿𝑋 = )$ )* 𝑋 = 𝑋(𝑑>:- ]. [𝑋(𝑑>) βˆ’ 𝑋(𝑑>:-)] Β¨ 𝑓(𝑋(𝑑>)) = 𝑓(𝑋(𝑑>:-)) + )$ )* . ([). 𝛿𝑋 + - . . )!$ )!! . ([). (𝛿𝑋). Β¨ 𝑓 𝑋 𝑑5 βˆ’ 𝑓 𝑋 𝑑6 = lim /β†’@ βˆ‘>3- >3/ {𝑓(𝑋(𝑑>)) βˆ’ 𝑓(𝑋(𝑑>:-))} Β¨ 𝑓 𝑋 𝑑5 βˆ’ 𝑓 𝑋 𝑑6 = lim /β†’@ βˆ‘>3- >3/ { )$ )* . ([). 𝛿𝑋 + - . . )!$ )!! . ([). (𝛿𝑋).} Β¨ 𝑓 𝑋 𝑑5 βˆ’ 𝑓 𝑋 𝑑6 = ∫+3+6 +3+5 )$ )* . ([). 𝑑𝑋(𝑑) + ∫+3+6 +3+5 - . . )!$ )!! . ([). (𝛿𝑋). Β¨ In the ”limit” of small time increments, this can be written formally as the Ito lemma: Β¨ 𝛿𝑓 = )$ )* . 𝛿𝑋 + - . . )!$ )!! . (𝛿𝑋). 71
Luc_Faucheux_2020 Chain Rule (Strato lemma)- I Β¨ We will now expand around the middle point Β¨ We still have: Β¨ 𝑓 𝑋 𝑑5 βˆ’ 𝑓 𝑋 𝑑6 = βˆ‘>3- >3/ {𝑓(𝑋(𝑑>)) βˆ’ 𝑓(𝑋(𝑑>:-))} Β¨ 𝑓 𝑋 𝑑5 βˆ’ 𝑓 𝑋 𝑑6 = lim /β†’@ βˆ‘>3- >3/ {𝑓(𝑋(𝑑>)) βˆ’ 𝑓(𝑋(𝑑>:-))} Β¨ We now write it a little differently by looking at expanding around the middle point: Β¨ 𝑓 𝑋 𝑑5 βˆ’ 𝑓 𝑋 𝑑6 = lim /β†’@ βˆ‘>3- >3/ {𝑓(𝑋(𝑑>)) βˆ’ 𝑓(𝑋(𝑑>:-))} Β¨ 𝑓 𝑋 𝑑5 βˆ’ 𝑓 𝑋 𝑑6 = lim /β†’@ βˆ‘>3- >3/ 𝑓 𝑋(𝑑> βˆ’ 𝑓 ∘ + 𝑓 ∘ βˆ’ 𝑓(𝑋(𝑑>:-))} Β¨ Where for sake of clarity we noted : 𝑓 ∘ = 𝑓 *(+" ? *(+"%$ . 72
Luc_Faucheux_2020 Chain Rule (Strato lemma)- II Β¨ We then have the two following expansions of the function around the middle point: Β¨ 𝑓(𝑋(𝑑>)) = 𝑓(∘) + )$ )* . (∘). ;* . + - . . )!$ )!! . (∘). ( ;* . ). Β¨ 𝑓(𝑋(𝑑>:-)) = 𝑓(∘) βˆ’ )$ )* . (∘). ;* . + - . . )!$ )!! . (∘). ( ;* . ). Β¨ 𝛿𝑋 = 𝑋(𝑑>) βˆ’ 𝑋(𝑑>:-) Β¨ 𝑋(𝑑>) βˆ’ *(+")?*(+"%$) . = ;* . Β¨ *(+")?*(+"%$) . βˆ’ 𝑋(𝑑>:-) = ;* . Β¨ 𝑓 𝑋 𝑑5 βˆ’ 𝑓 𝑋 𝑑6 = lim /β†’@ βˆ‘>3- >3/ 𝑓 𝑋(𝑑> βˆ’ 𝑓 ∘ + 𝑓 ∘ βˆ’ 𝑓(𝑋(𝑑>:-))} Β¨ 𝑓 𝑋 𝑑5 βˆ’ 𝑓 𝑋 𝑑6 = lim /β†’@ {βˆ‘>3- >3/ )$ )* . ∘ . ;* . + - . . )!$ )!! . ∘ . ;* . . βˆ’ (βˆ’ )$ )* . (∘). ;* . + - . . )!$ )!! . (∘). ( ;* . ).)} 73
Luc_Faucheux_2020 Chain Rule (Strato lemma)- III Β¨ 𝑓 𝑋 𝑑5 βˆ’ 𝑓 𝑋 𝑑6 = lim /β†’@ {βˆ‘>3- >3/ )$ )* . ∘ . ;* . + - . . )!$ )!! . ∘ . ;* . . βˆ’ (βˆ’ )$ )* . (∘). ;* . + - . . )!$ )!! . (∘). ( ;* . ).)} Β¨ 𝑓 𝑋 𝑑5 βˆ’ 𝑓 𝑋 𝑑6 = lim /β†’@ {βˆ‘>3- >3/ )$ )* . ∘ . ;* . + βˆ’(βˆ’ )$ )* . (∘). ;* . )} Β¨ 𝑓 𝑋 𝑑5 βˆ’ 𝑓 𝑋 𝑑6 = lim /β†’@ {βˆ‘>3- >3/ )$ )* . ∘ . 𝛿𝑋} Β¨ And so defining the integral with the Stratonovitch convention: Β¨ 𝑓 𝑋 𝑑5 βˆ’ 𝑓 𝑋 𝑑6 = ∫+3+6 +3+5 )$ )* . (∘). 𝑑𝑋(𝑑) Β¨ In the ”limit” of small time increments, this can be written formally as the Strato lemma: Β¨ 𝛿𝑓 = )$ )* . ∘ . 𝛿𝑋 Β¨ Please note that we are keeping the notation : ∘ 74
Luc_Faucheux_2020 Chain Rule (Strato lemma)- IV Β¨ Using the Stratonovitch definition of the stochastic integral, we can write : Β¨ 𝛿𝑓 = )$ )* . ∘ . 𝛿𝑋 Β¨ This is the usual chain rule Β¨ So formally in some textbooks, you will see the following statement: Β¨ β€œWe do not mean that the Stratonovitch stochastic integral is a classical (Riemann) integral. We only claim that the corresponding chain rules have a similar structure”. Mikosh p127 Β¨ It is important to note that BOTH the Ito and the Stratonovitch integrals are defined in a mathematically correct manner. Β¨ Ito rules of calculus are not the usual ones but the Ito integral is a martingale Β¨ Strato rules of calculus are the usual ones (FORMALLY) but the Strato integral is NOT a martingale Β¨ We still have to review how we treat SDE and PDE in those frameworks. 75
Luc_Faucheux_2020 Chain Rule (Ito and Strato lemma)- I Β¨ The crux of the matter is which interpretation of an integral do you want to use: Β¨ Because lim /β†’@ {βˆ‘>3- >3/ )$ )* . ∘ . 𝛿𝑋} Β¨ And Β¨ lim /β†’@ βˆ‘>3- >3/ { )$ )* . ([). 𝛿𝑋} Β¨ Do NOT converge to the same value, unlike a regular Riemann integral Β¨ In fact we just showed that Β¨ 𝑓 𝑋 𝑑5 βˆ’ 𝑓 𝑋 𝑑6 = lim /β†’@ {βˆ‘>3- >3/ )$ )* . ∘ . 𝛿𝑋} Β¨ 𝑓 𝑋 𝑑5 βˆ’ 𝑓 𝑋 𝑑6 = lim /β†’@ βˆ‘>3- >3/ { )$ )* . ([). 𝛿𝑋 + - . . )!9 )!! . ([). (𝛿𝑋).} 76
Luc_Faucheux_2020 Chain Rule (Ito and Strato lemma)- II Β¨ 𝑓 𝑋 𝑑5 βˆ’ 𝑓 𝑋 𝑑6 = lim /β†’@ {βˆ‘>3- >3/ )$ )* . ∘ . 𝛿𝑋} Β¨ 𝑓 𝑋 𝑑5 βˆ’ 𝑓 𝑋 𝑑6 = lim /β†’@ βˆ‘>3- >3/ { )$ )* . ([). 𝛿𝑋 + - . . )!9 )!! . ([). (𝛿𝑋).} Β¨ We formally define in integral form: Β¨ 𝑓 𝑋 𝑑5 βˆ’ 𝑓 𝑋 𝑑6 = ∫+3+& +3+' 𝑑𝐹(π‘₯ 𝑑 ) Β¨ And ITO: Β¨ ∫+3+6 +3+5 𝐹 𝑋 𝑑 . ([). 𝑑𝑋(𝑑) = lim /β†’@ {βˆ‘>3- >3/ 𝐹(𝑋(𝑑>)). [𝑋(𝑑>?-) βˆ’ 𝑋(𝑑>)]} Β¨ And STRATO: Β¨ ∫+3+6 +3+5 𝐹 𝑋 𝑑 . (∘). 𝑑𝑋(𝑑) = lim /β†’@ {βˆ‘>3- >3/ 𝐹 [𝑋(𝑑> + 𝑋(𝑑>?-)]/2). [𝑋(𝑑>?-) βˆ’ 𝑋(𝑑>)]} 77
Luc_Faucheux_2020 Chain Rule (Ito and Strato lemma)- III Β¨ We then have for the chain rule using the ITO interpretation of the integral: Β¨ 𝑓 𝑋 𝑑5 βˆ’ 𝑓 𝑋 𝑑6 = lim /β†’@ βˆ‘>3- >3/ { )$ )* . ([). 𝛿𝑋 + - . . )!9 )!! . ([). (𝛿𝑋).} Β¨ ∫+3+& +3+' 𝑑𝐹(𝑋 𝑑 ) = ∫+3+6 +3+5 )$ )* . ([). 𝑑𝑋(𝑑) + - . ∫+3+6 +3+5 )!9 )!! 𝑋 𝑑 . (𝑑𝑋). Β¨ Note that all integrals are on : ∫+3+& +3+' () Β¨ HOWEVER, the increment of integrands are different, 𝑑𝐹, 𝑑𝑋 and 𝑑𝑑 Β¨ BOTH 𝐹 and 𝑋 are functions of 𝑑 ultimately, 𝐹(𝑋 𝑑 ) and 𝑋(𝑑) Β¨ So only in a formal manner we write: Β¨ 𝛿𝑓 = )$ )* . 𝛿𝑋 + - . . )!9 )!! . (𝛿𝑋)., which is the celebrated Ito lemma (chain rule) 78
Luc_Faucheux_2020 Chain Rule (Ito and Strato lemma)- IV Β¨ 𝛿𝑓 = )$ )* . 𝛿𝑋 + - . . )!9 )!! . 𝑏.. 𝛿𝑑, which is the celebrated Ito lemma (chain rule) Β¨ We somehow convinced ourselves at first that this was just using regular Taylor expansion but just keeping higher order terms, in particular the second order in 𝑋, because: Β¨ 𝛿𝑋. 𝛿𝑋~𝑏.. 𝛿𝑑 Β¨ But even though it looks formally the same, the only rigorous manner in which to write it is: Β¨ ∫+3+& +3+' 𝑑𝐹(𝑋 𝑑 ) = ∫+3+6 +3+5 )$ )* . ([). 𝑑𝑋(𝑑) + - . ∫+3+6 +3+5 )!9 )!! 𝑋 𝑑 . 𝑏 𝑋 𝑑 , 𝑑 .. 𝑑𝑑 Β¨ With the ITO interpretation of the integral: Β¨ ∫+3+6 +3+5 𝐹 𝑋 𝑑 . ([). 𝑑𝑋(𝑑) = lim /β†’@ {βˆ‘>3- >3/ 𝐹(𝑋(𝑑>)). [𝑋(𝑑>?-) βˆ’ 𝑋(𝑑>)]} 79
Luc_Faucheux_2020 Chain Rule (Ito and Strato lemma)- V Β¨ We then have for the chain rule using the STRATO interpretation of the integral: Β¨ 𝑓 𝑋 𝑑5 βˆ’ 𝑓 𝑋 𝑑6 = lim /β†’@ {βˆ‘>3- >3/ )$ )* . ∘ . 𝛿𝑋} Β¨ ∫+3+& +3+' 𝑑𝐹(𝑋 𝑑 ) = ∫+3+6 +3+5 )$ )* . ∘ . 𝑑𝑋(𝑑) Β¨ Note that all integrals are on : ∫+3+& +3+' () Β¨ HOWEVER, the increment of integrands are different, 𝑑𝐹, 𝑑𝑋 and 𝑑𝑑 Β¨ BOTH 𝐹 and 𝑋 are functions of 𝑑 ultimately, 𝐹(𝑋 𝑑 ) and 𝑋(𝑑) Β¨ So only in a formal manner we write: Β¨ 𝛿𝑓 = )$ )* . 𝛿𝑋, which is the celebrated STRATO lemma (chain rule) 80
Luc_Faucheux_2020 Chain Rule (Ito and Strato lemma)- VI Β¨ 𝛿𝑓 = )$ )* . 𝛿𝑋, which is the celebrated STRATO lemma (chain rule) Β¨ HOWEVER, the only rigorous manner in which to write it is: Β¨ ∫+3+& +3+' 𝑑𝐹(𝑋 𝑑 ) = ∫+3+6 +3+5 )$ )* . ∘ . 𝑑𝑋(𝑑) Β¨ With the STRATO interpretation of the integral: Β¨ ∫+3+6 +3+5 𝐹 𝑋 𝑑 . (∘). 𝑑𝑋(𝑑) = lim /β†’@ {βˆ‘>3- >3/ 𝐹 [𝑋(𝑑> + 𝑋(𝑑>?-)]/2). [𝑋(𝑑>?-) βˆ’ 𝑋(𝑑>)]} 81
Luc_Faucheux_2020 Chain Rule (Ito and Strato lemma)- VII Β¨ So sometimes we will just stick to the notation: Β¨ Ito lemma: Β¨ 𝛿𝑓 = )$ )* . ([). 𝛿𝑋 + - . . )!9 )!! . ([). 𝛿𝑋. Β¨ Just to remind us that we are using stochastic calculus and that it is NOT a usual product Β¨ Similarly when using STRATO we will sometimes use: Β¨ 𝛿𝑓 = )$ )* . ∘ . 𝛿𝑋 Β¨ To remind us that even if it looks like the regular chain rule, we are in the stochastic world where things are a little weird. Β¨ Again the only rigorous manner to deal with those is to always go back to the integrals, and the interpretation of is as a limit of a sum, which is rigorous. 82
Luc_Faucheux_2020 Chain Rule (Ito and Strato lemma)- VIII Β¨ Note that if the function 𝐹(𝑋 𝑑 ) has an explicit dependency on time 𝐹(𝑋 𝑑 , 𝑑) Β¨ Then for the time part the regular chain rule applies and we will obtain terms of the expression: )9 )+ . 𝑑𝑑 in BOTH ITO and STRATO Β¨ Because integrating over time is a normal Riemann integral, and both sums converge to the same limit Β¨ ∫+3+6 +3+5 )9 )+ . ∘ . 𝑑𝑑 = ∫+3+6 +3+5 )9 )+ . [ . 𝑑𝑑 = ∫+3+6 +3+5 )9 )+ . 𝑑𝑑 83
Luc_Faucheux_2020 84 Leibniz rule
Luc_Faucheux_2020 Leibniz rule - I Β¨ In most textbooks, it is usually also presented as Β¨ Hey do a Taylor expansion Β¨ Just make sure to keep the higher order terms Β¨ And you good Β¨ Surprisingly it is formally the same expression Β¨ Even more surprisingly, if you were to be working in a Strato world, textbooks would say: Β¨ Hey just use regular calculus Β¨ Since we have seen now that stochastic calculus can be tricky, let’s spend some time on convincing ourselves that we can adapt the Leibniz rule in stochastic calculus (it will also be super useful when changing numeraires in the Numeraire deck) 85
Luc_Faucheux_2020 Leibniz rule - II Β¨ It will be easier to do it once we have a more general expression for the ITO and STRATO integrals, so for now we will just state them (Leibniz rule for first order) Β¨ Leibniz rule in the REGULAR calculus: Β¨ 𝛿 𝑓𝑔 = 𝑔 )$ )* . 𝛿𝑋 + 𝑓 )( )* . 𝛿𝑋 Β¨ Leibniz rule in the ITO calculus: Β¨ 𝛿(𝑓𝑔) = 𝑔 )$ )* . ([). 𝛿𝑋 + 𝑔 - . . )!$ )*! . ([). 𝛿𝑋. + 𝑓 )( )* . ([). 𝛿𝑋 + 𝑓 - . . )!( )*! . ([). 𝛿𝑋. + )( )* . )$ )* . ([). 𝛿𝑋. Β¨ Leibniz rule in the STRATO calculus: Β¨ 𝛿 𝑓𝑔 = 𝑔 )$ )* . ∘ . 𝛿𝑋 + 𝑓 )( )* . ∘ . 𝛿𝑋 86
Luc_Faucheux_2020 87 A simple example Integrating X
Luc_Faucheux_2020 A worked out example to gain some intuition Β¨ ∫+3+6 +3+5 𝑓 𝑋 𝑑 . ([). 𝑑𝑋(𝑑) = lim /β†’@ {βˆ‘>3- >3/ 𝑓(𝑋(𝑑>)). [𝑋(𝑑>?-) βˆ’ 𝑋(𝑑>)]} Β¨ This is the definition of the Ito integral Β¨ Let’s try with the simple case of the function 𝑓(𝑋(𝑑>)) = 𝑋(𝑑>) Β¨ The Riemann sum is: 𝑆/ = βˆ‘>3- >3/ 𝑓(𝑋(𝑑>)). [𝑋(𝑑>?-) βˆ’ 𝑋(𝑑>)] Β¨ For this specific case: 𝑆/ = βˆ‘>3- >3/ 𝑋(𝑑>). [𝑋(𝑑>?-) βˆ’ 𝑋(𝑑>)] 88
Luc_Faucheux_2020 In the β€œregular” case – Riemann integral Β¨ ∫!3!6 !3!5 𝑓 π‘₯ . 𝑑π‘₯ = lim /β†’@ {βˆ‘>3- >3/ 𝑓(π‘₯>). [(π‘₯>?-) βˆ’ (π‘₯>)]} Β¨ This is the usual β€œtriangle” representation 89
Luc_Faucheux_2020 In the β€œregular” case – Riemann integral - II Β¨ In the regular case, Β¨ (π‘₯>?-) βˆ’ (π‘₯>) = 𝛿π‘₯ = !':!& / = ;* / Β¨ Another way to look at it, is π‘₯ = 𝑑, so at every point in time, Ξ΄π‘₯ = 𝛿𝑑 Β¨ In particular, Ξ΄π‘₯. = δ𝑑. 90 Xi i+1i-1 F(X) to integrate
Luc_Faucheux_2020 In the stochastic case, we cannot draw that picture Β¨ We cannot write (π‘₯>?-) βˆ’ (π‘₯>) = 𝛿π‘₯ = !':!& / = ;* / Β¨ If anything, what we can write is (π‘₯>?-) βˆ’ (π‘₯>) = 𝛿π‘₯> = ±𝛿π‘₯ Β¨ And so βˆ‘(π‘₯>?-) βˆ’ (π‘₯>) = βˆ‘ 𝛿π‘₯> = 0 Β¨ And βˆ‘[(π‘₯>?-) βˆ’ (π‘₯>)].= βˆ‘ 𝛿π‘₯> . = βˆ‘ 𝛿π‘₯. = 𝑁. 𝛿π‘₯. 91 Xi i+1i-1
Luc_Faucheux_2020 Another way to think about the difference Β¨ The regular case, the horizontal axis is the regular non-stochastic variable Β¨ Any integration of function is the regular Riemann integral Β¨ In the stochastic case, the horizontal axis becomes time, and the vertical is the stochastic variable X, and we will need to integrate a function of it over the vertical axis, but whereas time is regular, a stochastic process cannot be such that each step is just the interval divided by the number of steps 92 X(t) is stochastic Time t is regular F(X(t))tointegrate
Luc_Faucheux_2020 The regular case revisited Β¨ ∫!3!6 !3!5 𝑓 π‘₯ ([)𝑑π‘₯ = lim /β†’@ {βˆ‘>3- >3/ 𝑓(π‘₯>). [(π‘₯>?-) βˆ’ (π‘₯>)]} Β¨ ∫!3!6 !3!5 𝑓 π‘₯ ([)𝑑π‘₯ = lim /β†’@ βˆ‘>3- >3/ 𝑓(π‘₯>). 𝛿π‘₯> = lim /β†’@ !':!& / . βˆ‘>3- >3/ 𝑓(π‘₯>) Β¨ In the case where 𝑓(π‘₯>) = π‘₯> = π‘˜. 𝛿π‘₯ = π‘˜. !':!& / Β¨ ∫!3!6 !3!5 π‘₯([)𝑑π‘₯ = lim /β†’@ !':!& / . !':!& / . βˆ‘>3- >3/ π‘˜ = lim /β†’@ !':!& / . !':!& / . /(/?-) . Β¨ ∫!3!6 !3!5 π‘₯([)𝑑π‘₯ = (π‘₯5 βˆ’ π‘₯6). lim /β†’@ - / . - / . /(/?-) . Β¨ Using π‘₯6 = 0 without losing any generality, Β¨ ∫!34 !3* π‘₯([)𝑑π‘₯ = - . 𝑋., which is the usual result 93
Luc_Faucheux_2020 The regular case revisited – II – Strato convention Β¨ ∫!3!6 !3!5 𝑓 π‘₯ (∘)𝑑π‘₯ = lim /β†’@ {βˆ‘>3- >3/ 𝑓( !"#$?!" . ). [(π‘₯>?-) βˆ’ (π‘₯>)]} Β¨ ∫!3!6 !3!5 𝑓 π‘₯ (∘)𝑑π‘₯ = lim /β†’@ βˆ‘>3- >3/ 𝑓( !"#$?!" . ). 𝛿π‘₯> = lim /β†’@ !':!& / . βˆ‘>3- >3/ 𝑓( !"#$?!" . ) Β¨ In the case where𝑓( !"#$?!" . ) = !"#$?!" . = (π‘˜ + - . ). 𝛿π‘₯ = (π‘˜ + - . ). !':!& / Β¨ ∫!3!6 !3!5 π‘₯(∘)𝑑π‘₯ = lim /β†’@ !':!& / . !':!& / . βˆ‘>3- >3/ (π‘˜ + - . ) = lim /β†’@ !':!& / . !':!& / . [ / /?- . + / . ] Β¨ ∫!3!6 !3!5 π‘₯(∘)𝑑π‘₯ = (π‘₯5 βˆ’ π‘₯6). lim /β†’@ - / . - / . [ / /?- . + / . ] Β¨ Using π‘₯6 = 0 without losing any generality, Β¨ ∫!34 !3* π‘₯(∘)𝑑π‘₯ = - . 𝑋., which is the usual result 94
Luc_Faucheux_2020 The regular case revisited – III Β¨ So in the regular case, the Riemann integral gives the same result, irrespective of where inside the small interval we pick the value of the function Β¨ ∫!34 !3* π‘₯([)𝑑π‘₯ = - . 𝑋. = ∫!34 !3* π‘₯(∘)𝑑π‘₯ Β¨ In essence, this is because the terms βˆ‘>3- >3/ . 𝛿π‘₯> scale like 1, and so any terms in βˆ‘>3- >3/ . 𝛿π‘₯> . will go to 0 in the limit 𝑁 β†’ ∞ Β¨ The β€œquadratic variation” βˆ‘>3- >3/ . 𝛿π‘₯> . does not add up to any finite number in the limit. Β¨ In the case of a stochastic variable, the terms βˆ‘>3- >3/ . 𝛿π‘₯> scale like 0, because 𝛿π‘₯> = ±𝛿π‘₯, and the terms like βˆ‘>3- >3/ . 𝛿π‘₯> . will actually scale like time. Β¨ The β€œquadratic variation” βˆ‘>3- >3/ . 𝛿π‘₯> . is said to scale linearly with time (some textbooks use the expression additive with time) and will not β€œdisappear” to 0 in the limit. 95
Luc_Faucheux_2020 Back to the worked-out example Β¨ The Riemann sum is: 𝑆/ = βˆ‘>3- >3/ 𝑋(𝑑>). [𝑋(𝑑>?-) βˆ’ 𝑋(𝑑>)] Β¨ We can expand: [𝑋(𝑑>?-) βˆ’ 𝑋(𝑑>)].= 𝑋(𝑑>?-). + 𝑋(𝑑>). βˆ’ 2. 𝑋(𝑑>). 𝑋(𝑑>?-) Β¨ And: 𝑋(𝑑>). [𝑋(𝑑>?-) βˆ’ 𝑋(𝑑>)] = 𝑋(𝑑>). 𝑋(𝑑>?-) βˆ’ 𝑋(𝑑>). Β¨ And : 𝑋(𝑑>). [𝑋(𝑑>?-) βˆ’ 𝑋(𝑑>)] = - . . [𝑋(𝑑>?-). + 𝑋(𝑑>). βˆ’ [𝑋(𝑑>?-) βˆ’ 𝑋(𝑑>)].βˆ’2. 𝑋(𝑑>).] Β¨ Or: 𝑋(𝑑>). [𝑋(𝑑>?-) βˆ’ 𝑋(𝑑>)] = - . . [𝑋(𝑑>?-). βˆ’ 𝑋(𝑑>).] βˆ’ - . . [𝑋(𝑑>?-) βˆ’ 𝑋(𝑑>)]. Β¨ So: 𝑆/ = βˆ‘>3- >3/ 𝑋(𝑑>). [𝑋(𝑑>?-) βˆ’ 𝑋(𝑑>)] = - . . 𝑋(𝑑/?-). βˆ’ - . βˆ‘>3- >3/ [𝑋(𝑑>?-) βˆ’ 𝑋(𝑑>)]. Β¨ The first term is the expected *! . Β¨ The second term would usually disappear when the variable X is regular, because the sum would scale as 𝑁. ( - / ).~ - / which will tend to 0 when 𝑁 β†’ ∞ 96
Luc_Faucheux_2020 Back to the worked-out example - II Β¨ 𝑆/ = βˆ‘>3- >3/ 𝑋(𝑑>). [𝑋(𝑑>?-) βˆ’ 𝑋(𝑑>)] = - . . 𝑋(𝑑/?-). βˆ’ - . βˆ‘>3- >3/ [𝑋(𝑑>?-) βˆ’ 𝑋(𝑑>)]. Β¨ We define 𝑄/ = βˆ‘>3- >3/ [𝑋(𝑑>?-) βˆ’ 𝑋(𝑑>)]. Β¨ We can use here the simple β€œbinary” assumption that [𝑋(𝑑>?-) βˆ’ 𝑋(𝑑>)].= (𝑑>?-βˆ’π‘‘>) Β¨ Note that this is capturing the essence of the matter Β¨ A more accurate and general way to go about this would be to assume the following: Β¨ {𝑋(𝑑>?-) βˆ’ 𝑋(𝑑>)} follows the 𝑁(0, (𝑑>?-βˆ’π‘‘>)) distribution Β¨ We would then estimate the expected value of 𝑄/ Β¨ 𝐸 𝑄/ = βˆ‘>3- >3/ 𝐸[𝑋(𝑑>?-) βˆ’ 𝑋(𝑑>)]. = βˆ‘>3- >3/ (𝑑>?-βˆ’π‘‘>) = 𝑑/?- Β¨ We would have to show then that: lim /β†’@ 𝑄/ = 𝐸[𝑄/] Β¨ This opens up the can of worms of how you define convergence (convergence in distribution, in probability, almost sure convergence, Lp-convergence). Will try to incorporate later, but trying to keep it simple to not hide the intuition behind notations 97
Luc_Faucheux_2020 What did we get? Β¨ ∫+34 +3= 𝑓 𝑋 𝑑 . ([). 𝑑𝑋(𝑑) = lim /β†’@ {βˆ‘>3- >3/ 𝑓(𝑋(𝑑>)). [𝑋(𝑑>?-) βˆ’ 𝑋(𝑑>)]} Β¨ In the simple case 𝑓 𝑋(𝑑> = 𝑋(𝑑>) Β¨ ∫+34 +3= 𝑋(𝑑). ([). 𝑑𝑋(𝑑) = - . 𝑋. βˆ’ - . 𝑇 Β¨ A couple of notes: Β¨ Ito integral does NOT recover the usual rules of calculus Β¨ Ito integral is called a martingale, meaning if X is a martingale (E[X]=0), then the Ito integral of a function f(X) is also a martingale Β¨ 𝐸 𝑋 = 0 and 𝐸[∫+34 +3= 𝑋 𝑑 . [). 𝑑𝑋 𝑑 = 𝐸 - . 𝑋. βˆ’ - . 𝑇 = 𝐸 - . 𝑋. βˆ’ - . 𝑇 = 0 Β¨ So 𝐸 𝑋. = 𝑇 Β¨ We recover the usual variance of the Brownian motion 98
Luc_Faucheux_2020 A couple more notes Β¨ Let’s recall the relationship between the Ito and Stratonovitch integral (we can also work it out from the Riemann sum) Β¨ ∫+3+6 +3+5 𝑓 𝑋 𝑑 . (∘). 𝑑𝑋(𝑑) = ∫+3+6 +3+5 𝑓 𝑋 𝑑 . ([). 𝑑𝑋(𝑑) + - . ∫+3+6 +3+5 𝑓′ 𝑋 𝑑 . 𝑑𝑑 Β¨ Here, 𝑓 𝑋 = 𝑋 so 𝑓% 𝑋 = 1 Β¨ ∫+34 +3= 𝑓 𝑋 𝑑 . (∘). 𝑑𝑋(𝑑) = ∫+34 +3= 𝑓 𝑋 𝑑 . ([). 𝑑𝑋(𝑑) + - . ∫+34 +3= 1. 𝑑𝑑 Β¨ And we have: ∫+34 +3= 𝑋(𝑑). ([). 𝑑𝑋(𝑑) = - . 𝑋. βˆ’ - . 𝑇 Β¨ So ∫+34 +3= 𝑓 𝑋 𝑑 . (∘). 𝑑𝑋(𝑑) = - . 𝑋. Β¨ This is the usual rule of calculus Β¨ The Stratonovitch integral preserves the usual rule of calculus 99
Luc_Faucheux_2020 100 What does the usual exponential even mean in stochastic calculus? The ITO exponential
Luc_Faucheux_2020 A couple more notes - II Β¨ The Stratonovitch integral is NOT a martingale (this makes sense since taking the mid point introduces correlation, and thus the terms in the product are NOT independent) Β¨ Because the Ito lemma does NOT recover the usual rules of calculus, what a function actually means has to be at times redefined. Β¨ For example, we all know the exponential function 𝑓 𝑑 = exp(𝑑) Β¨ It is such that : 𝑓% 𝑑 = 𝑓(𝑑) Β¨ However we would not expect this function to preserve the same property when dealing with a stochastic variable (another way to say it is that this function is convex, and so the Jensen inequality will introduce a convexity correction, see the lecture on options) Β¨ We know that Stratonovitch will preserve the rules of calculus Β¨ So ∫+34 +3= 𝑒π‘₯𝑝 𝑋 𝑑 . ∘ . 𝑑𝑋 𝑑 = exp(𝑋(𝑇)) Β¨ And ∫+34 +3= 𝑓 𝑋 𝑑 . (∘). 𝑑𝑋(𝑑) = ∫+34 +3= 𝑓 𝑋 𝑑 . ([). 𝑑𝑋(𝑑) + - . ∫+34 +3= 𝑓′ 𝑋 𝑑 . 𝑑𝑑 101
Luc_Faucheux_2020 The Ito exponential function Β¨ ∫+34 +3= 𝑓 𝑋 𝑑 . (∘). 𝑑𝑋(𝑑) = 𝑒π‘₯𝑝 𝑋 𝑇 = ∫+34 +3= 𝑓 𝑋 𝑑 . ([). 𝑑𝑋(𝑑) + - . ∫+34 +3= 𝑓′ 𝑋 𝑑 . 𝑑𝑑 Β¨ 𝑒π‘₯𝑝 𝑋 𝑇 = ∫+34 +3= 𝑓 𝑋 𝑑 . ([). 𝑑𝑋(𝑑) + - . ∫+34 +3= 𝑒π‘₯𝑝 𝑋 𝑑 . 𝑑𝑑 Β¨ Since the second term is always positive, we have Β¨ ∫+34 +3= 𝑒π‘₯𝑝 𝑋 𝑑 . ([). 𝑑𝑋(𝑑) <> 𝑒π‘₯𝑝 𝑋 𝑇 Β¨ So clearly under Ito, the usual exponential function does not verify 𝐹% 𝑋 = 𝐹(𝑋) Β¨ For that reason, the convention is to rename some of the functions by specifying β€œIto” in front of them Β¨ This can be confusing at times 102
Luc_Faucheux_2020 The Ito exponential function -II Β¨ Ito lemma has : 𝛿𝐹 = )9 )* . 𝛿𝑋 + - . . )!9 )*! . 𝛿𝑋. + )9 )+ . 𝛿𝑑 Β¨ With 𝐹 𝑋 = exp 𝑋 , 𝐹% 𝑋 = exp 𝑋 , 𝐹%% 𝑋 = exp(𝑋) Β¨ We get 𝛿𝐹 = exp 𝑋 . 𝛿𝑋 + - . . exp 𝑋 . 𝛿𝑋. = exp(𝑋). 𝛿𝑋 + - . . exp(𝑋). 𝛿𝑑 Β¨ With 𝐹 𝑋, 𝑑 = exp(𝑋 βˆ’ + . ) , )9 )* = exp 𝑋 βˆ’ + . , )!9 )*! = exp 𝑋 βˆ’ + . , )9 )+ = :- . . exp(𝑋 βˆ’ + . ) Β¨ We get 𝛿𝐹 = exp 𝑋 βˆ’ + . . 𝛿𝑋 + - . . exp 𝑋 βˆ’ + . . 𝛿𝑋. βˆ’ - . . exp 𝑋 βˆ’ + . . 𝛿𝑑 Β¨ Or: 𝛿𝐹 = exp 𝑋 βˆ’ + . . 𝛿𝑋 = 𝐹 𝑋, 𝑑 . 𝛿𝑋 Β¨ So you will see sometimes the function 𝐹 𝑋, 𝑑 = exp 𝑋 βˆ’ + . being referred to as the Ito- exponential, whereas the regular exponential is of course 𝐹 𝑋 = exp 𝑋 103
Luc_Faucheux_2020 The Ito exponential function -III Β¨ So this is another indication that we should be careful with using regular functions and some of their properties. Β¨ In regular calculus, the exponential function 𝑓 π‘₯ = exp(π‘₯) is such that : 𝑓% π‘₯ = 𝑓(π‘₯) Β¨ In ITO calculus, the β€œITO exponential function” that still verifies: 𝐹% 𝑋(𝑑) = 𝐹(𝑋(𝑑)) is given by: 𝐹 𝑋(𝑑), 𝑑 = exp 𝑋(𝑑) βˆ’ + . Β¨ In STRATO calculus, the β€œSTRATO exponential function” that still verifies: 𝐹% 𝑋(𝑑) = 𝐹(𝑋(𝑑)) is given by: 𝐹 𝑋(𝑑), 𝑑 = exp 𝑋(𝑑) Β¨ Note that we are somehow lucky that those functions are still local (i.e. depends only on the value of 𝑋(𝑑) and 𝑑. It is not clear that we could not have ended up with a more complicated function that depends on the history or the path ∫+34 +3= 𝑋 𝑑 . 𝑑𝑑 for example. 104
Luc_Faucheux_2020 A nice little summary Name of Integral RIEMANN ITO STRATONOVITCH Type Deterministic Stochastic Stochastic Points of integration Doesn’t matter LEFT MIDDLE Martingale NO YES NO Non-Anticipating NO YES NO Usual Calculus Rule YES NO YES(*) Main Usage Everywhere Finance Physics 105 Β¨ (*) ONLY from a formal point of view. This is still a stochastic integral and a nasty beast not to be messed with β€œΓ  la lΓ©gΓ¨re”
Luc_Faucheux_2020 106 Back to trying to β€œintegrate” X
Luc_Faucheux_2020 Another example: can you integrate X ? Β¨ Somewhat cultural side note: In French β€œto integrate” (or β€œintegrer”) is the same word to perform a mathematical integration or to be accepted in a school. Β¨ One of the most prestigious schools is called β€œPolytechnique” or β€œX” because of the logo on their hats of two crossed swords (it was created and is still technically a military school, a little like West Point here, but starts around the junior college level) Β¨ So anyways, after high school, there are special schools called β€œclasses preparatoires” where you just study for the entrance exam to those advanced schools (β€œhautes ecoles”), with names like X, Ecole Normale Superieure (Normal Superieure school) or others with names that are usually tied to their original concentration: Ponts et Chaussees (bridges and sidewalks), Mines (mining), Supelec (Superior Electricity), and so on Β¨ Usually students spend around 2 years studying in β€œclasses preparatoires” before taking the entrance exam (everything is an exam, there is no application, you take the exam, you get ranked and then the school offers you a spot in the order of ranking). Β¨ So you get the idea, the question is how many years does it take you to integrate X Β¨ 50 years later, people will still remember what β€œhalfs” they were, or they will lie about it 107
Luc_Faucheux_2020 Another example: can you integrate X ? - II Β¨ If you are a genius and if it takes you only one year to integrate X, the student is called a ”one half” because: ∫*34 *3- 𝑋. 𝑑𝑋 = [ *! . ]*34 *3- = - . Β¨ I have heard of β€œone-half” I have never met one. Β¨ This is not the same as β€œhalf-bloods” from Harry Potter, I have never met one of those either Β¨ If you are a decent student and if it takes you two years to integrate X, the student is called a ”three half” because: ∫*3- *3. 𝑋. 𝑑𝑋 = [ *! . ]*3- *3. = C . βˆ’ - . = D . Β¨ (I was a 3/2, but I did not integrate X, my father did, my brother did, I am the black sheep of the family) Β¨ If you are a decent student and if it takes you three years to integrate X, the student is called a ”five half” because: ∫*3. *3D 𝑋. 𝑑𝑋 = [ *! . ]*3. *3D = E . βˆ’ C . = F . Β¨ If it takes you four years to integrate X, the student is called a ”seven half” because: ∫*3D *3C 𝑋. 𝑑𝑋 = [ *! . ]*3D *3C = -G . βˆ’ E . = H . Β¨ Being called a β€œseven half” is an insult 108
Luc_Faucheux_2020 Another example: can you integrate X ? - III Β¨ Oh also if you integrate 𝑋 not in 𝑑𝑋 but in 𝑑𝑑, you are called β€œendette”, or in debt Β¨ OK, so that was a little digression Β¨ Let’s get back to the matter at hands. Β¨ If 𝑋 is not stochastic (also called deterministic), this is the usual calculus that we are used to, and so: Β¨ ∫ 𝑋. 𝑑𝑋 = *! . + 𝐢 Β¨ What happens if 𝑋 is now a stochastic variable? Β¨ The truth is that I have never met anyone who can integrate a stochastic variable (I have also never met a β€œone half”), and so usually people always treat the problem from the other way, you start with a guess of what the integral is, you use ITO lemma and then you check in an iterative manner 109
Luc_Faucheux_2020 Another example: can you integrate X ? - IV Β¨ If we have a function 𝐹(𝑋), Ito lemma is the following: Β¨ Ito (1951) guessed that you can write 𝑑π‘₯ = π‘Ž π‘₯, 𝑑 . 𝑑𝑑 + 𝑏 π‘₯, 𝑑 . 𝑑𝑧 as Ξ΄π‘₯ = π‘Ž π‘₯, 𝑑 . 𝛿𝑑 + 𝑏 π‘₯, 𝑑 . 𝛿𝑧 Β¨ If 𝐹(π‘₯) a funcyon of x, the corresponding SDE as a result of β€œTaylor” expansion in ITO is: Β¨ Bear in mind that this is really not a Taylor expansion, it is just ITO lemma that looks like a Taylor expansion where you kept some terms and not others Β¨ 𝛿𝐹 = )9 )! . 𝛿π‘₯ + - . . )!9 )!! . 𝑏.. 𝛿𝑑 Β¨ 𝛿𝐹 = )9 )* . 𝛿𝑋 + - . . )!9 )*! . 𝛿𝑋. Β¨ Let’s use 𝐹 𝑋 = 𝑋. as a good guess Β¨ 𝐹 𝑋 = 𝑋., )9 )* = 2𝑋, )!9 )*! = 2, we then get: 𝛿 𝑋. = 2𝑋. 𝛿𝑋 + - . . 2. 𝛿𝑋. 110
Luc_Faucheux_2020 Another example: can you integrate X ? - V Β¨ 𝛿 𝑋. = 2𝑋. 𝛿𝑋 + - . . 2. 𝛿𝑋. Β¨ In the case of the Geometric Brownian motion (Hull): 𝑑𝑋 = πœŽπ‘‹π‘‘π‘Š Β¨ 𝛿 𝑋. = 2𝑋. 𝛿𝑋 + - . . 2. (πœŽπ‘‹). 𝛿𝑑 Β¨ 𝛿 𝑋. = 2𝑋. 𝛿𝑋 + (πœŽπ‘‹). 𝛿𝑑 Β¨ And so: ∫ 𝑋. 𝑑𝑋 = ∫{ ; *! . βˆ’ - . . (πœŽπ‘‹). 𝛿𝑑} Β¨ ∫ 𝑋. 𝑑𝑋 = [ *! . ] βˆ’ - . ∫(πœŽπ‘‹). 𝛿𝑑 Β¨ With the usual convention that 𝑋 𝑑 = 0 = 0, in the ITO convention: Β¨ ∫+34 +3= 𝑋. 𝑑𝑋 = *(=)! . βˆ’ - . ∫+34 +3= (πœŽπ‘‹). 𝛿𝑑 111
Luc_Faucheux_2020 Another example: can you integrate X ? - VI Β¨ Things are a little different in the Stratonovitch convention: Β¨ Strato guessed that you can write 𝑑π‘₯ = π‘Ž π‘₯, 𝑑 . 𝑑𝑑 + 𝑏 π‘₯, 𝑑 . π‘‘π‘Š as Ξ΄π‘₯ = π‘Ž π‘₯, 𝑑 . 𝛿𝑑 + 𝑏 π‘₯ + 𝛿π‘₯/2, 𝑑 . π›Ώπ‘Š Β¨ The SDE for 𝐹(π‘₯) in Stratonovitch convention is (using what is known as ITO calculus): Β¨ 𝛿𝐹 = )9 )! . 𝛿π‘₯ + - . . )!9 )!! . 𝑏.. 𝛿𝑑 + - . . )9 )! . )5 )! . 𝑏. 𝛿𝑑 Β¨ We know this is wrong, but let’s just illustrate how wrong it is: Β¨ Rewriting it without the drift term we get: Β¨ 𝛿𝑋 = 𝑏 𝑋 + ;* . , 𝑑 . π›Ώπ‘Š = 𝑏 𝑋 . π›Ώπ‘Š + - . . )5 )! . 𝛿𝑋. π›Ώπ‘Š Β¨ 𝛿𝑋 = 𝑏 𝑋 + ;* . , 𝑑 . π›Ώπ‘Š = 𝑏 𝑋 . π›Ώπ‘Š + - . . )5 )! . 𝑏 𝑋 . 𝛿𝑑 Β¨ 𝛿𝑋 = 𝑏 𝑋 . π›Ώπ‘Š + - . . )5 )! . 𝑏 𝑋 . 𝛿𝑑 in Stratonovitch while Ito had: 𝛿𝑋 = 𝑏 𝑋 . π›Ώπ‘Š 112
Luc_Faucheux_2020 Another example: can you integrate X ? - VII Β¨ Redoing it wrong just to convince ourselves of how wrong it is: Β¨ 𝛿𝑋 = 𝑏 𝑋 ∘ π›Ώπ‘Š = 𝑏 𝑋 + ;* . , 𝑑 . π›Ώπ‘Š = 𝑏 𝑋 . π›Ώπ‘Š + - . . )5 )! . 𝑏 𝑋 . 𝛿𝑑 Β¨ 𝛿𝐹 = )9 )* . 𝛿𝑋 + - . . )!9 )*! . 𝑏.. 𝛿𝑑 + - . . )9 )* . )5 )* . 𝑏. 𝛿𝑑 Β¨ 𝛿𝐹 = )9 )* . 𝛿𝑋 + - . . )!9 )*! . 𝛿𝑋. Β¨ 𝛿𝐹 = )9 )* . 𝑏 𝑋 + ;* . . π›Ώπ‘Š + - . . )!9 )*! . 𝑏 𝑋 + ;* . , 𝑑 . . 𝛿𝑑 Β¨ 𝛿𝐹 = )9 )* . 𝑏 𝑋 . π›Ώπ‘Š + )9 )* . )5 )* . ;* . . π›Ώπ‘Š + - . . )!9 )*! . 𝑏 𝑋 + ;* . , 𝑑 . . 𝛿𝑑 113
Luc_Faucheux_2020 Another example: can you integrate X ? - IX Β¨ We get: Β¨ 𝛿𝐹 = )9 )* . 𝑏 𝑋 . π›Ώπ‘Š + )9 )* . )5 )* . ;* . . π›Ώπ‘Š + - . . )!9 )*! . 𝑏 𝑋 + ;* . . . 𝛿𝑑 Β¨ And : Β¨ 𝑏 𝑋 + ;* . . = 𝑏 𝑋 . + 2. 𝑏 𝑋 . )5 )* . ;* . Β¨ 𝛿𝐹 = )9 )* . 𝑏 𝑋 . π›Ώπ‘Š + )9 )* . )5 )* . ;* . . π›Ώπ‘Š + - . . )!9 )*! . 𝑏 𝑋 .. 𝛿𝑑 + - . . )!9 )*! . 2𝑏 𝑋 . )5 )* . ;* . 𝛿𝑑 Β¨ Keeping only the terms in order 1 in 𝛿𝑍 and 𝛿𝑑 Β¨ 𝛿𝐹 = )9 )* . 𝑏 𝑋 . π›Ώπ‘Š + - . . )!9 )*! . 𝑏 𝑋 .. 𝛿𝑑 + - . . )9 )* . )5 )* . 𝑏. 𝛿𝑑 114
Luc_Faucheux_2020 Another example: can you integrate X ? - X Β¨ 𝛿𝐹 = )9 )* . 𝑏 𝑋 . π›Ώπ‘Š + - . . )!9 )*! . 𝑏 𝑋 .. 𝛿𝑑 + - . . )9 )* . )5 )* . 𝑏. 𝛿𝑑 Β¨ In the case of the simple Geometric Brownian motion (Hull): 𝑑𝑋 = πœŽπ‘‹π‘‘π‘Š Β¨ Using the guess 𝐹 𝑋 = 𝑋. Β¨ 𝛿(𝑋.) = 2𝑋. πœŽπ‘‹. π›Ώπ‘Š + - . . 2. (πœŽπ‘‹). 𝛿𝑑𝛿𝑑 + - . . 2𝑋. 𝜎. πœŽπ‘‹. 𝛿𝑑 Β¨ 𝛿(𝑋.) = 2𝑋. 𝛿𝑋 + 2. (πœŽπ‘‹).. 𝛿𝑑 Β¨ In Ito we had : 𝛿 𝑋. = 2𝑋. 𝛿𝑋 + - . . 2. (πœŽπ‘‹).. 𝛿𝑑 Β¨ So using ITO: ∫+34 +3= 𝑋. 𝑑𝑋 = *(=)! . βˆ’ - . ∫+34 +3= (πœŽπ‘‹).. 𝛿𝑑 Β¨ Using Stratonovitch: ∫+34 +3= 𝑋. 𝑑𝑋 = *(=)! . βˆ’ ∫+34 +3= (πœŽπ‘‹).. 𝛿𝑑 115
Luc_Faucheux_2020 Another example: can you integrate X ? - Xa Β¨ If we were to look at the simple Brownian motion case: Β¨ 𝑑𝑋 = π‘‘π‘Š Β¨ ITO: 𝛿 𝑋. = 2𝑋. 𝛿𝑋 + - . . 2. 𝛿𝑋. Β¨ 𝛿 𝑋. = 2𝑋. 𝛿𝑋 + 𝛿𝑑 Β¨ And so: ∫ 𝑋. 𝑑𝑋 = ∫{ ; *! . βˆ’ - . . 𝛿𝑑} Β¨ ∫ 𝑋. 𝑑𝑋 = [ *! . ] βˆ’ - . ∫ 𝛿𝑑 Β¨ With the usual convention that 𝑋 𝑑 = 0 = 0, in the ITO convention: Β¨ ∫+34 +3= 𝑋. 𝑑𝑋 = *(=)! . βˆ’ - . 𝑇 116
Luc_Faucheux_2020 Another example: can you integrate X ? - Xb Β¨ ∫+34 +3= 𝑋. 𝑑𝑋 = *(=)! . βˆ’ - . 𝑇 Β¨ Note that we know this to be consistent since the ITO integral is a martingale (or a trading strategy) Β¨ We will go over this again at more length but we have: Β¨ 𝔼 ∫+34 +3= 𝑋. 𝑑𝑋 = 0 = 𝔼 * = ! . βˆ’ 𝔼 - . 𝑇 = 𝔼 * = ! . βˆ’ - . 𝑇 Β¨ And we recover the usual dispersion expectation for the simple Brownian motion: Β¨ 𝔼 𝑋 𝑇 . = 𝑇 117
Luc_Faucheux_2020 Another example: can you integrate X ? - Xc Β¨ If we were to look at the simple Brownian motion case: Β¨ 𝑑𝑋 = π‘‘π‘Š 𝑏 𝑋 = 1 so )5 )* = 0 Β¨ STRATO: 𝛿𝐹 = )9 )* . 𝑏 𝑋 . π›Ώπ‘Š + - . . )!9 )*! . 𝑏 𝑋 .. 𝛿𝑑 + - . . )9 )* . )5 )* . 𝑏. 𝛿𝑑 Β¨ 𝛿 𝑋. = 2𝑋. 𝛿𝑋 + 𝛿𝑑 This is exactly the same as ITO Β¨ So we get the strange result that in BOTH ITO and STRATO we will recover for the simple Brownian motion: Β¨ ∫+34 +3= 𝑋. 𝑑𝑋 = *(=)! . βˆ’ - . 𝑇 Β¨ This is clearly wrong, and once again the error was in β€œusing a STRATO convention in the ITO calculus”, and not even that, as we took β€œITO calculus” as β€œdoing regular Taylor expansions like in regular calculus and just keeping some terms and neglecting some other terms” 118
Luc_Faucheux_2020 Another example: can you integrate X ? - XI Β¨ HOWEVER, remember the formula in the simple Wiener case : 𝑋 = π‘Š Β¨ 𝑆𝑇𝑅𝐴𝑇𝑂(𝑓) = 𝐼𝑇𝑂 𝑓 + - . . 𝑅𝐼𝐸𝑀𝐴𝑁𝑁(𝑓%) Β¨ In the case of 𝑓 = 𝑋, 𝑓% = 1 Β¨ 𝑆𝑇𝑅𝐴𝑇𝑂 𝑋 = 𝐼𝑇𝑂 𝑋 + - . . 𝑅𝐼𝐸𝑀𝐴𝑁𝑁(1) with Β¨ 𝑅𝐼𝐸𝑀𝐴𝑁𝑁 1 = ∫+34 +3= 1. 𝑑𝑑 = 𝑇 Β¨ So Ito : ∫+34 +3= 𝑋. 𝑑𝑋 = *(=)! . βˆ’ - . ∫+34 +3= 1. 𝛿𝑑 or ∫+34 +3= 𝑋. 𝑑𝑋 = *(=)! . βˆ’ - . 𝑇 Β¨ Stratonovitch: ∫+34 +3= 𝑋 ∘ 𝑑𝑋 = ∫+34 +3= 𝑋. 𝑑𝑋 + - . 𝑇 = *(=)! . βˆ’ - . 𝑇 + - . 𝑇 = *(=)! . Β¨ Startonovitch as expected follows in a formal manner the usual rules of calculus 119
Luc_Faucheux_2020 Another example: can you integrate X ? - XII Β¨ So clearly we failed to integrate X, because we found the following wrong results: Β¨ In the Geometric Brownian motion case, Strato does not match formally the usual rules of calculus Β¨ In the simple Brownian motion case, BOTH ITO and STRATO returns the same result Β¨ This is not possible as we know that the ITO and STRATO integrals are different and we know what the difference is ( - . . 𝑅𝐼𝐸𝑀𝐴𝑁𝑁) Β¨ But again, apologies if so far the mistake was obvious, but NEVER rely on usual rules of calculus, or think that Stratanovitch is just β€œtaking the middle point” and that you can still use ITO lemma or Taylor expansion in a nested manner. If you choose STRATO you CANNOT use ITO calculus, you have to stay in STRATO calculus, and vice versa Β¨ Also note that ITO lemma is very generic in (𝛿𝑋). but when choosing a specific expression for𝑋(𝑑) as a function of the Brownian motion (Wiener process π‘Š(𝑑)), we end up with very different expressions, some tractable, some not so much 120
Luc_Faucheux_2020 Another example: can you integrate X ? - XIII Β¨ The right way to do it is of course: Β¨ ITO: 𝛿𝑓 = )$ )* . ([). 𝛿𝑋 + - . . )!9 )!! . ([). 𝛿𝑋. Β¨ The results we obtained in the preceding slides for ITO are still correct Β¨ STRATO: 𝛿𝑓 = )$ )* . ∘ . 𝛿𝑋 Β¨ And then we do get in the case of the simple Brownian motion: 𝑑𝑋 = π‘‘π‘Š Β¨ 𝐹 𝑋 = 𝑋. Β¨ 𝛿𝑓 = 2𝑋. ∘ . π›Ώπ‘Š Β¨ ∫+34 +3= 𝑋. ∘ . 𝑑𝑋 = - . ∫+34 +3= 2𝑋. ∘ . π›Ώπ‘Š = - . ∫+34 +3= 𝑑 𝑋. = [ *! . ]+34 +3= Β¨ As expected if STRATO follows formally the usual rules of calculus 121
Luc_Faucheux_2020 Quick notes Β¨ Usually when dealing with an SDE, people will try Β¨ 1) get back to an SDE where the stochastic scaling does not depend on the stochastic variable. For example, when dealing with 𝑑𝑆 = πœ‡. 𝑆. 𝑑𝑑 + 𝜎. 𝑆. π‘‘π‘Š, the natural road to explore is writing something like I< < = πœ‡. 𝑑𝑑 + 𝜎. π‘‘π‘Š, and then try to ”define” what I< < is. Β¨ It seems intuitive that I< < should have something to do with 𝑑(ln 𝑆 ), so we would love to write something like 𝑑(𝑙𝑛 𝑆 ) = πœ‡. 𝑑𝑑 + 𝜎. π‘‘π‘Š, but it turns out that is not quite right, because 𝑙𝑛 𝑆 is convex as a function of 𝑆, and 𝑆 is stochastic, not deterministic Β¨ So always re-derive ITO lemma to make sure you are not dropping terms in the β€œTaylor expansion” Β¨ 2) try to get rid of the term in from of the 𝑑𝑧, because then you are not dealing with a SDE anymore, but with a PDE. This is what Black Sholes did by adding a β€œdelta hedge”, and off to a Nobel prize they went 122
Luc_Faucheux_2020 Quick notes - II Β¨ We also see the famous β€œconvexity adjustment” that we looked at in the Options deck. Β¨ 𝑙𝑛 𝑆 is negatively convex as a function of the stochastic variable Β¨ So from an intuition point of view as we saw Β¨ < 𝑙𝑛 𝑆 > β‰  𝑙𝑛 < 𝑆 > Β¨ Actually we know that : < 𝑙𝑛 𝑆 > = 𝔼(𝑙𝑛 𝑆 ) ≀ 𝑙𝑛 𝔼(𝑆) = 𝑙𝑛 < 𝑆 > Β¨ So it would make sense that around the point 𝔼(𝑆) the function 𝑙𝑛 𝑆 would not behave as the β€œregular” function 𝑙𝑛 𝔼(𝑆) Β¨ Note that the distinction though, the convexity adjustment was coming from integrating over the possible outcomes at a given time of the stochastic variable. Here we are integrating over the time (over the path over time of the stochastic variable). They are related nonetheless. Β¨ Obviously if you have the expression that resulted from integrating over the path, you can now use it to integrate over the distribution 123
Luc_Faucheux_2020 Quick notes - III Β¨ Just to make it explicit. Β¨ ∫+3+& +3+' 𝑑𝐹(𝑋 𝑑 ) = ∫+3+6 +3+5 )9 )* . ([). 𝑑𝑋(𝑑) + - . ∫+3+6 +3+5 )!9 )!! 𝑋 𝑑 . (𝑑𝑋). Β¨ 𝛿𝐹 = )9 )* . 𝛿𝑋 + - . . )!9 )!! . (𝛿𝑋)., which is the celebrated Ito lemma (chain rule) Β¨ 𝐹 𝑋 = ln(𝑋), )9 )* = 1/𝑋, )!9 )*! = βˆ’1/𝑋. Β¨ In the case of the simple Brownian motion: 𝑑𝑋 = π‘‘π‘Š Β¨ 𝛿(π‘™π‘›π‘Š) = - 1 . π›Ώπ‘Š βˆ’ - . . - 1! . 𝛿𝑑 this is quite ugly to deal with Β¨ In the case of the Geometric Brownian motion (Hull) 𝑑𝑋 = 𝑋. π‘‘π‘Š Β¨ 𝛿 𝑙𝑛𝑋 = - * . 𝑋. π›Ώπ‘Š βˆ’ - . . - *! . 𝑋.. 𝛿𝑑 = π›Ώπ‘Š βˆ’ - . . 𝛿𝑑 this is quite nice and easy to deal with 124
Luc_Faucheux_2020 Quick notes - IV Β¨ So for the log function and in the case of a geometric Brownian motion, we have Β¨ 𝛿 𝑙𝑛𝑋 = π›Ώπ‘Š βˆ’ - . . 𝛿𝑑 Β¨ Or more rigorously: Β¨ ∫+3+& +3+' 𝑑𝐿𝑁(𝑋 𝑑 ) = 𝐿𝑛 𝑋 𝑑5 βˆ’ 𝐿𝑛(𝑋 𝑑6 ) = Β¨ 𝐿𝑛 𝑋 𝑑5 βˆ’ 𝐿𝑛 𝑋 𝑑6 = ∫+3+6 +3+5 π‘‘π‘Š 𝑑 βˆ’ - . ∫+3+6 +3+5 𝑑𝑑 = π‘Š 𝑑5 βˆ’ π‘Š 𝑑6 βˆ’ - . (𝑑5 βˆ’ 𝑑6) Β¨ One could be tempted to identify - . . 𝛿𝑑 as a convexity adjustment (which it is in some way, since if the function was not convex, i.e. )!9 )*! = 0, this term would not appear), but it is not exactly the one we are dealing with in the option deck, as that one also depends on the specific distribution being used. 125
Luc_Faucheux_2020 Quick notes - V Β¨ Still rather crudely (but rigorous, as we will show in section V on the GBM) Β¨ 𝑑𝑋 = 𝑋. π‘‘π‘Š careful, this is not the same as writing 𝑋 𝑑 = exp(π‘Š 𝑑 ) Β¨ 𝔼 𝑋 𝑑5 = 𝑋 𝑑6 Β¨ 𝐿𝑛 𝑋 𝑑5 βˆ’ 𝐿𝑛 𝑋 𝑑6 = ∫+3+6 +3+5 π‘‘π‘Š 𝑑 βˆ’ - . ∫+3+6 +3+5 𝑑𝑑 = π‘Š 𝑑5 βˆ’ π‘Š 𝑑6 βˆ’ - . (𝑑5 βˆ’ 𝑑6) Β¨ 𝔼 𝐿𝑛 𝑋 𝑑5 βˆ’ 𝐿𝑛 𝑋 𝑑6 = 𝔼{π‘Š 𝑑5 βˆ’ π‘Š 𝑑6 βˆ’ - . (𝑑5 βˆ’ 𝑑6)} Β¨ 𝔼 𝐿𝑛 𝑋 𝑑5 βˆ’ 𝐿𝑛 𝑋 𝑑6 = 𝔼 βˆ’ - . 𝑑5 βˆ’ 𝑑6 = βˆ’ - . (𝑑5 βˆ’ 𝑑6) Β¨ 𝔼 𝐿𝑛 𝑋 𝑑5 = 𝔼 𝐿𝑛 𝑋 𝑑6 βˆ’ - . 𝑑5 βˆ’ 𝑑6 = 𝐿𝑛 𝑋 𝑑6 βˆ’ - . (𝑑5 βˆ’ 𝑑6) Β¨ 𝔼 𝐿𝑛 𝑋 𝑑5 = 𝐿𝑛 𝔼 𝑋 𝑑5 βˆ’ - . (𝑑5 βˆ’ 𝑑6) Β¨ This is indeed the convexity adjustment (again makes sense, it is the second order) 126
Luc_Faucheux_2020 Quick notes - VI Β¨ Just to make it explicit. Β¨ ∫+3+& +3+' 𝑑𝐹(𝑋 𝑑 ) = ∫+3+6 +3+5 )9 )* . ([). 𝑑𝑋(𝑑) + - . ∫+3+6 +3+5 )!9 )!! 𝑋 𝑑 . (𝑑𝑋). Β¨ 𝛿𝐹 = )9 )* . 𝛿𝑋 + - . . )!9 )!! . (𝛿𝑋)., which is the celebrated Ito lemma (chain rule) Β¨ 𝐹 𝑋 = 𝑋., )9 )* = 2𝑋, )!9 )*! = 2 Β¨ In the case of the simple Brownian motion: 𝑑𝑋 = π‘‘π‘Š Β¨ 𝛿(π‘Š.) = 2π‘Š. π›Ώπ‘Š + - . . 2. 𝛿𝑑 Β¨ In the case of the Geometric Brownian motion (Hull) 𝑑𝑋 = 𝑋. π‘‘π‘Š Β¨ 𝛿 𝑋. = 2𝑋. 𝑋. π›Ώπ‘Š + - . . 2. 𝑋.. 𝛿𝑑 = 2 𝑋. π›Ώπ‘Š + (𝑋.). 𝛿𝑑 127
Luc_Faucheux_2020 Quick notes - VII Β¨ So for the square function and in the case of a simple Brownian motion, we have Β¨ 𝛿(π‘Š.) = 2π‘Š. π›Ώπ‘Š + - . . 2. 𝛿𝑑 Β¨ Or more rigorously: Β¨ ∫+3+& +3+' π‘‘π‘Š. = π‘Š.(𝑑5) βˆ’ π‘Š.(𝑑6) = Β¨ π‘Š. 𝑑5 βˆ’ π‘Š. 𝑑6 = ∫+3+6 +3+5 2π‘Š 𝑑 . [ . π‘‘π‘Š 𝑑 + ∫+3+6 +3+5 𝑑𝑑 Β¨ π‘Š. 𝑑5 βˆ’ π‘Š. 𝑑6 = ∫+3+6 +3+5 2π‘Š 𝑑 . ([). π‘‘π‘Š 𝑑 + (𝑑5 βˆ’ 𝑑6) 128
Luc_Faucheux_2020 Quick notes - VIII Β¨ Still rather crudely (but rigorous, as we will show in section V on the GBM) Β¨ 𝑑𝑋 = π‘‘π‘Š Β¨ 𝔼 π‘Š 𝑑5 = π‘Š 𝑑6 Β¨ π‘Š. 𝑑5 βˆ’ π‘Š. 𝑑6 = ∫+3+6 +3+5 2π‘Š 𝑑 . ([). π‘‘π‘Š 𝑑 + 𝑑5 βˆ’ 𝑑6 Β¨ 𝔼 π‘Š. 𝑑5 βˆ’ π‘Š. 𝑑6 = 𝔼{∫+3+6 +3+5 2π‘Š 𝑑 . ([). π‘‘π‘Š 𝑑 + 𝑑5 βˆ’ 𝑑6 } Β¨ 𝔼 π‘Š. 𝑑5 βˆ’ π‘Š. 𝑑6 = 𝔼 𝑑5 βˆ’ 𝑑6 = (𝑑5 βˆ’ 𝑑6) Β¨ 𝔼 π‘Š. 𝑑5 = 𝔼 π‘Š. 𝑑6 + 𝑑5 βˆ’ 𝑑6 = π‘Š. 𝑑6 + (𝑑5 βˆ’ 𝑑6) Β¨ 𝔼 π‘Š. 𝑑5 = 𝔼 π‘Š 𝑑5 . + (𝑑5 βˆ’ 𝑑6) Β¨ This is indeed the convexity adjustment (again makes sense, it is the second order) Β¨ Since the square function is positively convex we would expect the convexity adjustment to be positive 129
Luc_Faucheux_2020 130 From Ordinary Differential Equations to SDE and SIE
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Lf 2020 stochastic_calculus_ito-i

  • 1. Luc_Faucheux_2020 Stochastic Calculus - Part I Integrals - Ito lemma – SDEs and SIEs 1
  • 2. Luc_Faucheux_2020 What is this class, and what it is not Β¨ Not a formal, so please interrupt if any question Β¨ More of a pragmatic approach on how to approach stochastic calculus Β¨ I have tried as much as possible to alert when there is something to be careful about Β¨ I have also tried as much as possible to be as rigorous as possible, without getting lost in the notations, or being too formal just for the sake of being formal Β¨ Those notes are more β€œnotes of a practitioner”, and by no means I would dare to hope to substitute a robust course in stochastic calculus Β¨ Those slides originated from a class I taught in 2018 at the hedge fund DRW to their first year associate class, 40 students or so from various backgrounds. The class was a general Fixed-Income class over 8 full days. It was intense, exhilarating, and kept me on my toes the whole time Β¨ Starting from the usual Ito lemma as in most textbooks (Hull), play with it for a while, then start again from the proper formulation of the stochastic integrals 2
  • 3. Luc_Faucheux_2020 How those slides came about Β¨ Those slides originated from a class I taught in 2018 at the hedge fund DRW in the great city of Chicago. I have modified them and added to them over the past couple of years. Β¨ The class was composed of 40 students or so with various backgrounds, ranging from computer science students with almost no background in Finance, to recent graduates of Masters in Math, to some graduates of the prestigious MSCF (Master of Science in Computational Finance) at Carnegie Melon University under the guidance of Steven Shreve, to some who had almost no mathematical background in options, bond math, random processes but had advanced degrees in Economy, so it was rather a tricky bunch Β¨ The class was a general Fixed-Income class over 8 full days. It was intense, exhilarating, and kept me on my toes the whole time. It pushed me to realize what I had not understood about stochastic processes for 20 years or so, because I never bothered to ask the β€œwhat if” questions and took a lot for granted out of sheer intellectual laziness Β¨ The textbook we used was Hull so you will see pages reference to this book, as we used it in class as a starting point to further explorations of the derivative pricing theory. 3
  • 4. Luc_Faucheux_2020 How those slides came about - II Β¨ It is often said that the only person who learns anything out of a class is the teacher. Β¨ I hope that my DRW students got something out of it. Β¨ But I know for a fact that without those two weeks in Chicago teaching, I would not have gotten those slides off the ground, and most of the materials would still be in disparate pieces of papers flying around my desk. Β¨ Hopefully the end result is not complete garbage, at least I know that I greatly enjoyed putting those together. Β¨ So, even if the end result is not up to your standards, I am extremely grateful to DRW for having given me the opportunity to teach the associate class that summer of 2018, and even more so grateful to the students who during those two weeks, took me out of my comfort zone, and forced me to confront what I knew and what I realized I was rather ignorant of. Β¨ When you are teaching in front of a bunch of super smart people with different backgrounds for 8 days straight, there is no hiding behind the curtain Β¨ So thanks again to DRW and the 2018 associate class ! I miss you guys. 4
  • 5. Luc_Faucheux_2020 How those slides came about - III Β¨ Also for those of you who have spent more than 5 minutes with me, you will have noticed that I bring the Ito-Stratonovitch controversy a lot. Β¨ First of all that is sort of a hobby of mine, ever since my PhD thesis (Appendix B) Β¨ Second, I have found it to be quite enlightening, because everyone takes Ito for granted, and then you ask the question β€œwhat if”, and that forces you to make sure that your understanding of Ito was solid. So I am using the example of Stratonovitch to really test the fact that my understanding of ITO is solid Β¨ Apologies on the Powerpoint format, there are battles with Microsoft that you cannot win (including the fact that the Solver is a Macro and not a function, as opposed to some of the earlier spreadsheets like Wingz) Β¨ At times it might feel like going down the rabbit hole, or going up the river to meet Kurtz, but (at least in my experience), I have found those discursions to be useful. Β¨ So this is not really something great about Stratonovitch, it is using Stratonovitch to convince ourselves that we understand ITO 5
  • 6. Luc_Faucheux_2020 How those slides came about - IV Β¨ Again quite frankly those notes are more for me (sorry) Β¨ I had noticed that I had a number of handwritten notes flying around my desk, and when needed at times I would just rederive from scratch rather then finding the right one. Β¨ So this is trying to put in one place, in a format that I can copy/past easily, most of what I had to go through and still use. I have tried (and failed at many places) to keep the notation consistent Β¨ Again, this is from a β€œpractitioner” point of view, I have tried to be rigorous when I felt it was needed, and be less so when I felt that this was a detail that was not needed and at times would obscure the intuition. Β¨ Notations are hard to keep consistent. I have also find at times that the right notation can be illuminating, whereas the full explicit one can be cumbersome. So I have tried to adapt the notation to what was needed to grasp the concept, and at times be more careful about it to make sure that we do not fall into a trap. Β¨ I guess Godel knew that, the right notation can change completely the proof…. 6
  • 7. Luc_Faucheux_2020 If you really want to master stochastic calculus. Β¨ The uncontested bible in the field of stochastic calculus for Finance. Quite dense and concise. It sometimes take me 40 pages to understand what Steven Shreve does in 2 lines. 7
  • 8. Luc_Faucheux_2020 If you really want to master stochastic calculus - II Β¨ An absolute wonderful short book. The notations at times are infuriating, but an absolute must read 8
  • 9. Luc_Faucheux_2020 If you really want to master stochastic calculus - III Β¨ If you are like me coming from a Physics background, this book is still relevant today, a testament to the genius of Van Kampen 9
  • 10. Luc_Faucheux_2020 If you really want to master stochastic calculus - IV Β¨ You cannot ignore this book. Every sentence carries meaning, and is worth reading time and time again. 10
  • 11. Luc_Faucheux_2020 If you really want to master stochastic calculus - V Β¨ Another wonderful short book. The logic is clear, concise and beautiful. The exercises are worth going through. From one of the most respected options traders in the field. He also has quite a ferocious appetite for chocolate cakes. 11
  • 12. Luc_Faucheux_2020 If you really want to master stochastic calculus - VI Β¨ Quite applied. The appendix on SDEs is rather beautiful, and follows Mikosch. Great for practical applications in the field of Fixed-Income 12
  • 13. Luc_Faucheux_2020 If you really want to master stochastic calculus - VII Β¨ What not to say about this book? An absolute gem. The 1900 Ph.D. thesis of Louis Bachelier (in French!) with an amazing translation by Davis and Etheridge, and some great chapters about the history of modern finance. Bachelier did it all, 80 years or so before everyone else 13
  • 14. Luc_Faucheux_2020 If you really want to master stochastic calculus - VIII Β¨ I could not but not add this one here. It is a movie about the life of Vincent (Wolfgang) Doblin (Doebling) who essentially discovered Ito calculus at least 5 to 10 years before Ito in circumstances so incredible that they made a movie out of his life. Ito lemma and calculus is now usually referred to as Ito-Doblin lemma and calculus in recognition of Vincent’s incredible accomplishments 14
  • 15. Luc_Faucheux_2020 If you really want to master stochastic calculus - IX Β¨ Amazing book if you are coming from a Physics background 15
  • 16. Luc_Faucheux_2020 If you really want to master stochastic calculus - X Β¨ Found this book as I was almost finished with those slides, and thought about throwing them away, because this book has pretty much anything you want. Pretty heavy on the operators formalism, which is quite elegant once you get used to it, but that it a step that you need to go through 16
  • 17. Luc_Faucheux_2020 What is so hard about Stochastic Calculus? Β¨ It is quite recent Β¨ It is quite cumbersome Β¨ It is not intuitive Β¨ It is incomplete Β¨ No one really knows how to do it. Β¨ This presentation is trying to strike a balance between being practical and being rigorous, so apologies for the many terms in β€œβ€, whereas a rigorous math class will define what terms like β€œstable” or β€œgot to 0” or β€œgo to infinity” or β€œconverge” much more exactly. What we will say is β€œalmost” rigorous and works in practice 99% of the time Β¨ This is more of a practitioner’s point of view on how to use (or not) stochastic calculus, as it relates to finance and Physics 17
  • 18. Luc_Faucheux_2020 Stochastic Calculus is quite recent Β¨ Geometry ~ -6,000 BC (navigating looking at the stars, buildings, ships,…) Β¨ Fractions (music with scale, buildings,,) Β¨ Probabilities (~1,600 AD), Pascal triangle, combinatory analysis Β¨ Calculus (Newton, Leibniz, finite differences) (~1,600 AD) (we started shooting canons long range, Electricity, magnets, how do they work? The ICP is still asking) Β¨ Taylor Expansion (1720) Β¨ Brown (1890), Wiener (1940), Bachelier (1900), Einstein (1905),Langevin (1908), Doblin(1940), Ito (1950), Feynman-Kac (1950), Stratonovitch (1966), Black-Sholes (1972) Β¨ So let’s go a little easy on ourselves, shall we? Β¨ Quantum Stochastic Calculus (1980), random processes (diffusion) in fractal geometries 18
  • 19. Luc_Faucheux_2020 Stochastic calculus is a French thing (except Gauss) Β¨ Black-Sholes might have gotten a Nobel prize in 1972, but Louis Bachelier did it all in his Ph.D. thesis in 1900 (almost) Β¨ Ito might have been known until recently for the Ito calculus and Ito lemma, but Vincent Doelin wrote it all while on the Ardennes front in World War I. This is now being recognized and some textbooks use the term β€œIto-Doeblin” instead of β€œIto” Β¨ Paul Langevin also essentially wrote the textbook on SDEs Β¨ And for the math, all you need is Laplace, Fourier, Cauchy Β¨ Taylor expansion is the only non-French, but it is essentially L’Hospital rule, so again…French… Β¨ Note: L’Hopital rule is very powerful and often overlooked. Β¨ If two functions 𝑓(π‘₯) and 𝑔(π‘₯) and are differentiable, and lim !β†’# [ $%(!) (%(!) ] exists, then Β¨ lim !β†’# [ $(!) ((!) ] = lim !β†’# [ $%(!) (%(!) ] 19
  • 20. Luc_Faucheux_2020 The Ph.D. thesis of Louis Bachelier Β¨ Reading the original thesis (both in French if you can and the excellent translation by Mark Davis and Alison Etheridge) is humbling. Β¨ Without a strong well-developed theory of stochastic calculus (Ito lemma) that only came about in the 1960s or so Β¨ Without a strong theoretical footing of what is a numeraire and how to price a derivative in the risk-neutral probability associated to that numeraire (Pliska 1980 or so) Β¨ Without yet the strong connection between PDE (Partial Differential Equations) and SDE (Stochastic Differential Equations) that really came about from the Feynman-Kac formula (1950 roughly) Β¨ Louis Bachelier managed to not only built a theory of option pricing that is nowadays coming back in fashion with a vengeance, but perusing through the rather short thesis, one cannot but be amazed at the breadth of his genius, but also at his attention to details. Bachelier at times go through numerical examples with the same precision and clarity of thoughts that he displays in the other more theoretical parts of his thesis. 20
  • 21. Luc_Faucheux_2020 Stochastic Calculus is not intuitive Β¨ Regular calculus has usually to do with ”things that are around some other things” Β¨ Taylor expansion and derivatives (expansion around a value, local derivatives at or around a point) Β¨ Finite difference for integrating functions on an axis or a path (keep following the path in a continuous fashion) Β¨ Stochastic Calculus tries to address the problem of dealing with β€œthings around things that are not there”. What do I mean ? Β¨ Take the coin flipping problem (Head is +1, Tail is -1). The coin is either head or tail (+1 or - 1), never anything else. And yet we will try to calculate expansions, derivations, integrations of functions around the mean or average (0), which is NOT a possible state of the coin. Β¨ Without stating the obvious or oversimplifying, this is the crux of the problem with stochastic calculus, and also that we are so used to usual calculus that we take a lot of things for granted 21
  • 22. Luc_Faucheux_2020 Stochastic processes are β€œnon-differentiable” Β¨ A stochastic process essentially β€œflips a coin” at each point in time. Β¨ A β€œregular” process, meaning it is differentiable, would have a unique tangent for every point in time. If 𝑋 𝑑 is differentiable, there is a unique )*(+) )+ Β¨ The stochastic process does NOT have a unique tangent, because, using the coin flip idea, and being somewhat liberal with scaling, at each point in time, 𝑋 𝑑 goes to either {𝑋 𝑑 + 𝛿𝑋} or {𝑋 𝑑 βˆ’ 𝛿𝑋} with some probability (50% in the simplest case, or driftless case). Β¨ The tangent is NOT the average (0) of the two possible tangents. Β¨ So in essence for a stochastic process, writing something like )*(+) )+ is meaningless Β¨ In stochastic processes, the only real thing that we can do is integrals, almost never differential calculus (not surprising as it is not differentiable) Β¨ NOTE that non-differentiable does NOT mean not smooth or not continuous 22
  • 23. Luc_Faucheux_2020 Stochastic calculus is usually self-similar Β¨ We will go over this in more details, but essentially the simplest stochastic process is the Wiener process or also called the standard Brownian motion. We will start with it Β¨ π‘Š 𝑑 has a normal 𝑁(0, 𝑑) distribution function Β¨ π‘Š 𝑑 βˆ’ 𝑠 has obviously the same normal 𝑁(0, 𝑑 βˆ’ 𝑠) distribution function Β¨ {π‘Š 𝑑 βˆ’ π‘Š(𝑠)} has ALSO the same 𝑁(0, 𝑑 βˆ’ 𝑠) distribution Β¨ Note that they are NOT the same, writing π‘Š 𝑑 βˆ’ π‘Š 𝑠 = π‘Š(𝑑 βˆ’ 𝑠) is obviously wrong but you will find sometimes the notation: Β¨ π‘Š 𝑑 βˆ’ π‘Š 𝑠 ≝ π‘Š(𝑑 βˆ’ 𝑠), meaning distributional identity, NOT pathwise identity Β¨ The simple Brownian motion is self-similar with coefficient (1/2) Β¨ 𝑇, π‘Š 𝑑- , 𝑇, π‘Š 𝑑. , . . , 𝑇, π‘Š 𝑑/ ≝ π‘Š 𝑇𝑑- , π‘Š 𝑇𝑑. , . . , π‘Š 𝑇𝑑/ , with 𝐻 = 1/2 Β¨ This is sometimes used to numerically construct Brownian motion. Β¨ If you β€œscale” up the time scale by a factor 𝑇, the space scale gets only magnified by a factor 𝑇. 23
  • 24. Luc_Faucheux_2020 Stochastic Calculus is cumbersome Β¨ Usual knowledge and tricks of calculus do not apply anymore Β¨ Chain rule does not work Β¨ Functional derivation does not work : 𝑑 𝑙𝑛𝑆 β‰  ( ⁄𝑑𝑆 𝑆) ! Β¨ Integration and especially derivations are not well defined Β¨ Usual calculus π‘Š(𝑑), when (𝛿𝑑) β†’0, (π›Ώπ‘Š)~𝛿𝑑, (π›Ώπ‘Š).~(𝛿𝑑). Β¨ Stochastic calculus we still have (π›Ώπ‘Š) β†’ 0, BUT WE ALSO HAVE (π›Ώπ‘Š).~𝛿𝑑 so higher orders are mixed together Β¨ Coin toss (+1, -1). Average is 0, variance scales linearly with the number of flips Β¨ Can you think of processes where variance goes to 0 and we need to go to the next order? Β¨ Can you think of a process where (π›Ώπ‘Š). scales maybe as (𝛿𝑑)0, where π‘ˆ < 1 ? Β¨ Also, if you think about it, when π›Ώπ‘Šis stochastic, somehow (π›Ώπ‘Š). is now deterministic, or at least has a deterministic component, the inverse is NOT true 24
  • 25. Luc_Faucheux_2020 Always better to integrate than to differentiate Β¨ If we have a stochastic process π‘Š(𝑑), we would like to work with functions 𝑓(π‘Š) (please note that those functions are usually well behaved, meaning differentiable and such, without going into too much math) Β¨ 𝑓(π‘Š) is differentiable in π‘Š Β¨ π‘Š(𝑑) is NOT differentiable in 𝑑 Β¨ But we are really dealing with 𝑓(π‘Š 𝑑 ) so we would like to write things like Β¨ 𝛿𝑓 = )$ )1 . )1 )+ . 𝛿𝑑 which is one way to write the traditional β€œchain rule” Β¨ In integral form, the chain rule would read something like: Β¨ 𝑓 π‘Š 𝑑 βˆ’ 𝑓 π‘Š 0 = ∫234 23+ )$ )1 . )1 )2 . 𝑑𝑠 Β¨ The crux of the issue in stochastic calculus is that we do not know what )1 )2 means, and so we should NOT expect to be able to rely on the usual chain rule 25
  • 26. Luc_Faucheux_2020 The whole Ito-Stratanovitch thing Β¨ Essentially, ITO breakthrough was to find a way to define: Β¨ 𝑓 π‘Š 𝑑 βˆ’ 𝑓 π‘Š 0 = ∫234 23+ )$ )1 . )1 )2 . 𝑑𝑠 = ∫234 23+ )$ )1 . π‘‘π‘Š Β¨ ITO invented the field of stochastic integrals Β¨ The essence of it is that π‘‘π‘Š(𝑑) is a β€œjump” Β¨ ITO (1950) defines the ITO integral as a limit of a sum where the value of )$ )1 is taken β€œbefore the jump”. A consequence of this convention is that the usual rules of calculus (chain rule, Leibniz rule,..) do not apply. For example 𝑑 𝑙𝑛𝑆 β‰  ( ⁄𝑑𝑆 𝑆) ! On the other hand the ITO integral has the nice property to be a martingale (zero expected value) Β¨ Stratanovitch (STRATO) defines the STRATO integral as a limit of a sum where the value of )$ )1 is taken β€œin the middle of the jump”. A rather intriguing consequence is that in STRATO calculus the usual rules of calculus are (formally!) respected. 𝑑 𝑙𝑛𝑆 = ( ⁄𝑑𝑆 𝑆). HOWEVER the STRATO integral is NOT a martingale. 26
  • 27. Luc_Faucheux_2020 The whole Ito-Stratanovitch thing - II Β¨ So we will see how to not get confused between the two equally valid interpretations of the integrals (from a really theoretical point of view mathematicians prefer ITO because it is defined over a wider range of functions than STRATO, but really nothing we should concern ourselves at this point). Β¨ This will take us some time, first going over the Riemann integral in regular calculus Β¨ Then defining the ITO Β¨ Then looking at the relationship between ITO and STRATO integral Β¨ Then explicitly proving the ITO lemma, then the equivalent STRATO lemma Β¨ From then we look at the SDE (SIE), and we try to map the relations between a specific SDE and its associated PDE for the PDF Β¨ SDE: Stochastic Differential Equation Β¨ SIE: Stochastic Integral Equation Β¨ PDE: Partial Differential Equation (that is usual regular calculus, but not easy by any means) Β¨ PDF: Probability Density Function 27
  • 28. Luc_Faucheux_2020 The whole Ito-Stratanovitch thing - III Β¨ So apologies in advance if those slides feel pedestrian at time, but unfortunately in order to be somewhat rigorous without losing the intuition, and in order to convince ourselves that we are on somewhat firm ground to justify what we write (without having a 100% rigorous mathematical proof), we have to walk before we run. Β¨ Alternatively, you could be a genius like Vincent Doelin and bypass the entire theory of stochastic integrals and express everything as a Brownian time change, 40 years or so before everyone else, while fighting WWII in the Ardennes as a radio operator. Β¨ If I have time, part V of those decks will be on that Β¨ Also the ITO-STRATO controversy in the 1990s was linked to the concept of thermal ratchets, Brownian motors, biological motors, hence quite a few articles on the subject. Β¨ This is also linked to an individual that physicists refer to as the Maxwell’s demon Β¨ In deck III we will revisit this unsavory character, who prompted me spending a lot of time on ITO-STRATO over my life and as a PhD student. 28
  • 29. Luc_Faucheux_2020 The whole Ito-Stratanovitch thing - IV Β¨ The Maxwell demon putting the moves on Mr. Tompkins fiancΓ©e and trying to impress her with his tennis skills 29
  • 30. Luc_Faucheux_2020 The whole Ito-Stratanovitch thing - V Β¨ A more common representation of the demon 30
  • 31. Luc_Faucheux_2020 The whole Ito-Stratanovitch thing - VI Β¨ There are actually applications in Finance of the Maxell demon, known as the Parrondo paradox. Β¨ Two trading strategies (PM at a hedge fund) on average lose money (B and C, blue and green line) Β¨ However you can alternate between the two strategies to create one (A-red line) that on average will be profitable, like the thermal ratchet who is extracting work out of thermal noise, this Parrondo construct extract positive return out of random switches between two losing strategies 31
  • 32. Luc_Faucheux_2020 Just a taste of how powerful Vincent Doelin was Β¨ SDE have usually the form: Β¨ 𝑑𝑋 𝑑 = π‘Ž 𝑋 𝑑 , 𝑑 . 𝑑𝑑 + 𝑏 𝑋 𝑑 , 𝑑 . π‘‘π‘Š, which really should always be written as SIE: Β¨ 𝑋 𝑑5 βˆ’ 𝑋 𝑑6 = ∫+3+6 +3+5 𝑑𝑋 𝑑 = ∫+3+6 +3+5 π‘Ž 𝑋 𝑑 , 𝑑 . 𝑑𝑑) + ∫+3+6 +3+5 𝑏 𝑋 𝑑 , 𝑑 . π‘‘π‘Š(𝑑) Β¨ The whole theory of ITO is trying to define what exactly is : ∫+3+6 +3+5 𝑏 𝑋 𝑑 , 𝑑 . π‘‘π‘Š(𝑑) Β¨ What Doelin did essentially was to say: hey I do not need to define this integral, which is subject to the exact convention of β€œwhere to take the value of 𝑏 𝑋 𝑑 , 𝑑 before, during or after the jump π‘‘π‘Š(𝑑)”, and run into all sort of Ito-Stratanovitch confusion, because I am a genius, whatever that integral is, it is equal to : Β¨ ∫+3+6 +3+5 𝑏 𝑋 𝑑 , 𝑑 . π‘‘π‘Š(𝑑) ≝ π‘Š(∫+3+6 +3+5 𝑏 𝑋 𝑑 , 𝑑 .. 𝑑𝑑) which is perfectly well defined. Β¨ Boom. Microphone drop. Β¨ And by the way he burnt most of his research before killing himself to avoid capture by the Germans, so who knows what else he had discovered…. 32
  • 33. Luc_Faucheux_2020 ITO and DOELIN comparison in 2 pictures 33
  • 34. Luc_Faucheux_2020 Ito-Doblin and stochastic calculus is.. Β¨ β€œFirst and foremost defined as an integral calculus” Β¨ Really the only thing that works is integrating stochastic processes. Β¨ There are theories of differentiations for stochastic variables Β¨ Malliavin calculus (1980) Β¨ One day I will try to understand what that actually means. I still have no idea. But I should try after all Malliavin is also French Β¨ So even the Ito lemma as we know it is really better expressed in integral form Β¨ However most textbooks do present it in β€œdifferential” form like a Taylor expansion, or even deal with SDE quite liberally. This is sometimes for ease of notations and we will fall into that pattern also. Note that this is a FORMAL equivalence to regular calculus equations like Taylor expansion. Taking those literally leads to mistakes, as we will demonstrate 34
  • 35. Luc_Faucheux_2020 Why are PDEs so important, and why SDEs are terrifying Β¨ PDEs are Partial Differential Equations )!7(!,+) )!! = )7(!,+) )+ Β¨ Usually in the context of stochastic calculus they will appear for the PDF (Probability Density Function) of the stochastic variable 𝑋 (like a Gaussian), 𝑃(π‘₯, 𝑑) or for functions of that variable (call payoff or 𝐢(𝑋, 𝜎, 𝐾, 𝑇) for example) Β¨ PDEs are well defined and well known (since 1700), tons of knowledge on how to deal with those (Navier-Stokes in fluid mechanic, Maxwell equations in electro-magnetism, Fokker- Planck, Feynman-Kac,..) Β¨ Once you know the PDE, you know ALL the moments of the distribution in the case of a PDE Β¨ PDEs are complete, SDEs are incomplete Β¨ SDEs are Stochastic Differential equations 𝑑𝑆 = πœ‡. 𝑆. 𝑑𝑑 + 𝜎. 𝑆. π‘‘π‘Š Β¨ No one really knows how to deal with them, especially if the volatility is a function of the stochastic variable (and it always is, or when you look at functions of X) Β¨ But sometimes you can get rid of a SDE and use a related PDE (that was the trick that Black Sholes discovered and got a Nobel prize for), or Dupire equation Β¨ Also stay away from SDE, always look for the PDE 35
  • 36. Luc_Faucheux_2020 The structure of those slides Β¨ We will first start with the usual textbook (Hull) that presents essentially what are Ito SDEs and Ito lemma being a regular Taylor expansion just making sure that we go high enough to keep all the terms linear in time and linear in the stochastic driver Β¨ So in some ways, in the usual calculus π‘Š(𝑑), when (𝛿𝑑) β†’0, (π›Ώπ‘Š)~𝛿𝑑, (π›Ώπ‘Š).~(π›Ώπ‘Š). Β¨ Stochastic calculus we still have (π›Ώπ‘Š) β†’ 0, BUT WE ALSO HAVE (π›Ώπ‘Š).~𝛿𝑑 so higher orders are mixed together, but if we keep the right terms we should be fine Β¨ This is usually where most textbooks in finance stops Β¨ We will show by using something called Stratonovitch convention, that treating the SDEs and the Taylor expansion the way we would do in regular calculus is wrong. Β¨ In fact the Ito lemma and the Ito SDEs are just a formal manner to write Integrals and SIE, which is really the only thing that you can do in stochastic calculus Β¨ Remember, you always read that a stochastic process is NOT differentiable, yet somehow we all proceed happy to write Stochastic DIFFERENTIAL Equations, and writing Ito lemma as a Taylor expansion, without really thinking twice about it 36
  • 37. Luc_Faucheux_2020 The structure of those slides - II - Integrals Β¨ We then take a couple of steps back and go over a review of the regular integrals (Riemann) in the regular calculus Β¨ We extend this to the realm of stochastic calculus Β¨ We show that unlike the regular case, the point taken in the partition buckets (the mesh) actually matters. One convention is to take the starting point (ITO). Another one is to take the middle point (Stratonovitch) Β¨ We then derive the relationships between the Ito and the Strato integral Β¨ At this point there is no finance theory but it will be worth keeping in mind that the Ito integral will have the property of being a martingale (zero expected value) Β¨ Note that things like β€œpartition” or ”mesh” will not be super rigorously defined, but we leave that to mathematicians, we use those concepts assuming that they are well defined, and we can check that indeed they are over a certain range of β€œnon-pathological” functions or geometries 37
  • 38. Luc_Faucheux_2020 The structure of those slides - III - Lemma Β¨ From what we learned from looking at those integrals, we then look at the issue around making some operations on those integrals. This is where we look at Ito lemma Β¨ Ito lemma is crucial in finance Β¨ We will see that formally we can write the lemma, and perform operations on functions and integral, in a formal manner that looks like regular calculus, with the exceptions that you have to go β€œone more up” in any kind of derivations or Taylor expansion, in order to capture ALL the terms linear in time and linear in the stochastic driver Β¨ In doing so the β€œusual” rules of calculus (derivations, integrations, Leibniz,..) are no longer true in stochastic calculus using the Ito interpretation of the integral Β¨ We will compare this with the Strato integral Β¨ We will show that the ”usual” rules of calculus are still formally present in Strato calculus (remember this is a formal analogy). This can be useful when explicitly solving equations Β¨ We then show the relationship between the Ito and the Strato lemma 38
  • 39. Luc_Faucheux_2020 The structure of those slides - IV - SDE Β¨ From what we learned looking at the integrals and how to manipulate them, we can try to look at what would be SDE, Stochastic Differential Equations Β¨ Remember though, that just the same way we can formally write Ito and Strato lemma in β€œdifferential” form for ease of notation, those are ALWAYS simpler way to write down what would really be equations dealing with integrals Β¨ In stochastic calculus, I do not know what a differentiation would actually mean Β¨ All I can do really is to integrate Β¨ Like the integral and the lemma, we will show the relationship between the Ito SDE and the Strato SDE (or really more exactly the Ito SIE and the Strato SIE), and introduce the so-called β€œdrift” between the two representations Β¨ SIE: Stochastic Integral Equations 39
  • 40. Luc_Faucheux_2020 The structure of those slides - V - PDE Β¨ From the SIE, we then explore the correspondence between the SIE and the PDE (Partial Differential Equation) for the PDF (Probability Distribution Function) of the stochastic variable. Β¨ PDE are just part of the regular calculus, there is no issue there Β¨ SIE and SDE depends on the interpretation (Ito or Strato). Β¨ We will then by extension look at PDE that would correspond to a specific SDE under Ito, and similarly under Strato. Β¨ This is where it can get a little confusing (or even more confusing that it already is) Β¨ We will revisit our old friend the Maxwell demon and see why it was so confusing in the 1990s when applied to biological concepts of thermal ratchets or Brownian motors Β¨ Because a PDE is β€œexact” (once you know the PDE, and if you can solve it you have the PDF, so you have exactly all the moments of the stochastic variable) whereas an SDE only has the first two moments in most cases, and is also subject to interpretation, it is worth keeping in mind that fact: a PDE is not subject to interpretation. An SDE is subject to interpretation, and needs to be treated with great caution 40
  • 41. Luc_Faucheux_2020 The structure of those slides - VI - PDE Β¨ We will look at some cases of PDEs from the world of Physics in order to gain some more intuition on what is a firm ground to stand on (PDE), and a somewhat more recent and still a little shaky one (SDE) Β¨ This will be a somewhat indirect introduction to a fundamental theorem that links the world of β€œregular” well known calculus of PDE (400 years in the making) with the more recent one having to do with probability and stochastic calculus (only 100 years in the making), and still very new. Β¨ This year Abel prize was given to pioneers of the ergodic theory, that essentially to crudely oversimplify, try to find the solutions of PDEs by using properties of the associated SDEs Β¨ We can then go back to Black-Sholes with a renewed belief in how justified we are in our derivation, in particular we tried to β€œget away” using simple Taylor expansions and just keeping some higher orders, we showed however how wrong it can get very quickly. 41
  • 42. Luc_Faucheux_2020 Β¨ Since we will first fall into the trap of treating an SDE the way we would do usual calculus, a number of slides in this deck are WRONG. It is sometimes more useful to learn from mistakes than to follow a magistral correct demonstration. Β¨ So in order to not make the whole deck a complete garbage of my random wrong ramblings on stochastic calculus while going up the river to meet colonel Kurtz, I have put a stamp β€œWRONG” across those slides. Β¨ So do not read those slides at face value, but know that those are WRONG. Β¨ It might be an interesting exercise for you to convince yourself why those slides are wrong. Β¨ Again, I left them there because I think it is quite instructive to identify the fault in a derivation. Β¨ We could have kept all the slides in ITO calculus, but it is quite enlightening to also look at STRATANOVITCH calculus, if only to convince ourselves that we really understand ITO (If we understand the difference between two things, then most likely that increases our knowledge about each of those things) A note of caution 42
  • 43. Luc_Faucheux_2020 Why this whole thing about Ito calculus? Hull textbook Β¨ Hull – White chapters 13, 14 and 15 Β¨ People got excited about stock prices trading as a percentage (people expect a β€œreturn”), p.306, and so what mattered was the return of the stock 𝑆, or β„βˆ†π‘† 𝑆 Β¨ So then they started writing things like : 𝑑𝑆 = πœ‡. 𝑆. 𝑑𝑑 + 𝜎. 𝑆. π‘‘π‘Š, (p.307) Β¨ And then they got stuck, because 𝑑π‘₯ = π‘Ž π‘₯, 𝑑 . 𝑑𝑑 + 𝑏 π‘₯, 𝑑 . π‘‘π‘Š, where 𝑏 π‘₯, 𝑑 is a function of the stochastic variable π‘₯, is not something we know how to deal with (p.306, and no, it is NOT a β€œsmall approximation” as they claim) Β¨ So you need to use a ”guess” on how to deal with 𝑏 π‘₯, 𝑑 , which is why it is called a β€œlemma” Β¨ Ito (1951) guessed that you can write 𝑑π‘₯ = π‘Ž π‘₯, 𝑑 . 𝑑𝑑 + 𝑏 π‘₯, 𝑑 . π‘‘π‘Š as βˆ†π‘₯ = π‘Ž π‘₯, 𝑑 . βˆ†π‘‘ + 𝑏 π‘₯, 𝑑 . βˆ†π‘Š or Ξ΄π‘₯ = π‘Ž π‘₯, 𝑑 . 𝛿𝑑 + 𝑏 π‘₯, 𝑑 . π›Ώπ‘Š Β¨ That seems like a good guess but then the rules of calculus are no longer applicable, you can barely derive without making a mistake, and forget about trying to integrate (p. 311) Β¨ Now you get this weird thing where 𝑑 𝑙𝑛𝑆 = ( ⁄𝑑𝑆 𝑆) βˆ’ ( β„πœŽ. 2). 𝑑𝑑, (p.312) 43
  • 44. Luc_Faucheux_2020 Why chose Ito then? 44 Β¨ Someone else made a β€œbetter” guess, Stratonovitch (1966) Β¨ Strato guessed that you can write 𝑑π‘₯ = π‘Ž π‘₯, 𝑑 . 𝑑𝑑 + 𝑏 π‘₯, 𝑑 . π‘‘π‘Š as βˆ†π‘₯ = π‘Ž π‘₯, 𝑑 . βˆ†π‘‘ + 𝑏 π‘₯ + Ξ”π‘₯/2, 𝑑 . βˆ†π‘Š or Ξ΄π‘₯ = π‘Ž π‘₯, 𝑑 . 𝛿𝑑 + 𝑏 π‘₯ + 𝛿π‘₯/2, 𝑑 . π›Ώπ‘Š Β¨ Within Strato’s convention, the usual rules of calculus FORMALLY apply 𝑑 𝑙𝑛𝑆 = ( ⁄𝑑𝑆 𝑆), and the chain rule is verified in the usual manner Β¨ So this is super confusing Β¨ We will look at both Ito and Strato, and understand how to go from the SDE (Stochastic Differential Equation) to the PDE (Partial Differential Equation) and the correct PDF (Probability Density Function). Β¨ We will go over the binomial trees and binomial distribution, and its limit the Gaussian distribution (HW chapter 13, p.296-299) Β¨ We will show why the Gaussian distribution is so common and so central to everything (central limit theorem)
  • 45. Luc_Faucheux_2020 Actually it is not called Ito, but Ito-Doeblin Β¨ It was discovered quite recently (2000) that Vincent Doeblin, born Wolfgang Doeblin, Ph.D. at 23, and drafted in the French army in 1938, while posted in the Ardennes as a phone operator, essentially worked out Ito calculus on his own and sent his results to the Academie des Sciences β€œsous plis scelle”. Β¨ He shot himself after burning the rest of his notes when the German army was about to advance and take over his positions Β¨ So wo knows what else was in his notes? Malliavin calculus maybe ? 45
  • 46. Luc_Faucheux_2020 Ito and Stratonovitch β€œTaylor” expansion Β¨ Ito (1951) guessed that you can write 𝑑π‘₯ = π‘Ž π‘₯, 𝑑 . 𝑑𝑑 + 𝑏 π‘₯, 𝑑 . π‘‘π‘Š as Ξ΄π‘₯ = π‘Ž π‘₯, 𝑑 . 𝛿𝑑 + 𝑏 π‘₯, 𝑑 . π›Ώπ‘Š Β¨ If 𝐹(π‘₯) a function of x, the corresponding SDE as a result of Taylor expansion in ITO is: Β¨ 𝛿𝐹 = )9 )! . 𝛿π‘₯ + - . . )!9 )!! . 𝑏.. 𝛿𝑑 Β¨ That is the celebrated Ito lemma, or how the chain rule gets modified in Ito stochastic calculus Β¨ Strato guessed that you can write 𝑑π‘₯ = π‘Ž π‘₯, 𝑑 . 𝑑𝑑 + 𝑏 π‘₯, 𝑑 . π‘‘π‘Š as Ξ΄π‘₯ = π‘Ž π‘₯, 𝑑 . 𝛿𝑑 + 𝑏 π‘₯ + 𝛿π‘₯/2, 𝑑 . π›Ώπ‘Š Β¨ So you think that you could write something like this: the SDE for 𝐹(π‘₯) in Stratonovitch convention is: Β¨ 𝛿𝐹 = )9 )! . 𝛿π‘₯ + - . . )!9 )!! . 𝑏.. 𝛿𝑑 + - . . )9 )! . )5 )! . 𝑏. 𝛿𝑑 46
  • 47. Luc_Faucheux_2020 Ito and Stratonovitch β€œTaylor” expansion - II Β¨ We would like to write something like this for Stratonovitch, following the rule of expanding to the second order and keeping the terms linear in time and linear in the stochastic driver (so essentially in a way use Ito calculus in the Stratonovitch convention) Β¨ Bear in mind that this is absolutely wrong Β¨ It is quite insightful to go through it though and see where it is wrong Β¨ So we have a function 𝐹 of the stochastic variable π‘₯, the function 𝐹 in itself is nothing weird, it is a regular function that we assume to be differentiable 𝐹(π‘₯) Β¨ Ito calculus tells us that if we want to look at the variations of that function in terms of the stochastic variable π‘₯, assuming a process 𝑑π‘₯ = π‘Ž π‘₯, 𝑑 . 𝑑𝑑 + 𝑏 π‘₯, 𝑑 . π‘‘π‘Š Β¨ You can formally write within Ito calculus: Ξ΄π‘₯ = π‘Ž π‘₯, 𝑑 . 𝛿𝑑 + 𝑏 π‘₯, 𝑑 . π›Ώπ‘Š Β¨ 𝛿𝐹 = )9 )! . 𝛿π‘₯ + - . . )!9 )!! . 𝑏.. 𝛿𝑑 47
  • 48. Luc_Faucheux_2020 Ito and Stratonovitch β€œTaylor” expansion - III Β¨ More exactly Β¨ 𝛿𝐹 = )9 )! . 𝛿π‘₯ + - . . )!9 )!! . 𝛿π‘₯. Β¨ And : Ξ΄π‘₯ = π‘Ž π‘₯, 𝑑 . 𝛿𝑑 + 𝑏 π‘₯, 𝑑 . π›Ώπ‘Š Β¨ So keeping the terms linear in time and in the stochastic driver Β¨ 𝛿π‘₯. = 𝑏.. 𝛿𝑑 Β¨ And we get the usual Ito formula: 𝛿𝐹 = )9 )! . 𝛿π‘₯ + - . . )!9 )!! . 𝑏.. 𝛿𝑑 Β¨ The β€œcanonical” example is usually 𝐹 π‘₯ = 𝐿𝑛(π‘₯), with )9 )! = - ! , and )!9 )!! = :- !!, and with the stochastic process: Ξ΄π‘₯ = π‘₯. π›Ώπ‘Š, so 𝑏 = π‘₯, and 𝑏. = π‘₯. Β¨ And so we get: 𝛿 𝐿𝑁 π‘₯ = - ! . 𝛿π‘₯ + - . . :- !! . 𝑏.. 𝛿𝑑 = - ! . π‘₯. π›Ώπ‘Š + - . . :- !! . π‘₯.. 𝛿𝑑 48
  • 49. Luc_Faucheux_2020 Ito and Stratonovitch β€œTaylor” expansion - IV Β¨ Or again: Β¨ 𝛿 𝐿𝑁 π‘₯ = 𝛿𝑧 βˆ’ - . . 𝛿𝑑 = ;! ! βˆ’ - . . 𝛿𝑑 Β¨ Whereas the β€œusual rule of calculus would read : 𝛿 𝐿𝑁 π‘₯ = ;! ! Β¨ That is the usual example in most textbooks, especially in Finance Β¨ NOW comes the weird little Stratanovitch trick, where we assume that we can write something like : Β¨ Strato guessed that you can write 𝑑π‘₯ = π‘Ž π‘₯, 𝑑 . 𝑑𝑑 + 𝑏 π‘₯, 𝑑 . π‘‘π‘Š as Ξ΄π‘₯ = π‘Ž π‘₯, 𝑑 . 𝛿𝑑 + 𝑏 π‘₯ + 𝛿π‘₯/2, 𝑑 . π›Ώπ‘Š Β¨ So again we follow the β€œIto” rule of calculus by expanding in linear terms in time and in the stochastic driver 49
  • 50. Luc_Faucheux_2020 Ito and Stratonovitch β€œTaylor” expansion - V Β¨ 𝛿𝐹 = )9 )! . 𝛿π‘₯ + - . . )!9 )!! . 𝛿π‘₯. Β¨ And : Ξ΄π‘₯ = π‘Ž π‘₯, 𝑑 . 𝛿𝑑 + 𝑏 π‘₯ + 𝛿π‘₯/2, 𝑑 . π›Ώπ‘Š Β¨ Or: Ξ΄π‘₯ = π‘Ž π‘₯, 𝑑 . 𝛿𝑑 + 𝑏 π‘₯, 𝑑 . π›Ώπ‘Š + ;! . . )5 )! . π›Ώπ‘Š, which keeping only the terms linear in time and in the stochastic driver π›Ώπ‘Š, leads to: Β¨ Ξ΄π‘₯ = π‘Ž π‘₯, 𝑑 . 𝛿𝑑 + 𝑏 π‘₯, 𝑑 . π›Ώπ‘Š + - . . )5 )! . 𝑏. 𝛿𝑑 Β¨ And so: 𝛿𝐹 = )9 )! . 𝛿π‘₯ + - . . )!9 )!! . 𝛿π‘₯. Β¨ 𝛿𝐹 = )9 )! . 𝛿π‘₯ + - . . )!9 )!! . 𝑏.. 𝛿𝑑 + - . . )9 )! . )5 )! . 𝑏. 𝛿𝑑 50
  • 51. Luc_Faucheux_2020 Ito and Stratonovitch β€œTaylor” expansion - VI Β¨ So we think that within the Ito rules of calculus (doing a Taylor expansion and keeping only the terms linear in time and the stochastic driver) but following the Stratonovitch convention we can write something like: Β¨ 𝛿𝐹 = )9 )! . 𝛿π‘₯ + - . . )!9 )!! . 𝑏.. 𝛿𝑑 + - . . )9 )! . )5 )! . 𝑏. 𝛿𝑑 Β¨ The β€œcanonical” example again is 𝐹 π‘₯ = 𝐿𝑛(π‘₯), with )9 )! = - ! , and )!9 )!! = :- !!, and with the stochastic process: Ξ΄π‘₯ = π‘₯. π›Ώπ‘Š, so 𝑏 = π‘₯, and 𝑏. = π‘₯., and )5 )! = 1 Β¨ So we get: Β¨ 𝛿 𝐿𝑁 π‘₯ = - ! . 𝛿π‘₯ βˆ’ - . . - !! . 𝛿𝑑 + - . . - ! . 1. π‘₯. 𝛿𝑑 Β¨ The last two terms cancel out and we get: 𝛿 𝐿𝑁 π‘₯ = - ! . 𝛿π‘₯ Β¨ Which is the usual result in ”regular” (non stochastic) calculus, and we think that we now understood stochastic calculus because textbooks are telling us that the usual rules of calculus are preserved in Strato 51
  • 52. Luc_Faucheux_2020 Ito and Stratonovitch β€œTaylor” expansion - VII Β¨ So far we seem to be pretty happy because we found that for the canonical example, the β€œusual rules of calculus” are preserved when using the Stratonovitch convention within the Ito calculus (meaning that we are using formally a Taylor expansion, making sure to keep all the terms linear in time and in the stochastic driver, i.e. the Ito rule of calculus, but assuming that we are taking the β€œmid-point” of the jump for functions multiplying this jump, i.e. what we think to be what the Stratonovitch convention is) Β¨ We will show that nothing could be more wrong Β¨ Both Ito and Stratonovitch are calculus on their own on the same footing Β¨ We just cannot do the Taylor expansion within the Stratonovitch framework Β¨ Not super obvious, took me a long time to be confused about it, the point is to always go back to the fact that in stochastic calculus the only thing that you can write is an integral, and sometimes for ease of notation we write something that looks like a Taylor expansion or and SDE. But this is only a formal way of writing, and the fact that it looks like regular calculus does NOT allow us to use those equations β€œas is” 52
  • 53. Luc_Faucheux_2020 Ito and Stratonovitch β€œTaylor” expansion – VII - a Β¨ So for example, when in Ito calculus we write the chain rule as: Β¨ 𝛿𝐹 = )9 )1 . π›Ώπ‘Š + - . . )!9 )1! . π›Ώπ‘Š. Β¨ What we are really writing, (and what we should always write from time to remind us, since we do not have enough paper and ink to write it all the time at every step) is: Β¨ 𝑓 π‘Š 𝑑5 βˆ’ 𝑓 π‘Š 𝑑6 = ∫+3+6 +3+5 )$ )* . ([). π‘‘π‘Š(𝑑) + - . ∫+3+6 +3+5 )!9 )1! (π‘Š 𝑑 ). 𝑑𝑑 Β¨ In the ”limit” of small time increments, this can be written formally as the Ito lemma: Β¨ 𝛿𝑓 = )$ )* . π›Ώπ‘Š + - . . )!9 )1! . 𝛿𝑑 Β¨ We will go over it once we rebuild our knowledge of stochastic calculus around the integral Β¨ Here we are following mostly Thomas Mikosch β€œElementary Stochastic Calculus with Finance in View”, a wonderful little book that is at times frustrating for some of the weird notations that he uses. 53
  • 54. Luc_Faucheux_2020 Ito and Stratonovitch β€œTaylor” expansion - VIII Β¨ A very quick manner to realize how wrong we are is to use another example: Β¨ 𝑑π‘₯ = π‘Ž π‘₯, 𝑑 . 𝑑𝑑 + 𝑏 π‘₯, 𝑑 . π‘‘π‘Š Β¨ with 𝑏 π‘₯, 𝑑 = Οƒ. π‘₯ , )5 )! = 𝜎 , 𝑏. = 𝜎.. π‘₯. and π‘Ž π‘₯, 𝑑 = 0, Β¨ 𝐹 π‘₯ = π‘₯., )9 )! = 2π‘₯ , )!9 )!! = 2 Β¨ Using ITO, 𝛿𝐹 = )9 )! . 𝛿π‘₯ + - . . )!9 )!! . 𝑏.. 𝛿𝑑 = 2π‘₯. 𝛿π‘₯ + 𝜎.. 𝛿𝑑 Β¨ Using Strato, 𝛿𝐹 = )9 )! . 𝛿π‘₯ + - . . )!9 )!! . 𝑏.. 𝛿𝑑 + - . . )9 )! . )5 )! . 𝑏. 𝛿𝑑 Β¨ We then get: 𝛿𝐹 = 2π‘₯. 𝛿π‘₯ + 𝜎.. 𝛿𝑑 + - . . 2π‘₯. Οƒ. Οƒ. π‘₯. 𝛿𝑑 = 2π‘₯. 𝛿π‘₯ + 2.𝜎.. 𝛿𝑑 Β¨ So neither what we call Ito and what we call Strato do follow the usual rule of calculus. That should be a sign that we did something very wrong when we loosely used the Taylor expansion and applied it to the Stratonovitch case, because textbooks are telling us that the usual rules of calculus (chain rule) are ”preserved” (similar in form) in Strato 54
  • 55. Luc_Faucheux_2020 Ito and Stratonovitch β€œTaylor” expansion - IX Β¨ The answer is not completely obvious in identifying what we did wrong Β¨ Conversely, someone could say it is absolutely obvious, because stochastic process are NOT differentiable, and so any kind of Taylor expansion is wrong Β¨ As a note, even though in most textbooks in Finance the Ito lemma is expressed as a Taylor expansion using the formal rules of calculus, the rule is that with stochastic processes you can NEVER differentiate (at least in a manner that makes sense and is safe), you can ONLY integrate Β¨ And so a more formally correct formulation of the Ito Lemma is: Β¨ 𝐹 π‘Š 𝑑 = 𝑇 βˆ’ 𝐹 π‘Š 𝑑 = 𝑆 = ∫+3< +3= )9 )1 . π‘‘π‘Š + - . ∫+3< +3= )!9 )1! . 𝑑𝑑 Β¨ Or also: Β¨ ∫+3< +3= 𝑑𝐹(π‘Š 𝑑 ) = ∫+3< +3= )9 )1 . π‘‘π‘Š(𝑑) + - . ∫+3< +3= )!9 )1! . 𝑑𝑑 55
  • 56. Luc_Faucheux_2020 Ito and Stratonovitch β€œTaylor” expansion - X Β¨ What is then the correct formulation of Stratonovitch lemma? Β¨ Is that? Β¨ 𝐹 π‘₯ 𝑑 = 𝑇 βˆ’ 𝐹 π‘₯ 𝑑 = 𝑆 = ∫+3< +3= )9 )! . 𝑑π‘₯ + ∫+3< +3= [ - . )!9 )!! . 𝑏. + - . . )9 )! . )5 )! . 𝑏]. 𝑑𝑑 Β¨ Or: Β¨ ∫+3< +3= 𝑑𝐹(π‘₯ 𝑑 ) = ∫+3< +3= )9 )! . 𝑑π‘₯ + ∫+3< +3= [ - . )!9 )!! . 𝑏. + - . . )9 )! . )5 )! . 𝑏]. 𝑑𝑑 Β¨ Or to identify the distinctions between the two: Β¨ STRATO(∫+3< +3= 𝑑𝐹(π‘₯ 𝑑 ))=ITO(∫+3< +3= 𝑑𝐹(π‘₯ 𝑑 ))+ - . ∫+3< +3= )9 )! . )5 )! . 𝑏. 𝑑𝑑 Β¨ This is still wrong, as we mixed Taylor expansion using the usual rules of calculus (Ito rules) with something completely different. We will now show what is the correct way to look at it using integrals. 56
  • 57. Luc_Faucheux_2020 Simple rules of stochastic calculus (cheat sheet) Β¨ NEVER EVER EVER work with processes where the volatility is a function of the stochastic variable Β¨ If you do or have no choice, transform it into a constant volatility equation (Hull p. 320), or find a way to set the volatility term to 0 (p.330), or give up and come up with something different (SABR model) Β¨ If you are working in Finance (and ”discrete processes) -> use ITO Β¨ If you are working with a DIGITAL computer -> use ITO Β¨ If you are working with an ANALOG computer -> use STRATO Β¨ If you are working in physics, and you do not like to break the time invariance, and you are also not that smart, so you want the usual rules of calculus -> use STRATO Β¨ In ALL cases, especially when working with the SDE, and discrete computer simulations, ALWAYS check that your drift has not been β€œpolluted” by the variable diffusion (β€œspurious” drift, Ryter 1980) 57
  • 58. Luc_Faucheux_2020 58 Building our knowledge of stochastic calculus around the integral
  • 59. Luc_Faucheux_2020 We have to start from the basics Β¨ Because clearly using ITO lemma as a Taylor expansion where we keep certain terms and using still the usual rules of calculus is wrong. Β¨ As we hinted at it, we should never write something that looks like a differential but always as an integral. Β¨ Let’s now go through that derivation, and start with the usual Riemann integral in the usual calculus Β¨ We then show that extending that concept to a stochastic variable is not well defined, and needs a convention, or interpretation of the integral. ITO is one interpretation, STRATO is another one. Β¨ We show the correspondence between the two interpretations. Β¨ Extending the concept of the integral being a limit of sums subject to an interpretation, we then derive the ITO lemma as well as the STRATO lemma 59
  • 60. Luc_Faucheux_2020 β€œRegular” Riemann integrals (definite integrals) Β¨ Riemann integrals are the regular integrals Β¨ Interval [a,b] on regular ”continuous” variable t Β¨ N sections of width (𝑏 βˆ’ π‘Ž)/𝑁, left side 𝐿>, right side 𝑅>, and middle 𝑀> Β¨ The Riemann integral ∫+36 +35 𝑓 𝑑 . 𝑑𝑑 is the limit when (𝑁 β†’ ∞) of the Riemann sums Β¨ LEFT Riemann Sum: 𝐿𝑅𝑆 = 5:6 / βˆ‘>3- >3/ 𝑓(𝐿>) Β¨ RIGHT Riemann Sum: 𝑅𝑅𝑆 = 5:6 / βˆ‘>3- >3/ 𝑓(𝑅>) Β¨ MIDDLE Riemann Sum: 𝑀𝑅𝑆 = 5:6 / βˆ‘>3- >3/ 𝑓(𝑀>) Β¨ SOMETHING Riemann Sum: 𝑆𝑅𝑆 = 5:6 / βˆ‘>3- >3/ 𝑓(𝑆>) where 𝑆> is somewhere in the section indexed by k, we could also define irregular partitions or β€œmesh” if we like Β¨ All those different sums converge to the same integral 60
  • 61. Luc_Faucheux_2020 61 ITO and STRATO integrals in the simple case W(t)
  • 62. Luc_Faucheux_2020 Stochastic Integrals Β¨ When not integrating over a β€œregular” continuous variable t but over a stochastic variable X, those sums do NOT converge to the same value. What does that even look like? Β¨ So first of all we would like to write something like this ∫*3*6 *3*5 𝑓 𝑋 . 𝑑𝑋 Β¨ The problem is that this is not well defined, what is the path over 𝑋 that we will integrate over? Remember 𝑋 is really 𝑋(𝑑) and is stochastic (jumps all over the place) Β¨ So really the only integration we can do is over 𝑑 Β¨ So we are looking for something like this: ∫+3+6 +3+5 𝑓 𝑋 𝑑 . 𝑑𝑋(𝑑) Β¨ When 𝑑 increases by a small amount 𝛿𝑑, 𝑋(𝑑) jumps by a small amount δ𝑋(𝑑) Β¨ So, breaking the time interval into 𝑁 sections of small time increment 𝛿𝑑 = (+5:+6) / Β¨ We looking at something like : SOMETHING = βˆ‘>3- >3/ 𝑓(𝑋(𝑑>)). [𝑋(𝑑>?-) βˆ’ 𝑋(𝑑>)] 62
  • 63. Luc_Faucheux_2020 Ito Integral Β¨ That something is the ITO sum, which converges to the ITO integral. Note that at this point it is just how we define it Β¨ ∫+3+6 +3+5 𝑓 𝑋 𝑑 . ([). 𝑑𝑋(𝑑) = lim /β†’@ {βˆ‘>3- >3/ 𝑓(𝑋(𝑑>)). [𝑋(𝑑>?-) βˆ’ 𝑋(𝑑>)]} Β¨ Note in the integral the usual (.) is replaced by ([) to explicitly indicate ITO convention Β¨ This is to indicate that we take for 𝑓(𝑋(𝑑>) the value of 𝑓 𝑋 BEFORE the jump Β¨ This is to be compared to the ITO lemma where 𝑑π‘₯ = π‘Ž π‘₯, 𝑑 . 𝑑𝑑 + 𝑏 π‘₯, 𝑑 . π‘‘π‘Š gets written in discrete manner as βˆ†π‘₯ = π‘Ž π‘₯, 𝑑 . βˆ†π‘‘ + 𝑏 π‘₯, 𝑑 . βˆ†π‘Š or Ξ΄π‘₯ = π‘Ž π‘₯, 𝑑 . 𝛿𝑑 + 𝑏 π‘₯, 𝑑 . π›Ώπ‘Š Β¨ Note also that to be fairly rigorous there are a lot of conditions that need to be verified for that β€œSOMETHING” to converge in a well defined manner to a well defined function 63
  • 64. Luc_Faucheux_2020 Stratonovitch Integral Β¨ Similar to the Stratonovitch lemma, where 𝑑π‘₯ = π‘Ž π‘₯, 𝑑 . 𝑑𝑑 + 𝑏 π‘₯, 𝑑 . π‘‘π‘Š gets written in the discrete manner as βˆ†π‘₯ = π‘Ž π‘₯, 𝑑 . βˆ†π‘‘ + 𝑏 π‘₯ + Ξ”π‘₯/2, 𝑑 . βˆ†π‘Š or Ξ΄π‘₯ = π‘Ž π‘₯, 𝑑 . 𝛿𝑑 + 𝑏 π‘₯ + 𝛿π‘₯/2, 𝑑 . π›Ώπ‘Š Β¨ The Stratonovitch integral is defined as: Β¨ ∫+3+6 +3+5 𝑓 𝑋 𝑑 . (∘). 𝑑𝑋(𝑑) = lim /β†’@ {βˆ‘>3- >3/ 𝑓 [𝑋(𝑑> + 𝑋(𝑑>?-)]/2). [𝑋(𝑑>?-) βˆ’ 𝑋(𝑑>)]} Β¨ If ITO was the β€œleft side Riemann”, Stratonovitch is the β€œmiddle Riemann” Β¨ Surprise surprise, they do NOT converge to the same value Β¨ Notice in the integral the usual (.) is replaced by (∘) to indicate that we take the middle point 64
  • 65. Luc_Faucheux_2020 Reverse ITO integral Β¨ Similarly we can also define a Reverse ITO (OTI?) integral as Β¨ ∫+3+6 +3+5 𝑓 𝑋 𝑑 . (]). 𝑑𝑋(𝑑) = lim /β†’@ {βˆ‘>3- >3/ 𝑓(𝑋(𝑑>?-)). [𝑋(𝑑>?-) βˆ’ 𝑋(𝑑>)]} Β¨ Notice in the integral the usual (.) is replaced by (]) to indicate that we take the right side Β¨ The Reverse Ito lemma would then expand 𝑑π‘₯ = π‘Ž π‘₯, 𝑑 . 𝑑𝑑 + 𝑏 π‘₯, 𝑑 . π‘‘π‘Š in the discrete manner as βˆ†π‘₯ = π‘Ž π‘₯, 𝑑 . βˆ†π‘‘ + 𝑏 π‘₯ + Ξ”π‘₯, 𝑑 . βˆ†π‘Š or Ξ΄π‘₯ = π‘Ž π‘₯, 𝑑 . 𝛿𝑑 + 𝑏 π‘₯ + 𝛿π‘₯, 𝑑 . π›Ώπ‘Š Β¨ This would be equivalent as β€œreading into the future” because we would take the value after the jump, but we do not know yet what the jump will be Β¨ Stratonovitch is also sometimes said to be β€œsimilar”, as in you need to know the jump ahead of time to evaluate the function, even though you do not know the jump ahead of time Β¨ ITO is well adapted to finance and to β€œMartingales” (you do not know the future, expected value is the current value). The technical term is that the ITO integral is β€œnon-anticipating” 65
  • 66. Luc_Faucheux_2020 Couple of notes here Β¨ It would seem that ITO would be well suited for processes that are β€œdiscontinuous” or β€œdiscrete” in nature (a very poor choice of words on my part), like: Β¨ Discrete computer simulation (on a digital computer) Β¨ Finance, gambling, games of chance Β¨ Radioactive decay (number of particles is discrete and the rate only depends on the previous state) Β¨ On the other hand STRATO seems to be better suited for β€œcontinuous” processes like most processes in Physics and Biology (diffusion, advection, chemotaxis, Brownian motors,..) Β¨ Part of the confusion will arise when trying to model in a discrete fashion a β€œcontinuous” process 66
  • 67. Luc_Faucheux_2020 Conversion between ITO and Stratonovitch Β¨ ∫+3+6 +3+5 𝑓 𝑋 𝑑 . (∘). 𝑑𝑋(𝑑) = lim /β†’@ {βˆ‘>3- >3/ 𝑓 [𝑋(𝑑> + 𝑋(𝑑>?-)]/2). [𝑋(𝑑>?-) βˆ’ 𝑋(𝑑>)]} Β¨ ∫+3+6 +3+5 𝑓 𝑋 𝑑 . (∘). 𝑑𝑋(𝑑) = lim /β†’@ {βˆ‘>3- >3/ 𝑓 𝑋(𝑑> + [*(+"#$):*(+")] . ). [𝑋(𝑑>?-) βˆ’ 𝑋(𝑑>)]} Β¨ ∫+3+6 +3+5 𝑓 𝑋 𝑑 . (∘). 𝑑𝑋(𝑑) = lim /β†’@ {βˆ‘>3- >3/ 𝑓 𝑋(𝑑> ). [𝑋(𝑑>?-) βˆ’ 𝑋(𝑑>)]} + lim /β†’@ {βˆ‘>3- >3/ 𝑓′ 𝑋(𝑑> . [*(+"#$):*(+")] . ). [𝑋(𝑑>?-) βˆ’ 𝑋(𝑑>)]} Β¨ ∫+3+6 +3+5 𝑓 𝑋 𝑑 . (∘). 𝑑𝑋(𝑑) = ∫+3+6 +3+5 𝑓 𝑋 𝑑 . ([). 𝑑𝑋(𝑑) + lim /β†’@ {βˆ‘>3- >3/ 𝑓′ 𝑋(𝑑> . ( [*(+"#$):*(+")] . ). [𝑋(𝑑>?-) βˆ’ 𝑋(𝑑>)]} Β¨ 𝑆𝑇𝑅𝐴𝑇𝑂(𝑓) = 𝐼𝑇𝑂(𝑓) + lim /β†’@ {βˆ‘>3- >3/ 𝑓′ 𝑋(𝑑> . ( [*(+"#$):*(+")] . ). [𝑋(𝑑>?-) βˆ’ 𝑋(𝑑>)]} 67
  • 68. Luc_Faucheux_2020 Conversion between ITO and Stratonovitch 2 Β¨ 𝑆𝑇𝑅𝐴𝑇𝑂(𝑓) = 𝐼𝑇𝑂(𝑓) + lim /β†’@ {βˆ‘>3- >3/ 𝑓′ 𝑋(𝑑> . ( [*(+"#$):*(+")] . ). [𝑋(𝑑>?-) βˆ’ 𝑋(𝑑>)]} Β¨ NOW this is a limit of a Riemann sum, because (𝛿𝑋).~𝛿𝑑 and is now deterministic (in the case of a simple Wiener process for 𝑋 = π‘Š) Β¨ In the more general case of 𝑑𝑋 𝑑 = π‘Ž 𝑋 𝑑 , 𝑑 . 𝑑𝑑 + 𝑏 𝑋 𝑑 , 𝑑 . π‘‘π‘Š(𝑑) we will show that Β¨ (𝛿𝑋).~𝑏 𝑋 𝑑 , 𝑑 . 𝛿𝑑 but for now let’s keep 𝑋 𝑑 = π‘Š(𝑑) the simple Brownian motion Β¨ That Riemann sum converges to the definite Riemann integral - . ∫+3+6 +3+5 𝑓′ π‘Š 𝑑 . 𝑑𝑑 Β¨ ∫+3+6 +3+5 𝑓 π‘Š 𝑑 . (∘). π‘‘π‘Š(𝑑) = ∫+3+6 +3+5 𝑓 π‘Š 𝑑 . ([). π‘‘π‘Š(𝑑) + - . ∫+3+6 +3+5 𝑓′ π‘Š 𝑑 . 𝑑𝑑 Β¨ 𝑆𝑇𝑅𝐴𝑇𝑂(𝑓) = 𝐼𝑇𝑂 𝑓 + - . ∫+3+6 +3+5 𝑓′ π‘Š 𝑑 . 𝑑𝑑 Β¨ 𝑆𝑇𝑅𝐴𝑇𝑂(𝑓) = 𝐼𝑇𝑂 𝑓 + - . . 𝑅𝐼𝐸𝑀𝐴𝑁𝑁(𝑓%) 68
  • 70. Luc_Faucheux_2020 Chain Rule (Ito lemma)- I Β¨ ITO integral: Β¨ ∫+3+6 +3+5 𝑓 𝑋 𝑑 . ([). 𝑑𝑋(𝑑) = lim /β†’@ {βˆ‘>3- >3/ 𝑓(𝑋(𝑑>)). [𝑋(𝑑>?-) βˆ’ 𝑋(𝑑>)]} Β¨ Another way to express it is the following: Β¨ 𝑓 𝑋 𝑑5 βˆ’ 𝑓 𝑋 𝑑6 = βˆ‘>3- >3/ {𝑓(𝑋(𝑑>)) βˆ’ 𝑓(𝑋(𝑑>:-))} Β¨ 𝑓 𝑋 𝑑5 βˆ’ 𝑓 𝑋 𝑑6 = lim /β†’@ βˆ‘>3- >3/ {𝑓(𝑋(𝑑>)) βˆ’ 𝑓(𝑋(𝑑>:-))} Β¨ As long as we have defined a reasonable β€œmesh” for the sequence of 𝑋(𝑑>) Β¨ 𝑓(𝑋(𝑑>)) = 𝑓(𝑋(𝑑>:-)) + )$ )* . ([). 𝛿𝑋 + - . . )!$ )!! . ([). (𝛿𝑋). Β¨ Where we use the ([) notation to reflect the β€œleft-hand” Riemann. Β¨ )$ )* . ([). 𝛿𝑋 = )$ )* 𝑋 = 𝑋(𝑑>:- ]. [𝑋(𝑑>) βˆ’ 𝑋(𝑑>:-)] 70
  • 71. Luc_Faucheux_2020 Chain Rule (Ito lemma)- II Β¨ - . . )!9 )!! . ([). 𝛿𝑋 . = - . . )!$ )!! [𝑋 = 𝑋(𝑑>:-)]. [𝑋(𝑑>) βˆ’ 𝑋(𝑑>:-)]. Β¨ )$ )* . ([). 𝛿𝑋 = )$ )* 𝑋 = 𝑋(𝑑>:- ]. [𝑋(𝑑>) βˆ’ 𝑋(𝑑>:-)] Β¨ 𝑓(𝑋(𝑑>)) = 𝑓(𝑋(𝑑>:-)) + )$ )* . ([). 𝛿𝑋 + - . . )!$ )!! . ([). (𝛿𝑋). Β¨ 𝑓 𝑋 𝑑5 βˆ’ 𝑓 𝑋 𝑑6 = lim /β†’@ βˆ‘>3- >3/ {𝑓(𝑋(𝑑>)) βˆ’ 𝑓(𝑋(𝑑>:-))} Β¨ 𝑓 𝑋 𝑑5 βˆ’ 𝑓 𝑋 𝑑6 = lim /β†’@ βˆ‘>3- >3/ { )$ )* . ([). 𝛿𝑋 + - . . )!$ )!! . ([). (𝛿𝑋).} Β¨ 𝑓 𝑋 𝑑5 βˆ’ 𝑓 𝑋 𝑑6 = ∫+3+6 +3+5 )$ )* . ([). 𝑑𝑋(𝑑) + ∫+3+6 +3+5 - . . )!$ )!! . ([). (𝛿𝑋). Β¨ In the ”limit” of small time increments, this can be written formally as the Ito lemma: Β¨ 𝛿𝑓 = )$ )* . 𝛿𝑋 + - . . )!$ )!! . (𝛿𝑋). 71
  • 72. Luc_Faucheux_2020 Chain Rule (Strato lemma)- I Β¨ We will now expand around the middle point Β¨ We still have: Β¨ 𝑓 𝑋 𝑑5 βˆ’ 𝑓 𝑋 𝑑6 = βˆ‘>3- >3/ {𝑓(𝑋(𝑑>)) βˆ’ 𝑓(𝑋(𝑑>:-))} Β¨ 𝑓 𝑋 𝑑5 βˆ’ 𝑓 𝑋 𝑑6 = lim /β†’@ βˆ‘>3- >3/ {𝑓(𝑋(𝑑>)) βˆ’ 𝑓(𝑋(𝑑>:-))} Β¨ We now write it a little differently by looking at expanding around the middle point: Β¨ 𝑓 𝑋 𝑑5 βˆ’ 𝑓 𝑋 𝑑6 = lim /β†’@ βˆ‘>3- >3/ {𝑓(𝑋(𝑑>)) βˆ’ 𝑓(𝑋(𝑑>:-))} Β¨ 𝑓 𝑋 𝑑5 βˆ’ 𝑓 𝑋 𝑑6 = lim /β†’@ βˆ‘>3- >3/ 𝑓 𝑋(𝑑> βˆ’ 𝑓 ∘ + 𝑓 ∘ βˆ’ 𝑓(𝑋(𝑑>:-))} Β¨ Where for sake of clarity we noted : 𝑓 ∘ = 𝑓 *(+" ? *(+"%$ . 72
  • 73. Luc_Faucheux_2020 Chain Rule (Strato lemma)- II Β¨ We then have the two following expansions of the function around the middle point: Β¨ 𝑓(𝑋(𝑑>)) = 𝑓(∘) + )$ )* . (∘). ;* . + - . . )!$ )!! . (∘). ( ;* . ). Β¨ 𝑓(𝑋(𝑑>:-)) = 𝑓(∘) βˆ’ )$ )* . (∘). ;* . + - . . )!$ )!! . (∘). ( ;* . ). Β¨ 𝛿𝑋 = 𝑋(𝑑>) βˆ’ 𝑋(𝑑>:-) Β¨ 𝑋(𝑑>) βˆ’ *(+")?*(+"%$) . = ;* . Β¨ *(+")?*(+"%$) . βˆ’ 𝑋(𝑑>:-) = ;* . Β¨ 𝑓 𝑋 𝑑5 βˆ’ 𝑓 𝑋 𝑑6 = lim /β†’@ βˆ‘>3- >3/ 𝑓 𝑋(𝑑> βˆ’ 𝑓 ∘ + 𝑓 ∘ βˆ’ 𝑓(𝑋(𝑑>:-))} Β¨ 𝑓 𝑋 𝑑5 βˆ’ 𝑓 𝑋 𝑑6 = lim /β†’@ {βˆ‘>3- >3/ )$ )* . ∘ . ;* . + - . . )!$ )!! . ∘ . ;* . . βˆ’ (βˆ’ )$ )* . (∘). ;* . + - . . )!$ )!! . (∘). ( ;* . ).)} 73
  • 74. Luc_Faucheux_2020 Chain Rule (Strato lemma)- III Β¨ 𝑓 𝑋 𝑑5 βˆ’ 𝑓 𝑋 𝑑6 = lim /β†’@ {βˆ‘>3- >3/ )$ )* . ∘ . ;* . + - . . )!$ )!! . ∘ . ;* . . βˆ’ (βˆ’ )$ )* . (∘). ;* . + - . . )!$ )!! . (∘). ( ;* . ).)} Β¨ 𝑓 𝑋 𝑑5 βˆ’ 𝑓 𝑋 𝑑6 = lim /β†’@ {βˆ‘>3- >3/ )$ )* . ∘ . ;* . + βˆ’(βˆ’ )$ )* . (∘). ;* . )} Β¨ 𝑓 𝑋 𝑑5 βˆ’ 𝑓 𝑋 𝑑6 = lim /β†’@ {βˆ‘>3- >3/ )$ )* . ∘ . 𝛿𝑋} Β¨ And so defining the integral with the Stratonovitch convention: Β¨ 𝑓 𝑋 𝑑5 βˆ’ 𝑓 𝑋 𝑑6 = ∫+3+6 +3+5 )$ )* . (∘). 𝑑𝑋(𝑑) Β¨ In the ”limit” of small time increments, this can be written formally as the Strato lemma: Β¨ 𝛿𝑓 = )$ )* . ∘ . 𝛿𝑋 Β¨ Please note that we are keeping the notation : ∘ 74
  • 75. Luc_Faucheux_2020 Chain Rule (Strato lemma)- IV Β¨ Using the Stratonovitch definition of the stochastic integral, we can write : Β¨ 𝛿𝑓 = )$ )* . ∘ . 𝛿𝑋 Β¨ This is the usual chain rule Β¨ So formally in some textbooks, you will see the following statement: Β¨ β€œWe do not mean that the Stratonovitch stochastic integral is a classical (Riemann) integral. We only claim that the corresponding chain rules have a similar structure”. Mikosh p127 Β¨ It is important to note that BOTH the Ito and the Stratonovitch integrals are defined in a mathematically correct manner. Β¨ Ito rules of calculus are not the usual ones but the Ito integral is a martingale Β¨ Strato rules of calculus are the usual ones (FORMALLY) but the Strato integral is NOT a martingale Β¨ We still have to review how we treat SDE and PDE in those frameworks. 75
  • 76. Luc_Faucheux_2020 Chain Rule (Ito and Strato lemma)- I Β¨ The crux of the matter is which interpretation of an integral do you want to use: Β¨ Because lim /β†’@ {βˆ‘>3- >3/ )$ )* . ∘ . 𝛿𝑋} Β¨ And Β¨ lim /β†’@ βˆ‘>3- >3/ { )$ )* . ([). 𝛿𝑋} Β¨ Do NOT converge to the same value, unlike a regular Riemann integral Β¨ In fact we just showed that Β¨ 𝑓 𝑋 𝑑5 βˆ’ 𝑓 𝑋 𝑑6 = lim /β†’@ {βˆ‘>3- >3/ )$ )* . ∘ . 𝛿𝑋} Β¨ 𝑓 𝑋 𝑑5 βˆ’ 𝑓 𝑋 𝑑6 = lim /β†’@ βˆ‘>3- >3/ { )$ )* . ([). 𝛿𝑋 + - . . )!9 )!! . ([). (𝛿𝑋).} 76
  • 77. Luc_Faucheux_2020 Chain Rule (Ito and Strato lemma)- II Β¨ 𝑓 𝑋 𝑑5 βˆ’ 𝑓 𝑋 𝑑6 = lim /β†’@ {βˆ‘>3- >3/ )$ )* . ∘ . 𝛿𝑋} Β¨ 𝑓 𝑋 𝑑5 βˆ’ 𝑓 𝑋 𝑑6 = lim /β†’@ βˆ‘>3- >3/ { )$ )* . ([). 𝛿𝑋 + - . . )!9 )!! . ([). (𝛿𝑋).} Β¨ We formally define in integral form: Β¨ 𝑓 𝑋 𝑑5 βˆ’ 𝑓 𝑋 𝑑6 = ∫+3+& +3+' 𝑑𝐹(π‘₯ 𝑑 ) Β¨ And ITO: Β¨ ∫+3+6 +3+5 𝐹 𝑋 𝑑 . ([). 𝑑𝑋(𝑑) = lim /β†’@ {βˆ‘>3- >3/ 𝐹(𝑋(𝑑>)). [𝑋(𝑑>?-) βˆ’ 𝑋(𝑑>)]} Β¨ And STRATO: Β¨ ∫+3+6 +3+5 𝐹 𝑋 𝑑 . (∘). 𝑑𝑋(𝑑) = lim /β†’@ {βˆ‘>3- >3/ 𝐹 [𝑋(𝑑> + 𝑋(𝑑>?-)]/2). [𝑋(𝑑>?-) βˆ’ 𝑋(𝑑>)]} 77
  • 78. Luc_Faucheux_2020 Chain Rule (Ito and Strato lemma)- III Β¨ We then have for the chain rule using the ITO interpretation of the integral: Β¨ 𝑓 𝑋 𝑑5 βˆ’ 𝑓 𝑋 𝑑6 = lim /β†’@ βˆ‘>3- >3/ { )$ )* . ([). 𝛿𝑋 + - . . )!9 )!! . ([). (𝛿𝑋).} Β¨ ∫+3+& +3+' 𝑑𝐹(𝑋 𝑑 ) = ∫+3+6 +3+5 )$ )* . ([). 𝑑𝑋(𝑑) + - . ∫+3+6 +3+5 )!9 )!! 𝑋 𝑑 . (𝑑𝑋). Β¨ Note that all integrals are on : ∫+3+& +3+' () Β¨ HOWEVER, the increment of integrands are different, 𝑑𝐹, 𝑑𝑋 and 𝑑𝑑 Β¨ BOTH 𝐹 and 𝑋 are functions of 𝑑 ultimately, 𝐹(𝑋 𝑑 ) and 𝑋(𝑑) Β¨ So only in a formal manner we write: Β¨ 𝛿𝑓 = )$ )* . 𝛿𝑋 + - . . )!9 )!! . (𝛿𝑋)., which is the celebrated Ito lemma (chain rule) 78
  • 79. Luc_Faucheux_2020 Chain Rule (Ito and Strato lemma)- IV Β¨ 𝛿𝑓 = )$ )* . 𝛿𝑋 + - . . )!9 )!! . 𝑏.. 𝛿𝑑, which is the celebrated Ito lemma (chain rule) Β¨ We somehow convinced ourselves at first that this was just using regular Taylor expansion but just keeping higher order terms, in particular the second order in 𝑋, because: Β¨ 𝛿𝑋. 𝛿𝑋~𝑏.. 𝛿𝑑 Β¨ But even though it looks formally the same, the only rigorous manner in which to write it is: Β¨ ∫+3+& +3+' 𝑑𝐹(𝑋 𝑑 ) = ∫+3+6 +3+5 )$ )* . ([). 𝑑𝑋(𝑑) + - . ∫+3+6 +3+5 )!9 )!! 𝑋 𝑑 . 𝑏 𝑋 𝑑 , 𝑑 .. 𝑑𝑑 Β¨ With the ITO interpretation of the integral: Β¨ ∫+3+6 +3+5 𝐹 𝑋 𝑑 . ([). 𝑑𝑋(𝑑) = lim /β†’@ {βˆ‘>3- >3/ 𝐹(𝑋(𝑑>)). [𝑋(𝑑>?-) βˆ’ 𝑋(𝑑>)]} 79
  • 80. Luc_Faucheux_2020 Chain Rule (Ito and Strato lemma)- V Β¨ We then have for the chain rule using the STRATO interpretation of the integral: Β¨ 𝑓 𝑋 𝑑5 βˆ’ 𝑓 𝑋 𝑑6 = lim /β†’@ {βˆ‘>3- >3/ )$ )* . ∘ . 𝛿𝑋} Β¨ ∫+3+& +3+' 𝑑𝐹(𝑋 𝑑 ) = ∫+3+6 +3+5 )$ )* . ∘ . 𝑑𝑋(𝑑) Β¨ Note that all integrals are on : ∫+3+& +3+' () Β¨ HOWEVER, the increment of integrands are different, 𝑑𝐹, 𝑑𝑋 and 𝑑𝑑 Β¨ BOTH 𝐹 and 𝑋 are functions of 𝑑 ultimately, 𝐹(𝑋 𝑑 ) and 𝑋(𝑑) Β¨ So only in a formal manner we write: Β¨ 𝛿𝑓 = )$ )* . 𝛿𝑋, which is the celebrated STRATO lemma (chain rule) 80
  • 81. Luc_Faucheux_2020 Chain Rule (Ito and Strato lemma)- VI Β¨ 𝛿𝑓 = )$ )* . 𝛿𝑋, which is the celebrated STRATO lemma (chain rule) Β¨ HOWEVER, the only rigorous manner in which to write it is: Β¨ ∫+3+& +3+' 𝑑𝐹(𝑋 𝑑 ) = ∫+3+6 +3+5 )$ )* . ∘ . 𝑑𝑋(𝑑) Β¨ With the STRATO interpretation of the integral: Β¨ ∫+3+6 +3+5 𝐹 𝑋 𝑑 . (∘). 𝑑𝑋(𝑑) = lim /β†’@ {βˆ‘>3- >3/ 𝐹 [𝑋(𝑑> + 𝑋(𝑑>?-)]/2). [𝑋(𝑑>?-) βˆ’ 𝑋(𝑑>)]} 81
  • 82. Luc_Faucheux_2020 Chain Rule (Ito and Strato lemma)- VII Β¨ So sometimes we will just stick to the notation: Β¨ Ito lemma: Β¨ 𝛿𝑓 = )$ )* . ([). 𝛿𝑋 + - . . )!9 )!! . ([). 𝛿𝑋. Β¨ Just to remind us that we are using stochastic calculus and that it is NOT a usual product Β¨ Similarly when using STRATO we will sometimes use: Β¨ 𝛿𝑓 = )$ )* . ∘ . 𝛿𝑋 Β¨ To remind us that even if it looks like the regular chain rule, we are in the stochastic world where things are a little weird. Β¨ Again the only rigorous manner to deal with those is to always go back to the integrals, and the interpretation of is as a limit of a sum, which is rigorous. 82
  • 83. Luc_Faucheux_2020 Chain Rule (Ito and Strato lemma)- VIII Β¨ Note that if the function 𝐹(𝑋 𝑑 ) has an explicit dependency on time 𝐹(𝑋 𝑑 , 𝑑) Β¨ Then for the time part the regular chain rule applies and we will obtain terms of the expression: )9 )+ . 𝑑𝑑 in BOTH ITO and STRATO Β¨ Because integrating over time is a normal Riemann integral, and both sums converge to the same limit Β¨ ∫+3+6 +3+5 )9 )+ . ∘ . 𝑑𝑑 = ∫+3+6 +3+5 )9 )+ . [ . 𝑑𝑑 = ∫+3+6 +3+5 )9 )+ . 𝑑𝑑 83
  • 85. Luc_Faucheux_2020 Leibniz rule - I Β¨ In most textbooks, it is usually also presented as Β¨ Hey do a Taylor expansion Β¨ Just make sure to keep the higher order terms Β¨ And you good Β¨ Surprisingly it is formally the same expression Β¨ Even more surprisingly, if you were to be working in a Strato world, textbooks would say: Β¨ Hey just use regular calculus Β¨ Since we have seen now that stochastic calculus can be tricky, let’s spend some time on convincing ourselves that we can adapt the Leibniz rule in stochastic calculus (it will also be super useful when changing numeraires in the Numeraire deck) 85
  • 86. Luc_Faucheux_2020 Leibniz rule - II Β¨ It will be easier to do it once we have a more general expression for the ITO and STRATO integrals, so for now we will just state them (Leibniz rule for first order) Β¨ Leibniz rule in the REGULAR calculus: Β¨ 𝛿 𝑓𝑔 = 𝑔 )$ )* . 𝛿𝑋 + 𝑓 )( )* . 𝛿𝑋 Β¨ Leibniz rule in the ITO calculus: Β¨ 𝛿(𝑓𝑔) = 𝑔 )$ )* . ([). 𝛿𝑋 + 𝑔 - . . )!$ )*! . ([). 𝛿𝑋. + 𝑓 )( )* . ([). 𝛿𝑋 + 𝑓 - . . )!( )*! . ([). 𝛿𝑋. + )( )* . )$ )* . ([). 𝛿𝑋. Β¨ Leibniz rule in the STRATO calculus: Β¨ 𝛿 𝑓𝑔 = 𝑔 )$ )* . ∘ . 𝛿𝑋 + 𝑓 )( )* . ∘ . 𝛿𝑋 86
  • 88. Luc_Faucheux_2020 A worked out example to gain some intuition Β¨ ∫+3+6 +3+5 𝑓 𝑋 𝑑 . ([). 𝑑𝑋(𝑑) = lim /β†’@ {βˆ‘>3- >3/ 𝑓(𝑋(𝑑>)). [𝑋(𝑑>?-) βˆ’ 𝑋(𝑑>)]} Β¨ This is the definition of the Ito integral Β¨ Let’s try with the simple case of the function 𝑓(𝑋(𝑑>)) = 𝑋(𝑑>) Β¨ The Riemann sum is: 𝑆/ = βˆ‘>3- >3/ 𝑓(𝑋(𝑑>)). [𝑋(𝑑>?-) βˆ’ 𝑋(𝑑>)] Β¨ For this specific case: 𝑆/ = βˆ‘>3- >3/ 𝑋(𝑑>). [𝑋(𝑑>?-) βˆ’ 𝑋(𝑑>)] 88
  • 89. Luc_Faucheux_2020 In the β€œregular” case – Riemann integral Β¨ ∫!3!6 !3!5 𝑓 π‘₯ . 𝑑π‘₯ = lim /β†’@ {βˆ‘>3- >3/ 𝑓(π‘₯>). [(π‘₯>?-) βˆ’ (π‘₯>)]} Β¨ This is the usual β€œtriangle” representation 89
  • 90. Luc_Faucheux_2020 In the β€œregular” case – Riemann integral - II Β¨ In the regular case, Β¨ (π‘₯>?-) βˆ’ (π‘₯>) = 𝛿π‘₯ = !':!& / = ;* / Β¨ Another way to look at it, is π‘₯ = 𝑑, so at every point in time, Ξ΄π‘₯ = 𝛿𝑑 Β¨ In particular, Ξ΄π‘₯. = δ𝑑. 90 Xi i+1i-1 F(X) to integrate
  • 91. Luc_Faucheux_2020 In the stochastic case, we cannot draw that picture Β¨ We cannot write (π‘₯>?-) βˆ’ (π‘₯>) = 𝛿π‘₯ = !':!& / = ;* / Β¨ If anything, what we can write is (π‘₯>?-) βˆ’ (π‘₯>) = 𝛿π‘₯> = ±𝛿π‘₯ Β¨ And so βˆ‘(π‘₯>?-) βˆ’ (π‘₯>) = βˆ‘ 𝛿π‘₯> = 0 Β¨ And βˆ‘[(π‘₯>?-) βˆ’ (π‘₯>)].= βˆ‘ 𝛿π‘₯> . = βˆ‘ 𝛿π‘₯. = 𝑁. 𝛿π‘₯. 91 Xi i+1i-1
  • 92. Luc_Faucheux_2020 Another way to think about the difference Β¨ The regular case, the horizontal axis is the regular non-stochastic variable Β¨ Any integration of function is the regular Riemann integral Β¨ In the stochastic case, the horizontal axis becomes time, and the vertical is the stochastic variable X, and we will need to integrate a function of it over the vertical axis, but whereas time is regular, a stochastic process cannot be such that each step is just the interval divided by the number of steps 92 X(t) is stochastic Time t is regular F(X(t))tointegrate
  • 93. Luc_Faucheux_2020 The regular case revisited Β¨ ∫!3!6 !3!5 𝑓 π‘₯ ([)𝑑π‘₯ = lim /β†’@ {βˆ‘>3- >3/ 𝑓(π‘₯>). [(π‘₯>?-) βˆ’ (π‘₯>)]} Β¨ ∫!3!6 !3!5 𝑓 π‘₯ ([)𝑑π‘₯ = lim /β†’@ βˆ‘>3- >3/ 𝑓(π‘₯>). 𝛿π‘₯> = lim /β†’@ !':!& / . βˆ‘>3- >3/ 𝑓(π‘₯>) Β¨ In the case where 𝑓(π‘₯>) = π‘₯> = π‘˜. 𝛿π‘₯ = π‘˜. !':!& / Β¨ ∫!3!6 !3!5 π‘₯([)𝑑π‘₯ = lim /β†’@ !':!& / . !':!& / . βˆ‘>3- >3/ π‘˜ = lim /β†’@ !':!& / . !':!& / . /(/?-) . Β¨ ∫!3!6 !3!5 π‘₯([)𝑑π‘₯ = (π‘₯5 βˆ’ π‘₯6). lim /β†’@ - / . - / . /(/?-) . Β¨ Using π‘₯6 = 0 without losing any generality, Β¨ ∫!34 !3* π‘₯([)𝑑π‘₯ = - . 𝑋., which is the usual result 93
  • 94. Luc_Faucheux_2020 The regular case revisited – II – Strato convention Β¨ ∫!3!6 !3!5 𝑓 π‘₯ (∘)𝑑π‘₯ = lim /β†’@ {βˆ‘>3- >3/ 𝑓( !"#$?!" . ). [(π‘₯>?-) βˆ’ (π‘₯>)]} Β¨ ∫!3!6 !3!5 𝑓 π‘₯ (∘)𝑑π‘₯ = lim /β†’@ βˆ‘>3- >3/ 𝑓( !"#$?!" . ). 𝛿π‘₯> = lim /β†’@ !':!& / . βˆ‘>3- >3/ 𝑓( !"#$?!" . ) Β¨ In the case where𝑓( !"#$?!" . ) = !"#$?!" . = (π‘˜ + - . ). 𝛿π‘₯ = (π‘˜ + - . ). !':!& / Β¨ ∫!3!6 !3!5 π‘₯(∘)𝑑π‘₯ = lim /β†’@ !':!& / . !':!& / . βˆ‘>3- >3/ (π‘˜ + - . ) = lim /β†’@ !':!& / . !':!& / . [ / /?- . + / . ] Β¨ ∫!3!6 !3!5 π‘₯(∘)𝑑π‘₯ = (π‘₯5 βˆ’ π‘₯6). lim /β†’@ - / . - / . [ / /?- . + / . ] Β¨ Using π‘₯6 = 0 without losing any generality, Β¨ ∫!34 !3* π‘₯(∘)𝑑π‘₯ = - . 𝑋., which is the usual result 94
  • 95. Luc_Faucheux_2020 The regular case revisited – III Β¨ So in the regular case, the Riemann integral gives the same result, irrespective of where inside the small interval we pick the value of the function Β¨ ∫!34 !3* π‘₯([)𝑑π‘₯ = - . 𝑋. = ∫!34 !3* π‘₯(∘)𝑑π‘₯ Β¨ In essence, this is because the terms βˆ‘>3- >3/ . 𝛿π‘₯> scale like 1, and so any terms in βˆ‘>3- >3/ . 𝛿π‘₯> . will go to 0 in the limit 𝑁 β†’ ∞ Β¨ The β€œquadratic variation” βˆ‘>3- >3/ . 𝛿π‘₯> . does not add up to any finite number in the limit. Β¨ In the case of a stochastic variable, the terms βˆ‘>3- >3/ . 𝛿π‘₯> scale like 0, because 𝛿π‘₯> = ±𝛿π‘₯, and the terms like βˆ‘>3- >3/ . 𝛿π‘₯> . will actually scale like time. Β¨ The β€œquadratic variation” βˆ‘>3- >3/ . 𝛿π‘₯> . is said to scale linearly with time (some textbooks use the expression additive with time) and will not β€œdisappear” to 0 in the limit. 95
  • 96. Luc_Faucheux_2020 Back to the worked-out example Β¨ The Riemann sum is: 𝑆/ = βˆ‘>3- >3/ 𝑋(𝑑>). [𝑋(𝑑>?-) βˆ’ 𝑋(𝑑>)] Β¨ We can expand: [𝑋(𝑑>?-) βˆ’ 𝑋(𝑑>)].= 𝑋(𝑑>?-). + 𝑋(𝑑>). βˆ’ 2. 𝑋(𝑑>). 𝑋(𝑑>?-) Β¨ And: 𝑋(𝑑>). [𝑋(𝑑>?-) βˆ’ 𝑋(𝑑>)] = 𝑋(𝑑>). 𝑋(𝑑>?-) βˆ’ 𝑋(𝑑>). Β¨ And : 𝑋(𝑑>). [𝑋(𝑑>?-) βˆ’ 𝑋(𝑑>)] = - . . [𝑋(𝑑>?-). + 𝑋(𝑑>). βˆ’ [𝑋(𝑑>?-) βˆ’ 𝑋(𝑑>)].βˆ’2. 𝑋(𝑑>).] Β¨ Or: 𝑋(𝑑>). [𝑋(𝑑>?-) βˆ’ 𝑋(𝑑>)] = - . . [𝑋(𝑑>?-). βˆ’ 𝑋(𝑑>).] βˆ’ - . . [𝑋(𝑑>?-) βˆ’ 𝑋(𝑑>)]. Β¨ So: 𝑆/ = βˆ‘>3- >3/ 𝑋(𝑑>). [𝑋(𝑑>?-) βˆ’ 𝑋(𝑑>)] = - . . 𝑋(𝑑/?-). βˆ’ - . βˆ‘>3- >3/ [𝑋(𝑑>?-) βˆ’ 𝑋(𝑑>)]. Β¨ The first term is the expected *! . Β¨ The second term would usually disappear when the variable X is regular, because the sum would scale as 𝑁. ( - / ).~ - / which will tend to 0 when 𝑁 β†’ ∞ 96
  • 97. Luc_Faucheux_2020 Back to the worked-out example - II Β¨ 𝑆/ = βˆ‘>3- >3/ 𝑋(𝑑>). [𝑋(𝑑>?-) βˆ’ 𝑋(𝑑>)] = - . . 𝑋(𝑑/?-). βˆ’ - . βˆ‘>3- >3/ [𝑋(𝑑>?-) βˆ’ 𝑋(𝑑>)]. Β¨ We define 𝑄/ = βˆ‘>3- >3/ [𝑋(𝑑>?-) βˆ’ 𝑋(𝑑>)]. Β¨ We can use here the simple β€œbinary” assumption that [𝑋(𝑑>?-) βˆ’ 𝑋(𝑑>)].= (𝑑>?-βˆ’π‘‘>) Β¨ Note that this is capturing the essence of the matter Β¨ A more accurate and general way to go about this would be to assume the following: Β¨ {𝑋(𝑑>?-) βˆ’ 𝑋(𝑑>)} follows the 𝑁(0, (𝑑>?-βˆ’π‘‘>)) distribution Β¨ We would then estimate the expected value of 𝑄/ Β¨ 𝐸 𝑄/ = βˆ‘>3- >3/ 𝐸[𝑋(𝑑>?-) βˆ’ 𝑋(𝑑>)]. = βˆ‘>3- >3/ (𝑑>?-βˆ’π‘‘>) = 𝑑/?- Β¨ We would have to show then that: lim /β†’@ 𝑄/ = 𝐸[𝑄/] Β¨ This opens up the can of worms of how you define convergence (convergence in distribution, in probability, almost sure convergence, Lp-convergence). Will try to incorporate later, but trying to keep it simple to not hide the intuition behind notations 97
  • 98. Luc_Faucheux_2020 What did we get? Β¨ ∫+34 +3= 𝑓 𝑋 𝑑 . ([). 𝑑𝑋(𝑑) = lim /β†’@ {βˆ‘>3- >3/ 𝑓(𝑋(𝑑>)). [𝑋(𝑑>?-) βˆ’ 𝑋(𝑑>)]} Β¨ In the simple case 𝑓 𝑋(𝑑> = 𝑋(𝑑>) Β¨ ∫+34 +3= 𝑋(𝑑). ([). 𝑑𝑋(𝑑) = - . 𝑋. βˆ’ - . 𝑇 Β¨ A couple of notes: Β¨ Ito integral does NOT recover the usual rules of calculus Β¨ Ito integral is called a martingale, meaning if X is a martingale (E[X]=0), then the Ito integral of a function f(X) is also a martingale Β¨ 𝐸 𝑋 = 0 and 𝐸[∫+34 +3= 𝑋 𝑑 . [). 𝑑𝑋 𝑑 = 𝐸 - . 𝑋. βˆ’ - . 𝑇 = 𝐸 - . 𝑋. βˆ’ - . 𝑇 = 0 Β¨ So 𝐸 𝑋. = 𝑇 Β¨ We recover the usual variance of the Brownian motion 98
  • 99. Luc_Faucheux_2020 A couple more notes Β¨ Let’s recall the relationship between the Ito and Stratonovitch integral (we can also work it out from the Riemann sum) Β¨ ∫+3+6 +3+5 𝑓 𝑋 𝑑 . (∘). 𝑑𝑋(𝑑) = ∫+3+6 +3+5 𝑓 𝑋 𝑑 . ([). 𝑑𝑋(𝑑) + - . ∫+3+6 +3+5 𝑓′ 𝑋 𝑑 . 𝑑𝑑 Β¨ Here, 𝑓 𝑋 = 𝑋 so 𝑓% 𝑋 = 1 Β¨ ∫+34 +3= 𝑓 𝑋 𝑑 . (∘). 𝑑𝑋(𝑑) = ∫+34 +3= 𝑓 𝑋 𝑑 . ([). 𝑑𝑋(𝑑) + - . ∫+34 +3= 1. 𝑑𝑑 Β¨ And we have: ∫+34 +3= 𝑋(𝑑). ([). 𝑑𝑋(𝑑) = - . 𝑋. βˆ’ - . 𝑇 Β¨ So ∫+34 +3= 𝑓 𝑋 𝑑 . (∘). 𝑑𝑋(𝑑) = - . 𝑋. Β¨ This is the usual rule of calculus Β¨ The Stratonovitch integral preserves the usual rule of calculus 99
  • 100. Luc_Faucheux_2020 100 What does the usual exponential even mean in stochastic calculus? The ITO exponential
  • 101. Luc_Faucheux_2020 A couple more notes - II Β¨ The Stratonovitch integral is NOT a martingale (this makes sense since taking the mid point introduces correlation, and thus the terms in the product are NOT independent) Β¨ Because the Ito lemma does NOT recover the usual rules of calculus, what a function actually means has to be at times redefined. Β¨ For example, we all know the exponential function 𝑓 𝑑 = exp(𝑑) Β¨ It is such that : 𝑓% 𝑑 = 𝑓(𝑑) Β¨ However we would not expect this function to preserve the same property when dealing with a stochastic variable (another way to say it is that this function is convex, and so the Jensen inequality will introduce a convexity correction, see the lecture on options) Β¨ We know that Stratonovitch will preserve the rules of calculus Β¨ So ∫+34 +3= 𝑒π‘₯𝑝 𝑋 𝑑 . ∘ . 𝑑𝑋 𝑑 = exp(𝑋(𝑇)) Β¨ And ∫+34 +3= 𝑓 𝑋 𝑑 . (∘). 𝑑𝑋(𝑑) = ∫+34 +3= 𝑓 𝑋 𝑑 . ([). 𝑑𝑋(𝑑) + - . ∫+34 +3= 𝑓′ 𝑋 𝑑 . 𝑑𝑑 101
  • 102. Luc_Faucheux_2020 The Ito exponential function Β¨ ∫+34 +3= 𝑓 𝑋 𝑑 . (∘). 𝑑𝑋(𝑑) = 𝑒π‘₯𝑝 𝑋 𝑇 = ∫+34 +3= 𝑓 𝑋 𝑑 . ([). 𝑑𝑋(𝑑) + - . ∫+34 +3= 𝑓′ 𝑋 𝑑 . 𝑑𝑑 Β¨ 𝑒π‘₯𝑝 𝑋 𝑇 = ∫+34 +3= 𝑓 𝑋 𝑑 . ([). 𝑑𝑋(𝑑) + - . ∫+34 +3= 𝑒π‘₯𝑝 𝑋 𝑑 . 𝑑𝑑 Β¨ Since the second term is always positive, we have Β¨ ∫+34 +3= 𝑒π‘₯𝑝 𝑋 𝑑 . ([). 𝑑𝑋(𝑑) <> 𝑒π‘₯𝑝 𝑋 𝑇 Β¨ So clearly under Ito, the usual exponential function does not verify 𝐹% 𝑋 = 𝐹(𝑋) Β¨ For that reason, the convention is to rename some of the functions by specifying β€œIto” in front of them Β¨ This can be confusing at times 102
  • 103. Luc_Faucheux_2020 The Ito exponential function -II Β¨ Ito lemma has : 𝛿𝐹 = )9 )* . 𝛿𝑋 + - . . )!9 )*! . 𝛿𝑋. + )9 )+ . 𝛿𝑑 Β¨ With 𝐹 𝑋 = exp 𝑋 , 𝐹% 𝑋 = exp 𝑋 , 𝐹%% 𝑋 = exp(𝑋) Β¨ We get 𝛿𝐹 = exp 𝑋 . 𝛿𝑋 + - . . exp 𝑋 . 𝛿𝑋. = exp(𝑋). 𝛿𝑋 + - . . exp(𝑋). 𝛿𝑑 Β¨ With 𝐹 𝑋, 𝑑 = exp(𝑋 βˆ’ + . ) , )9 )* = exp 𝑋 βˆ’ + . , )!9 )*! = exp 𝑋 βˆ’ + . , )9 )+ = :- . . exp(𝑋 βˆ’ + . ) Β¨ We get 𝛿𝐹 = exp 𝑋 βˆ’ + . . 𝛿𝑋 + - . . exp 𝑋 βˆ’ + . . 𝛿𝑋. βˆ’ - . . exp 𝑋 βˆ’ + . . 𝛿𝑑 Β¨ Or: 𝛿𝐹 = exp 𝑋 βˆ’ + . . 𝛿𝑋 = 𝐹 𝑋, 𝑑 . 𝛿𝑋 Β¨ So you will see sometimes the function 𝐹 𝑋, 𝑑 = exp 𝑋 βˆ’ + . being referred to as the Ito- exponential, whereas the regular exponential is of course 𝐹 𝑋 = exp 𝑋 103
  • 104. Luc_Faucheux_2020 The Ito exponential function -III Β¨ So this is another indication that we should be careful with using regular functions and some of their properties. Β¨ In regular calculus, the exponential function 𝑓 π‘₯ = exp(π‘₯) is such that : 𝑓% π‘₯ = 𝑓(π‘₯) Β¨ In ITO calculus, the β€œITO exponential function” that still verifies: 𝐹% 𝑋(𝑑) = 𝐹(𝑋(𝑑)) is given by: 𝐹 𝑋(𝑑), 𝑑 = exp 𝑋(𝑑) βˆ’ + . Β¨ In STRATO calculus, the β€œSTRATO exponential function” that still verifies: 𝐹% 𝑋(𝑑) = 𝐹(𝑋(𝑑)) is given by: 𝐹 𝑋(𝑑), 𝑑 = exp 𝑋(𝑑) Β¨ Note that we are somehow lucky that those functions are still local (i.e. depends only on the value of 𝑋(𝑑) and 𝑑. It is not clear that we could not have ended up with a more complicated function that depends on the history or the path ∫+34 +3= 𝑋 𝑑 . 𝑑𝑑 for example. 104
  • 105. Luc_Faucheux_2020 A nice little summary Name of Integral RIEMANN ITO STRATONOVITCH Type Deterministic Stochastic Stochastic Points of integration Doesn’t matter LEFT MIDDLE Martingale NO YES NO Non-Anticipating NO YES NO Usual Calculus Rule YES NO YES(*) Main Usage Everywhere Finance Physics 105 Β¨ (*) ONLY from a formal point of view. This is still a stochastic integral and a nasty beast not to be messed with β€œΓ  la lΓ©gΓ¨re”
  • 106. Luc_Faucheux_2020 106 Back to trying to β€œintegrate” X
  • 107. Luc_Faucheux_2020 Another example: can you integrate X ? Β¨ Somewhat cultural side note: In French β€œto integrate” (or β€œintegrer”) is the same word to perform a mathematical integration or to be accepted in a school. Β¨ One of the most prestigious schools is called β€œPolytechnique” or β€œX” because of the logo on their hats of two crossed swords (it was created and is still technically a military school, a little like West Point here, but starts around the junior college level) Β¨ So anyways, after high school, there are special schools called β€œclasses preparatoires” where you just study for the entrance exam to those advanced schools (β€œhautes ecoles”), with names like X, Ecole Normale Superieure (Normal Superieure school) or others with names that are usually tied to their original concentration: Ponts et Chaussees (bridges and sidewalks), Mines (mining), Supelec (Superior Electricity), and so on Β¨ Usually students spend around 2 years studying in β€œclasses preparatoires” before taking the entrance exam (everything is an exam, there is no application, you take the exam, you get ranked and then the school offers you a spot in the order of ranking). Β¨ So you get the idea, the question is how many years does it take you to integrate X Β¨ 50 years later, people will still remember what β€œhalfs” they were, or they will lie about it 107
  • 108. Luc_Faucheux_2020 Another example: can you integrate X ? - II Β¨ If you are a genius and if it takes you only one year to integrate X, the student is called a ”one half” because: ∫*34 *3- 𝑋. 𝑑𝑋 = [ *! . ]*34 *3- = - . Β¨ I have heard of β€œone-half” I have never met one. Β¨ This is not the same as β€œhalf-bloods” from Harry Potter, I have never met one of those either Β¨ If you are a decent student and if it takes you two years to integrate X, the student is called a ”three half” because: ∫*3- *3. 𝑋. 𝑑𝑋 = [ *! . ]*3- *3. = C . βˆ’ - . = D . Β¨ (I was a 3/2, but I did not integrate X, my father did, my brother did, I am the black sheep of the family) Β¨ If you are a decent student and if it takes you three years to integrate X, the student is called a ”five half” because: ∫*3. *3D 𝑋. 𝑑𝑋 = [ *! . ]*3. *3D = E . βˆ’ C . = F . Β¨ If it takes you four years to integrate X, the student is called a ”seven half” because: ∫*3D *3C 𝑋. 𝑑𝑋 = [ *! . ]*3D *3C = -G . βˆ’ E . = H . Β¨ Being called a β€œseven half” is an insult 108
  • 109. Luc_Faucheux_2020 Another example: can you integrate X ? - III Β¨ Oh also if you integrate 𝑋 not in 𝑑𝑋 but in 𝑑𝑑, you are called β€œendette”, or in debt Β¨ OK, so that was a little digression Β¨ Let’s get back to the matter at hands. Β¨ If 𝑋 is not stochastic (also called deterministic), this is the usual calculus that we are used to, and so: Β¨ ∫ 𝑋. 𝑑𝑋 = *! . + 𝐢 Β¨ What happens if 𝑋 is now a stochastic variable? Β¨ The truth is that I have never met anyone who can integrate a stochastic variable (I have also never met a β€œone half”), and so usually people always treat the problem from the other way, you start with a guess of what the integral is, you use ITO lemma and then you check in an iterative manner 109
  • 110. Luc_Faucheux_2020 Another example: can you integrate X ? - IV Β¨ If we have a function 𝐹(𝑋), Ito lemma is the following: Β¨ Ito (1951) guessed that you can write 𝑑π‘₯ = π‘Ž π‘₯, 𝑑 . 𝑑𝑑 + 𝑏 π‘₯, 𝑑 . 𝑑𝑧 as Ξ΄π‘₯ = π‘Ž π‘₯, 𝑑 . 𝛿𝑑 + 𝑏 π‘₯, 𝑑 . 𝛿𝑧 Β¨ If 𝐹(π‘₯) a funcyon of x, the corresponding SDE as a result of β€œTaylor” expansion in ITO is: Β¨ Bear in mind that this is really not a Taylor expansion, it is just ITO lemma that looks like a Taylor expansion where you kept some terms and not others Β¨ 𝛿𝐹 = )9 )! . 𝛿π‘₯ + - . . )!9 )!! . 𝑏.. 𝛿𝑑 Β¨ 𝛿𝐹 = )9 )* . 𝛿𝑋 + - . . )!9 )*! . 𝛿𝑋. Β¨ Let’s use 𝐹 𝑋 = 𝑋. as a good guess Β¨ 𝐹 𝑋 = 𝑋., )9 )* = 2𝑋, )!9 )*! = 2, we then get: 𝛿 𝑋. = 2𝑋. 𝛿𝑋 + - . . 2. 𝛿𝑋. 110
  • 111. Luc_Faucheux_2020 Another example: can you integrate X ? - V Β¨ 𝛿 𝑋. = 2𝑋. 𝛿𝑋 + - . . 2. 𝛿𝑋. Β¨ In the case of the Geometric Brownian motion (Hull): 𝑑𝑋 = πœŽπ‘‹π‘‘π‘Š Β¨ 𝛿 𝑋. = 2𝑋. 𝛿𝑋 + - . . 2. (πœŽπ‘‹). 𝛿𝑑 Β¨ 𝛿 𝑋. = 2𝑋. 𝛿𝑋 + (πœŽπ‘‹). 𝛿𝑑 Β¨ And so: ∫ 𝑋. 𝑑𝑋 = ∫{ ; *! . βˆ’ - . . (πœŽπ‘‹). 𝛿𝑑} Β¨ ∫ 𝑋. 𝑑𝑋 = [ *! . ] βˆ’ - . ∫(πœŽπ‘‹). 𝛿𝑑 Β¨ With the usual convention that 𝑋 𝑑 = 0 = 0, in the ITO convention: Β¨ ∫+34 +3= 𝑋. 𝑑𝑋 = *(=)! . βˆ’ - . ∫+34 +3= (πœŽπ‘‹). 𝛿𝑑 111
  • 112. Luc_Faucheux_2020 Another example: can you integrate X ? - VI Β¨ Things are a little different in the Stratonovitch convention: Β¨ Strato guessed that you can write 𝑑π‘₯ = π‘Ž π‘₯, 𝑑 . 𝑑𝑑 + 𝑏 π‘₯, 𝑑 . π‘‘π‘Š as Ξ΄π‘₯ = π‘Ž π‘₯, 𝑑 . 𝛿𝑑 + 𝑏 π‘₯ + 𝛿π‘₯/2, 𝑑 . π›Ώπ‘Š Β¨ The SDE for 𝐹(π‘₯) in Stratonovitch convention is (using what is known as ITO calculus): Β¨ 𝛿𝐹 = )9 )! . 𝛿π‘₯ + - . . )!9 )!! . 𝑏.. 𝛿𝑑 + - . . )9 )! . )5 )! . 𝑏. 𝛿𝑑 Β¨ We know this is wrong, but let’s just illustrate how wrong it is: Β¨ Rewriting it without the drift term we get: Β¨ 𝛿𝑋 = 𝑏 𝑋 + ;* . , 𝑑 . π›Ώπ‘Š = 𝑏 𝑋 . π›Ώπ‘Š + - . . )5 )! . 𝛿𝑋. π›Ώπ‘Š Β¨ 𝛿𝑋 = 𝑏 𝑋 + ;* . , 𝑑 . π›Ώπ‘Š = 𝑏 𝑋 . π›Ώπ‘Š + - . . )5 )! . 𝑏 𝑋 . 𝛿𝑑 Β¨ 𝛿𝑋 = 𝑏 𝑋 . π›Ώπ‘Š + - . . )5 )! . 𝑏 𝑋 . 𝛿𝑑 in Stratonovitch while Ito had: 𝛿𝑋 = 𝑏 𝑋 . π›Ώπ‘Š 112
  • 113. Luc_Faucheux_2020 Another example: can you integrate X ? - VII Β¨ Redoing it wrong just to convince ourselves of how wrong it is: Β¨ 𝛿𝑋 = 𝑏 𝑋 ∘ π›Ώπ‘Š = 𝑏 𝑋 + ;* . , 𝑑 . π›Ώπ‘Š = 𝑏 𝑋 . π›Ώπ‘Š + - . . )5 )! . 𝑏 𝑋 . 𝛿𝑑 Β¨ 𝛿𝐹 = )9 )* . 𝛿𝑋 + - . . )!9 )*! . 𝑏.. 𝛿𝑑 + - . . )9 )* . )5 )* . 𝑏. 𝛿𝑑 Β¨ 𝛿𝐹 = )9 )* . 𝛿𝑋 + - . . )!9 )*! . 𝛿𝑋. Β¨ 𝛿𝐹 = )9 )* . 𝑏 𝑋 + ;* . . π›Ώπ‘Š + - . . )!9 )*! . 𝑏 𝑋 + ;* . , 𝑑 . . 𝛿𝑑 Β¨ 𝛿𝐹 = )9 )* . 𝑏 𝑋 . π›Ώπ‘Š + )9 )* . )5 )* . ;* . . π›Ώπ‘Š + - . . )!9 )*! . 𝑏 𝑋 + ;* . , 𝑑 . . 𝛿𝑑 113
  • 114. Luc_Faucheux_2020 Another example: can you integrate X ? - IX Β¨ We get: Β¨ 𝛿𝐹 = )9 )* . 𝑏 𝑋 . π›Ώπ‘Š + )9 )* . )5 )* . ;* . . π›Ώπ‘Š + - . . )!9 )*! . 𝑏 𝑋 + ;* . . . 𝛿𝑑 Β¨ And : Β¨ 𝑏 𝑋 + ;* . . = 𝑏 𝑋 . + 2. 𝑏 𝑋 . )5 )* . ;* . Β¨ 𝛿𝐹 = )9 )* . 𝑏 𝑋 . π›Ώπ‘Š + )9 )* . )5 )* . ;* . . π›Ώπ‘Š + - . . )!9 )*! . 𝑏 𝑋 .. 𝛿𝑑 + - . . )!9 )*! . 2𝑏 𝑋 . )5 )* . ;* . 𝛿𝑑 Β¨ Keeping only the terms in order 1 in 𝛿𝑍 and 𝛿𝑑 Β¨ 𝛿𝐹 = )9 )* . 𝑏 𝑋 . π›Ώπ‘Š + - . . )!9 )*! . 𝑏 𝑋 .. 𝛿𝑑 + - . . )9 )* . )5 )* . 𝑏. 𝛿𝑑 114
  • 115. Luc_Faucheux_2020 Another example: can you integrate X ? - X Β¨ 𝛿𝐹 = )9 )* . 𝑏 𝑋 . π›Ώπ‘Š + - . . )!9 )*! . 𝑏 𝑋 .. 𝛿𝑑 + - . . )9 )* . )5 )* . 𝑏. 𝛿𝑑 Β¨ In the case of the simple Geometric Brownian motion (Hull): 𝑑𝑋 = πœŽπ‘‹π‘‘π‘Š Β¨ Using the guess 𝐹 𝑋 = 𝑋. Β¨ 𝛿(𝑋.) = 2𝑋. πœŽπ‘‹. π›Ώπ‘Š + - . . 2. (πœŽπ‘‹). 𝛿𝑑𝛿𝑑 + - . . 2𝑋. 𝜎. πœŽπ‘‹. 𝛿𝑑 Β¨ 𝛿(𝑋.) = 2𝑋. 𝛿𝑋 + 2. (πœŽπ‘‹).. 𝛿𝑑 Β¨ In Ito we had : 𝛿 𝑋. = 2𝑋. 𝛿𝑋 + - . . 2. (πœŽπ‘‹).. 𝛿𝑑 Β¨ So using ITO: ∫+34 +3= 𝑋. 𝑑𝑋 = *(=)! . βˆ’ - . ∫+34 +3= (πœŽπ‘‹).. 𝛿𝑑 Β¨ Using Stratonovitch: ∫+34 +3= 𝑋. 𝑑𝑋 = *(=)! . βˆ’ ∫+34 +3= (πœŽπ‘‹).. 𝛿𝑑 115
  • 116. Luc_Faucheux_2020 Another example: can you integrate X ? - Xa Β¨ If we were to look at the simple Brownian motion case: Β¨ 𝑑𝑋 = π‘‘π‘Š Β¨ ITO: 𝛿 𝑋. = 2𝑋. 𝛿𝑋 + - . . 2. 𝛿𝑋. Β¨ 𝛿 𝑋. = 2𝑋. 𝛿𝑋 + 𝛿𝑑 Β¨ And so: ∫ 𝑋. 𝑑𝑋 = ∫{ ; *! . βˆ’ - . . 𝛿𝑑} Β¨ ∫ 𝑋. 𝑑𝑋 = [ *! . ] βˆ’ - . ∫ 𝛿𝑑 Β¨ With the usual convention that 𝑋 𝑑 = 0 = 0, in the ITO convention: Β¨ ∫+34 +3= 𝑋. 𝑑𝑋 = *(=)! . βˆ’ - . 𝑇 116
  • 117. Luc_Faucheux_2020 Another example: can you integrate X ? - Xb Β¨ ∫+34 +3= 𝑋. 𝑑𝑋 = *(=)! . βˆ’ - . 𝑇 Β¨ Note that we know this to be consistent since the ITO integral is a martingale (or a trading strategy) Β¨ We will go over this again at more length but we have: Β¨ 𝔼 ∫+34 +3= 𝑋. 𝑑𝑋 = 0 = 𝔼 * = ! . βˆ’ 𝔼 - . 𝑇 = 𝔼 * = ! . βˆ’ - . 𝑇 Β¨ And we recover the usual dispersion expectation for the simple Brownian motion: Β¨ 𝔼 𝑋 𝑇 . = 𝑇 117
  • 118. Luc_Faucheux_2020 Another example: can you integrate X ? - Xc Β¨ If we were to look at the simple Brownian motion case: Β¨ 𝑑𝑋 = π‘‘π‘Š 𝑏 𝑋 = 1 so )5 )* = 0 Β¨ STRATO: 𝛿𝐹 = )9 )* . 𝑏 𝑋 . π›Ώπ‘Š + - . . )!9 )*! . 𝑏 𝑋 .. 𝛿𝑑 + - . . )9 )* . )5 )* . 𝑏. 𝛿𝑑 Β¨ 𝛿 𝑋. = 2𝑋. 𝛿𝑋 + 𝛿𝑑 This is exactly the same as ITO Β¨ So we get the strange result that in BOTH ITO and STRATO we will recover for the simple Brownian motion: Β¨ ∫+34 +3= 𝑋. 𝑑𝑋 = *(=)! . βˆ’ - . 𝑇 Β¨ This is clearly wrong, and once again the error was in β€œusing a STRATO convention in the ITO calculus”, and not even that, as we took β€œITO calculus” as β€œdoing regular Taylor expansions like in regular calculus and just keeping some terms and neglecting some other terms” 118
  • 119. Luc_Faucheux_2020 Another example: can you integrate X ? - XI Β¨ HOWEVER, remember the formula in the simple Wiener case : 𝑋 = π‘Š Β¨ 𝑆𝑇𝑅𝐴𝑇𝑂(𝑓) = 𝐼𝑇𝑂 𝑓 + - . . 𝑅𝐼𝐸𝑀𝐴𝑁𝑁(𝑓%) Β¨ In the case of 𝑓 = 𝑋, 𝑓% = 1 Β¨ 𝑆𝑇𝑅𝐴𝑇𝑂 𝑋 = 𝐼𝑇𝑂 𝑋 + - . . 𝑅𝐼𝐸𝑀𝐴𝑁𝑁(1) with Β¨ 𝑅𝐼𝐸𝑀𝐴𝑁𝑁 1 = ∫+34 +3= 1. 𝑑𝑑 = 𝑇 Β¨ So Ito : ∫+34 +3= 𝑋. 𝑑𝑋 = *(=)! . βˆ’ - . ∫+34 +3= 1. 𝛿𝑑 or ∫+34 +3= 𝑋. 𝑑𝑋 = *(=)! . βˆ’ - . 𝑇 Β¨ Stratonovitch: ∫+34 +3= 𝑋 ∘ 𝑑𝑋 = ∫+34 +3= 𝑋. 𝑑𝑋 + - . 𝑇 = *(=)! . βˆ’ - . 𝑇 + - . 𝑇 = *(=)! . Β¨ Startonovitch as expected follows in a formal manner the usual rules of calculus 119
  • 120. Luc_Faucheux_2020 Another example: can you integrate X ? - XII Β¨ So clearly we failed to integrate X, because we found the following wrong results: Β¨ In the Geometric Brownian motion case, Strato does not match formally the usual rules of calculus Β¨ In the simple Brownian motion case, BOTH ITO and STRATO returns the same result Β¨ This is not possible as we know that the ITO and STRATO integrals are different and we know what the difference is ( - . . 𝑅𝐼𝐸𝑀𝐴𝑁𝑁) Β¨ But again, apologies if so far the mistake was obvious, but NEVER rely on usual rules of calculus, or think that Stratanovitch is just β€œtaking the middle point” and that you can still use ITO lemma or Taylor expansion in a nested manner. If you choose STRATO you CANNOT use ITO calculus, you have to stay in STRATO calculus, and vice versa Β¨ Also note that ITO lemma is very generic in (𝛿𝑋). but when choosing a specific expression for𝑋(𝑑) as a function of the Brownian motion (Wiener process π‘Š(𝑑)), we end up with very different expressions, some tractable, some not so much 120
  • 121. Luc_Faucheux_2020 Another example: can you integrate X ? - XIII Β¨ The right way to do it is of course: Β¨ ITO: 𝛿𝑓 = )$ )* . ([). 𝛿𝑋 + - . . )!9 )!! . ([). 𝛿𝑋. Β¨ The results we obtained in the preceding slides for ITO are still correct Β¨ STRATO: 𝛿𝑓 = )$ )* . ∘ . 𝛿𝑋 Β¨ And then we do get in the case of the simple Brownian motion: 𝑑𝑋 = π‘‘π‘Š Β¨ 𝐹 𝑋 = 𝑋. Β¨ 𝛿𝑓 = 2𝑋. ∘ . π›Ώπ‘Š Β¨ ∫+34 +3= 𝑋. ∘ . 𝑑𝑋 = - . ∫+34 +3= 2𝑋. ∘ . π›Ώπ‘Š = - . ∫+34 +3= 𝑑 𝑋. = [ *! . ]+34 +3= Β¨ As expected if STRATO follows formally the usual rules of calculus 121
  • 122. Luc_Faucheux_2020 Quick notes Β¨ Usually when dealing with an SDE, people will try Β¨ 1) get back to an SDE where the stochastic scaling does not depend on the stochastic variable. For example, when dealing with 𝑑𝑆 = πœ‡. 𝑆. 𝑑𝑑 + 𝜎. 𝑆. π‘‘π‘Š, the natural road to explore is writing something like I< < = πœ‡. 𝑑𝑑 + 𝜎. π‘‘π‘Š, and then try to ”define” what I< < is. Β¨ It seems intuitive that I< < should have something to do with 𝑑(ln 𝑆 ), so we would love to write something like 𝑑(𝑙𝑛 𝑆 ) = πœ‡. 𝑑𝑑 + 𝜎. π‘‘π‘Š, but it turns out that is not quite right, because 𝑙𝑛 𝑆 is convex as a function of 𝑆, and 𝑆 is stochastic, not deterministic Β¨ So always re-derive ITO lemma to make sure you are not dropping terms in the β€œTaylor expansion” Β¨ 2) try to get rid of the term in from of the 𝑑𝑧, because then you are not dealing with a SDE anymore, but with a PDE. This is what Black Sholes did by adding a β€œdelta hedge”, and off to a Nobel prize they went 122
  • 123. Luc_Faucheux_2020 Quick notes - II Β¨ We also see the famous β€œconvexity adjustment” that we looked at in the Options deck. Β¨ 𝑙𝑛 𝑆 is negatively convex as a function of the stochastic variable Β¨ So from an intuition point of view as we saw Β¨ < 𝑙𝑛 𝑆 > β‰  𝑙𝑛 < 𝑆 > Β¨ Actually we know that : < 𝑙𝑛 𝑆 > = 𝔼(𝑙𝑛 𝑆 ) ≀ 𝑙𝑛 𝔼(𝑆) = 𝑙𝑛 < 𝑆 > Β¨ So it would make sense that around the point 𝔼(𝑆) the function 𝑙𝑛 𝑆 would not behave as the β€œregular” function 𝑙𝑛 𝔼(𝑆) Β¨ Note that the distinction though, the convexity adjustment was coming from integrating over the possible outcomes at a given time of the stochastic variable. Here we are integrating over the time (over the path over time of the stochastic variable). They are related nonetheless. Β¨ Obviously if you have the expression that resulted from integrating over the path, you can now use it to integrate over the distribution 123
  • 124. Luc_Faucheux_2020 Quick notes - III Β¨ Just to make it explicit. Β¨ ∫+3+& +3+' 𝑑𝐹(𝑋 𝑑 ) = ∫+3+6 +3+5 )9 )* . ([). 𝑑𝑋(𝑑) + - . ∫+3+6 +3+5 )!9 )!! 𝑋 𝑑 . (𝑑𝑋). Β¨ 𝛿𝐹 = )9 )* . 𝛿𝑋 + - . . )!9 )!! . (𝛿𝑋)., which is the celebrated Ito lemma (chain rule) Β¨ 𝐹 𝑋 = ln(𝑋), )9 )* = 1/𝑋, )!9 )*! = βˆ’1/𝑋. Β¨ In the case of the simple Brownian motion: 𝑑𝑋 = π‘‘π‘Š Β¨ 𝛿(π‘™π‘›π‘Š) = - 1 . π›Ώπ‘Š βˆ’ - . . - 1! . 𝛿𝑑 this is quite ugly to deal with Β¨ In the case of the Geometric Brownian motion (Hull) 𝑑𝑋 = 𝑋. π‘‘π‘Š Β¨ 𝛿 𝑙𝑛𝑋 = - * . 𝑋. π›Ώπ‘Š βˆ’ - . . - *! . 𝑋.. 𝛿𝑑 = π›Ώπ‘Š βˆ’ - . . 𝛿𝑑 this is quite nice and easy to deal with 124
  • 125. Luc_Faucheux_2020 Quick notes - IV Β¨ So for the log function and in the case of a geometric Brownian motion, we have Β¨ 𝛿 𝑙𝑛𝑋 = π›Ώπ‘Š βˆ’ - . . 𝛿𝑑 Β¨ Or more rigorously: Β¨ ∫+3+& +3+' 𝑑𝐿𝑁(𝑋 𝑑 ) = 𝐿𝑛 𝑋 𝑑5 βˆ’ 𝐿𝑛(𝑋 𝑑6 ) = Β¨ 𝐿𝑛 𝑋 𝑑5 βˆ’ 𝐿𝑛 𝑋 𝑑6 = ∫+3+6 +3+5 π‘‘π‘Š 𝑑 βˆ’ - . ∫+3+6 +3+5 𝑑𝑑 = π‘Š 𝑑5 βˆ’ π‘Š 𝑑6 βˆ’ - . (𝑑5 βˆ’ 𝑑6) Β¨ One could be tempted to identify - . . 𝛿𝑑 as a convexity adjustment (which it is in some way, since if the function was not convex, i.e. )!9 )*! = 0, this term would not appear), but it is not exactly the one we are dealing with in the option deck, as that one also depends on the specific distribution being used. 125
  • 126. Luc_Faucheux_2020 Quick notes - V Β¨ Still rather crudely (but rigorous, as we will show in section V on the GBM) Β¨ 𝑑𝑋 = 𝑋. π‘‘π‘Š careful, this is not the same as writing 𝑋 𝑑 = exp(π‘Š 𝑑 ) Β¨ 𝔼 𝑋 𝑑5 = 𝑋 𝑑6 Β¨ 𝐿𝑛 𝑋 𝑑5 βˆ’ 𝐿𝑛 𝑋 𝑑6 = ∫+3+6 +3+5 π‘‘π‘Š 𝑑 βˆ’ - . ∫+3+6 +3+5 𝑑𝑑 = π‘Š 𝑑5 βˆ’ π‘Š 𝑑6 βˆ’ - . (𝑑5 βˆ’ 𝑑6) Β¨ 𝔼 𝐿𝑛 𝑋 𝑑5 βˆ’ 𝐿𝑛 𝑋 𝑑6 = 𝔼{π‘Š 𝑑5 βˆ’ π‘Š 𝑑6 βˆ’ - . (𝑑5 βˆ’ 𝑑6)} Β¨ 𝔼 𝐿𝑛 𝑋 𝑑5 βˆ’ 𝐿𝑛 𝑋 𝑑6 = 𝔼 βˆ’ - . 𝑑5 βˆ’ 𝑑6 = βˆ’ - . (𝑑5 βˆ’ 𝑑6) Β¨ 𝔼 𝐿𝑛 𝑋 𝑑5 = 𝔼 𝐿𝑛 𝑋 𝑑6 βˆ’ - . 𝑑5 βˆ’ 𝑑6 = 𝐿𝑛 𝑋 𝑑6 βˆ’ - . (𝑑5 βˆ’ 𝑑6) Β¨ 𝔼 𝐿𝑛 𝑋 𝑑5 = 𝐿𝑛 𝔼 𝑋 𝑑5 βˆ’ - . (𝑑5 βˆ’ 𝑑6) Β¨ This is indeed the convexity adjustment (again makes sense, it is the second order) 126
  • 127. Luc_Faucheux_2020 Quick notes - VI Β¨ Just to make it explicit. Β¨ ∫+3+& +3+' 𝑑𝐹(𝑋 𝑑 ) = ∫+3+6 +3+5 )9 )* . ([). 𝑑𝑋(𝑑) + - . ∫+3+6 +3+5 )!9 )!! 𝑋 𝑑 . (𝑑𝑋). Β¨ 𝛿𝐹 = )9 )* . 𝛿𝑋 + - . . )!9 )!! . (𝛿𝑋)., which is the celebrated Ito lemma (chain rule) Β¨ 𝐹 𝑋 = 𝑋., )9 )* = 2𝑋, )!9 )*! = 2 Β¨ In the case of the simple Brownian motion: 𝑑𝑋 = π‘‘π‘Š Β¨ 𝛿(π‘Š.) = 2π‘Š. π›Ώπ‘Š + - . . 2. 𝛿𝑑 Β¨ In the case of the Geometric Brownian motion (Hull) 𝑑𝑋 = 𝑋. π‘‘π‘Š Β¨ 𝛿 𝑋. = 2𝑋. 𝑋. π›Ώπ‘Š + - . . 2. 𝑋.. 𝛿𝑑 = 2 𝑋. π›Ώπ‘Š + (𝑋.). 𝛿𝑑 127
  • 128. Luc_Faucheux_2020 Quick notes - VII Β¨ So for the square function and in the case of a simple Brownian motion, we have Β¨ 𝛿(π‘Š.) = 2π‘Š. π›Ώπ‘Š + - . . 2. 𝛿𝑑 Β¨ Or more rigorously: Β¨ ∫+3+& +3+' π‘‘π‘Š. = π‘Š.(𝑑5) βˆ’ π‘Š.(𝑑6) = Β¨ π‘Š. 𝑑5 βˆ’ π‘Š. 𝑑6 = ∫+3+6 +3+5 2π‘Š 𝑑 . [ . π‘‘π‘Š 𝑑 + ∫+3+6 +3+5 𝑑𝑑 Β¨ π‘Š. 𝑑5 βˆ’ π‘Š. 𝑑6 = ∫+3+6 +3+5 2π‘Š 𝑑 . ([). π‘‘π‘Š 𝑑 + (𝑑5 βˆ’ 𝑑6) 128
  • 129. Luc_Faucheux_2020 Quick notes - VIII Β¨ Still rather crudely (but rigorous, as we will show in section V on the GBM) Β¨ 𝑑𝑋 = π‘‘π‘Š Β¨ 𝔼 π‘Š 𝑑5 = π‘Š 𝑑6 Β¨ π‘Š. 𝑑5 βˆ’ π‘Š. 𝑑6 = ∫+3+6 +3+5 2π‘Š 𝑑 . ([). π‘‘π‘Š 𝑑 + 𝑑5 βˆ’ 𝑑6 Β¨ 𝔼 π‘Š. 𝑑5 βˆ’ π‘Š. 𝑑6 = 𝔼{∫+3+6 +3+5 2π‘Š 𝑑 . ([). π‘‘π‘Š 𝑑 + 𝑑5 βˆ’ 𝑑6 } Β¨ 𝔼 π‘Š. 𝑑5 βˆ’ π‘Š. 𝑑6 = 𝔼 𝑑5 βˆ’ 𝑑6 = (𝑑5 βˆ’ 𝑑6) Β¨ 𝔼 π‘Š. 𝑑5 = 𝔼 π‘Š. 𝑑6 + 𝑑5 βˆ’ 𝑑6 = π‘Š. 𝑑6 + (𝑑5 βˆ’ 𝑑6) Β¨ 𝔼 π‘Š. 𝑑5 = 𝔼 π‘Š 𝑑5 . + (𝑑5 βˆ’ 𝑑6) Β¨ This is indeed the convexity adjustment (again makes sense, it is the second order) Β¨ Since the square function is positively convex we would expect the convexity adjustment to be positive 129