- The document discusses modeling interest rates and term structure models. It provides a taxonomy of different rate models. - Key concepts covered include specifying the payoff function, fixing time, and payment time for interest rate products. The discount curve and forward rates are defined. - Common models include short-rate models where the short rate follows some stochastic process, and lattice models where the term structure is modeled on a tree.

1. Luc_Faucheux_2020
THE RATES WORLD – Part IV
Starting to look at modeling rates, a taxonomy
of models…
1

2. Luc_Faucheux_2020
Couple of notes on those slides
¨ In this deck we continue our exploration of the interest rate modeling world
¨ We go over the summary of Part I-III of the Rates
¨ We explain the general principles of Term Structure modeling
¨ We use what we saw on the deck on trees to explain local versus global arbitrage
¨ We use the section on Stochastic Calculus to go over some of the common models, and
attract attention to the fact that you should NEVER write an SDE (Stochastic Differential
Equation), always an SIE (Stochastic Integral Equation), especially if the volatility is itself a
function of rates (not only a constant or a time dependent only function)
¨ Again, by no means this is meant to be a textbook with linear acquisition of knowledge, but
somewhat of a bunch of circular meandering around Term Structure modeling, so that you
can read a textbook without hopefully being too confused, or work/interact with a Fixed-
Income desk and understand some of the issues at stake
¨ So again, I have tried to keep the formalism to a minimum to preserve the intuition but not
lose the rigor when needed
2

3. Luc_Faucheux_2020
SUMMARY OF PART III
3

4. Luc_Faucheux_2020
Summary - I
¨ When looking at payoffs, we should ALWAYS specify the following: What is the payoff
function, when is it fixed, when is it paid, at what time are we trying to compute its value
¨ 𝑉(𝑡) = 𝑉 𝑡, $𝐻(𝑡), 𝑡!, 𝑡"
4
𝑃𝑎𝑖𝑑 𝑎𝑡 𝑡𝑖𝑚𝑒 𝑡"
𝐹𝑖𝑥𝑒𝑑 𝑜𝑟 𝑠𝑒𝑡 𝑎𝑡 𝑡𝑖𝑚𝑒 𝑡!
𝐺𝑒𝑛𝑒𝑟𝑎𝑙 𝑃𝑎𝑦𝑜𝑓𝑓 𝐻 𝑡 𝑖𝑛 𝑐𝑢𝑟𝑟𝑒𝑛𝑐𝑦 $
𝑉𝑎𝑙𝑢𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑝𝑎𝑦𝑜𝑓𝑓 𝑜𝑏𝑠𝑒𝑟𝑣𝑒𝑑 𝑎𝑡 𝑡𝑖𝑚𝑒 𝑡

5. Luc_Faucheux_2020
Summary – I -a
¨ 𝑉(𝑡) = 𝑉 𝑡, $𝐻(𝑡), 𝑡!, 𝑡"
¨ Most simple payoffs $𝐻(𝑡) are a function of random variables that gets fixed at the same
time 𝑡!, hence why I isolated 𝑡!
¨ However (say SOFR or OIS), the function $𝐻(𝑡) could be as complicated as it can be, and in
the case of averaging indices, could be an integral or a discrete sum over a number of
observations point.
¨ It could also be the MAX or MIN over a given period, or a range accrual
¨ So the possibilities are endless in order to customize this function, making the observation
time 𝑡! meaningless in the very general case
¨ Again, a lot of the simple payoffs have a single discrete time 𝑡! for “fixing”, which is generally
different from the payment time 𝑡", hence the reason why I explicitly kept it as a variable on
its own
5

6. Luc_Faucheux_2020
Summary – I -b
¨ In some ways, this is why quantitative finance can be so tricky for people used to simple
stochastic processes.
¨ Usually we deal with random variables 𝑋(𝑡), which are observed at time 𝑡
¨ HOWEVER in finance, we are looking at random payoff that are observed at time 𝑡! and PAID
at time 𝑡!, where those two points in time usually do not align
¨ This is what usually creates most of the confusion because the deferred payment is actually
a big deal as soon as we introduce volatility (non-deterministic) and correlation between the
payoffs and the Zero discount factors
¨ So ALWAYS explicitly describe the actual payoff and especially WHEN it is paid out
¨ A perfect example of the consequence of this timing difference is the Libor in arrears / in
advance trade or the CMS versus swap rate
¨ BTW, those trades are not that common, but you see in most textbooks, because they were
famous at the time, but also they are a great way to check our understanding and
knowledge, to make sure that we do not get tricked.
6

7. Luc_Faucheux_2020
Summary - II
¨ At each point in time 𝑡, we observe the discount curve 𝑧𝑐 𝑡, 𝑡, 𝑡"
7
𝑂𝑏𝑠𝑒𝑟𝑣𝑒𝑑 𝑎𝑡 𝑡𝑖𝑚𝑒 𝑡
𝑆𝑡𝑎𝑟𝑡 𝑜𝑓 𝑡ℎ𝑒 𝑝𝑒𝑟𝑖𝑜𝑑
𝐸𝑛𝑑 𝑜𝑓 𝑡ℎ𝑒 𝑝𝑒𝑟𝑖𝑜𝑑

8. Luc_Faucheux_2020
Summary - III
¨ At each point in time 𝑡, we observe the discount curve 𝑧𝑐 𝑡, 𝑡, 𝑡"
¨ 𝑧𝑐 𝑡, 𝑡, 𝑡" is the price at time 𝑡 of a contract that will pay $1 at time 𝑡"
¨ At that point in time 𝑡 one can define the “then-spot simply compounded rate” as:
¨ 𝑧𝑐 𝑡, 𝑡, 𝑡" =
#
#$% &,&,&! .) &,&,&!
¨ For any point 𝑡! such that 𝑡 < 𝑡! < 𝑡" we can bootstrap the following discount factors:
¨ 𝑧𝑐 𝑡, 𝑡, 𝑡" = 𝑧𝑐 𝑡, 𝑡, 𝑡! ∗ 𝑧𝑐 𝑡, 𝑡!, 𝑡"
¨ We can then also define the “then-forward simply compounded rate” as:
¨ 𝑧𝑐 𝑡, 𝑡!, 𝑡" =
#
#$% &,&",&! .) &,&",&!
8

9. Luc_Faucheux_2020
Summary - IV
¨ Lower case means that the value is known, or fixed or observed
¨ Upper case means the random variable
¨ At each point in time 𝑡, we observe the discount curve 𝑧𝑐 𝑡, 𝑡, 𝑡"
¨ At each point in time 𝑡, we observe the bootstrapped discount curve 𝑧𝑐 𝑡, 𝑡!, 𝑡"
¨ The discount factors 𝑍𝐶 𝑡, 𝑡!, 𝑡" evolve randomly in time 𝑡 for a given period [𝑡!, 𝑡"]
¨ The corresponding rates we defined as:
¨ 𝐿 𝑡, 𝑡!, 𝑡" =
#
) &,&",&!
. [
#
*+ &,&",&!
− 1]
¨ Also evolves randomly in time 𝑡 for a given period [𝑡!, 𝑡"]
¨ Note that we have not yet defined any dynamics (normal, lognormal,..) of those variables
yet
9

10. Luc_Faucheux_2020
Summary - V
¨ 𝐿 𝑡, 𝑡!, 𝑡" =
#
) &,&",&!
. [
#
*+ &,&",&!
− 1]
¨ 𝐿 𝑡, 𝑡!, 𝑡" . 𝜏 𝑡, 𝑡!, 𝑡" =
#
*+ &,&",&!
. [1 − 𝑍𝐶 𝑡, 𝑡!, 𝑡" ]
¨ When 𝑡 reaches 𝑡!, the random rate 𝐿 𝑡, 𝑡!, 𝑡" gets fixed to 𝑙 𝑡 = 𝑡!, 𝑡!, 𝑡"
¨ (The forward rate becomes fixed to the spot rate)
¨ When 𝑡 reaches 𝑡!, the random discount 𝑍𝐶 𝑡, 𝑡!, 𝑡" gets fixed to 𝑧𝑐 𝑡 = 𝑡!, 𝑡!, 𝑡"
¨ Random variables are observed at a given point in time
¨ HOWEVER what matters in Finance is not only the observation (“fixing”) time, but WHEN a
particular payoff function of those random variables is paid.
¨ The fixing time and the payment time do not have to be the same
¨ In fact most of the time they are not
10

11. Luc_Faucheux_2020
Summary - VI
¨ A very common and useful numeraire is the Zero Discount factor whose period end is the
payment date for the payoff.
¨ The value of a claim that pays on the payment date, normalized by the Zeros, is a
martingale.
¨ The measure under which we compute expectations, that is associated to the Zeros whose
period end is the payment date is often referred to as the Terminal measure of Forward
measure
¨ You are free to choose another numeraire or another measure of course (see the deck on
Numeraire), it is a matter of what makes the computation convenient without obscuring the
intuition.
¨ In particular if the claim always pays $1 at time 𝑡"
¨
, &,$#,&",&!
./ &,&,&!
= 𝔼&!
*+ , &!,$#,&",&!
*+ &!,&!,&!
|𝔉(𝑡) = 𝔼&!
*+ , &!,$#,&",&!
#
|𝔉(𝑡) = 𝔼&!
*+ #
#
|𝔉(𝑡) = 1
¨ 𝑉 𝑡, $1, 𝑡!, 𝑡" = 𝑧𝑐 𝑡, 𝑡, 𝑡"
11

12. Luc_Faucheux_2020
Summary - VII
¨ We have derived a couple of useful formulas in part III
¨ Zero coupons:
¨ 𝔼&"
*+ 𝑉 𝑡!, $1 𝑡 , 𝑡!, 𝑡! |𝔉(𝑡) = 1
¨
, &,$#,&",&"
./ &,&,&"
= 𝔼&"
*+ , &",$# & ,&",&"
*+ &",&",&"
|𝔉(𝑡) = 1
¨ 𝑉 𝑡, $1, 𝑡!, 𝑡! = 𝑧𝑐 𝑡, 𝑡, 𝑡!
¨
, &,$#,&",&!
./ &,&,&!
= 𝔼&!
*+ , &",$# & ,&",&!
*+ &!,&!,&!
|𝔉(𝑡) = 1
¨ 𝑉 𝑡, $1, 𝑡!, 𝑡" = 𝑧𝑐 𝑡, 𝑡, 𝑡"
12

13. Luc_Faucheux_2020
Summary - VIII
¨ Deferred premium
¨ 𝔼&"
*+ 𝑉 𝑡!, $1 𝑡 , 𝑡!, 𝑡" |𝔉(𝑡) = 𝔼&"
*+ 𝑉 𝑡!, $𝑍𝐶(𝑡, 𝑡!, 𝑡"), 𝑡!, 𝑡! |𝔉(𝑡) = 𝑧𝑐(𝑡, 𝑡!, 𝑡")
¨
, &,$#,&",&"
./ &,&,&"
= 𝔼&"
*+ , &",$# & ,&",&"
*+ &",&",&"
|𝔉(𝑡) = 1
¨
, &,$#,&",&!
./ &,&,&!
= 𝔼&!
*+ , &",$# & ,&",&!
*+ &!,&!,&!
|𝔉(𝑡) = 1
¨
, &,$#,&",&!
./ &,&,&"
= 𝔼&"
*+ , &",$# & ,&",&!
*+ &",&",&"
|𝔉(𝑡) = 𝔼&"
*+ 𝑉 𝑡!, $𝑍𝐶(𝑡, 𝑡!, 𝑡"), 𝑡!, 𝑡! |𝔉(𝑡) = 𝑧𝑐(𝑡, 𝑡!, 𝑡")
¨ 𝑉 𝑡, $1, 𝑡!, 𝑡" = 𝑧𝑐 𝑡, 𝑡, 𝑡! . 𝑧𝑐 𝑡, 𝑡!, 𝑡" = 𝑧𝑐 𝑡, 𝑡, 𝑡"
¨ If the general claim $𝐻 𝑡 is fixed at time 𝑡!
¨ 𝔼&"
*+
𝑉 𝑡!, $𝐻 𝑡 , 𝑡!, 𝑡" |𝔉(𝑡) = 𝔼&"
*+
𝑉 𝑡!, $𝐻 𝑡 . 𝑍𝐶(𝑡, 𝑡!, 𝑡"), 𝑡!, 𝑡! |𝔉(𝑡)
13

14. Luc_Faucheux_2020
Summary - IX
¨ 𝔼&"
*+
𝑉 𝑡!, $1 𝑡 , 𝑡!, 𝑡" |𝔉(𝑡) = 𝔼&"
*+
𝑉 𝑡!, $𝑍𝐶(𝑡, 𝑡!, 𝑡"), 𝑡!, 𝑡! |𝔉(𝑡) = 𝑧𝑐(𝑡, 𝑡!, 𝑡")
¨ 𝔼&"
*+ 𝑉 𝑡!, $𝐻 𝑡 , 𝑡!, 𝑡" |𝔉(𝑡) = 𝔼&"
*+ 𝑉 𝑡!, $𝐻 𝑡 . 𝑍𝐶(𝑡, 𝑡!, 𝑡"), 𝑡!, 𝑡! |𝔉(𝑡)
¨ Note that in the case of a general claim that could be a function of the 𝑍𝐶(𝑡, 𝑡!, 𝑡"), we cannot
split the expectation of the products into a product of expectation
¨ But we can use the covariance formula, which is a useful trick used in Tuckmann book, especially
when computing the forward-future convexity adjustment
¨ 𝐶𝑜𝑣𝑎𝑟 𝑋, 𝑌 = 𝔼{𝑋 − 𝔼 𝑋 }. 𝔼{𝑌 − 𝔼[𝑌]}
¨ 𝐶𝑜𝑣𝑎𝑟 𝑋, 𝑌 = 𝔼[𝑋. 𝑌] − 𝔼 𝑋 . 𝔼 𝑌
¨ So in the above, something we should start getting used to:
¨ 𝔼&"
*+ 𝑉 𝑡!, $𝐻 𝑡 . 𝑍𝐶(𝑡, 𝑡!, 𝑡"), 𝑡!, 𝑡! |𝔉(𝑡) =
𝔼&"
*+ 𝑉 𝑡!, $𝐻 𝑡 , 𝑡!, 𝑡! |𝔉(𝑡) . 𝔼&"
*+ 𝑉 𝑡!, $𝑍𝐶(𝑡, 𝑡!, 𝑡"), 𝑡!, 𝑡! |𝔉(𝑡) +
𝐶𝑂𝑉𝐴𝑅{𝑉 𝑡!, $𝐻 𝑡 , 𝑍𝐶(𝑡, 𝑡!, 𝑡"), 𝑡!, 𝑡! |𝔉(𝑡)}
14

15. Luc_Faucheux_2020
Summary - X
¨ 𝔼&"
*+
𝑉 𝑡!, $𝐻 𝑡 . 𝑍𝐶(𝑡, 𝑡!, 𝑡"), 𝑡!, 𝑡! |𝔉(𝑡) =
𝔼&"
*+ 𝑉 𝑡!, $𝐻 𝑡 , 𝑡!, 𝑡! |𝔉(𝑡) . 𝔼&"
*+ 𝑉 𝑡!, $𝑍𝐶(𝑡, 𝑡!, 𝑡"), 𝑡!, 𝑡! |𝔉(𝑡) +
𝐶𝑂𝑉𝐴𝑅{𝑉 𝑡!, $𝐻 𝑡 , 𝑍𝐶(𝑡, 𝑡!, 𝑡"), 𝑡!, 𝑡! |𝔉(𝑡)}
¨ 𝔼&"
*+
𝑉 𝑡!, $𝐻 𝑡 . 𝑍𝐶(𝑡, 𝑡!, 𝑡"), 𝑡!, 𝑡! |𝔉(𝑡) = 𝔼&"
*+
𝑉 𝑡!, $𝐻 𝑡 , 𝑡!, 𝑡! |𝔉(𝑡) . 𝑧𝑐 𝑡, 𝑡!, 𝑡" +
𝐶𝑂𝑉𝐴𝑅{𝑉 𝑡!, $𝐻 𝑡 , 𝑍𝐶(𝑡, 𝑡!, 𝑡"), 𝑡!, 𝑡! |𝔉(𝑡)}
¨ This looks like we just replaced something by something more complicated, but it highlights
the fact that if the claim is NOT correlated with the discount 𝑍𝐶(𝑡, 𝑡!, 𝑡")
¨ Then:
¨ 𝐶𝑂𝑉𝐴𝑅 𝑉 𝑡!, $𝐻 𝑡 , 𝑍𝐶 𝑡, 𝑡!, 𝑡" , 𝑡!, 𝑡! 𝔉 𝑡 = 0
¨ And:
¨ 𝔼&"
*+ 𝑉 𝑡!, $𝐻 𝑡 . 𝑍𝐶(𝑡, 𝑡!, 𝑡"), 𝑡!, 𝑡! |𝔉(𝑡) = 𝑧𝑐 𝑡, 𝑡!, 𝑡" . 𝔼&"
*+ 𝑉 𝑡!, $𝐻 𝑡 , 𝑡!, 𝑡! |𝔉(𝑡)
15

16. Luc_Faucheux_2020
Summary - XI
¨ When there is NO correlation between the claim and the Zeros
¨ 𝔼&"
*+ 𝑉 𝑡!, $𝐻 𝑡 . 𝑍𝐶(𝑡, 𝑡!, 𝑡"), 𝑡!, 𝑡! |𝔉(𝑡) = 𝑧𝑐 𝑡, 𝑡!, 𝑡" . 𝔼&"
*+ 𝑉 𝑡!, $𝐻 𝑡 , 𝑡!, 𝑡! |𝔉(𝑡)
¨ If the general claim $𝐻 𝑡 is fixed at time 𝑡!
¨ 𝔼&"
*+
𝑉 𝑡!, $𝐻 𝑡 , 𝑡!, 𝑡" |𝔉(𝑡) = 𝔼&"
*+
𝑉 𝑡!, $𝐻 𝑡 . 𝑍𝐶(𝑡, 𝑡!, 𝑡"), 𝑡!, 𝑡! |𝔉(𝑡)
¨ 𝔼&"
*+
𝑉 𝑡!, $𝐻 𝑡 , 𝑡!, 𝑡" |𝔉(𝑡) = 𝑧𝑐 𝑡, 𝑡!, 𝑡" . 𝔼&"
*+
𝑉 𝑡!, $𝐻 𝑡 , 𝑡!, 𝑡! |𝔉(𝑡)
¨
, &,$0(&),&",&!
./ &,&,&"
= 𝔼&"
*+ , &",$0 & ,&",&!
*+ &",&",&"
|𝔉(𝑡) = 𝔼&"
*+ 𝑉 𝑡!, $𝐻 𝑡 , 𝑡!, 𝑡" |𝔉(𝑡)
¨
, &,$0(&),&",&!
./ &,&,&"
= 𝑧𝑐 𝑡, 𝑡!, 𝑡" . 𝔼&"
*+
𝑉 𝑡!, $𝐻 𝑡 , 𝑡!, 𝑡! |𝔉(𝑡)
¨ 𝑉 𝑡, $𝐻(𝑡), 𝑡!, 𝑡" = 𝑧𝑐 𝑡, 𝑡, 𝑡! . 𝑧𝑐 𝑡, 𝑡!, 𝑡" . 𝔼&"
*+
𝑉 𝑡!, $𝐻 𝑡 , 𝑡!, 𝑡! |𝔉(𝑡)
¨ 𝑉 𝑡, $𝐻(𝑡), 𝑡!, 𝑡" = 𝑧𝑐 𝑡, 𝑡, 𝑡" . 𝔼&"
*+ 𝑉 𝑡!, $𝐻 𝑡 , 𝑡!, 𝑡! |𝔉(𝑡)
16

17. Luc_Faucheux_2020
Summary - XI
¨ 𝑉 𝑡, $𝐻(𝑡), 𝑡!, 𝑡" = 𝑧𝑐 𝑡, 𝑡, 𝑡" . 𝔼&"
*+ 𝑉 𝑡!, $𝐻 𝑡 , 𝑡!, 𝑡! |𝔉(𝑡)
¨ Note again that the above is ONLY true if there is no correlation between the claim and the
discount
¨ If there is, the Covariance term will appear, (this will be the famed convexity adjustment)
¨ Expressing the convexity adjustment as a covariance term sometimes makes it easier to
compute (Tuckmann book) but also put front and center the fact that if you value a claim
that is a function of the Zeros, and the timing is not the regular timing for the payment
(value a LIBOR in ARREARS trade for example), or that function is not a linear combination of
the Zeros (value a LIBOR square trade for example) YOU WILL HAVE a convexity adjustment
to take into account
¨ IF CORRELATION
¨ 𝑉 𝑡, $𝐻(𝑡), 𝑡!, 𝑡" = 𝑧𝑐 𝑡, 𝑡, 𝑡" . 𝔼&"
*+
𝑉 𝑡!, $𝐻 𝑡 , 𝑡!, 𝑡! |𝔉(𝑡) +
𝑧𝑐 𝑡, 𝑡, 𝑡! . 𝐶𝑂𝑉𝐴𝑅&"
*+
𝑉 𝑡!, $𝐻 𝑡 , 𝑍𝐶 𝑡, 𝑡!, 𝑡" , 𝑡!, 𝑡! 𝔉 𝑡
17

18. Luc_Faucheux_2020
Summary - XII
¨ If the payoff has no correlation, you can “move” the payment up and down the curve as per
the deterministic zeros (lower case), like you would on a swap desk
¨ If the payoff has ANY correlation with the zeros, go talk to the option desk because there is
some convexity
¨ There are however some special payoffs that ARE function of the zeros but for which the
convexity magically disappear, and you can price them in the deterministic world of lower
case, and go talk to the swap trader (hint: those payoffs are the regular swaps).
¨ Those are in the next slide
¨ The magic trick is usually (1 = 1), or (𝑋 = 𝑋), or (𝑋 − 𝑋 = 0) or (
3
3
= 1) or (1 − 1 = 0)
18

19. Luc_Faucheux_2020
¨ 𝑍𝐶 𝑡, 𝑡!, 𝑡" =
#
#$4 &,&",&! .)
and 𝑧𝑐 𝑡, 𝑡!, 𝑡" =
#
#$% &,&",&! .)
¨ $𝐻 𝑡 = $𝐿 𝑡, 𝑡!, 𝑡" = $
#
)
(
#
*+ &,&",&!
− 1)
¨ 𝔼&!
*+ 𝑉 𝑡", $𝐿 𝑡, 𝑡!, 𝑡" , 𝑡!, 𝑡" |𝔉(𝑡) = 𝑙 𝑡, 𝑡!, 𝑡" =
#
)
(
#
./ &,&",&!
− 1)
¨ 𝔼&"
*+
𝑉 𝑡!, $
4 &,&",&! .)
#$4 &,&",&! .)
, 𝑡!, 𝑡! |𝔉(𝑡) =
% &,&",&! .)
#$% &,&",&! .)
¨ 𝔼&"
*+
𝑉 𝑡!, $
#
#$4 &,&",&! .)
, 𝑡!, 𝑡! |𝔉(𝑡) =
#
#$% &,&",&! .)
= 𝑧𝑐 𝑡, 𝑡!, 𝑡" =
./ &,&,&!
./ &,&,&"
¨ 𝔼&"
*+ 𝑉 𝑡!, $𝑍𝐶 𝑡, 𝑡!, 𝑡" , 𝑡!, 𝑡! |𝔉(𝑡) = 𝑧𝑐 𝑡, 𝑡!, 𝑡" = 𝔼&"
*+ 𝑉 𝑡!, $1 𝑡 , 𝑡!, 𝑡" |𝔉(𝑡)
¨ THIS is why you can value a swap in the deterministic world (lower case, no volatility, no
convexity, no dynamics, no option trader involved, just a swap trader and one discount
curve)
¨ All right that was a good summary
Summary - XIII
19

20. Luc_Faucheux_2020
Modeling Stochastic Rates
20

21. Luc_Faucheux_2020
Modeling stochastic rates
¨ So essentially that is it:
¨ Model a random process for 𝐿 𝑡, 𝑡!, 𝑡" over time 𝑡
¨ (So assume some dynamics for the process).
¨ ”Calibrate” your model:
¨ “Calibrate the drift” (Chapter 9 Tuckmann): recover at least some of the arbitrage-free
constraints (I say some because based on the actual model you might not be to fullfill all of
them, example BDT we saw in the Trees deck only fullfill them for 𝑡 = 0, we will go over
that)
¨ Essentially those arbitrage-free constraints are always given by something like:
¨ 𝔼&"
*+
𝑉 𝑡!, $
#
#$4 &,&",&! .)
, 𝑡!, 𝑡! |𝔉(𝑡) =
#
#$% &,&",&! .)
= 𝑧𝑐 𝑡, 𝑡!, 𝑡"
¨ “Calibrate the volatility” (Chapter 10 Tuckmann): again like we saw in the Trees deck if you
have some market information about the distribution of some rates based securities, find a
way to calibrate your model to recover the market price
21

22. Luc_Faucheux_2020
Modeling stochastic rates-II
¨ You do not have to model 𝐿 𝑡, 𝑡!, 𝑡" , you could choose the model 𝑍𝐶 𝑡, 𝑡!, 𝑡"
¨ Because generating random processes for all the 𝐿 𝑡, 𝑡!, 𝑡" and enforcing the arbitrage
conditions can be quite cumbersome, in practice the problem is reduced to a simpler one (a
little like when we were looking at a flat curve in part II).
¨ You reduce the dimensionality/complexity of the problem, it becomes more tractable and
easy to use, but you might lose some essential features you need to manage your book.
¨ For example you have a book of Curve options (options on the spread between two swaps
of different tenors), you will need a term structure model with a least 2 factors to properly
price and risk manage
¨ You only have a book of regular swaps, you actually do not even care about volatility and
diffusion, you can price and risk manage in the “deterministic” zero-vol world of using only a
discount curve
22

23. Luc_Faucheux_2020
Modeling stochastic rates-III
¨ However if your “regular” swap becomes more complicated, for example
¨ Libor is in arrears
¨ CSA has an embedded zero floor option in it (in the case of negative yield on the securities
put up as collateral against the mark-to-market of the swap, no accrued interest payment is
made)
¨ CSA is in a different currency than your swap
¨ In all of those case, the swap will exhibit some added convexity, and becomes an option,
which price will be sensitive to the specific model you use and how it was calibrated to
market, and to which instrument
¨ Example: a callable model calibrated to European Swaptions will NOT recover the market
price of callable swaps, see the deck on Structured products
23

24. Luc_Faucheux_2020
Modeling stochastic rates-IV
¨ By the way, a quick note on convexity
¨ What do you mean “regular swaps do not have convexity”?
¨ I learnt that bonds and swaps have convexity
¨ Answer: yes they do, but to the WRONG variable (x-axis)
¨ Bonds and swaps in practice are priced and risk managed on the yield curve, and when the
yield move, the price of swaps does move in a non-linear manner (not a straight line) and
indeed will have a non-zero second derivative (Gamma) to the yield
¨ HOWEVER, the yields are the wrong “measure”
¨ What matters are the Zeros
¨ A regular swap and a bond are a linear combination of fixed cashflows, hence a linear
combination of zeros. Linear -> no convexity
24

25. Luc_Faucheux_2020
Modeling stochastic rates-V
¨ So that added even more to the confusion, because we all learn about bond convexity and
swap convexity, but that is looking at the price of a swap or a bond against the WRONG x-
axis (the WRONG variable).
¨ Because from the option deck, you should be saying, wait a second, a swap has convexity, so
the average of the function is not the function of the average (Jensen inequality), so
..boom..to value a swap I need to know something about the dynamics of something.
¨ You would be right, however, you need to define “function of what”.
¨ Again, because a swap or a bond is a linear function of Zeros, they are not convex as a
function of the Zeros, and so the expected value is today’s value, and you are ok
¨ So for a regular swap you are ok to only price it on a yield curve (really a discount curve)
¨ What you should really look at is bumping the Zeros, not the yield (so have the Zeros on the
x-axis). In that case the present value is a linear function (straight line), there is no convexity,
you do not care about the dynamics, you pretty happy
25

26. Luc_Faucheux_2020
Modeling stochastic rates-VI
¨ This is why a swap desk will usually do not have a Vega limit (because it is not an option
desk).
¨ Note however that in some places (Salomon in the good old days, GS also I think), the
discount curve was produced from a term structure model that had some volatility or
correlation.
¨ So bumping the yield curve was done through bumping volatility, so the yield curve had
some Vega in it, so a swap desk had Vega. Usually in that construct the option and swap
desk were combined into one unit
¨ Note also that even if you just had a swap desk, with a regular bootstrapped yield curve, and
you were just using that curve as is (so there is no Vega), if the swap desk was using
Eurodollar Future as a hedge, those contracts are future contracts and do exhibit some Vega
(we will explicitly compute it at the end of this deck for a given model), and so that desk
would need to have Vega limit, which is essentially a Vega limit on the convexity adjustment.
¨ Note that because people did not really understand convexity, I know of many places in the
1990s where swap desks had futures but no Vega limits, and when computed their Vega
position actually did dwarf the position of the option desk
26

27. Luc_Faucheux_2020
Modeling stochastic rates-VII
¨ There is also a famous story of a desk who sold some 5x7 caps in order to sell volatility, but
hedged the delta exposure by selling ED futures (purples and oranges), thus getting long
volatility (being short a ED future is being long vol, trust me on that one, we will derive that
again), and on an overall net basis being long volatility.
¨ They had the right idea at the time (selling vol) but because they did not realize the Vega
coming from the ED futures, they ended up losing quite a lot of money, as they had put on
the trade on rather large size.
¨ Remind me to go over that trade in details again at the end.
¨ That was quite a famous trade because also the trader at the time was wearing some
distinct ear jewelry, and was quite arrogant…so yeah..karma is a ….
¨ Also because the market moved away from the strike on their caps, they lost their short
Vega, but the Vega coming from the ED future convexity adjustment is strikeless (profile as a
function of rate is not the Gaussian Bell curve centered around the strike), so they ended up
being long Vega….good thinking….
27

28. Luc_Faucheux_2020
Modeling stochastic rates-VIII
¨ Model a random process for 𝐿 𝑡, 𝑡!, 𝑡" over time 𝑡
¨ That is essentially saying that we are writing something like:
¨ 𝑑𝐿 𝑡, 𝑡!, 𝑡" = 𝐴 𝑡, 𝑡!, 𝑡" . 𝑑𝑡 + 𝐵 𝑡, 𝑡!, 𝑡" . 𝑑𝑊 𝑡, 𝑡!, 𝑡"
¨ Where the time 𝑡 is the time variable that we are used to from the Stochastic calculus deck,
and the other two time are constant and fixed indices
¨ Remember, IF 𝐵 𝑡, 𝑡!, 𝑡" is a function of 𝐿 𝑡, 𝑡!, 𝑡" or 𝑊 𝑡, 𝑡!, 𝑡" , then we get into the
whole ITO versus Stratanovitch issue, and we should not write an SDE, but an SIE anyways,
as the stochastic integral is the only thing that we know how to use, NOT a stochastic
differentiation (since most random processes are NOT differentiable)
¨ If we put ourselves in the ITO framework
¨ 𝑑𝐿 𝑡, 𝑡!, 𝑡" = 𝐴 𝑡, 𝑡!, 𝑡" . 𝑑𝑡 + 𝐵 𝑡, 𝑡!, 𝑡" . ([). 𝑑𝑊 𝑡, 𝑡!, 𝑡" or more exactly in SIE form:
¨ 𝐿 𝑡 + 𝛿𝑡, 𝑡!, 𝑡" − 𝐿 𝑡, 𝑡!, 𝑡" = ∫&
&$5&
𝐴 𝑠, 𝑡!, 𝑡" . 𝑑𝑠 + ∫&
&$5&
𝐵 𝑠, 𝑡!, 𝑡" . ([). 𝑑𝑊 𝑠, 𝑡!, 𝑡"
28

29. Luc_Faucheux_2020
Modeling stochastic rates-IX
¨ 𝑑𝐿 𝑡, 𝑡!, 𝑡" = 𝐴 𝑡, 𝑡!, 𝑡" . 𝑑𝑡 + 𝐵 𝑡, 𝑡!, 𝑡" . ([). 𝑑𝑊 𝑡, 𝑡!, 𝑡" or more exactly in SIE form:
¨ 𝐿 𝑡 + 𝛿𝑡, 𝑡!, 𝑡" − 𝐿 𝑡, 𝑡!, 𝑡" = ∫&
&$5&
𝐴 𝑠, 𝑡!, 𝑡" . 𝑑𝑠 + ∫&
&$5&
𝐵 𝑠, 𝑡!, 𝑡" . ([). 𝑑𝑊 𝑠, 𝑡!, 𝑡"
¨ Where ([) is there to remind us that when we define the ITO integral as a limit of a sum, in
the sum we take the value for 𝐵 𝑠, 𝑡!, 𝑡" on the LHS (Left hand side) of the little partition
¨ Stratonovitch would take the middle and we would note it (∘)
¨ ∫&6&7
&6&8
𝑓 𝑋 𝑡 . ([). 𝑑𝑋(𝑡) = lim
9→;
{∑<6#
<69
𝑓(𝑋(𝑡<)). [𝑋(𝑡<$#) − 𝑋(𝑡<)]}
¨ ∫&6&7
&6&8
𝑓 𝑋 𝑡 . (∘). 𝑑𝑋(𝑡) = lim
9→;
{∑<6#
<69
𝑓 [𝑋(𝑡< + 𝑋(𝑡<$#)]/2). [𝑋(𝑡<$#) − 𝑋(𝑡<)]}
¨ In regular calculus, those two integrals are the usual Riemann integral because the two sums
converge to the same value
¨ In stochastic calculus they do NOT, as that is one of the crux of the difficulties dealing with
random variables, usually Finance textbooks are quite liberal with notations on that subject,
only Mercurio I think actually brings up the issue in one of the appendix
29

30. Luc_Faucheux_2020
Modeling stochastic rates-X
¨ But seriously folks, that is kind of it.
¨ Any model out there in any textbook is a simplification of :
¨ 𝐿 𝑡 + 𝛿𝑡, 𝑡!, 𝑡" − 𝐿 𝑡, 𝑡!, 𝑡" = ∫&
&$5&
𝐴 𝑠, 𝑡!, 𝑡" . 𝑑𝑠 + ∫&
&$5&
𝐵 𝑠, 𝑡!, 𝑡" . ([). 𝑑𝑊 𝑠, 𝑡!, 𝑡"
¨ Subject to the arbitrage free conditions (if possible) which will constrain the drift or
advection term 𝐴 𝑠, 𝑡!, 𝑡" and you can choose some measure like the early one:
¨ 𝔼&"
*+ 𝑉 𝑡!, $
#
#$4 &,&",&! .)
, 𝑡!, 𝑡! |𝔉(𝑡) =
#
#$% &,&",&! .)
= 𝑧𝑐 𝑡, 𝑡!, 𝑡"
¨ And when possible calibrated to some market information on the volatility that will
constrain the diffusion term 𝐵 𝑠, 𝑡!, 𝑡"
¨ So if you want we can call it a day and go outside enjoy the nice Chicago summer weather or
stay inside and start exploring some reduction and simplification of the problem above in
some more tractable and common models.
¨ Note that those are the “old” models without stochastic volatility
30

31. Luc_Faucheux_2020
Modeling stochastic rates-XI
¨ We will obviously run again into the issue of continuous description versus discrete, knowing
that in Finance the daily period is a natural discrete block of time.
¨ Overnight Rates are defined over one day, we do not have such a thing as “borrowing for 2
hours”
¨ Note, do not be confused, there is such a thing as “buying a bond and selling it a few
millisecond later” that did not mean that you borrowed a rate over a few milliseconds.
¨ The Bond is still a security that is defined using rates. Actually at the core, it is not even
rates, bond is a security that is defined using discount factors. And in practice the smallest
time increment is daily.
¨ All curves usually gets constructed with a minimum time step of one day (again that does
not mean that curves are fixed for the day, they move all the time, the anchor points to build
the curve are usually never smaller than one day)
¨ Note that volatility is not the same as it is a parameter used to price options, at Citi we had
the “TimeWarp” which put different weights on different parts of the day, say more weights
around 8:30am during Employment Friday, less so during a full Saturday
31

32. Luc_Faucheux_2020
Modeling stochastic rates-XII
¨ The “TimeWarp” is for pricing and hedging an option book
¨ The discount curve and yield curve still had minimum daily time increments
¨ The TimeWarp is also a super catchy song, my most unconventional conventionists….
32

33. Luc_Faucheux_2020
Modeling stochastic rates-XIII
¨ Again seriously I am not kidding, all there is about modeling rates is to model the dynamics
of the 𝐿 𝑡, 𝑡!, 𝑡" subject to the arbitrage free conditions (if possible) which will constrain the
drift or advection term
¨ 𝔼&"
*+
𝑉 𝑡!, $
#
#$4 &,&",&! .)
, 𝑡!, 𝑡! |𝔉(𝑡) =
#
#$% &,&",&! .)
= 𝑧𝑐 𝑡, 𝑡!, 𝑡"
¨ I could have started with that instead of trying to slowly build up the formalism to this, but
that would not have helped build intuition, would be quite of douchy, and would not follow
the history of how the field of quantitative finance got built
¨ All the rest is just trying to find some simplification or easy way to incorporate the arbitrage
free a-priori in the dynamics of rates as opposed to an a-posteriori calibration.
¨ Essentially the whole field of rates modeling is confusing because a lot of people are trying
to make it more complicated than it really is, in order to do two things: show everyone else
how smart they are, and also that they deserve a job. Yours truly has essentially done this
for the past 25 years, and keep falling into that pattern of behavior.
¨ I will try in the next couple of slides to summarize the field of models
¨ We will then spend some time digging into some of them in more details
33

34. Luc_Faucheux_2020
A taxonomy of term structure models
¨ To the best of my ability, am sure I will offend a lot of people.
34

35. Luc_Faucheux_2020
Taxonomy of models
¨ Write the dynamics on 𝐿 𝑡, 𝑡!, 𝑡" as Lognormal, with one stochastic factor per 𝐿 𝑡, 𝑡!, 𝑡" ,and no
stochastic volatility, et voila ! and you have what is sometimes called:
¨ BGM (Brace – Gatarek – Musiela) : 1997, also called (with minimal tweaks)
¨ LMM : Libor Market Model, LFM : Lognormal Forward – Libor Model, or FMM : Forward Market
Model
¨ Boom, that’s it, the complication is dealing with a lot of equations and enforcing the arbitrage
relationship. My career advice to you would be to stick to a generalized BGM, make it multi-factor
both in rates and in volatility, and then use the deep learning / AI / Machine Learning / Neural
Network / big data of the cloud to find the right calibration that respect the arbitrage and
calibrates to the market
¨ No one will really understand what you are doing, CPU is cheap, and you will be guaranteed a job
¨ 𝐿 𝑡 + 𝛿𝑡, 𝑡#, 𝑡$ − 𝐿 𝑡, 𝑡#, 𝑡$ = ∫%
%&'%
𝜇 𝑠, 𝑡#, 𝑡$ . 𝐿 𝑠, 𝑡#, 𝑡$ . 𝑑𝑠 + ∫%
%&'%
𝜎 𝑠, 𝑡#, 𝑡$ . 𝐿 𝑠, 𝑡#, 𝑡$ . ([). 𝑑𝑊 𝑠, 𝑡#, 𝑡$
¨ 𝑑𝐿 𝑡, 𝑡!, 𝑡" = 𝜇 𝑡, 𝑡!, 𝑡" . 𝐿 𝑡, 𝑡!, 𝑡" . 𝑑𝑡 + 𝜎 𝑡, 𝑡!, 𝑡" . 𝐿 𝑡, 𝑡!, 𝑡" . ([). 𝑑𝑊 𝑡, 𝑡!, 𝑡"
35

36. Luc_Faucheux_2020
Taxonomy of models - II
¨ If you say, 𝐿 𝑡, 𝑡!, 𝑡" is too complicated, I will reduce the dimensionality and write the
dynamics on 𝐿 𝑡, 𝑡!, 𝑡! as normal diffusion with only one factor, no stochastic volatility, and
only one stochastic driver for all different maturities, then BOOM you have what is called:
¨ HJM (Heath-Jarrow-Morton): 1992
¨ Essentially, write something like this:
¨ 𝐿 𝑡 + 𝛿𝑡, 𝑡!, 𝑡! − 𝐿 𝑡, 𝑡!, 𝑡! = ∫&
&$5&
𝐴 𝑠, 𝑡!, 𝑡! . 𝑑𝑠 + ∫&
&$5&
𝐵 𝑠, 𝑡!, 𝑡! . ([). 𝑑𝑊 𝑠
¨ The trick is to show that the arbitrage free relationship do impose the following constraint
on the drift:
¨ 𝐴 𝑠, 𝑡!, 𝑡! = 𝐵 𝑠, 𝑡!, 𝑡! . ∫&
&"
𝐵 𝑠, 𝑢, 𝑢 . 𝑑𝑢
¨ That is not trivial, if we have time we will derive it.
¨ But it follows the general principal that “the arbitrage-free relationships impose a constraint
on the drift of the diffusive process"
36

37. Luc_Faucheux_2020
Taxonomy of models - III
¨ If you say 𝐿 𝑡, 𝑡!, 𝑡" is too complicated, and then even 𝐿 𝑡, 𝑡!, 𝑡! is too complicated, the you
just write the dynamics for 𝐿 𝑡, 𝑡, 𝑡 , you call it 𝑟 𝑡 = 𝐿 𝑡, 𝑡, 𝑡 , and BOOM voila you have
the whole family of SHORT RATE models (following Mercurio p.49)
¨ You get the Dothan model (1978) if you write
¨ 𝑟 𝑡 + 𝛿𝑡 − 𝑟 𝑡 = 𝑎. ∫&
&$5&
𝑟(𝑠). 𝑑𝑠 + 𝜎 ∫&
&$5&
𝑟(𝑠). ([). 𝑑𝑊 𝑠
¨ You get the Vasicek model (1977) if you write
¨ 𝑟 𝑡 + 𝛿𝑡 − 𝑟 𝑡 = 𝑘. ∫&
&$5&
[𝜃 − 𝑟 𝑠 ]. 𝑑𝑠 + 𝜎 ∫&
&$5&
1. ([). 𝑑𝑊 𝑠
¨ You get the Cox-Ingersoll-Ross model (1985) if you write
¨ 𝑟 𝑡 + 𝛿𝑡 − 𝑟 𝑡 = 𝑘. ∫&
&$5&
[𝜃 − 𝑟 𝑠 ]. 𝑑𝑠 + 𝜎 ∫&
&$5&
𝑟(𝑠). ([). 𝑑𝑊 𝑠
37

38. Luc_Faucheux_2020
Taxonomy of models - IV
¨ You get the Exponential Vasicek model (1985) if you write
¨ 𝑟 𝑡 + 𝛿𝑡 − 𝑟 𝑡 = ∫&
&$5&
𝑟 𝑠 . [𝜂 − 𝑎. ln(𝑟 𝑠 ]. 𝑑𝑠 + 𝜎. ∫&
&$5&
𝑟(𝑠). ([). 𝑑𝑊 𝑠
¨ You get the Hull-White model (1990) if you write
¨ 𝑟 𝑡 + 𝛿𝑡 − 𝑟 𝑡 = ∫&
&$5&
[𝜃(𝑠) − 𝑘. 𝑟 𝑠 ]. 𝑑𝑠 + 𝜎. ∫&
&$5&
1. ([). 𝑑𝑊 𝑠
¨ You get the Black-Karasinski model (1991) if you write
¨ 𝑟 𝑡 + 𝛿𝑡 − 𝑟 𝑡 = ∫&
&$5&
𝑟 𝑠 . [𝜂(𝑠) − 𝑎. ln(𝑟 𝑠 ]. 𝑑𝑠 + 𝜎. ∫&
&$5&
𝑟(𝑠). ([). 𝑑𝑊 𝑠
38

39. Luc_Faucheux_2020
Taxonomy of models - V
¨ You get the Mercurio – Moraleda model (2000) if you write
¨ 𝑟 𝑡 + 𝛿𝑡 − 𝑟 𝑡 = ∫&
&$5&
𝑟 𝑠 . [𝜂(𝑠) − (𝜆 −
=
#$=>
). ln(𝑟 𝑠 ]. 𝑑𝑠 + 𝜎. ∫&
&$5&
𝑟(𝑠). ([). 𝑑𝑊 𝑠
¨ You get the Cox-Ingersoll-Ross ++ model (1985) if you write
¨ 𝑟 𝑡 + 𝛿𝑡 − 𝑟 𝑡 = 𝜑 𝑡 + 𝛿𝑡 − 𝜑 𝑡 + 𝑘. ∫&
&$5&
[𝜃 − 𝑟 𝑠 ]. 𝑑𝑠 + 𝜎 ∫&
&$5&
𝑟(𝑠). ([). 𝑑𝑊 𝑠
¨ You get the Ho-Lee model (1985) if you write
¨ 𝑟 𝑡 + 𝛿𝑡 − 𝑟 𝑡 = ∫&
&$5&
𝜃 𝑠 . 𝑑𝑠 + 𝜎 ∫&
&$5&
1. ([). 𝑑𝑊 𝑠
39

40. Luc_Faucheux_2020
Taxonomy of models - VI
¨ You get the original Salomon Brothers model (1970) if you write
¨ 𝑟 𝑡 + 𝛿𝑡 − 𝑟 𝑡 = ∫&
&$5&
𝜃 𝑠 . 𝑑𝑠 + 𝜎 ∫&
&$5&
𝑟(𝑠). ([). 𝑑𝑊 𝑠
¨ You get the Brennan-Schwartz model (1980) if you write
¨ 𝑟 𝑡 + 𝛿𝑡 − 𝑟 𝑡 = ∫&
&$5&
[𝛼 + 𝛽. 𝑟 𝑠 . ] 𝑑𝑠 + 𝜎 ∫&
&$5&
𝑟(𝑠). ([). 𝑑𝑊 𝑠
¨ You get the “Constant elasticity of Variance” model (John Cox, 1975) if you write
¨ 𝑟 𝑡 + 𝛿𝑡 − 𝑟 𝑡 = ∫&
&$5&
𝛽. 𝑟 𝑠 . 𝑑𝑠 + 𝜎 ∫&
&$5&
𝑟(𝑠)=. ([). 𝑑𝑊 𝑠
40

41. Luc_Faucheux_2020
Taxonomy of models – VI - a
¨ You can also implement the dynamics on a function of the short rate instead of on the short
rate itself, a common one coming from equity and trying to avoid negative rates is to
implement the dynamics on the log of the rate, but in that case be careful about ITO lemma
¨ You would then have the Black-Karasinsky (1991) model:
¨ ln(𝑟 𝑡 + 𝛿𝑡 ) − ln(𝑟 𝑡 ) = ∫&
&$5&
[𝜃(𝑠) − 𝑘. ln(𝑟 𝑠 )]. 𝑑𝑠 +. ∫&
&$5&
𝜎. ([). 𝑑𝑊 𝑠
¨ Which is kind of the Hull-White but on ln(𝑟 𝑡 ) instead of 𝑟 𝑡
¨ Or if you implement numerically as we did in the Tree deck, the BDT (Black-Derman-Toy
1990) model:
¨ ln(𝑟 𝑡 + 𝛿𝑡 ) − ln(𝑟 𝑡 ) = ∫&
&$5&
[𝜃(𝑠) −
?@(>)
?(>)
. ln(𝑟 𝑠 )]. 𝑑𝑠 +. ∫&
&$5&
𝜎(𝑠). ([). 𝑑𝑊 𝑠
¨ The possible combinations are quite endless and guaranteed a lot of jobs on Wall Street for
physicists like me for a while
41

42. Luc_Faucheux_2020
Taxonomy of models – VI - b
¨ It is trivia time again as we are getting close to the 50 slides point and am starting to feel like
I am losing you again
¨ Why are there so many physicists on Wall Street building so many derivatives, that are
arguably according to Buffet weapons of financial destruction, and also arguably one of the
main cause, if not an aggravating factor of the 2008 great financial crisis?
¨ As often, you have to thank the US congress and point out to two specific dates:
¨ October 19th, 1993: House refused funding for the Texas SSC (Superconducting Super
Collider)
¨ October 30th, 1993: Bill was signed cancelling the SSC
¨ Overnight, an entire generation of physicist were essentially without a career and looked
around, and saw that they could use their knowledge of stochastic processes, diffusion,
statistics, and parlayed that knowledge into getting a job on Wall Street by using fancy terms
like convexity, arbitrage, measure, affine models, OU models, and built the entire field of
derivatives with crazy payoffs like Libor square, Libor in arrears, callable curve steepeners,
callable non-inversion notes, callable snowball, thunderballs, CMBS, ABS, ….
42

43. Luc_Faucheux_2020
Taxonomy of models – VI - b1
¨ Just a note, in all of the derivatives like ABS, MBS, CMBS, …
¨ The “BS” stands for “Based Security”, as in “Asset Based Security”
¨ Yeah I know this is confusing, you might have easily thought that the “BS” did stand for
something else….
43

44. Luc_Faucheux_2020
Taxonomy of models – VI - c
¨ Defunding the SSC did not really create the 2008 crisis (arguably it was a mispricing of
individual credit with real estate housing as collateral), but it certainly made things more
complicated and resulted in a financial disaster that we are still dealing with these days.
¨ The overall cost of the SSC was estimated at the time to be around 5bn
¨ The overall cost of the 2008 crisis is very hard to quantify but a ball park would be around
multiple of trillions (yep.. Trillions….)
¨ https://hbr.org/2018/09/the-social-and-political-costs-of-the-financial-crisis-10-years-later
¨ Oh and also that let the European built the Geneva LHC (Large Hadron Collider) and discover
the God’s particle (Higgs boson), which is essentially the black goo that you need in order to
build a time machine….
¨ So yeahh… great job once again US congress…..
44

45. Luc_Faucheux_2020
Taxonomy of models – VI - d
¨ The Higgs boson black goo you need to build a time machine
45

46. Luc_Faucheux_2020
Taxonomy of models – VI - e
¨ The US congress killing the Texas SSC to save 5bn and engineering the massive creation of
derivatives and “weapons of financial destruction” (W. Buffet), with the 2008 crisis resulting
in losses estimated to be in the trillions…..penny wise…pound foolish….? Pennywise ?
¨ An term sheet example of such derivative weapon of financial destruction in the following
slide (from the Structured deck) from a fictitious dealer in order to offend no one
46

47. Luc_Faucheux_2020
LIFT Notes (Laddered Inverse Floaters) / Snowball
Achieve enhanced yield while expressing bullish rate view
TitleTitleTitle5nc3mo Lift Note Sample Lift Note Termsheet
Snowball Coupon Structure
Note Details:
Structure: 5yr nc 3mo
Issuer: Lehman Brothers
Currency: USD
Deal Size: TBD
CUSIP: TBD
Maturity Date: 5 years (subject to call)
First Call Date: 3 months
Coupon: Yr 1: 8.50% fixed
Yr 2: previous coupon + 5.0% - 6m LIBOR (in arrears)
Yr 3: previous coupon + 6.0% - 6m LIBOR (in arrears)
Yr 4: previous coupon + 7.0% - 6m LIBOR (in arrears)
Yr 5: previous coupon + 8.0% - 6m LIBOR (in arrears)
Frequency/Basis: Quarterly, 30/360, unadjusted
Denominations: $1,000
Selling Points:
u Above market year 1 coupon
u Potential yield pick-up over bullets or vanilla callables
u Coupons “snowball” if bullish rate view realized
u Note can be customized to multiple rate views/ bearish alternatives available
upon request
36 47

48. Luc_Faucheux_2020
Taxonomy of models - VII
¨ If you get bored with one-factor short rate models, you can using multi-factors short rate
models.
¨ If you are a genius like Craig Fithian and worked at Salomon in 1972, you write (am using the
SIE form to be more compact) what got to be known worldwide as the 2+ IRMA model
¨ q
𝑑𝑥 = −𝑘A. 𝑥. 𝑑𝑡 + 𝜎A. ([). 𝑑𝑊A
𝑑𝑦 = −𝑘B. 𝑦. 𝑑𝑡 + 𝜎B. ([). 𝑑𝑊B
𝑑𝑧 = −𝑘.. (𝑥 + 𝑦 − 𝑧). 𝑑𝑡 + 𝜎.. ([). 𝑑𝑊.
¨ With: < 𝑑𝑊A. 𝑑𝑊A >= 𝜌. 𝑑𝑡
¨ And : 𝑟 𝑡 = 𝐼𝑅𝑀𝐴[𝑧 𝑡 + 𝜇 𝑡 ]
¨ Where 𝐼𝑅𝑀𝐴(𝑟) is the IRMA function (Interest Rate Mapping I think) which created an
incredible stable skew, they had historical data on skew going back to the 1960
¨ IRMA was not named after the hurricane from 2017
48

49. Luc_Faucheux_2020
Taxonomy of models - VII-a
¨ The Salomon IRMA function was NOT named after hurricane IRMA (2017)
49

50. Luc_Faucheux_2020
Taxonomy of models - VIII
¨ I recalled the day when working there I got my hands on the original code (which I think was
in FORTRAN) from 1972.
¨ Thinks about it, when Black Sholes came out, Salomon Brothers was running its swap and
option desk with a 3-factor short rate model with IRMA skew !!
¨ The trick was the calibration of the IRMA mapping.
¨ It was defined with 3 variables originally (we then extended to 4 to account for negative
rates), but the original three variables were:
¨ v
Intercept 𝐼
Slope 𝑆
Regime Change 𝑅
50

51. Luc_Faucheux_2020
Taxonomy of models - IX
¨ The IRMA function 𝐼𝑅𝑀𝐴 𝑟 = 𝑓(𝑟) was actually defined from plotting
CD
CE
as a function of 𝑓
and was implemented numerically
51
𝑓(𝑟)
𝑓′(𝑟)
Regime Change 𝑅
Intercept 𝐼
Slope 𝑆

52. Luc_Faucheux_2020
Taxonomy of models - X
¨ Above the Regime Change 𝑅, the function IRMA is a straight line of slope 𝑆
¨ If that straight line was continued below the regime change 𝑅 it would intercept the y-axis at
the Intercept 𝐼
¨ Below the Regime Change 𝑅, the function IRMA is a quadratic function that connect with
the straight line at the point (𝑅, 𝐼 + 𝑆. 𝑅) and goes through the origin (0,0)
¨ Note, when I got there in 2002, that function was still used throughout the firm and
matched the observed skew in a very stable and remarkable manner. We dabbled into
tweaking it for negative rates by essentially adding a parameter similar to the shifted
lognormal model, so that the above sentence got changed to:
¨ Below the Regime Change 𝑅, the function IRMA is a quadratic function that connect with
the straight line at the point (𝑅, 𝐼 + 𝑆. 𝑅) and goes through a point (−𝑁, 0) left of the origin
52

53. Luc_Faucheux_2020
Taxonomy of models - XI
¨ You can see the beauty of this: 𝑟 𝑡 = 𝐼𝑅𝑀𝐴[𝑧 𝑡 + 𝜇 𝑡 ]
¨ IF (𝑆 = 0, 𝑅 = 0, 𝐼 = 𝑐𝑡𝑒) then we have: 𝑓@ 𝑟 = 𝐼, and so 𝑓 𝑟 = 𝐼. 𝑟 + 𝐵
¨ 𝑟 𝑡 is a linear function of the Gaussian variable 𝑧 𝑡 , the model will produce a NORMAL
skew
¨ IF (𝑆 = 𝑐𝑡𝑒, 𝑅 = 0, 𝐼 = 0) then we have: 𝑓@ 𝑟 = 𝑆. 𝑓(𝑟), and so 𝑓 𝑟 = exp(𝑆. 𝑟)
¨ 𝑟 𝑡 is an exponential function of the Gaussian variable 𝑧 𝑡 , the model will produce a
LOGNORMAL skew
¨ The quadratic part under the Regime Change 𝑅 when non-zero will fold the distribution of
𝑧 𝑡 back on positive rates, so the model avoids negative rates (which for a while was
deemed to be a good thing, unless things changed)
53

54. Luc_Faucheux_2020
Taxonomy of models - XII
¨ The 3 parameters had been calibrated to historical data for the skew covering like 40 years
of historical market moves or so, which was in itself amazing (the fact that Salomon had a
clean database that you could use that was going back so far)
¨ The 3 parameters were surprisingly stable, and essentially produced something that was
getting Lognormal at low rates below the Regime Change 𝑅, and closer to Normal above the
Regime Change 𝑅
¨ Ask anyone who worked on the options desk there and worked with 2+IRMA, and they
might still remember by heart those parameters
¨ v
Intercept 𝐼 = 0.06
Slope 𝑆=1
Regime Change 𝑅 = 0.01
54

55. Luc_Faucheux_2020
Taxonomy of models - XIII
¨ If you are bored of model that do not have stochastic volatility, you can add some by also
describing the dynamics of the diffusion, you would get what some people call SVBGM
(Stochastic Volatility BGM model) if you use BGM and write something like this:
¨ 𝐿 𝑡 + 𝛿𝑡, 𝑡!, 𝑡" − 𝐿 𝑡, 𝑡!, 𝑡" = ∫&
&$5&
𝜇 𝑠, 𝑡!, 𝑡" . 𝐿 𝑠, 𝑡!, 𝑡" . 𝑑𝑠 +
∫&
&$5&
Σ 𝑠, 𝑡!, 𝑡" . 𝐿 𝑠, 𝑡!, 𝑡" . ([). 𝑑𝑊 𝑠, 𝑡!, 𝑡"
¨ Where now Σ 𝑠, 𝑡!, 𝑡" is also a stochastic variable:
¨ Σ 𝑡 + 𝛿𝑡, 𝑡!, 𝑡" − Σ 𝑡, 𝑡!, 𝑡" = ∫&
&$5&
𝐶 𝑠, 𝑡!, 𝑡" . 𝑑𝑠 + ∫&
&$5&
𝐷 𝑠, 𝑡!, 𝑡" . ([). 𝑑𝑊F 𝑠, 𝑡!, 𝑡"
¨ Where 𝐶 𝑠, 𝑡!, 𝑡" and 𝐷 𝑠, 𝑡!, 𝑡" could be function of Σ 𝑡, 𝑡!, 𝑡"
¨ And also where the drivers 𝑑𝑊F 𝑠, 𝑡!, 𝑡" and 𝑑𝑊 𝑠, 𝑡!, 𝑡" could be correlated
55

56. Luc_Faucheux_2020
Taxonomy of models - XIV
¨ Similarly, if you want to reduce the dimensions of the problem and work in the SHORT RATE
model, you can introduce stochastic volatility there too, for example if you write below
(Kwok p.407), you get the Fong-Vasicek model (1991)
¨ ~
𝑑𝑟 = −𝛼. (𝑟 − ̅𝑟). 𝑑𝑡 + 𝑣. ([). 𝑑𝑊E
𝑑𝑣 = −𝛾. 𝑣 − ̅𝑣 . 𝑑𝑡 + 𝜉. 𝑣. ([). 𝑑𝑊G
¨ Where:
¨ < 𝑑𝑊E. 𝑑𝑊G >= 𝜌. 𝑑𝑡
¨ And so on and so forth as someone I knew used to say…as you can see if you know your way
around Stochastic Differential Equations, there is a lot you can do (or again just write the
discrete dynamics, and let Machine Learning figure out the calibration for you by crunching
CPU like you are mining bitcoins)
56

57. Luc_Faucheux_2020
Taxonomy of models - XV
¨ Oh, one final note before we start looking at some of the models in greater details:
¨ You get the Hull-White model (1990), one of the most commonly used, if you write
¨ 𝑟 𝑡 + 𝛿𝑡 − 𝑟 𝑡 = ∫&
&$5&
[𝜃(𝑠) − 𝑘. 𝑟 𝑠 ]. 𝑑𝑠 + 𝜎. ∫&
&$5&
1. ([). 𝑑𝑊 𝑠
¨ 𝑑𝑟(𝑡) = [𝜃(𝑡) − 𝑘. 𝑟 𝑡 ] + 𝜎. 𝑑𝑊(𝑡)
¨ 𝑑𝑟 𝑡 = 𝜃 𝑡 . 𝑑𝑡 − 𝑘. 𝑟 𝑡 . 𝑑𝑡 + 𝜎. 𝑑𝑊(𝑡)
¨ Without the added drift 𝜃 𝑡 the equation becomes:
¨ 𝑑𝑟 𝑡 = −𝑘. 𝑟 𝑡 . 𝑑𝑡 + 𝜎. 𝑑𝑊(𝑡)
¨ Remember our good friend the Langevin equation from 1908?
¨ 𝑑𝑉 𝑡 = −𝑘. 𝑉(𝑡). 𝑑𝑡 + 𝜎. 𝑑𝑊
¨ Yep, that’s the one. Goes to show you that Quohelet was right, there is not much that is
new under the sun. The good piece of news is that people have been using the Langevin
equation since 1908, so there are tons of results that we can easily transfer to the Hull-
White model for example
57

58. Luc_Faucheux_2020
Taxonomy of models - XVI
¨ So we will use a lot when looking at Hull-White the results we derived on the Langevin deck
¨ That goes to show you that nothing beats the wisdom of King Salomon (not affiliated with
Salomon Brothers)
58

59. Luc_Faucheux_2020
The Art of Term Structure Modeling:
The Art of the Drift(*)
¨ (*) Bruce Tuckman, “Fixed-Income Securities”
59

60. Luc_Faucheux_2020
¨ Move that to part V
60

61. Luc_Faucheux_2020
Some notes about trees,
Respecting the Arbitrage relationships locally
and everywhere
61

62. Luc_Faucheux_2020
Local arbitrage and global arbitrage
¨ We saw in the “tree” deck when building the BDT model that the only relationships that we
were enforcing were the “global” arbitrage free relationships as viewed from the origin
(when we calibrated the “k”s in order to recover the discount factors
62

63. Luc_Faucheux_2020
Local arbitrage and global arbitrage - II
¨ This is because in general a recombining binomial tree does not have enough degrees of
freedom in order to respect the arbitrage “at every node in the tree”. (except in some very
simple cases like the Ho-Lee model or BDT, see further in this deck).
¨ And so we really enforce the “global” arbitrage relationships, essentially calibrating the tree
so that we recover the discount factors from the initial discount curve
¨ Instead of enforcing all the possible constraints:
¨ 𝔼&"
*+
𝑉 𝑡!, $
#
#$4 &,&",&! .)
, 𝑡!, 𝑡! |𝔉(𝑡) =
#
#$% &,&",&! .)
= 𝑧𝑐 𝑡, 𝑡!, 𝑡"
¨ We only enforce (calibrate) the ones from the origin of the tree for example like we did in
BDT:
¨ 𝔼&"
*+
𝑉 𝑡!, $
#
#$4 &,&",&! .)
, 𝑡!, 𝑡! |𝔉(𝑡 = 0) =
#
#$% &6H,&",&! .)
= 𝑧𝑐 𝑡 = 0, 𝑡!, 𝑡"
63

64. Luc_Faucheux_2020
Local arbitrage and global arbitrage - III
¨ In general for a non constant volatility the BDT tree will get distorted to look something like
below. The average on each slices of the zeros are such that they are the value from the
initial discount curve. If you look at a specific node inside the tree, the arbitrage constraints
will be violated, and there is not much that you will be able to do about it.
64

65. Luc_Faucheux_2020
Local arbitrage and global arbitrage - IV
¨ Because recombining binomial trees are computationally attractive (especially when we did
not have cloud computing in the 1990s, am dating myself), Richard Robb my boss at the
time and I tried for a while to come up with a bunch of tricks to try to enforce arbitrage
everywhere in the BDT framework we had, trying to find ways to even allow for a minimal
amount of arbitrage, when in the end we realized that it would actually be easier to
implement a non-recombining tree and essentially do the backward valuation pass using a
regression method, what is known in the literature as the Longstaff-Schwartz method, but
we did not know that at the time
65

66. Luc_Faucheux_2020
Local arbitrage and global arbitrage - V
¨ Note HOWEVER that in short rate model like BDT, even though the tree looks distorted, it
will not break the arbitrage locally, as in essence the short rate is the only information that
you have, so any tree that you rebuild locally will be arbitrage-free.
¨ It is when you look at the evolution of more than one forward on the curve in a binomial
recombining tree that you will end up always breaking the arbitrage locally (and a numerical
precision will grow exponentially, the best I could ever achieve was 12 steps or so before
observing a local arbitrage that was not respected)
¨ Bear in mind that looking at say 2 forwards in a binomial recombining tree is still a one
factor model.
¨ It is not the same thing as say just a short rate model that is multi-factor
¨ Again, maybe obvious to most of you, but worth pointing out.
66

67. Luc_Faucheux_2020
Local arbitrage and global arbitrage - VI
¨ From the Tree deck, inside a one factor BDT model, the tree and every other small trees
inside will be arbitrage free by construction (because the only information that you have is
the short rate, or daily zeros between nodes, and every curve can and has to be
reconstructed from that).
67
NEW_D 0.9901 0.985235 0.980507 0.975929 0.971767 0.97038 0.97009 0.970699 0.97308 0.975147 0.974849 0.974541
0.985206 0.980392 0.975717 0.971325 0.969655 0.968852 0.969484 0.972015 0.974261 0.974104 0.97399
0.980277 0.975503 0.970877 0.968913 0.967564 0.968221 0.970909 0.973345 0.973338 0.973427
0.975288 0.970422 0.968153 0.966224 0.966907 0.969761 0.972396 0.972549 0.972853
0.969961 0.967375 0.964831 0.96554 0.968568 0.971416 0.971738 0.972266
0.966579 0.963383 0.96412 0.967331 0.970401 0.970904 0.971667
0.961877 0.962643 0.966046 0.969351 0.970045 0.971056
0.961107 0.964713 0.968266 0.969163 0.970432
0.963329 0.967143 0.968255 0.969794
0.965982 0.967321 0.969144
0.966361 0.96848
0.967802
𝐴 “𝑠𝑙𝑖𝑐𝑒” 𝑖𝑠 𝑐𝑎𝑙𝑖𝑏𝑟𝑎𝑡𝑒𝑑 𝑓𝑟𝑜𝑚
𝑡ℎ𝑒 𝑖𝑛𝑖𝑡𝑖𝑎𝑙 𝑑𝑖𝑠𝑐𝑜𝑢𝑛𝑡 𝑐𝑢𝑟𝑣𝑒
𝑤𝑖𝑡ℎ 𝑡ℎ𝑒 𝑎𝑠𝑠𝑢𝑚𝑝𝑡𝑖𝑜𝑛𝑠
𝑜𝑛 𝑡ℎ𝑒 𝑑𝑦𝑛𝑎𝑚𝑖𝑐𝑠 𝑜𝑓 𝑡ℎ𝑒 𝑓𝑜𝑟𝑤𝑎𝑟𝑑 𝑟𝑎𝑡𝑒
𝐴𝑛𝑦 𝑡𝑟𝑒𝑒 𝑖𝑛𝑠𝑖𝑑𝑒 𝑡ℎ𝑒 𝑔𝑙𝑜𝑏𝑎𝑙 𝑡𝑟𝑒𝑒
𝑤𝑖𝑙𝑙 𝑠𝑡𝑖𝑙𝑙 𝑏𝑒 𝑎𝑟𝑏𝑖𝑡𝑟𝑎𝑔𝑒 𝑓𝑟𝑒𝑒
𝑎𝑠 𝑦𝑜𝑢 𝑜𝑛𝑙𝑦 ℎ𝑎𝑣𝑒 𝑜𝑛𝑒 𝑓𝑎𝑐𝑡𝑜𝑟 𝑡𝑜 𝑔𝑜 𝑏𝑦

68. Luc_Faucheux_2020
Local arbitrage and global arbitrage - VII
¨ Bear in mind that not every short rate model can be implemented in a binomial recombining
tree (we will see in the HJM drift section that in general the short rate is non Markovian for a
generic volatility surface as an input)
¨ All that I am saying here, is that the BDT implementation we looked at in the “Tree” deck is
binomial, recombining, based on the short rate (overnight zeros) and was calibrated to the
initial discount curve (enforce the initial arbitrage free relationships). By construction, any
subset of the tree that you look at, because the only information that you have at this point
is the discrete zeros, will be also arbitrage free locally.
¨ The BDT has a very specific dynamics, that we showed to be:
¨ ln(𝑟 𝑡 + 𝛿𝑡 ) − ln(𝑟 𝑡 ) = ∫&
&$5&
[𝜃(𝑠) −
?@(>)
?(>)
. ln(𝑟 𝑠 )]. 𝑑𝑠 +. ∫&
&$5&
𝜎(𝑠). ([). 𝑑𝑊 𝑠
68

69. Luc_Faucheux_2020
Towards a continuous description
Instantaneous Forward Rates
69

70. Luc_Faucheux_2020
Instantaneous rates
¨ There is only one discount curve
¨ From the unique discount curve you can define many different rates of many different
tenors
¨ Simply compounded rates
¨
#
#$4 &,&",&! .) &,&",&!
= 𝑍𝐶 𝑡, 𝑡!, 𝑡"
¨ Continuously compounded
¨ Annually compounded
¨ K-times per year compounded.
¨ The point is that all those definitions converge to the same limit when 𝜏 𝑡, 𝑡!, 𝑡" → 0, or
equivalently 𝑡" → 𝑡!
¨ This leads to the concept of “instantaneous rates”
¨ You need that formalism for HJM, and simpler term structure models
70

71. Luc_Faucheux_2020
Summary of the rates notation we had in Part II and Part III
¨ We are now ready to revisit the notations and definitions that we had in part II and part III
with the more generic notation that is now rigorous
¨ Not saying that the ones before were not, but usually in textbooks they start with the simple
ones go through the simple models, and then start introducing more complicated notations
¨ Here we sort of started with the simple notations as you will find them in every textbooks,
went through why we need the more general ones, and then are now reducing the
complexity of the modeling making assumptions
¨ I think it is always better to have a general framework and work out specific simple cases of
it rather than getting stuck at the bottom level
¨ Here were the slides we had in Part II and Part III
71

72. Luc_Faucheux_2020
Summary of the definitions in the “simple”
notation
72

73. Luc_Faucheux_2020
Notations and conventions in the rates world -IV
¨ Continuously compounded spot interest rate:
¨ 𝑟 𝑡, 𝑇 = −
IJ(./(&,K))
)(&,K)
¨ Where 𝜏(𝑡, 𝑇) is the year fraction, using whatever convention (ACT/360, ACT/365, 30/360,
30/250,..) and possible holidays calendar we want. In the simplest case:
¨ 𝜏 𝑡, 𝑇 = 𝑇 − 𝑡
¨ 𝑧𝑐 𝑡, 𝑇 . exp 𝑟 𝑡, 𝑇 . 𝜏 𝑡, 𝑇 = 1
¨ 𝑧𝑐 𝑡, 𝑇 = exp −𝑟 𝑡, 𝑇 . 𝜏 𝑡, 𝑇
¨ In the deterministic case:
¨ 𝑧𝑐 𝑡, 𝑇 = exp −𝑟 𝑡, 𝑇 . 𝜏 𝑡, 𝑇 = 𝐷 𝑡, 𝑇 =
L(&)
L(K)
= exp(− ∫&
K
𝑅 𝑠 . 𝑑𝑠)
¨ 𝑟 𝑡, 𝑇 =
#
) &,K
. ∫&
K
𝑅 𝑠 . 𝑑𝑠
73

74. Luc_Faucheux_2020
Notations and conventions in the rates world - V
¨ Simply compounded spot interest rate
¨ 𝑙 𝑡, 𝑇 =
#
)(&,K)
.
#M./(&,K)
./(&,K)
¨ Or alternatively, in the bootstrap form
¨ 𝜏 𝑡, 𝑇 . 𝑙 𝑡, 𝑇 . 𝑧𝑐 𝑡, 𝑇 = 1 − 𝑧𝑐(𝑡, 𝑇)
¨ 1 + 𝜏 𝑡, 𝑇 . 𝑙 𝑡, 𝑇 . 𝑧𝑐 𝑡, 𝑇 = 1
¨ 𝑧𝑐 𝑡, 𝑇 =
#
#$) &,K .% &,K
¨ In the deterministic case:
¨ 𝑧𝑐 𝑡, 𝑇 =
#
#$) &,K .% &,K
= 𝐷 𝑡, 𝑇 =
L(&)
L(K)
= exp(− ∫&
K
𝑅 𝑠 . 𝑑𝑠)
¨ 𝑙 𝑡, 𝑇 =
#
) &,K
. [1 − exp − ∫&
K
𝑅 𝑠 . 𝑑𝑠 ]
74

75. Luc_Faucheux_2020
Notations and conventions in the rates world - VI
¨ Annually compounded spot interest rate
¨ 𝑦 𝑡, 𝑇 =
#
./(&,K)(/*(,,.) − 1
¨ Or alternatively, in the bootstrap form
¨ (1 + 𝑦 𝑡, 𝑇 ). 𝑧𝑐 𝑡, 𝑇 #/) &,K = 1
¨ (1 + 𝑦 𝑡, 𝑇 )) &,K . 𝑧𝑐 𝑡, 𝑇 = 1
¨ 𝑧𝑐 𝑡, 𝑇 =
#
(#$B &,K )* ,,.
¨ In the deterministic case:
¨ 𝑧𝑐 𝑡, 𝑇 =
#
(#$B &,K )* ,,. = 𝐷 𝑡, 𝑇 =
L(&)
L(K)
= exp(− ∫&
K
𝑅 𝑠 . 𝑑𝑠)
75

76. Luc_Faucheux_2020
Notations and conventions in the rates world - VII
¨ 𝑞-times per year compounded spot interest rate
¨ 𝑦O 𝑡, 𝑇 =
O
./(&,K)(/0*(,,.) − 𝑞
¨ Or alternatively, in the bootstrap form
¨ (1 +
#
O
𝑦O 𝑡, 𝑇 ). 𝑧𝑐 𝑡, 𝑇 #/O) &,K = 1
¨ (1 +
#
O
𝑦O 𝑡, 𝑇 )O.) &,K . 𝑧𝑐 𝑡, 𝑇 = 1
¨ 𝑧𝑐 𝑡, 𝑇 =
#
(#$
(
0
.B0 &,K )0.* ,,.
¨ In the deterministic case:
¨ 𝑧𝑐 𝑡, 𝑇 =
#
(#$
(
0
.B0 &,K )0.* ,,.
= 𝐷 𝑡, 𝑇 =
L(&)
L(K)
= exp(− ∫&
K
𝑅 𝑠 . 𝑑𝑠)
76

77. Luc_Faucheux_2020
Notations and conventions in the rates world - VIII
¨ In bootstrap form which is the intuitive way:
¨ Continuously compounded spot: 𝑧𝑐 𝑡, 𝑇 = exp −𝑟 𝑡, 𝑇 . 𝜏 𝑡, 𝑇
¨ Simply compounded spot: 𝑧𝑐 𝑡, 𝑇 =
#
#$) &,K .% &,K
¨ Annually compounded spot: 𝑧𝑐 𝑡, 𝑇 =
#
(#$B &,K )* ,,.
¨ 𝑞-times per year compounded spot 𝑧𝑐 𝑡, 𝑇 =
#
(#$
(
0
.B0 &,K )0.* ,,.
77

78. Luc_Faucheux_2020
Notations and conventions in the rates world - IX
¨ In the small 𝜏 𝑡, 𝑇 → 0 limit (also if the rates themselves are such that they are <<1)
¨ In bootstrap form which is the intuitive way:
¨ Continuously compounded spot: 𝑧𝑐 𝑡, 𝑇 = 1 − 𝑟 𝑡, 𝑇 . 𝜏 𝑡, 𝑇 + 𝒪(𝜏P. 𝑟P)
¨ Simply compounded spot: 𝑧𝑐 𝑡, 𝑇 = 1 − 𝑙 𝑡, 𝑇 . 𝜏 𝑡, 𝑇 + 𝒪(𝜏P. 𝑙P)
¨ Annually compounded spot: 𝑧𝑐 𝑡, 𝑇 = 1 − 𝑦 𝑡, 𝑇 . 𝜏 𝑡, 𝑇 + 𝒪(𝜏P. 𝑦P)
¨ 𝑞-times per year compounded spot 𝑧𝑐 𝑡, 𝑇 = 1 − 𝑦O 𝑡, 𝑇 . 𝜏 𝑡, 𝑇 + 𝒪(𝜏P. 𝑦O
P)
¨ So in the limit of small 𝜏 𝑡, 𝑇 (and also small rates), in particular when 𝑇 → 𝑡, all rates
converge to the same limit we call
¨ 𝐿𝑖𝑚 𝑇 → 𝑡 = lim
K→&
(
#M./ &,K
) &,K
)
78

79. Luc_Faucheux_2020
Notations and conventions in the rates world - X
¨ In the deterministic case using the continuously compounded spot rate for example:
¨ 𝑧𝑐 𝑡, 𝑇 = exp −𝑟 𝑡, 𝑇 . 𝜏 𝑡, 𝑇 = 𝐷 𝑡, 𝑇 =
L(&)
L(K)
= exp(− ∫&
K
𝑅 𝑠 . 𝑑𝑠)
¨ 𝑟 𝑡, 𝑇 =
#
) &,K
. ∫&
K
𝑅 𝑠 . 𝑑𝑠
¨ When 𝑇 → 𝑡, 𝑟 𝑡, 𝑇 → 𝑅(𝑡)
¨ So: 𝐿𝑖𝑚 𝑇 → 𝑡 = lim
K→&
(
#M./ &,K
) &,K
) = 𝑅(𝑡)
¨ So 𝑅(𝑡) can be seen as the limit of all the different rates defined above.
¨ You can also do this using any of the rates defined previously
79

80. Luc_Faucheux_2020
Instantaneous rates - II
¨ We are reviewing the definitions of part II and III with the new notation:
¨ Continuously compounded spot interest rate:
¨ 𝑟 𝑡, 𝑇 = −
IJ(./(&,K))
)(&,K)
¨ 𝑧𝑐 𝑡, 𝑇 . exp 𝑟 𝑡, 𝑇 . 𝜏 𝑡, 𝑇 = 1
¨ 𝑧𝑐 𝑡, 𝑇 = exp −𝑟 𝑡, 𝑇 . 𝜏 𝑡, 𝑇
¨ With our more generic notation in the case of the stochastic variable this reads:
¨ 𝑅 𝑡, 𝑡!, 𝑡" = −
IJ(*+ &,&",&! )
) &,&",&!
¨ CAREFUL that now you are dealing with stochastic variable, and within the ITO calculus, the
functions Log and exp cannot be used as in the regular calculus
¨ ALSO be careful when trying to do any derivation or differentiation
80

81. Luc_Faucheux_2020
Instantaneous rates – III
¨ In the deterministic case:
¨ 𝑧𝑐 𝑡, 𝑇 = exp −𝑟 𝑡, 𝑇 . 𝜏 𝑡, 𝑇 = 𝐷 𝑡, 𝑇 =
L(&)
L(K)
= exp(− ∫&
K
𝑅 𝑠 . 𝑑𝑠)
¨ 𝑟 𝑡, 𝑇 =
#
) &,K
. ∫&
K
𝑅 𝑠 . 𝑑𝑠
¨ This now reads:
¨ 𝑧𝑐 𝑡, 𝑡!, 𝑡" = exp −𝑟 𝑡, 𝑡!, 𝑡" . 𝜏 𝑡, 𝑡!, 𝑡" =
./ &,&,&"
./ &,&,&!
¨ We are not yet dealing with the equation: 𝑟 𝑡, 𝑇 =
#
) &,K
. ∫&
K
𝑅 𝑠 . 𝑑𝑠
¨ We are essentially dropping the notation 𝐷 𝑡, 𝑇 , you find in some textbooks but I found it
to be confusing and useless since you have the 𝑍𝐶 𝑡, 𝑡!, 𝑡" and their fixed value 𝑧𝑐 𝑡, 𝑡!, 𝑡"
for the specific observation of the discount curve at time 𝑡
81

82. Luc_Faucheux_2020
Instantaneous rates – IV
¨ Simply compounded spot interest rate
¨ 𝑙 𝑡, 𝑇 =
#
)(&,K)
.
#M./(&,K)
./(&,K)
¨ Or alternatively, in the bootstrap form
¨ 𝜏 𝑡, 𝑇 . 𝑙 𝑡, 𝑇 . 𝑧𝑐 𝑡, 𝑇 = 1 − 𝑧𝑐(𝑡, 𝑇)
¨ 1 + 𝜏 𝑡, 𝑇 . 𝑙 𝑡, 𝑇 . 𝑧𝑐 𝑡, 𝑇 = 1
¨ 𝑧𝑐 𝑡, 𝑇 =
#
#$) &,K .% &,K
¨ This is familiar to us now since we have done most of Part II and Part III using the simply
compounded rate. The equations above should of course read:
¨ 𝐿 𝑡, 𝑡!, 𝑡" =
#
) &,&",&!
.
#M*+ &,&",&!
*+ &,&",&!
¨ 𝑍𝐶 𝑡, 𝑡!, 𝑡" =
#
#$) &,&",&! .4 &,&",&!
82

83. Luc_Faucheux_2020
Instantaneous rates – V
¨ Annually compounded spot interest rate
¨ 𝑦 𝑡, 𝑇 =
#
./(&,K)(/*(,,.) − 1
¨ Or alternatively, in the bootstrap form
¨ (1 + 𝑦 𝑡, 𝑇 ). 𝑧𝑐 𝑡, 𝑇 #/) &,K = 1
¨ (1 + 𝑦 𝑡, 𝑇 )) &,K . 𝑧𝑐 𝑡, 𝑇 = 1
¨ 𝑧𝑐 𝑡, 𝑇 =
#
(#$B &,K )* ,,.
¨ Same, this now reads:
¨ 𝑍𝐶 𝑡, 𝑡!, 𝑡" =
#
(#$Q &,&",&! )
* ,,,",,!
and the one for the observed value at time 𝑡
¨ 𝑧𝑐 𝑡, 𝑡!, 𝑡" =
#
(#$B &,&",&! )
* ,,,",,!
83

84. Luc_Faucheux_2020
Instantaneous rates – VI
¨ 𝑞-times per year compounded spot interest rate
¨ 𝑦O 𝑡, 𝑇 =
O
./(&,K)(/0*(,,.) − 𝑞
¨ Or alternatively, in the bootstrap form
¨ (1 +
#
O
𝑦O 𝑡, 𝑇 ). 𝑧𝑐 𝑡, 𝑇 #/O) &,K = 1
¨ (1 +
#
O
𝑦O 𝑡, 𝑇 )O.) &,K . 𝑧𝑐 𝑡, 𝑇 = 1
¨ 𝑧𝑐 𝑡, 𝑇 =
#
(#$
(
0
.B0 &,K )0.* ,,.
¨ Becomes : 𝑍𝐶 𝑡, 𝑡!, 𝑡" =
#
(#$
(
0
.B0 &,&",&! )
0.* ,,,",,!
84

85. Luc_Faucheux_2020
Instantaneous rates – VII
¨ In bootstrap form which is the intuitive way:
¨ Continuously compounded spot: 𝑧𝑐 𝑡, 𝑇 = exp −𝑟 𝑡, 𝑇 . 𝜏 𝑡, 𝑇
¨ Simply compounded spot: 𝑧𝑐 𝑡, 𝑇 =
#
#$) &,K .% &,K
¨ Annually compounded spot: 𝑧𝑐 𝑡, 𝑇 =
#
(#$B &,K )* ,,.
¨ 𝑞-times per year compounded spot 𝑧𝑐 𝑡, 𝑇 =
#
(#$
(
0
.B0 &,K )0.* ,,.
85

86. Luc_Faucheux_2020
Instantaneous rates – VIII
¨ The slide before becomes
¨ In bootstrap form which is the intuitive way:
¨ Continuously compounded : 𝑍𝐶 𝑡, 𝑡!, 𝑡" = exp −𝑅 𝑡, 𝑡!, 𝑡" . 𝜏 𝑡, 𝑡!, 𝑡"
¨ Simply compounded : 𝑍𝐶 𝑡, 𝑡!, 𝑡" =
#
#$) &,&",&! .4 &,&",&!
¨ Annually compounded : 𝑍𝐶 𝑡, 𝑡!, 𝑡" =
#
(#$Q &,&",&! )
* ,,,",,!
¨ 𝑞-times per year compounded: 𝑍𝐶 𝑡, 𝑡!, 𝑡" =
#
(#$
(
0
.Q0 &,&",&! )
0.* ,,,",,!
86

87. Luc_Faucheux_2020
Instantaneous rates – IX
¨ In the small 𝜏 𝑡, 𝑡!, 𝑡" → 0 limit (also if the rates themselves are such that they are <<1)
¨ In bootstrap form which is the intuitive way:
¨ Continuously compounded spot: 𝑍𝐶 𝑡, 𝑡!, 𝑡" = 1 − 𝑅 𝑡, 𝑡!, 𝑡" . 𝜏 + 𝒪(𝜏P. 𝑅P)
¨ Simply compounded spot: 𝑍𝐶 𝑡, 𝑡!, 𝑡" = 1 − 𝐿 𝑡, 𝑡!, 𝑡" . 𝜏 + 𝒪(𝜏P. 𝑙P)
¨ Annually compounded spot: 𝑍𝐶 𝑡, 𝑡!, 𝑡" = 1 − 𝑌 𝑡, 𝑡!, 𝑡" . 𝜏 + 𝒪(𝜏P. 𝑦P)
¨ 𝑞-times per year compounded spot 𝑍𝐶 𝑡, 𝑡!, 𝑡" = 1 − 𝑌O 𝑡, 𝑡!, 𝑡" . 𝜏 + 𝒪(𝜏P. 𝑦O
P)
¨ So in the limit of small 𝜏 𝑡, 𝑡!, 𝑡" (and also small rates), in particular when: 𝑡" → 𝑡!, all rates
converge to the same limit we call
¨ 𝐿𝑖𝑚 𝑡" → 𝑡! = lim
&!→&"
(
#M*+ &,&",&!
) &,&",&!
) that we will note Instantaneous Forward Rate
87

88. Luc_Faucheux_2020
Backup of slide X
¨ Backup of slide X (sorry for that, Powerpoint file somehow got corrupted, and kept dropping
that slide and replacing it with the Master header, time to stop working on part IV and start
part V)
¨ 𝐿𝑖𝑚 𝑡" → 𝑡! = lim
&!→&"
(
#M*+ &,&",&!
) &,&",&!
) that we will note Instantaneous Forward Rate
¨ 𝐿𝑖𝑚 𝑡" → 𝑡!, 𝑡" → 𝑡 = lim
&!→&",&!→&
(
#M*+ &,&",&!
) &,&",&!
) that we will note Instantaneous SHORT RATE
¨ We already have the notation 𝑆𝑅 for Swap Rate. Also 𝑆 could stand for short, swap, spot, a
lot of different things
¨ Some textbooks use the lower case 𝑟 for short rate. This is confusing, especially since we
would like to keep lower case for values that are fixed or observed, and upper case for
random variables for which we compute expectations
88

89. Luc_Faucheux_2020
Backup of slide X-b
¨ So just to not be too confused, we will use the notation:
¨ 𝐼𝐹𝑤𝑅 𝑡, 𝑡!, 𝑡" = 𝐼𝐹𝑤𝑅 𝑡, 𝑡!, 𝑡!$ = 𝐼𝐹𝑤𝑅 𝑡, 𝑡! = lim
&!→&"
(
#M*+ &,&",&!
) &,&",&!
)
¨ 𝐼𝑆ℎ𝑅 𝑡, 𝑡!, 𝑡" = 𝐼𝑆ℎ𝑅 𝑡, 𝑡+, 𝑡 + = 𝐼𝑆ℎ𝑅 𝑡 = lim
&!→&",&!→&
(
#M*+ &,&",&!
) &,&",&!
)
89

90. Luc_Faucheux_2020
Instantaneous rates – XI
¨ 𝐼𝐹𝑤𝑅 𝑡, 𝑡!, 𝑡" = 𝐼𝐹𝑤𝑅 𝑡, 𝑡!, 𝑡!$ = 𝐼𝐹𝑤𝑅 𝑡, 𝑡! = lim
&!→&"
(
#M*+ &,&",&!
) &,&",&!
)
¨ 𝐼𝑆ℎ𝑅 𝑡, 𝑡!, 𝑡" = 𝐼𝑆ℎ𝑅 𝑡, 𝑡+, 𝑡 + = 𝐼𝑆ℎ𝑅 𝑡 = lim
&!→&",&!→&
(
#M*+ &,&",&!
) &,&",&!
)
¨ 𝐼𝑆ℎ𝑅 𝑡 = lim
&"→&
𝐼𝐹𝑤𝑅 𝑡, 𝑡!
¨ For now let’s keep those notations for a while
¨ Note also that
¨ 𝐼𝑆ℎ𝑅 𝑡 = lim
&!→&
(
#M*+ &,&,&!
) &,&,&!
)
90

91. Luc_Faucheux_2020
Instantaneous rates – XII
¨ Some other terms that you sometimes find in the literature:
¨ 𝐼𝑆ℎ𝑅 𝑡 is sometimes noted 𝑟(𝑡) and called the “instantaneous risk-free spot rate, or short
rate, it is the rate at which an associated money market (or bank) account accrues
continuously starting from $1 at time 𝑡 = 0)
¨ It is a crucial concept as most models developed at the beginning were “SHORT RATE
MODELS”, meaning that instead of modeling the 𝑍𝐶 𝑡, 𝑡!, 𝑡" , the only variable we are
modeling is the short rate 𝐼𝑆ℎ𝑅 𝑡
¨ We will go through the taxonomy of all those models but it is crucial to note that the “short
rate models” are reduction of the general framework
¨ It is also quite DANGEROUS to reduce 𝑅 𝑡, 𝑡!, 𝑡" to 𝑟 𝑡 = 𝑅 𝑡, 𝑡! = 𝑡, 𝑡" = 𝑡 because you
lose track of which time variable is the “Brownian” one (the first one) and which one is the
“Newtonian” one (the second one and the third one) to use the analogy from the Baxter
book. We use that book a lot in the deck on Numeraire and Measures. If you want to really
understand Girsanov’s theorem, that was the only book that did it for me.
91

92. Luc_Faucheux_2020
Instantaneous rates – XII - a
92

93. Luc_Faucheux_2020
Instantaneous rates – XIII
¨ Some more intuition around those ”instantaneous” rates.
¨ “Yield-to-Maturity”
¨ This is the corresponding rate for a 𝑍𝐶 𝑡, 𝑡, 𝑡" up until time 𝑡"
¨ In the case of the continuously compounded rate
¨ 𝑍𝐶 𝑡, 𝑡!, 𝑡" = exp −𝑅 𝑡, 𝑡!, 𝑡" . 𝜏 𝑡, 𝑡!, 𝑡"
¨ 𝑍𝐶 𝑡, 𝑡, 𝑡" = exp −𝑅 𝑡, 𝑡, 𝑡" . 𝜏 𝑡, 𝑡, 𝑡"
¨ 𝑅 𝑡, 𝑡, 𝑡" = −
#
) &,&,&!
. ln(𝑍𝐶 𝑡, 𝑡, 𝑡" )
¨ 𝑅 𝑡, 𝑡!, 𝑡" = −
#
) &,&",&!
. ln 𝑍𝐶 𝑡, 𝑡!, 𝑡" = −
#
) &,&",&!
. ln(
*+ &,&,&!
*+ &,&,&"
)
¨ 𝑅 𝑡, 𝑡!, 𝑡" = −
#
) &,&",&!
. {ln𝑍𝐶 𝑡, 𝑡, 𝑡" − ln𝑍𝐶 𝑡, 𝑡, 𝑡! }
93

94. Luc_Faucheux_2020
Instantaneous rates – XIV
¨ 𝑅 𝑡, 𝑡!, 𝑡" = −
#
) &,&",&!
. {ln𝑍𝐶 𝑡, 𝑡, 𝑡" − ln𝑍𝐶 𝑡, 𝑡, 𝑡! }
¨ 𝐼𝐹𝑤𝑅 𝑡, 𝑡!, 𝑡" = 𝐼𝐹𝑤𝑅 𝑡, 𝑡!, 𝑡!$ = 𝐼𝐹𝑤𝑅 𝑡, 𝑡! = lim
&!→&"
(
#M*+ &,&",&!
) &,&",&!
)
¨ 𝐼𝐹𝑤𝑅 𝑡, 𝑡! = lim
&!→&"
(
#M*+ &,&",&!
) &,&",&!
) = lim
&!→&"
(𝑅 𝑡, 𝑡!, 𝑡" )
¨ 𝐼𝐹𝑤𝑅 𝑡, 𝑡! = lim
&!→&"
(
#M*+ &,&",&!
) &,&",&!
) = lim
&!→&"
(𝐿 𝑡, 𝑡!, 𝑡" )
¨ 𝐼𝐹𝑤𝑅 𝑡, 𝑡! = lim
&!→&"
(
#M*+ &,&",&!
) &,&",&!
) = lim
&!→&"
(𝑌 𝑡, 𝑡!, 𝑡" )
¨ 𝐼𝐹𝑤𝑅 𝑡, 𝑡! = lim
&!→&"
(
#M*+ &,&",&!
) &,&",&!
) = lim
&!→&"
(𝑌O 𝑡, 𝑡!, 𝑡" )
94

95. Luc_Faucheux_2020
Instantaneous rates – XV
¨ 𝑅 𝑡, 𝑡!, 𝑡" = −
#
) &,&",&!
. {ln𝑍𝐶 𝑡, 𝑡, 𝑡" − ln𝑍𝐶 𝑡, 𝑡, 𝑡! }
¨ 𝐼𝐹𝑤𝑅 𝑡, 𝑡! = lim
&!→&"
(
#M*+ &,&",&!
) &,&",&!
) = lim
&!→&"
(𝑅 𝑡, 𝑡!, 𝑡" ) = lim
) &,&",&! →H
(𝑅 𝑡, 𝑡!, 𝑡" )
¨ For small 𝜏 𝑡, 𝑡!, 𝑡" , we have 𝜏 𝑡, 𝑡!, 𝑡" = 𝛼. (𝑡" − 𝑡!)
¨ The 𝛼 depends on the actual daycount fraction used to compute 𝜏 𝑡, 𝑡!, 𝑡"
¨ 𝜏 𝑡, 𝑡!, 𝑡" = 𝛼. 𝑡" − 𝑡! = 𝛼. 𝛿𝑡
¨ 𝑡" = 𝑡! +
) &,&",&!
R
= 𝑡! + 𝛿𝑡
¨ 𝐼𝐹𝑤𝑅 𝑡, 𝑡! = lim
5&→H
(𝑅 𝑡, 𝑡!, 𝑡! + 𝛿𝑡 )
95

96. Luc_Faucheux_2020
Instantaneous rates – XVI
¨ 𝐼𝐹𝑤𝑅 𝑡, 𝑡! = 𝐼𝐹𝑤𝑅 𝑡, 𝑡!, 𝑡!$ = lim
5&→H
(𝑅 𝑡, 𝑡!, 𝑡! + 𝛿𝑡 )
¨ 𝑅 𝑡, 𝑡!, 𝑡" = −
#
) &,&",&!
. {ln𝑍𝐶 𝑡, 𝑡, 𝑡" − ln𝑍𝐶 𝑡, 𝑡, 𝑡! }
¨ 𝜏 𝑡, 𝑡!, 𝑡" = 𝛼. 𝑡" − 𝑡! = 𝛼. 𝛿𝑡
¨ 𝐼𝐹𝑤𝑅 𝑡, 𝑡! = 𝐼𝐹𝑤𝑑𝑅 𝑡, 𝑡!, 𝑡!$ = lim
5&→H
(𝑅 𝑡, 𝑡!, 𝑡! + 𝛿𝑡 )
¨ 𝐼𝐹𝑤𝑅 𝑡, 𝑡! = lim
5&→H
(−
#
) &,&",&!
. {ln𝑍𝐶 𝑡, 𝑡, 𝑡" − ln𝑍𝐶 𝑡, 𝑡, 𝑡! } )
¨ 𝐼𝐹𝑤𝑅 𝑡, 𝑡! = lim
5&→H
(−
#
R.5&
. {ln𝑍𝐶 𝑡, 𝑡, 𝑡! + 𝛿𝑡 − ln𝑍𝐶 𝑡, 𝑡, 𝑡! } )
¨ Using Taylor expansion:
¨ ln𝑍𝐶 𝑡, 𝑡, 𝑡! + 𝛿𝑡 = ln𝑍𝐶 𝑡, 𝑡, 𝑡! + 𝛿𝑡.
SIJ*+ &,&,&"
S&"
+ 𝒪(𝛿𝑡P)
96

97. Luc_Faucheux_2020
Instantaneous rates – XVII
¨ ln𝑍𝐶 𝑡, 𝑡, 𝑡! + 𝛿𝑡 = ln𝑍𝐶 𝑡, 𝑡, 𝑡! + 𝛿𝑡.
SIJ*+ &,&,&"
S&"
+ 𝒪(𝛿𝑡P)
¨ ln𝑍𝐶 𝑡, 𝑡, 𝑡! + 𝛿𝑡 = ln𝑍𝐶 𝑡, 𝑡, 𝑡! + 𝛿𝑡.
#
*+ &,&,&"
.
S*+ &,&,&"
S&"
+ 𝒪(𝛿𝑡P)
¨ Because:
¨
SIJ(D(A)
SA
=
#
A
SIJ(D(A)
SA
¨ 𝐼𝐹𝑤𝑅 𝑡, 𝑡! = 𝐼𝐹𝑤𝑑𝑅 𝑡, 𝑡!, 𝑡!$ = lim
5&→H
(𝑅 𝑡, 𝑡!, 𝑡! + 𝛿𝑡 )
¨ 𝐼𝐹𝑤𝑅 𝑡, 𝑡! = lim
5&→H
(−
#
R.5&
. {ln𝑍𝐶 𝑡, 𝑡, 𝑡! + 𝛿𝑡 − ln𝑍𝐶 𝑡, 𝑡, 𝑡! } )
¨ 𝐼𝐹𝑤𝑅 𝑡, 𝑡! =
#
R
. (−
#
*+ &,&,&"
)
S*+ &,&,&"
S&"
)
97

98. Luc_Faucheux_2020
Instantaneous rates
Comparison of our notation with Kwok,
Mercurio, Hull, Piterbarg
98

99. Luc_Faucheux_2020
Instantaneous rates – XVIII
¨ 𝐼𝐹𝑤𝑅 𝑡, 𝑡! = −
#
R
.
#
*+ &,&,&"
.
S*+ &,&,&"
S&"
= −
#
R
.
SIJ(*+ &,&,&" )
S&"
¨ If you read Mercurio, that formula shows up page 12 as:
¨ lim
T→K$
𝐹 𝑡, 𝑇, 𝑆 = 𝑓 𝑡, 𝑇 = −
S IJ U &,K
SK
= −
#
U &,K
.
SU &,K
SK
¨ “where we use our convention that 𝜏 𝑡, 𝑡!, 𝑡" = 𝑡" − 𝑡!”, so using (𝛼 = 1).
¨ Not super obvious that you can set (𝛼 = 1), it really depends what daycount fraction you
use, and what are the units you use for time (do you measure time in days, years,..?)
¨ In that notation 𝑆 is the end of the period (also not super intuitive, usually 𝑆 stands for Start)
¨ 𝐹 𝑡, 𝑇, 𝑆 = 𝐿 𝑡, 𝑇, 𝑆
¨ 𝑓 𝑡, 𝑇 is what Mercurio calls the Instantaneous forward interest rate
¨ 𝑓 𝑡, 𝑇 = 𝐼𝐹𝑤𝑅 𝑡, 𝑇
¨ 𝑟 𝑡 = 𝑓 𝑡, 𝑡 = 𝐼𝑆ℎ𝑅 𝑡
99

100. Luc_Faucheux_2020
Instantaneous rates – XVIII - a
100

101. Luc_Faucheux_2020
Instantaneous rates – XIX
¨ 𝐼𝐹𝑤𝑅 𝑡, 𝑡! = −
#
R
.
#
*+ &,&,&"
.
S*+ &,&,&"
S&"
= −
#
R
.
SIJ(*+ &,&,&" )
S&"
¨ If you read Kwok, that formula shows up page 388 as:
¨ lim
∆K→H
𝑓 𝑡, 𝑇, 𝑇 + ∆𝑇 = 𝐹 𝑡, 𝑇 = −
S IJ L &,K
SK
= −
#
L &,K
.
SL &,K
SK
¨ So there again assuming that 𝜏 𝑡, 𝑡!, 𝑡" = 𝑡" − 𝑡!, so using (𝛼 = 1).
¨ AGAIN, Not super obvious that you can set (𝛼 = 1), it really depends what daycount fraction
you use, and what are the units you use for time (do you measure time in days, years,..?)
¨ EQUATIONS ARE ALWAYS GREAT UNTIL YOU NEED TO PUT SOME UNITS ON THEM
¨ In that notation (𝑇 + ∆𝑇) is the end of the period (more intuitive than Mercurio)
¨ 𝑓 𝑡, 𝑇, 𝑆 = 𝐿 𝑡, 𝑇, 𝑆
¨ 𝐹 𝑡, 𝑇 = 𝐼𝐹𝑤𝑅 𝑡, 𝑇
¨ 𝑟 𝑡 = 𝐹 𝑡, 𝑡 = 𝐼𝑆ℎ𝑅 𝑡
¨ So upper case and lower case notations are reversed between Kwok and Mercurio…arghh…
101

102. Luc_Faucheux_2020
Instantaneous rates – XIX-a
102

103. Luc_Faucheux_2020
Instantaneous rates – XX
¨ 𝐼𝐹𝑤𝑅 𝑡, 𝑡! = −
#
R
.
#
*+ &,&,&"
.
S*+ &,&,&"
S&"
= −
#
R
.
SIJ(*+ &,&,&" )
S&"
¨ If you read Hull, that formula shows up page 398 (2nd edition) as:
¨ lim
K2→K(
𝑓 𝑡, 𝑇#, 𝑇P = 𝐹 𝑡, 𝑇# = −
S IJ U &,K(
SK(
= −
#
U &,K(
.
SU &,K(
SK(
¨ So there again assuming that 𝜏 𝑡, 𝑡!, 𝑡" = 𝑡" − 𝑡!, so using (𝛼 = 1).
¨ AGAIN, Not super obvious that you can set (𝛼 = 1), it really depends what daycount fraction
you use, and what are the units you use for time (do you measure time in days, years,..?)
¨ EQUATIONS ARE ALWAYS GREAT UNTIL YOU NEED TO PUT SOME UNITS ON THEM
¨ In that notation the period is [𝑇#, 𝑇P]
¨ 𝑓 𝑡, 𝑇#, 𝑇P = 𝐿 𝑡, 𝑇#, 𝑇P
¨ 𝐹 𝑡, 𝑇# = 𝐼𝐹𝑤𝑅 𝑡, 𝑇# and 𝑟 𝑡 = 𝐹 𝑡, 𝑡 = 𝐼𝑆ℎ𝑅 𝑡
¨ So close to Mercurio, I also like the fact that Hull use a different lower and upper case for
𝑡, 𝑇#, 𝑇P, we will see in the next couple of slides why that makes sense and is quite nice
103

104. Luc_Faucheux_2020
Instantaneous rates – XX-a
104

105. Luc_Faucheux_2020
Instantaneous rates – XX-b
105

106. Luc_Faucheux_2020
Instantaneous rates – XX-c
¨ It’s trivia time because I think that I am losing you, and we getting close to 100 slides.
¨ Do you know what the picture represents on the cover of the 2nd edition of the Hull book?
¨ Hint: we are in Chicago
106

107. Luc_Faucheux_2020
Instantaneous rates – XXI
¨ 𝐼𝐹𝑤𝑅 𝑡, 𝑡! = −
#
R
.
#
*+ &,&,&"
.
S*+ &,&,&"
S&"
= −
#
R
.
SIJ(*+ &,&,&" )
S&"
¨ If you read Piterbarg, that formula shows up page 169 (volume I) as:
¨ lim
)→H
𝐿 𝑡, 𝑇, 𝑇 + 𝜏 = 𝑓 𝑡, 𝑇 = −
S IJ U &,K
SK
= −
#
U &,K
.
SU &,K
SK
¨ So there again assuming that 𝜏 𝑡, 𝑡!, 𝑡" = 𝑡" − 𝑡!, so using (𝛼 = 1).
¨ AGAIN, Not super obvious that you can set (𝛼 = 1), it really depends what daycount fraction
you use, and what are the units you use for time (do you measure time in days, years,..?)
¨ EQUATIONS ARE ALWAYS GREAT UNTIL YOU NEED TO PUT SOME UNITS ON THEM
¨ In that notation the period is [𝑇, 𝑇 + 𝜏]
¨ 𝐿 𝑡, 𝑇, 𝑇 + 𝜏 = 𝐿 𝑡, 𝑇, 𝑇 + 𝜏
¨ 𝑓 𝑡, 𝑇 = 𝐼𝐹𝑤𝑅 𝑡, 𝑇 and 𝑟 𝑡 = 𝑓 𝑡, 𝑡 = 𝐼𝑆ℎ𝑅 𝑡
107

108. Luc_Faucheux_2020
Instantaneous rates – XXII
108

109. Luc_Faucheux_2020
Instantaneous rates – XXIII
¨ So again, one of the reason why Finance is so much more complicated than Physics I think, is
that you cannot find two textbooks with the same notation.
¨ They seem to use lower case and capital letters whenever they so please
¨ And do not get me started on Mercurio who sometimes uses 𝜏 as a daycount fraction, and
sometimes as a time variable (p.38).
¨ As Godel found out, if you have the right notation, that helps a lot. The formalism and the
right choice can be illuminating.
¨ So I have tried to slowly come up with a notation that is complete enough so that you do not
get trapped by Libor in arrears, but also tries to not be too overwhelming
¨ I found that it usually works for me, whenever I read some textbooks or research paper, I
usually spend some time “translating” the formulas back into what I know and have been
using, something that I did not use to do in Physics, and that translation exercise usually
tends to be in itself a worthy thing to do
109

110. Luc_Faucheux_2020
Instantaneous rates – XXIV
¨ “The right notation is 95% of the work”.
¨ “Die richtige Notation macht 95% der Arbeit aus” (*)
¨ Kurt Godel, also known for the following:
¨ Prov∗(x)=def∃y[PrfF(y,x)∧∀z<y(¬PrfF(z,neg(x)))],
¨ Or if you read the beautiful book by Nagel and Newman:
¨ ~(∃x) Dem (x, Sub(n,17,n))
¨ (*) Am quite certain that Godel actually never uttered that quote, I made it up, but I think it
would make for a great urban legend.
110

111. Luc_Faucheux_2020
Instantaneous rates – XXV
¨ Kurt Godel on the right with an unidentified German / Swiss peasant on the left.
111

112. Luc_Faucheux_2020
Instantaneous rates
Doing some integration
112

113. Luc_Faucheux_2020
Instantaneous rates – XXI
¨ 𝐼𝐹𝑤𝑅 𝑡, 𝑡! = −
#
R
.
#
*+ &,&,&"
.
S*+ &,&,&"
S&"
= −
#
R
.
SIJ(*+ &,&,&" )
S&"
¨ 𝐼𝑆ℎ𝑅 𝑡 = lim
&"→&
𝐼𝐹𝑤𝑅 𝑡, 𝑡!
¨ If we choose to express time in units of years, then (𝛼 = 1), which is the assumption (even
if they do not tell you) in most textbooks. That simplifies a little
¨ 𝐼𝐹𝑤𝑅 𝑡, 𝑡! = −
SIJ(*+ &,&,&" )
S&"
¨ ∫W6&
W6&"
𝐼𝐹𝑤𝑅 𝑡, 𝑢 . 𝑑𝑢 = −ln(𝑍𝐶 𝑡, 𝑡, 𝑡! )
¨ exp − ∫W6&
W6&"
𝐼𝐹𝑤𝑅 𝑡, 𝑢 . 𝑑𝑢 = 𝑍𝐶 𝑡, 𝑡, 𝑡!
113

114. Luc_Faucheux_2020
Instantaneous rates – XXII
¨ This is quite the famous formula:
¨ 𝑍𝐶 𝑡, 𝑡, 𝑡! = exp − ∫W6&
W6&"
𝐼𝐹𝑤𝑅 𝑡, 𝑢 . 𝑑𝑢
¨ In Mercurio notation, that reads:
¨ 𝑃 𝑡, 𝑡! = exp − ∫W6&
W6&"
𝑓 𝑡, 𝑢 . 𝑑𝑢
¨ This is useful because the HJM framework for example, uses the Instantaneous Forward
Rates 𝐼𝐹𝑤𝑅 𝑡, 𝑢 as the basis for the dynamics of the rate
¨ We can then express the quantities 𝑍𝐶 𝑡, 𝑡, 𝑡!
¨ We can then enforce the arbitrage free relationships
¨ 𝔼&"
*+
𝑉 𝑡!, $𝑍𝐶 𝑡, 𝑡!, 𝑡" , 𝑡!, 𝑡! |𝔉(𝑡) =
#
#$% &,&",&! .)
= 𝑧𝑐 𝑡, 𝑡!, 𝑡"
114

115. Luc_Faucheux_2020
Instantaneous rates – XXII
¨ WAIT A SECOND !!!
¨ You told us that we could not do differentiation and integration just like in regular calculus
when dealing with stochastic variables, that we had to use this complicated ITO calculus ?
¨ And the you go happy go lucky taking integrals and such ?
¨ Am happy that you are reacting, that means that the first couple hundred slides on
stochastic calculus were not in vain
¨ It is also a really good question.
¨ This is also why I like the Hull notation but did not take because I already have the indices
¨ Hull: 𝑓 𝑡, 𝑇#, 𝑇P = 𝐿 𝑡, 𝑇#, 𝑇P
¨ Hull: 𝐹 𝑡, 𝑇# = 𝐼𝐹𝑤𝑅 𝑡, 𝑇#
¨ Our notation: 𝐿 𝑡, 𝑡!, 𝑡"
115

116. Luc_Faucheux_2020
Instantaneous rates – XXIII
¨ The point to realize is that 𝑡! and 𝑡" are indices, denoting the period on the curve [𝑡!, 𝑡"]
¨ 𝐿 𝑡, 𝑡!, 𝑡" does NOT evolve in time with 𝑡! and 𝑡" in a random manner
¨ In some ways, think of 𝑡! and 𝑡" as being “fixed”, they denote on the curve a fixed portion
[𝑡!, 𝑡"]
¨ 𝐿 𝑡, 𝑡!, 𝑡" evolves in time in a random manner ONLY with the first variable 𝑡
¨ So even though 𝑡, 𝑡! and 𝑡" are all time variable, only 𝑡 is the real time (the one that passes
by).
¨ The other ones are just to indicate some portion of the curve
¨ As such, we are totally ok doing regular calculus and manipulate 𝑡! and 𝑡", perform integrals
and differentiate as we see fit (assuming some regularity for the yield curve and such
obviously). Those time variables are what Baxter refers to as the “Newtonian” ones.
¨ It is ONLY when dealing with 𝑡 that we will have to be careful and use ITO calculus (if we so
desire), that is the one that Baxter refers to as the “Brownian” one,
116

117. Luc_Faucheux_2020
Instantaneous rates – XXIV
¨ So worth taking a moment here and convincing ourselves which one of the time variable is
the “time that goes by” and for which we will have to use stochastic calculus rules, and
which one are just “indexing a curve” and we can perform regular calculus on those
¨ It is crucial because usual rules of calculus do NOT apply in the stochastic world
117
𝑂𝑏𝑠𝑒𝑟𝑣𝑒𝑑 𝑎𝑡 𝑡𝑖𝑚𝑒 𝑡,
STOCHASTIC CALCULUS RULES APPLY
“BROWNIAN”
𝑆𝑡𝑎𝑟𝑡 𝑜𝑓 𝑡ℎ𝑒 𝑝𝑒𝑟𝑖𝑜𝑑 𝑜𝑛 𝑡ℎ𝑒 𝑐𝑢𝑟𝑣𝑒,
REGULAR CALCULUS RULES APPLY
“NEWTONIAN”
𝑍𝐶 𝑡, 𝑡!, 𝑡"
𝐸𝑛𝑑 𝑜𝑓 𝑡ℎ𝑒 𝑝𝑒𝑟𝑖𝑜𝑑 𝑜𝑛 𝑡ℎ𝑒 𝑐𝑢𝑟𝑣𝑒,
REGULAR CALCULUS RULES APPLY
“NEWTONIAN”

118. Luc_Faucheux_2020
Instantaneous rates – XXV
¨ Some more intuition on Instantaneous Forward Rates
¨ 𝑍𝐶 𝑡, 𝑡, 𝑡! = exp − ∫W6&
W6&"
𝐼𝐹𝑤𝑅 𝑡, 𝑢 . 𝑑𝑢
¨ We also have by definition in the case of the continuously compounded rate
¨ 𝑍𝐶 𝑡, 𝑡!, 𝑡" = exp −𝑅 𝑡, 𝑡!, 𝑡" . 𝜏 𝑡, 𝑡!, 𝑡"
¨ In the case where (𝛼 = 1), which is equivalent of choosing to express the time in variable in
units of years (1 year = 1) and assuming what we could call an ACT/ACT daycount fraction,
𝜏 𝑡, 𝑡!, 𝑡" = 𝑡" − 𝑡!
¨ In particular: 𝑍𝐶 𝑡, 𝑡, 𝑡! = exp −𝑅 𝑡, 𝑡, 𝑡! . 𝜏 𝑡, 𝑡, 𝑡!
¨ 𝑅 𝑡, 𝑡, 𝑡! =
#
&"M&
∫W6&
W6&"
𝐼𝐹𝑤𝑅 𝑡, 𝑢 . 𝑑𝑢
118

119. Luc_Faucheux_2020
Instantaneous rates – XXV
¨ 𝑅 𝑡, 𝑡, 𝑡! =
#
&"M&
∫W6&
W6&"
𝐼𝐹𝑤𝑅 𝑡, 𝑢 . 𝑑𝑢
¨ The continuously compounded then-spot rate 𝑅 𝑡, 𝑡, 𝑡! is the time average of the
instantaneous forward rate 𝐼𝐹𝑤𝑅 𝑡, 𝑢 between 𝑢 = 𝑡 and 𝑢 = 𝑡!
¨ Equivalently:
¨ 𝑅 𝑡, 𝑡, 𝑡! . 𝑡! − 𝑡 = ∫W6&
W6&"
𝐼𝐹𝑤𝑅 𝑡, 𝑢 . 𝑑𝑢
¨ 𝐼𝐹𝑤𝑅 𝑡, 𝑡! =
S
S&"
𝑅 𝑡, 𝑡, 𝑡! . 𝑡! − 𝑡 = 𝑅 𝑡, 𝑡, 𝑡! + 𝑡! − 𝑡 .
S
S&"
𝑅 𝑡, 𝑡, 𝑡!
¨ 𝑍𝐶 𝑡, 𝑡, 𝑡! = exp −𝑅 𝑡, 𝑡, 𝑡! . 𝜏 𝑡, 𝑡, 𝑡! = exp[ −𝑅 𝑡, 𝑡, 𝑡! . 𝑡! − 𝑡 ]
¨ 𝑅 𝑡, 𝑡, 𝑡! = −
#
&"M&
. ln(𝑍𝐶 𝑡, 𝑡, 𝑡! )
119

120. Luc_Faucheux_2020
Instantaneous rates – XXVI
¨ 𝑍𝐶 𝑡, 𝑡, 𝑡! = exp −𝑅 𝑡, 𝑡, 𝑡! . 𝜏 𝑡, 𝑡, 𝑡! = exp[−𝑅 𝑡, 𝑡, 𝑡! . 𝑡! − 𝑡 ]
¨
S
S&"
. 𝑍𝐶 𝑡, 𝑡, 𝑡! = − exp −𝑅 𝑡, 𝑡, 𝑡! . 𝑡! − 𝑡
S
S&"
[𝑅 𝑡, 𝑡, 𝑡! . 𝑡! − 𝑡 ]
¨
S
S&"
. 𝑍𝐶 𝑡, 𝑡, 𝑡! = − exp −𝑅 𝑡, 𝑡, 𝑡! . 𝑡! − 𝑡 {𝑅 𝑡, 𝑡, 𝑡! + 𝑡! − 𝑡 .
S
S&"
𝑅 𝑡, 𝑡, 𝑡! }
¨
S
S&"
. 𝑍𝐶 𝑡, 𝑡, 𝑡! = − exp −𝑅 𝑡, 𝑡, 𝑡! . 𝑡! − 𝑡 . {𝐼𝐹𝑤𝑅 𝑡, 𝑡! }
¨ 𝑅 𝑡, 𝑡, 𝑡! = −
#
&"M&
. ln(𝑍𝐶 𝑡, 𝑡, 𝑡! )
¨ exp −𝑅 𝑡, 𝑡, 𝑡! . 𝑡! − 𝑡 = 𝑍𝐶 𝑡, 𝑡, 𝑡!
¨
#
*+ &,&,&"
S
S&"
. 𝑍𝐶 𝑡, 𝑡, 𝑡! = −𝐼𝐹𝑤𝑅 𝑡, 𝑡! =
S
S&"
. ln(𝑍𝐶 𝑡, 𝑡, 𝑡! )
¨ 𝐼𝐹𝑤𝑅 𝑡, 𝑡! =
S
S&"
𝑅 𝑡, 𝑡, 𝑡! . 𝑡! − 𝑡 = 𝑅 𝑡, 𝑡, 𝑡! + 𝑡! − 𝑡 .
S
S&"
𝑅 𝑡, 𝑡, 𝑡!
120

121. Luc_Faucheux_2020
Instantaneous rates – XXVII
¨ So we are quite happy and this is all consistent, and we have proven that we still can
manage simple regular calculus without getting lost in the notations
¨ In some textbooks you will see the following statements on the shape of the curves
¨ 𝐼𝐹𝑤𝑅 𝑡, 𝑡! =
S
S&"
𝑅 𝑡, 𝑡, 𝑡! . 𝑡! − 𝑡 = 𝑅 𝑡, 𝑡, 𝑡! + 𝑡! − 𝑡 .
S
S&"
𝑅 𝑡, 𝑡, 𝑡!
¨ The forward curve is the plot of 𝐼𝐹𝑤𝑅 𝑡, 𝑡! against 𝑡! at a given time 𝑡
¨ The yield curve is the plot of 𝑅 𝑡, 𝑡, 𝑡! against 𝑡! at a given time 𝑡
¨ The dependence of the yield curve on the variable 𝑡! − 𝑡 is called TERM STRUCTURE
¨ The yield curve is upward sloping (increasing) if
S
S&"
𝑅 𝑡, 𝑡, 𝑡! > 0
¨ The yield curve is downward sloping (decreasing) if
S
S&"
𝑅 𝑡, 𝑡, 𝑡! < 0
¨ Note that also the assumption in most textbooks is that rates are positive
121

122. Luc_Faucheux_2020
Instantaneous rates – XXVIII
¨ 𝐼𝐹𝑤𝑅 𝑡, 𝑡! =
S
S&"
𝑅 𝑡, 𝑡, 𝑡! . 𝑡! − 𝑡 = 𝑅 𝑡, 𝑡, 𝑡! + 𝑡! − 𝑡 .
S
S&"
𝑅 𝑡, 𝑡, 𝑡!
¨ The forward curve will be above the yield curve if the yield curve is upward sloping
¨ The forward curve will be below the yield curve if the yield curve is downward sloping
¨ Those are the generally accepted terms.
¨ Note that really to be exact,
¨ yield curve = then spot continuously compounded spot rate
¨ Forward curve = plot of the instantaneous forward 𝐼𝐹𝑤𝑅 𝑡, 𝑡! against 𝑡!
¨ 𝐼𝐹𝑤𝑅 𝑡, 𝑡! = lim
&!→&"
(
#M*+ &,&",&!
) &,&",&!
) = lim
&!→&"
(𝑅 𝑡, 𝑡!, 𝑡" )
¨ WE DO NOT HAVE FOR EXAMPLE: 𝐼𝐹𝑤𝑅 𝑡, 𝑡! =
S
S&"
𝑅 𝑡, 𝑡, 𝑡!
¨ So the forward is NOT the first derivative of the spot, using that terminology
122

123. Luc_Faucheux_2020
Instantaneous rates – XXIX
¨ Again the point above might be subtle or completely obvious, but you need to pay attention
to exact notation there.
¨ 𝐼𝐹𝑤𝑅 𝑡, 𝑡! = lim
&!→&"
(𝑅 𝑡, 𝑡!, 𝑡" )
¨ 𝐼𝐹𝑤𝑅 𝑡, 𝑡! =
S
S&"
𝑅 𝑡, 𝑡, 𝑡! . 𝑡! − 𝑡
¨ 𝑅 𝑡, 𝑡!, 𝑡" = −
#
) &,&",&!
. ln𝑍𝐶 𝑡, 𝑡, 𝑡" − ln𝑍𝐶 𝑡, 𝑡, 𝑡! = −
#
) &,&",&!
. ln(
*+ &,&,&!
*+ &,&,&"
)
¨ 𝑅 𝑡, 𝑡, 𝑡" = −
#
) &,&,&!
. ln
*+ &,&,&!
*+ &,&,&
= −
#
) &,&,&!
. ln(𝑍𝐶 𝑡, 𝑡, 𝑡" )
¨ And in most textbook we assume 𝜏 𝑡, 𝑡!, 𝑡" = 𝑡" − 𝑡!
123

124. Luc_Faucheux_2020
Instantaneous rates – XXX
¨ All we can say is that:
¨ 𝐼𝐹𝑤𝑅 𝑡, 𝑡! =
S
S&"
𝑅 𝑡, 𝑡, 𝑡! . 𝑡! − 𝑡
¨ ONLY when applied to 𝑅 𝑡, 𝑡, 𝑡! , the then-continuously compounded spot rate (and with the
convention that 𝛼 = 1, so time is expressed in units of years and the daycount is ACT/ACT
¨ In all other cases,
¨ 𝐼𝐹𝑤𝑅 𝑡, 𝑡! = lim
&!→&"
(𝑅 𝑡, 𝑡!, 𝑡" ) = lim
&!→&"
(𝐿 𝑡, 𝑡!, 𝑡" ) = lim
&!→&"
(𝑌 𝑡, 𝑡!, 𝑡" ) = lim
&!→&"
(𝑌O 𝑡, 𝑡!, 𝑡" )
124

125. Luc_Faucheux_2020
After
Instantaneous Forward Rates
Now
Instantaneous Short Rates
125

126. Luc_Faucheux_2020
Instantaneous short rate
¨ 𝐼𝐹𝑤𝑅 𝑡, 𝑡!, 𝑡" = 𝐼𝐹𝑤𝑅 𝑡, 𝑡!, 𝑡!$ = 𝐼𝐹𝑤𝑅 𝑡, 𝑡! = lim
&!→&"
(
#M*+ &,&",&!
) &,&",&!
)
¨ 𝐼𝑆ℎ𝑅 𝑡, 𝑡!, 𝑡" = 𝐼𝑆ℎ𝑅 𝑡, 𝑡+, 𝑡 + = 𝐼𝑆ℎ𝑅 𝑡 = lim
&!→&",&!→&
(
#M*+ &,&",&!
) &,&",&!
)
¨ 𝐼𝑆ℎ𝑅 𝑡 = lim
&"→&
𝐼𝐹𝑤𝑅 𝑡, 𝑡!
¨ For now let’s keep those notations for a while
¨ Note also that
¨ 𝐼𝑆ℎ𝑅 𝑡 = lim
&!→&
(
#M*+ &,&,&!
) &,&,&!
)
¨ 𝐼𝐹𝑤𝑅 𝑡, 𝑡! = lim
&!→&"
(𝑅 𝑡, 𝑡!, 𝑡" ) = lim
&!→&"
(𝐿 𝑡, 𝑡!, 𝑡" ) = lim
&!→&"
(𝑌 𝑡, 𝑡!, 𝑡" ) = lim
&!→&"
(𝑌O 𝑡, 𝑡!, 𝑡" )
126

127. Luc_Faucheux_2020
Instantaneous short rate - I
¨ So in the textbooks you will see those limits expressed as:
¨ 𝐼𝐹𝑤𝑅 𝑡, 𝑡! = lim
&!→&"
(𝑅 𝑡, 𝑡!, 𝑡" ) = lim
&!→&"
(𝐿 𝑡, 𝑡!, 𝑡" ) = lim
&!→&"
(𝑌 𝑡, 𝑡!, 𝑡" ) = lim
&!→&"
(𝑌O 𝑡, 𝑡!, 𝑡" )
¨ 𝐼𝐹𝑤𝑅 𝑡, 𝑡! = 𝑅 𝑡, 𝑡!, 𝑡! = 𝐿 𝑡, 𝑡!, 𝑡! = 𝑌 𝑡, 𝑡!, 𝑡! = 𝑌O 𝑡, 𝑡!, 𝑡!
¨ 𝐼𝑆ℎ𝑅 𝑡 = lim
&"→&
𝐼𝐹𝑤𝑅 𝑡, 𝑡!
¨ 𝐼𝑆ℎ𝑅 𝑡 = 𝐼𝐹𝑤𝑅 𝑡, 𝑡 = 𝑅 𝑡, 𝑡, 𝑡 = 𝐿 𝑡, 𝑡, 𝑡 = 𝑌 𝑡, 𝑡, 𝑡 = 𝑌O 𝑡, 𝑡, 𝑡
¨ Terminology is usually “short rate 𝑟(𝑡)”
¨ “Short term risk free interest rate at time 𝑡”
¨ “Instantaneous risk-free rate at time 𝑡”
127

128. Luc_Faucheux_2020
Instantaneous short rate - II
¨ Why do we care?
¨ Turns out most of the models pre-2000 and pre-HJM and pre-stochastic vol, were actually
models on the “short-rate”.
¨ Hence why they are usually called “SHORT RATE MODEL”
¨ Usually the first few chapter of the texbooks on rates modeling
¨ This is in some way the simplest case, and the most we can reduce the problem of modeling
the dynamics
¨ Going from 𝐿 𝑡, 𝑡!, 𝑡" to only one variable in time 𝐼𝑆ℎ𝑅 𝑡
128

129. Luc_Faucheux_2020
Instantaneous short rate - III
¨ If there is only one stochastic driver, meaning you writing something like this
¨ 𝐼𝑆ℎ𝑅 𝑡 + 𝛿𝑡 − 𝐼𝑆ℎ𝑅 𝑡 = ∫&
&$5&
𝐴 𝑠 . 𝑑𝑠 + ∫&
&$5&
𝐵 𝑠 . ([). 𝑑𝑊 𝑠
¨ Where 𝐴 𝑠 and 𝐵 𝑠 could be function of the rate, so
¨ 𝐴 𝑠 = 𝐴 𝑠, 𝐼𝑆ℎ𝑅 𝑠
¨ 𝐵 𝑠 = 𝐵 𝑠, 𝐼𝑆ℎ𝑅 𝑠
¨ This would be called a “ONE FACTOR SHORT RATE MODEL”
¨ Instead of the many [𝑡!, 𝑡"] indexed set of equations on the 𝐿 𝑡, 𝑡!, 𝑡"
¨ 𝐿 𝑡 + 𝛿𝑡, 𝑡!, 𝑡" − 𝐿 𝑡, 𝑡!, 𝑡" = ∫&
&$5&
𝐴 𝑠, 𝑡!, 𝑡" . 𝑑𝑠 + ∫&
&$5&
𝐵 𝑠, 𝑡!, 𝑡" . ([). 𝑑𝑊 𝑠, 𝑡!, 𝑡"
¨ Note that above as I wrote it is still one factor for each [𝑡!, 𝑡"]
129

130. Luc_Faucheux_2020
Instantaneous Rates
and
Great Expectations
130

131. Luc_Faucheux_2020
Instantaneous rates and expectations
¨ Remember what we had under the terminal measure (forward measure).
¨ 𝔼&!
*+ 𝑉 𝑡", $𝐿 𝑡, 𝑡!, 𝑡" , 𝑡!, 𝑡" |𝔉(𝑡) = 𝑙 𝑡, 𝑡!, 𝑡"
¨ “Any simply compounded forward rate spanning a time interval ending in 𝑡" is a martingale
under the 𝑡"-forward measure also called 𝑡"-terminal measure, associated with the Zero
coupon numeraire 𝑍𝐶 𝑡, 𝑡, 𝑡" ” (roughly speaking Mercurio p34).
¨ We have of course:
¨ 𝑖𝑓𝑤𝑟 𝑡, 𝑡! = lim
&!→&"
(𝑙 𝑡, 𝑡!, 𝑡" ) = 𝑙 𝑡, 𝑡!, 𝑡!
¨ Those are not the random variables, those are the values “fixed” on time 𝑡 curve.
¨ 𝐼𝐹𝑤𝑅 𝑡, 𝑡! = lim
&!→&"
(𝐿 𝑡, 𝑡!, 𝑡" ) = 𝐿 𝑡, 𝑡!, 𝑡!
¨ This applies to the random variables still evolving in time 𝑡 before “dying” as Mercurio would
say or fixing at time 𝑡!
131