deck on FX

1. Luc_Faucheux_2021
THE RATES WORLD – Part VIII
Let’s have some fun with the FX market
1

2. Luc_Faucheux_2021
That deck
2
¨ We now look at some concepts in Foreign Exchange markets (FX)
¨ This is at the same time surprisingly easy and surprisingly difficult
¨ When teaching my students at Fairfield University, the FX chapters are usually the ones that
surprisingly offer the most challenges
¨ Everyone has travelled or has seen movies about exchanging one currency for another
¨ Yet once you start introducing volatility for example you get things like the Siegel paradox, or
two-currency paradox
¨ Once you start introducing the issue of the funding (in which currency is your PL actually
counted and matters), you get people confused on the what is the PL of a simple FX trade
for example
¨ You can also get confused with the quanto effect when pricing some derivatives
¨ Oh also I just got let go of my current job at Natixis, so that will give me some more time to
work on those presentations. But if you know of any job for me, am interested.

3. Luc_Faucheux_2021
That deck - II
¨ In a way we needed all the tools of the Rates world (Bank Account numeraire, deflated
Zeros, change of measure, IOT Leibniz,..) to start really dealing with more than one currency
¨ Also, as Godel rightly pointed out, notations sometimes can be a pain in the neck
¨ (he never said that, but I would like to start that rumor)
¨ In any case, a lot of FX textbooks have example where they use DEM or USD or GBP or YEN,
showing how old they are, and it does not help with the confusion
¨ So I will try to be extra careful on the notation
¨ In particular, even good textbooks like Hull on the Siegel paradox, essentially use the same
notation for two different processes, essentially negating the whole point that they were
trying to make
¨ So, to quote Kurt Godel, “careful on the notation, the notation is 95% of the work”
¨ (again, he did not say that, but starting that rumor is my pet peeve)
3

4. Luc_Faucheux_2021
That deck - III
¨ Yours truly teaching FX this semester at the Fairfield University Dolan School of Business
4

5. Luc_Faucheux_2021
FX notations
5

6. Luc_Faucheux_2021
FX notations
¨ We are going to try to be a little rigorous here.
¨ Instead of using USD, DEM, EUR, JPY,…we are going to index the currencies by integers
¨ So we have currencies 𝑖, 𝑗, 𝑘 …
¨ We have to come up with a notation for the SPOT Foreign Exchange
¨ Usual market convention is 𝑋
¨ Again, am not sure I like that as usually this is reserved for a variable, but I do not want to
have 2 letters and use 𝐹𝑋, so we will stick with 𝑋 for now
¨ NOW, we will choose for convention:
¨ 𝑋!,# = 𝑋!←# is the value in currency (𝑖) of 1 unit of currency (𝑗)
¨ 𝑋!,# = 𝑋!←# translates into currency (𝑖) from currency (𝑗) the value of whatever you
multiply it by (cup of coffee, BigMac, barrel of oil, anything)
6

7. Luc_Faucheux_2021
FX notations - II
¨ 𝑋!,# = 𝑋!←# translates into currency (𝑖) from currency (𝑗) the value of whatever you
multiply it by (cup of coffee, BigMac, barrel of oil, anything)
¨ So say a cup of coffee in London is 4 GBP (four quids for a cup of joe mate ?!)
¨ And then the exchange rate is 1.3841
¨ Which exchange rate ?
¨ Well that is usually where a lot of the confusion occurs.
¨ People do not like numbers smaller than 1, so historically, they have decided to quote the
exchange rate for that currency pair in terms of whatever order gives the higher number
¨ For (YEN,USD) pair, it is around 108, and it is the number of the YEN that you get for 1 USD
¨ For (GBP,USD) pair, is it around 1.4, and it is the number of USD that you get for 1 GBP
¨ For (USD,CAD) pair, it is around 1.25, and it is the number of CAD that you get for 1 USD
7

8. Luc_Faucheux_2021
FX notations - III
¨ The FX “cross”, or FXC in Bloomberg
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9. Luc_Faucheux_2021
FX notations - IV
¨ When I was working at FujiCap, I went for a week in London and I was sitting right next to
the FX desk there, and I guess “An American Werewolf in London” was still a big thing at the
time.
¨ For some reason the guys on the FX desk kept yelling all day long
¨ “Watch the cross”
¨ “Watch the moon”
¨ “Stick to the roads”
¨ “Keep off the moors”
¨ Fun times….
9

10. Luc_Faucheux_2021
FX notations - IV
¨ The usual market convention for the (USD,EUR) currency pair exchange rate
¨ People like to deal with numbers greater than 1
10

11. Luc_Faucheux_2021
FX notations - V
¨ You can also get the inverse if you want, and start wondering what happens to a stochastic
process when you take the inverse of it (hint: convexity!)
11

12. Luc_Faucheux_2021
FX notations - VI
¨ In Bloomberg, the convention is that:
¨ EURUSD is the value in USD of 1 unit of EUR
¨ USDEUR is the value in EUR of 1 unit of USD
¨ GBPUSD is the value in USD of 1 unit of GBP
¨ Back to that 4 quids cup of mocha
¨ 1 cup = 4 GBP = GBP 4 = £4
¨ 1 cup = (4GBP)*(GBPUSD)=(4)*(1.3841)=5.5364USD=USD 5.5364 = $5.5364
¨ So for a lad living in London (William), 1 cup = £4
¨ For a Yankee bloke living in NYC (Mike), that same cup is worth $5.5364
¨ NOW of course, this is somewhat artificial, because there is no way that Mike could access
that same exact cup like William can, but it is nonetheless a true statement
12

13. Luc_Faucheux_2021
FX notations - VI
¨ Going from the Bloomberg notation to ours:
¨ (GBPUSD) = 𝑋%&',()* = 𝑋%&'←()*
¨ GBPUSD is the value in USD of 1 unit of GBP
¨ GBPUSD is the value of 1 unit of GBP in USD
¨ {coffee_cup}GPB is the value of 1 coffee cup in GBP
¨ {coffee_cup}USD is the value of 1 coffee cup in USD
¨ coffee_cup USD = coffee_cup GBP. (GBPUSD)
¨
+,--.._+01
%&'
=
+,--..!"#
()*
. GBPUSD =
+,--.._+01
()*
.
()*
%&'
¨ coffee_cup USD = 𝑋%&'←()*. coffee_cup GBP
¨ coffee_cup USD = 𝑋%&',()*. coffee_cup GBP
13

14. Luc_Faucheux_2021
FX notations - VII
¨ 𝑋!←# =
2
3$←&
¨ 𝑋!,# =
2
3$,&
¨ 𝑋!←# = 𝑋!←4 ∗ 𝑋4←#
¨ 𝑋!,# = 𝑋!,4 ∗ 𝑋4,#
¨ (GBPUSD)=(GBPEUR)*(EURUSD)
14

15. Luc_Faucheux_2021
FX notations - VIII
¨ GBPUSD=1.3841
¨ GBPEUR=1.1617
¨ EURUSD=1.1915
¨ (GBPUSD)=(GBPEUR)*(EURUSD)=1.1617*1.1915=1.3841
15

16. Luc_Faucheux_2021
FX notations - IX
¨ So the usual way it is quoted in the market is artificial just to make it easier
¨ Remember it was the same for Eurodollar future, the quote price is not the market price
¨ Let’s write Q(USD,GBP)=Q(BGP,USD) the usual way the exchange rate for the currency pair
(GBP,USD) is quoted
¨ Q(GBP,USD) = Q(USD,GBP) = GBPUSD = 𝑋%&',()* = 𝑋%&'←()*
¨ Q(USD,YEN) = Q(YEN,USD) = USDYEN = 𝑋567,%&' = 𝑋567←%&'
¨ Q(CAD,USD) = Q(USD,CAD) = USDCAD= 𝑋89',%&' = 𝑋89'←%&'
¨ Q(USD,EUR) = Q(EUR,USD) = EURUSD = 𝑋%&',6%: = 𝑋%&'←6%:
16

17. Luc_Faucheux_2021
FX notations - X
¨ Just to make it a little more lively than 𝑖, 𝑗, 𝑘 and also because we already use those a lot for
the Zero Coupon Bonds, we will go with 1,2,3 most of the time
¨ In the currency 1
¨ We have the usual Bank Account numeraire associated to the Risk free measure
¨ 𝑊ℚ2 𝑡 is the Brownian motion associated to the risk-free measure {ℚ1} which is
associated with the rolling numeraire 𝐵2 𝑡 = exp[∫
<=>
<=?
𝑅2 𝑠, 𝑠, 𝑠 . 𝑑𝑠]
¨ We have in the currency 1 the Zero coupon bonds: 𝑍2 𝑡, 𝑡!, 𝑡#
¨ We will also have in the currency 1 assets (stocks for example, or cups of coffee),
DENOMINATED and tradeable in the currency 1, and we will note those 𝐴2 𝑡
17

18. Luc_Faucheux_2021
FX notations - XI
¨ As you might have guessed, once we have a process for the quantities 𝐴@ 𝑡 and 𝑍@ 𝑡, 𝑡!, 𝑡#
in the (currency 2 world), we will crank the ITO Leibniz handle quite a lot in order to come
up with something useful to say for the processes:
¨ [𝑋2←@. 𝐴@ 𝑡 ]
¨ And:
¨ [𝑋2←@. 𝑍@ 𝑡, 𝑡!, 𝑡# ]
¨ And yes, you guessed it right once again, the ever important deflated quantities:
¨
[3(←).C) ?,?&,?$ ]
E( ?
and
[3(←).F) ? ]
E( ?
¨ As you can also guess, we are going to say that some of those quantities are going to be
martingale (driftless process) under the appropriate measure, and then crank back up the
Ito Leibniz handle back up to the original processes for 𝐴@ 𝑡 and 𝑍@ 𝑡, 𝑡!, 𝑡#
¨ But first, some cool geometric tricks on correlation and volatilities
18

19. Luc_Faucheux_2021
The FX correlation triangle
19

20. Luc_Faucheux_2021
The FX correlation triangle
¨ Essentially it is:
¨ 𝑋!,# = 𝑋!,4 ∗ 𝑋4,#
¨ 𝑋2,G = 𝑋2,@ ∗ 𝑋@,G
¨ 𝑋2,@ = 𝑋2,G/𝑋@,G
¨ Suppose that I know something about the stochastic process for 𝑋2,G and 𝑋@,G
¨ Is there something useful that I can say about the stochastic process for 𝑋2,@ ?
¨ In particular about the variance of that process (leaving out the complicated issue of the
drift aside)
20

21. Luc_Faucheux_2021
The FX correlation triangle - II
¨ First some results on 𝑋 and (
2
3
)
¨ Let’s pick some 𝑋 = 𝑋!,# and
2
3
=
2
3&,$
= 𝑋#,!
¨ The usual market convention for FX is to work with GBM (geometric Brownian Motion).
¨ Again there is an assumption that Spot FX cannot become negative
¨ This assumption sounds a little more grounded that the one in rates (which turned out to be
wrong, you can have negative rates). It is a little harder to imagine a negative spot FX
¨ So usually the process is the following:
¨
H3&,$
3&,$
= 𝜇 𝑋!,# . 𝑑𝑡 + 𝜎 𝑋!,# . [ . 𝑑𝑊3&,$(𝑡)
¨
H3
3
= 𝜇 𝑋 . 𝑑𝑡 + 𝜎 𝑋 . [ . 𝑑𝑊3(𝑡)
21

22. Luc_Faucheux_2021
The FX correlation triangle - III
¨
H3&,$
3&,$
= 𝜇 𝑋!,# . 𝑑𝑡 + 𝜎 𝑋!,# . [ . 𝑑𝑊3&,$(𝑡)
¨
H3&←$
3&←$
= 𝜇 𝑋!←# . 𝑑𝑡 + 𝜎 𝑋!←# . [ . 𝑑𝑊3&←$(𝑡)
¨
H3&,$
3&,$
= 𝜇 𝑋!,# . 𝑑𝑡 + 𝜎 𝑋!,# . [ . 𝑑𝑊3&,$(𝑡)
¨
H3$,&
3$,&
= 𝜇 𝑋#,! . 𝑑𝑡 + 𝜎 𝑋#,! . [ . 𝑑𝑊3$,&(𝑡)
¨ In that case we can use the very useful relation that is :
¨ 𝜎 𝑋!,# = 𝜎 𝑋#,!
¨ 𝜎 𝑋!,# = 𝜎 1/𝑋!,# = 𝜎
2
3&,$
22

23. Luc_Faucheux_2021
ITO lemma on (1/X)
23

24. Luc_Faucheux_2021
The FX correlation triangle - IV
¨ We can use our good old friend the ITO lemma on:
¨ 𝑓 𝑋 = 1/𝑋 Stochastic Variable
¨ 𝑓 𝑥 = 1/𝑥 Regular “Newtonian” variable with well defined partial derivatives
¨ 𝛿𝑓 =
IJ
IK
. 𝛿𝑋 +
2
@
.
I)J
IK) . (𝛿𝑋)@
¨
IJ
IK
=
IJ
IK
|K=3 ?
¨
I)J
IK) =
I)J
IK) |K=3 ?
24

25. Luc_Faucheux_2021
The FX correlation triangle - V
¨ 𝑓 𝑥 = 1/𝑥
¨
IJ
IK
=
L2
K)
¨
I)J
IK) =
@
K*
¨ 𝛿𝑓 =
IJ
IK
. 𝛿𝑋 +
2
@
.
I)J
IK) . (𝛿𝑋)@
¨ 𝛿𝑓 =
IJ
IK
|K=3 ? . 𝛿𝑋 +
2
@
.
I)J
IK) |K=3 ? . (𝛿𝑋)@
¨ 𝛿(
2
3
) =
L2
3) . 𝛿𝑋 +
2
@
.
@
3* . (𝛿𝑋)@
¨ 𝑑(
2
3
) =
L2
3) . 𝑑𝑋 +
2
@
.
@
3* . (𝑑𝑋)@
25

26. Luc_Faucheux_2021
The FX correlation triangle - VI
¨ 𝑑(
2
3
) =
L2
3) . 𝑑𝑋 +
2
@
.
@
3* . (𝑑𝑋)@
¨
H3$,&
3$,&
= 𝜇 𝑋#,! . 𝑑𝑡 + 𝜎 𝑋#,! . [ . 𝑑𝑊3$,&(𝑡)
¨
H3
3
= 𝜇 𝑋 . 𝑑𝑡 + 𝜎 𝑋 . [ . 𝑑𝑊3(𝑡)
¨ 𝑑𝑋 = 𝜇 𝑋 . 𝑋. 𝑑𝑡 + 𝜎 𝑋 . 𝑋. [ . 𝑑𝑊3(𝑡)
¨ 𝑑𝑋(𝑡) = 𝜇 𝑋 𝑡 . 𝑋(𝑡). 𝑑𝑡 + 𝜎 𝑋 𝑡 . 𝑋(𝑡). [ . 𝑑𝑊3(𝑡)
¨ 𝑑𝑋 = 𝜇 𝑋 . 𝑋. 𝑑𝑡 + 𝜎 𝑋 . 𝑋. [ . 𝑑𝑊3(𝑡)
¨ (𝑑𝑋)@= (𝜎 𝑋 . 𝑋)@. 𝑑𝑡 from the quadradic variation property of the Brownian motion
26

27. Luc_Faucheux_2021
The FX correlation triangle - VII
¨ 𝑑(
2
3
) =
L2
3) . 𝑑𝑋 +
2
@
.
@
3* . (𝑑𝑋)@
¨ 𝑑𝑋 = 𝜇 𝑋 . 𝑋. 𝑑𝑡 + 𝜎 𝑋 . 𝑋. [ . 𝑑𝑊3(𝑡)
¨ (𝑑𝑋)@= (𝜎 𝑋 . 𝑋)@. 𝑑𝑡
¨ 𝑑(
2
3
) =
L2
3) . (𝜇 𝑋 . 𝑋. 𝑑𝑡 + 𝜎 𝑋 . 𝑋. [ . 𝑑𝑊3(𝑡) ) +
2
@
.
@
3* . (𝜎 𝑋 . 𝑋)@. 𝑑𝑡
¨ 𝑑
2
3
= − 𝜇 𝑋 − 𝜎 𝑋 @ .
2
3
. 𝑑𝑡 − 𝜎 𝑋 . (
2
3
). [ . 𝑑𝑊3(𝑡)
¨
H
(
+
(
+
= − 𝜇 𝑋 − 𝜎 𝑋 @ . 𝑑𝑡 − 𝜎 𝑋 . [ . 𝑑𝑊3(𝑡)
¨
H3
3
= 𝜇 𝑋 . 𝑑𝑡 + 𝜎 𝑋 . [ . 𝑑𝑊3(𝑡)
27

28. Luc_Faucheux_2021
The FX correlation triangle - VIII
¨
H
(
+
(
+
= − 𝜇 𝑋 − 𝜎 𝑋 @ . 𝑑𝑡 − 𝜎 𝑋 . [ . 𝑑𝑊3(𝑡)
¨
H3
3
= 𝜇 𝑋 . 𝑑𝑡 + 𝜎 𝑋 . [ . 𝑑𝑊3(𝑡)
¨
H
(
+
(
+
= 𝜇
2
3
. 𝑑𝑡 + 𝜎
2
3
. [ . 𝑑𝑊
(
+ (𝑡)
28

29. Luc_Faucheux_2021
The FX correlation triangle - IX
¨
H
(
+
(
+
= − 𝜇 𝑋 − 𝜎 𝑋 @ . 𝑑𝑡 − 𝜎 𝑋 . [ . 𝑑𝑊3(𝑡)
¨
H
(
+
(
+
= 𝜇
2
3
. 𝑑𝑡 + 𝜎
2
3
. [ . 𝑑𝑊
(
+ (𝑡)
¨ Note that at this point, all that we can say is that:
¨ (𝑑𝑊3(𝑡))@= 𝑑𝑡
¨ (𝑑𝑊
(
+ (𝑡) )@= 𝑑𝑡
¨ And that is it, but that is enough for now
29

30. Luc_Faucheux_2021
The FX correlation triangle - X
¨
H
(
+
(
+
= − 𝜇 𝑋 − 𝜎 𝑋 @ . 𝑑𝑡 − 𝜎 𝑋 . [ . 𝑑𝑊3(𝑡)
¨ [
H
(
+
(
+
]@= [𝑑
2
3
]@. [
2
(
+
]@= [
2
(
+
]@. [
2
3
. 𝜎 𝑋 . [ . 𝑑𝑊3(𝑡)]@= [𝜎 𝑋 ]@. 𝑑𝑡
¨
H
(
+
(
+
= 𝜇
2
3
. 𝑑𝑡 + 𝜎
2
3
. [ . 𝑑𝑊
(
+ (𝑡)
¨ [
H
(
+
(
+
]@= [𝑑
2
3
]@. [
2
(
+
]@= [
2
(
+
]@. [
2
3
. 𝜎
2
3
. [ . 𝑑𝑊
(
+ (𝑡) ]@= [𝜎
2
3
]@. 𝑑𝑡
30

31. Luc_Faucheux_2021
The FX correlation triangle - XI
¨ [
H
(
+
(
+
]@= [𝑑
2
3
]@. [
2
(
+
]@= [
2
(
+
]@. [
2
3
. 𝜎 𝑋 . [ . 𝑑𝑊3(𝑡)]@= [𝜎 𝑋 ]@. 𝑑𝑡
¨ [
H
(
+
(
+
]@= [𝑑
2
3
]@. [
2
(
+
]@= [
2
(
+
]@. [
2
3
. 𝜎
2
3
. [ . 𝑑𝑊
(
+ (𝑡) ]@= [𝜎
2
3
]@. 𝑑𝑡
¨ [𝜎 𝑋 ]@. 𝑑𝑡 = [𝜎
2
3
]@. 𝑑𝑡
¨ If we also make the assumption, which seems quite reasonable, that we are dealing with
positive volatilities,
¨ 𝜎 𝑋 = 𝜎
2
3
31

32. Luc_Faucheux_2021
The FX correlation triangle - XII
¨ 𝜎 𝑋 = 𝜎
2
3
¨ 𝜎 𝑋#,! = 𝜎
2
3$,&
= 𝜎 𝑋!,#
¨ 𝜎 𝑋#,! = 𝜎 𝑋!,#
¨ 𝜎 𝑋#←! = 𝜎 𝑋!←#
¨ So that is pretty cool, and will be useful when we derive the correlation triangle
¨ Note that this is ONLY true when we define a GBM:
¨
H3&,$
3&,$
= 𝜇 𝑋!,# . 𝑑𝑡 + 𝜎 𝑋!,# . [ . 𝑑𝑊3&,$(𝑡)
32

33. Luc_Faucheux_2021
The FX correlation triangle - XIII
¨ As always, if you do not trust me, you can always resort to Excel
¨ I have a feeling that if Excel, and cheap CPU had come out say maybe 100 years ago, we
would have just reduced everything to a massive GoalSeek, and not spend the time to
derive equations and such.
¨ Hey, that is kind of what we are doing now with the big data /AI/ML/DL/cloud stuff.
33

34. Luc_Faucheux_2021
The FX correlation triangle - XIV
¨ Calculating the Expected Value, Variance and all that good stuff for 𝑋!,# in a discrete binomial
example
¨
H3&,$
3&,$
= 𝜇 𝑋!,# . 𝑑𝑡 + 𝜎 𝑋!,# . [ . 𝑑𝑊3&,$(𝑡)
¨ 𝑀 𝑋!,# = 𝐸 𝑋!,# =< 𝑋!,# > = ∑ 𝑝𝑟𝑜𝑏𝑎 𝑠 . 𝑋!,#(𝑠) where 𝑠 index of the outcomes
¨ 𝑉 𝑋!,# = 𝐸 (𝑋!,# − 𝑀[𝑋!,#])@ =< (𝑋!,# − < 𝑋!,# >)@> = ∑ 𝑝𝑟𝑜𝑏𝑎 𝑠 . (𝑋!,#(𝑠) − 𝑀[𝑋!,#])@
¨ 𝑉 𝑋!,# = (𝜎 𝑋!,# . 𝑋!,#)@
¨ 𝜎 𝑋!,# =
2
3&,$
. 𝑉 𝑋!,#
34

35. Luc_Faucheux_2021
The FX correlation triangle - XV
35

36. Luc_Faucheux_2021
The FX correlation triangle - XVI
¨ Then we do it for
2
3&,$
…It works !!!..... 𝜎 𝑋#,! = 𝜎
2
3$,&
= 𝜎 𝑋!,#
36

37. Luc_Faucheux_2021
ITO Leibniz on (𝑋!,# = 𝑋!,$ ∗ 𝑋$,#)
37

38. Luc_Faucheux_2021
The FX correlation triangle - XVII
¨ Ok, so we have not even started talking about correlations…
¨ So let’s get to it, and we will stick to 1,2,3 instead of 𝑖, 𝑗, 𝑘 for ease of notation
¨ 𝑋2,@ = 𝑋2,G ∗ 𝑋G,@
¨ We use our good old friend the ITO Leibniz
¨ 𝛿𝑓 𝑋, 𝑌 =
IJ
I3
. 𝛿𝑋 +
IJ
IM
. 𝛿𝑌 +
2
@
.
I)J
I3) . 𝛿𝑋@ +
2
@
.
I)J
IM) . 𝛿𝑌@ +
I)J
I3IM
. 𝛿𝑋. 𝛿𝑌
¨ 𝑓 𝑋, 𝑌 is really noted 𝑓 𝑥 = 𝑋(𝑡), 𝑦 = 𝑌(𝑡) and all the partial derivatives are for example:
¨
I)J
I3IM
=
I)J
IKIN
|K=3 ? ,N=M(?)
¨ Where 𝑓 𝑥, 𝑦 is a nice function operating in the usual Newtonian calculus, where taking
partial derivatives is well defined and makes sense (remember that a stochastic process is
NOT differentiable)
38

39. Luc_Faucheux_2021
The FX correlation triangle - XVIII
¨ 𝑋2,@ = 𝑋2,G ∗ 𝑋G,@
¨ 𝛿𝑓 𝑋, 𝑌 =
IJ
I3
. 𝛿𝑋 +
IJ
IM
. 𝛿𝑌 +
2
@
.
I)J
I3) . 𝛿𝑋@ +
2
@
.
I)J
IM) . 𝛿𝑌@ +
I)J
I3IM
. 𝛿𝑋. 𝛿𝑌
¨ 𝛿 𝑋,,- ∗ 𝑋-,. =
/0
/1!,#
. 𝛿𝑋,,- +
/0
/1#,$
. 𝛿𝑋-,. +
,
.
.
/$0
/1!,#
$ . 𝛿𝑋,,-
.
+
,
.
.
/$0
/1#,$
$ . 𝛿𝑋-,.
.
+
/$0
/1!,#/1#,$
. 𝛿𝑋,,-. 𝛿𝑋-,.
¨ 𝑓 𝑋,,-, 𝑋-,. = 𝑋,,- ∗ 𝑋-,.
¨
/0
/1!,#
= 𝑋-,.
¨
/$0
/1!,#
$ = 0
¨
/0
/1#,$
= 𝑋,,-
¨
/$0
/1#,$
$ = 0
¨
/$0
/1!,#/1#,$
= 1
39

40. Luc_Faucheux_2021
The FX correlation triangle - XIX
¨ 𝛿 𝑋,,- ∗ 𝑋-,. =
/0
/1!,#
. 𝛿𝑋,,- +
/0
/1#,$
. 𝛿𝑋-,. +
,
.
.
/$0
/1!,#
$ . 𝛿𝑋,,-
.
+
,
.
.
/$0
/1#,$
$ . 𝛿𝑋-,.
.
+
/$0
/1!,#/1#,$
. 𝛿𝑋,,-. 𝛿𝑋-,.
¨ 𝛿 𝑋,,- ∗ 𝑋-,. = 𝑋-,.. 𝛿𝑋,,- + 𝑋,,-. 𝛿𝑋-,. +
,
.
. 0. 𝛿𝑋,,-
.
+
,
.
. 0. 𝛿𝑋-,.
.
+ 1. 𝛿𝑋,,-. 𝛿𝑋-,.
¨ 𝛿 𝑋2,G ∗ 𝑋G,@ = 𝑋G,@. 𝛿𝑋2,G + 𝑋2,G. 𝛿𝑋G,@ + 𝛿𝑋2,G. 𝛿𝑋G,@
¨ 𝑑 𝑋2,G ∗ 𝑋G,@ = 𝑋G,@. 𝑑𝑋2,G + 𝑋2,G. 𝑑𝑋G,@ + 𝑑𝑋2,G. 𝑑𝑋G,@
¨ We still always operate in the GBM framework:
¨
H3&,$
3&,$
= 𝜇 𝑋!,# . 𝑑𝑡 + 𝜎 𝑋!,# . [ . 𝑑𝑊3&,$(𝑡)
¨ 𝑑 𝑋,,- ∗ 𝑋-,. = 𝑋-,.. 𝑋,,-. 𝜇 𝑋,,- . 𝑑𝑡 + 𝜎 𝑋,,- . [ . 𝑑𝑊1!,# 𝑡 + 𝑋,,-. 𝑋-,.. 1
2
𝜇 𝑋-,. . 𝑑𝑡 +
𝜎 𝑋-,. . [ . 𝑑𝑊1#,$ 𝑡 + 𝑋-,.. 𝑋,,-. 𝜇 𝑋,,- . 𝑑𝑡 + 𝜎 𝑋,,- . [ . 𝑑𝑊1!,# 𝑡 . 1
2
𝜇 𝑋-,. . 𝑑𝑡 +
𝜎 𝑋-,. . [ . 𝑑𝑊1#,$ 𝑡
40

41. Luc_Faucheux_2021
The FX correlation triangle - XX
¨ 𝑑 𝑋2,G ∗ 𝑋G,@ = 𝑋G,@. 𝑋2,G. 𝜇 𝑋2,G . 𝑑𝑡 + 𝜎 𝑋2,G . [ . 𝑑𝑊3(,* 𝑡 + 𝑋2,G. 𝑋G,@. `
a
𝜇 𝑋G,@ . 𝑑𝑡 +
𝜎 𝑋G,@ . [ . 𝑑𝑊3*,) 𝑡 + 𝑋G,@. 𝑋2,G. 𝜇 𝑋2,G . 𝑑𝑡 + 𝜎 𝑋2,G . [ . 𝑑𝑊3(,* 𝑡 . `
a
𝜇 𝑋G,@ . 𝑑𝑡 +
𝜎 𝑋G,@ . [ . 𝑑𝑊3*,) 𝑡
¨ This is quite cumbersome
¨ Luckily for us, we are after computing the variance, so we can neglect all the terms that are not
in first order of the Brownian motion
¨ 𝑑 𝑋,,- ∗ 𝑋-,. = 𝑋-,.. 𝑋,,-. 𝜎 𝑋,,- . [ . 𝑑𝑊1!,# 𝑡 + 𝑋,,-. 𝑋-,.. 𝜎 𝑋-,. . [ . 𝑑𝑊1#,$ 𝑡 + 𝑆𝑜𝑚𝑒𝑡ℎ𝑖𝑛𝑔
¨ 𝑆𝑜𝑚𝑒𝑡ℎ𝑖𝑛𝑔 = 𝑆𝑜𝑚𝑒𝑡𝑒𝑟𝑚𝑠𝑖𝑛. 𝑑𝑡 + 𝑠𝑜𝑚𝑒𝑜𝑡ℎ𝑒𝑟𝑡𝑒𝑟𝑚𝑠𝑖𝑛 . 𝑑𝑡. 𝑑𝑊
41

42. Luc_Faucheux_2021
The FX correlation triangle - XXI
¨ 𝑑 𝑋,,- ∗ 𝑋-,. = 𝑋-,.. 𝑋,,-. 𝜎 𝑋,,- . [ . 𝑑𝑊1!,# 𝑡 + 𝑋,,-. 𝑋-,.. 𝜎 𝑋-,. . [ . 𝑑𝑊1#,$ 𝑡 + 𝑆𝑜𝑚𝑒𝑡ℎ𝑖𝑛𝑔
¨ We also have:
¨ (𝑑𝑊3&,$(𝑡) )@= 𝑑𝑡
¨ 𝑑𝑊3&,$ 𝑡 . 𝑑𝑊32,3 𝑡 = 𝜌 𝑋!,#; 𝑋Q,R . 𝑑𝑡
¨ Of course:
¨ 𝜌 𝑋!,#; 𝑋Q,R = 𝜌 𝑋Q,R; 𝑋!,#
¨ 𝜌 𝑋!,#; 𝑋!,# = 1
¨ Question: is it that obvious that 𝜌 < 1 ?
¨ Will leave that for another note at some point later in time
42

43. Luc_Faucheux_2021
The FX correlation triangle - XXII
¨ 𝑑 𝑋,,- ∗ 𝑋-,. = 𝑋-,.. 𝑋,,-. 𝜎 𝑋,,- . [ . 𝑑𝑊1!,# 𝑡 + 𝑋,,-. 𝑋-,.. 𝜎 𝑋-,. . [ . 𝑑𝑊1#,$ 𝑡 + 𝑆𝑜𝑚𝑒𝑡ℎ𝑖𝑛𝑔
¨ 𝑆𝑜𝑚𝑒𝑡ℎ𝑖𝑛𝑔 = 𝑆𝑜𝑚𝑒𝑡𝑒𝑟𝑚𝑠𝑖𝑛. 𝑑𝑡 + 𝑠𝑜𝑚𝑒𝑜𝑡ℎ𝑒𝑟𝑡𝑒𝑟𝑚𝑠𝑖𝑛 . 𝑑𝑡. 𝑑𝑊
¨ We also have:
¨ (𝑑𝑊3&,$(𝑡) )@= 𝑑𝑡
¨ 𝑑𝑊3&,$ 𝑡 . 𝑑𝑊32,3 𝑡 = 𝜌 𝑋!,#; 𝑋Q,R . 𝑑𝑡
¨ (𝑑 𝑋2,G ∗ 𝑋G,@ )@= (𝑋G,@. 𝑋2,G)@. {𝜎 𝑋2,G
@
. 𝑑𝑡 + 𝜎 𝑋G,@
@
. 𝑑𝑡 + 2. 𝜌 𝑋2,G; 𝑋G,@ 𝜎 𝑋2,G . 𝜎 𝑋G,@ . 𝑑𝑡}
¨ Since all the other terms are in higher order than just (𝑑𝑡)
¨ (𝑑 𝑋2,G ∗ 𝑋G,@ )@= (𝑑𝑋2,@)@
¨ (𝑑 𝑋2,G ∗ 𝑋G,@ )@= (𝑑𝑋2,@)@= (𝑋2,@)@. 𝜎 𝑋2,@
@
. 𝑑𝑡
43

44. Luc_Faucheux_2021
The FX correlation triangle - XXIII
¨ (𝑑 𝑋2,G ∗ 𝑋G,@ )@= (𝑋G,@. 𝑋2,G)@. {𝜎 𝑋2,G
@
. 𝑑𝑡 + 𝜎 𝑋G,@
@
. 𝑑𝑡 + 2. 𝜌 𝑋2,G; 𝑋G,@ 𝜎 𝑋2,G . 𝜎 𝑋G,@ . 𝑑𝑡}
¨ (𝑑 𝑋2,G ∗ 𝑋G,@ )@= (𝑑𝑋2,@)@
¨ (𝑑 𝑋2,G ∗ 𝑋G,@ )@= (𝑑𝑋2,@)@= (𝑋2,@)@. 𝜎 𝑋2,@
@
. 𝑑𝑡
¨ 𝜎 𝑋2,@
@
= 𝜎 𝑋2,G
@
+ 𝜎 𝑋G,@
@
+ 2. 𝜌 𝑋2,G; 𝑋G,@ . 𝜎 𝑋2,G . 𝜎 𝑋G,@
¨ OK, that is a good start, we should be able to start doing a lot with that relation.
44

45. Luc_Faucheux_2021
The FX correlation triangle - XXIV
¨ 𝜎 𝑋2,@
@
= 𝜎 𝑋2,G
@
+ 𝜎 𝑋G,@
@
+ 2. 𝜌 𝑋2,G; 𝑋G,@ . 𝜎 𝑋2,G . 𝜎 𝑋G,@
¨ We also know that we can change the order in the volatility:
¨ 𝜎 𝑋!,# = 𝜎 𝑋#,!
¨ The real question is inside the correlation exponent:
¨ What can we say about for example 𝜌 𝑋2,G; 𝑋G,@ if we know 𝜌 𝑋2,G; 𝑋@,G ?
¨ Let’s derive the relation in another way to illustrate the difference:
45

46. Luc_Faucheux_2021
ITO Leibniz on (𝑋!,# =
%!,#
%$,#
)
46

47. Luc_Faucheux_2021
The FX correlation triangle - XXV
¨ 𝑋2,@ = 𝑋2,G ∗ 𝑋G,@
¨ 𝑋2,@ =
3(,*
3),*
¨ 𝛿𝑓 𝑋, 𝑌 =
IJ
I3
. 𝛿𝑋 +
IJ
IM
. 𝛿𝑌 +
2
@
.
I)J
I3) . 𝛿𝑋@ +
2
@
.
I)J
IM) . 𝛿𝑌@ +
I)J
I3IM
. 𝛿𝑋. 𝛿𝑌
¨ 𝛿
1!,#
1$,#
=
/0
/1!,#
. 𝛿𝑋,,- +
/0
/1$,#
. 𝛿𝑋.,- +
,
.
.
/$0
/1!,#
$ . 𝛿𝑋,,-
.
+
,
.
.
/$0
/1$,#
$ . 𝛿𝑋.,-
.
+
/$0
/1!,#/1$,#
. 𝛿𝑋,,-. 𝛿𝑋.,-
47

48. Luc_Faucheux_2021
The FX correlation triangle - XXVI
¨ 𝑓 𝑋2,G, 𝑋G,@ =
3(,*
3),*
¨
IJ
I3(,*
=
2
3),*
¨
I)J
I3(,*
) = 0
¨
IJ
I3),*
=
L3(,*
3),*
)
¨
I)J
I3),*
) =
@.3(,*
3),*
*
¨
I)J
I3(,*I3),*
=
L2
3),*
)
48

49. Luc_Faucheux_2021
The FX correlation triangle - XXVII
¨ 𝛿
1!,#
1$,#
=
/0
/1!,#
. 𝛿𝑋,,- +
/0
/1$,#
. 𝛿𝑋.,- +
,
.
.
/$0
/1!,#
$ . 𝛿𝑋,,-
.
+
,
.
.
/$0
/1$,#
$ . 𝛿𝑋.,-
.
+
/$0
/1!,#/1$,#
. 𝛿𝑋,,-. 𝛿𝑋.,-
¨ 𝛿
1!,#
1$,#
=
,
1$,#
. 𝛿𝑋,,- +
41!,#
1$,#
$ . 𝛿𝑋.,- +
,
.
. 0. 𝛿𝑋,,-
.
+
,
.
.
..1!,#
1$,#
# . 𝛿𝑋.,-
.
+
4,
1$,#
$ . 𝛿𝑋,,-. 𝛿𝑋.,-
¨ 𝑑
1!,#
1$,#
=
,
1$,#
. 𝑑𝑋,,- +
41!,#
1$,#
$ . 𝑑𝑋.,- +
,
.
. 0. 𝑑𝑋,,-
.
+
,
.
.
..1!,#
1$,#
# . 𝑑𝑋.,-
.
+
4,
1$,#
$ . 𝑑𝑋,,-. 𝑑𝑋.,-
¨ We still always operate in the GBM framework:
¨
61%,&
1%,&
= 𝜇 𝑋7,8 . 𝑑𝑡 + 𝜎 𝑋7,8 . [ . 𝑑𝑊1%,& (𝑡)
¨ As we did before, focusing only on the terms in first order in the stochastic Brownian motion:
¨ 𝑑
1!,#
1$,#
=
,
1$,#
. 𝑋,,-. 𝜎 𝑋,,- . [ . 𝑑𝑊1!,# 𝑡 +
41!,#
1$,#
$ . 𝑋.,-. 𝜎 𝑋.,- . [ . 𝑑𝑊1$,# 𝑡 + 𝒪 𝑑𝑡 + 𝒪 𝑑𝑡. 𝑑𝑊
49

50. Luc_Faucheux_2021
The FX correlation triangle - XXVIII
¨ 𝑑
1!,#
1$,#
=
,
1$,#
. 𝑋,,-. 𝜎 𝑋,,- . [ . 𝑑𝑊1!,# 𝑡 +
41!,#
1$,#
$ . 𝑋.,-. 𝜎 𝑋.,- . [ . 𝑑𝑊1$,# 𝑡 + 𝒪 𝑑𝑡 + 𝒪 𝑑𝑡. 𝑑𝑊
¨ 𝑑
1!,#
1$,#
=
1!,#
1$,#
. 𝜎 𝑋,,- . [ . 𝑑𝑊1!,# 𝑡 −
1!,#
1$,#
. 𝜎 𝑋.,- . [ . 𝑑𝑊1$,# 𝑡 + 𝒪 𝑑𝑡 + 𝒪 𝑑𝑡. 𝑑𝑊
¨ 𝑋,,. =
1!,#
1$,#
¨ 𝑑 𝑋,,. = 𝑋,,. . 𝜎 𝑋,,- . [ . 𝑑𝑊1!,# 𝑡 − 𝑋,,. . 𝜎 𝑋.,- . [ . 𝑑𝑊1$,# 𝑡 + 𝒪 𝑑𝑡 + 𝒪 𝑑𝑡. 𝑑𝑊
¨
H3(,)
3(,)
= 𝜎 𝑋2,G . [ . 𝑑𝑊3(,* 𝑡 − 𝜎 𝑋@,G . [ . 𝑑𝑊3),* 𝑡 + 𝒪 𝑑𝑡 + 𝒪 𝑑𝑡. 𝑑𝑊
¨ (𝑑𝑊1%,& (𝑡) ).
= 𝑑𝑡
¨ 𝑑𝑊1%,& 𝑡 . 𝑑𝑊1',( 𝑡 = 𝜌 𝑋7,8; 𝑋9,: . 𝑑𝑡
50

51. Luc_Faucheux_2021
The FX correlation triangle - XXIX
¨
H3(,)
3(,)
= 𝜎 𝑋2,G . [ . 𝑑𝑊3(,* 𝑡 − 𝜎 𝑋@,G . [ . 𝑑𝑊3),* 𝑡 + 𝒪 𝑑𝑡 + 𝒪 𝑑𝑡. 𝑑𝑊
¨ (𝑑𝑊1%,& (𝑡) ).
= 𝑑𝑡
¨ 𝑑𝑊1%,& 𝑡 . 𝑑𝑊1',( 𝑡 = 𝜌 𝑋7,8; 𝑋9,: . 𝑑𝑡
¨ (
61!,$
1!,$
).
=
61!,$
$
1!,$
$ = 𝜎 𝑋,,-
.
. 𝑑𝑡 + 𝜎 𝑋.,-
.
. 𝑑𝑡 − 2. 𝜌 𝑋,,-; 𝑋.,- 𝜎 𝑋,,- . 𝜎 𝑋.,- . 𝑑𝑡 + 𝒪 𝑑𝑡.
+ 𝒪 𝑑𝑡. 𝑑𝑊
¨
H3(,)
3(,)
= 𝜇 𝑋2,@ . 𝑑𝑡 + 𝜎 𝑋2,@ . [ . 𝑑𝑊3(,)(𝑡)
¨ (
H3(,)
3(,)
)@=
H3(,)
)
3(,)
) = 𝜎 𝑋2,@
@
. 𝑑𝑡
51

52. Luc_Faucheux_2021
The FX correlation triangle - XXX
¨ (
H3(,)
3(,)
)@=
H3(,)
)
3(,)
) = 𝜎 𝑋2,G
@
. 𝑑𝑡 + 𝜎 𝑋@,G
@
. 𝑑𝑡 − 2. 𝜌 𝑋2,G; 𝑋@,G 𝜎 𝑋2,G . 𝜎 𝑋@,G . 𝑑𝑡
¨ (
H3(,)
3(,)
)@=
H3(,)
)
3(,)
) = 𝜎 𝑋2,@
@
. 𝑑𝑡
¨ 𝜎 𝑋2,@
@
= 𝜎 𝑋2,G
@
+ 𝜎 𝑋@,G
@
− 2. 𝜌 𝑋2,G; 𝑋@,G . 𝜎 𝑋2,G . 𝜎 𝑋@,G
¨ This one with the minus sign is the most commonly used correlation formula because of the
analogy with a triangle that we will look at shortly
52

53. Luc_Faucheux_2021
The FX correlation triangle - XXXI
¨ 𝜎 𝑋2,@
@
= 𝜎 𝑋2,G
@
+ 𝜎 𝑋@,G
@
− 2. 𝜌 𝑋2,G; 𝑋@,G . 𝜎 𝑋2,G . 𝜎 𝑋@,G
¨ And we had before:
¨ 𝜎 𝑋2,@
@
= 𝜎 𝑋2,G
@
+ 𝜎 𝑋G,@
@
+ 2. 𝜌 𝑋2,G; 𝑋G,@ . 𝜎 𝑋2,G . 𝜎 𝑋G,@
¨ We know that:
¨ 𝜎 𝑋@,G = 𝜎 𝑋G,@
¨ So we get:
¨ 𝜎 𝑋2,@
@
= 𝜎 𝑋2,G
@
+ 𝜎 𝑋G,@
@
− 2. 𝜌 𝑋2,G; 𝑋@,G . 𝜎 𝑋2,G . 𝜎 𝑋G,@
¨ 𝜎 𝑋2,@
@
= 𝜎 𝑋2,G
@
+ 𝜎 𝑋G,@
@
+ 2. 𝜌 𝑋2,G; 𝑋G,@ . 𝜎 𝑋2,G . 𝜎 𝑋G,@
53

54. Luc_Faucheux_2021
The FX correlation triangle - XXXII
¨ 𝜎 𝑋2,@
@
= 𝜎 𝑋2,G
@
+ 𝜎 𝑋G,@
@
− 2. 𝜌 𝑋2,G; 𝑋@,G . 𝜎 𝑋2,G . 𝜎 𝑋G,@
¨ 𝜎 𝑋2,@
@
= 𝜎 𝑋2,G
@
+ 𝜎 𝑋G,@
@
+ 2. 𝜌 𝑋2,G; 𝑋G,@ . 𝜎 𝑋2,G . 𝜎 𝑋G,@
¨ So we get:
¨ 𝜌 𝑋2,G; 𝑋G,@ = −𝜌 𝑋2,G; 𝑋@,G
¨ 𝜌 𝑋2,G; 𝑋G,@ = −𝜌 𝑋2,G;
2
3*,)
¨ With more general stochastic variables 𝑋 and 𝑌
¨ 𝜌 𝑌; 𝑋 = −𝜌 𝑌;
2
3
¨ Which is general, we did not need the specific derivation above to get to that result, we
could have plugged directly once we got the first equation, but it is worth doing it just to
have the pleasure of doing ITO Leibniz, truly a marvel of stochastic calculus.
54

55. Luc_Faucheux_2021
The FX correlation triangle - XXXIII
¨ Why go through all that trouble ?
¨ First of all it is fun
¨ Second of all it provides constraints on the FX currency pairs volatilities, so if in the market
you observe volatilities that deviate from the equation, you might be thinking about putting
a trade on (careful that this is not a hard arbitrage per se, as most likely you will be putting
an option trade against another option trade)
¨ 𝜎 𝑋2,@
@
= 𝜎 𝑋2,G
@
+ 𝜎 𝑋G,@
@
− 2. 𝜌 𝑋2,G; 𝑋@,G . 𝜎 𝑋2,G . 𝜎 𝑋G,@
¨ 𝜎 𝑋!,#
@
= 𝜎 𝑋!,4
@
+ 𝜎 𝑋4,#
@
− 2. 𝜌 𝑋!,#; 𝑋#,4 . 𝜎 𝑋!,# . 𝜎 𝑋4,#
¨ 𝜎 𝑋!,# = 𝜎 𝑋#,!
¨ 𝜌 𝑋!,4; 𝑋4,# = −𝜌 𝑋!,4; 𝑋#,4
55

56. Luc_Faucheux_2021
The FX correlation triangle - XXXIV
¨ And thirdly, it is usually botched up in a number of textbooks, because they don’t pay
attention to either the notation (Uncle Godel will not be happy), or even worse in some
cases, they derive the relation by using regular calculus and completely sweeping under the
rug the fact that the exchange rates are stochastic, which is ironic, because they are talking
about variance, and usually also right after they go gently into talking about “FX options
modeling).
¨ I will not name any textbooks/publication, but sometimes you encounter something like the
next couples of slides, which do get indeed the right result at the end, but going completely
willy-nilly about differentiating ratios of stochastic process as if we were in regular
Newtonian calculus and like we never heard about ITO lemma or ITO Leibniz
¨ Turns out that the end result is correct as we are concerning ourselves with the variance
which means that we are not concerning ourselves with the higher order terms, but still a
little rigor never hurt anyone
56

57. Luc_Faucheux_2021
The FX correlation triangle - XXXV
57

58. Luc_Faucheux_2021
The FX correlation triangle - XXXVI
58

59. Luc_Faucheux_2021
The FX correlation triangle - XXXVII
¨ All right Robin, enough equations, let’s draw some nice graphs, and use the power of
analogies.
59

60. Luc_Faucheux_2021
To the triangle Robin !
Or the power of analogies
60

61. Luc_Faucheux_2021
The triangle – the power of analogies - I
¨ So we are getting the formula:
¨ 𝜎 𝑋2,@
@
= 𝜎 𝑋2,G
@
+ 𝜎 𝑋G,@
@
− 2. 𝜌 𝑋2,G; 𝑋@,G . 𝜎 𝑋2,G . 𝜎 𝑋G,@
¨ There is a beautiful way to express this in a geometric manner
¨ We used to be really good at geometry, when we were building houses and using the stars
to navigate the seas
¨ Not so much anymore, now I guess we are getting good at coding (and maybe hopefully
stochastic calculus).
61

62. Luc_Faucheux_2021
The triangle – the power of analogies - II
¨ 𝜎 𝑋2,@
@
= 𝜎 𝑋2,G
@
+ 𝜎 𝑋G,@
@
− 2. 𝜌 𝑋2,G; 𝑋@,G . 𝜎 𝑋2,G . 𝜎 𝑋G,@
¨ 𝜌 𝑋2,G; 𝑋@,G =
S 3(,)
)
L S 3(,*
)
L S 3*,)
)
@.S 3(,* .S 3*,)
¨ Again, not so obvious that 𝜌 𝑋2,G; 𝑋@,G has to be smaller than 1 in absolute value, but we
will check that in later notes
¨ Not super obvious.
¨ So not super obvious that there is a name for it, the Cauchy-Schwartz inequality
¨ Back to the triangle for now
62

63. Luc_Faucheux_2021
The triangle – the power of analogies - III
¨ Let’s look at the following triangle and let’s note 1,2,3 the corners, and the length of the
sides by 𝐿(𝑖, 𝑗)
¨ We can easily verify the first rule of triangles: it is impossible to draw a triangle that does not
look like a special case (right, isosceles,..)
63
1
2
3
𝐿(1,2)
𝐿(1,3)
𝐿(3,2)
𝜃(3)

64. Luc_Faucheux_2021
The triangle – the power of analogies - IV
¨ So first of all, we have obviously
¨ 𝐿 𝑖, 𝑗 = 𝐿(𝑗, 𝑖)
¨ Which kinds of reminds us of:
¨ 𝜎 𝑋!,# = 𝜎 𝑋#,!
¨ So we are kind of thinking that maybe if there is an analogy there, the length of the sides
will be the volatilities.
¨ Let’s try to derive a relation between the length of the sides in the triangle
64

65. Luc_Faucheux_2021
The triangle – the power of analogies - V
¨ Let’s compute 𝐿 1,2
¨ Since we are not that great in geometry (and neither have we as a specie for a thousand
years or so), let me use the power of calculus by computing the coordinates of corner 3.
¨ I choose corner 1 to be the origin
65
1
2
3
𝐿(1,2)
𝐿(1,3)
𝐿(3,2)
𝜃(3)
𝑦
𝑥

66. Luc_Faucheux_2021
The triangle – the power of analogies - VI
¨ The coordinates are then:
¨ 𝐶 1 = (0,0)
¨ 𝐶 3 = (𝐿(1,3), 0)
¨ 𝐶 2 = (𝑥@, 𝑦@)
¨ We have :
¨ 𝑥@ = 𝑥G − cos 𝜃 3 ∗ 𝐿(3,2)
¨ 𝑦@ = 𝑦G + sin 𝜃 3 ∗ 𝐿(3,2)
¨ With
¨ 𝑥G = 𝐿 1,3
¨ 𝑦G = 0
66

67. Luc_Faucheux_2021
The triangle – the power of analogies - VII
¨ 𝑥@ = 𝐿(1,3) − cos 𝜃 3 ∗ 𝐿(3,2)
¨ 𝑦@ = sin 𝜃 3 ∗ 𝐿(3,2)
¨ And we have:
¨ 𝐿(1,2)@ = 𝑥@
@ + 𝑦@
@
¨ 𝐿(1,2)@ = (𝐿(1,3) − cos 𝜃 3 ∗ 𝐿(3,2))@+(sin 𝜃 3 ∗ 𝐿(3,2) )@
¨ 𝐿(1,2).
= 𝐿(1,3).
− 2. 𝐿 1,3 . cos 𝜃 3 . 𝐿 3,2 + 𝐿 3,2 .
. cos 𝜃 3
.
+ (sin 𝜃 3 ∗ 𝐿(3,2) ).
¨ 𝐿(1,2)@ = 𝐿(1,3)@ − 2. 𝐿 1,3 . cos 𝜃 3 . 𝐿 3,2 + 𝐿 3,2 @. {cos 𝜃 3
@
+ sin 𝜃 3
@
}
¨ 𝐿(1,2)@ = 𝐿(1,3)@ − 2. 𝐿 1,3 . cos 𝜃 3 . 𝐿 3,2 + 𝐿 3,2 @. {1}
¨ 𝐿(1,2)@ = 𝐿(1,3)@ + 𝐿 3,2 @ − 2. 𝐿 1,3 . cos 𝜃 3 . 𝐿 3,2
¨ Now we getting somewhere
67

68. Luc_Faucheux_2021
The triangle – the power of analogies - VIII
¨ 𝐿(1,2)@ = 𝐿(1,3)@ + 𝐿 3,2 @ − 2. 𝐿 1,3 . cos 𝜃 3 . 𝐿 3,2
¨ 𝐿(1,2)@ = 𝐿(1,3)@ + 𝐿 3,2 @ − 2. cos 𝜃 3 . 𝐿 1,3 . 𝐿 3,2
¨ 𝜎 𝑋2,@
@
= 𝜎 𝑋2,G
@
+ 𝜎 𝑋G,@
@
− 2. 𝜌 𝑋2,G; 𝑋@,G . 𝜎 𝑋2,G . 𝜎 𝑋G,@
¨ So we have the analogy, or the mapping between the two problems:
¨ 𝐿 𝑖, 𝑗 ↔ 𝜎 𝑋!,#
¨ cos 𝜃 𝑘 ↔ 𝜌 𝑋!,4; 𝑋#,4
68

69. Luc_Faucheux_2021
The triangle – the power of analogies - IX
¨ cos 𝜃 3 = 𝜌 𝑋2,G; 𝑋@,G
¨ This is also another strong indication that: −1 ≤ 𝜌 𝑋2,G; 𝑋@,G ≤ (1)
69
1
2
3
𝐿 3,2 = 𝜎(3,2)
𝜃(3)
𝐿 1,2 = 𝜎(1,2)
𝐿 1,3 = 𝜎(1,3)

70. Luc_Faucheux_2021
The triangle – the power of analogies - X
¨ This is one of the beautiful analogies that one can find in Maths or Physics or science in
general.
¨ Here is a little random thread that keeps popping into my mind every time I look at the
correlation triangle in FX
¨ As you know I really like the Godel’s theorem of Incompleteness
¨ Every couple of years or so, to prove to myself that my brain is still semi-functioning, I tried
to go through the simplified derivation that you could find in a couple of great books (I never
attempted going through the actual derivation from the original Godel’s paper yet, maybe
one day when I get really bored).
¨ One of those great books is the one by Nagel and Newman:
70

71. Luc_Faucheux_2021
The triangle – the power of analogies - XI
71

72. Luc_Faucheux_2021
The triangle – the power of analogies - XII
¨ In that book, as an example so that we can familiarize ourselves with the formal mapping
that Godel achieved in his famous paper, they use the triangle example (page 15 and 16):
72

73. Luc_Faucheux_2021
The triangle – the power of analogies - XIII
¨ Later in the book they revisit the concept of mapping on a triangle and the crucial concept
of duality with the theorem of Pappus:
73

74. Luc_Faucheux_2021
The triangle – the power of analogies - XIV
¨ Now, chance would have it that the preface to this beautiful little book is written by no other
than Douglas Hofstadter, who went on to write the very successful and seminal “GEB” on
Godel, Escher and Bach, centered around the Godel’s theorem, but using analogies with
music and drawings, biology and a lot of other stuff.
74

75. Luc_Faucheux_2021
The triangle – the power of analogies - XV
¨ He also wrote a more recent one about the immense power of analogies in thinking:
75

76. Luc_Faucheux_2021
The triangle – the power of analogies - XVI
¨ So this is quite funny and circular at the same time.
¨ A couple of additional notes:
¨ Another great book (which was partially rewritten during the Covid pandemic) is the one by
Peter Smith:
76

77. Luc_Faucheux_2021
The triangle – the power of analogies - XVII
¨ Also in Physics, conformal mapping is a very powerful tool.
¨ Finally the triangle is quite powerful. As we know now from Dark, the triangle is essential
for time travel:
77

78. Luc_Faucheux_2021
The triangle – the power of analogies - XVIII
¨ So that was my little “Heart of darkness” random stream of consciousness on triangles and
the power of analogies
¨ Now onto some more ITO Leibniz to look at changing measures when looking at assets
¨ But before, a blast from the past..
78

79. Luc_Faucheux_2021
A blast from the past
79

80. Luc_Faucheux_2021
A blast from the past
¨ I was not lying to you when I was telling you that I am doing those decks mostly for myself
¨ I have all those notes flying around the house on legal pads, and it is time to put them in a
more formal format and in the cloud
¨ What is sad is that I am struggling now to do in an hour what I could do in like 5 minutes
then. Maybe back then I was not really understanding deeply what I was doing, and now I
am much wiser?
¨ I doubt it
¨ There is a study that says that after 30 years old, on average for every year that passes by,
the speed of your tennis serve goes down by 1mph
¨ I used to clock those at 110mph, not anymore, so I would say that study sounds right
¨ Maybe it is the same for the brain, and the ability to do ITO Leibniz, every year it takes 1
more minute to do it
¨ Check out the date on those notes !
80

81. Luc_Faucheux_2021
A blast from the past - II
81

82. Luc_Faucheux_2021
A blast from the past - III
82

83. Luc_Faucheux_2021
A blast from the past - IV
83

84. Luc_Faucheux_2021
Some more notes on measures in the foreign
and domestic world
84

85. Luc_Faucheux_2021
Measures in the foreign and domestic world
¨ Essentially it is putting in Powerpoint and many many slides the handwritten notes on
yellow legal pad from more than 25 years ago.
85

86. Luc_Faucheux_2021
Measures in the foreign and domestic world - II
¨ Let’s assume that currency 1 is the domestic currency
¨ We will note currency 2 to be the foreign currency
¨ 𝑋2,@ = 𝑋2←@ is the value in currency (1) of 1 unit of currency (2)
¨ 𝐵2 𝑡 = exp[∫
<=>
<=?
𝑅2 𝑠, 𝑠, 𝑠 . 𝑑𝑠] is the Bank Account Numeraire in the domestic world of
currency (1).
¨ 𝐵2 𝑡 = exp[∫
<=>
<=?
𝑅2 𝑠, 𝑠, 𝑠 . 𝑑𝑠] is associated to the Risk free measure in the domestic
currency (1) that we will note 𝑊ℚ2 𝑡
¨ 𝐵@ 𝑡 = exp[∫
<=>
<=?
𝑅@ 𝑠, 𝑠, 𝑠 . 𝑑𝑠] is the Bank Account Numeraire in the foreign world of
currency (2).
¨ 𝐵@ 𝑡 = exp[∫
<=>
<=?
𝑅@ 𝑠, 𝑠, 𝑠 . 𝑑𝑠] is associated to the Risk free measure in the foreign
currency (2) that we will note 𝑊ℚ@ 𝑡
86

87. Luc_Faucheux_2021
Measures in the foreign and domestic world - III
¨ The Zero Coupon bonds in the domestic world of currency (1) are noted 𝑍2 𝑡, 𝑡!, 𝑡#
¨ The Zero Coupon bonds in the foreign world of currency (2) are noted 𝑍@ 𝑡, 𝑡!, 𝑡#
¨ 𝐴2 𝑡 is a tradeable asset in the domestic world of currency (1) (like a stock)
¨ 𝐴@ 𝑡 is a tradeable asset in the foreign world of currency (2) (like a stock)
¨ We are going to assume the following processes:
¨
HF(
F(
= 𝜇 𝐴2 . 𝑑𝑡 + 𝜎 𝐴2 . [ . 𝑑𝑊F((𝑡)
¨
HF)
F)
= 𝜇 𝐴@ . 𝑑𝑡 + 𝜎 𝐴@ . [ . 𝑑𝑊F)(𝑡)
¨
H3(,)
3(,)
= 𝜇 𝑋2,@ . 𝑑𝑡 + 𝜎 𝑋2,@ . [ . 𝑑𝑊3(,)(𝑡)
87

88. Luc_Faucheux_2021
Measures in the foreign and domestic world - IV
¨ The trick is going to be essentially:
¨ Create in the domestic world the quantities:
¨ [𝑋2←@. 𝐴@ 𝑡 ]
¨ And:
¨ [𝑋2←@. 𝑍@ 𝑡, 𝑡!, 𝑡# ]
¨ And yes, you guessed it right once again, the ever important deflated quantities:
¨
[3(←).C) ?,?&,?$ ]
E( ?
and
[3(←).F) ? ]
E( ?
¨ As you can also guess, we are going to say that some of those quantities are going to be
martingale (driftless process) under the appropriate measure, and then crank back up the
Ito Leibniz handle back up to the original processes for 𝐴@ 𝑡 and 𝑍@ 𝑡, 𝑡!, 𝑡#
¨ Hopefully we can discover something interesting, because it is going to be rather tedious, I
warn you, but here we go….
88

89. Luc_Faucheux_2021
Measures in the foreign and domestic world - V
¨ Just a quick note before we start
¨ 𝐴@ 𝑡 is a tradeable asset in the foreign world of currency (2) (like a stock)
¨ [𝑋2←@. 𝐴@ 𝑡 ] is a tradeable asset in the domestic world of currency (1)
¨ That does NOT mean that:
¨ [𝑋2←@. 𝐴@ 𝑡 ] = 𝐴2 𝑡
¨ Rather obvious but it is sometimes easy to get confused
89

90. Luc_Faucheux_2021
ITO Leibniz on [𝑋!←#. 𝐴# 𝑡 ]
90

91. Luc_Faucheux_2021
Measures in the foreign and domestic world - VI
¨ All right here we go, let’s do ITO Leibniz on [𝑋2←@. 𝐴@ 𝑡 ]
¨ 𝛿𝑓 𝑋, 𝑌 =
IJ
I3
. 𝛿𝑋 +
IJ
IM
. 𝛿𝑌 +
2
@
.
I)J
I3) . 𝛿𝑋@ +
2
@
.
I)J
IM) . 𝛿𝑌@ +
I)J
I3IM
. 𝛿𝑋. 𝛿𝑌
¨ 𝑓 𝑋, 𝑌 is really noted 𝑓 𝑥 = 𝑋(𝑡), 𝑦 = 𝑌(𝑡) and all the partial derivatives are for example:
¨
I)J
I3IM
=
I)J
IKIN
|K=3 ? ,N=M(?)
¨ Where 𝑓 𝑥, 𝑦 is a nice function operating in the usual Newtonian calculus, where taking
partial derivatives is well defined and makes sense (remember that a stochastic process is
NOT differentiable)
91

92. Luc_Faucheux_2021
Measures in the foreign and domestic world - VII
¨ 𝑓 = 𝑋2←@. 𝐴@ 𝑡 = 𝑋2,@. 𝐴@ 𝑡
¨ 𝛿𝑓 𝑋2,@, 𝐴@ =
IJ
I3(,)
. 𝛿𝑋2,@ +
IJ
IF)
. 𝛿𝐴@ +
2
@
.
I)J
I3(,)
) . 𝛿𝑋2,@
@
+
2
@
.
I)J
IF)
) . 𝛿𝐴@
@
+
I)J
I3(,)IF)
. 𝛿𝑋2,@. 𝛿𝐴@
¨ 𝑓 𝑋,,.. 𝐴. 𝑡 = 𝑋,,.. 𝐴. 𝑡
¨
/0
/1!,$
= 𝐴.
¨
/$0
/1!,$
$ = 0
¨
/0
/;$
= 𝑋,,.
¨
/$0
/;$
$ = 0
¨
/$0
/;$/1!,$
= 1
92

93. Luc_Faucheux_2021
Measures in the foreign and domestic world - VIII
¨ 𝛿𝑓 𝑋2,@, 𝐴@ =
IJ
I3(,)
. 𝛿𝑋2,@ +
IJ
IF)
. 𝛿𝐴@ +
2
@
.
I)J
I3(,)
) . 𝛿𝑋2,@
@
+
2
@
.
I)J
IF)
) . 𝛿𝐴@
@
+
I)J
I3(,)IF)
. 𝛿𝑋2,@. 𝛿𝐴@
¨ 𝛿𝑓 𝑋2,@, 𝐴@ = 𝐴@. 𝛿𝑋2,@ + 𝑋2,@. 𝛿𝐴@ +
2
@
. 0. 𝛿𝑋2,@
@
+
2
@
. 0. 𝛿𝐴@
@
+ 1. 𝛿𝑋2,@. 𝛿𝐴@
¨ 𝑑𝑓 𝑋2,@, 𝐴@ = 𝐴@. 𝑑𝑋2,@ + 𝑋2,@. 𝑑𝐴@ +
2
@
. 0. 𝑑𝑋2,@
@
+
2
@
. 0. 𝑑𝐴@
@
+ 1. 𝑑𝑋2,@. 𝑑𝐴@
¨ 𝑑𝑓 𝑋2,@, 𝐴@ = 𝑑 𝑋2,@. 𝐴@ = 𝐴@. 𝑑𝑋2,@ + 𝑋2,@. 𝑑𝐴@ + 𝑑𝑋2,@. 𝑑𝐴@
¨
HF)
F)
= 𝜇 𝐴@ . 𝑑𝑡 + 𝜎 𝐴@ . [ . 𝑑𝑊F)(𝑡)
¨
H3(,)
3(,)
= 𝜇 𝑋2,@ . 𝑑𝑡 + 𝜎 𝑋2,@ . [ . 𝑑𝑊3(,)(𝑡)
¨
6 1!,$.;$
1!,$.;$
= 𝜇 𝑋,,. . 𝑑𝑡 + 𝜎 𝑋,,. . [ . 𝑑𝑊1!,$ 𝑡 + {𝜇 𝐴. . 𝑑𝑡 + 𝜎 𝐴. . [ . 𝑑𝑊;$ (𝑡)} +
61!,$.6;$
1!,$.;$
93

94. Luc_Faucheux_2021
Measures in the foreign and domestic world - IX
¨
6 1!,$.;$
1!,$.;$
= 𝜇 𝑋,,. . 𝑑𝑡 + 𝜎 𝑋,,. . [ . 𝑑𝑊1!,$ 𝑡 + {𝜇 𝐴. . 𝑑𝑡 + 𝜎 𝐴. . [ . 𝑑𝑊;$ (𝑡)} +
61!,$.6;$
1!,$.;$
¨
6 1!,$.;$
1!,$.;$
= {𝜇 𝑋,,. + 𝜇 𝐴. +
,
6<
.
61!,$.6;$
1!,$.;$
}. 𝑑𝑡 + 𝜎 𝑋,,. . [ . 𝑑𝑊1!,$ 𝑡 + 𝜎 𝐴. . [ . 𝑑𝑊;$ (𝑡)
¨ We define the correlation in the usual manner:
¨ 𝑑𝑋2,@. 𝑑𝐴@ = 𝑋2,@. 𝐴@. 𝜎 𝐴@ . 𝜎 𝑋2,@ . 𝜌 𝑋2,@; 𝐴@ . 𝑑𝑡
¨ 𝑑𝑊F) 𝑡 . 𝑑𝑊3(,) 𝑡 = 𝜌 𝑋2,@; 𝐴@ . 𝑑𝑡
¨
6 1!,$.;$
1!,$.;$
= 𝜇 𝑋,,. + 𝜇 𝐴. + 𝜎 𝐴. . 𝜎 𝑋,,. . 𝜌 𝑋,,.; 𝐴. . 𝑑𝑡 + 𝜎 𝑋,,. . [ . 𝑑𝑊1!,$ 𝑡 + 𝜎 𝐴. . [ . 𝑑𝑊;$ (𝑡)
¨ All right let’s do the same thing for [𝑋2←@. 𝐵@ 𝑡 ]
94

95. Luc_Faucheux_2021
ITO Leibniz on [𝑋!←#. 𝐵# 𝑡 ]
95

96. Luc_Faucheux_2021
Measures in the foreign and domestic world - X
¨ 𝐵@ 𝑡 = exp[∫
<=>
<=?
𝑅@ 𝑠, 𝑠, 𝑠 . 𝑑𝑠]
¨ 𝑑𝐵@ 𝑡 = 𝑅@ 𝑡, 𝑡, 𝑡 . 𝐵@ 𝑡 . 𝑑𝑡
¨ This works because even though 𝑅@ 𝑡, 𝑡, 𝑡 is stochastic, the equation for 𝐵@ 𝑡 or [𝑙𝑛𝐵@ 𝑡 ]
has only terms in 𝑑𝑡
¨ We could if we wanted to, do an ITO lemma:
¨ 𝐵@ 𝑡 = exp[∫
<=>
<=?
𝑅@ 𝑠, 𝑠, 𝑠 . 𝑑𝑠]
¨ 𝑙𝑛𝐵@ 𝑡 = ∫
<=>
<=?
𝑅@ 𝑠, 𝑠, 𝑠 . 𝑑𝑠
¨ 𝑑𝑙𝑛𝐵@ 𝑡 = 𝑙𝑛𝐵@ 𝑡 + 𝑑𝑡 − 𝑙𝑛𝐵@ 𝑡 = 𝑅@ 𝑡, 𝑡, 𝑡 . 𝑑𝑡
¨ 𝛿𝑓 =
IJ
IK
. 𝛿𝑋 +
2
@
.
I)J
IK) . (𝛿𝑋)@
¨ 𝑑𝐵. 𝑡 = 𝑑(exp 𝑙𝑛𝐵. 𝑡 ) = exp 𝑙𝑛𝐵. 𝑡 ) . 𝑑𝑙𝑛𝐵. 𝑡 +
,
.
. exp 𝑙𝑛𝐵. 𝑡 ) . (𝑑 𝑙𝑛𝐵. 𝑡 )).
96

97. Luc_Faucheux_2021
Measures in the foreign and domestic world - XI
¨ 𝑑𝐵. 𝑡 = 𝑑(exp 𝑙𝑛𝐵. 𝑡 ) = exp 𝑙𝑛𝐵. 𝑡 ) . 𝑑𝑙𝑛𝐵. 𝑡 +
,
.
. exp 𝑙𝑛𝐵. 𝑡 ) . (𝑑 𝑙𝑛𝐵. 𝑡 )).
¨ exp 𝑙𝑛𝐵@ 𝑡 ) = 𝐵@ 𝑡
¨ (𝑑 𝑙𝑛𝐵@ 𝑡 ))@= 0
¨ 𝑑𝐵@ 𝑡 = 𝑑(exp 𝑙𝑛𝐵@ 𝑡 ) = 𝐵@ 𝑡 . 𝑑𝑙𝑛𝐵@ 𝑡
¨ 𝑑𝑙𝑛𝐵@ 𝑡 = 𝑙𝑛𝐵@ 𝑡 + 𝑑𝑡 − 𝑙𝑛𝐵@ 𝑡 = 𝑅@ 𝑡, 𝑡, 𝑡 . 𝑑𝑡
¨ 𝑑𝐵@ 𝑡 = 𝑑(exp 𝑙𝑛𝐵@ 𝑡 ) = 𝐵@ 𝑡 . 𝑅@ 𝑡, 𝑡, 𝑡 . 𝑑𝑡
¨ 𝑑𝐵@ 𝑡 = 𝐵@ 𝑡 . 𝑅@ 𝑡, 𝑡, 𝑡 . 𝑑𝑡
¨ 𝐵@ 𝑡 = exp[∫
<=>
<=?
𝑅@ 𝑠, 𝑠, 𝑠 . 𝑑𝑠]
97

98. Luc_Faucheux_2021
Measures in the foreign and domestic world - XII
¨ 𝑓 = 𝑋2←@. 𝐵@ 𝑡 = 𝑋2,@. 𝐵@ 𝑡
¨ 𝛿𝑓 𝑋,,., 𝐵. =
/0
/1!,$
. 𝛿𝑋,,. +
/0
/=$
. 𝛿𝐵. +
,
.
.
/$0
/1!,$
$ . 𝛿𝑋,,.
.
+
,
.
.
/$0
/=$
$ . 𝛿𝐵.
.
+
/$0
/1!,$/=$
. 𝛿𝑋,,.. 𝛿𝐵.
¨ 𝑓 𝑋,,.. 𝐵. 𝑡 = 𝑋,,.. 𝐵. 𝑡
¨
IJ
I3(,)
= 𝐵@
¨
I)J
I3(,)
) = 0
¨
IJ
IE)
= 𝑋2,@
¨
I)J
IE)
) = 0
¨
I)J
IE)I3(,)
= 1
98

99. Luc_Faucheux_2021
Measures in the foreign and domestic world - XIII
¨ 𝛿𝑓 𝑋,,., 𝐵. =
/0
/1!,$
. 𝛿𝑋,,. +
/0
/=$
. 𝛿𝐵. +
,
.
.
/$0
/1!,$
$ . 𝛿𝑋,,.
.
+
,
.
.
/$0
/=$
$ . 𝛿𝐵.
.
+
/$0
/1!,$/=$
. 𝛿𝑋,,.. 𝛿𝐵.
¨ 𝛿𝑓 𝑋2,@, 𝐵@ = 𝐵@. 𝛿𝑋2,@ + 𝑋2,@. 𝛿𝐵@ +
2
@
. 0. 𝛿𝑋2,@
@
+
2
@
. 0. 𝛿𝐵@
@
+ 1. 𝛿𝑋2,@. 𝛿𝐵@
¨ 𝑑𝑓 𝑋2,@, 𝐵@ = 𝐵@. 𝑑𝑋2,@ + 𝑋2,@. 𝑑𝐵@ +
2
@
. 0. 𝑑𝑋2,@
@
+
2
@
. 0. 𝑑𝐵@
@
+ 1. 𝑑𝑋2,@. 𝑑𝐵@
¨ 𝑑𝑓 𝑋2,@, 𝐵@ = 𝑑 𝑋2,@. 𝐵@ = 𝐵@. 𝑑𝑋2,@ + 𝑋2,@. 𝑑𝐵@ + 𝑑𝑋2,@. 𝑑𝐵@
¨ Since:
¨ 𝑑𝐵@ 𝑡 = 𝐵@ 𝑡 . 𝑅@ 𝑡, 𝑡, 𝑡 . 𝑑𝑡
¨ We have:
¨ 𝑑𝑋2,@. 𝑑𝐵@ = 0
¨ 𝑑 𝑋2,@. 𝐵@ = 𝐵@. 𝑑𝑋2,@ + 𝑋2,@. 𝑑𝐵@
99

100. Luc_Faucheux_2021
Measures in the foreign and domestic world - XIV
¨ 𝑑 𝑋2,@. 𝐵@ = 𝐵@. 𝑑𝑋2,@ + 𝑋2,@. 𝑑𝐵@
¨ 𝑑𝐵@ 𝑡 = 𝐵@ 𝑡 . 𝑅@ 𝑡, 𝑡, 𝑡 . 𝑑𝑡
¨
H3(,)
3(,)
= 𝜇 𝑋2,@ . 𝑑𝑡 + 𝜎 𝑋2,@ . [ . 𝑑𝑊3(,)(𝑡)
¨
H(3(,).E))
3(,).E)
= 𝜇 𝑋2,@ + 𝑅@ 𝑡, 𝑡, 𝑡 . 𝑑𝑡 + 𝜎 𝑋2,@ . [ . 𝑑𝑊3(,)(𝑡)
¨ All right, we like a third of the way there.
¨ We are now going to look at the usual deflated quantities:
¨
[3(←).E) ? ]
E( ?
= r
[𝑋2←@. 𝐵@ 𝑡 ] and
[3(←).F) ? ]
E( ?
= r
[𝑋2,@. 𝐴@ 𝑡 ]
¨ Again, let’s crank the ITO Leibniz handle on those bad boys
100

101. Luc_Faucheux_2021
ITO Leibniz on
[%!←$.)$ * ]
,! *
101

102. Luc_Faucheux_2021
Measures in the foreign and domestic world - XV
¨ 𝑑 r
[𝑋2,@. 𝐴@] = 𝑑(
3(,).F)
E(
)
¨ Where we neglected to indicate the dependence on time for sake of clarity of notations
¨
6 1!,$.;$
1!,$.;$
= 𝜇 𝑋,,. + 𝜇 𝐴. + 𝜎 𝐴. . 𝜎 𝑋,,. . 𝜌 𝑋,,.; 𝐴. . 𝑑𝑡 + 𝜎 𝑋,,. . [ . 𝑑𝑊1!,$ 𝑡 + 𝜎 𝐴. . [ . 𝑑𝑊;$ (𝑡)
102

103. Luc_Faucheux_2021
Measures in the foreign and domestic world - XVI
¨ 𝑓 𝑋2,@. 𝐴@ 𝑡 , 𝐵2 𝑡 =
3(,).F) ?
E( ?
=
3(,).F)
E(
¨ 𝛿𝑓 𝑋!,#. 𝐴#, 𝐵! =
$%
$ &!,#.(#
. 𝛿(𝑋!,#. 𝐴#) +
$%
$)!
. 𝛿𝐵! +
!
#
.
$#%
$ &!,#.(#
# . 𝛿 𝑋!,#. 𝐴#
#
+
!
#
.
$#%
$)!
# . 𝛿𝐵!
#
+
$#%
$ &!,#.(# $)!
. 𝛿 𝑋!,#. 𝐴# . 𝛿𝐵!
¨
IJ
I 3(,).F)
=
2
E(
¨
I)J
I 3(,).F)
) = 0
¨
IJ
IE(
=
L 3(,).F)
E(
)
¨
I)J
IE(
) =
@ 3(,).F)
E(
*
¨
I)J
I 3(,).F) IE(
=
L2
E(
)
103

104. Luc_Faucheux_2021
Measures in the foreign and domestic world - XVII
¨ 𝛿𝑓 𝑋!,#. 𝐴#, 𝐵! =
$%
$ &!,#.(#
. 𝛿(𝑋!,#. 𝐴#) +
$%
$)!
. 𝛿𝐵! +
!
#
.
$#%
$ &!,#.(#
# . 𝛿 𝑋!,#. 𝐴#
#
+
!
#
.
$#%
$)!
# . 𝛿𝐵!
#
+
$#%
$ &!,#.(# $)!
. 𝛿 𝑋!,#. 𝐴# . 𝛿𝐵!
¨ 𝑑𝑓 𝑋!,#. 𝐴#, 𝐵! =
$%
$ &!,#.(#
. 𝑑(𝑋!,#. 𝐴#) +
$%
$)!
. 𝑑𝐵! +
!
#
.
$#%
$ &!,#.(#
# . 𝑑 𝑋!,#. 𝐴#
#
+
!
#
.
$#%
$)!
# . 𝑑𝐵!
#
+
$#%
$ &!,#.(# $)!
. 𝑑 𝑋!,#. 𝐴# . 𝑑𝐵!
¨ Now, since:
¨ 𝑑𝐵@ 𝑡 = 𝐵@ 𝑡 . 𝑅@ 𝑡, 𝑡, 𝑡 . 𝑑𝑡
¨ 𝑑𝐵2 𝑡 = 𝐵2 𝑡 . 𝑅2 𝑡, 𝑡, 𝑡 . 𝑑𝑡
¨
6 1!,$.;$
1!,$.;$
= 𝜇 𝑋,,. + 𝜇 𝐴. + 𝜎 𝐴. . 𝜎 𝑋,,. . 𝜌 𝑋,,.; 𝐴. . 𝑑𝑡 + 𝜎 𝑋,,. . [ . 𝑑𝑊1!,$ 𝑡 + 𝜎 𝐴. . [ . 𝑑𝑊;$ (𝑡)
¨ We have:
¨ 𝑑𝐵2
@
= 0
¨ 𝑑 𝑋2,@. 𝐴@ 𝑡 . 𝑑𝐵2 = 0
104

105. Luc_Faucheux_2021
Measures in the foreign and domestic world - XVIII
¨ 𝑑𝑓 𝑋!,#. 𝐴#, 𝐵! =
$%
$ &!,#.(#
. 𝑑(𝑋!,#. 𝐴#) +
$%
$)!
. 𝑑𝐵! +
!
#
.
$#%
$ &!,#.(#
# . 𝑑 𝑋!,#. 𝐴#
#
+
!
#
.
$#%
$)!
# . 𝑑𝐵!
#
+
$#%
$ &!,#.(# $)!
. 𝑑 𝑋!,#. 𝐴# . 𝑑𝐵!
¨ 𝑑𝑓 𝑋2,@. 𝐴@, 𝐵2 =
IJ
I 3(,).F)
. 𝑑(𝑋2,@. 𝐴@) +
IJ
IE(
. 𝑑𝐵2 +
2
@
.
I)J
I 3(,).F)
) . 𝑑 𝑋2,@. 𝐴@
@
¨ 𝑑𝑓 𝑋2,@. 𝐴@, 𝐵2 =
2
E(
. 𝑑(𝑋2,@. 𝐴@) + (
L 3(,).F)
E(
) ). 𝑑𝐵2 +
2
@
. 0. 𝑑 𝑋2,@. 𝐴@
@
¨ 𝑑𝑓 𝑋2,@. 𝐴@, 𝐵2 = 𝑑
3(,).F)
E(
=
2
E(
. 𝑑(𝑋2,@. 𝐴@) + (
L 3(,).F)
E(
) ). 𝑑𝐵2
105

106. Luc_Faucheux_2021
Measures in the foreign and domestic world - XIX
¨ 𝑑𝑓 𝑋2,@. 𝐴@, 𝐵2 = 𝑑
3(,).F)
E(
=
2
E(
. 𝑑(𝑋2,@. 𝐴@) + (
L 3(,).F)
E(
) ). 𝑑𝐵2
¨ 𝑑𝐵2 𝑡 = 𝐵2 𝑡 . 𝑅2 𝑡, 𝑡, 𝑡 . 𝑑𝑡
¨
6 1!,$.;$
1!,$.;$
= 𝜇 𝑋,,. + 𝜇 𝐴. + 𝜎 𝐴. . 𝜎 𝑋,,. . 𝜌 𝑋,,.; 𝐴. . 𝑑𝑡 + 𝜎 𝑋,,. . [ . 𝑑𝑊1!,$ 𝑡 + 𝜎 𝐴. . [ . 𝑑𝑊;$ (𝑡)
¨ 𝑑
3(,).F)
E(
=
2
E( ?
. 𝑑(𝑋2,@. 𝐴@) + (
L 3(,).F) ?
E( ? ) ). 𝐵2 𝑡 . 𝑅2 𝑡, 𝑡, 𝑡 . 𝑑𝑡
¨
H
+(,).>)
?(
(
+(,).>)
?(
)
= (𝑋2,@. 𝐴@). 𝑑(𝑋2,@. 𝐴@) − 𝑅2 𝑡, 𝑡, 𝑡 . 𝑑𝑡
¨
* &!,#.(#
&!,#.(#
= 𝜇 𝑋!,# + 𝜇 𝐴# + 𝜎 𝐴# . 𝜎 𝑋!,# . 𝜌 𝑋!,#; 𝐴# . 𝑑𝑡 + 𝜎 𝑋!,# . [ . 𝑑𝑊&!,# 𝑡 + 𝜎 𝐴# . [ . 𝑑𝑊(#(𝑡)
106

107. Luc_Faucheux_2021
Measures in the foreign and domestic world - XX
¨
H
+(,).>)
?(
(
+(,).>)
?(
)
= (𝑋2,@. 𝐴@). 𝑑(𝑋2,@. 𝐴@) − 𝑅2 𝑡, 𝑡, 𝑡 . 𝑑𝑡
¨
* &!,#.(#
&!,#.(#
= 𝜇 𝑋!,# + 𝜇 𝐴# + 𝜎 𝐴# . 𝜎 𝑋!,# . 𝜌 𝑋!,#; 𝐴# . 𝑑𝑡 + 𝜎 𝑋!,# . [ . 𝑑𝑊&!,# 𝑡 + 𝜎 𝐴# . [ . 𝑑𝑊(#(𝑡)
¨
*
$!,#.&#
'!
(
$!,#.&#
'!
)
= 𝜇 𝑋!,# + 𝜇 𝐴# + 𝜎 𝐴# . 𝜎 𝑋!,# . 𝜌 𝑋!,#; 𝐴# − 𝑅! 𝑡, 𝑡, 𝑡 . 𝑑𝑡 + 𝜎 𝑋!,# . [ . 𝑑𝑊&!,# 𝑡 + 𝜎 𝐴# . [ . 𝑑𝑊(#(𝑡)
¨ All right, almost there, let’s now do it for
[3(,).E)]
E(
= r
[𝑋2@. 𝐵@]
107

108. Luc_Faucheux_2021
ITO Leibniz on
[%!←$.,$ * ]
,! *
108

109. Luc_Faucheux_2021
Measures in the foreign and domestic world - XXI
¨ 𝑓 𝑋2,@. 𝐵@, 𝐵2 𝑡 =
3(,).E)
E(
=
3(,).E)
E(
¨ We can just replace in the previous slides 𝐴@ by 𝐵@
¨ 𝑑𝑓 𝑋2,@. 𝐴@, 𝐵2 = 𝑑
3(,).F)
E(
=
2
E(
. 𝑑(𝑋2,@. 𝐴@) + (
L 3(,).F)
E(
) ). 𝑑𝐵2
¨ Becomes:
¨ 𝑑𝑓 𝑋2,@. 𝐵@, 𝐵2 = 𝑑
3(,).E)
E(
=
2
E(
. 𝑑(𝑋2,@. 𝐵@) + (
L 3(,).E)
E(
) ). 𝑑𝐵2
¨ If you do not believe me, you can also redo the ITO Leibniz from scratch
109

110. Luc_Faucheux_2021
Measures in the foreign and domestic world - XXII
¨ 𝑑𝑓 𝑋2,@. 𝐵@, 𝐵2 = 𝑑
3(,).E)
E(
=
2
E(
. 𝑑(𝑋2,@. 𝐵@) + (
L 3(,).E)
E(
) ). 𝑑𝐵2
¨ 𝑑𝐵2 𝑡 = 𝐵2 𝑡 . 𝑅2 𝑡, 𝑡, 𝑡 . 𝑑𝑡
¨
H(3(,).E))
3(,).E)
= 𝜇 𝑋2,@ + 𝑅@ 𝑡, 𝑡, 𝑡 . 𝑑𝑡 + 𝜎 𝑋2,@ . [ . 𝑑𝑊3(,)(𝑡)
¨ 𝑑𝑓 𝑋2,@. 𝐵@, 𝐵2 = 𝑑
3(,).E)
E(
=
2
E(
. 𝑑(𝑋2,@. 𝐵@) + (
L 3(,).E)
E(
) ). 𝑑𝐵2
¨
H
+(,).?)
?(
+(,).?)
?(
= 𝜇 𝑋2,@ + 𝑅@ 𝑡, 𝑡, 𝑡 . 𝑑𝑡 + 𝜎 𝑋2,@ . [ . 𝑑𝑊3(,) 𝑡 − 𝑅2 𝑡, 𝑡, 𝑡 . 𝑑𝑡
¨
H
+(,).?)
?(
+(,).?)
?(
= 𝜇 𝑋2,@ + 𝑅@ 𝑡, 𝑡, 𝑡 − 𝑅2 𝑡, 𝑡, 𝑡 . 𝑑𝑡 + 𝜎 𝑋2,@ . [ . 𝑑𝑊3(,) 𝑡
110

111. Luc_Faucheux_2021
Measures in the foreign and domestic world - XXIII
¨ We have:
¨
*
$!,#.&#
'!
(
$!,#.&#
'!
)
= 𝜇 𝑋!,# + 𝜇 𝐴# + 𝜎 𝐴# . 𝜎 𝑋!,# . 𝜌 𝑋!,#; 𝐴# − 𝑅! 𝑡, 𝑡, 𝑡 . 𝑑𝑡 + 𝜎 𝑋!,# . [ . 𝑑𝑊&!,# 𝑡 + 𝜎 𝐴# . [ . 𝑑𝑊(#(𝑡)
¨
H
+(,).?)
?(
+(,).?)
?(
= 𝜇 𝑋2,@ + 𝑅@ 𝑡, 𝑡, 𝑡 − 𝑅2 𝑡, 𝑡, 𝑡 . 𝑑𝑡 + 𝜎 𝑋2,@ . [ . 𝑑𝑊3(,) 𝑡
¨
3(,).F)
E(
= r
(𝑋2,@. 𝐴@) using the notation for deflated variables
¨
3(,).E)
E(
= r
(𝑋2,@. 𝐵@)
111

112. Luc_Faucheux_2021
Measures in the foreign and domestic world – XXIV
¨ All right, we are like two thirds of the way there.
¨
* -
(&!,#.(#)
-
(&!,#.(#)
= 𝜇 𝑋!,# + 𝜇 𝐴# + 𝜎 𝐴# . 𝜎 𝑋!,# . 𝜌 𝑋!,#; 𝐴# − 𝑅! 𝑡, 𝑡, 𝑡 . 𝑑𝑡 + 𝜎 𝑋!,# . [ . 𝑑𝑊&!,# 𝑡 + 𝜎 𝐴# . [ . 𝑑𝑊(#(𝑡)
¨
H d
(3(,).E))
d
(3(,).E))
= 𝜇 𝑋2,@ + 𝑅@ 𝑡, 𝑡, 𝑡 − 𝑅2 𝑡, 𝑡, 𝑡 . 𝑑𝑡 + 𝜎 𝑋2,@ . [ . 𝑑𝑊3(,) 𝑡
¨ 𝑑𝑊3(,) 𝑡 and 𝑑𝑊F)(𝑡) are the “physical” Brownian motions associated to their respective
variables:
¨
H3(,)
3(,)
= 𝜇 𝑋2,@ . 𝑑𝑡 + 𝜎 𝑋2,@ . [ . 𝑑𝑊3(,)(𝑡)
¨
HF)
F)
= 𝜇 𝐴@ . 𝑑𝑡 + 𝜎 𝐴@ . [ . 𝑑𝑊F)(𝑡)
¨ We are now going to perform a change of measure, which we know consists in defining a new
Brownian motion with a drift (from the Girsanov theorem)
112

113. Luc_Faucheux_2021
Finding new Brownian motions so that the
deflated processes are martingales
113

114. Luc_Faucheux_2021
Measures in the foreign and domestic world – XXV
¨ So we are picking another Brownian motion such that the variance of the process is not
affected, only the expected value.
¨ We are going to define:
¨ 𝑑𝑊
ℚ
3(,)
𝑡 = 𝑑𝑊3(,) 𝑡 + 𝜑 𝑋2,@ . 𝑑𝑡
¨ 𝑑𝑊
ℚ
F)
𝑡 = 𝑑𝑊F) 𝑡 + 𝜑 𝐴@ . 𝑑𝑡
¨ The reason why we are denoting those with ℚ like the risk free measure will become soon
apparent, but for now we can just say that we are just picking another Brownian motion
which is defined in relationship to the initial one with an additive drift.
¨ We will actually solve for that drift so that the deflated processes are martingales under the
new measures.
114

115. Luc_Faucheux_2021
Measures in the foreign and domestic world – XXVI
¨
H d
(3(,).E))
d
(3(,).E))
= 𝜇 𝑋2,@ + 𝑅@ 𝑡, 𝑡, 𝑡 − 𝑅2 𝑡, 𝑡, 𝑡 . 𝑑𝑡 + 𝜎 𝑋2,@ . [ . 𝑑𝑊3(,) 𝑡
¨ 𝑑𝑊
ℚ
3(,)
𝑡 = 𝑑𝑊3(,) 𝑡 + 𝜑 𝑋2,@ . 𝑑𝑡
¨ So using the new Brownian motion we get:
¨
6 @
(1!,$.=$)
@
(1!,$.=$)
= 𝜇 𝑋,,. + 𝑅. 𝑡, 𝑡, 𝑡 − 𝑅, 𝑡, 𝑡, 𝑡 − 𝜑 𝑋,,. . 𝜎 𝑋,,. . 𝑑𝑡 + 𝜎 𝑋,,. . [ . 𝑑𝑊
ℚ
1!,$
𝑡
¨ We would want the risk measure associated to 𝑊
ℚ
3(,)
to be the risk free measure, meaning
that tradeable instruments deflated by the Bank Account numeraire are martingale in this
measure, meaning they are driftless, or:
¨ 𝜇 𝑋2,@ + 𝑅@ 𝑡, 𝑡, 𝑡 − 𝑅2 𝑡, 𝑡, 𝑡 − 𝜑 𝑋2,@ . 𝜎 𝑋2,@ = 0
115

116. Luc_Faucheux_2021
Measures in the foreign and domestic world – XXVII
¨ 𝜇 𝑋2,@ + 𝑅@ 𝑡, 𝑡, 𝑡 − 𝑅2 𝑡, 𝑡, 𝑡 − 𝜑 𝑋2,@ . 𝜎 𝑋2,@ = 0
¨ 𝜑 𝑋2,@ =
e 3(,) fg) ?,?,? Lg( ?,?,?
S 3(,)
¨ 𝜑 𝑋2,@ =
e 3(,) L(g( ?,?,? Lg) ?,?,? )
S 3(,)
¨ This expression should be familiar, and reminds us of the market price of risk that we looked
at in the single currency case
¨ The difference here is that the rate 𝑅 𝑡, 𝑡, 𝑡 is actually the differential of the the domestic
and the foreign rate:
¨ 𝑅 𝑡, 𝑡, 𝑡 = 𝑅2 𝑡, 𝑡, 𝑡 − 𝑅@ 𝑡, 𝑡, 𝑡
¨ 𝜑 𝑋2,@ =
e 3(,) Lg ?,?,?
S 3(,)
116

117. Luc_Faucheux_2021
Measures in the foreign and domestic world – XXVIII
¨
* -
(&!,#.(#)
-
(&!,#.(#)
= 𝜇 𝑋!,# + 𝜇 𝐴# + 𝜎 𝐴# . 𝜎 𝑋!,# . 𝜌 𝑋!,#; 𝐴# − 𝑅! 𝑡, 𝑡, 𝑡 . 𝑑𝑡 + 𝜎 𝑋!,# . [ . 𝑑𝑊&!,# 𝑡 + 𝜎 𝐴# . [ . 𝑑𝑊(#(𝑡)
¨ 𝑑𝑊
ℚ
F)
𝑡 = 𝑑𝑊F) 𝑡 + 𝜑 𝐴@ . 𝑑𝑡
¨ 𝑑𝑊
ℚ
3(,)
𝑡 = 𝑑𝑊3(,) 𝑡 + 𝜑 𝑋2,@ . 𝑑𝑡
¨ So using the new Brownian motions we get:
¨
6 @
(1!,$.;$)
@
(1!,$.;$)
= 𝜇 𝑋,,. + 𝜇 𝐴. + 𝜎 𝐴. . 𝜎 𝑋,,. . 𝜌 𝑋,,.; 𝐴. − 𝑅, . 𝑑𝑡 + 𝜎 𝑋,,. . 𝑑𝑊1!,$ + 𝜎 𝐴. . 𝑑𝑊;$
¨
6 @
(1!,$.;$)
@
(1!,$.;$)
= 𝜇 𝑋,,. + 𝜇 𝐴. + 𝜎 𝐴. . 𝜎 𝑋,,. . 𝜌 𝑋,,.; 𝐴. − 𝑅, − 𝜑 𝑋,,. . 𝜎 𝑋,,. − 𝜑 𝐴. . 𝜎 𝐴. . 𝑑𝑡 +
𝜎 𝑋,,. . 𝑑𝑊
ℚ
1!,$
+ 𝜎 𝐴. . 𝑑𝑊
ℚ
;$
¨ For this process to be driftless we require:
¨ 𝜇 𝑋2,@ + 𝜇 𝐴@ + 𝜎 𝐴@ . 𝜎 𝑋2,@ . 𝜌 𝑋2,@; 𝐴@ − 𝑅2 − 𝜑 𝑋2,@ . 𝜎 𝑋2,@ − 𝜑 𝐴@ . 𝜎 𝐴@ = 0
117

118. Luc_Faucheux_2021
Measures in the foreign and domestic world – XXIX
¨ So we have the set of two equations:
¨ 𝜇 𝑋2,@ + 𝑅@ 𝑡, 𝑡, 𝑡 − 𝑅2 𝑡, 𝑡, 𝑡 − 𝜑 𝑋2,@ . 𝜎 𝑋2,@ = 0
¨ 𝜇 𝑋2,@ + 𝜇 𝐴@ + 𝜎 𝐴@ . 𝜎 𝑋2,@ . 𝜌 𝑋2,@; 𝐴@ − 𝑅2 − 𝜑 𝑋2,@ . 𝜎 𝑋2,@ − 𝜑 𝐴@ . 𝜎 𝐴@ = 0
¨ Let’s plug:
¨ 𝜑 𝑋2,@ =
e 3(,) fg) ?,?,? Lg( ?,?,?
S 3(,)
¨ Into the second equation
¨ 𝜇 𝑋2,@ + 𝜇 𝐴@ + 𝜎 𝐴@ . 𝜎 𝑋2,@ . 𝜌 𝑋2,@; 𝐴@ − 𝑅2 − 𝜑 𝑋2,@ . 𝜎 𝑋2,@ − 𝜑 𝐴@ . 𝜎 𝐴@ = 0
¨ 𝜇 𝑋!,# + 𝜇 𝐴# + 𝜎 𝐴# . 𝜎 𝑋!,# . 𝜌 𝑋!,#; 𝐴# − 𝑅! − {𝜇 𝑋!,# + 𝑅# 𝑡, 𝑡, 𝑡 − 𝑅! 𝑡, 𝑡, 𝑡 } − 𝜑 𝐴# . 𝜎 𝐴# = 0
¨ 𝜇 𝐴@ + 𝜎 𝐴@ . 𝜎 𝑋2,@ . 𝜌 𝑋2,@; 𝐴@ − {𝑅@ 𝑡, 𝑡, 𝑡 } − 𝜑 𝐴@ . 𝜎 𝐴@ = 0
118

119. Luc_Faucheux_2021
Measures in the foreign and domestic world – XXX
¨ So the conditions on the drift for the processes to be martingales in the new measures
associated with the new Brownian motions are:
¨ 𝜑 𝑋2,@ . 𝜎 𝑋2,@ = 𝜇 𝑋2,@ + 𝑅@ 𝑡, 𝑡, 𝑡 − 𝑅2 𝑡, 𝑡, 𝑡
¨ 𝜑 𝐴@ . 𝜎 𝐴@ = 𝜇 𝐴@ + 𝜎 𝐴@ . 𝜎 𝑋2,@ . 𝜌 𝑋2,@; 𝐴@ − 𝑅@ 𝑡, 𝑡, 𝑡
¨ The processes with those conditions now read:
¨
H d
(3(,).E))
d
(3(,).E))
= 𝜎 𝑋2,@ . [ . 𝑑𝑊
ℚ
3(,)
𝑡
¨
H d
(3(,).F))
d
(3(,).F))
= 𝜎 𝑋2,@ . 𝑑𝑊
ℚ
3(,)
+ 𝜎 𝐴@ . 𝑑𝑊
ℚ
F)
¨ I know that in some other decks I had the ℚ on top and not at the bottom, but I would have
needed to start with 𝑊3(,)
𝑡 . Next time I rewrite this deck, will try to make it look nicer
and more consistent with the previous decks, sorry for that
119

120. Luc_Faucheux_2021
Measures in the foreign and domestic world – XXXI
¨
H d
(3(,).E))
d
(3(,).E))
= 𝜎 𝑋2,@ . [ . 𝑑𝑊
ℚ
3(,)
𝑡
¨
H d
(3(,).F))
d
(3(,).F))
= 𝜎 𝑋2,@ . 𝑑𝑊
ℚ
3(,)
+ 𝜎 𝐴@ . 𝑑𝑊
ℚ
F)
¨ All right, now we need to unfold back the deflated processes back to the original processes
that we started with: 𝑋2,@ and 𝐴@
¨ Time to crank the ITO Leibniz handle again
120

121. Luc_Faucheux_2021
Measures in the foreign and domestic world – XXXII
¨
H d
(3(,).E))
d
(3(,).E))
= 𝜎 𝑋2,@ . [ . 𝑑𝑊
ℚ
3(,)
𝑡
¨
3(,).E)
E(
= r
(𝑋2,@. 𝐵@)
¨ 𝑋2,@. 𝐵@ = 𝐵2. r
(𝑋2,@. 𝐵@)
¨ 𝑑𝐵2 𝑡 = 𝐵2 𝑡 . 𝑅2 𝑡, 𝑡, 𝑡 . 𝑑𝑡
¨ We will do Ito Leibniz on 𝑓 𝐵2, r
𝑋2,@. 𝐵@ = 𝐵2. r
(𝑋2,@. 𝐵@)
121

122. Luc_Faucheux_2021
Going back up one level from the dream
ITO Leibniz on:
𝐵!. ,
𝑋!,#. 𝐵# = 𝐵!.
%!,$.,$
,!
= 𝑋!,#. 𝐵#
122

123. Luc_Faucheux_2021
Measures in the foreign and domestic world – XXXIII
¨ 𝑓 𝐵2, r
𝑋2,@. 𝐵@ = 𝐵2. r
(𝑋2,@. 𝐵@)
¨ 𝑑𝑓 𝐵(, @
𝑋(,). 𝐵) =
*+
* ,
-!,#..#
. 𝑑 @
𝑋(,). 𝐵) +
*+
*.!
. 𝑑𝐵( +
(
)
.
*#+
* ,
-!,#..#
# . 𝑑 @
𝑋(,). 𝐵)
)
+
(
)
.
*#+
*.!
# . 𝑑𝐵(
)
+
*#+
* ,
-!,#..# *.!
. 𝑑 @
𝑋(,). 𝐵) . 𝑑𝐵(
¨
IJ
I d
3(,).E)
= 𝐵2
¨
I)J
I d
3(,).E)
) = 0
¨
IJ
IE(
= r
(𝑋2,@. 𝐵@)
¨
I)J
IE(
) = 0
¨
I)J
I d
3(,).E) IE(
= 1
123

124. Luc_Faucheux_2021
Measures in the foreign and domestic world – XXXIV
¨ 𝑑𝑓 𝐵!, d
𝑋!,#. 𝐵# =
$%
$ -
&!,#.)#
. 𝑑 d
𝑋!,#. 𝐵# +
$%
$)!
. 𝑑𝐵! +
!
#
.
$#%
$ -
&!,#.)#
# . 𝑑 d
𝑋!,#. 𝐵#
#
+
!
#
.
$#%
$)!
# . 𝑑𝐵!
#
+
$#%
$ -
&!,#.)# $)!
. 𝑑 d
𝑋!,#. 𝐵# . 𝑑𝐵!
¨ 𝑑𝑓 𝐵!, d
𝑋!,#. 𝐵# = 𝐵!. 𝑑 d
𝑋!,#. 𝐵# + d
(𝑋!,#. 𝐵#). 𝑑𝐵! +
!
#
. 0. 𝑑 d
𝑋!,#. 𝐵#
#
+
!
#
. 0. 𝑑𝐵!
#
+ 1. 𝑑 d
𝑋!,#. 𝐵# . 𝑑𝐵!
¨ 𝑑𝑓 𝐵2, r
𝑋2,@. 𝐵@ = 𝐵2. 𝑑 r
𝑋2,@. 𝐵@ + r
(𝑋2,@. 𝐵@). 𝑑𝐵2 + 1. 𝑑 r
𝑋2,@. 𝐵@ . 𝑑𝐵2
¨ And we have:
¨ 𝑑𝐵2 𝑡 = 𝐵2 𝑡 . 𝑅2 𝑡, 𝑡, 𝑡 . 𝑑𝑡
¨
H d
(3(,).E))
d
(3(,).E))
= 𝜎 𝑋2,@ . [ . 𝑑𝑊
ℚ
3(,)
𝑡
¨ So we have 𝑑 r
𝑋2,@. 𝐵@ . 𝑑𝐵2 = 0
¨ 𝑑𝑓 𝐵2, r
𝑋2,@. 𝐵@ = 𝐵2. 𝑑 r
𝑋2,@. 𝐵@ + r
(𝑋2,@. 𝐵@). 𝑑𝐵2
124

125. Luc_Faucheux_2021
Measures in the foreign and domestic world – XXXV
¨ 𝑑𝑓 𝐵2, r
𝑋2,@. 𝐵@ = 𝑑 𝐵2. r
𝑋2,@. 𝐵@ = 𝐵2. 𝑑 r
𝑋2,@. 𝐵@ + r
(𝑋2,@. 𝐵@). 𝑑𝐵2
¨ 𝑑𝐵2 𝑡 = 𝐵2 𝑡 . 𝑅2 𝑡, 𝑡, 𝑡 . 𝑑𝑡
¨
H d
(3(,).E))
d
(3(,).E))
= 𝜎 𝑋2,@ . [ . 𝑑𝑊
ℚ
3(,)
𝑡
¨ 𝑑 𝐵2. r
𝑋2,@. 𝐵@ = 𝐵2. r
𝑋2,@. 𝐵@ . 𝜎 𝑋2,@ . [ . 𝑑𝑊
ℚ
3(,)
𝑡 + r
(𝑋2,@. 𝐵@). 𝐵2 𝑡 . 𝑅2 𝑡, 𝑡, 𝑡 . 𝑑𝑡
¨ And since:
¨
3(,).E)
E(
= r
(𝑋2,@. 𝐵@)
¨ 𝐵2. r
𝑋2,@. 𝐵@ = 𝐵2.
3(,).E)
E(
= 𝑋2,@. 𝐵@
¨ 𝑑 𝑋2,@. 𝐵@ = 𝑋2,@. 𝐵@. 𝜎 𝑋2,@ . [ . 𝑑𝑊
ℚ
3(,)
𝑡 + 𝑋2,@. 𝐵@. 𝑅2 𝑡, 𝑡, 𝑡 . 𝑑𝑡
125

126. Luc_Faucheux_2021
Measures in the foreign and domestic world – XXXVI
¨
H d
(3(,).E))
d
(3(,).E))
= 𝜎 𝑋2,@ . [ . 𝑑𝑊
ℚ
3(,)
𝑡
¨
H(3(,).E))
(3(,).E))
= 𝑅2 𝑡, 𝑡, 𝑡 . 𝑑𝑡 + 𝜎 𝑋2,@ . [ . 𝑑𝑊
ℚ
3(,)
𝑡
¨ We should be accustomed to this by now, going between the deflated process and the
process in the risk free measure amounts to just a drift 𝑅2 𝑡, 𝑡, 𝑡 . 𝑑𝑡
¨ Let’s now go one step further unfolding our processes:
¨
H(3(,).E))
(3(,).E))
= 𝑅2 𝑡, 𝑡, 𝑡 . 𝑑𝑡 + 𝜎 𝑋2,@ . [ . 𝑑𝑊
ℚ
3(,)
𝑡
¨ 𝑑𝐵@ 𝑡 = 𝐵@ 𝑡 . 𝑅@ 𝑡, 𝑡, 𝑡 . 𝑑𝑡
¨ We will do Ito Leibniz on: 𝑓 𝐵@, (𝑋2,@. 𝐵@) =
2
E)
. (𝑋2,@. 𝐵@)
126

127. Luc_Faucheux_2021
Going back up one level from the dream
ITO Leibniz on:
1
𝐵#
. (𝑋!,#. 𝐵#) = 𝑋!,#
127

128. Luc_Faucheux_2021
Measures in the foreign and domestic world – XXXVII
¨ 𝑓 𝐵@, (𝑋2,@. 𝐵@) =
2
E)
. (𝑋2,@. 𝐵@)
¨ 𝑑𝑓 𝐵#, (𝑋!,#. 𝐵#) =
$%
$(&!,#.)#)
. 𝑑(𝑋!,#. 𝐵#) +
$%
$)#
. 𝑑𝐵# +
!
#
.
$#%
$(&!,#.)#)# . 𝑑(𝑋!,#. 𝐵#)# +
!
#
.
$#%
$)#
# . 𝑑𝐵#
#
+
$#%
$(&!,#.)#).$)#
. 𝑑(𝑋!,#. 𝐵#). 𝑑𝐵#
¨
IJ
I(3(,).E))
=
2
E)
¨
I)J
I(3(,).E))) = 0
¨
IJ
IE)
=
L2
E)
) . (𝑋2,@. 𝐵@)
¨
I)J
IE)
) =
@
E)
* . (𝑋2,@. 𝐵@)
¨
I)J
I(3(,).E)).IE)
=
L2
E)
)
128

129. Luc_Faucheux_2021
Measures in the foreign and domestic world – XXXVIII
¨ 𝑑𝑓 𝐵#, (𝑋!,#. 𝐵#) =
$%
$(&!,#.)#)
. 𝑑(𝑋!,#. 𝐵#) +
$%
$)#
. 𝑑𝐵# +
!
#
.
$#%
$(&!,#.)#)# . 𝑑(𝑋!,#. 𝐵#)# +
!
#
.
$#%
$)#
# . 𝑑𝐵#
#
+
$#%
$(&!,#.)#).$)#
. 𝑑(𝑋!,#. 𝐵#). 𝑑𝐵#
¨ 𝑑𝑓 𝐵#, (𝑋!,#. 𝐵#) =
!
)#
. 𝑑(𝑋!,#. 𝐵#) + (
.!
)#
# . (𝑋!,#. 𝐵#)). 𝑑𝐵# +
!
#
. 0. 𝑑(𝑋!,#. 𝐵#)#
+
!
#
.
#
)#
/ . (𝑋!,#. 𝐵#). 𝑑𝐵#
#
+ (
.!
)#
#). 𝑑(𝑋!,#. 𝐵#). 𝑑𝐵#
¨ 𝑑𝑓 𝐵#, (𝑋!,#. 𝐵#) =
!
)#
. 𝑑(𝑋!,#. 𝐵#) + (
.!
)#
# . (𝑋!,#. 𝐵#)). 𝑑𝐵# +
!
#
.
#
)#
/ . (𝑋!,#. 𝐵#). 𝑑𝐵#
#
+ (
.!
)#
#). 𝑑(𝑋!,#. 𝐵#). 𝑑𝐵#
¨
H(3(,).E))
(3(,).E))
= 𝑅2 𝑡, 𝑡, 𝑡 . 𝑑𝑡 + 𝜎 𝑋2,@ . [ . 𝑑𝑊
ℚ
3(,)
𝑡
¨ 𝑑𝐵@ 𝑡 = 𝐵@ 𝑡 . 𝑅@ 𝑡, 𝑡, 𝑡 . 𝑑𝑡
¨ And so we also have:
¨ 𝑑(𝑋2,@. 𝐵@). 𝑑𝐵@ = 0
¨ 𝑑𝐵@
@
= 0
¨ 𝑑𝑓 𝐵@, (𝑋2,@. 𝐵@) =
2
E)
. 𝑑(𝑋2,@. 𝐵@) + (
L2
E)
) . (𝑋2,@. 𝐵@)). 𝑑𝐵@
129

130. Luc_Faucheux_2021
Measures in the foreign and domestic world – XXXIX
¨ 𝑑𝑓 𝐵@, (𝑋2,@. 𝐵@) = 𝑑(
2
E)
. (𝑋2,@. 𝐵@)) =
2
E)
. 𝑑(𝑋2,@. 𝐵@) + (
L2
E)
) . (𝑋2,@. 𝐵@)). 𝑑𝐵@
¨ 𝑑(
2
E)
. (𝑋2,@. 𝐵@)) = 𝑑𝑋2,@ =
2
E)
. 𝑑(𝑋2,@. 𝐵@) + (
L2
E)
) . (𝑋2,@. 𝐵@)). 𝑑𝐵@
¨
H(3(,).E))
(3(,).E))
= 𝑅2 𝑡, 𝑡, 𝑡 . 𝑑𝑡 + 𝜎 𝑋2,@ . [ . 𝑑𝑊
ℚ
3(,)
𝑡
¨ 𝑑𝐵@ 𝑡 = 𝐵@ 𝑡 . 𝑅@ 𝑡, 𝑡, 𝑡 . 𝑑𝑡
¨ And so we get:
¨ 𝑑𝑋,,. =
(1!,$.=$)
=$
. {𝑅, 𝑡, 𝑡, 𝑡 . 𝑑𝑡 + 𝜎 𝑋,,. . [ . 𝑑𝑊
ℚ
1!,$
𝑡 } + (
4,
=$
$ . (𝑋,,.. 𝐵.)). 𝐵. 𝑡 . 𝑅. 𝑡, 𝑡, 𝑡 . 𝑑𝑡
¨ 𝑑𝑋,,. = 𝑋,,.. 𝑅, 𝑡, 𝑡, 𝑡 . 𝑑𝑡 + 𝜎 𝑋,,. . [ . 𝑑𝑊
ℚ
1!,$
𝑡 − 𝑋,,.. 𝑅. 𝑡, 𝑡, 𝑡 . 𝑑𝑡
¨
61!,$
1!,$
= {𝑅, 𝑡, 𝑡, 𝑡 − 𝑅. 𝑡, 𝑡, 𝑡 }. 𝑑𝑡 + 𝜎 𝑋,,. . [ . 𝑑𝑊
ℚ
1!,$
𝑡
130

131. Luc_Faucheux_2021
Measures in the foreign and domestic world – XXXX
¨
H3(,)
3(,)
= {𝑅2 𝑡, 𝑡, 𝑡 − 𝑅@ 𝑡, 𝑡, 𝑡 }. 𝑑𝑡 + 𝜎 𝑋2,@ . [ . 𝑑𝑊
ℚ
3(,)
𝑡
¨ Let’s stop here for a moment
¨ We started with:
¨
H3(,)
3(,)
= 𝜇 𝑋2,@ . 𝑑𝑡 + 𝜎 𝑋2,@ . [ . 𝑑𝑊3(,)(𝑡)
¨ Through change of measure and martingale / driftless process, we showed that using the
Brownian motion associated with the risk free measure in the domestic world:
¨
H3(,)
3(,)
= {𝑅2 𝑡, 𝑡, 𝑡 − 𝑅@ 𝑡, 𝑡, 𝑡 }. 𝑑𝑡 + 𝜎 𝑋2,@ . [ . 𝑑𝑊
ℚ
3(,)
𝑡
¨ This seems like an awful lot of slides and math to recover something that actually makes a
lot of sense
131

132. Luc_Faucheux_2021
Measures in the foreign and domestic world – XXXXI
¨
H3(,)
3(,)
= {𝑅2 𝑡, 𝑡, 𝑡 − 𝑅@ 𝑡, 𝑡, 𝑡 }. 𝑑𝑡 + 𝜎 𝑋2,@ . [ . 𝑑𝑊
ℚ
3(,)
𝑡
¨ A lot of time, it is super helpful to check our results against our intuition, especially using the
deterministic case of zero volatility
¨ 𝜎 𝑋2,@ = 0
¨
H3(,)
3(,)
= {𝑅2 𝑡, 𝑡, 𝑡 − 𝑅@ 𝑡, 𝑡, 𝑡 }. 𝑑𝑡
¨ This is actually what I teach in the undergraduate class on Money, Banking and Financial
Markets
¨ Oh also just learnt that I was let go of Natixis this morning
¨ So if any of you know of any interesting job around NYC, Chicago, Miami or San Diego, drop
me a line .
132

133. Luc_Faucheux_2021
Measures in the foreign and domestic world – XXXXII
¨ In any case, here is the textbook that we use in class. It is really good!
133

134. Luc_Faucheux_2021
Measures in the foreign and domestic world – XXXXIII
¨
H3(,)
3(,)
= {𝑅2 𝑡, 𝑡, 𝑡 − 𝑅@ 𝑡, 𝑡, 𝑡 }. 𝑑𝑡
¨ This actually makes sense using the concept of Purchasing Power Parity (p.246 in Cecchetti),
also saying that the real exchange rate should be equal to 1
¨ If 𝑃2 is the price in currency (1) of a basket of goods (coffee, burger, computer, basis goods
that are somewhat transportable or comparable), and 𝑃@ the price in currency (2) of that
same (or very similar) basket of goods, and if people in both the foreign and domestic world
essentially value equally that basket of goods (which is a big assumption), then the nominal
exchange rate is given by:
¨ 𝑋2,@ = 𝑋2←@ =
m(
m)
¨ 𝑃2 = 𝑋2←@. 𝑃@
134

135. Luc_Faucheux_2021
Measures in the foreign and domestic world – XXXXIV
¨ 𝑃2 = 𝑋2←@. 𝑃@
¨ That also makes sense when I use the “plane travel” example when I teach that class.
¨ You live in the foreign country with foreign currency (2)
¨ You have a basket of goods 𝑃@
¨ You sell it to get an amount of currency (2)
¨ You come to the domestic country with currency (1) (either walking or flying or swimming,
let’s assume that there is no cost to travel, no restrictions due to pandemic or political
differences between the countries,…), with your handful bills or coins of currency (2), go to
exchange them for another handful of bills or coins of currency (1), and then proceed to buy
an equivalent basket of goods 𝑃2 so that you can settle and live in that new country of yours.
¨ If things are sort of the same, then 𝑃2 = 𝑋2←@. 𝑃@
¨ Note that before I made sure to note that 𝐴2 ≠ 𝑋2←@. 𝐴@ and 𝐵2 ≠ 𝑋2←@. 𝐵@
135

136. Luc_Faucheux_2021
Measures in the foreign and domestic world – XXXXV
¨ 𝑃2 = 𝑋2←@. 𝑃@
¨ 𝑋2,@ = 𝑋2←@ =
m(
m)
¨ In the classroom, we then usually equate inflation to the return of the basket of goods (to
some approximation), and say that over the long run, the return on the nominal exchange
rate is equal to the return on the basket of goods 𝑃2 minus the return on the basket of
goods 𝑃@
¨ In the deterministic world, we are on the safe and firm ground of Newtonian calculus
¨ 𝑋2,@ = 𝑋2←@ =
m(
m)
¨ 𝑑𝑋2,@ =
2
m)
. 𝑑𝑃2 −
m(
m)
) . 𝑑𝑃@
136

137. Luc_Faucheux_2021
Measures in the foreign and domestic world – XXXXVI
¨ 𝑋2,@ = 𝑋2←@ =
m(
m)
¨ 𝑑𝑋2,@ =
2
m)
. 𝑑𝑃2 −
m(
m)
) . 𝑑𝑃@
¨ If we assume that the inflation in each country is noted 𝑅2 and 𝑅@, the inflation is then the
rate of return of each basket of goods (again, that makes sense)
¨ 𝑑𝑃2 = 𝑅2. 𝑃2. 𝑑𝑡
¨ 𝑑𝑃@ = 𝑅@. 𝑃@. 𝑑𝑡
¨ 𝑑𝑋2,@ =
2
m)
. 𝑑𝑃2 −
m(
m)
) . 𝑑𝑃@
¨ 𝑑𝑋2,@ =
2
m)
. 𝑅2. 𝑃2. 𝑑𝑡 −
m(
m)
) . 𝑅@. 𝑃@. 𝑑𝑡 =
m(
m)
. 𝑅2 − 𝑅@ . 𝑑𝑡 = 𝑋2,@. 𝑅2 − 𝑅@ . 𝑑𝑡
137

138. Luc_Faucheux_2021
Measures in the foreign and domestic world – XXXXVII
¨ 𝑑𝑋2,@ = 𝑋2,@. 𝑅2 − 𝑅@ . 𝑑𝑡
¨
H3(,)
3(,)
= (𝑅2−𝑅@). 𝑑𝑡
¨ To be compared to what we had obtained after a couple thousand slides of stochastic
calculus:
¨
H3(,)
3(,)
= {𝑅2 𝑡, 𝑡, 𝑡 − 𝑅@ 𝑡, 𝑡, 𝑡 }. 𝑑𝑡
138

139. Luc_Faucheux_2021
Measures in the foreign and domestic world – XXXXVIII
¨ This is what Cecchetti and Schoenholtz explains on p.246, making sur to point out that such
arguments tend to be true over the long run, and that short term behaviors of rates and FX
are much more random and sometimes violent in nature, and driven by supply and demand.
139

140. Luc_Faucheux_2021
Measures in the foreign and domestic world - IL
¨ Ahhh !! I got you !
¨ The Roman numeral for 49 is NOT IL, because I can only be subtracted from V and X
¨ Wanted to see if you guys were following.
¨ Do not trust me ? To the Google Robin !
¨ But if you follow superbowl and such, you know all about Roman numerals
140

141. Luc_Faucheux_2021
Measures in the foreign and domestic world - L
¨ OK, so now here is an amazing graph (although to be fully transparent I think that they
should have plotted it on a logarithmic scale, not doing it is a tad ethnocentric if you ask
me).
¨ But in any case, they looks at returns over 1980 to 2010 on the USD exchange rate for a
number of countries (that is easy to do)
¨ Then they looked at the difference in annualized inflation rate between that country and the
US (that is a little harder to do, because how you measure inflation is not super easy, is that
CPI, is that PPI, does the basket change over time, like computer percentage in the basket, or
rentals, or gas,..). But anyways they did their homework and produced a graph that is quite
impressive !
¨ Over the long run, and with some common sense assumptions, we do have indeed:
¨
H3(,)
3(,)
= {𝑅2 𝑡, 𝑡, 𝑡 − 𝑅@ 𝑡, 𝑡, 𝑡 }. 𝑑𝑡 + 𝜎 𝑋2,@ . [ . 𝑑𝑊
ℚ
3(,)
𝑡
141

142. Luc_Faucheux_2021
Measures in the foreign and domestic world - LI
142
𝑑𝑋!,#
𝑋!,#
= {𝑅! 𝑡, 𝑡, 𝑡 − 𝑅# 𝑡, 𝑡, 𝑡 }. 𝑑𝑡
IT WORKS !!!!

143. Luc_Faucheux_2021
Measures in the foreign and domestic world - LII
¨ When the real exchange rate deviates from 1, then things get a little weirder. Let’s see if you
could answer on of the questions I ask the class every week:
143

144. Luc_Faucheux_2021
Going from X to (1/X)
144

145. Luc_Faucheux_2021
Going from X to (1/X)
¨ Before we finish the deck, let’s go over some quick notes on going from X to (1/X)
¨
H3(,)
3(,)
= {𝑅2 𝑡, 𝑡, 𝑡 − 𝑅@ 𝑡, 𝑡, 𝑡 }. 𝑑𝑡 + 𝜎 𝑋2,@ . [ . 𝑑𝑊
ℚ
3(,)
𝑡
¨ Now of course by symmetry we have:
¨
H3),(
3),(
= {𝑅@ 𝑡, 𝑡, 𝑡 − 𝑅2 𝑡, 𝑡, 𝑡 }. 𝑑𝑡 + 𝜎 𝑋@,2 . [ . 𝑑𝑊
ℚ
3),(
𝑡
¨ We already know that:
¨ 𝑋@,2 =
2
3(,)
¨ And that:
¨ 𝜎 𝑋2,@ = 𝜎 𝑋@,2
145

146. Luc_Faucheux_2021
Going from X to (1/X) - II
¨ Let’s use good old friend ITO lemma on 𝑓 𝑋2,@ = (
2
3(,)
)
¨ 𝛿𝑓 =
IJ
IK
. 𝛿𝑋 +
2
@
.
I)J
IK) . (𝛿𝑋)@
¨
IJ
I3(,)
=
L2
3(,)
)
¨
I)J
I3(,)
) =
@
3(,)
*
¨ 𝑑𝑋@,2 = 𝑑
2
3(,)
= 𝑑𝑓(𝑋2,@) =
IJ
I3(,)
. 𝑑𝑋2,@ +
2
@
.
I)J
I3(,)
) . (𝑑𝑋2,@)@
¨ Note that again, we really are dealing with a regular function 𝑓 𝑥 = 1/𝑥
¨ Which is nicely differentiable and where the rules of regular (Newtonian) calculus do apply
146

147. Luc_Faucheux_2021
Going from X to (1/X) - III
¨ So really we should be writing to be rigorous:
¨ 𝑑𝑓(𝑋) =
IJ
IK
|K=3 ? . 𝑑𝑋 +
2
@
.
I)J
IK) |K=3 ? . (𝑑𝑋)@
¨ Which is usually abbreviated for sake of simplicity to:
¨ 𝑑𝑓(𝑋) =
IJ
I3
. 𝑑𝑋 +
2
@
.
I)J
I3) . (𝑑𝑋)@
¨ But bear in mind that this is the whole point of stochastic processes, is that they are not
differentiable.
¨ So writing something like
IJ
I3
is fraught with peril
147

148. Luc_Faucheux_2021
Going from X to (1/X) - IV
¨ 𝑑𝑋@,2 = 𝑑
2
3(,)
= 𝑑𝑓(𝑋2,@) =
IJ
I3(,)
. 𝑑𝑋2,@ +
2
@
.
I)J
I3(,)
) . (𝑑𝑋2,@)@
¨
IJ
I3(,)
=
L2
3(,)
)
¨
I)J
I3(,)
) =
@
3(,)
*
¨ 𝑑𝑋@,2 = 𝑑
2
3(,)
= 𝑑𝑓(𝑋2,@) =
L2
3(,)
) . 𝑑𝑋2,@ +
2
@
.
@
3(,)
* . (𝑑𝑋2,@)@
¨
H3(,)
3(,)
= {𝑅2 𝑡, 𝑡, 𝑡 − 𝑅@ 𝑡, 𝑡, 𝑡 }. 𝑑𝑡 + 𝜎 𝑋2,@ . [ . 𝑑𝑊
ℚ
3(,)
𝑡
¨ (𝑑𝑋2,@)@= 𝜎 𝑋2,@
@
. 𝑋2,@
@
. 𝑑𝑡
148

149. Luc_Faucheux_2021
Going from X to (1/X) - V
¨ 𝑑𝑋@,2 = 𝑑
2
3(,)
= 𝑑𝑓(𝑋2,@) =
L2
3(,)
) . 𝑑𝑋2,@ +
2
@
.
@
3(,)
* . (𝑑𝑋2,@)@
¨
H3(,)
3(,)
= {𝑅2 𝑡, 𝑡, 𝑡 − 𝑅@ 𝑡, 𝑡, 𝑡 }. 𝑑𝑡 + 𝜎 𝑋2,@ . [ . 𝑑𝑊
ℚ
3(,)
𝑡
¨ (𝑑𝑋2,@)@= 𝜎 𝑋2,@
@
. 𝑋2,@
@
. 𝑑𝑡
¨ 𝑑𝑋#,! =
.!
&!,#
# . 𝑋!,#. {{𝑅! 𝑡, 𝑡, 𝑡 − 𝑅# 𝑡, 𝑡, 𝑡 }. 𝑑𝑡 + 𝜎 𝑋!,# . [ . 𝑑𝑊
ℚ
&!,#
𝑡 } +
!
#
.
#
&!,#
/ . 𝜎 𝑋!,#
#
. 𝑋!,#
#
. 𝑑𝑡
¨ 𝑑𝑋.,, = 𝑋.,, . {{−𝑅, 𝑡, 𝑡, 𝑡 + 𝑅. 𝑡, 𝑡, 𝑡 + 𝜎 𝑋,,.
.
}. 𝑑𝑡 − 𝜎 𝑋,,. . [ . 𝑑𝑊
ℚ
1!,$
𝑡 }
¨
H3),(
3),(
= −𝑅2 𝑡, 𝑡, 𝑡 + 𝑅@ 𝑡, 𝑡, 𝑡 + 𝜎 𝑋2,@
@
. 𝑑𝑡 − 𝜎 𝑋2,@ . [ . 𝑑𝑊
ℚ
3(,)
𝑡
149

150. Luc_Faucheux_2021
Going from X to (1/X) - VI
¨ So we have :
¨
H3(,)
3(,)
= {𝑅2 𝑡, 𝑡, 𝑡 − 𝑅@ 𝑡, 𝑡, 𝑡 }. 𝑑𝑡 + 𝜎 𝑋2,@ . [ . 𝑑𝑊
ℚ
3(,)
𝑡
¨
H3),(
3),(
= {𝑅@ 𝑡, 𝑡, 𝑡 − 𝑅2 𝑡, 𝑡, 𝑡 }. 𝑑𝑡 + 𝜎 𝑋@,2 . [ . 𝑑𝑊
ℚ
3),(
𝑡
¨
H3),(
3),(
= −𝑅2 𝑡, 𝑡, 𝑡 + 𝑅@ 𝑡, 𝑡, 𝑡 + 𝜎 𝑋2,@
@
. 𝑑𝑡 − 𝜎 𝑋2,@ . [ . 𝑑𝑊
ℚ
3(,)
𝑡
¨
H3),(
3),(
= 𝑅@ 𝑡, 𝑡, 𝑡 − 𝑅2 𝑡, 𝑡, 𝑡 + 𝜎 𝑋2,@
@
. 𝑑𝑡 − 𝜎 𝑋2,@ . [ . 𝑑𝑊
ℚ
3(,)
𝑡
¨
H3),(
3),(
= 𝑅@ 𝑡, 𝑡, 𝑡 − 𝑅2 𝑡, 𝑡, 𝑡 + 𝜎 𝑋2,@
@
. 𝑑𝑡 − 𝜎 𝑋@,2 . [ . 𝑑𝑊
ℚ
3(,)
𝑡
150

151. Luc_Faucheux_2021
Going from X to (1/X) - VII
¨
H3),(
3),(
= 𝑅@ 𝑡, 𝑡, 𝑡 − 𝑅2 𝑡, 𝑡, 𝑡 + 𝜎 𝑋2,@
@
. 𝑑𝑡 − 𝜎 𝑋@,2 . [ . 𝑑𝑊
ℚ
3(,)
𝑡
¨
H3),(
3),(
= {𝑅@ 𝑡, 𝑡, 𝑡 − 𝑅2 𝑡, 𝑡, 𝑡 }. 𝑑𝑡 + 𝜎 𝑋@,2 . [ . 𝑑𝑊
ℚ
3),(
𝑡
¨ So we have:
¨ 𝑑𝑊
ℚ
3),(
𝑡 = −𝑑𝑊
ℚ
3(,)
𝑡 − 𝜎 𝑋@,2 . 𝑑𝑡
¨ This if you want explain the Siegel paradox for 2 currencies.
¨ We saw in the first part of the deck (even using Excel) that 𝜎 𝑋@,2 = 𝜎 𝑋2,@ but at the
time we did not look in details at the drift, and in particular the difference in drift between
an exchange rate and its inverse
151

152. Luc_Faucheux_2021
The arrival of correlation in the drift
152

153. Luc_Faucheux_2021
Going back up one level from the dream
ITO Leibniz on:
𝐵!. ,
𝑋!,#. 𝐴# = 𝐵!.
%!,$.)$
,!
= 𝑋!,#. 𝐴#
153

154. Luc_Faucheux_2021
The arrival of correlation in the drift
¨ All right, we almost there, we have one more couple of ITO Leibniz to do to go back from the
deflated to the original process
¨
H d
(3(,).F))
d
(3(,).F))
= 𝜎 𝑋2,@ . 𝑑𝑊
ℚ
3(,)
+ 𝜎 𝐴@ . 𝑑𝑊
ℚ
F)
¨ r
(𝑋2,@. 𝐴@) =
3(,).F)
E(
¨ 𝑑𝐵2 𝑡 = 𝐵2 𝑡 . 𝑅2 𝑡, 𝑡, 𝑡 . 𝑑𝑡
¨ By now we should be super familiar with going from deflated to reflated, you can do ITO
Leibniz or just trust me by now
¨
H d
(3(,).F))
d
(3(,).F))
= 𝜎 𝑋2,@ . 𝑑𝑊
ℚ
3(,)
+ 𝜎 𝐴@ . 𝑑𝑊
ℚ
F)
¨
H(3(,).F))
(3(,).F))
= 𝑅2 𝑡, 𝑡, 𝑡 . 𝑑𝑡 + 𝜎 𝑋2,@ . 𝑑𝑊
ℚ
3(,)
+ 𝜎 𝐴@ . 𝑑𝑊
ℚ
F)
154

155. Luc_Faucheux_2021
The arrival of correlation in the drift - II
¨
H(3(,).F))
(3(,).F))
= 𝑅2 𝑡, 𝑡, 𝑡 . 𝑑𝑡 + 𝜎 𝑋2,@ . 𝑑𝑊
ℚ
3(,)
+ 𝜎 𝐴@ . 𝑑𝑊
ℚ
F)
¨ 𝐴@ =
3(,).F)
3(,)
¨ All right so by now we are quite familiar with the ITO Leibniz handle cranking
¨ It is a little different than from we unfolded 𝑋2,@ because then we were dealing with:
¨ 𝑓 𝐵@, (𝑋2,@. 𝐵@) =
2
E)
. (𝑋2,@. 𝐵@)
¨ Here we are going to be dealing with:
¨ 𝑓 𝑋2,@, (𝑋2,@. 𝐴@) =
2
3(,)
. (𝑋2,@. 𝐴@)
¨ So instead of just an 𝑅@ 𝑡, 𝑡, 𝑡 term popping in the drift, we will have some non zero crosses
between the driver of 𝑋2,@ and the driver of (𝑋2,@. 𝐴@). All right, let’s do it, the finish line is
almost there
155

156. Luc_Faucheux_2021
Going back up one level from the dream
ITO Leibniz on:
1
𝑋!,#
. (𝑋!,#. 𝐴#) = 𝐴#
156

157. Luc_Faucheux_2021
The arrival of correlation in the drift - III
¨ 𝑓 𝑋2,@, (𝑋2,@. 𝐴@) =
2
3(,)
. (𝑋2,@. 𝐴@)
¨ 𝑑𝑓 𝑋!,#, (𝑋!,#. 𝐴#) = 𝑑𝑓
!
&!,#
. (𝑋!,#. 𝐴#) = 𝑑 𝐴# = 𝑑𝐴#
¨ 𝑑𝐴# =
$%
$(&!,#.(#)
. 𝑑(𝑋!,#. 𝐴#) +
$%
$&!,#
. 𝑑𝑋!,# +
!
#
.
$#%
$(&!,#.(#)# . 𝑑(𝑋!,#. 𝐴#)# +
!
#
.
$#%
$&!,#
# . 𝑑𝑋!,#
#
+
$#%
$(&!,#.(#).$&!,#
. 𝑑(𝑋!,#. 𝐴#). 𝑑𝑋!,#
¨
/0
/(1!,$.;$)
=
,
1!,$
¨
/$0
/(1!,$.;$)$ = 0
¨
/0
/1!,$
=
4,
1!,$
$ . (𝑋,,.. 𝐴.)
¨
/$0
/1!,$
$ =
.
1!,$
# . (𝑋,,.. 𝐴.)
¨
/$0
/(1!,$.;$)./1!,$
=
4,
1!,$
$
157

158. Luc_Faucheux_2021
The arrival of correlation in the drift - IV
¨ 𝑑𝐴# =
$%
$(&!,#.(#)
. 𝑑(𝑋!,#. 𝐴#) +
$%
$&!,#
. 𝑑𝑋!,# +
!
#
.
$#%
$(&!,#.(#)# . 𝑑(𝑋!,#. 𝐴#)# +
!
#
.
$#%
$&!,#
# . 𝑑𝑋!,#
#
+
$#%
$(&!,#.(#).$&!,#
. 𝑑(𝑋!,#. 𝐴#). 𝑑𝑋!,#
¨ 𝑑𝐴# =
!
&!,#
. 𝑑(𝑋!,#. 𝐴#) +
.!
&!,#
# . (𝑋!,#. 𝐴#). 𝑑𝑋!,# +
!
#
. 0. 𝑑(𝑋!,#. 𝐴#)#
+
!
#
.
#
&!,#
/ . (𝑋!,#. 𝐴#). 𝑑𝑋!,#
#
+
.!
&!,#
# . 𝑑(𝑋!,#. 𝐴#). 𝑑𝑋!,#
¨ 𝑑𝐴# =
!
&!,#
. 𝑑(𝑋!,#. 𝐴#) +
.!
&!,#
# . (𝑋!,#. 𝐴#). 𝑑𝑋!,# +
!
#
.
#
&!,#
/ . (𝑋!,#. 𝐴#). 𝑑𝑋!,#
#
+
.!
&!,#
# . 𝑑(𝑋!,#. 𝐴#). 𝑑𝑋!,#
¨ And we now have:
¨
H3(,)
3(,)
= {𝑅2 𝑡, 𝑡, 𝑡 − 𝑅@ 𝑡, 𝑡, 𝑡 }. 𝑑𝑡 + 𝜎 𝑋2,@ . [ . 𝑑𝑊
ℚ
3(,)
𝑡
¨
H(3(,).F))
(3(,).F))
= 𝑅2 𝑡, 𝑡, 𝑡 . 𝑑𝑡 + 𝜎 𝑋2,@ . 𝑑𝑊
ℚ
3(,)
+ 𝜎 𝐴@ . 𝑑𝑊
ℚ
F)
¨ (𝑑𝑋2,@)@= 𝑋2,@
@
. 𝜎 𝑋2,@
@
. 𝑑𝑡
¨ 𝑑(𝑋2,@. 𝐴@). 𝑑𝑋2,@ = 𝑋2,@. 𝑋2,@. 𝐴@. {𝜎 𝑋2,@ . 𝜎 𝑋2,@ . 𝑑𝑡 + 𝜎 𝑋2,@ . 𝜎 𝐴@ . 𝑑𝑊
ℚ
3(,)
. 𝑑𝑊
ℚ
F)
}
158

159. Luc_Faucheux_2021
The arrival of correlation in the drift - V
¨ 𝑑𝑊
ℚ
3(,)
. 𝑑𝑊
ℚ
F)
= 𝜌ℚ 𝑋2,@, 𝐴@ . 𝑑𝑡
¨ Let’s keep the notation 𝜌ℚ for now as this is one of our first forays in the wonderful world of
correlations and such.
¨ 𝑑𝐴. =
,
1!,$
. 𝑑(𝑋,,.. 𝐴.) +
4,
1!,$
$ . (𝑋,,.. 𝐴.). 𝑑𝑋,,. +
,
.
.
.
1!,$
# . (𝑋,,.. 𝐴.). 𝑑𝑋,,.
.
+
4,
1!,$
$ . 𝑑(𝑋,,.. 𝐴.). 𝑑𝑋,,.
¨
61!,$
1!,$
= {𝑅, 𝑡, 𝑡, 𝑡 − 𝑅. 𝑡, 𝑡, 𝑡 }. 𝑑𝑡 + 𝜎 𝑋,,. . [ . 𝑑𝑊
ℚ
1!,$
𝑡
¨
6(1!,$.;$)
(1!,$.;$)
= 𝑅, 𝑡, 𝑡, 𝑡 . 𝑑𝑡 + 𝜎 𝑋,,. . 𝑑𝑊
ℚ
1!,$
+ 𝜎 𝐴. . 𝑑𝑊
ℚ
;$
¨ (𝑑𝑋,,.).
= 𝑋,,.
.
. 𝜎 𝑋,,.
.
. 𝑑𝑡
¨ 𝑑(𝑋,,.. 𝐴.). 𝑑𝑋,,. = 𝑋,,.. 𝑋,,.. 𝐴.. {𝜎 𝑋,,. . 𝜎 𝑋,,. . 𝑑𝑡 + 𝜎 𝑋,,. . 𝜎 𝐴. . 𝑑𝑊
ℚ
1!,$
. 𝑑𝑊
ℚ
;$
}
¨ 𝑑(𝑋,,.. 𝐴.). 𝑑𝑋,,. = 𝑋,,.. 𝑋,,.. 𝐴.. {𝜎 𝑋,,. . 𝜎 𝑋,,. . 𝑑𝑡 + 𝜎 𝑋,,. . 𝜎 𝐴. . 𝜌ℚ 𝑋,,., 𝐴. . 𝑑𝑡}
159

160. Luc_Faucheux_2021
The arrival of correlation in the drift - VI
¨ 𝑑𝐴. =
,
1!,$
. 𝑑(𝑋,,.. 𝐴.) +
4,
1!,$
$ . (𝑋,,.. 𝐴.). 𝑑𝑋,,. +
,
.
.
.
1!,$
# . (𝑋,,.. 𝐴.). 𝑑𝑋,,.
.
+
4,
1!,$
$ . 𝑑(𝑋,,.. 𝐴.). 𝑑𝑋,,.
¨ 𝑑𝐴# =
!
&!,#
. 𝑋!,#. 𝐴# . {𝑅! 𝑡, 𝑡, 𝑡 . 𝑑𝑡 + 𝜎 𝑋!,# . 𝑑𝑊
ℚ
&!,#
+ 𝜎 𝐴# . 𝑑𝑊
ℚ
(#
} +
.!
&!,#
# . (𝑋!,#. 𝐴#). 𝑋!,#. {{𝑅! 𝑡, 𝑡, 𝑡 −
𝑅# 𝑡, 𝑡, 𝑡 }. 𝑑𝑡 + 𝜎 𝑋!,# . [ . 𝑑𝑊
ℚ
&!,#
𝑡 } +
!
#
.
#
&!,#
/ . (𝑋!,#. 𝐴#). 𝑋!,#
#
. 𝜎 𝑋!,#
#
. 𝑑𝑡 +
.!
&!,#
# . {𝑋!,#. 𝑋!,#. 𝐴#. {𝜎 𝑋!,# . 𝜎 𝑋!,# . 𝑑𝑡 + 𝜎 𝑋!,# . 𝜎 𝐴# . 𝜌ℚ 𝑋!,#, 𝐴# . 𝑑𝑡}}
¨ That is quite a formidable equation.
¨ Luckily for us, there are some simplifications
¨ Let’s first simplify all the
2
3(,)
. 𝑋2,@. 𝐴@ = 𝐴@ and such
160

161. Luc_Faucheux_2021
The arrival of correlation in the drift - VII
¨ 𝑑𝐴@ =
2
3(,)
. 𝑋2,@. 𝐴@ . {𝑅2 𝑡, 𝑡, 𝑡 . 𝑑𝑡 + 𝜎 𝑋2,@ . 𝑑𝑊
ℚ
3(,)
+ 𝜎 𝐴@ . 𝑑𝑊
ℚ
F)
} +
L2
3(,)
) . (𝑋2,@. 𝐴@). 𝑋2,@. {{𝑅2 𝑡, 𝑡, 𝑡 − 𝑅@ 𝑡, 𝑡, 𝑡 }. 𝑑𝑡 + 𝜎 𝑋2,@ . [ . 𝑑𝑊
ℚ
3(,)
𝑡 } +
2
@
.
@
3(,)
* . (𝑋2,@. 𝐴@). 𝑋2,@
@
. 𝜎 𝑋2,@
@
. 𝑑𝑡 +
L2
3(,)
) . {𝑋2,@. 𝑋2,@. 𝐴@. {𝜎 𝑋2,@ . 𝜎 𝑋2,@ . 𝑑𝑡 +
𝜎 𝑋2,@ . 𝜎 𝐴@ . 𝜌ℚ 𝑋2,@, 𝐴@ . 𝑑𝑡}}
¨ 𝑑𝐴@ = 𝐴@ . 𝑅2 𝑡, 𝑡, 𝑡 . 𝑑𝑡 + 𝜎 𝑋2,@ . 𝑑𝑊
ℚ
3(,)
+ 𝜎 𝐴@ . 𝑑𝑊
ℚ
F)
− (𝐴@). {{𝑅2 𝑡, 𝑡, 𝑡 −
𝑅@ 𝑡, 𝑡, 𝑡 }. 𝑑𝑡 + 𝜎 𝑋2,@ . [ . 𝑑𝑊
ℚ
3(,)
𝑡 } + (𝐴@). 𝜎 𝑋2,@
@
. 𝑑𝑡 +
L2
2
. {𝐴@. 𝜎 𝑋2,@ . 𝜎 𝑋2,@ . 𝑑𝑡 + 𝜎 𝑋2,@ . 𝜎 𝐴@ . 𝜌ℚ 𝑋2,@, 𝐴@ . 𝑑𝑡 }
161

162. Luc_Faucheux_2021
The arrival of correlation in the drift - VIII
¨
HF)
F)
= 𝑅2 𝑡, 𝑡, 𝑡 . 𝑑𝑡 + 𝜎 𝑋2,@ . 𝑑𝑊
ℚ
3(,)
+ 𝜎 𝐴@ . 𝑑𝑊
ℚ
F)
− v
w
𝑅2 𝑡, 𝑡, 𝑡 − 𝑅@ 𝑡, 𝑡, 𝑡 . 𝑑𝑡 +
𝜎 𝑋2,@ . [ . 𝑑𝑊
ℚ
3(,)
𝑡 + 𝜎 𝑋2,@
@
. 𝑑𝑡 − `
a
𝜎 𝑋2,@ . 𝜎 𝑋2,@ . 𝑑𝑡 +
𝜎 𝑋2,@ . 𝜎 𝐴@ . 𝜌ℚ 𝑋2,@, 𝐴@ . 𝑑𝑡 }
¨ There are now a number of terms that “magically” disappear (it is not magic, it is the fact
that we are unfolding back the ITO Leibniz
¨ That reminds me of the turtle in GEB, will put that in the next slide, hopefully we did not
mess upby being off one level going back down and back up in the different levels of the
dream…
162

163. Luc_Faucheux_2021
The arrival of correlation in the drift – VIII-a
¨ Every time I go down and back up those ITO Leibniz derivations, I am thinking about the
tortoise and Achilles going back down and up in the book Godel Escher Bach…and am always
scared that I missed one level, or that I am not back at the right level…
163

164. Luc_Faucheux_2021
The arrival of correlation in the drift – VIII-b
¨ Or it is also like that Chris Nolan movie, where I am going back up from the levels of the
dream, but not quite sure where I end up, and I might have gotten lost on the
way…terrifying stuff..
164

165. Luc_Faucheux_2021
The arrival of correlation in the drift - IX
¨
HF)
F)
= 𝑅2 𝑡, 𝑡, 𝑡 . 𝑑𝑡 + 𝜎 𝑋2,@ . 𝑑𝑊
ℚ
3(,)
+ 𝜎 𝐴@ . 𝑑𝑊
ℚ
F)
− v
w
𝑅2 𝑡, 𝑡, 𝑡 − 𝑅@ 𝑡, 𝑡, 𝑡 . 𝑑𝑡 +
𝜎 𝑋2,@ . [ . 𝑑𝑊
ℚ
3(,)
𝑡 + 𝜎 𝑋2,@
@
. 𝑑𝑡 − `
a
𝜎 𝑋2,@ . 𝜎 𝑋2,@ . 𝑑𝑡 +
𝜎 𝑋2,@ . 𝜎 𝐴@ . 𝜌ℚ 𝑋2,@, 𝐴@ . 𝑑𝑡
¨
HF)
F)
= 𝜎 𝐴@ . 𝑑𝑊
ℚ
F)
− 0 − 𝑅@ 𝑡, 𝑡, 𝑡 . 𝑑𝑡 + 𝜎 𝑋2,@
@
. 𝑑𝑡 − `
a
𝜎 𝑋2,@ . 𝜎 𝑋2,@ . 𝑑𝑡 +
𝜎 𝑋2,@ . 𝜎 𝐴@ . 𝜌ℚ 𝑋2,@, 𝐴@ . 𝑑𝑡
¨
HF)
F)
= 𝜎 𝐴@ . 𝑑𝑊
ℚ
F)
+ 𝑅@ 𝑡, 𝑡, 𝑡 . 𝑑𝑡 − 𝜎 𝑋2,@ . 𝜎 𝐴@ . 𝜌ℚ 𝑋2,@, 𝐴@ . 𝑑𝑡
165

166. Luc_Faucheux_2021
The arrival of correlation in the drift - X
¨
HF)
F)
= 𝑅@ 𝑡, 𝑡, 𝑡 − 𝜎 𝑋2,@ . 𝜎 𝐴@ . 𝜌ℚ 𝑋2,@, 𝐴@ . 𝑑𝑡 + 𝜎 𝐴@ . 𝑑𝑊
ℚ
F)
¨ We are all the way back in the first level of the dream (or are we?)
¨ We started with:
¨
HF)
F)
= 𝜇 𝐴@ . 𝑑𝑡 + 𝜎 𝐴@ . [ . 𝑑𝑊F)(𝑡)
¨ We ended up with:
¨
HF)
F)
= 𝑅@ 𝑡, 𝑡, 𝑡 − 𝜎 𝑋2,@ . 𝜎 𝐴@ . 𝜌ℚ 𝑋2,@, 𝐴@ . 𝑑𝑡 + 𝜎 𝐴@ . 𝑑𝑊
ℚ
F)
¨ So the first term in 𝑅@ 𝑡, 𝑡, 𝑡 should not surprise us, when we go into the risk free measure,
the drift from the physical measure 𝜇 𝐴@ is replaced by the drift 𝑅@ 𝑡, 𝑡, 𝑡 from the
Numeraire associated to the risk free measure, meaning the bank account
¨ The second term in the drift −𝜎 𝑋2,@ . 𝜎 𝐴@ . 𝜌ℚ 𝑋2,@, 𝐴@ is new to us
166

167. Luc_Faucheux_2021
The arrival of correlation in the drift - XI
¨ As usual we note that the variance of the process is not affected by the change of measure
that we performed.
¨ Changing the measure changes the drift, it does not change the variance
¨ Let us ponder a little that new adjustment to the drift that we have encountered
¨
HF)
F)
= 𝑅@ 𝑡, 𝑡, 𝑡 − 𝜎 𝑋2,@ . 𝜎 𝐴@ . 𝜌ℚ 𝑋2,@, 𝐴@ . 𝑑𝑡 + 𝜎 𝐴@ . 𝑑𝑊
ℚ
F)
¨ Note that if we were only in the domestic world of the currency (2), and if we had done the
usual change of measure from the physical to the risk free for an asset 𝐴@, we would have
obtained:
¨
HF)
F)
= 𝑅@ 𝑡, 𝑡, 𝑡 . 𝑑𝑡 + 𝜎 𝐴@ . 𝑑𝑊
ℚ
F)
¨ So what is wrong ?
167

168. Luc_Faucheux_2021
Careful about the notations again
168

169. Luc_Faucheux_2021
The arrival of correlation in the drift - XII
¨ There is nothing wrong, we just need to be a little careful with the notation
¨ Remember, all we did up to here was to bring the foreign asset (currency 2) back into the
domestic world (currency 1), deflate it by the Bank Account:
¨ 𝐵2 𝑡 = exp[∫
<=>
<=?
𝑅2 𝑠, 𝑠, 𝑠 . 𝑑𝑠]
¨ Express it as martingale under a new Brownian motion associated with that asset “IN THE
DOMESTIC WORLD 1”
¨ So really (we did not do it for sake of notation, but now is the time to do it), when we were
writing ℚ it should really have been ℚ2
169

170. Luc_Faucheux_2021
The arrival of correlation in the drift - XIII
¨ To be rigorous on the Exchange Rate:
¨
H3(,)
3(,)
= {𝑅2 𝑡, 𝑡, 𝑡 − 𝑅@ 𝑡, 𝑡, 𝑡 }. 𝑑𝑡 + 𝜎 𝑋2,@ . [ . 𝑑𝑊
ℚ(
3(,)
𝑡
¨ Now of course by symmetry we have:
¨
H3),(
3),(
= {𝑅@ 𝑡, 𝑡, 𝑡 − 𝑅2 𝑡, 𝑡, 𝑡 }. 𝑑𝑡 + 𝜎 𝑋@,2 . [ . 𝑑𝑊
ℚ)
3),(
𝑡
¨ And doing ITO lemma on 𝑓 𝑥 = (
2
K
) in the first equation led to :
¨
H3),(
3),(
= 𝑅@ 𝑡, 𝑡, 𝑡 − 𝑅2 𝑡, 𝑡, 𝑡 + 𝜎 𝑋2,@
@
. 𝑑𝑡 − 𝜎 𝑋@,2 . [ . 𝑑𝑊
ℚ(
3(,)
𝑡
¨
H3),(
3),(
= {𝑅@ 𝑡, 𝑡, 𝑡 − 𝑅2 𝑡, 𝑡, 𝑡 }. 𝑑𝑡 + 𝜎 𝑋@,2 . [ . 𝑑𝑊
ℚ)
3),(
𝑡
170

171. Luc_Faucheux_2021
The arrival of correlation in the drift - XIV
¨ So we have:
¨ 𝑑𝑊
ℚ)
3),(
𝑡 = −𝑑𝑊
ℚ(
3(,)
𝑡 − 𝜎 𝑋@,2 . 𝑑𝑡
¨ 𝑊
ℚ)
3),(
𝑡 = −𝑊
ℚ(
3(,)
𝑡 − 𝜎 𝑋@,2 . 𝑡
171

172. Luc_Faucheux_2021
The arrival of correlation in the drift - XV
¨ Similar for the asset:
¨
HF)
F)
= 𝑅@ 𝑡, 𝑡, 𝑡 − 𝜎 𝑋2,@ . 𝜎 𝐴@ . 𝜌ℚ 𝑋2,@, 𝐴@ . 𝑑𝑡 + 𝜎 𝐴@ . 𝑑𝑊
ℚ
F)
¨ IS really:
¨
HF)
F)
= 𝑅@ 𝑡, 𝑡, 𝑡 − 𝜎 𝑋2,@ . 𝜎 𝐴@ . 𝜌ℚ(
𝑋2,@, 𝐴@ . 𝑑𝑡 + 𝜎 𝐴@ . 𝑑𝑊
ℚ(
F)
¨ By symmetry of course we will have:
¨
HF(
F(
= 𝑅2 𝑡, 𝑡, 𝑡 − 𝜎 𝑋@,2 . 𝜎 𝐴2 . 𝜌ℚ)
𝑋@,2, 𝐴2 . 𝑑𝑡 + 𝜎 𝐴2 . 𝑑𝑊
ℚ)
F(
¨ And if we were to only stick to the domestic world:
¨
HF(
F(
= 𝑅2 𝑡, 𝑡, 𝑡 . 𝑑𝑡 + 𝜎 𝐴2 . 𝑑𝑊
ℚ(
F(
172

173. Luc_Faucheux_2021
The arrival of correlation in the drift – XV-a
¨
HF(
F(
= 𝑅2 𝑡, 𝑡, 𝑡 . 𝑑𝑡 + 𝜎 𝐴2 . 𝑑𝑊
ℚ(
F(
¨
Hn
F(
n
F(
= 𝜎 𝐴2 . 𝑑𝑊
ℚ(
F(
¨ x
𝐴2 =
F(
E(
173

174. Luc_Faucheux_2021
The arrival of correlation in the drift - XVI
¨ So we have:
¨
HF(
F(
= 𝑅2 𝑡, 𝑡, 𝑡 − 𝜎 𝑋@,2 . 𝜎 𝐴2 . 𝜌ℚ)
𝑋@,2, 𝐴2 . 𝑑𝑡 + 𝜎 𝐴2 . 𝑑𝑊
ℚ)
F(
¨
HF(
F(
= 𝑅2 𝑡, 𝑡, 𝑡 . 𝑑𝑡 + 𝜎 𝐴2 . 𝑑𝑊
ℚ(
F(
¨ Which leads to another useful equation:
¨ 𝑑𝑊
ℚ(
F(
= 𝑑𝑊
ℚ)
F(
− 𝜎 𝑋@,2 . 𝜌ℚ)
𝑋@,2, 𝐴2 . 𝑑𝑡
¨ 𝜌ℚ)
𝑋@,2, 𝐴2 . 𝑑𝑡 = 𝑑𝑊
ℚ)
3),(
. 𝑑𝑊
ℚ)
F(
¨ 𝑊
ℚ(
F(
= 𝑊
ℚ)
F(
− 𝜎 𝑋@,2 . 𝜌ℚ)
𝑋@,2, 𝐴2 . 𝑡
174

175. Luc_Faucheux_2021
The arrival of correlation in the drift - XVII
¨ And then also by symmetry:
¨ 𝑑𝑊
ℚ(
F(
= 𝑑𝑊
ℚ)
F(
− 𝜎 𝑋@,2 . 𝜌ℚ)
𝑋@,2, 𝐴2 . 𝑑𝑡
¨ 𝑑𝑊
ℚ)
F)
= 𝑑𝑊
ℚ(
F)
− 𝜎 𝑋2,@ . 𝜌ℚ(
𝑋2,@, 𝐴@ . 𝑑𝑡
¨ 𝜌ℚ)
𝑋@,2, 𝐴2 . 𝑑𝑡 = 𝑑𝑊
ℚ)
3),(
. 𝑑𝑊
ℚ)
F(
¨ 𝜌ℚ(
𝑋2,@, 𝐴@ . 𝑑𝑡 = 𝑑𝑊
ℚ(
3(,)
. 𝑑𝑊
ℚ(
F)
175

176. Luc_Faucheux_2021
Things to still do in FX
176

177. Luc_Faucheux_2021
Things to still do in FX
¨ Expand on the quanto adjustment
¨ Redo the quanto adjustment using the Radon Nikodym derivative
¨ Draw more figures and examples on the correlation triangle
¨ Explain how FX options are traded in practice
¨ Some more slides on the correlations
¨ Build some examples rom the quanto drift
¨ Link quanto to bi-curve valuations in the swap world
177

178. Luc_Faucheux_2021
So at least for now…..
178