More notes on FX - going over the LOOP (Law Of One Price) and the connection with Radon-Nikodym, change of measures and FX

1. Luc_Faucheux_2021
THE RATES WORLD – Part VIII_a
More fun with the FX market, and some stuff
on change of measures, and the LOOP (Law Of
One Price)
1

2. Luc_Faucheux_2021
That deck
2

3. Luc_Faucheux_2021
That deck - I
¨ We continue looking at the FX world
¨ We revisit things like change of measures, Radon-Nikodym and Girsanov theorem after
having seen some of those concepts in FX
¨ In particular in most textbooks the probabilities are always ℙ and ℚ, and that ends up being
confusing which one is which, or when dealing with FX you sometimes see the foreign and
domestic measures ℚ! and ℚ"
¨ Now that we have one deck under our belt of being used to indices like:
¨ 𝑋#,% = 𝑋#←% is the value in currency (𝑖) of 1 unit of currency (𝑗)
¨ We will revisit a couple of things about change of measure and RN derivative using a less
confusing notation (at least for me)ℙ' and ℙ(
3

4. Luc_Faucheux_2021
What we had from the previous deck
4

5. 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:
¨ 𝐵' 𝑡 = exp[∫
)*+
)*,
𝑅' 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 ℚ'
5

6. Luc_Faucheux_2021
The arrival of correlation in the drift - XIII
¨ To be rigorous on the Exchange Rate:
¨
"-!,#
-!,#
= {𝑅' 𝑡, 𝑡, 𝑡 − 𝑅( 𝑡, 𝑡, 𝑡 }. 𝑑𝑡 + 𝜎 𝑋',( . [ . 𝑑𝑊
ℚ!
-!,#
𝑡
¨ Now of course by symmetry we have:
¨
"-#,!
-#,!
= {𝑅( 𝑡, 𝑡, 𝑡 − 𝑅' 𝑡, 𝑡, 𝑡 }. 𝑑𝑡 + 𝜎 𝑋(,' . [ . 𝑑𝑊
ℚ#
-#,!
𝑡
¨ And doing ITO lemma on 𝑓 𝑥 = (
'
/
) in the first equation led to :
¨
"-#,!
-#,!
= 𝑅( 𝑡, 𝑡, 𝑡 − 𝑅' 𝑡, 𝑡, 𝑡 + 𝜎 𝑋',(
(
. 𝑑𝑡 − 𝜎 𝑋(,' . [ . 𝑑𝑊
ℚ!
-!,#
𝑡
¨
"-#,!
-#,!
= {𝑅( 𝑡, 𝑡, 𝑡 − 𝑅' 𝑡, 𝑡, 𝑡 }. 𝑑𝑡 + 𝜎 𝑋(,' . [ . 𝑑𝑊
ℚ#
-#,!
𝑡
6

7. Luc_Faucheux_2021
The arrival of correlation in the drift - XIV
¨ So we have:
¨ 𝑑𝑊
ℚ#
-#,!
𝑡 = −𝑑𝑊
ℚ!
-!,#
𝑡 + 𝜎 𝑋(,' . 𝑑𝑡
¨ In the simple case of a constant 𝜎 𝑋(,'
¨ 𝑊
ℚ#
-#,!
𝑡 = −𝑊
ℚ!
-!,#
𝑡 + 𝜎 𝑋(,' . 𝑡
7

8. Luc_Faucheux_2021
The arrival of correlation in the drift - XV
¨ Similar for the asset:
¨
"0#
0#
= 𝑅( 𝑡, 𝑡, 𝑡 − 𝜎 𝑋',( . 𝜎 𝐴( . 𝜌ℚ 𝑋',(, 𝐴( . 𝑑𝑡 + 𝜎 𝐴( . 𝑑𝑊
ℚ
0#
¨ IS really:
¨
"0#
0#
= 𝑅( 𝑡, 𝑡, 𝑡 − 𝜎 𝑋',( . 𝜎 𝐴( . 𝜌ℚ!
𝑋',(, 𝐴( . 𝑑𝑡 + 𝜎 𝐴( . 𝑑𝑊
ℚ!
0#
¨ By symmetry of course we will have:
¨
"0!
0!
= 𝑅' 𝑡, 𝑡, 𝑡 − 𝜎 𝑋(,' . 𝜎 𝐴' . 𝜌ℚ#
𝑋(,', 𝐴' . 𝑑𝑡 + 𝜎 𝐴' . 𝑑𝑊
ℚ#
0!
¨ And if we were to only stick to the domestic world:
¨
"0!
0!
= 𝑅' 𝑡, 𝑡, 𝑡 . 𝑑𝑡 + 𝜎 𝐴' . 𝑑𝑊
ℚ!
0!
8

9. Luc_Faucheux_2021
The arrival of correlation in the drift – XV-a
¨
"0!
0!
= 𝑅' 𝑡, 𝑡, 𝑡 . 𝑑𝑡 + 𝜎 𝐴' . 𝑑𝑊
ℚ!
0!
¨
"1
0!
1
0!
= 𝜎 𝐴' . 𝑑𝑊
ℚ!
0!
¨ @
𝐴' =
0!
2!
9

10. Luc_Faucheux_2021
The arrival of correlation in the drift - XVI
¨ So we have:
¨
"0!
0!
= 𝑅' 𝑡, 𝑡, 𝑡 − 𝜎 𝑋(,' . 𝜎 𝐴' . 𝜌ℚ#
𝑋(,', 𝐴' . 𝑑𝑡 + 𝜎 𝐴' . 𝑑𝑊
ℚ#
0!
¨
"0!
0!
= 𝑅' 𝑡, 𝑡, 𝑡 . 𝑑𝑡 + 𝜎 𝐴' . 𝑑𝑊
ℚ!
0!
¨ Which leads to another useful equation:
¨ 𝑑𝑊
ℚ!
0!
= 𝑑𝑊
ℚ#
0!
− 𝜎 𝑋(,' . 𝜌ℚ#
𝑋(,', 𝐴' . 𝑑𝑡
¨ 𝜌ℚ#
𝑋(,', 𝐴' . 𝑑𝑡 = 𝑑𝑊
ℚ#
-#,!
. 𝑑𝑊
ℚ#
0!
¨ In the simple case of constant parameters 𝜎 𝑋(,' and 𝜌ℚ#
𝑋(,', 𝐴'
¨ 𝑊
ℚ!
0!
= 𝑊
ℚ#
0!
− 𝜎 𝑋(,' . 𝜌ℚ#
𝑋(,', 𝐴' . 𝑡
10

11. Luc_Faucheux_2021
The arrival of correlation in the drift - XVII
¨ And then also by symmetry:
¨ 𝑑𝑊
ℚ!
0!
= 𝑑𝑊
ℚ#
0!
− 𝜎 𝑋(,' . 𝜌ℚ#
𝑋(,', 𝐴' . 𝑑𝑡
¨ 𝑑𝑊
ℚ#
0#
= 𝑑𝑊
ℚ!
0#
− 𝜎 𝑋',( . 𝜌ℚ!
𝑋',(, 𝐴( . 𝑑𝑡
¨ 𝜌ℚ#
𝑋(,', 𝐴' . 𝑑𝑡 = 𝑑𝑊
ℚ#
-#,!
. 𝑑𝑊
ℚ#
0!
¨ 𝜌ℚ!
𝑋',(, 𝐴( . 𝑑𝑡 = 𝑑𝑊
ℚ!
-!,#
. 𝑑𝑊
ℚ!
0#
11

12. Luc_Faucheux_2021
Things to still do in FX
12

13. 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
13

14. Luc_Faucheux_2021
Back to some notes on FX
14

15. Luc_Faucheux_2021
Some notes on FX - I
¨ Let’s recap what we have:
15

16. Luc_Faucheux_2021
Some notes on FX - II
¨ Let’s assume that currency 1 is the domestic currency
¨ We will note currency 2 to be the foreign currency
¨ 𝑋',( = 𝑋'←( is the value in currency (1) of 1 unit of currency (2)
¨ 𝐵' 𝑡 = exp[∫
)*+
)*,
𝑅' 𝑠, 𝑠, 𝑠 . 𝑑𝑠] is the Bank Account Numeraire in the domestic world of
currency (1).
¨ 𝐵' 𝑡 = exp[∫
)*+
)*,
𝑅' 𝑠, 𝑠, 𝑠 . 𝑑𝑠] is associated to the Risk free measure in the domestic
currency (1) that we will note 𝑊ℚ' 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
16

17. Luc_Faucheux_2021
Some notes on FX - III
¨ The Zero Coupon bonds in the domestic world of currency (1) are noted 𝑍' 𝑡, 𝑡#, 𝑡%
¨ The Zero Coupon bonds in the foreign world of currency (2) are noted 𝑍( 𝑡, 𝑡#, 𝑡%
¨ 𝐴' 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:
¨
"0!
0!
= 𝜇 𝐴' . 𝑑𝑡 + 𝜎 𝐴' . [ . 𝑑𝑊0!(𝑡)
¨
"0#
0#
= 𝜇 𝐴( . 𝑑𝑡 + 𝜎 𝐴( . [ . 𝑑𝑊0#(𝑡)
¨
"-!,#
-!,#
= 𝜇 𝑋',( . 𝑑𝑡 + 𝜎 𝑋',( . [ . 𝑑𝑊-!,#(𝑡)
17

18. Luc_Faucheux_2021
Some notes on FX - IV
¨ Using the argument that the deflated tradable assets are martingale under the Risk Free
measure, we simplified the equations to the following:
¨
"-!,#
-!,#
= {𝑅' 𝑡, 𝑡, 𝑡 − 𝑅( 𝑡, 𝑡, 𝑡 }. 𝑑𝑡 + 𝜎 𝑋',( . [ . 𝑑𝑊
ℚ!
-!,#
𝑡
¨
"0#
0#
= 𝑅( 𝑡, 𝑡, 𝑡 − 𝜎 𝑋',( . 𝜎 𝐴( . 𝜌ℚ!
𝑋',(, 𝐴( . 𝑑𝑡 + 𝜎 𝐴( . 𝑑𝑊
ℚ!
0#
¨
"0#
0#
= 𝑅( 𝑡, 𝑡, 𝑡 . 𝑑𝑡 + 𝜎 𝐴( . 𝑑𝑊
ℚ#
0#
¨ 𝑑𝑊
ℚ!
0#
𝑡 = 𝑑𝑊0# 𝑡 + 𝜑 𝐴( . 𝑑𝑡
¨ 𝑑𝑊
ℚ!
-!,#
𝑡 = 𝑑𝑊-!,# 𝑡 + 𝜑 𝑋',( . 𝑑𝑡
¨ 𝜑 𝑋',( . 𝜎 𝑋',( = 𝜇 𝑋',( + 𝑅( 𝑡, 𝑡, 𝑡 − 𝑅' 𝑡, 𝑡, 𝑡
¨ 𝜑 𝐴( . 𝜎 𝐴( = 𝜇 𝐴( + 𝜎 𝐴( . 𝜎 𝑋',( . 𝜌 𝑋',(; 𝐴( − 𝑅( 𝑡, 𝑡, 𝑡
18

19. Luc_Faucheux_2021
Some notes on FX - V
¨ Using symmetry argument we had the following relations:
¨ 𝑑𝑊
ℚ#
-#,!
𝑡 = −𝑑𝑊
ℚ!
-!,#
𝑡 + 𝜎 𝑋(,' . 𝑑𝑡
¨ 𝑑𝑊
ℚ!
0!
= 𝑑𝑊
ℚ#
0!
− 𝜎 𝑋(,' . 𝜌ℚ#
𝑋(,', 𝐴' . 𝑑𝑡
¨ 𝑑𝑊
ℚ#
0#
= 𝑑𝑊
ℚ!
0#
− 𝜎 𝑋',( . 𝜌ℚ!
𝑋',(, 𝐴( . 𝑑𝑡
¨ 𝜌ℚ#
𝑋(,', 𝐴' . 𝑑𝑡 = 𝑑𝑊
ℚ#
-#,!
. 𝑑𝑊
ℚ#
0!
¨ 𝜌ℚ!
𝑋',(, 𝐴( . 𝑑𝑡 = 𝑑𝑊
ℚ!
-!,#
. 𝑑𝑊
ℚ!
0#
19

20. Luc_Faucheux_2021
Some notes on FX – VI
¨ Now, remember that changing the measure does NOT change the correlation
¨ Changing the measure does NOT change the variance
¨ Changing the measure only changes the drift (expected value)
¨ Changing the measure is adding a drift term to the Brownian motion.
¨ Note that all we did on the FX deck has been relying on GBM (Geometric Brownian Motion).
A lot of the results would NOT apply say if we were to start writing the equations in a
Normal model. Careful to keep that in mind.
¨ In particular:
¨ 𝜎 𝑋#,% = 𝜎 𝑋%,# = 𝜎
'
-$,%
¨ 𝜌 𝑋#,3; 𝑋3,% = −𝜌 𝑋#,3; 𝑋%,3 = −𝜌 𝑋#,3;
'
-&,%
20

21. Luc_Faucheux_2021
Some notes on FX – VII
¨ Because changing the measure does NOT change the correlation:
¨ 𝜌ℚ!
𝑋',(, 𝐴( . 𝑑𝑡 = 𝑑𝑊
ℚ!
-!,#
. 𝑑𝑊
ℚ!
0#
= 𝑑𝑊-!,#. 𝑑𝑊0# = 𝜌 𝑋',(, 𝐴( . 𝑑𝑡
¨ 𝜌ℚ!
𝑋',(, 𝐴( = 𝜌 𝑋',(, 𝐴(
21

22. Luc_Faucheux_2021
Some notes on FX – VIII
¨ Why is it so important that we can derive relationships between the Brownian motions ?
¨ Remember (again, shut up you millennials), most of that was done at a time where CPU and
memory was super expensive, we did not have AWS to run anything and everything that we
wanted to on the cloud, so we had to “pre-digest” a lot of the equations to be fed to the
machines. In some ways we were actually working more for the machines then than now, as
ironic as it sounds.
¨ The big thing about the fact that you can express 𝑑𝑊
ℚ#
-#,!
𝑡 as a function of 𝑑𝑊
ℚ!
-!,#
𝑡 and
same for 𝑑𝑊
ℚ#
0#
and 𝑑𝑊
ℚ!
0#
, is that you are REDUCING THE DIMENSIONALITY of the problem.
¨ So say you run a Monte Carlo simulations, instead of 2 different Brownian motions (where
the computations go as 𝑁4 (in the non-recombining case, say you are doing MC but not on a
grid, and not on a recombining tree), if you now have a simple drift adjustment between the
two Brownian motions, your computations get cut down to 𝑁( (again in the most brute
force case where you do a full blown non recombining MC).
22

23. Luc_Faucheux_2021
Some notes on FX – IX
¨ This is why for FX this simple model with the adjustment has been around for so long, and
why in the industry we have tried to preserve it and tweak it rather than jump onto a more
complicated model to start with
¨ In the single currency world, the jump from simple short rate models to LMM or BGM was
not as computationally daunting, this is why it happened sooner
¨ In the equity world, where the stochastic volatility is of paramount importance, same story,
jumping from the simple Black-Scholes model to Heston for example, was not such a big
jump in dimensions, and so was done rather early.
¨ The FX world remains that super-hard bastion of keeping a simple model around and
tweaking it as much as possible, as people have not done the jump to a full “ab initio”
modeling of the FX, because at the crux of the matter there are too many variables to deal
with.
¨ This is why the current model with its adjustment is so popular and has been kept around,
with mostly ad-hoc” adjustments to try to take into account volatility skew and such.
23

24. Luc_Faucheux_2021
Law of one Price in FX
24

25. Luc_Faucheux_2021
Law of one price in FX - I
¨ Also called the FTAP
¨ Fundamental Theorem of Asset Pricing
¨ I like it better as “Law of one price”
¨ Essentially if there is an arbitrage, there is an assumption that arbitrageurs will arbitrage it
away
¨ That assumption is a famous widow maker
¨ “Show me an arbitrage I will show you a squeeze” unknown
¨ Also another great quote about arbitrage:
¨ “I buy money, I sell money. Some people call it arbitrage. I call it making a living”
25

26. Luc_Faucheux_2021
Law of one price in FX - II
¨ Anyhow, let’s assume that we can rely on the Law of One Price.
¨ You have to start somewhere, but make sure that you never have to critically rely on your
assumptions.
¨ We have shown in the previous decks the relations between the Radon-Nikodym as a ratio
of numeraire
¨ In the FX world it is a little more complicated because you need to bring back the asset into
a common currency.
¨ We are just for now tie up the LOOP (Law Of One Price) with the Radon-Nikodym, and then
later in this deck revisit it with improved notations (following Baxter p. 104-105)
26

27. Luc_Faucheux_2021
Law of one price in FX – II-a
¨ The Law Of One Price, I like to refer to as the LOOP
¨ There are tons of tales from the Loop…
27

28. Luc_Faucheux_2021
Law of one price in FX – II-b
¨ Not to be confused with the Chicago loop…plenty of tales there also…courtesy of the Blues
Brothers…
28

29. Luc_Faucheux_2021
Law of one price in FX - III
¨ 𝐴( 𝑡 is a tradeable asset in the foreign world of currency (2) (like a stock)
¨ The Zero Coupon bonds in the foreign world of currency (2) are noted 𝑍( 𝑡, 𝑡#, 𝑡%
¨ The foreign Risk Free measure (currency 2) has for numeraire the foreign MMA (Money
Market Account)
¨ 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
¨ The deflated asset is a martingale under the Risk Free measure
¨
0# ,
2# ,
is a martingale under the foreign Risk free measure ℚ(, with the adapted Brownian
motion 𝑊
ℚ#
0#
29

30. Luc_Faucheux_2021
Law of one price in FX - IV
¨
0# ,
2# ,
is a martingale under the foreign Risk free measure ℚ(
¨
0# )
2# )
= 𝔼,
ℚ#
{
0# ,
2# ,
|𝔉 𝑠 }
¨ NOW we have shown that the tradeable asset in the domestic currency 1 is (𝑋',( 𝑡 . 𝐴( 𝑡 )
¨ In the domestic currency 1 risk free measure ℚ', the deflated tradeable asset is a martingale
¨
-!,# ) .0# )
2! )
= 𝔼,
ℚ!
{
-!,# , .0# ,
2! ,
|𝔉 𝑠 }
¨ The LOOP essentially tells you that 𝐴( 𝑠 = 𝐴( 𝑠 , or more exactly the value 𝐴( 𝑠 in the
expectation equation in the foreign currency is the same 𝐴( 𝑠 in the expectation formula in
the domestic
30

31. Luc_Faucheux_2021
Law of one price in FX - V
¨
0# )
2# )
= 𝔼,
ℚ#
{
0# ,
2# ,
|𝔉 𝑠 }
¨ 𝐴( 𝑠 = 𝐵( 𝑠 . 𝔼,
ℚ# 0# ,
2# ,
𝔉 𝑠 = 𝔼,
ℚ#
𝐵( 𝑠 .
0# ,
2# ,
𝔉 𝑠
¨
-!,# ) .0# )
2! )
= 𝔼,
ℚ!
{
-!,# , .0# ,
2! ,
|𝔉 𝑠 }
¨ 𝐴( 𝑠 = 𝐵' 𝑠 .
'
-!,# )
. 𝔼,
ℚ! -!,# , .0# ,
2! ,
𝔉 𝑠 = 𝔼,
ℚ!
{𝐵' 𝑠 .
'
-!,# )
.
-!,# , .0# ,
2! ,
|𝔉 𝑠 }
¨ 𝔼,
ℚ#
𝐵( 𝑠 .
0# ,
2# ,
𝔉 𝑠 = 𝔼,
ℚ!
{𝐵' 𝑠 .
'
-!,# )
.
-!,# , .0# ,
2! ,
|𝔉 𝑠 }
¨ 𝔼,
ℚ# 2# )
2# ,
. 𝐴( 𝑡 𝔉 𝑠 = 𝔼,
ℚ!
{
2! ) .-!,# ,
-!,# ) .2! ,
. 𝐴( 𝑡 |𝔉 𝑠 }
31

32. Luc_Faucheux_2021
Law of one price in FX - VI
¨ 𝔼,
ℚ# 2# )
2# ,
. 𝐴( 𝑡 𝔉 𝑠 = 𝔼,
ℚ!
{
2! ) .-!,# ,
-!,# ) .2! ,
. 𝐴( 𝑡 |𝔉 𝑠 }
¨ 𝔼,
ℚ# 0# ,
2# ,
𝔉 𝑠 =
'
2# )
. 𝔼,
ℚ!
{
2! ) .-!,# ,
-!,# ) .2! ,
. 𝐴( 𝑡 |𝔉 𝑠 }
¨ 𝔼,
ℚ# 0# ,
2# ,
𝔉 𝑠 =. 𝔼,
ℚ!
{
2! )
2# )
.
-!,# ,
-!,# )
.
'
2! ,
. 𝐴( 𝑡 |𝔉 𝑠 }
¨ And there we rely once again on the most celebrated and useful equation ever:
¨ 𝑎𝑛𝑦𝑡ℎ𝑖𝑛𝑔 = 𝑎𝑛𝑦𝑡ℎ𝑖𝑛𝑔
¨ And its corollary
¨
0678), 09:,;#9<
0678), 09:,;#9<
= 1
¨ Let’s apply it to 𝐵( 𝑡
32

33. Luc_Faucheux_2021
Law of one price in FX - VII
¨ 𝔼,
ℚ# 0# ,
2# ,
𝔉 𝑠 =. 𝔼,
ℚ!
{
2! )
2# )
.
-!,# ,
-!,# )
.
'
2! ,
. 𝐴( 𝑡 |𝔉 𝑠 }
¨ 𝔼,
ℚ# 0# ,
2# ,
𝔉 𝑠 =. 𝔼,
ℚ!
{
2! )
2# )
.
-!,# ,
-!,# )
.
'
2! ,
.
2# ,
2# ,
𝐴( 𝑡 |𝔉 𝑠 }
¨ 𝔼,
ℚ# 0# ,
2# ,
𝔉 𝑠 =. 𝔼,
ℚ!
{
2! )
2# )
.
-!,# ,
-!,# )
.
2# ,
2! ,
.
0# ,
2# ,
|𝔉 𝑠 }
¨ Let’s define a new function
¨ ℜ 𝑡 =
2! )
2# )
.
-!,# ,
-!,# )
.
2# ,
2! ,
¨ 𝔼,
ℚ# 0# ,
2# ,
𝔉 𝑠 =. 𝔼,
ℚ!
{ℜ 𝑡 .
0# ,
2# ,
|𝔉 𝑠 }
33

34. Luc_Faucheux_2021
Law of one price in FX - VIII
¨ 𝔼,
ℚ# 0# ,
2# ,
𝔉 𝑠 = 𝔼,
ℚ!
{ℜ 𝑡 .
0# ,
2# ,
|𝔉 𝑠 }
¨ That looks an awful lot like when we change measures and the definition of the Radon-
Nikodym derivative.
¨ 𝔼,
ℚ
𝑋 𝑡 𝔉 0 = 𝔼,
ℙ{
"ℚ
"ℙ
𝑡 . 𝑋(𝑡)|𝔉 0 }
¨ We are quite tempted to write:
¨ ℜ 𝑡 =
2! )
2# )
.
-!,# ,
-!,# )
.
2# ,
2! ,
=
"ℚ#
"ℚ!
𝑡
¨ Let’s come up with some reassuring notations on ℜ 𝑡
¨ First, we like the fact that it is a capital letter, because if it is indeed a Radon-Nikodym
derivative, it is a martingale itself, and a stochastic process
¨ 𝑅 is already taken for the rates, 𝑅𝑁 has 2 letters, so let’s keep ℜ 𝑡
34

35. Luc_Faucheux_2021
Law of one price in FX - IX
¨ 𝔼,
ℚ# 0# ,
2# ,
𝔉 𝑠 = 𝔼,
ℚ!
{ℜ 𝑡 .
0# ,
2# ,
|𝔉 𝑠 }
¨ ℜ 𝑡 =
2! )
2# )
.
-!,# ,
-!,# )
.
2# ,
2! ,
=
"ℚ#
"ℚ!
𝑡
¨ Let’s actually go one step further and write it as :
¨ ℜ 𝑡 =
2! )
2# )
.
-!,# ,
-!,# )
.
2# ,
2! ,
=
"ℚ#
"ℚ!
𝑡 = ℜ(,' 𝑡 = ℜ(←'(𝑡)
¨ Because it “brings” the expectation calculation from the 𝔼,
ℚ!
into the 𝔼,
ℚ#
¨ 𝔼,
ℚ#
𝑋(𝑡) 𝔉 𝑠 = 𝔼,
ℚ!
{ℜ(←'(𝑡). 𝑋(𝑡)|𝔉 𝑠 }
¨ ℜ(,' 𝑡 = ℜ(←' 𝑡 =
"ℚ#
"ℚ!
𝑡
35

36. Luc_Faucheux_2021
Law of one price in FX - X
¨ If you recall the deck on measures, we had the following between two measures associated
with two different numeraires:
¨
"ℚ
"ℙ
𝑡 =
>ℚ(,)
>ℙ(,)
/
>ℚ(+)
>ℙ(+)
¨ If we normalize the numeraires by their time 𝑡 = 0 values:
¨ @
𝑁ℚ 𝑡 = 𝑁ℚ(𝑡)/𝑁ℚ(0)
¨ @
𝑁ℙ 𝑡 = 𝑁ℙ(𝑡)/𝑁ℙ(0)
¨ We obtain the celebrated formula:
¨
"ℚ
"ℙ
𝑡 =
1
>ℚ ,
1
>ℙ ,
¨ The Radon-Nikodym derivative at time 𝑡 is given by the ratio of the numeraires normalized
by their time 𝑡 = 0 values.
36

37. Luc_Faucheux_2021
Law of one price in FX - XI
¨
"ℚ
"ℙ
𝑡 =
>ℚ(,)
>ℙ(,)
/
>ℚ(+)
>ℙ(+)
=
>ℚ(,)
>ℚ(+)
/
>ℙ ,
>ℙ(+)
¨ ℜ(,' 𝑡 = ℜ(←' 𝑡 =
"ℚ#
"ℚ!
𝑡 =
2! )
2# )
.
-!,# ,
-!,# )
.
2# ,
2! ,
¨ 𝐵' 𝑡 is the numeraire in the domestic currency 1 associated with the domestic risk free
measure
¨ 𝐵' 𝑡 = 𝑁ℚ!
(𝑡)
¨ 𝐵( 𝑡 is the numeraire in the domestic currency 2 associated with the domestic risk free
measure
¨ 𝐵( 𝑡 = 𝑁ℚ#
(𝑡)
¨ ℜ(,' 𝑡 = ℜ(←' 𝑡 =
"ℚ#
"ℚ!
𝑡 =
2! )
2# )
.
-!,# ,
-!,# )
.
2# ,
2! ,
=
-!,# ,
-!,# )
.
>ℚ#(,)
ℚ#())
/
>ℚ! ,
>ℚ!())
37

38. Luc_Faucheux_2021
Law of one price in FX - XII
¨
"ℚ
"ℙ
𝑡 =
>ℚ(,)
>ℙ(,)
/
>ℚ(+)
>ℙ(+)
=
>ℚ(,)
>ℚ(+)
/
>ℙ ,
>ℙ(+)
¨ However in the world you have to deal with the currency conversion:
¨ ℜ(,' 𝑡 = ℜ(←' 𝑡 =
"ℚ#
"ℚ!
𝑡 =
2! )
2# )
.
-!,# ,
-!,# )
.
2# ,
2! ,
=
-!,# ,
-!,# )
.
>ℚ#(,)
>ℚ#())
/
>ℚ! ,
>ℚ!())
¨ Alternatively you can define in the domestic currency:
¨ 𝑁′ℚ#
𝑡 = 𝑁ℚ#
𝑡 . 𝑋',( 𝑡
¨ ℜ(,' 𝑡 = ℜ(←' 𝑡 =
"ℚ#
"ℚ!
𝑡 =
2! )
2# )
.
-!,# ,
-!,# )
.
2# ,
2! ,
=
>Aℚ#(,)
>Aℚ#())
/
>ℚ! ,
>ℚ!())
¨ Where everything is expressed in the common domestic currency 1
38

39. Luc_Faucheux_2021
Law of one price in FX - XIII
¨ We are starting to see how we can formalize a little the change of measure
¨
"ℚ
"ℙ
𝑡 =
>ℚ(,)
>ℙ(,)
/
>ℚ(+)
>ℙ(+)
=
>ℚ(,)
>ℚ(+)
/
>ℙ ,
>ℙ(+)
¨
"ℙ#
"ℙ!
𝑡 =
>ℙ#(,)
>ℙ!(,)
/
>ℙ#())
>ℙ!())
=
>ℙ#(,)
>ℙ#())
/
>ℙ! ,
>ℙ!())
¨ ℜ(←' 𝑡 = ℜℙ#←ℙ!
𝑡 =
"ℙ#
"ℙ!
𝑡
¨ 𝔼,
ℙ#
𝑋 𝑡 𝔉 0 = 𝔼,
ℙ! "ℙ#
"ℙ!
𝑡 . 𝑋 𝑡 𝔉 0 = 𝔼,
ℙ!
{ℜℙ#←ℙ!
𝑡 . 𝑋(𝑡)|𝔉 0 }
39

40. Luc_Faucheux_2021
Law of one price in FX - XIV
¨ So, in the same currency between 2 different measures (like the T-forward and the risk free)
¨ ℜℙ#←ℙ!
𝑡 =
"ℙ#
"ℙ!
𝑡 =
>ℙ#(,)
>ℙ!(,)
/
>ℙ#())
>ℙ!())
=
>ℙ#(,)
>ℙ#())
/
>ℙ! ,
>ℙ!())
¨ 𝔼,
ℙ#
𝑋 𝑡 𝔉 0 = 𝔼,
ℙ! "ℙ#
"ℙ!
𝑡 . 𝑋 𝑡 𝔉 0 = 𝔼,
ℙ!
{ℜℙ#←ℙ!
𝑡 . 𝑋(𝑡)|𝔉 0 }
¨ In the FX world you can still use a similar expression, but you have to remember to bring
every quantity into the common currency
¨ ℜ(,' 𝑡 = ℜℚ#←ℚ!
𝑡 =
"ℚ#
"ℚ!
𝑡 =
2! )
2# )
.
-!,# ,
-!,# )
.
2# ,
2! ,
=
>Aℚ#(,)
>Aℚ#())
/
>ℚ! ,
>ℚ!())
40

41. Luc_Faucheux_2021
Law of one price in FX - XV
¨ Note that it does not matter in which currency you bring it
¨ ℜ(,' 𝑡 = ℜℚ#←ℚ!
𝑡 =
"ℚ#
"ℚ!
𝑡 =
2! )
2# )
.
-!,# ,
-!,# )
.
2# ,
2! ,
=
>Aℚ#(,)
>Aℚ#())
/
>ℚ! ,
>ℚ!())
¨ If you bring it into the domestic currency 1
¨ ℜ(,' 𝑡 = ℜℚ#←ℚ!
𝑡 =
"ℚ#
"ℚ!
𝑡 =
2! )
2# )
.
-!,# ,
-!,# )
.
2# ,
2! ,
=
>ℚ#(,)
>ℚ#())
/
>Aℚ! ,
>Aℚ!())
¨ If you bring it into the foreign currency 2
¨ 𝑁′ℚ#
𝑡 = 𝑁ℚ#
𝑡 . 𝑋',( 𝑡 = 𝑋',( 𝑡 . 𝑁ℚ#
𝑡 = 𝑋'←( 𝑡 . 𝑁ℚ#
𝑡 = 𝑋'←( 𝑡 . 𝐵( 𝑡
¨ 𝑁′ℚ!
𝑡 = 𝑁ℚ!
𝑡 . 𝑋(,' 𝑡 = 𝑋(,' 𝑡 . 𝑁ℚ!
(𝑡) = 𝑋(←' 𝑡 . 𝑁ℚ!
(𝑡) = 𝐵' 𝑡 /𝑋'←( 𝑡
41

42. Luc_Faucheux_2021
Law of one price in FX - XVI
¨ ℜ(,' 𝑡 = ℜℚ#←ℚ!
𝑡 =
"ℚ#
"ℚ!
𝑡 =
2! )
2# )
.
-!,# ,
-!,# )
.
2# ,
2! ,
¨ ℜ(,' 𝑡 = ℜℚ#←ℚ!
𝑡 =
"ℚ#
"ℚ!
𝑡 = (
-!,# , .2# ,
-!,# ) .2# )
)/(
2! ,
2! )
)
¨ ℜ(,' 𝑡 = ℜℚ#←ℚ!
𝑡 =
"ℚ#
"ℚ!
𝑡 =
>Aℚ#(,)
>Aℚ#())
/
>ℚ! ,
>ℚ!())
¨ ℜ(,' 𝑡 = ℜℚ#←ℚ!
𝑡 =
"ℚ#
"ℚ!
𝑡 =
2! )
2# )
.
-!,# ,
-!,# )
.
2# ,
2! ,
¨ ℜ(,' 𝑡 = ℜℚ#←ℚ!
𝑡 =
"ℚ#
"ℚ!
𝑡 =
2! ) /-!,# )
2! , /-!,# ,
.
2# ,
2# )
¨ ℜ(,' 𝑡 = ℜℚ#←ℚ!
𝑡 =
"ℚ#
"ℚ!
𝑡 =
-#,! ) .2! )
-!,# , .2! ,
.
2# ,
2# )
= (
2# ,
2# )
)/(
-!,# , .2! ,
-#,! ) .2! )
)
¨ ℜ(,' 𝑡 = ℜℚ#←ℚ!
𝑡 =
"ℚ#
"ℚ!
𝑡 =
>ℚ#(,)
>ℚ#())
/
>Aℚ! ,
>Aℚ!())
42

43. Luc_Faucheux_2021
Law of one price in FX - XVII
¨ Where the (‘) indicates that you are bringing the variable into the other currency (am
assuming for now only two currencies, if more we will have to revisit that notation)
¨ 𝑁′ℚ#
𝑡 = 𝑁ℚ#
𝑡 . 𝑋',( 𝑡 = 𝑋',( 𝑡 . 𝑁ℚ#
𝑡 = 𝑋'←( 𝑡 . 𝑁ℚ#
𝑡 = 𝑋'←( 𝑡 . 𝐵( 𝑡
¨ 𝑁′ℚ!
𝑡 = 𝑁ℚ!
𝑡 . 𝑋(,' 𝑡 = 𝑋(,' 𝑡 . 𝑁ℚ!
(𝑡) = 𝑋(←' 𝑡 . 𝑁ℚ!
(𝑡) = 𝐵' 𝑡 /𝑋'←( 𝑡
¨ ℜ(,' 𝑡 = ℜℚ#←ℚ!
𝑡 =
"ℚ#
"ℚ!
𝑡 =
2! )
2# )
.
-!,# ,
-!,# )
.
2# ,
2! ,
¨ But the interesting thing about this equation is that we know something about 𝑋',( 𝑡
¨ Bear in mind again that this is in the GBM case based on the assumptions that we made in
the previous deck on FX
43

44. Luc_Faucheux_2021
Law of one price in FX - XVIII
¨ 𝐵' 𝑡 = exp[∫
)*+
)*,
𝑅' 𝑠, 𝑠, 𝑠 . 𝑑𝑠] is the Bank Account Numeraire in the domestic world of
currency (1).
¨ 𝐵' 𝑡 = exp[∫
)*+
)*,
𝑅' 𝑠, 𝑠, 𝑠 . 𝑑𝑠] is associated to the Risk free measure in the domestic
currency (1) that we will note 𝑊ℚ' 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
44

45. Luc_Faucheux_2021
Law of one price in FX - XIX
¨
"0!
0!
= 𝜇 𝐴' . 𝑑𝑡 + 𝜎 𝐴' . [ . 𝑑𝑊0!(𝑡)
¨
"0#
0#
= 𝜇 𝐴( . 𝑑𝑡 + 𝜎 𝐴( . [ . 𝑑𝑊0#(𝑡)
¨
"-!,#
-!,#
= 𝜇 𝑋',( . 𝑑𝑡 + 𝜎 𝑋',( . [ . 𝑑𝑊-!,#(𝑡)
¨ 𝐵' 𝑡 = exp[∫
)*+
)*,
𝑅' 𝑠, 𝑠, 𝑠 . 𝑑𝑠]
¨ 𝑑𝐵' 𝑡 = 𝐵' 𝑡 . 𝑅' 𝑡, 𝑡, 𝑡 . 𝑑𝑡
¨ 𝐵( 𝑡 = exp[∫
)*+
)*,
𝑅( 𝑠, 𝑠, 𝑠 . 𝑑𝑠]
¨ 𝑑𝐵( 𝑡 = 𝐵( 𝑡 . 𝑅( 𝑡, 𝑡, 𝑡 . 𝑑𝑡
45

46. Luc_Faucheux_2021
Law of one price in FX - XX
¨ We also have:
¨
"-!,#
-!,#
= {𝑅' 𝑡, 𝑡, 𝑡 − 𝑅( 𝑡, 𝑡, 𝑡 }. 𝑑𝑡 + 𝜎 𝑋',( . [ . 𝑑𝑊
ℚ!
-!,#
𝑡
¨
"-#,!
-#,!
= {𝑅( 𝑡, 𝑡, 𝑡 − 𝑅' 𝑡, 𝑡, 𝑡 }. 𝑑𝑡 + 𝜎 𝑋(,' . [ . 𝑑𝑊
ℚ#
-#,!
𝑡
¨ So there we can do it explicitely, or go back to the previous deck where we had done the
derivation in the deflated variables:
¨
" C
(-!,#.2#)
C
(-!,#.2#)
= 𝜎 𝑋',( . [ . 𝑑𝑊
ℚ!
-!,#
𝑡
¨
-!,#.2#
2!
= T
(𝑋',(. 𝐵()
¨ ℜ(,' 𝑡 = ℜℚ#←ℚ!
𝑡 =
"ℚ#
"ℚ!
𝑡 =
2! )
2# )
.
-!,# ,
-!,# )
.
2# ,
2! ,
=
C
-!,#.2# (,)
C
-!,#.2# ())
46

47. Luc_Faucheux_2021
Law of one price in FX - XXI
¨
" C
(-!,#.2#)
C
(-!,#.2#)
= 𝜎 𝑋',( . [ . 𝑑𝑊
ℚ!
-!,#
𝑡
¨
-!,#.2#
2!
= T
(𝑋',(. 𝐵()
¨ ℜ(,' 𝑡 = ℜℚ#←ℚ!
𝑡 =
"ℚ#
"ℚ!
𝑡 =
2! )
2# )
.
-!,# ,
-!,# )
.
2# ,
2! ,
=
C
-!,#.2# (,)
C
-!,#.2# ())
¨ T
𝑋',(. 𝐵( 𝑡 = T
𝑋',(. 𝐵( 𝑠 . ℜ(,' 𝑡
¨ 𝑑 T
𝑋',(. 𝐵( 𝑡 = T
𝑋',(. 𝐵( 𝑠 . 𝑑ℜ(,' 𝑡
¨
"ℜ#,! ,
ℜ#,! ,
=
"ℜℚ#←ℚ! ,
ℜℚ#←ℚ! ,
= 𝜎 𝑋',( . [ . 𝑑𝑊
ℚ!
-!,#
𝑡
¨ ℜ(,' 𝑡 = ℜℚ#←ℚ!
𝑡 =
"ℚ#
"ℚ!
𝑡
47

48. Luc_Faucheux_2021
Law of one price in FX - XXII
¨ If you think about it for a second, this is frigging awesome (or completely obvious)
¨ ℜ(,' 𝑡 = ℜℚ#←ℚ!
𝑡 =
"ℚ#
"ℚ!
𝑡 =
2! )
2# )
.
-!,# ,
-!,# )
.
2# ,
2! ,
=
C
-!,#.2# (,)
C
-!,#.2# ())
¨ The Radon-Nikodym derivative between the two Risk free measures is actually up to a
constant the deflated tradeable foreign numeraire brought back into the domestic currency
¨
"ℜ#,! ,
ℜ#,! ,
=
"ℜℚ#←ℚ! ,
ℜℚ#←ℚ! ,
= 𝜎 𝑋',( . [ . 𝑑𝑊
ℚ!
-!,#
𝑡
¨ This is where we can relate to the Dade-Doleans exponential:
¨ The solution of:𝑑ℜ(,' 𝑡 = 𝜎 𝑋',( . ℜ(,' 𝑡 . [ . 𝑑𝑊
ℚ!
-!,#
𝑡
¨ Is:
¨ ℜ(,' 𝑡 = ℜℚ#←ℚ!
𝑡 = ℜℚ#←ℚ!
𝑠 . ℇ(∫
E*)
E*,
𝜎 𝑋',( . [ . 𝑑𝑊
ℚ!
-!,#
𝑢 )
48

49. Luc_Faucheux_2021
Law of one price in FX - XXIII
¨ ℜ(,' 𝑡 = ℜℚ#←ℚ!
𝑡 = ℜℚ#←ℚ!
𝑠 . ℇ(∫
E*)
E*,
𝜎 𝑋',( . [ . 𝑑𝑊
ℚ!
-!,#
𝑢 )
¨ This works in the general case where 𝜎 𝑋',( = 𝜎 𝑋',( (𝑡)
¨ And in the simple case where 𝜎 𝑋',( = 𝑐𝑡𝑒
¨ ℜ(,' 𝑡 = ℜℚ#←ℚ!
𝑡 = ℜℚ#←ℚ!
𝑠 . ℇ(∫
E*)
E*,
𝜎 𝑋',( . [ . 𝑑𝑊
ℚ!
-!,#
𝑢 )
¨ ℜ(,' 𝑡 = ℜℚ#←ℚ!
𝑡 = ℜℚ#←ℚ!
𝑠 . ℇ(𝜎 𝑋',( . 𝑊
ℚ!
-!,#
𝑡 )
¨ ℇ(∫
E*)
E*,
𝜎 𝑋',( . [ . 𝑑𝑊
ℚ!
-!,#
𝑢 ) = exp(∫
E*)
E*,
𝜎 𝑋',( . [ . 𝑑𝑊
ℚ!
-!,#
𝑢 −
'
(
. ∫
E*)
E*,
𝜎 𝑋',(
(
. 𝑑𝑢)
¨ And in the simple case where 𝜎 𝑋',( = 𝑐𝑡𝑒
¨ ℇ(𝜎 𝑋',( . 𝑊
ℚ!
-!,#
𝑡 ) = exp(𝜎 𝑋',( . 𝑊
ℚ!
-!,#
𝑡 −
'
(
. 𝜎 𝑋',(
(
. 𝑡)
49

50. Luc_Faucheux_2021
Law of one price in FX - XXIV
¨ Subject to the Novikov condition..
¨ All right am going to do the joke again.
¨ The Novikov condition has nothing to do with the fancy Mayfair place in London where
plenty of beautiful models and hedge funds people spend most of their evenings….
¨ However if you hang around Novikov too long, you might catch a condition…the Novikov
condition….am so funny I cannot help myself…
50

51. Luc_Faucheux_2021
Law of one price in FX - XXV
¨ All right, back to business.
¨ So assuming that the Novikov condition is respected (essentially things are not blowing up
and you do not have any divergence in the variance)
¨ ℜ(,' 𝑡 = ℜℚ#←ℚ!
𝑡 =
"ℚ#
"ℚ!
𝑡 =
2! )
2# )
.
-!,# ,
-!,# )
.
2# ,
2! ,
=
C
-!,#.2# (,)
C
-!,#.2# ())
¨ 𝔼,
ℚ#
𝑋(𝑡) 𝔉 𝑠 = 𝔼,
ℚ!
{ℜ(←'(𝑡). 𝑋(𝑡)|𝔉 𝑠 }
¨ Novikov condition:
¨ 𝔼,
ℚ!
exp
'
(
. ∫
E*)
E*,
𝜎 𝑋',(
(
. 𝑑𝑢 𝔉 𝑠 < ∞
51

52. Luc_Faucheux_2021
Law of one price in FX - XXVI
¨ ℜ(,' 𝑡 = ℜℚ#←ℚ!
𝑡 =
"ℚ#
"ℚ!
𝑡 = ℜℚ#←ℚ!
𝑠 . ℇ(∫
E*)
E*,
𝜎 𝑋',( . [ . 𝑑𝑊
ℚ!
-!,#
𝑢 )
¨ That means from the CMG (Cameron-Martin-Girsanov) theorem that:
¨ 𝑑𝑊
ℚ#
-!,#
𝑡 = 𝑑𝑊
ℚ!
-!,#
𝑡 − 𝜎 𝑋',( . 𝑑𝑡
¨ Guess what ?
¨ That is exactly what we had derived before
¨ 𝑑𝑊
ℚ#
-#,!
𝑡 = −𝑑𝑊
ℚ!
-!,#
𝑡 + 𝜎 𝑋(,' . 𝑑𝑡
¨ And by symmetry
¨ 𝑑𝑊
ℚ!
-!,#
𝑡 = −𝑑𝑊
ℚ#
-#,!
𝑡 + 𝜎 𝑋',( . 𝑑𝑡
¨ Let’s spend some time on that to make sure that we did not screw up a minus sign
somewhere
52

53. Luc_Faucheux_2021
Law of one price in FX - XXVII
¨ 𝑑𝑊
ℚ#
-#,!
𝑡 = −𝑑𝑊
ℚ!
-!,#
𝑡 + 𝜎 𝑋(,' . 𝑑𝑡
¨ 𝑋(,' 𝑡 = 1/𝑋',((𝑡)
¨
"-!,#
-!,#
= {𝑅' 𝑡, 𝑡, 𝑡 − 𝑅( 𝑡, 𝑡, 𝑡 }. 𝑑𝑡 + 𝜎 𝑋',( . [ . 𝑑𝑊
ℚ!
-!,#
𝑡
¨ Now of course by symmetry we have:
¨
"-#,!
-#,!
= {𝑅( 𝑡, 𝑡, 𝑡 − 𝑅' 𝑡, 𝑡, 𝑡 }. 𝑑𝑡 + 𝜎 𝑋(,' . [ . 𝑑𝑊
ℚ#
-#,!
𝑡
¨ And doing ITO lemma on 𝑓 𝑥 = (
'
/
) in the first equation led to :
¨
"-#,!
-#,!
= 𝑅( 𝑡, 𝑡, 𝑡 − 𝑅' 𝑡, 𝑡, 𝑡 + 𝜎 𝑋',(
(
. 𝑑𝑡 − 𝜎 𝑋(,' . [ . 𝑑𝑊
ℚ!
-!,#
𝑡
¨
"-#,!
-#,!
= {𝑅( 𝑡, 𝑡, 𝑡 − 𝑅' 𝑡, 𝑡, 𝑡 }. 𝑑𝑡 + 𝜎 𝑋(,' . [ . 𝑑𝑊
ℚ#
-#,!
𝑡
53

54. Luc_Faucheux_2021
Law of one price in FX - XXVIII
¨
"-#,!
-#,!
= 𝑅( 𝑡, 𝑡, 𝑡 − 𝑅' 𝑡, 𝑡, 𝑡 + 𝜎 𝑋',(
(
. 𝑑𝑡 − 𝜎 𝑋(,' . [ . 𝑑𝑊
ℚ!
-!,#
𝑡
¨
"-#,!
-#,!
= {𝑅( 𝑡, 𝑡, 𝑡 − 𝑅' 𝑡, 𝑡, 𝑡 }. 𝑑𝑡 + 𝜎 𝑋(,' . [ . 𝑑𝑊
ℚ#
-#,!
𝑡
¨ That is how we derived:
¨ 𝑑𝑊
ℚ#
-#,!
𝑡 = −𝑑𝑊
ℚ!
-!,#
𝑡 + 𝜎 𝑋(,' . 𝑑𝑡
¨ We need now to relate 𝑑𝑊
ℚ#
-#,!
𝑡 with 𝑑𝑊
ℚ#
-!,#
𝑡
¨ This is again cranking the ITO lemma handle on 𝑋 → (
'
-
)
¨ Let’s get a couple of slides from the previous deck to refresh our memory and make sure
that we are not getting confused.
54

55. Luc_Faucheux_2021
ITO lemma on (1/X)
from the previous deck
55

56. 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 =
F!
F/
. 𝛿𝑋 +
'
(
.
F#!
F/# . (𝛿𝑋)(
¨
F!
F/
=
F!
F/
|/*- ,
¨
F#!
F/# =
F#!
F/# |/*- ,
56

57. Luc_Faucheux_2021
The FX correlation triangle - V
¨ 𝑓 𝑥 = 1/𝑥
¨
F!
F/
=
G'
/#
¨
F#!
F/# =
(
/*
¨ 𝛿𝑓 =
F!
F/
. 𝛿𝑋 +
'
(
.
F#!
F/# . (𝛿𝑋)(
¨ 𝛿𝑓 =
F!
F/
|/*- , . 𝛿𝑋 +
'
(
.
F#!
F/# |/*- , . (𝛿𝑋)(
¨ 𝛿(
'
-
) =
G'
-# . 𝛿𝑋 +
'
(
.
(
-* . (𝛿𝑋)(
¨ 𝑑(
'
-
) =
G'
-# . 𝑑𝑋 +
'
(
.
(
-* . (𝑑𝑋)(
57

58. Luc_Faucheux_2021
The FX correlation triangle - VI
¨ 𝑑(
'
-
) =
G'
-# . 𝑑𝑋 +
'
(
.
(
-* . (𝑑𝑋)(
¨
"-%,$
-%,$
= 𝜇 𝑋%,# . 𝑑𝑡 + 𝜎 𝑋%,# . [ . 𝑑𝑊-%,$(𝑡)
¨
"-
-
= 𝜇 𝑋 . 𝑑𝑡 + 𝜎 𝑋 . [ . 𝑑𝑊-(𝑡)
¨ 𝑑𝑋 = 𝜇 𝑋 . 𝑋. 𝑑𝑡 + 𝜎 𝑋 . 𝑋. [ . 𝑑𝑊-(𝑡)
¨ 𝑑𝑋(𝑡) = 𝜇 𝑋 𝑡 . 𝑋(𝑡). 𝑑𝑡 + 𝜎 𝑋 𝑡 . 𝑋(𝑡). [ . 𝑑𝑊-(𝑡)
¨ 𝑑𝑋 = 𝜇 𝑋 . 𝑋. 𝑑𝑡 + 𝜎 𝑋 . 𝑋. [ . 𝑑𝑊-(𝑡)
¨ (𝑑𝑋)(= (𝜎 𝑋 . 𝑋)(. 𝑑𝑡 from the quadradic variation property of the Brownian motion
58

59. Luc_Faucheux_2021
The FX correlation triangle - VII
¨ 𝑑(
'
-
) =
G'
-# . 𝑑𝑋 +
'
(
.
(
-* . (𝑑𝑋)(
¨ 𝑑𝑋 = 𝜇 𝑋 . 𝑋. 𝑑𝑡 + 𝜎 𝑋 . 𝑋. [ . 𝑑𝑊-(𝑡)
¨ (𝑑𝑋)(= (𝜎 𝑋 . 𝑋)(. 𝑑𝑡
¨ 𝑑(
'
-
) =
G'
-# . (𝜇 𝑋 . 𝑋. 𝑑𝑡 + 𝜎 𝑋 . 𝑋. [ . 𝑑𝑊-(𝑡) ) +
'
(
.
(
-* . (𝜎 𝑋 . 𝑋)(. 𝑑𝑡
¨ 𝑑
'
-
= − 𝜇 𝑋 − 𝜎 𝑋 ( .
'
-
. 𝑑𝑡 − 𝜎 𝑋 . (
'
-
). [ . 𝑑𝑊-(𝑡)
¨
"
!
+
!
+
= − 𝜇 𝑋 − 𝜎 𝑋 ( . 𝑑𝑡 − 𝜎 𝑋 . [ . 𝑑𝑊-(𝑡)
¨
"-
-
= 𝜇 𝑋 . 𝑑𝑡 + 𝜎 𝑋 . [ . 𝑑𝑊-(𝑡)
59

60. Luc_Faucheux_2021
The FX correlation triangle - VIII
¨
"
!
+
!
+
= − 𝜇 𝑋 − 𝜎 𝑋 ( . 𝑑𝑡 − 𝜎 𝑋 . [ . 𝑑𝑊-(𝑡)
¨
"-
-
= 𝜇 𝑋 . 𝑑𝑡 + 𝜎 𝑋 . [ . 𝑑𝑊-(𝑡)
¨
"
!
+
!
+
= 𝜇
'
-
. 𝑑𝑡 + 𝜎
'
-
. [ . 𝑑𝑊
!
+ (𝑡)
¨ 𝜎
'
-
= 𝜎 𝑋
¨ 𝜇
'
-
= − 𝜇 𝑋 − 𝜎 𝑋 ( = −𝜇 𝑋 + 𝜎 𝑋 (
¨ 𝑑𝑊
!
+ 𝑡 = −𝑑𝑊-(𝑡)
60

61. Luc_Faucheux_2021
Law of one price in FX - XXIX
¨ So going back to our notations:
¨ 𝑑𝑊
ℚ#
-#,!
𝑡 = −𝑑𝑊
ℚ!
-!,#
𝑡 + 𝜎 𝑋(,' . 𝑑𝑡
¨ 𝑑𝑊
ℚ#
-#,!
𝑡 = −𝑑𝑊
ℚ#
-!,#
𝑡
¨ 𝑑𝑊
ℚ#
-!,#
𝑡 = 𝑑𝑊
ℚ!
-!,#
𝑡 − 𝜎 𝑋(,' . 𝑑𝑡
¨ 𝜎 𝑋(,' = 𝜎 𝑋',(
¨ 𝑑𝑊
ℚ#
-!,#
𝑡 = 𝑑𝑊
ℚ!
-!,#
𝑡 − 𝜎 𝑋',( . 𝑑𝑡 and we had:
¨ ℜ(,' 𝑡 = ℜℚ#←ℚ!
𝑡 =
"ℚ#
"ℚ!
𝑡 = ℜℚ#←ℚ!
𝑠 . ℇ(∫
E*)
E*,
𝜎 𝑋',( . [ . 𝑑𝑊
ℚ!
-!,#
𝑢 )
¨ 𝑑𝑊
ℚ#
-!,#
𝑡 = 𝑑𝑊
ℚ!
-!,#
𝑡 − 𝜎 𝑋',( . 𝑑𝑡
61

62. Luc_Faucheux_2021
Law of one price in FX - XXX
¨ So it worked !! We did not screw up a minus sign anywhere.
¨ Quite an achievement. Let’s recap, with what we hope are rigorous enough notation to
avoid any confusion.
¨ 𝑑𝑊
ℚ#
-#,!
𝑡 = −𝑑𝑊
ℚ#
-!,#
𝑡
¨ 𝑑𝑊
ℚ!
-#,!
𝑡 = −𝑑𝑊
ℚ!
-!,#
𝑡
¨ 𝑑𝑊
ℚ#
-!,#
𝑡 = 𝑑𝑊
ℚ!
-!,#
𝑡 − 𝜎 𝑋',( . 𝑑𝑡
¨ 𝑑𝑊
ℚ!
-#,!
𝑡 = 𝑑𝑊
ℚ#
-#,!
𝑡 − 𝜎 𝑋(,' . 𝑑𝑡
¨ −𝑑𝑊
ℚ!
-!,#
𝑡 = −𝑑𝑊
ℚ#
-!,#
𝑡 − 𝜎 𝑋(,' . 𝑑𝑡
¨ 𝑑𝑊
ℚ!
-!,#
𝑡 = 𝑑𝑊
ℚ#
-!,#
𝑡 + 𝜎 𝑋(,' . 𝑑𝑡
62

63. Luc_Faucheux_2021
Some more notes on
ℜ!,# 𝑡 = ℜℚ!←ℚ"
𝑡 =
𝑑ℚ!
𝑑ℚ#
𝑡
63

64. Luc_Faucheux_2021
Some more notes on ℜ!,# 𝑡 - I
¨ Let’s revisit some of the facts on the mysterious Radon-Nikodym derivative ℜ(,' 𝑡
¨ So first of all ℜ(,' 𝑡 is a martingale in the domestic risk free measure:
¨
"ℜ#,! ,
ℜ#,! ,
=
"ℜℚ#←ℚ! ,
ℜℚ#←ℚ! ,
= 𝜎 𝑋',( . [ . 𝑑𝑊
ℚ!
-!,#
𝑡
¨ ℜ(,' 𝑡 = ℜℚ#←ℚ!
𝑡 =
"ℚ#
"ℚ!
𝑡
¨ ℜ(,' 𝑠 = ℜℚ#←ℚ!
𝑠 =
"ℚ#
"ℚ!
𝑠 = 𝔼,
ℚ!
{ℜ(,' 𝑡 |𝔉 𝑠 }
¨
"ℜ#,! ,
ℜ#,! ,
= 𝑠𝑜𝑚𝑒𝑡ℎ𝑖𝑛𝑔 . [ . 𝑑𝑊ℚ!
𝑡
¨ In the case of the FX market:
¨
"ℜ#,! ,
ℜ#,! ,
= 𝜎 𝑋',( . [ . 𝑑𝑊
ℚ!
-!,#
𝑡
64

65. Luc_Faucheux_2021
Some more notes on ℜ!,# 𝑡 - II
¨ ℜ(,' 𝑡 is itself a stochastic process, and has a PDF, which will be a solution of the FP PDE
¨ PDF: Probability Distribution Function
¨ PDE: Partial Differential Equations
¨ FP: Fokker-Plank
¨ In the case of FX:
¨
"ℜ#,! ,
ℜ#,! ,
= 𝜎 𝑋',( . [ . 𝑑𝑊
ℚ!
-!,#
𝑡
¨ 𝑑ℜ(,' 𝑡 = 𝜎 𝑋',( . ℜ(,' 𝑡 . [ . 𝑑𝑊
ℚ!
-!,#
𝑡
¨ Let’s remind us of the SDE->PDE for ITO
65

66. Luc_Faucheux_2021
We need a nice summary to avoid any confusion - IV
¨ ITO SDE is: 𝑑𝑋 𝑡 = 𝑎 𝑡, 𝑋 𝑡 . 𝑑𝑡 + 𝑏 𝑡, 𝑋 𝑡 . ([). 𝑑𝑊
¨ This implies that the PDF follows the FORWARD ITO Kolmogorov PDE
¨
FH(/,,|-,,,,)
F,
= −
F
F-
𝑎 𝑋 𝑡 , 𝑡 . 𝑝 𝑥, 𝑡 𝑋J, 𝑡J −
F
F-
[
'
(
. [𝑏(𝑋 𝑡 , 𝑡)( . 𝑝(𝑥, 𝑡|𝑋J, 𝑡J)]
¨ The PDF ALSO follows the BACKWARD ITO Kolmogorov PDE:
¨
FH(/,,|-,,,,)
F,,
= −𝑎 𝑋J, 𝑡J
F
F-,
𝑝 𝑥, 𝑡 𝑋J, 𝑡J −
'
(
. 𝑏(𝑋J, 𝑡J)( F#
F-,
# 𝑝(𝑥, 𝑡|𝑋J, 𝑡J)
¨
FH
F,,
= −𝑎
FH
F-,
−
'
(
. 𝑏( F#H
F-,
#
¨
FH
F,,
= −𝑎
FH
F-,
− 𝐷
F#H
F-,
#
¨ Where I have explicitly kept the notation 𝑡J and 𝑋J to indicate the fact that this is a
BACKWARD PDE (expectation of a payoff at maturity)
66

67. Luc_Faucheux_2021
Some more notes on ℜ!,# 𝑡 - III
¨ ℜ(,' 𝑡 is a stochastic process
¨ 𝓇(,' is a regular variable (Newtonian calculus)
¨ We can define a Distribution Function such that:
¨ PDF Probability Density Function: 𝑝ℜ#,!
(𝓇(,', 𝑡)
¨ Distribution function : 𝑃ℜ#,!
(𝓇(,', 𝑡)
¨ 𝑃ℜ#,!
𝓇(,', 𝑡 = 𝑃𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 ℜ(,' ≤ 𝓇(,', 𝑡 = ∫
:*GK
:*𝓇#,!
𝑝ℜ#,!
(𝑦, 𝑡) . 𝑑𝑦
¨ 𝑝ℜ#,!
𝓇(,', 𝑡 =
F
F𝓇#,!
𝑃ℜ#,!
(𝓇(,', 𝑡)
67

68. Luc_Faucheux_2021
Some more notes on ℜ!,# 𝑡 - IV
¨ 𝑃ℜ#,!
𝓇(,', 𝑡 = 𝑃𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 ℜ(,' ≤ 𝓇(,', 𝑡 = ∫
:*GK
:*𝓇#,!
𝑝ℜ#,!
(𝑦, 𝑡) . 𝑑𝑦
¨ 𝑃ℜ#,!
𝓇(,', 𝑡|𝔉 𝑠 = 𝑃𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 ℜ(,' ≤ 𝓇(,', 𝑡|𝔉 𝑠 = ∫
:*GK
:*𝓇#,!
𝑝ℜ#,!
(𝑦, 𝑡|𝔉 𝑠 ) . 𝑑𝑦
¨ 𝑃ℜ#,!
𝓇(,', 𝑡|𝔉 𝑠 = 𝑃𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 ℜ(,' ≤ 𝓇(,', 𝑡|𝔉 𝑠
¨ 𝑃ℜ#,!
𝓇(,', 𝑡|𝔉 𝑠 = 𝔼,
ℚ!
{1{ℜ#,!N𝓇#,!}|𝔉 𝑠 }
68

69. Luc_Faucheux_2021
Some more notes on ℜ!,# 𝑡 - V
¨ ITO SDE is: 𝑑𝑋 𝑡 = 𝑎 𝑡, 𝑋 𝑡 . 𝑑𝑡 + 𝑏 𝑡, 𝑋 𝑡 . ([). 𝑑𝑊
¨ This implies that the PDF follows the FORWARD ITO Kolmogorov PDE
¨
FH(/,,|-,,,,)
F,
= −
F
F-
𝑎 𝑋 𝑡 , 𝑡 . 𝑝 𝑥, 𝑡 𝑋J, 𝑡J −
F
F-
[
'
(
. [𝑏(𝑋 𝑡 , 𝑡)( . 𝑝(𝑥, 𝑡|𝑋J, 𝑡J)]
¨ Becomes in our case:
¨ 𝑑ℜ(,' 𝑡 = 𝜎 𝑋',( . ℜ(,' 𝑡 . [ . 𝑑𝑊
ℚ!
-!,#
𝑡
¨ ITO SDE is: dℜ(,' 𝑡 = 0. 𝑑𝑡 + 𝜎 𝑋',( . ℜ(,' 𝑡 . ([). 𝑑𝑊
ℚ!
-!,#
𝑡
¨ This implies that the PDF follows the FORWARD ITO Kolmogorov PDE
¨
FHℜ#,!(𝓇#,!,,|𝔉(,,))
F,
=
F
F𝓇#,!
F
F𝓇#,!
[
'
(
. [(𝜎 𝑋',( . 𝓇(,')( . 𝑝ℜ#,!
𝓇(,', 𝑡 𝔉(𝑡J )]
69

70. Luc_Faucheux_2021
Some more notes on ℜ!,# 𝑡 - VI
¨
FHℜ#,!(𝓇#,!,,|𝔉(,,))
F,
=
F
F𝓇#,!
F
F𝓇#,!
[
'
(
. [(𝜎 𝑋',( . 𝓇(,')( . 𝑝ℜ#,!
(𝓇(,', 𝑡|𝔉(𝑡J))]
¨
FHℜ#,!(𝓇#,!,,|𝔉(,,))
F,
=
'
(
.
F#
F𝓇#,!
# . [(𝜎 𝑋',( . 𝓇(,')( . 𝑝ℜ#,!
(𝓇(,', 𝑡|𝔉(𝑡J))]
¨ For ease of notation:
¨ ℜ is the Radon-Nikodym derivative
¨ ℜ is a stochastic process that is a martingale in the measure on the right (of ℜ(,')
¨ ℜ will follow a distribution which is a solution of the PDE:
¨
FH(𝓇,,|𝔉(,,))
F,
=
'
(
.
F#
F𝓇# . [(𝑏(𝓇 𝑡 , 𝑡)( . 𝑝(𝓇, 𝑡|𝔉(𝑡J))]
¨ In the case of FX: 𝑏(𝓇 𝑡 , 𝑡 = 𝜎 𝑋',( . 𝓇 𝑡 , where 𝜎 𝑋',( could be a function and not
always a constant
70

71. Luc_Faucheux_2021
Some more notes on ℜ!,# 𝑡 - VII
¨ In the simple case where 𝜎 𝑋',( is a constant, we have the following which is actually
pretty cool:
¨ ℜ(,' 𝑡 = ℜℚ#←ℚ!
𝑡 =
"ℚ#
"ℚ!
𝑡 = ℜℚ#←ℚ!
𝑠 . ℇ(∫
E*)
E*,
𝜎 𝑋',( . [ . 𝑑𝑊
ℚ!
-!,#
𝑢 )
¨ ℜ(,' 𝑡 = ℜℚ#←ℚ!
𝑡 =
"ℚ#
"ℚ!
𝑡 = ℜℚ#←ℚ!
𝑠 . ℇ(𝜎 𝑋',( . 𝑊
ℚ!
-!,#
𝑡 )
¨ Assuming that the starting position is 𝑊
ℚ!
-!,#
𝑠 = 0
¨ We then also have then: ℜℚ#←ℚ!
𝑠 = 1
¨ ℜ(,' 𝑡 = ℜℚ#←ℚ!
𝑡 =
"ℚ#
"ℚ!
𝑡 = ℇ(𝜎 𝑋',( . 𝑊
ℚ!
-!,#
𝑡 )
71

72. Luc_Faucheux_2021
Some more notes on ℜ!,# 𝑡 - VIII
¨ ℜ(,' 𝑡 = ℜℚ#←ℚ!
𝑡 =
"ℚ#
"ℚ!
𝑡 = ℇ(𝜎 𝑋',( . 𝑊
ℚ!
-!,#
𝑡 )
¨ We also have:
¨ 𝑑𝑊
ℚ!
-!,#
𝑡 = 𝑑𝑊
ℚ#
-!,#
𝑡 + 𝜎 𝑋(,' . 𝑑𝑡
¨ In the simple case of a constant:
¨ 𝑊
ℚ!
-!,#
𝑡 = 𝑊
ℚ#
-!,#
𝑡 + 𝜎 𝑋(,' . 𝑡
¨ 𝜎 𝑋(,' =
Qℚ!
+!,#
, GQℚ#
+!,#
,
,
¨ ℜ(,' 𝑡 = ℜℚ#←ℚ!
𝑡 =
"ℚ#
"ℚ!
𝑡 = ℇ(
Qℚ!
+!,#
, GQℚ#
+!,#
,
,
. 𝑊
ℚ!
-!,#
𝑡 )
¨ Super cool equation, but not sure if really useful
72

73. Luc_Faucheux_2021
Siegel paradox
73

74. Luc_Faucheux_2021
Siegel paradox - I
74

75. Luc_Faucheux_2021
So at least for now…..
75