we start playing with measures and changes of measures. We will need to derive the LMM (Libor Market Model) in the next deck.

1. Luc_Faucheux_2021
THE RATES WORLD – Part VII
More on measures and change of measures
1

2. Luc_Faucheux_2021
That deck
2
¨ After a bunch of decks, we take here a breather to revisit some of the assumptions/results,
and finish up a number of sections that we had left unfinished
¨ Something to say about the notation / progression of those decks.
¨ I tried very hard to do it in a progressive manner, and so the formalism and notations
became more complicated but also more complete as we went on.
¨ So in many ways the ”simple” notation that I used at the beginning were potentially
confusing. Many apologies for that, but that was intended in order to demonstrate as we go
along the need for more complicated notation, as opposed to just dump it at the beginning
in a very formal manner
¨ Hopefully you will have found the journey interesting and enlightning, and maybe more alive
than a formal class, which again this is not. This is merely a bunch of notes that I put down
in a Powerpoint in a selfish purpose so that I can more easily find them and retrieve them,
and hopefully this helps you reading and understanding real serious and formal textbooks on
the subject.

3. Luc_Faucheux_2021
That deck - II
¨ Here we start playing with measures and change of measure
¨ We revisit Ho-Lee and understand why the deflated zeros were not only a martingale, but
could be expressed as the Radon-Nikodym derivative between two measures.
¨ We start looking at drift between measures and start laying down the foundations of what
we will need to derive the LMM (Libor Market Model), hopefully in deck VIII
¨ Also, a section on some career advice
3

4. Luc_Faucheux_2021
A couple of useful tools
4

5. Luc_Faucheux_2021
Useful tools
¨ As you go through those slides, it is quite apparent that there are some relations or
properties that we keep using over and over again, or that are worth mentioning.
¨ I tried to put all of them together in a quick summary section here
¨ I still need to work on a notation section, maybe once I get my book deal
¨ Would love to get your feedback on this section, if there are tools that you tend to use a lot
and find useful, just drop me a note and I would be happy to include those
5

6. Luc_Faucheux_2021
Useful tools – ITO LEMMA
¨ The ITO lemma is revered in stochastic calculus.
¨ In the somewhat misleading “differential” form it reads:
¨ 𝛿𝑓 =
!"
!#
. 𝛿𝑋 +
$
%
.
!!"
!#! . (𝛿𝑋)%
¨ It should really only be expressed as:
¨ 𝑓 𝑋 𝑡& − 𝑓 𝑋 𝑡' = ∫
()("
()(# !"
!*
. ([). 𝑑𝑋(𝑡) + ∫
()("
()(# $
%
.
!!"
!#! . ([). (𝛿𝑋)%
¨ The ITO convention for the ITO integral is that we take the “LHS” (Left Hand side) in the
partition as noted by: ([)
¨ And the definition of the integral is:
¨ ∫
()("
()(#
𝑓 𝑋(𝑡) . 𝑑𝑊 𝑡 = lim
+→-
∑.)/
.)+
𝑓 𝑋(𝑡.) . {𝑊 𝑡.0$ − 𝑊(𝑡.)}
¨ Where we assume that we do not choose a pathological mesh and the the function is
relatively well behaved
6

7. Luc_Faucheux_2021
Useful tools – ITO LEMMA - II
¨ Be careful that stochastic calculus in many ways has NOTHING to do with regular calculus
¨ So it is quite dangerous to write:
¨ 𝛿𝑓 =
!"
!#
. 𝛿𝑋 +
$
%
.
!!"
!#! . (𝛿𝑋)%
¨ And say “ oh well stochastic calculus is the same as regular calculus, it is just when I do
Taylor expansion I should really go up one more order in order to go up to all the orders that
are at least linear in time”
¨ Again, this is ONLY a formal correspondence, or a way to write down two things that are
almost completely different
¨ Stochastic processes are NOT differentiable, so do not even think of using a “Taylor
expansion on a stochastic process”
¨ ALWAYS go back to the integral, always try to use the SIE format (Stochastic Integral
Equation), never the SDE format (Stochastic Differential Equation)
7

8. Luc_Faucheux_2021
Useful tools – ITO Leibniz
¨ Again, for ease of notation, we use the “differential” form, but by now we know better than
to trust is:
¨ 𝛿𝑓 𝑋, 𝑌 =
!"
!*
. 𝛿𝑋 +
!"
!1
. 𝛿𝑌 +
$
%
.
!!"
!*! . 𝛿𝑋% +
$
%
.
!!"
!1! . 𝛿𝑌% +
!!"
!*!1
. 𝛿𝑋. 𝛿𝑌
¨ Note: should really be written as:
¨ 𝛿𝑓 𝑋, 𝑌 =
!"
!#
. 𝛿𝑋 +
!"
!2
. 𝛿𝑌 +
$
%
.
!!"
!#! . 𝛿𝑋% +
$
%
.
!!"
!2! . 𝛿𝑌% +
!!"
!#!2
. 𝛿𝑋. 𝛿𝑌
¨ Lower case 𝑥 is a regular variable
¨ Upper case 𝑋 is a stochastic variable
¨ 𝑓 𝑋, 𝑌 is really noted 𝑓 𝑥 = 𝑋, 𝑦 = 𝑌 and all the partial derivatives are for example:
¨
!!"
!#!2
=
!!"
!#!2
|#)* ( ,2)1(()
8

9. Luc_Faucheux_2021
Useful tools – ITO and STRATO correspondence
¨ ITO integral is defined as LHS (Left Hand Side)
¨ ∫
()('
()(&
𝐹 𝑋 𝑡 . ([). 𝑑𝑋(𝑡) = lim
6→-
{∑7)$
7)6
𝐹(𝑋(𝑡7)). [𝑋(𝑡70$) − 𝑋(𝑡7)]}
¨ STRATO integral is defined as M (Middle)
¨ ∫
()('
()(&
𝐹 𝑋 𝑡 . (∘). 𝑑𝑋(𝑡) = lim
6→-
{∑7)$
7)6
𝐹(
*(($%& 0*(($)]
%
). [𝑋(𝑡70$) − 𝑋(𝑡7)]}
¨ For a simple Brownian motion
¨ ∫
()('
()(&
𝑓 𝑊 𝑡 . (∘). 𝑑𝑊(𝑡) = ∫
()('
()(&
𝑓 𝑊 𝑡 . ([). 𝑑𝑊(𝑡) +
$
%
∫
()('
()(& !"
!9
|9):((). 𝑑𝑡
¨ The integral in time ∫
()('
()(& !"
!9
|9):((). 𝑑𝑡 is the usual Riemann integral defined as
¨ ∫
()('
()(&
𝐹 𝑋 𝑡 . 𝑑𝑡 = lim
6→-
{∑7)$
7)6
𝐹(𝑋(𝜑[𝑡7, 𝑡70$])). [𝑡70$ − 𝑡7]}
9

10. Luc_Faucheux_2021
Useful tools – ITO and STRATO correspondence - II
¨ Where 𝜑[𝑡7, 𝑡70$] is a function that takes some point within the mesh (does not matter
where, LHS, RIHS, middle, anywhere, could also varies from one bucket to the next, that is
the beauty of the Riemann integral in regular, or Newtonian, calculus, is that you do not
have all those pesky differences between ITO or Stratonovitch,…)
¨ For a more complicated stochastic process
¨ 𝑑𝑋 𝑡 = 𝑎 𝑡, 𝑋 𝑡 . 𝑑𝑡 + 𝑏 𝑡, 𝑋 𝑡 . 𝑑𝑊
¨ We have:
¨ ∫
()('
()(&
𝑓 𝑋 𝑡 . ∘ . 𝑑𝑊 𝑡 = ∫
()('
()(&
𝑓 𝑋 𝑡 . ([). 𝑑𝑊(𝑡) + ∫
()('
()(& $
%
. 𝑏 𝑡, 𝑋 𝑡 .
!"
!#
|#)*((). 𝑑𝑡
10

11. Luc_Faucheux_2021
Useful tools – ITO integral is a martingale
¨ This is super useful
¨ For a Brownian motion 𝑊 𝑠 associated to the measure
¨ 𝔼{∫
;)/
;)(
𝑓 𝑠 . 𝑑𝑊 𝑠 } = 𝔼{lim
+→-
∑.)/
.)+
𝑓 𝑠. . {𝑊 𝑠.0$ − 𝑊(𝑠.)} }
¨ 𝔼{∫
;)/
;)(
𝑓 𝑠 . 𝑑𝑊 𝑠 } = lim
+→-
∑.)/
.)+
𝑓 𝑠. . 𝔼{𝑊 𝑠.0$ − 𝑊(𝑠.)}
¨ 𝔼{∫
;)/
;)(
𝑓 𝑠 . 𝑑𝑊 𝑠 } = lim
+→-
∑.)/
.)+
𝑓 𝑠. . 0 = 0
¨ 𝔼 ∫
;)/
;)(
𝑓 𝑠 . 𝑑𝑊 𝑠 = 0
¨ 𝔼 ∫
;)/
;)(
𝑓 𝑊 𝑠 , 𝑠 . 𝑑𝑊 𝑠 = 0
11

12. Luc_Faucheux_2021
Useful tools – Isometry of the ITO integral
¨ 𝔼{ ∫
;)/
;)(
𝑓 𝑠 . 𝑑𝑊 𝑠
%
} = ∫
;)/
;)(
𝑓 𝑠 %. 𝑑𝑠
¨ 𝔼{ ∫
;)/
;)(
𝑓 𝑊 𝑠 , 𝑠 . 𝑑𝑊 𝑠
%
} = ∫
;)/
;)(
𝑓 𝑊 𝑠 , 𝑠 %. 𝑑𝑠
12

13. Luc_Faucheux_2021
Useful tools – A martingale is driftless, a driftless process is a
martingale
¨ 𝔼:{𝑋(𝑡)|)|𝔉(𝑠)} = 0
¨ 𝑑𝑋 𝑡 = 𝑎 𝑡, 𝑋 𝑡 . 𝑑𝑡 + 𝑏 𝑡, 𝑋 𝑡 . 𝑑𝑊
¨ 𝑎 𝑡, 𝑋 𝑡 = 0
¨ 𝑑𝑋 𝑡 = 𝑏 𝑡, 𝑋 𝑡 . 𝑑𝑊
¨ No advection, no drift for a martingale
¨ 𝑋 𝑡 = ∫
;)/
;)(
𝑏 𝑡, 𝑋 𝑡 . 𝑑𝑊 𝑠
¨ Again the ITO integral is a martingale
¨ 𝔼 ∫
;)/
;)(
𝑏 𝑡, 𝑋 𝑡 . 𝑑𝑊 𝑠 = 0
¨ 𝔼:{𝑋(𝑡)|)|𝔉(𝑠)} = 0
13

14. Luc_Faucheux_2021
Useful tools – useful relationship
¨ 𝔼 exp ∫
;)/
;)(
𝑓 𝑠 . 𝑑𝑊 𝑠 = exp[∫
;)/
;)( $
%
𝑓 𝑠 %. 𝑑𝑠]
14

15. Luc_Faucheux_2021
Useful tools – expected value of the exponential
¨ 𝔼 exp 𝑋 = exp 𝔼 𝑋 . exp
$
%
𝔼 (𝑋 − 𝔼 𝑋 )%
¨ 𝑀 𝑋 𝑡 = 𝔼 𝑋
¨ 𝑉 𝑋 𝑡 = 𝔼 (𝑋(𝑡) − 𝑀 𝑋 𝑡 )%
¨ 𝔼 exp 𝑋 = exp 𝑀[𝑋(𝑡)] . exp
$
%
𝑉[𝑋(𝑡)]
¨ 𝔼 exp 𝑋 = exp[𝑀] . exp
$
%
𝑉
15

16. Luc_Faucheux_2021
Useful tools - Fubini
¨ 𝑋 = ∫
;)(
;)('
{∫
<)(
<);
𝑓(𝑢). ([). 𝑑𝑊(𝑢)}. 𝑑𝑠
16
s
𝑠 = 𝑡!
u
s
u
𝑠 = 𝑡 𝑠 = 𝑡!
𝑠 = 𝑡
𝑋 = J
;)(
;)('
{ J
<)(
<);
𝑓(𝑢). ([). 𝑑𝑊(𝑢)}. 𝑑𝑠 𝑋 = J
<)(
<)('
{ J
;)<
;)('
𝑓(𝑠). 𝑑𝑠}. ([). 𝑑𝑊(𝑢)

17. Luc_Faucheux_2021
Useful tools – how to always create a martingale
¨ We use here the Tower property:
¨ For any process 𝑋 𝑡 , we create:
¨ 𝑁 𝑡 = 𝔼=
:
{𝑋(𝑇)|𝔉(𝑡)}
¨ 𝔼=
:
𝑁 𝑡 𝔉 𝑠 = 𝔼=
:
𝔼=
:
𝑋 𝑇 𝔉 𝑡 𝔉 𝑠 = 𝔼=
:
𝑋 𝑇 𝔉 𝑠 = 𝑁(𝑠)
¨ Because conditioning firstly on information back to time 𝑡 then back to time 𝑠 is just the
same as conditioning back to time 𝑠 to start with.
¨ 𝔼=
:
𝑁 𝑡 𝔉 𝑠 = 𝑁(𝑠)
¨ So 𝑁 𝑡 = 𝔼=
:
{𝑋(𝑇)|𝔉(𝑡)} is by construction a 𝑊-martingale
¨ That is a neat little trick to always create a martingale process (Baxter p. 77)
17

18. Luc_Faucheux_2021
Useful tools – Radon-Nikodym as expectation
¨ 𝔼ℙ is the measure associated to the Brownian motion 𝑊ℙ(𝑡)
¨ 𝔼ℚ is the measure associated to the Brownian motion 𝑊ℚ(𝑡)
¨ The Radon-Nikodym
@ℚ
@ℙ
(𝑡) is such that:
¨ 𝔼(
ℚ
𝑋 𝑡 𝔉 0 = 𝔼(
ℙ{
@ℚ
@ℙ
𝑡 . 𝑋(𝑡)|𝔉 0 }
¨ We also have this beautiful equation (Baxter p.68)
¨
@ℚ
@ℙ
𝑡 = 𝔼=
ℙ
{
@ℚ
@ℙ
𝑇 |𝔉 𝑡 } for 𝑇 > 𝑡
¨ The Radon-Nikodym derivative
@ℚ
@ℙ
𝑡 is a martingale under the ℙ-measure 𝔼ℙ
¨ In particular:
@ℚ
@ℙ
0 = 1
¨ 𝔼=
ℙ @ℚ
@ℙ
𝑇 𝔉 0 = 1
18

19. Luc_Faucheux_2021
Useful tools – Radon-Nikodym as expectation -II
¨ 𝔼=
ℙ @ℚ
@ℙ
𝑇 𝔉 0 = 1
¨ We had derived this in the deck V-b using the “useful formula” starting from:
¨
@ℚ
@ℙ
= exp[− ∫
;)/
;)(
𝜉 𝑠 . 𝑑𝑊 𝑠 −
$
%
∫
;)/
;)(
𝜉 𝑠 %. 𝑑𝑠]
¨ 𝔼(
ℙ exp ∫
;)/
;)(
𝑓 𝑠 . 𝑑𝑊 𝑠 −
$
%
∫
;)/
;)(
𝑓 𝑠 %. 𝑑𝑠 |𝔉 0 = 1
¨ 𝔼(
ℙ @ℚ
@ℙ
(𝑡)|𝔉 0 = 1
¨ Note that this should not be too surprising since the definition of the derivative is:
¨ 𝔼ℚ 𝑋 𝑡 𝔉 0 = 𝔼ℙ{
@ℚ
@ℙ
𝑋(𝑡)|𝔉 0 }, replacing 𝑋 𝑡 = 1
¨ 𝔼ℚ 1 𝔉 0 = 1 = 𝔼ℙ{
@ℚ
@ℙ
|𝔉 0 } so we get: 𝔼(
ℙ @ℚ
@ℙ
(𝑡)|𝔉 0 = 1
19

20. Luc_Faucheux_2021
Useful tools – Radon-Nikodym as expectation -III
¨ 𝔼(
ℚ
𝑋 𝑡 𝔉 0 = 𝔼(
ℙ{
@ℚ
@ℙ
𝑡 . 𝑋(𝑡)|𝔉 0 }
¨
@ℚ
@ℙ
𝑡 = 𝔼=
ℙ{
@ℚ
@ℙ
𝑇 |𝔉 𝑡 } for 𝑇 > 𝑡
¨ 𝔼(
ℚ
𝑋 𝑡 𝔉 𝑠 =
$
(ℚ
(ℙ
;
. 𝔼(
ℙ{
@ℚ
@ℙ
𝑡 . 𝑋(𝑡)|𝔉 𝑠 } for 𝑠 < 𝑡
¨ 𝔼(
ℙ @ℚ
@ℙ
𝑡 . 𝑋 𝑡 𝔉 𝑠 = 𝔼(
ℚ @ℚ
@ℙ
𝑠 . 𝑋 𝑡 𝔉 𝑠 =
@ℚ
@ℙ
𝑠 . 𝔼(
ℚ
𝑋 𝑡 𝔉 𝑠
¨ 𝔼(
ℚ
𝑋 𝑡 𝔉 𝑠 = 𝔼(
ℙ{
(ℚ
(ℙ
(
(ℚ
(ℙ
;
. 𝑋(𝑡)|𝔉 𝑠 }
¨ 𝔼(
ℚ
𝑋 𝑡 𝔉 0 = 𝔼(
ℙ
(ℚ
(ℙ
(
(ℚ
(ℙ
/
. 𝑋 𝑡 𝔉 0 = 𝔼(
ℙ{
@ℚ
@ℙ
𝑡 . 𝑋(𝑡)|𝔉 0 }
20

21. Luc_Faucheux_2021
Useful tools – most stupid equation ever
¨ 𝑍 0,0, 𝑡 = exp(− ∫
;)/
;)(
𝔼;
ℤ(;)
𝑅(𝑠, 𝑠, 𝑠) 𝔉 0 . 𝑑𝑠)
¨ 𝑍𝐶 0,0, 𝑡 = 𝔼(
ℚ
exp[− ∫
;)/
;)(
𝑅 𝑠, 𝑠, 𝑠 . 𝑑𝑠]|𝔉(0)
¨ 𝑉 0, $1, 𝑡, 𝑡 = 𝑍 0,0, 𝑡
¨ 𝑉 0, $1, 𝑡, 𝑡 = exp(− ∫
;)/
;)(
𝔼;
ℤ(;)
𝑉(𝑠, $𝑅 𝑠, 𝑠, 𝑠 , 𝑠, 𝑠) 𝔉 0 . 𝑑𝑠)
¨ 𝑉 0, $1, 𝑡, 𝑡 = 𝔼(
ℚ
exp[− ∫
;)/
;)(
𝑉(𝑠, $𝑅 𝑠, 𝑠, 𝑠 , 𝑠, 𝑠). 𝑑𝑠]|𝔉(0)
¨
𝔼+
ℚ
CDE[G ∫
,-.
,-+
I(;,$K ;,;,; ,;,;).@;]|𝔉(/)
CDE(G ∫
,-.
,-+
𝔼,
ℤ(,)
𝑉(𝑠, $𝑅 𝑠, 𝑠, 𝑠 , 𝑠, 𝑠) 𝔉 0 .@;)
= 1
21

22. Luc_Faucheux_2021
Useful tools – most stupid equation ever - II
¨
𝔼+
ℚ
CDE[G ∫
,-.
,-+
I(;,$K ;,;,; ,;,;).@;]|𝔉(/)
CDE(G ∫
,-.
,-+
𝔼,
ℤ(,)
𝑉(𝑠, $𝑅 𝑠, 𝑠, 𝑠 , 𝑠, 𝑠) 𝔉 0 .@;)
= 1
¨ The ratio of ;
the expectation at time 𝑡 under the risk-neutral measure 𝔼ℚ associated to the rolling
numeraire 𝐵 𝑠 = exp[∫
<)/
<);
𝑅 𝑢, 𝑢, 𝑢 . 𝑑𝑢], subject to the filtration 𝔉 0 , of the exponential of
the opposite of the integral over the time 𝑠 from time 𝑠 = 0 to time 𝑠 = 𝑡 of the claim valued at
time 𝑠 that pays at time 𝑠 the instantaneous short rate 𝑅 𝑠, 𝑠, 𝑠 set at time 𝑠;
to the exponential of the opposite of the integral over the time 𝑠 from time 𝑠 = 0 to
time 𝑠 = 𝑡 of the expectations at time 𝑠 under the terminal measures 𝔼ℤ(;), subject to the same
filtration 𝔉 0 , associated to the Zeros 𝑍(𝑢, 𝑢, 𝑠), of the same claim valued at time 𝑠 that pays
at time 𝑠 the instantaneous short rate 𝑅 𝑠, 𝑠, 𝑠 set at time 𝑠,
is…..equal to 1
¨ There are on the internet a number of post about making 1=1 as complicated as possible.
22

23. Luc_Faucheux_2021
Change of Numeraire and change of Measure
23

24. Luc_Faucheux_2021
Change of numeraire - I
¨ 𝔼ℙ is the measure associated to the Brownian motion 𝑊ℙ(𝑡)
¨ 𝑁ℙ(𝑡) is the numeraire associated to that measure 𝔼ℙ
¨ 𝔼ℚ is the measure associated to the Brownian motion 𝑊ℚ(𝑡)
¨ 𝑁ℚ(𝑡) is the numeraire associated to that measure 𝔼ℚ
¨ A general claim 𝑉(𝑡) is such that:
¨ 𝔼=
ℙ I(=)
6ℙ(=)
𝔉 𝑡 =
I(()
6ℙ(()
¨ 𝔼=
ℚ I(=)
6ℚ(=)
𝔉 𝑡 =
I(()
6ℚ(()
24

25. Luc_Faucheux_2021
Change of numeraire - II
¨
@ℚ
@ℙ
𝑡 = 𝔼=
ℙ
{
@ℚ
@ℙ
𝑇 |𝔉 𝑡 } for 𝑇 > 𝑡
¨ More generally from the trick of always creating a martingale we know that:
¨ 𝑁 𝑡 = 𝔼=
:
{𝑋(𝑇)|𝔉(𝑡)} is by construction a 𝑊-martingale meaning that:
¨ 𝔼=
:
𝑁 𝑡 𝔉 𝑠 = 𝑁(𝑠)
¨ Let’s check it for the specific case 𝑁 𝑡 =
@ℚ
@ℙ
𝑡 in the ℙ-measure
¨ 𝔼=
ℙ @ℚ
@ℙ
𝑡 𝔉 𝑠 = 𝔼=
ℙ 𝔼=
ℙ @ℚ
@ℙ
𝑇 𝔉 𝑡 𝔉 𝑠 = 𝔼=
ℙ @ℚ
@ℙ
𝑇 𝔉 𝑠 =
@ℚ
@ℙ
𝑠
25

26. Luc_Faucheux_2021
Change of numeraire - III
¨ 𝔼=
ℙ I(=)
6ℙ(=)
𝔉 𝑡 =
I(()
6ℙ(()
¨ 𝔼=
ℚ I(=)
6ℚ(=)
𝔉 𝑡 =
I(()
6ℚ(()
¨ 𝔼(
ℚ
𝑋 𝑡 𝔉 𝑠 = 𝔼(
ℙ{
(ℚ
(ℙ
(
(ℚ
(ℙ
;
. 𝑋(𝑡)|𝔉 𝑠 }
¨ 𝔼=
ℚ
𝑋 𝑇 𝔉 𝑡 = 𝔼=
ℙ
{
(ℚ
(ℙ
=
(ℚ
(ℙ
(
. 𝑋(𝑇)|𝔉 𝑡 }
¨ We apply this to the specific case of :
¨ 𝑋 𝑇 =
I(=)
6ℚ(=)
26

27. Luc_Faucheux_2021
Change of numeraire - IV
¨
I(()
6ℚ(()
= 𝔼=
ℚ I(=)
6ℚ(=)
𝔉 𝑡 = 𝔼=
ℙ{
(ℚ
(ℙ
=
(ℚ
(ℙ
(
.
I(=)
6ℚ(=)
|𝔉 𝑡 }
¨ 𝔼=
ℙ I(=)
6ℙ(=)
𝔉 𝑡 =
I(()
6ℙ(()
¨
I(()
6ℚ(()
=
I(()
6ℙ(()
.
6ℙ(()
6ℚ(()
= 𝔼=
ℙ I(=)
6ℙ(=)
𝔉 𝑡 .
6ℙ(()
6ℚ(()
= 𝔼=
ℙ
{
(ℚ
(ℙ
=
(ℚ
(ℙ
(
.
I(=)
6ℚ(=)
|𝔉 𝑡 }
¨ 𝔼=
ℙ I(=)
6ℙ(=)
𝔉 𝑡 .
6ℙ(()
6ℚ(()
= 𝔼=
ℙ{
(ℚ
(ℙ
=
(ℚ
(ℙ
(
.
I(=)
6ℚ(=)
|𝔉 𝑡 }
¨ 𝔼=
ℙ I(=)
6ℙ(=)
.
6ℙ(()
6ℚ(()
𝔉 𝑡 = 𝔼=
ℙ{
(ℚ
(ℙ
=
(ℚ
(ℙ
(
.
I(=)
6ℚ(=)
|𝔉 𝑡 }
27

28. Luc_Faucheux_2021
Change of numeraire - V
¨ 𝔼=
ℙ I(=)
6ℙ(=)
.
6ℙ(()
6ℚ(()
𝔉 𝑡 = 𝔼=
ℙ{
(ℚ
(ℙ
=
(ℚ
(ℙ
(
.
I(=)
6ℚ(=)
|𝔉 𝑡 }
¨ 𝔼=
ℙ I(=)
6ℙ(=)
.
6ℙ(()
6ℚ(()
𝔉 𝑡 = 𝔼=
ℙ
{
(ℚ
(ℙ
=
(ℚ
(ℙ
(
.
I(=)
6ℙ(=)
.
6ℙ(=)
6ℚ(=)
|𝔉 𝑡 }
¨ Since this has to hold for any and every possible and imaginable claim 𝑉(𝑡):
¨
6ℙ(()
6ℚ(()
=
(ℚ
(ℙ
=
(ℚ
(ℙ
(
.
6ℙ(=)
6ℚ(=)
¨
6ℙ(()
6ℚ(()
.
@ℚ
@ℙ
𝑡 =
6ℙ(=)
6ℚ(=)
.
@ℚ
@ℙ
𝑇
28

29. Luc_Faucheux_2021
Change of numeraire - VI
¨
6ℙ(()
6ℚ(()
.
@ℚ
@ℙ
𝑡 =
6ℙ(=)
6ℚ(=)
.
@ℚ
@ℙ
𝑇
¨ And this has to be valid for every 𝑡 < 𝑇
¨ In particular for 𝑡 = 0
¨
@ℚ
@ℙ
0 = 1
¨
6ℙ(()
6ℚ(()
.
@ℚ
@ℙ
𝑡 =
6ℙ(/)
6ℚ(/)
¨ And so we finally obtain for the change of measure from a change of numeraire:
¨
@ℚ
@ℙ
𝑡 =
6ℙ(/)
6ℚ(/)
/
6ℙ(()
6ℚ(()
29

30. Luc_Faucheux_2021
Change of numeraire - VII
¨
@ℚ
@ℙ
𝑡 =
6ℙ(/)
6ℚ(/)
/
6ℙ(()
6ℚ(()
¨
@ℚ
@ℙ
𝑡 =
6ℚ(()
6ℙ(()
/
6ℚ(/)
6ℙ(/)
¨ If we normalize the numeraires by their time 𝑡 = 0 values:
¨ X
𝑁ℚ 𝑡 = 𝑁ℚ(𝑡)/𝑁ℚ(0)
¨ X
𝑁ℙ 𝑡 = 𝑁ℙ(𝑡)/𝑁ℙ(0)
¨ We obtain the celebrated formula:
¨
@ℚ
@ℙ
𝑡 =
O
6ℚ (
O
6ℙ (
¨ The Radon-Nikodym derivative at time 𝑡 is given by the ratio of the numeraires normalized
by their time 𝑡 = 0 values.
30

31. Luc_Faucheux_2021
Another way to look at Ho-Lee
31

32. Luc_Faucheux_2021
Another way to look at Ho-Lee
¨ We noticed in the previous deck that we had the expression:
¨ ℰ ∫
/
(
𝜉 𝑠 . 𝑑𝑊 𝑠 = exp(∫
/
(
𝜉 𝑠 . 𝑑𝑊(𝑠) − ∫
/
( $
%
𝜉 𝑠 %. 𝑑𝑠)
¨ Z
𝑍 𝑡, 𝑡, 𝑡. = Z
𝑍 0,0, 𝑡. . exp ∫
<)/
<)(
𝜎. (𝑡. − 𝑢). 𝑑𝑊(𝑢) −
$
%
. ∫
<)/
<)(
{𝜎. (𝑡. − 𝑢)}%. 𝑑𝑢
¨ Z
𝑍 𝑡, 𝑡, 𝑡. = Z
𝑍 0,0, 𝑡. . ℰ ∫
<)/
<)(
𝜎. (𝑡. − 𝑢). 𝑑𝑊(𝑢)
¨ We could not help but notice a rather strong connection between the deflated Zeros and the
expression of a Radon-Nikodym derivative.
¨ Z
𝑍 𝑡, 𝑡, 𝑡. =
P (,(,('
Q (
¨ Let’s illustrate here why this is not a coincidence
32

33. Luc_Faucheux_2021
Another way to look at Ho-Lee - II
¨ If ℚ$ is a measure with an associated 𝑊ℚ&(𝑡)Brownian motion (a ℚ$-Brownian motion)
¨ If ℚ% is a measure with an associated 𝑊ℚ!(𝑡)Brownian motion (a ℚ%-Brownian motion)
¨ We have (under the famous Novikov condition..)
¨ If there is a process 𝜉 𝑡 such that it is reasonably well-behaved
¨ 𝔼(
ℚ&
exp ∫
/
( $
%
𝜉 𝑠 %. 𝑑𝑠 |𝔉(0) < 0
¨ Then , following Baxter p.74:
¨ ℚ% is equivalent to ℚ$
¨
@ℚ!
@ℚ&
= exp − ∫
/
(
𝜉 𝑠 . 𝑑𝑊 𝑠 − ∫
/
( $
%
𝜉 𝑠 %. 𝑑𝑠 = ℰ ∫
/
(
(−𝜉 𝑠 ). 𝑑𝑊 𝑠
¨ 𝑊ℚ! 𝑡 = 𝑊ℚ& 𝑡 + ∫
/
(
𝜉 𝑠 . 𝑑𝑠
33

34. Luc_Faucheux_2021
Another way to look at Ho-Lee - III
¨ Under the Risk Neutral measure noted ℚ, associated to the Brownian motion 𝑊ℚ 𝑡 , the
SDE for the instantaneous forward rate is:
¨ 𝑑𝑅 𝑡, 𝑡., 𝑡. = 𝜎%. 𝑡. − 𝑡 . 𝑑𝑡 − 𝜎. ([). 𝑑𝑊ℚ 𝑡
¨ The SIE is:
¨ 𝑅 𝑡, 𝑡., 𝑡. − 𝑅 0, 𝑡., 𝑡. = 𝜎%. 𝑡. 𝑡. −
$
%
𝑡 − 𝜎. ([). 𝑊ℚ 𝑡
¨ The numeraire associated with the risk free measure is the rolling discount:
¨ 𝐵 𝑡 = exp[∫
;)/
;)(
𝑅 𝑠, 𝑠, 𝑠 . 𝑑𝑠]
34

35. Luc_Faucheux_2021
Another way to look at Ho-Lee - IV
¨ Under the Terminal (forward measure) noted ℤ(𝑡.), associated with the Brownian motion
that we note by: 𝑊ℤ((') 𝑡
¨ The instantaneous forward is a martingale under this measure
¨ 𝐿 𝑡, 𝑡., 𝑡R = 𝔼(2
ℤ((2)
𝑉(𝑡R, $𝐿 𝑡., 𝑡., 𝑡R , 𝑡., 𝑡R) 𝔉 𝑡
¨ lim
(2→('
𝐿 𝑡, 𝑡., 𝑡R = 𝑅 𝑡, 𝑡., 𝑡.
¨ lim
(2→('
[𝔼(2
ℤ((2)
𝑉(𝑡R, $𝐿 𝑡., 𝑡., 𝑡R , 𝑡., 𝑡R) 𝔉 𝑡 ] = 𝔼('
ℤ((')
𝑉(𝑡., $𝐿 𝑡., 𝑡., 𝑡. , 𝑡., 𝑡.) 𝔉 𝑡
¨ 𝑅 𝑡, 𝑡., 𝑡. = 𝔼('
ℤ((')
𝑉(𝑡., $𝐿 𝑡., 𝑡., 𝑡. , 𝑡., 𝑡.) 𝔉 𝑡 = 𝔼('
ℤ((')
𝑅(𝑡., 𝑡., 𝑡.) 𝔉 𝑡
¨ 𝑅 𝑡, 𝑡., 𝑡. = 𝔼('
ℤ((')
𝑅(𝑡., 𝑡., 𝑡.) 𝔉 𝑡
35

36. Luc_Faucheux_2021
Another way to look at Ho-Lee - V
¨ 𝑅 𝑡, 𝑡., 𝑡. = 𝔼('
ℤ((')
𝑅(𝑡., 𝑡., 𝑡.) 𝔉 𝑡
¨ 𝑅 𝑡, 𝑡., 𝑡. is a martingale under the terminal measure ℤ(𝑡.)
¨ 𝑅 𝑡, 𝑡., 𝑡. is a driftless process
¨ 𝑑𝑅 𝑡, 𝑡., 𝑡. = 0 . 𝑑𝑡 + 𝑠𝑜𝑚𝑒𝑡ℎ𝑖𝑛𝑔 . 𝑑𝑊ℤ((') 𝑡
¨ The numeraire associated to the terminal measure is the Zero 𝑍 𝑡, 𝑡, 𝑡.
¨ Let’s compare:
¨ 𝑑𝑅 𝑡, 𝑡., 𝑡. = 0 . 𝑑𝑡 + 𝑠𝑜𝑚𝑒𝑡ℎ𝑖𝑛𝑔 . 𝑑𝑊ℤ((') 𝑡
¨ 𝑑𝑅 𝑡, 𝑡., 𝑡. = 𝜎%. 𝑡. − 𝑡 . 𝑑𝑡 − 𝜎. ([). 𝑑𝑊ℚ 𝑡
36

37. Luc_Faucheux_2021
Another way to look at Ho-Lee - VI
¨ 𝑑𝑅 𝑡, 𝑡., 𝑡. = 0 . 𝑑𝑡 + 𝑠𝑜𝑚𝑒𝑡ℎ𝑖𝑛𝑔 . 𝑑𝑊ℤ((') 𝑡
¨ 𝑑𝑅 𝑡, 𝑡., 𝑡. = 𝜎%. 𝑡. − 𝑡 . 𝑑𝑡 − 𝜎. ([). 𝑑𝑊ℚ 𝑡
¨ For the two processes to have the same variance we need:
¨ 𝑠𝑜𝑚𝑒𝑡ℎ𝑖𝑛𝑔 = −𝜎
¨ So we have:
¨ 𝑑𝑅 𝑡, 𝑡., 𝑡. = 0 . 𝑑𝑡 − 𝜎. ([). 𝑑𝑊ℤ((') 𝑡
¨ 𝑑𝑅 𝑡, 𝑡., 𝑡. = 𝜎%. 𝑡. − 𝑡 . 𝑑𝑡 − 𝜎. ([). 𝑑𝑊ℚ 𝑡
37

38. Luc_Faucheux_2021
Another way to look at Ho-Lee - VII
¨ 𝑑𝑅 𝑡, 𝑡., 𝑡. = 0 . 𝑑𝑡 − 𝜎. ([). 𝑑𝑊ℤ((') 𝑡
¨ 𝑑𝑅 𝑡, 𝑡., 𝑡. = 𝜎%. 𝑡. − 𝑡 . 𝑑𝑡 − 𝜎. ([). 𝑑𝑊ℚ 𝑡
¨ Which leads to:
¨ 𝑑𝑊ℤ((') 𝑡 = 𝑑𝑊ℚ 𝑡 − 𝜎. 𝑡. − 𝑡 . 𝑑𝑡
38

39. Luc_Faucheux_2021
Another way to look at Ho-Lee - VIII
¨ We could also do it from the SIE:
¨ 𝑑𝑅 𝑡, 𝑡., 𝑡. = 0 . 𝑑𝑡 − 𝜎. ([). 𝑑𝑊ℤ((') 𝑡
¨ 𝑑𝑅 𝑡, 𝑡., 𝑡. = 𝜎%. 𝑡. − 𝑡 . 𝑑𝑡 − 𝜎. ([). 𝑑𝑊ℚ 𝑡
¨ 𝑅 𝑡, 𝑡., 𝑡. − 𝑅 0, 𝑡., 𝑡. = 0 . 𝑑𝑡 − 𝜎. ([). 𝑊ℤ((') 𝑡
¨ 𝑅 𝑡, 𝑡., 𝑡. − 𝑅 0, 𝑡., 𝑡. = 𝜎%. 𝑡. 𝑡. −
$
%
𝑡 − 𝜎. ([). 𝑊ℚ 𝑡
¨ Which leads to:
¨ 𝑊ℤ((') 𝑡 = 𝑊ℚ 𝑡 − 𝜎. 𝑡. 𝑡. −
$
%
𝑡
39

40. Luc_Faucheux_2021
Another way to look at Ho-Lee - IX
¨ 𝑊ℤ((') 𝑡 = 𝑊ℚ 𝑡 − 𝜎. 𝑡. 𝑡. −
$
%
𝑡
¨ 𝑑𝑊ℤ((') 𝑡 = 𝑑𝑊ℚ 𝑡 − 𝜎. 𝑡. −
$
%
𝑡 . 𝑑𝑡 − 𝜎. 𝑡. −
$
%
. 𝑑𝑡
¨ 𝑑𝑊ℤ((') 𝑡 = 𝑑𝑊ℚ 𝑡 − 𝜎. 𝑡. − 𝑡 . 𝑑𝑡
40

41. Luc_Faucheux_2021
Another way to look at Ho-Lee - X
¨ OK, let’s recap:
¨ In the risk free measure ℚ with the numeraire 𝐵 𝑡 and the Brownian motion 𝑊ℚ 𝑡 , the
Ho-Lee model is:
¨ 𝑑𝑅 𝑡, 𝑡., 𝑡. = 𝜎%. 𝑡. − 𝑡 . 𝑑𝑡 − 𝜎. ([). 𝑑𝑊ℚ 𝑡
¨ In the terminal measure ℤ(𝑡.) with the numeraire 𝑍 𝑡, 𝑡., 𝑡. and the Brownian motion
𝑊ℤ((') 𝑡 , the Ho-Lee model is:
¨ 𝑑𝑅 𝑡, 𝑡., 𝑡. = 0 . 𝑑𝑡 − 𝜎. ([). 𝑑𝑊ℤ((') 𝑡
¨ 𝑑𝑊ℤ((') 𝑡 = 𝑑𝑊ℚ 𝑡 − 𝜎. 𝑡. − 𝑡 . 𝑑𝑡
41

42. Luc_Faucheux_2021
Another way to look at Ho-Lee - XI
¨ We know from the change of numeraire section that:
¨
@ℚ
@ℙ
𝑡 =
6ℚ(()
6ℙ(()
/
6ℚ(/)
6ℙ(/)
¨
@ℤ((')
@ℚ
𝑡 =
6ℤ(+')(()
6ℚ(()
/
6ℤ(+')(/)
6ℚ(/)
¨ 𝑁ℚ 𝑡 = 𝐵(𝑡)
¨ 𝑁ℚ 0 = 𝐵(0)
¨ 𝑁ℤ ('
𝑡 = 𝑍 𝑡, 𝑡, 𝑡.
¨ 𝑁ℤ ('
0 = 𝑍 0,0, 𝑡.
42

43. Luc_Faucheux_2021
Another way to look at Ho-Lee - XII
¨
@ℤ((')
@ℚ
𝑡 =
6ℤ(+')(()
6ℚ(()
/
6ℤ(+')(/)
6ℚ(/)
¨
@ℤ((')
@ℚ
𝑡 =
P (,(',('
Q(()
/
P /,(',('
Q(/)
¨
@ℤ((')
@ℚ
𝑡 = Z
𝑍 𝑡, 𝑡, 𝑡. / Z
𝑍 0,0, 𝑡.
¨ So if we know the expression for the Radon-Nikodym derivative
@ℤ((')
@ℚ
𝑡 , we will know the
expression for the deflated Zeros
¨ Z
𝑍 𝑡, 𝑡, 𝑡. = Z
𝑍 0,0, 𝑡. .
@ℤ((')
@ℚ
𝑡
¨ So now the question is can we know what is the expression for :
@ℤ((')
@ℚ
𝑡
43

44. Luc_Faucheux_2021
Another way to look at Ho-Lee - XIII
¨ 𝑑𝑊ℤ((') 𝑡 = 𝑑𝑊ℚ 𝑡 − 𝜎. 𝑡. − 𝑡 . 𝑑𝑡
¨
@ℚ!
@ℚ&
= exp − ∫
/
(
𝜉 𝑠 . 𝑑𝑊 𝑠 − ∫
/
( $
%
𝜉 𝑠 %. 𝑑𝑠 = ℰ ∫
/
(
(−𝜉 𝑠 ). 𝑑𝑊 𝑠
¨ 𝑊ℚ! 𝑡 = 𝑊ℚ& 𝑡 + ∫
/
(
𝜉 𝑠 . 𝑑𝑠
¨ 𝑑𝑊ℚ! 𝑡 = 𝑑𝑊ℚ& 𝑡 + 𝜉 𝑡 . 𝑑𝑡
¨ ℚ% = ℤ(𝑡.)
¨ ℚ$ = ℚ
¨ 𝜉 𝑡 = −𝜎. 𝑡. − 𝑡
¨
@ℚ!
@ℚ&
=
@ℤ((')
@ℚ
= ℰ ∫
;)/
;)(
(−𝜉 𝑠 ). 𝑑𝑊 𝑠 = ℰ ∫
;)/
;)(
(𝜎. 𝑡. − 𝑠 ). 𝑑𝑊 𝑠
44

45. Luc_Faucheux_2021
Another way to look at Ho-Lee - XIV
¨ We also know that:
¨ Z
𝑍 𝑡, 𝑡, 𝑡. = Z
𝑍 0,0, 𝑡. .
@ℤ((')
@ℚ
𝑡
¨
@ℤ ('
@ℚ
(𝑡) = ℰ ∫
;)/
;)(
(𝜎. 𝑡. − 𝑠 ). 𝑑𝑊 𝑠
¨ And so we ”retrouve” the expression that did perplex us in the previous deck:
¨ Z
𝑍 𝑡, 𝑡, 𝑡. = Z
𝑍 0,0, 𝑡. . ℰ ∫
<)/
<)(
𝜎. (𝑡. − 𝑢). 𝑑𝑊(𝑢)
¨ Thus this is no coincidence that the deflated Zeros are ALSO the Radon-Nikodym derivative
45

46. Luc_Faucheux_2021
Another way to look at Ho-Lee - XV
¨ The deflated Zeros are the ratio of the Zeros to the rolling numeraire
¨ Z
𝑍 𝑡, 𝑡, 𝑡. =
P (,(,('
Q(()
¨ The deflated Zeros is the ratio of the Terminal measure ℤ(𝑡.) numeraire to the risk-free
measure ℚ numeraire
¨ The ratio of numeraires is related to the Radon-Nikodym derivative by the following:
¨
@ℤ((')
@ℚ
𝑡 =
6ℤ(+')(()
6ℚ(()
/
6ℤ(+')(/)
6ℚ(/)
¨
@ℤ((')
@ℚ
𝑡 =
P (,(,('
Q(()
/
P /,/,('
Q(/)
¨
@ℤ((')
@ℚ
𝑡 = Z
𝑍 𝑡, 𝑡, 𝑡. / Z
𝑍 0,0, 𝑡.
46

47. Luc_Faucheux_2021
Another way to look at Ho-Lee - XVI
¨ So it makes sense that the deflated Zeros can be casted as a function of a Radon-Nikodym
derivative:
¨ Z
𝑍 𝑡, 𝑡, 𝑡. = Z
𝑍 0,0, 𝑡. .
@ℤ((')
@ℚ
𝑡
¨ And now to obtain the RN derivative we just need to know the relation between the two
Brownian motions 𝑊ℤ((') 𝑡 and 𝑊ℚ 𝑡
¨ We know that in the terminal measure ℤ(𝑡.) the instantaneous forward rate 𝑅 𝑡, 𝑡., 𝑡. is a
martingale and thus is a driftless process:
¨ 𝑑𝑅 𝑡, 𝑡., 𝑡. = 0 . 𝑑𝑡 + 𝑠𝑜𝑚𝑒𝑡ℎ𝑖𝑛𝑔 . 𝑑𝑊ℤ((') 𝑡
¨ We just need to know the SDE for the instantaneous forward rate 𝑅 𝑡, 𝑡., 𝑡. in the risk free
measure in the Ho-Lee model (or another model)
47

48. Luc_Faucheux_2021
Another way to look at Ho-Lee - XVII
¨ In the case of Ho-Lee:
¨ 𝑑𝑅 𝑡, 𝑡., 𝑡. = 𝜎%. 𝑡. − 𝑡 . 𝑑𝑡 − 𝜎. ([). 𝑑𝑊ℚ 𝑡
¨ 𝑑𝑅 𝑡, 𝑡., 𝑡. = 0 . 𝑑𝑡 + 𝑠𝑜𝑚𝑒𝑡ℎ𝑖𝑛𝑔 . 𝑑𝑊ℤ((') 𝑡
¨ So:
¨ 𝑑𝑊ℤ((') 𝑡 = 𝑑𝑊ℚ 𝑡 − 𝜎. 𝑡. − 𝑡 . 𝑑𝑡
¨ And:
¨
@ℤ((')
@ℚ
= ℰ ∫
;)/
;)(
(𝜎. 𝑡. − 𝑠 ). 𝑑𝑊 𝑠
48

49. Luc_Faucheux_2021
Another way to look at Ho-Lee - XVIII
¨ Let’s also note that since the RN derivative is a martingale under the reference measure, so
will be the deflated Zeros:
¨ Z
𝑍 𝑡, 𝑡, 𝑡. = Z
𝑍 0,0, 𝑡. .
@ℤ((')
@ℚ
𝑡
¨
@ℤ((')
@ℚ
𝑡 is a martingale under the ℚ-measure
¨
@ℤ ('
@ℚ
(𝑡) = 𝔼=
ℚ
{
@ℤ((')
@ℚ
𝑇 |𝔉 𝑡 } for 𝑇 > 𝑡
¨ So the process for
@ℤ((')
@ℚ
𝑡 is driftless using the 𝑊ℚ 𝑡 Brownian motion
¨ 𝑑
@ℤ ('
@ℚ
𝑡 = 0 . 𝑑𝑡 + 𝑠𝑜𝑚𝑒𝑡ℎ𝑖𝑛𝑔 . 𝑑𝑊ℚ 𝑡
49

50. Luc_Faucheux_2021
Another way to look at Ho-Lee - XIX
¨ 𝑑
@ℤ ('
@ℚ
𝑡 = 0 . 𝑑𝑡 + 𝑠𝑜𝑚𝑒𝑡ℎ𝑖𝑛𝑔 . 𝑑𝑊ℚ 𝑡
¨ Z
𝑍 𝑡, 𝑡, 𝑡. = Z
𝑍 0,0, 𝑡. .
@ℤ((')
@ℚ
𝑡
¨ 𝑑
S
P (,(,('
S
P /,/,('
= 0 . 𝑑𝑡 + 𝑠𝑜𝑚𝑒𝑡ℎ𝑖𝑛𝑔 . 𝑑𝑊ℚ 𝑡
¨ We also know that :
¨ 𝑑𝑌 𝑡 = 𝜉 𝑡 . 𝑌 𝑡 . [ . 𝑑𝑊 𝑡
¨ Is driftless, and the solution of it is:
¨ 𝑌 𝑡 = 𝑌 0 . exp ∫
;)/
;)(
𝜉 𝑠 . [ . 𝑑𝑊 𝑠 −
$
%
∫
;)/
;)(
𝜉 𝑠 %. 𝑑𝑠 = 𝑌 0 . ℰ ∫
;)/
;)(
𝜉 𝑠 . 𝑑𝑊 𝑠
50

51. Luc_Faucheux_2021
Another way to look at Ho-Lee - XX
¨ Finally, we know that
¨
S
P (,(,('
S
P /,/,('
=
@ℤ((')
@ℚ
𝑡 and thus has to be a RN (radon-Nikodym) derivative.
¨ 𝑑𝑊ℤ((') 𝑡 = 𝑑𝑊ℚ 𝑡 + 𝜉 𝑡 . 𝑑𝑡
¨
@ℤ ('
@ℚ
(𝑡) = ℰ ∫
;)/
;)(
(−𝜉 𝑠 ). 𝑑𝑊ℚ 𝑠
¨ Cranking the ITO handle back down to the SDE will return:
¨ 𝑑
@ℤ ('
@ℚ
𝑡 = −𝜉 𝑡 . {
@ℤ ('
@ℚ
(𝑡)}. [ . 𝑑𝑊ℚ 𝑡
¨ 𝑑
S
P (,(,('
S
P /,/,('
= −𝜉 𝑡 . {
S
P (,(,('
S
P /,/,('
(𝑡)}. [ . 𝑑𝑊ℚ 𝑡
51

52. Luc_Faucheux_2021
Another way to look at Ho-Lee - XXI
¨ 𝑑
S
P (,(,('
S
P /,/,('
= −𝜉 𝑡 . {
S
P (,(,('
S
P /,/,('
(𝑡)}. [ . 𝑑𝑊ℚ 𝑡
¨ 𝑑 Z
𝑍 𝑡, 𝑡, 𝑡. = −𝜉 𝑡 . Z
𝑍 𝑡, 𝑡, 𝑡. . [ . 𝑑𝑊ℚ 𝑡
¨ Where: 𝑑𝑊ℤ((') 𝑡 = 𝑑𝑊ℚ 𝑡 + 𝜉 𝑡 . 𝑑𝑡
¨ This is somewhat of a general result:
¨ The deflated Zeros are a martingale in the Risk-free measure
¨ The SDE for the deflated Zeros is of the form:
¨ 𝑑 Z
𝑍 𝑡, 𝑡, 𝑡. = −𝜉 𝑡 . Z
𝑍 𝑡, 𝑡, 𝑡. . [ . 𝑑𝑊ℚ 𝑡
¨ The volatility coefficient for the deflated Zeros is the drift between the risk-free ℚ-Brownian
motion and the terminal ℤ(𝑡.)-Brownian motion
¨ 𝑑𝑊ℤ((') 𝑡 = 𝑑𝑊ℚ 𝑡 + 𝜉 𝑡 . 𝑑𝑡
52

53. Luc_Faucheux_2021
Another way to look at Ho-Lee - XXII
¨ In the case of Ho-Lee:
¨ 𝑑𝑊ℤ((') 𝑡 = 𝑑𝑊ℚ 𝑡 − 𝜎. 𝑡. − 𝑡 . 𝑑𝑡
¨ 𝑑𝑊ℤ((') 𝑡 = 𝑑𝑊ℚ 𝑡 + 𝜉 𝑡 . 𝑑𝑡
¨ 𝜉 𝑡 = −𝜎. 𝑡. − 𝑡
¨ 𝑑 Z
𝑍 𝑡, 𝑡, 𝑡. = −𝜉 𝑡 . Z
𝑍 𝑡, 𝑡, 𝑡. . [ . 𝑑𝑊ℚ 𝑡
¨ 𝑑 Z
𝑍 𝑡, 𝑡, 𝑡. = 𝜎. 𝑡. − 𝑡 . Z
𝑍 𝑡, 𝑡, 𝑡. . [ . 𝑑𝑊ℚ 𝑡
53

54. Luc_Faucheux_2021
Another way to look at Ho-Lee – XXIII
¨ Let’s see if we can do something more general with this (it seems that we sould be able to
do it).
¨ The deflated Zeros are the ratio of two numeraires.
¨ So they are also the RN derivative between the two measures
¨ In particular, they are always a martingale (driftless process) in the measure associated to
the bottom (denominator) numeraire.
¨ Because the Zeros are the numeraire of the terminal measure under which the
instantaneous forward rate is martingale (driftless process), that leaves us as “degrees of
freedom” what kind of process we can write for the instantaneous forward rates in the risk-
free measure (since we do not have much choice in the terminal measure, it is driftless).
¨ We follow here somewhat Baxter p.144 see if we can write something a little more general
than the specific Ho-Lee case.
54

55. Luc_Faucheux_2021
Something a little more general about the
Deflated Zeros and the Instantaneous
Forward Rates
55

56. Luc_Faucheux_2021
Another way to look at Ho-Lee – a little more general - I
¨ Let’s see if we can do something more general with this (it seems that we sould be able to
do it).
¨ The deflated Zeros are the ratio of two numeraires.
¨ So they are also the RN derivative between the two measures
¨ In particular, they are always a martingale (driftless process) in the measure associated to
the bottom (denominator) numeraire.
¨ Because the Zeros are the numeraire of the terminal measure under which the
instantaneous forward rate is martingale (driftless process), that leaves us as “degrees of
freedom” what kind of process we can write for the instantaneous forward rates in the risk-
free measure (since we do not have much choice in the terminal measure, it is driftless).
¨ We follow here somewhat Baxter p.144 see if we can write something a little more general
than the specific Ho-Lee case.
56

57. Luc_Faucheux_2021
Another way to look at Ho-Lee – a little more general - II
¨ In the terminal measure:
¨ 𝑑𝑅 𝑡, 𝑡., 𝑡. = 0 . 𝑑𝑡 + 𝑠𝑜𝑚𝑒𝑡ℎ𝑖𝑛𝑔 . 𝑑𝑊ℤ((') 𝑡
¨ In the risk free measure, let’s assume that we can write something like:
¨ 𝑑𝑅 𝑡, 𝑡., 𝑡. = 𝑎. 𝑑𝑡 + 𝑏. 𝑑𝑊ℚ 𝑡
¨ Where:
¨ 𝑎 = 𝑎(𝔉 𝑡 , 𝑡.)
¨ 𝑏 = 𝑏(𝔉 𝑡 , 𝑡.)
¨ Those functions (advection/drift and volatilities) can depend on the history of the Brownian
motion 𝑊ℚ 𝑡 and the rates themselves up to time 𝑡, and also depends on the terminal
time 𝑡.
¨ Note that here we are dealing with a single factor model for ease of notation
57

58. Luc_Faucheux_2021
Another way to look at Ho-Lee – a little more general - III
¨ Couple of “mathy” conditions on the advection and the volatility, from the original HJM
paper, essentially, this is ensuring ”well-behaved” functions, and somewhat simplified:
¨ 𝑎 = 𝑎(𝔉 𝑡 , 𝑡.)
¨ 𝑏 = 𝑏(𝔉 𝑡 , 𝑡.)
¨ ∫
;)/
;)(
|𝑎(𝔉 𝑠 , 𝑡.) | . 𝑑𝑠 < ∞
¨ ∫
;)/
;)(
𝑏(𝔉 𝑠 , 𝑡.) % . 𝑑𝑠 < ∞
¨ ∫
;)/
;)(
|𝑅(0, 𝑠, 𝑠, )| . 𝑑𝑠 < ∞
¨ ∫
;)/
;)(
𝑑𝑠 {∫
<)/
<);
|𝑎(𝔉 𝑢 , 𝑡.) | . 𝑑𝑢} < ∞ meaning that we can do Fubini
¨ 𝔼(
ℚ
{∫
;)/
;)(
𝑑𝑠 {∫
<)/
<);
𝑏 𝔉 𝑢 , 𝑡. . ({). 𝑑𝑊ℚ 𝑢 }|𝔉 0 } < ∞
58

59. Luc_Faucheux_2021
Another way to look at Ho-Lee – a little more general - IV
¨ Actually am trying to keep those decks under 150 slides or so, so will move that section in
the next deck
59

60. Luc_Faucheux_2021
A little quiz on martingales
60

61. Luc_Faucheux_2021
A little quiz on martingales – prep before the quiz
¨ 𝑅 𝑡, 𝑡., 𝑡. = 𝔼('
ℤ((')
𝑅(𝑡., 𝑡., 𝑡.) 𝔉 𝑡
¨ 𝑅 𝑡, 𝑡., 𝑡. is a martingale under the terminal measure ℤ(𝑡.)
¨ 𝑅 𝑡, 𝑡., 𝑡. is a driftless process
¨ 𝑑𝑅 𝑡, 𝑡., 𝑡. = 0 . 𝑑𝑡 + 𝑠𝑜𝑚𝑒𝑡ℎ𝑖𝑛𝑔 . 𝑑𝑊ℤ((') 𝑡
¨ The numeraire associated to the terminal measure is the Zero 𝑍 𝑡, 𝑡, 𝑡.
61

62. Luc_Faucheux_2021
A little quiz on martingales - II
¨ The deflated Zeros are the ratio of the Zeros to the rolling numeraire
¨ Z
𝑍 𝑡, 𝑡, 𝑡. =
P (,(,('
Q(()
¨
S
P (,(,('
S
P /,/,('
is a martingale under the Risk-Free measure ℚ
¨ {
S
P (,(,('
S
P /,/,('
}G$ is a martingale under the Terminal measure ℤ(𝑡.)
62

63. Luc_Faucheux_2021
A little quiz on martingales - III
¨ 𝐿 𝑡, 𝑡., 𝑡R = 𝔼(2
ℤ((2)
𝑉(𝑡R, $𝐿 𝑡., 𝑡., 𝑡R , 𝑡., 𝑡R) 𝔉 𝑡
¨ 𝑍 𝑡, 𝑡., 𝑡R = 𝔼('
ℤ((')
𝑉 𝑡., $1, 𝑡., 𝑡R 𝔉 𝑡 = 𝔼('
ℤ((')
𝑍(𝑡., 𝑡., 𝑡R) 𝔉 𝑡
¨ Simply compounded FORWARD at time 𝑡: 𝑍𝐶 𝑡, 𝑡., 𝑡R =
$
$0T (,(',(2 .U (,(',(2
¨ Simply compounded FORWARD at time 𝑡. : 𝑍𝐶 𝑡., 𝑡., 𝑡R =
$
$0T (',(',(2 .U (',(',(2
63

64. Luc_Faucheux_2021
A little quiz on martingales - IV
¨ 𝐿 𝑡, 𝑡., 𝑡R = 𝔼(2
ℤ((2)
𝑉(𝑡R, $𝐿 𝑡., 𝑡., 𝑡R , 𝑡., 𝑡R) 𝔉 𝑡
¨ The simply compounded forward 𝐿 𝑡, 𝑡., 𝑡R spanning the period [𝑡., 𝑡R] is a martingale in the
forward (𝑡R -terminal measure) ℤ(𝑡R)
64

65. Luc_Faucheux_2021
A little quiz on martingales - V
¨ 𝑍 𝑡, 𝑡., 𝑡R = 𝔼('
ℤ((')
𝑉 𝑡., $1, 𝑡., 𝑡R 𝔉 𝑡 = 𝔼('
ℤ((')
𝑍(𝑡., 𝑡., 𝑡R) 𝔉 𝑡
¨ The Zeros 𝑍 𝑡, 𝑡., 𝑡R spanning the period [𝑡., 𝑡R] is a martingale in the early/tree/discount
(𝑡. - Terminal measure) ℤ(𝑡.)
¨ The fixed swaplet payment {
$
$0T (,(',(2 .U (,(',(2
} is a martingale in the early/tree/discount (𝑡. -
Terminal measure) ℤ(𝑡.)
¨ The floating swaplet payment {
T (,(',(2 .U (,(',(2
$0T (,(',(2 .U (,(',(2
} is a martingale in the early/tree/discount
(𝑡. - Terminal measure) ℤ(𝑡.)
65

66. Luc_Faucheux_2021
A little quiz on martingales - VI
¨ Just to be a little safe but maybe pedantic, because in Finance, it is always claims that we are
looking at and WHEN they are paid (this is the first principle of the time value of money)
¨ 𝑍 𝑡, 𝑡., 𝑡R = 𝔼('
ℤ((')
𝑉 𝑡., $1, 𝑡., 𝑡R 𝔉 𝑡 = 𝔼('
ℤ((')
𝑍(𝑡., 𝑡., 𝑡R) 𝔉 𝑡
¨ 𝑍 𝑡, 𝑡., 𝑡R = 𝔼('
ℤ((')
𝑉 𝑡., $1, 𝑡., 𝑡R 𝔉 𝑡 = 𝔼('
ℤ((')
𝑉 𝑡., $𝑍(𝑡, 𝑡., 𝑡R), 𝑡., 𝑡. 𝔉 𝑡
¨ {
$
$0T (,(',(2 .U (,(',(2
} = 𝔼('
ℤ((')
𝑉 𝑡., ${
$
$0T (,(',(2 .U (,(',(2
}, 𝑡., 𝑡. 𝔉 𝑡
¨ {
T (,(',(2 .U (,(',(2
$0T (,(',(2 .U (,(',(2
} = 𝔼('
ℤ((')
𝑉 𝑡., ${
T (,(',(2 .U (,(',(2
$0T (,(',(2 .U (,(',(2
}, 𝑡., 𝑡. 𝔉 𝑡
66

67. Luc_Faucheux_2021
A little quiz on martingales – VI-a
¨ I think that part of the confusion comes from going between a variable and a claim.
¨ Saying that 𝑋(𝑡) is a martingale under the measure ℚ$ associated 𝑊ℚ&(𝑡)Brownian motion
¨ 𝑑𝑋 𝑡 = 0 . 𝑑𝑡 + 𝑠𝑜𝑚𝑒𝑡ℎ𝑖𝑛𝑔 . [ . 𝑑𝑊ℚ&(𝑡)
¨ Or using the expectation formalism:
¨ 𝑋 𝑠 = 𝔼(
ℚ&
𝑋 𝑡 𝔉 𝑠
¨ When going to a claim, the only thing that you can build from above is the value at time 𝑡 of
a claim that pays 𝑋 𝑡 set at time 𝑡 and paid at time 𝑡. This is really the only thing that you
can deduce on the valuation of claims from the fact that a variable is a martingale. Anything
gets out of sync (fixing time, payment time, valuation time of the claim) and you really
cannot say anything at all
¨ 𝑉(𝑠, $𝑋 𝑠 , 𝑠, 𝑠) = 𝔼(
ℚ&
𝑉(𝑡, $𝑋 𝑡 , 𝑡, 𝑡) 𝔉 𝑠
67

68. Luc_Faucheux_2021
A little quiz on martingales - VII
¨ We have to do a little refresher on the notation (because remember unlike in Physics, what
matters really in Finance is WHEN you get paid, not when you observe/fix/set the payment)
¨ 𝑉(𝑡) = 𝑉 𝑡, $𝐻(𝑡), 𝑡., 𝑡R
68
𝑃𝑎𝑖𝑑 𝑎𝑡 𝑡𝑖𝑚𝑒 𝑡R
𝐹𝑖𝑥𝑒𝑑 𝑜𝑟 𝑠𝑒𝑡 𝑎𝑡 𝑡𝑖𝑚𝑒 𝑡.
𝐺𝑒𝑛𝑒𝑟𝑎𝑙 𝑃𝑎𝑦𝑜𝑓𝑓 𝐻 𝑡 𝑖𝑛 𝑐𝑢𝑟𝑟𝑒𝑛𝑐𝑦 $
𝑉𝑎𝑙𝑢𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑝𝑎𝑦𝑜𝑓𝑓 𝑜𝑏𝑠𝑒𝑟𝑣𝑒𝑑 𝑎𝑡 𝑡𝑖𝑚𝑒 𝑡

69. Luc_Faucheux_2021
A little quiz on martingales - VIII
¨ 𝐿 𝑡, 𝑡., 𝑡R = 𝔼(2
ℤ((2)
𝑉(𝑡R, $𝐿 𝑡., 𝑡., 𝑡R , 𝑡., 𝑡R) 𝔉 𝑡
¨ 𝐿 𝑡, 𝑡., 𝑡R = 𝔼(2
ℤ((2)
𝑉(𝑡R, $𝐿 𝑡, 𝑡., 𝑡R , 𝑡., 𝑡R) 𝔉 𝑡
¨ 𝐿 𝑡, 𝑡., 𝑡R = 𝔼<
ℤ((2)
𝑉(𝑢, $𝐿 𝑢, 𝑡., 𝑡R , 𝑡., 𝑡R) 𝔉 𝑡 with 𝑡 < 𝑢 < 𝑡. < 𝑡R
¨ BUT
¨ 𝐿 𝑡, 𝑡., 𝑡R ≠ 𝔼<
ℤ((2)
𝑉(𝑢, $𝐿 𝑢, 𝑡., 𝑡R , 𝑡., 𝑡.) 𝔉 𝑡
¨ The simply compounded forward rates HAS to be paid AT THE END of the period [𝑡., 𝑡R] at
the time 𝑡R
¨ OTHERWISE that is a LIBOR-IN-ARREARS trade
69

70. Luc_Faucheux_2021
A little quiz on martingales - IX
¨ 𝑍 𝑡, 𝑡., 𝑡R = 𝔼('
ℤ((')
𝑉 𝑡., $1, 𝑡., 𝑡R 𝔉 𝑡 = 𝔼('
ℤ((')
𝑍(𝑡., 𝑡., 𝑡R) 𝔉 𝑡
¨ 𝑍 𝑡, 𝑡., 𝑡R = 𝔼<
ℤ((')
𝑉 𝑢, $1, 𝑡., 𝑡R 𝔉 𝑡 = 𝔼<
ℤ((')
𝑍(𝑢, 𝑡., 𝑡R) 𝔉 𝑡
¨ with 𝑡 < 𝑢 < 𝑡. < 𝑡R
¨ 𝑍 𝑢, 𝑢, 𝑡R = 𝑍 𝑢, 𝑢, 𝑡. . 𝑍(𝑢, 𝑡., 𝑡R)
70

71. Luc_Faucheux_2021
Let’s do the quiz
71

72. Luc_Faucheux_2021
Quiz, let’s see how you do..some notations first…
¨ 𝑊ℤ((') 𝑡 is the Brownian motion associated to the terminal measure ℤ(𝑡.) which is
associated with the Zero 𝑍 𝑡, 𝑡, 𝑡.
¨ 𝑊ℚ 𝑡 is the Brownian motion associated to the risk-free measure ℚ which is associated
with the rolling numeraire 𝐵 𝑡 = exp[∫
;)/
;)(
𝑅 𝑠, 𝑠, 𝑠 . 𝑑𝑠]
¨ ℚ$ is a measure with an associated 𝑊ℚ&(𝑡)Brownian motion
¨ ℚ% is a measure with an associated 𝑊ℚ!(𝑡)Brownian motion
¨ The simply compounded forward is defined through the usual bootstrapping formula:
¨ 𝑍 𝑡, 𝑡., 𝑡.0$ =
P (,(,('%&
P (,(,('
=
$
$0T (,(',('%& .U (,(',('%&
¨ The deflated Zeros are:
¨ Z
𝑍 𝑡, 𝑡, 𝑡. =
P (,(,('
Q (
72

73. Luc_Faucheux_2021
Quiz time ….right/wrong…
¨ 𝑋 𝑡 = 1 is a martingale under the risk free measure ℚ
¨ 𝑊ℚ 𝑡 is a martingale under the risk free measure ℚ
¨ {𝑊ℚ 𝑡 }% is a martingale under the risk free measure ℚ
¨ {𝑊ℚ 𝑡 }%V0$ is a martingale under the risk free measure ℚ
¨ 𝐼 𝑡 = ∫
;)/
;)(
𝑑𝑊ℚ 𝑠 is a martingale under the risk free measure ℚ
¨ 𝐼 𝑡 = ∫
;)/
;)(
𝜉 𝑠 . ([). 𝑑𝑊ℚ 𝑠 is a martingale under the risk free measure ℚ
¨ 𝐼 𝑡 = ∫
;)/
;)(
𝑊ℚ 𝑠 . ([). 𝑑𝑊ℚ 𝑠 is a martingale under the risk free measure ℚ
¨ 𝐼 𝑡 = ∫
;)/
;)(
{𝑊ℚ 𝑡 }%. ([). 𝑑𝑊ℚ 𝑠 is a martingale under the risk free measure ℚ
¨ 𝐼 𝑡 = ∫
;)/
;)(
{𝑊ℚ 𝑡 }7. ([). 𝑑𝑊ℚ 𝑠 is a martingale under the risk free measure ℚ
73

74. Luc_Faucheux_2021
Quiz time ….right/wrong…II
¨
@ℚ!
@ℚ&
𝑡 is a martingale under the ℚ$-measure
¨
@ℚ&
@ℚ!
𝑡 is a martingale under the ℚ%-measure
¨
@ℚ!
@ℚ&
𝑡 is a martingale under the ℚ%-measure
¨
@ℚ&
@ℚ!
𝑡 is a martingale under the ℚ$-measure
74

75. Luc_Faucheux_2021
Quiz time ….right/wrong…III
¨ 𝐿 𝑡, 𝑡., 𝑡.0$ is a martingale under the ℤ(𝑡.0$) measure
¨ 𝐿 𝑡, 𝑡., 𝑡.0$ is a martingale under the ℤ(𝑡.) measure
¨ 𝐿 𝑡, 𝑡., 𝑡.0$ is a martingale under the ℤ(𝑡) measure
¨ 𝐿 𝑡, 𝑡., 𝑡.0$ is a martingale under the risk free measure ℚ
¨ {𝐿 𝑡, 𝑡., 𝑡.0$ }% is a martingale under the ℤ(𝑡.0$) measure
¨
$
$0T'.U (,(',('%&
is a martingale under the ℤ(𝑡.) measure
¨
T'.U (,(',('%&
$0T'.U (,(',('%&
is a martingale under the ℤ(𝑡.) measure
¨ 𝑍 𝑡, 𝑡., 𝑡.0$ is a martingale under the ℤ(𝑡.) measure
¨ 𝑍 𝑡, 𝑡., 𝑡.0$
G$ is a martingale under the ℤ(𝑡.0$) measure
75

76. Luc_Faucheux_2021
Quiz time ….right/wrong…IV
¨ Z
𝑍 𝑡, 𝑡, 𝑡. is a martingale under the ℤ(𝑡.0$) measure
¨ Z
𝑍 𝑡, 𝑡, 𝑡. is a martingale under the risk free measure ℚ
¨ Z
𝑍 𝑡, 𝑡, 𝑡. is a martingale under the ℤ(𝑡.) measure
¨
$
S
P (,(,('
is a martingale under the ℤ(𝑡.0$) measure
¨
$
S
P (,(,('
is a martingale under the risk free measure ℚ
¨
$
S
P (,(,('
is a martingale under the ℤ(𝑡.) measure
76

77. Luc_Faucheux_2021
Quiz time ….right/wrong…V
¨ 𝑍 𝑡, 𝑡., 𝑡.0$ is a martingale under the ℤ(𝑡.) measure
¨ 𝑍 𝑡, 𝑡, 𝑡. is a martingale under the ℤ(𝑡.) measure
¨ 𝑍 𝑡, 𝑡, 𝑡.0$ is a martingale under the ℤ(𝑡.) measure
¨ 𝑍 𝑡, 𝑡, 𝑡.0$ is a martingale under the ℤ(𝑡.0$) measure
¨ 𝑍 𝑡, 𝑡, 𝑡.0$ is a martingale under the ℤ(𝑡) measure
¨
$
P (,(',('%&
is a martingale under the ℤ(𝑡.0$) measure
¨
$
P (,(',('%&
is a martingale under the ℤ(𝑡.) measure
¨
$
P (,(',('%&
is a martingale under the risk free measure ℚ
77

78. Luc_Faucheux_2021
Quiz time ….right/wrong…VI
¨
T'.U (,(',('%&
$0T'.U (,(',('%&
is a martingale under the ℤ(𝑡.) measure
¨
T'.U (,(',('%&
$0T'.U (,(',('%&
is a martingale under the ℤ(𝑡.0$) measure
¨
{T'.U (,(',('%& }!
$0T'.U (,(',('%&
is a martingale under the ℤ(𝑡.) measure
¨
{T'.U (,(',('%& }!
$0T'.U (,(',('%&
is a martingale under the ℤ(𝑡.0$) measure
¨
T'.U (,(',('%&
$0T'.U (,(',('%&
is a martingale under the risk free measure ℚ
78

79. Luc_Faucheux_2021
Bootstrapping the measures
79

80. Luc_Faucheux_2021
Bootstrapping the measures.
¨ 𝑍 𝑡, 𝑡., 𝑡R = 𝔼('
ℤ((')
𝑉 𝑡., $1, 𝑡., 𝑡R 𝔉 𝑡 = 𝔼('
ℤ((')
𝑍(𝑡., 𝑡., 𝑡R) 𝔉 𝑡
¨ The Zeros 𝑍 𝑡, 𝑡., 𝑡R spanning the period [𝑡., 𝑡R] is a martingale in the early/tree/discount
(𝑡. - Terminal measure) ℤ(𝑡.)
¨ Since we are going to use those to value successive swaplets, we will encounter first the
important case 𝑗 = 𝑖 + 1
¨ Here we assume that we have indexed the time buckets say along the periods of a swap or
derivative that we want to value.
¨ 𝑍 𝑡, 𝑡., 𝑡.0$ =
P (,(,('%&
P (,(,('
=
$
$0T (,(',('%& .U (,(',('%&
¨ 𝐿 𝑡, 𝑡., 𝑡.0$ is a martingale under the terminal (forward) measure ℤ(𝑡.0$)
¨ So the SDE looks something like:
¨ 𝑑𝐿 𝑡, 𝑡., 𝑡.0$ = 0 . 𝑑𝑡 + 𝑠𝑜𝑚𝑒𝑡ℎ𝑖𝑛𝑔 . 𝑑𝑊ℤ(('%&) 𝑡
80

81. Luc_Faucheux_2021
Bootstrapping the measures - II
¨ 𝐿 𝑡, 𝑡., 𝑡.0$ is a martingale under the terminal (forward) measure ℤ(𝑡.0$)
¨ 𝑑𝐿 𝑡, 𝑡., 𝑡.0$ = 0 . 𝑑𝑡 + 𝑠𝑜𝑚𝑒𝑡ℎ𝑖𝑛𝑔_𝐿 . 𝑑𝑊ℤ(('%&) 𝑡
¨ 𝑍 𝑡, 𝑡., 𝑡.0$ is a martingale in the the terminal (forward) measure ℤ(𝑡.)
¨ 𝑑𝑍 𝑡, 𝑡., 𝑡.0$ = 0 . 𝑑𝑡 + 𝑠𝑜𝑚𝑒𝑡ℎ𝑖𝑛𝑔_𝑍 . 𝑑𝑊ℤ((') 𝑡
¨ And we have the relationship:
¨ 𝑍 𝑡, 𝑡., 𝑡.0$ =
P (,(,('%&
P (,(,('
=
$
$0T (,(',('%& .U (,(',('%&
¨ So the idea is that we should be able to say something about how to go from ℤ(𝑡.0$) to
ℤ(𝑡.) because we need to verify those 3 relationships.
¨ That is the idea behind the LMM (Libor Market Model)
81

82. Luc_Faucheux_2021
Bootstrapping the measures - III
¨ 𝑍 𝑡, 𝑡., 𝑡.0$ =
P (,(,('%&
P (,(,('
=
$
$0T (,(',('%& .U (,(',('%&
¨ We use ITO lemma on this.
¨ 𝛿𝑓 =
!"
!#
. 𝛿𝑋 +
$
%
.
!!"
!#! . (𝛿𝑋)%
¨ With: 𝑋 = 𝐿 𝑡, 𝑡., 𝑡.0$ and 𝑓(𝑥) = 1/(1 + 𝜏 𝑡, 𝑡., 𝑡.0$ . 𝑥)
¨
!"
!#
=
GT (,(',('%&
($0T (,(',('%& .#)!
¨
!!"
!#! =
%.T (,(',('%&
!
($0T (,(',('%& .#)3
¨ 𝑑𝑍 𝑡, 𝑡., 𝑡.0$ =
GT (,(',('%&
($0T (,(',('%& .U (,(',('%& )! . 𝑑𝐿 𝑡, 𝑡., 𝑡.0$ +
$
%
.
%.T (,(',('%&
!
($0T (,(',('%& .#)3 . (𝑑𝐿 𝑡, 𝑡., 𝑡.0$ )%
82

83. Luc_Faucheux_2021
Bootstrapping the measures - IV
¨ 𝑑𝑍 𝑡, 𝑡., 𝑡.0$ =
GT (,(',('%&
($0T (,(',('%& .U (,(',('%& )! . 𝑑𝐿 𝑡, 𝑡., 𝑡.0$ +
$
%
.
%.T (,(',('%&
!
($0T (,(',('%& .#)3 . (𝑑𝐿 𝑡, 𝑡., 𝑡.0$ )%
¨ 𝑑𝐿 𝑡, 𝑡., 𝑡.0$ = 0 . 𝑑𝑡 + 𝑠𝑜𝑚𝑒𝑡ℎ𝑖𝑛𝑔_𝐿 . 𝑑𝑊ℤ(('%&) 𝑡
¨ If we choose a function :
¨ 𝑠𝑜𝑚𝑒𝑡ℎ𝑖𝑛𝑔_𝐿 = 𝑏{𝐿 𝑡, 𝑡., 𝑡.0$ , 𝔉 𝑡 } that could be function of the rate at all times prior to
time 𝑡, that we note 𝑏 just for ease of notation
¨ 𝑑𝐿 𝑡, 𝑡., 𝑡.0$ = 0 . 𝑑𝑡 + 𝑏. 𝑑𝑊ℤ(('%&) 𝑡
¨ (𝑑𝐿 𝑡, 𝑡., 𝑡.0$ )%= 𝑏%. 𝑑𝑡
¨ Note that if the function 𝑏{𝐿 𝑡, 𝑡., 𝑡.0$ , 𝔉 𝑡 } has dependency on the stochastic variable, we
need to make sure that we are working in the ITO calculus
¨ 𝑑𝐿 𝑡, 𝑡., 𝑡.0$ = 0 . 𝑑𝑡 + 𝑏. [ . 𝑑𝑊ℤ(('%&) 𝑡
¨ (𝑑𝐿 𝑡, 𝑡., 𝑡.0$ )%= 𝑏%. 𝑑𝑡
83

84. Luc_Faucheux_2021
Bootstrapping the measures - V
¨ 𝑑𝑍 𝑡, 𝑡., 𝑡.0$ =
GT (,(',('%&
($0T (,(',('%& .U (,(',('%& )! . 𝑑𝐿 𝑡, 𝑡., 𝑡.0$ +
$
%
.
%.T (,(',('%&
!
($0T (,(',('%& .#)3 . (𝑑𝐿 𝑡, 𝑡., 𝑡.0$ )%
¨ 𝑑𝐿 𝑡, 𝑡., 𝑡.0$ = 𝑏. [ . 𝑑𝑊ℤ(('%&) 𝑡
¨ (𝑑𝐿 𝑡, 𝑡., 𝑡.0$ )%= 𝑏%. 𝑑𝑡
¨ 𝑑𝑍 𝑡, 𝑡., 𝑡.0$ =
GT (,(',('%&
($0T (,(',('%& .U (,(',('%& )! . 𝑏. [ . 𝑑𝑊ℤ(('%&) 𝑡 +
$
%
.
%.T (,(',('%&
!
($0T (,(',('%& .U (,(',('%& )3 . 𝑏%. 𝑑𝑡
¨ Following Piterbarg (p. 595), let’s just simplify a little the daycount fraction by writing:
¨ 𝜏 𝑡, 𝑡., 𝑡.0$ = 𝜏.
¨ 𝑑𝑍 𝑡, 𝑡., 𝑡.0$ =
GT'
($0T'.U (,(',('%& )! . 𝑏. [ . 𝑑𝑊ℤ(('%&) 𝑡 +
$
%
.
%.T'
!
($0T'.U (,(',('%& )3 . 𝑏%. 𝑑𝑡
¨ 𝑑𝑍 𝑡, 𝑡., 𝑡.0$ =
GT'.&
($0T'.U (,(',('%& )! . { [ . 𝑑𝑊ℤ ('%& 𝑡 −
T'
($0T'.U (,(',('%& )
. 𝑏. 𝑑𝑡}
84

85. Luc_Faucheux_2021
Bootstrapping the measures - VI
¨ 𝑑𝑍 𝑡, 𝑡., 𝑡.0$ =
GT'.&
($0T'.U (,(',('%& )! . { [ . 𝑑𝑊ℤ ('%& 𝑡 −
T'
($0T'.U (,(',('%& )
. 𝑏. 𝑑𝑡}
¨ But we also know that:
¨ 𝑑𝑍 𝑡, 𝑡., 𝑡.0$ = 𝑠𝑜𝑚𝑒𝑡ℎ𝑖𝑛𝑔_𝑍 . 𝑑𝑊ℤ((') 𝑡
¨ So the only way that works is:
¨ 𝑠𝑜𝑚𝑒𝑡ℎ𝑖𝑛𝑔_𝑍 =
GT'.&
($0T'.U (,(',('%& )!
¨ 𝑑𝑊ℤ((') 𝑡 = 𝑑𝑊ℤ ('%& 𝑡 −
T'
($0T'.U (,(',('%& )
. 𝑏. 𝑑𝑡
¨ So this is how we can go from the ℤ 𝑡.0$ terminal measure to the ℤ(𝑡.) terminal measure
85

86. Luc_Faucheux_2021
Bootstrapping the measures - VII
¨ 𝑑𝐿 𝑡, 𝑡., 𝑡.0$ = 𝑏{𝐿 𝑡, 𝑡., 𝑡.0$ , 𝔉 𝑡 }. [ . 𝑑𝑊ℤ(('%&) 𝑡
¨ 𝑑𝑊ℤ((') 𝑡 = 𝑑𝑊ℤ ('%& 𝑡 −
T'
($0T'.U (,(',('%& )
. 𝑏{𝐿 𝑡, 𝑡., 𝑡.0$ , 𝔉 𝑡 }. 𝑑𝑡
¨ 𝑑𝐿 𝑡, 𝑡.G$, 𝑡. = 𝑏{𝐿 𝑡, 𝑡.G$, 𝑡. , 𝔉 𝑡 }. [ . 𝑑𝑊ℤ((') 𝑡
¨ 𝑑𝐿 𝑡, 𝑡.G$, 𝑡. = 𝑏 𝐿 𝑡, 𝑡.G$, 𝑡. , 𝔉 𝑡 . [ . {𝑑𝑊ℤ ('%& 𝑡 −
T'.&{U (,(',('%& ,𝔉 ( }
($0T'.U (,(',('%& )
. 𝑑𝑡}
¨ This illustrates that if 𝐿 𝑡, 𝑡.G$, 𝑡. is a martingale in the ℤ(𝑡.) terminal measure, it is NOT a
martingale in the ℤ(𝑡.0$) terminal measure, and the drift can be quite a complicated formula:
¨ 𝑑𝐿 𝑡, 𝑡.G$, 𝑡. = −
T'.& U (,(',('%& ,𝔉 ( .& U (,('4&,(' ,𝔉 (
$0T'.U (,(',('%&
. 𝑑𝑡 + 𝑏 𝐿 𝑡, 𝑡.G$, 𝑡. , 𝔉 𝑡 . [ 𝑑𝑊ℤ ('%& 𝑡
86

87. Luc_Faucheux_2021
Bootstrapping the measures - VIII
¨ 𝑑𝑊ℤ((') 𝑡 = 𝑑𝑊ℤ ('%& 𝑡 −
T'
($0T'.U (,(',('%& )
. 𝑏. 𝑑𝑡
¨ 𝑑𝑊ℤ ('%& 𝑡 = 𝑑𝑊ℤ((') 𝑡 +
T'
($0T'.U (,(',('%& )
. 𝑏. 𝑑𝑡
¨ Remember that:
¨ 𝑏 = 𝑏 𝐿 𝑡, 𝑡., 𝑡.0$ , 𝔉 𝑡 = 𝑏(𝑖)
¨ We can go down by recurrence
¨ 𝑑𝑊ℤ ('%& 𝑡 = 𝑑𝑊ℤ((') 𝑡 +
T'
$0T'.U (,(',('%&
. 𝑏(𝑖). 𝑑𝑡
¨ 𝑑𝑊ℤ (' 𝑡 = 𝑑𝑊ℤ(('4&) 𝑡 +
T'4&
$0T'4&.U (,('4&,('
. 𝑏(𝑖 − 1). 𝑑𝑡
¨ 𝑑𝑊ℤ (2 𝑡 = 𝑑𝑊ℤ((') 𝑡 + ∑7).
7)RG$ T$4&
$0T$4&.U (,($4&,($
. 𝑏(𝑘 − 1). 𝑑𝑡
87

88. Luc_Faucheux_2021
Bootstrapping the measures – VIII - a
¨ Let’s leave this one like that for now, but note that we are laying down the foundations to
derive the LMM (Libor Market Model). Hopefully will do that in deck VIII.
88

89. Luc_Faucheux_2021
Bootstrapping the measures - IX
¨ 𝑑𝑊ℤ((') 𝑡 = 𝑑𝑊ℤ ('%& 𝑡 −
T'
($0T'.U (,(',('%& )
. 𝑏. 𝑑𝑡
¨ Let’s tie that to the RN derivative
89

90. Luc_Faucheux_2021
Bootstrapping the measures - X
¨ If ℚ$ is a measure with an associated 𝑊ℚ&(𝑡)Brownian motion (a ℚ$-Brownian motion)
¨ If ℚ% is a measure with an associated 𝑊ℚ!(𝑡)Brownian motion (a ℚ%-Brownian motion)
¨ We have (under the famous Novikov condition..)
¨ If there is a process 𝜉 𝑡 such that it is reasonably well-behaved
¨
@ℚ!
@ℚ&
= exp − ∫
/
(
𝜉 𝑠 . 𝑑𝑊 𝑠 − ∫
/
( $
%
𝜉 𝑠 %. 𝑑𝑠 = ℰ ∫
/
(
(−𝜉 𝑠 ). 𝑑𝑊 𝑠
¨ 𝑊ℚ! 𝑡 = 𝑊ℚ& 𝑡 + ∫
/
(
𝜉 𝑠 . 𝑑𝑠
¨ 𝑑𝑊ℚ! 𝑡 = 𝑑𝑊ℚ& 𝑡 + 𝜉 𝑡 . 𝑑𝑡
¨ 𝑑𝑊ℤ((') 𝑡 = 𝑑𝑊ℤ ('%& 𝑡 −
T'
($0T'.U (,(',('%& )
. 𝑏. 𝑑𝑡
90

91. Luc_Faucheux_2021
Bootstrapping the measures - XI
¨ 𝑑𝑊ℚ! 𝑡 = 𝑑𝑊ℚ& 𝑡 + 𝜉 𝑡 . 𝑑𝑡
¨ 𝑑𝑊ℤ((') 𝑡 = 𝑑𝑊ℤ ('%& 𝑡 −
T'
($0T'.U (,(',('%& )
. 𝑏. 𝑑𝑡
¨ The ℚ% measure here is the ℤ(𝑡.), associated with the numeraire 𝑍 𝑡, 𝑡, 𝑡.
¨ The ℚ$ measure here is the ℤ(𝑡.0$), associated with the numeraire 𝑍 𝑡, 𝑡, 𝑡.0$
¨ 𝜉 𝑡 = −
T'
($0T'.U (,(',('%& )
. 𝑏
¨
@ℚ!
@ℚ&
= exp − ∫
/
(
𝜉 𝑠 . 𝑑𝑊 𝑠 − ∫
/
( $
%
𝜉 𝑠 %. 𝑑𝑠 = ℰ ∫
/
(
(−𝜉 𝑠 ). 𝑑𝑊 𝑠
91

92. Luc_Faucheux_2021
Bootstrapping the measures - XII
¨ X
𝑁ℚ&
𝑡 = 𝑁ℚ&
(𝑡)/𝑁ℚ&
(0)
¨ X
𝑁ℚ!
𝑡 = 𝑁ℚ!
(𝑡)/𝑁ℚ!
(0)
¨ We obtain the celebrated formula:
¨
@ℚ!
@ℚ&
𝑡 =
Y
6ℚ! (
Y
6ℚ& (
¨ 𝑁ℚ!
(𝑡) = 𝑍 𝑡, 𝑡, 𝑡.
¨ 𝑁ℚ&
(𝑡) = 𝑍 𝑡, 𝑡, 𝑡.0$
¨
@ℚ!
@ℚ&
𝑡 =
@ℤ((')
@ℤ(('%&)
𝑡 =
Y
6ℚ! (
Y
6ℚ& (
=
P (,(,(' /P /,/,('
P (,(,('%& /P /,/,('%&
¨
@ℤ((')
@ℤ(('%&)
𝑡 =
P (,(,(' /P /,/,('
P (,(,('%& /P /,/,('%&
=
P /,/,('%&
P /,/,('
. {1 + 𝜏.. 𝐿 𝑡, 𝑡., 𝑡.0$ }
92

93. Luc_Faucheux_2021
Bootstrapping the measures - XIII
¨
@ℤ((')
@ℤ(('%&)
𝑡 =
P /,/,('%&
P /,/,('
. {1 + 𝜏.. 𝐿 𝑡, 𝑡., 𝑡.0$ }
¨ We also know that:
¨
@ℚ!
@ℚ&
𝑡 is a martingale under the ℚ$-measure
¨
@ℚ&
@ℚ!
𝑡 is a martingale under the ℚ%-measure
¨ And since the RN derivative is a ratio of numeraire:
¨
@ℚ&
@ℚ!
𝑡 = {
@ℚ!
@ℚ&
𝑡 }G$
93

94. Luc_Faucheux_2021
Bootstrapping the measures - XIV
¨ The ℚ% = ℤ(𝑡.), associated with the numeraire 𝑍 𝑡, 𝑡, 𝑡.
¨ The ℚ$ = ℤ(𝑡.0$), associated with the numeraire 𝑍 𝑡, 𝑡, 𝑡.0$
¨
@ℤ((')
@ℤ(('%&)
𝑡 =
P /,/,('%&
P /,/,('
. {1 + 𝜏.. 𝐿 𝑡, 𝑡., 𝑡.0$ }
¨
@ℚ!
@ℚ&
𝑡 is a martingale under the ℚ$-measure
¨ BECOMES:
¨
@ℤ((')
@ℤ(('%&)
𝑡 =
P /,/,('%&
P /,/,('
. {1 + 𝜏.. 𝐿 𝑡, 𝑡., 𝑡.0$ } is a martingale under the ℤ(𝑡.0$) measure
¨ And thus:
¨ 𝐿 𝑡, 𝑡., 𝑡.0$ is a martingale under the ℤ(𝑡.0$) measure
94

95. Luc_Faucheux_2021
Bootstrapping the measures - XV
¨
@ℚ&
@ℚ!
𝑡 is a martingale under the ℚ%-measure
¨ BECOMES:
¨
@ℤ(('%&)
@ℤ((')
𝑡 = {
@ℤ((')
@ℤ(('%&)
𝑡 }G$= {
P /,/,('%&
P /,/,('
. {1 + 𝜏.. 𝐿 𝑡, 𝑡., 𝑡.0$ }}G$ is a martingale under
the ℤ(𝑡.) measure
¨ And thus:
¨
$
$0T'.U (,(',('%&
is a martingale under the ℤ(𝑡.) measure
¨
T'.U (,(',('%&
$0T'.U (,(',('%&
is a martingale under the ℤ(𝑡.) measure
¨ 𝑍 𝑡, 𝑡., 𝑡.0$ is a martingale under the ℤ(𝑡.) measure
95

96. Luc_Faucheux_2021
Bootstrapping the measures - XVI
¨ So we could have really started from that angle (like in some textbooks) if we knew a lot to
start with RN derivative, and changes of measures, we could have said:
¨ The ℤ(𝑡.)-measure is associated with the numeraire 𝑍 𝑡, 𝑡, 𝑡.
¨ The ℤ(𝑡.0$)-measure is associated with the numeraire 𝑍 𝑡, 𝑡, 𝑡.0$
¨ So by construction, since 𝑍 𝑡, 𝑡., 𝑡.0$ =
P (,(,('%&
P (,(,('
=
$
$0T (,(',('%& .U (,(',('%&
¨ 𝐿 𝑡, 𝑡., 𝑡.0$ is a martingale under the ℤ(𝑡.0$) measure
¨
$
$0T'.U (,(',('%&
is a martingale under the ℤ(𝑡.) measure
¨
T'.U (,(',('%&
$0T'.U (,(',('%&
is a martingale under the ℤ(𝑡.) measure
¨ 𝑍 𝑡, 𝑡., 𝑡.0$ is a martingale under the ℤ(𝑡.) measure
¨ 𝑍 𝑡, 𝑡., 𝑡.0$
G$ is a martingale under the ℤ(𝑡.0$) measure
96

97. Luc_Faucheux_2021
Bootstrapping the measures - XVII
¨ That would have been so much easier but maybe not that intuitive
¨ AS ALWAYS in Finance, when dealing with Terminal measures, when we say that 𝑋(𝑡)is a
martingale, when talking about the value of a claim, that is the value at time 𝑡 that pays
$𝑋 𝑡 at time 𝑇 associated with that measure (when the Numeraire is 𝑍 𝑡, 𝑡, 𝑇 =
𝑍 𝑇, 𝑇, 𝑇 = 1)
¨ 𝐿 𝑡, 𝑡., 𝑡.0$ is a martingale under the ℤ(𝑡.0$) measure
¨ When talking about a payment equal to it:
¨ 𝑉(𝑡, $𝐿 𝑡, 𝑡., 𝑡.0$ , 𝑡., 𝑡.0$) is a martingale under the ℤ(𝑡.0$) measure
¨
$
$0T'.U (,(',('%&
is a martingale under the ℤ(𝑡.) measure
¨ When talking about a payment equal to it:
¨ 𝑉(𝑡, $
$
$0T'.U (,(',('%&
, 𝑡., 𝑡.) is a martingale under the ℤ(𝑡.) measure
97

98. Luc_Faucheux_2021
Bootstrapping the measures - XVIII
¨ So in a sense we always start with the claim, and then when the timing coincide we can say
something about the actual variable:
¨ 𝐿 𝑡, 𝑡., 𝑡R = 𝔼(2
ℤ((2)
𝑉(𝑡R, $𝐿 𝑡., 𝑡., 𝑡R , 𝑡., 𝑡R) 𝔉 𝑡
¨ 𝑍 𝑡, 𝑡., 𝑡R = 𝔼('
ℤ((')
𝑉 𝑡., $1, 𝑡., 𝑡R 𝔉 𝑡 = 𝔼('
ℤ((')
𝑍(𝑡., 𝑡., 𝑡R) 𝔉 𝑡
¨ Simply compounded FORWARD at time 𝑡: 𝑍𝐶 𝑡, 𝑡., 𝑡R =
$
$0T (,(',(2 .U (,(',(2
¨ Simply compounded FORWARD at time 𝑡. : 𝑍𝐶 𝑡., 𝑡., 𝑡R =
$
$0T (',(',(2 .U (',(',(2
98

99. Luc_Faucheux_2021
Bootstrapping the measures - XIX
¨ In the next deck, we will revisit those slides, and start building the LMM model, where we
will explicitly derive the drift for all the forwards 𝐿 𝑡, 𝑡., 𝑡R in the same measure, following
the derivation from Piterbarg and also Tuckman which builds more intuition.
99

100. Luc_Faucheux_2021
Career advices from Uncle Luc
100

101. Luc_Faucheux_2021
Career advices from Uncle Luc
¨ So, not sure if this is the New Moon or Mercury going retrograde, or if it is the season of
bonuses, but Uncle Luc is feeling extra gloomy and wants to impart some of his wisdom on
the young generations on what they should do for their careers in finance
¨ So gather around the campfire (am the one with the laptop in the picture below)
101

102. Luc_Faucheux_2021
Never ever work in Rates Derivatives on the Sell side
¨ It is too hard
¨ There is no juice in it
¨ The reporting requirements for regulators are quite stringent
¨ The need for computing power is quite big
¨ The theory is really complicated. Most textbooks start with equity, and they either say that
the rate is equal to zero or a constant, not even a deterministic function of time. It is only
after some serious math that you can start talking about modeling rates
¨ Unlike Equity for example, the arbitrage-free conditions that you need to enforce create
some non-trivial constraints on the curve construction and stochastic processes
102

103. Luc_Faucheux_2021
Never ever work in Rates Derivatives on the Sell side - II
¨ There is no juice in it.
¨ People get excited about basis points.
¨ 1 basis point = 1% / 100
¨ 1% = 1/100
¨ 1 basis point = 1/100/100 = 1/10,000
¨ Bid offer is usually around a quarter of a basis point if that.
¨ Daily move is usually a couple of basis points if that.
¨ People would arb each other for half a basis point.
103

104. Luc_Faucheux_2021
Never ever work in Rates Derivatives on the Sell side - III
¨ Peter Carr in the jacket for the Pieterbarg book said it quite nicely:
¨ “In the complex and highly liquid interest rate derivatives market, the requirements for
model accuracy and realism are inordinately demanding.”
¨ Yeah, could not agree more, the need for precision, computing power, accuracy, speed,
complexity of the modeling to accurately capture the markets, with the reporting
requirements, the trading requirements (on a SEF, not on a SEF, MAT, not MAT, cleared, not
cleared, bla bla bla..) are INORDINATELY DEMANDING !!! And all that for not even half a
basis point, and for a trade that most likely is going to stay on your books for the whole
duration, eating up VAR, RWA, Cost of Risk, cost to run the risk, CPUs, emails,….
¨ So trust me, if that was to do it again I would not.
¨ I was Quixotic in my youth trying to understand what I thought was complicated and worthy
of my time and effort. I know better now, and hopefully you will heed my advice.
104

105. Luc_Faucheux_2021
Never ever work in Rates Derivatives on the Sell side - IV
¨ To be fair when you run any kind of financial endeavior, it all comes down to funding and
time value of money, so interest rates are at the core of any bank, and you cannot really
avoid it.
¨ So this is a necessary evil, and usually once you have done Interest-Rate derivatives, there is
really nothing that you cannot branch into. So I might be a tad overboard on the
gloominess.
105

106. Luc_Faucheux_2021
Never ever work in Rates Derivatives on the Sell side - V
¨ People get overly excited over changes in BASIS POINTS
¨ https://www.risk.net/derivatives/7739631/funds-steering-clear-of-bets-on-libor-timeline-
after-losses
¨ Here is the graph everyone is getting excited about: a 4 basis point move at most
106

107. Luc_Faucheux_2021
Never ever work in Rates Derivatives on the Sell side - VI
¨ Meanwhile on the equity side:
107

108. Luc_Faucheux_2021
Never ever work in Rates Derivatives on the Sell side - VII
¨ Or on the currency side:
108

109. Luc_Faucheux_2021
Never ever work in Rates Derivatives on the Sell side - VIII
¨ So let’s recap.
¨ On the Interest-Rates side, people get super excited about a move of 4 basis points, when
valuing the difference over 5 years of a swap paying 3month-LIBOR on one side, set
quarterly, compounded flat and paid semi-annually versus 6month-LIBOR on the other side,
set semi-annually and paid semi-annually. This is super hard to model and value.
¨ 4 basis points.
¨ Fine I will be nice to you and give you 10 basis points.
¨ 10 basis points = (10/100) % = 0.1%
¨ On the equity side, GameStop moved from $49 to $490 in a couple of days.
¨ That is a move of 1,000% !
¨ That is a move 10,000 greater than the 5year 3s6s basis, for a security that is easy to book
and needs no curve construction, bi-curve, discounting, rates modeling or such
109

110. Luc_Faucheux_2021
Never ever work in Rates Derivatives on the Sell side - IX
¨ Ok, on the currency side, since Bitcoin is as legitimate a currency as any fiat currency
according to Elon:
¨ Bitcoin moved from $10,000 to $50,000 in a little less than 5 months.
¨ That is a 500% move, again for something that does not require the full
HJM/measure/Girsanov theorem/arbitrage-free/martingale/thousands of powerpoint slides
before you can make sense of anything/army of Russian and French PhD to compute even
the simplest future contract or convexity adjustment
¨ So yeah congrats…no juice in it for an “inordinate” amount of effort and time..
110

111. Luc_Faucheux_2021
My career advice #1
¨ In my next life I want to be a baseball player
¨ You get paid a lot
¨ You live a pretty healthy lifestyle (they force you to exercise)
¨ You do not travel as much and as often as tennis players, basketball players, you do not have
jetlags as usually a series stays in one town for a week or so
¨ You are part of a team so you share expenses unlike golf players
¨ You do not damage your body like football soccer tennis
¨ You can play until you have grandkids
¨ It starts to rain, snow, or get too cold, you stop playing
¨ Spring ”training” is on Florida…yeahhh
¨ So yeah…problem is that it is quite boring, but hey that is the only minus I see…
111

112. Luc_Faucheux_2021
My career advice #2
¨ If you cannot make it as a baseball player, I highly recommend being a credit trader on the
sell side and selling all the credit protection that you can
¨ This is viewed as patriotic because you are bullish your clients and the market
¨ If something blows up, it is because someone else screwed up, not you. Let me elaborate.
¨ You pay fixed in 10year swap and the market rally, you lost money, you get yelled at
¨ You sell protection on ENRON, Parmalat, Worldcom, Wework, Theranos,….and then the
company goes belly-up, that is not something you did, it is fraud/accounting/wrong
management AT the company, certainly not something that you did wrong, so you do not
get yelled at as much, because you were supporting a key client of the bank. Psychologically
subtle but true…
¨ Oh hey also when you blow up, everyone usually blow up together you relatively speaking
you are still doing OK
¨ You also get paid for the carry before the blow-up, with usually no reserve whatsoever, so
life is good. Also there is more juice in Credit, moves in points, not in basis points
112

113. Luc_Faucheux_2021
My career advice #3
¨ If you cannot do #1 or #2, am starting to feel sorry for you.
¨ I have some other advice
¨ Become an equity option trader and sell all the long dated options that you can.
¨ Similar to credit, you are fulfilling a patriotic duty to support the market and key clients of
the bank
¨ Similar to credit, when you blow up, chances are everyone is blowing up at the same time,
so you can find another job, in the meantime you collected a nice carry, “clipping the
coupons” as they say
113

114. Luc_Faucheux_2021
My career advice #4
¨ All right so you could not do any of the above so far.
¨ I have one for you…Big data and Machine Learning
¨ CPUs is cheaper every year
¨ You have tons of data to play with
¨ So far the field of big data / ML / AI is just a big Excel GoalSeek, nothing more.
¨ Am still waiting for the qualitative jump that Douglas Hoffstadter predicted in the field of AI.
¨ So far no singularity, no emergence, no qualitative jump, just a lot more of number
crunching and burning of CPUs
¨ So use words like virtuous vortex of connectivity, deep learning, ML on BigData cloud based,
make sure that your project is way too ambitious to ever be measurable against the goal…et
voila !! You get yourself a nice cushy job, and while the CPUs that you are burning gently
warm up the planet, maybe you have some time to write some Powerpoint slides on more
eternal and timeless issues like Ito versus Stratanovitch
114

115. Luc_Faucheux_2021
Things I still want to do
115

116. Luc_Faucheux_2021
Things I still want to do
¨ Redo the Ho-Lee deck with the following models
¨ Ho-Lee with time-dependent volatility:
¨ 𝑑𝑅 𝑡, 𝑡, 𝑡 = 𝜃 𝑡 . 𝑑𝑡 − 𝜎(𝑡). ([). 𝑑𝑊(𝑡)
¨ Hull-White:
¨ 𝑑𝑅 𝑡, 𝑡, 𝑡 = {𝜃 𝑡 − 𝑘. 𝑅(𝑡, 𝑡, 𝑡)}. 𝑑𝑡 − 𝜎(𝑡). ([). 𝑑𝑊(𝑡)
¨ Langevin equation:
¨ 𝑑𝑉 𝑡 = −𝑘. 𝑉(𝑡). 𝑑𝑡 + 𝜎. ([). 𝑑𝑊(𝑡)
¨ So we can use a lot of the materials of the Langevin deck.
116

117. Luc_Faucheux_2021
Things I still want to do - II
¨ Caplet numeraire
¨ Swaption numeraire
¨ Normal BS derivation
¨ Finish the binary section
¨ Derive Gaussian from MaxEnt principle and Lagrange multipliers
¨ CLT
¨ Master equation -> Gaussian (Van Kampen book)
¨ Numeraire change
¨ Arcsin law
¨ Eris swap future contract
¨ CMS convexity adjustment
117

118. Luc_Faucheux_2021
Things I still want to do - III
¨ Add to yield curve section the work of Tom Coleman, the Sultan of Spline
¨ Do the efficient frontier and CAPM line
¨ C=int(delta,dS) -> P=C-delta.S -> E(P)=0
¨ Add to forward versus spot risk
¨ Expand with spreadsheet the MPT example in part I
¨ Add fast curve / slow curve section
¨ Expand on “it is it 0 at time t it is 0 at all time” wrong for CMS and Libor in arrrears
¨ Section on local versus global arbitrage in trees
¨ Derive the LMM drifts in both terminal, spot and risk free measures
¨ Tie out Tuckman with Piterbarg (the art of drift)
118

119. Luc_Faucheux_2021
So at least for now…..
119