part deux of Ho-Lee

1. Luc_Faucheux_2021
THE RATES WORLD – Part V_b
Summary of part IV, some notes on Ho-Lee
model
1

2. Luc_Faucheux_2021
That deck
2
¨ Could have been named “Everything that you ever wanted to know about Ho-Lee but were
too afraid to ask”
¨ Using Ho Lee as a working example to introduce a lot of concepts, mostly within the HJM
framework, but also illustrating some of the properties of the affine models
¨ Reached 280 slides and 82M on part V-a, so I had to split it into two sections
¨ Apologies for that.
¨ This is the second part
¨ It also answers the question: “how many slides does it take Luc to beat up a dead horse?”

3. Luc_Faucheux_2021
Quick summary so far of Ho-Lee
3

4. Luc_Faucheux_2021
Summary, using the affine formalism
¨ 𝑍𝐶 𝑡, 𝑡, 𝑡! = 𝑍𝐶 0, 𝑡, 𝑡! . exp{−[𝑅 𝑡, 𝑡, 𝑡 − 𝑅 0, 𝑡, 𝑡 ]. 𝑡! − 𝑡 −
"!
#
𝑡 𝑡! − 𝑡 #}
¨ 𝑅 𝑡, 𝑡, 𝑡! = −
$
% &,&,&"
. ln(𝑍𝐶 𝑡, 𝑡, 𝑡! )
¨ 𝑅 𝑡, 𝑡, 𝑡! − 𝑅 0, 𝑡, 𝑡! = {𝑅 𝑡, 𝑡, 𝑡 − 𝑅 0, 𝑡, 𝑡 } +
"!
#
𝑡. 𝑡! − 𝑡
¨ 𝑅 𝑡, 𝑡!, 𝑡( =
)$
% &,&",&#
. ln(
*+ &,&,&#
*+ &,&,&"
)
¨ 𝑅 𝑡, 𝑡$, 𝑡% = 𝑅 0, 𝑡$, 𝑡% + 𝑅 𝑡, 𝑡, 𝑡 − 𝑅 0, 𝑡, 𝑡 .
('!('")
* ','",'!
+
,#
-
𝑡 𝑡% − 𝑡
-
−
,#
-
𝑡 𝑡$ − 𝑡 -
.
.
* ','",'!
¨ 𝑅 𝑡, 𝑡!, 𝑡! = lim
&#→&"
𝑅 𝑡, 𝑡!, 𝑡( = −
-./(*+ &,&,&"
-&"
¨ 𝑅 𝑡, 𝑡!, 𝑡! − 𝑅 0, 𝑡!, 𝑡! = 𝑅 𝑡, 𝑡, 𝑡 − 𝑅 0, 𝑡, 𝑡 + 𝜎#. 𝑡. 𝑡! − 𝑡
4

5. Luc_Faucheux_2021
Summary, using the SIE formalism
¨ 𝑅 𝑡, 𝑡, 𝑡 − 𝑅 0, 𝑡, 𝑡 =
$
#
. 𝜎# 𝑡# − 𝜎. ([). 𝑊 𝑡
¨ 𝑅 𝑡, 𝑡!, 𝑡! − 𝑅 0, 𝑡!, 𝑡! = 𝑅 𝑡, 𝑡, 𝑡 − 𝑅 0, 𝑡, 𝑡 + 𝜎#. 𝑡. 𝑡! − 𝑡
¨ 𝑅 𝑡, 𝑡!, 𝑡! − 𝑅 0, 𝑡!, 𝑡! =
$
#
. 𝜎# 𝑡# − 𝜎. ([). 𝑊 𝑡 + 𝜎#. 𝑡. 𝑡! − 𝑡
¨ 𝑅 𝑡, 𝑡!, 𝑡! − 𝑅 0, 𝑡!, 𝑡! = 𝜎#. 𝑡. 𝑡! −
$
#
𝑡 − 𝜎. ([). 𝑊 𝑡
¨ 𝑅 𝑡, 𝑡, 𝑡! − 𝑅 0, 𝑡, 𝑡! = {𝑅 𝑡, 𝑡, 𝑡 − 𝑅 0, 𝑡, 𝑡 } +
"!
#
𝑡. 𝑡! − 𝑡
¨ 𝑅 𝑡, 𝑡, 𝑡! − 𝑅 0, 𝑡, 𝑡! = {
$
#
. 𝜎# 𝑡# − 𝜎. ([). 𝑊 𝑡 } +
"!
#
𝑡. 𝑡! − 𝑡
¨ 𝑅 𝑡, 𝑡, 𝑡! − 𝑅 0, 𝑡, 𝑡! =
"!
#
𝑡. 𝑡! − 𝜎. ([). 𝑊 𝑡
5

6. Luc_Faucheux_2021
Summary, using the SIE formalism - II
¨ 𝑅 𝑡, 𝑡, 𝑡 − 𝑅 0, 𝑡, 𝑡 =
$
#
. 𝜎# 𝑡# − 𝜎. ([). 𝑊 𝑡
¨ 𝑍𝐶 𝑡, 𝑡, 𝑡! = 𝑍𝐶 0, 𝑡, 𝑡! . exp{−[𝑅 𝑡, 𝑡, 𝑡 − 𝑅 0, 𝑡, 𝑡 ]. 𝑡! − 𝑡 −
"!
#
𝑡 𝑡! − 𝑡 #}
¨ 𝑍𝐶 𝑡, 𝑡, 𝑡! = 𝑍𝐶 0, 𝑡, 𝑡! . exp{−[
$
#
. 𝜎# 𝑡# − 𝜎. ([). 𝑊 𝑡 ]. 𝑡! − 𝑡 −
"!
#
𝑡 𝑡! − 𝑡 #}
¨ 𝑍𝐶 𝑡, 𝑡, 𝑡! = 𝑍𝐶 0, 𝑡, 𝑡! . exp{−[
$
#
. 𝜎#. 𝑡. 𝑡! − 𝑡 . 𝑡! − 𝜎. 𝑡! − 𝑡 . ([). 𝑊 𝑡 ]}
6

7. Luc_Faucheux_2021
Summary, using the SDE formalism
¨ 𝑅 𝑡, 𝑡, 𝑡 − 𝑅 0, 𝑡, 𝑡 =
$
#
. 𝜎# 𝑡# − 𝜎. ([). 𝑊 𝑡
¨ 𝑑𝑅 𝑡, 𝑡, 𝑡 =
-
-&
𝑅 0, 𝑡, 𝑡 + 𝜎#. 𝑡 − 𝜎. ([). 𝑑𝑊 𝑡
¨ 𝑅 𝑡, 𝑡!, 𝑡! − 𝑅 0, 𝑡!, 𝑡! = 𝜎#. 𝑡. 𝑡! −
$
#
𝑡 − 𝜎. ([). 𝑊 𝑡
¨ 𝑑𝑅 𝑡, 𝑡!, 𝑡! =
-
-&
{𝜎#. 𝑡. 𝑡! −
$
#
𝑡 } − 𝜎. ([). 𝑑𝑊 𝑡
¨ 𝑑𝑅 𝑡, 𝑡!, 𝑡! = 𝜎#. 𝑡! − 𝑡 − 𝜎. ([). 𝑑𝑊 𝑡
¨ 𝑅 𝑡, 𝑡, 𝑡! − 𝑅 0, 𝑡, 𝑡! =
"!
#
𝑡. 𝑡! − 𝜎. ([). 𝑊 𝑡
¨ 𝑑𝑅 𝑡, 𝑡, 𝑡! =
-
-&
𝑅 0, 𝑡, 𝑡! +
"!
#
𝑡. −𝜎. ([). 𝑑𝑊 𝑡
7

8. Luc_Faucheux_2021
Summary, using the SDE formalism - II
¨ Remember that SDEs are ferocious beasts not to be dealt with lightly, so always better to
either use the SIE formalism or the exact solution.
8

9. Luc_Faucheux_2021
A useful relationship
(material from the Langevin deck)
9

10. Luc_Faucheux_2021
A useful relationship
¨ 𝔼 𝑒"1 & = 𝑒
/!
!
&
¨ 𝔼 exp(𝜎𝑊 𝑡 ) = exp
"!
#
𝑡
¨ 𝔼 𝜎𝑊 𝑡 = 0
¨ exp 𝔼 𝜎𝑊 𝑡 = 1
¨ 𝔼 exp(𝜎𝑊 𝑡 ) ≠ exp 𝔼 𝜎𝑊 𝑡 because the exponential function is positively convex
¨ 𝔼 exp(𝜎𝑊 𝑡 ) > exp 𝔼 𝜎𝑊 𝑡
¨ 𝔼 exp(𝜎𝑊 𝑡 ) = exp 𝔼 𝜎𝑊 𝑡 + [exp
"!
#
𝑡 − 1] convexity adjustment
10

11. Luc_Faucheux_2021
A useful relationship - II
¨ The Langevin equation is quite commonly used when modeling interest rates.
¨ Since interest rates are the “speed” or “velocity” of the Money Market Numeraire, it is quite
natural to have thought about using the Langevin equation which represents the “speed” of
a Brownian particle.
¨ As a result, a number of quantities in Finance are related to the exponential of the integral
over time of the short-term rate (instantaneous spot rate)
¨ For example (Fabio Mercurio p. 3), the stochastic discount factor 𝐷(𝑡, 𝑇) between two time
instants 𝑡 and 𝑇 is the amount at time 𝑡 that is “equivalent” to one unit of currency payable
at time 𝑇, and is equal to
¨ 𝐷 𝑡, 𝑇 =
2(&)
2(4)
= exp(− ∫&
4
𝑟 𝑠 . 𝑑𝑠)
¨ The Bank account (Money-market account) is such that:
¨ 𝑑𝐵 𝑡 = 𝑟 𝑡 . 𝐵 𝑡 . 𝑑𝑡 with 𝐵 𝑡 = 0 = 1
¨ 𝐵 𝑡 = exp(∫5
&
𝑟 𝑠 . 𝑑𝑠)
11

12. Luc_Faucheux_2021
A useful relationship - III
¨ Since we will most likely be looking at equations like:
¨ 𝑑𝑉 𝑡 = −𝑘𝑉(𝑡). 𝑑𝑡 + 𝜎. 𝑑𝑊
¨ 𝑑𝑟 𝑡 = −𝑘𝑟(𝑡). 𝑑𝑡 + 𝜎. 𝑑𝑊
¨ Which has a formal solution:
¨ 𝑉 𝑡6 = exp −𝑘𝑡6 . {exp 𝑘𝑡7 . 𝑉 𝑡7 + ∫&8&7
&8&6
exp 𝑘𝑡 . 𝜎. ([). 𝑑𝑊 𝑡 }
¨ 𝑟 𝑡6 = exp −𝑘𝑡6 . {exp 𝑘𝑡7 . 𝑟 𝑡7 + ∫&8&7
&8&6
exp 𝑘𝑡 . 𝜎. ([). 𝑑𝑊 𝑡 }
¨ So it looks like we will be taking exponentials of integrals of the Wiener process times a
function of time
12

13. Luc_Faucheux_2021
A useful relationship - IV
¨ We had in the simple case: 𝔼 𝑒"1 & = 𝑒
/!
!
&
¨ We would like to find a relation between:
¨ 𝔼 exp[∫5
&
𝑓 𝑠 . 𝑑𝑊(𝑠)]
¨ And
¨ exp[∫5
& $
#
𝑓 𝑠 #. 𝑑𝑠]
¨ Because that seems to work for 𝑓 𝑠 = 𝜎
¨ 𝔼 exp[∫5
&
𝑓 𝑠 . 𝑑𝑊(𝑠)] = 𝔼 exp[∫5
&
𝜎. 𝑑𝑊(𝑠)] = 𝔼 exp[𝜎. 𝑊 𝑡 − 𝑊 0 ] = 𝔼 𝑒"1 &
¨ exp ∫5
& $
#
𝑓 𝑠 #. 𝑑𝑠 = exp ∫5
& $
#
𝜎#. 𝑑𝑠 = exp
$
#
𝜎#
∫5
&
𝑑𝑠 = exp
$
#
𝜎# 𝑡 = 𝑒
/!
!
&
13

14. Luc_Faucheux_2021
A useful relationship - V
¨ Let’s define:
¨ 𝑋 𝑡 = ∫5
&
𝑓 𝑠 . 𝑑𝑊(𝑠) − ∫5
& $
#
𝑓 𝑠 #. 𝑑𝑠
¨ And : 𝑌 𝑡 = exp(𝑋 𝑡 )
¨ Applying Ito lemma:
¨ 𝑓 𝑋 𝑡6 − 𝑓 𝑋 𝑡7 = ∫&8&7
&8&6 -9
-:
. ([). 𝑑𝑋(𝑡) + ∫&8&7
&8&6 $
#
.
-!9
-;! . ([). (𝛿𝑋)#
¨ In the ”limit” of small [me increments, this can be wrien formally as the Ito lemma:
¨ 𝛿𝑓 =
-9
-;
. ([). 𝛿𝑋 +
$
#
.
-!9
-;! . (𝛿𝑋)#
¨ To the function: 𝑓 𝑥 = 𝑋 𝑡 = 𝑌 𝑡 = exp(𝑥 = 𝑋 𝑡 )
14

15. Luc_Faucheux_2021
A useful relationship - VI
¨ 𝑑𝑌 =
-<
-:
. ([). 𝑑𝑋 +
$
#
.
-!9
-:! . (𝑑𝑋)#
¨ 𝑌 𝑡 = exp(𝑋 𝑡 ),
-<
-:
= exp(𝑋 𝑡 ),
-!9
-:! = exp(𝑋 𝑡 )
¨ 𝑋 𝑡 = ∫5
&
𝑓 𝑠 . 𝑑𝑊(𝑠) − ∫5
& $
#
𝑓 𝑠 #. 𝑑𝑠
¨ 𝑑𝑋 𝑡 = 𝑓 𝑡 . 𝑑𝑊(𝑡) −
$
#
𝑓 𝑡 #. 𝑑𝑡
¨ (𝑑𝑋)#= 𝑓 𝑡 #. 𝑑𝑡
¨ 𝑑𝑌 =
-<
-:
. ([). 𝑑𝑋 +
$
#
.
-!9
-:! . (𝑑𝑋)#
¨ 𝑑𝑌 = exp(𝑋 𝑡 ). ([). (𝑓 𝑡 . 𝑑𝑊(𝑡) −
$
#
𝑓 𝑡 #. 𝑑𝑡) +
$
#
. exp(𝑋 𝑡 ). 𝑓 𝑡 #. 𝑑𝑡
¨ 𝑑𝑌 = exp(𝑋 𝑡 ). ([). 𝑓 𝑡 . 𝑑𝑊(𝑡)
15

16. Luc_Faucheux_2021
A useful relationship - VII
¨ 𝑑𝑌 = exp(𝑋 𝑡 ). ([). 𝑓 𝑡 . 𝑑𝑊(𝑡)
¨ 𝑌 𝑡6 − 𝑌 𝑡7 = ∫&8&7
&8&6
𝑑𝑌(𝑡)
¨ 𝑌 𝑡6 − 𝑌 𝑡7 = ∫&8&7
&8&6
exp(𝑋 𝑡 ). ([). 𝑓 𝑡 . 𝑑𝑊(𝑡)
¨ Because the ITO integral is a martingale, 𝑌 𝑡 is also a martingale
¨ 𝔼 𝑌(𝑡) = 𝑌 𝑡 = 0 = 1
¨ 𝑋 𝑡 = ∫5
&
𝑓 𝑠 . 𝑑𝑊(𝑠) − ∫5
& $
#
𝑓 𝑠 #. 𝑑𝑠
¨ 𝑌 𝑡 = exp(𝑋 𝑡 )
¨ 𝔼 exp(𝑋 𝑡 ) = 1
¨ 𝔼 exp(∫5
&
𝑓 𝑠 . 𝑑𝑊(𝑠) − ∫5
& $
#
𝑓 𝑠 #. 𝑑𝑠) = 1
16

17. Luc_Faucheux_2021
A useful relationship - VIII
¨ Making the steps explicit here in order to convince ourselves of the validity (also when we
will want to expand this to a function 𝑓 𝑋 𝑠 , 𝑠 )
¨ 𝔼 exp(∫5
&
𝑓 𝑠 . 𝑑𝑊(𝑠) − ∫5
& $
#
𝑓 𝑠 #. 𝑑𝑠) = 1
¨ 𝔼 exp ∫5
&
𝑓 𝑠 . 𝑑𝑊 𝑠 . exp[− ∫5
& $
#
𝑓 𝑠 #. 𝑑𝑠)] = 1
¨ exp[− ∫5
& $
#
𝑓 𝑠 #. 𝑑𝑠)]. 𝔼 exp ∫5
&
𝑓 𝑠 . 𝑑𝑊 𝑠 = 1
¨ 𝔼 exp ∫5
&
𝑓 𝑠 . 𝑑𝑊 𝑠 = exp[∫5
& $
#
𝑓 𝑠 #. 𝑑𝑠)]
¨ This is indeed a beautiful relationship, and will be quite useful when dealing with interest-
rates modeling.
17

18. Luc_Faucheux_2021
A useful relationship - IX
¨ Under something called the “Novikov condition” (essentially none of the quantities diverge
to infinity and everything is well behaved)
¨ It can be shown that:
¨ exp(∫5
&
𝑋 𝑠 . 𝑑𝑊(𝑠) − ∫5
& $
#
𝑋 𝑠 #. 𝑑𝑠) is a martingale
¨ Sometimes the above function is referred to the Doleans-Dade exponential in memory of
Catherine Doleans-Dade
¨ ℰ ∫5
&
𝑋 𝑠 . 𝑑𝑊 𝑠 = exp(∫5
&
𝑋 𝑠 . 𝑑𝑊(𝑠) − ∫5
& $
#
𝑋 𝑠 #. 𝑑𝑠)
¨ In Finance it is related to the Radon-Nykodym derivative in the Girsanov theorem (in the
deck on numeraire and numeraire change)
¨ One moves from the historical measure ℙ to the risk neutral measure ℚ using the Radon-
Nykodym derivative:
¨
=ℚ
=ℙ
= ℰ ∫5
&
(
@ A )B
"
). 𝑑𝑊 𝑠
18

19. Luc_Faucheux_2021
A useful relationship - X
¨
=ℚ
=ℙ
= ℰ ∫5
&
(
@ A )B
"
). 𝑑𝑊 𝑠
¨ Where 𝑟 𝑠 is the instantaneous risk-free rate, 𝜇 the asset drift and 𝜎 its volatility
¨ Just wanted to mention this here as we will see it in the deck on numeraire.
¨ Novikov is also very famous in financial markets as it is the name of a very fancy restaurant
in the Mayfair section of London always crowded with hedge fund managers and beautiful
models (not the interest rates model type). You might also get a “Novikov condition”
hanging there too long, but it is not related to the one we just mentioned….
19

20. Luc_Faucheux_2021
A useful relationship - XI
¨ In any case, for a function of time only, we have shown a very useful relationship when
dealing with interest rate models:
¨ 𝔼 exp ∫A85
A8&
𝑓 𝑠 . 𝑑𝑊 𝑠 = exp[∫A85
A8& $
#
𝑓 𝑠 #. 𝑑𝑠]
20

21. Luc_Faucheux_2021
Let’s play a game.
Let’s see if you can spot the mistakes in the
next section
(*) Gilles Franchini found them under 2 minutes
21

22. Luc_Faucheux_2021
Another useful relationship
𝔼 𝑒!
= 𝑒 𝔼{!}
. 𝑒
%
&
𝔼{!!}
22

23. Luc_Faucheux_2021
Another useful relationship for a Gaussian process
¨ From the Bachelier deck in the case of a Gaussian process we had the following:
¨ For the regular Gaussian ℎ 𝑥, 𝑡 =
$
#C"!&
. exp(
);!
#"!&
) we have 𝛼 =
$
#"!&
¨ 𝐼D = ∫)E
FE
𝑒)D;!
. 𝑑𝑥 =
C
D
¨ 𝑃D 𝑥 =
$
G0
. 𝑒)D;!
is the normalized probability distribution
¨ < 𝑥H > = ∫)E
FE
𝑥H. 𝑃D 𝑥 . 𝑑𝑥
¨ < 𝑥#H > = 𝛼)H. ∏(8$
(8H
(
$
#
+ 𝑗 − 1)
¨ < 𝑥#HF$ > = 0
23

24. Luc_Faucheux_2021
Another useful relationship for a Gaussian process - II
¨ A couple of side notes first on 𝐼D = ∫)E
FE
𝑒)D;!
. 𝑑𝑥 =
C
D
¨ 𝑃D 𝑥 =
$
G0
. 𝑒)D;!
is the normalized probability distribution
¨ < 𝑥 > = ∫)E
FE
𝑥. 𝑃D 𝑥 . 𝑑𝑥 = 0 because 𝑥. 𝑃D 𝑥 is an odd function
¨ < 𝑥H > = ∫)E
FE
𝑥H. 𝑃D 𝑥 . 𝑑𝑥 = 0 because (𝑥H. 𝑃D 𝑥 ) is an odd function if 𝑘 is odd
24

25. Luc_Faucheux_2021
Kolmogorov equation: Bachelier thesis (page 35) IX
¨ 𝑃 𝑥, 𝑡 =
I
&
. exp{−
CI!;!
&
} is already normalized
¨ We still need to solve for the value of 𝐻
¨ A couple of side notes first on 𝐼D = ∫)E
FE
𝑒)D;!
. 𝑑𝑥 =
C
D
¨ 𝑃D 𝑥 =
$
G0
. 𝑒)D;!
is the normalized probability distribution
¨ < 𝑥 > = ∫)E
FE
𝑥. 𝑃D 𝑥 . 𝑑𝑥 = 0 because 𝑥. 𝑃D 𝑥 is an odd function
¨ < 𝑥H > = ∫)E
FE
𝑥H. 𝑃D 𝑥 . 𝑑𝑥 = 0 because (𝑥H. 𝑃D 𝑥 ) is an odd function if 𝑘 is odd
25

26. Luc_Faucheux_2021
Kolmogorov equation: Bachelier thesis (page 35) X
¨ < 𝑥# > = ∫)E
FE
𝑥#. 𝑃D 𝑥 . 𝑑𝑥 =
C
D
. ∫)E
FE
𝑥#. 𝑒)D;!
. 𝑑𝑥
¨ Now:
-
-D
𝑒)D;!
= −𝑥# 𝑒)D;!
¨ So: < 𝑥# > =
C
D
. ∫)E
FE
𝑥#. 𝑒)D;!
. 𝑑𝑥 =
C
D
. ∫)E
FE )-
-D
𝑒)D;!
. 𝑑𝑥 =
C
D
.
)-
-D
[∫)E
FE
𝑒)D;!
. 𝑑𝑥]
¨ A little more formally:
¨ < 𝑥# > =
)$
G0
.
-G0
-D
¨ Replacing 𝐼D =
C
D
, we get < 𝑥# > =
)$
1
0
.
-
1
0
-D
= − 𝛼.
-
-D
𝛼 ⁄23
! =
$
#
. 𝛼 ⁄3
!. 𝛼 ⁄24
! =
$
#D
26

27. Luc_Faucheux_2021
Kolmogorov equation: Bachelier thesis (page 35) XI
¨ < 𝑥# > =
$
#D
¨ < 𝑥#H > =
$
G0
. ∫)E
FE
𝑥#H. 𝑒)D;!
. 𝑑𝑥 and again 𝐼D = ∫)E
FE
𝑒)D;!
. 𝑑𝑥 =
C
D
¨ It is easy to see that (𝑥#H. 𝑒)D;!
) =
-5
-D5 𝑒)D;!
. (−1)H
¨ < 𝑥#H > =
$
G0
. ∫)E
FE -5
-D5 𝑒)D;!
. (−1)H . 𝑑𝑥 =
$
G0
.
=5
=D5 𝐼D . (−1)H
¨ < 𝑥#H > =
$
G0
.
=5
=D5 𝐼D . (−1)H=
$
G0
.
=5
=D5 𝐼D . (−1)H= 𝛼 ⁄3
!.
=5
=D5 𝛼 ⁄23
! . (−1)H
¨
=5
=D5 𝛼 ⁄23
! = 𝛼 ⁄23
!. 𝛼)H. ∏(8$
(8H
(
$
#
+ 𝑗 − 1) . (−1)H
¨ < 𝑥#H > = 𝛼)H. ∏(8$
(8H
(
$
#
+ 𝑗 − 1), with 𝑘 = 1, we recover indeed < 𝑥# > =
$
#D
27

28. Luc_Faucheux_2021
Kolmogorov equation: Bachelier thesis (page 35) XII
¨ 𝐼D = ∫)E
FE
𝑒)D;!
. 𝑑𝑥 =
C
D
¨ 𝑃D 𝑥 =
$
G0
. 𝑒)D;!
is the normalized probability distribution
¨ < 𝑥H > = ∫)E
FE
𝑥H. 𝑃D 𝑥 . 𝑑𝑥
¨ < 𝑥#H > = 𝛼)H. ∏(8$
(8H
(
$
#
+ 𝑗 − 1)
¨ < 𝑥#HF$ > = 0
¨ We will also look like Bachelier did at the positive part of the price distribution
¨ < (𝑥H|𝑥 > 0) > = ∫5
FE
𝑥H. 𝑃D 𝑥 . 𝑑𝑥
28

29. Luc_Faucheux_2021
Kolmogorov equation: Bachelier thesis (page 35) XIII
¨ < 𝑥#H > = 𝛼)H. ∏(8$
(8H
(
$
#
+ 𝑗 − 1)
¨ For the regular Gaussian ℎ 𝑥, 𝑡 =
$
#C"!&
. exp(
);!
#"!&
) we have 𝛼 =
$
#"!&
¨ A somewhat useful notation:
¨ 𝑘! = ∏(8$
(8H
𝑗 is the usual factorial
¨ 𝑘!! = ∏(8$
(8H
𝑗 is called the “double factorial” and only includes in the product the terms that
have the SAME parity as 𝑘
¨ In our specific case we can rewrite ∏(8$
(8H
(
$
#
+ 𝑗 − 1) as:
¨ ∏(8$
(8H
(
$
#
+ 𝑗 − 1) = ∏(8$
(8H
(
#()$
#
) = 2)H ∏(8$
(8H
(2𝑗 − 1) = 2)H. 2𝑘 − 1 ‼
¨ < 𝑥#H > = 𝛼)H. 2)H. 2𝑘 − 1 ‼
29

30. Luc_Faucheux_2021
Kolmogorov equation: Bachelier thesis (page 35) XIV
¨ < 𝑥#H > = 𝛼)H. 2)H. 2𝑘 − 1 ‼
¨ In the case of the Gaussian, 𝛼 =
$
#"!&
, so < 𝑥#H > = (2𝜎# 𝑡)H. 2)H. 2𝑘 − 1 ‼
¨ So : < 𝑥#H > = (𝜎# 𝑡)H. 2𝑘 − 1 ‼ and < 𝑥#HF$ > = 0
¨ Another cute way to express it is the following:
¨ < 𝑥K > = (𝜎 𝑡)K. 𝑛 − 1 ‼ if 𝑛 is even, 0 otherwise
¨ This is quite compact and beautiful
30

31. Luc_Faucheux_2021
Another useful relationship for a Gaussian process - III
¨ We put ourselves in the Gaussian case with 0 drift
¨ ℎ 𝑥, 𝑡 =
$
#C"!&
. exp(
);!
#"!&
) we have 𝛼 =
$
#"!&
¨ 𝔼 𝑋 = ∫)E
FE
ℎ 𝑥, 𝑡 . 𝑥 . 𝑑𝑥 = 0 because ℎ 𝑥, 𝑡 is an even function of 𝑥
¨ 𝔼 𝑋# = ∫)E
FE
ℎ 𝑥, 𝑡 . 𝑥# . 𝑑𝑥 = 𝜎# 𝑡
¨ 𝔼 𝑋#L = ∫)E
FE
ℎ 𝑥, 𝑡 . 𝑥#L . 𝑑𝑥 = 𝜎# 𝑡 L. 2𝑝 − 1 ‼
¨ 𝔼 𝑋#LF$ = ∫)E
FE
ℎ 𝑥, 𝑡 . 𝑥#LF! . 𝑑𝑥 = 0
¨ Using the Taylor expansion we can get:
¨ exp
$
#
𝔼 𝑋# = ∑H85
H8E $
H!
.
$
#
𝔼 𝑋#
H
= ∑H85
H8E $
H!
.
"!&
#
H
31

32. Luc_Faucheux_2021
Another useful relationship for a Gaussian process - IV
¨ exp
$
#
𝔼 𝑋# = ∑H85
H8E $
H!
.
$
#
𝔼 𝑋#
H
= ∑H85
H8E $
H!
.
"!&
#
H
¨ 𝔼 exp[𝑋] = ∫)E
FE
ℎ 𝑥, 𝑡 . exp(𝑥) . 𝑑𝑥
¨ 𝔼 exp[𝑋] = ∫)E
FE
ℎ 𝑥, 𝑡 . ∑H85
H8E $
H!
. 𝑥 H . 𝑑𝑥
¨ 𝔼 exp[𝑋] = ∑H85
H8E $
H!
. ∫)E
FE
ℎ 𝑥, 𝑡 . 𝑥H. 𝑑𝑥
¨ 𝔼 exp[𝑋] = ∑H85
H8E $
H!
. 𝔼{𝑋H} and only the terms even in 𝑘 are non zero, so we can rewrite
using 𝑘 = 2𝑝
¨ 𝔼 exp[𝑋] = ∑L85
L8E $
#L !
. 𝔼{𝑋#L}
¨ 𝔼 exp[𝑋] = ∑L85
L8E $
#L !
. 𝜎# 𝑡 L. 2𝑝 − 1 ‼
32

33. Luc_Faucheux_2021
Another useful relationship for a Gaussian process - V
¨ 𝔼 exp[𝑋] = ∑L85
L8E $
#L !
. 𝜎# 𝑡 L. 2𝑝 − 1 ‼
¨ exp
$
#
𝔼 𝑋# = ∑H85
H8E $
H!
.
$
#
𝔼 𝑋#
H
= ∑H85
H8E $
H!
.
"!&
#
H
¨ Almost there.
¨
#L)$ ‼
#L !
=
#L)$ . #L)O . #L)P …..P.O.$
#L . #L)$ . #L)# . #L)O ….P.R.O.#.$
¨
#L)$ ‼
#L !
=
$
#L . #L)# . #L)R …..R.#.$
= (
$
#
)L.
$
L. L)$ . L)# …O.#.$
=
$
#6.L!
¨ 𝔼 exp[𝑋] = ∑L85
L8E $
#L !
. 𝜎# 𝑡 L. 2𝑝 − 1 ‼ = ∑L85
L8E $
#6.L!
. 𝜎# 𝑡 L = ∑L85
L8E $
L!
.
"!&
#
L
¨ 𝔼 exp[𝑋] = exp
$
#
𝔼 𝑋#
33

34. Luc_Faucheux_2021
Another useful relationship for a Gaussian process - VI
¨ 𝔼 exp[𝑋] = exp
$
#
𝔼 𝑋#
¨ In the case of no drift (zero mean).
¨ In the case of a non-zero drift, we define:
¨ 𝑌 = 𝑋− < 𝑋 > = 𝑋 − 𝔼 𝑋
¨ 𝑌(𝑡) = 𝑋(𝑡)− < 𝑋 >& = 𝑋(𝑡) − 𝔼& 𝑋|𝔉(0)
¨ 𝔼 𝑌# = 𝔼 (𝑋(𝑡)− < 𝑋 >&)# = 𝔼 𝑋(𝑡)# +< 𝑋 >&
#
− 2. 𝑋 𝑡 . < 𝑋 >&
¨ 𝔼 𝑌# = 𝔼 𝑋(𝑡)# + 𝔼 < 𝑋 >&
#
− 2. 𝔼 𝑋 𝑡 . < 𝑋 >&
¨ 𝔼 𝑌# = 𝔼 𝑋(𝑡)# +< 𝑋 >&
#
− 2. < 𝑋 >&. 𝔼 𝑋 𝑡
¨ 𝔼 𝑌# = 𝔼 𝑋(𝑡)# +< 𝑋 >&
#
− 2. < 𝑋 >&. < 𝑋 >&
¨ 𝔼 𝑌# = 𝔼 𝑋(𝑡)# +< 𝑋 >&
#
− 2. < 𝑋 >&
#
¨ 𝔼 𝑌# = 𝔼 𝑋(𝑡)# −< 𝑋 >&
#
34

35. Luc_Faucheux_2021
Another useful relationship for a Gaussian process - VII
¨ 𝑌 = 𝑋− < 𝑋 > = 𝑋 − 𝔼 𝑋
¨ 𝔼 𝑌 = 𝔼 𝑋 − 𝔼 𝑋 = 𝔼 𝑋 − 𝔼 𝑋 = 0
¨ 𝔼 𝑌# = 𝔼 𝑋(𝑡)# −< 𝑋 >&
#
¨ 𝔼 𝑌# = 𝔼 (𝑋(𝑡)− < 𝑋 >&)#
¨ If 𝑋 𝑡 is Gaussian centered around 𝑋(𝑡), then 𝑌(𝑡) is also Gaussian centered around 0
¨ ℎ 𝑥, 𝑡 =
$
#C"!&
. exp(
)(;)𝔼 : )!
#"!&
)
¨ ℎ 𝑦, 𝑡 =
$
#C"!&
. exp(
)T!
#"!&
)
¨ So we have then:
¨ 𝔼 exp[𝑌] = exp
$
#
𝔼 𝑌#
35

36. Luc_Faucheux_2021
Another useful relationship for a Gaussian process - VIII
¨ 𝔼 exp[𝑌] = exp
$
#
𝔼 𝑌#
¨ 𝑌 = 𝑋− < 𝑋 > = 𝑋 − 𝔼 𝑋
¨ 𝔼 exp[𝑌] = 𝔼 exp[𝑋 − 𝔼 𝑋 ] = 𝔼 exp 𝑋 . exp[−𝔼 𝑋 ]
¨ 𝔼 exp[𝑌] = exp −𝔼 𝑋 . 𝔼 exp 𝑋
¨ 𝔼 𝑌# = 𝔼 (𝑋(𝑡)− < 𝑋 >&)#
¨ 𝔼 exp 𝑋 = exp 𝔼 𝑋 . 𝔼 exp[𝑌]
¨ 𝔼 exp 𝑋 = exp 𝔼 𝑋 . exp
$
#
𝔼 𝑌#
¨ 𝔼 exp 𝑋 = exp 𝔼 𝑋 . exp
$
#
𝔼 (𝑋(𝑡)− < 𝑋 >&)#
36

37. Luc_Faucheux_2021
Another useful relationship for a Gaussian process - IX
¨ 𝔼 exp 𝑋 = exp 𝔼 𝑋 . exp
$
#
𝔼 (𝑋(𝑡)− < 𝑋 >&)#
¨ In some textbooks you find the following notation:
¨ 𝔼 (𝑋(𝑡)− < 𝑋 >&)# = 𝑉[𝑋(𝑡)] for the Variance
¨ 𝔼 𝑋 = 𝑀[𝑋(𝑡)] for the Mean
¨ 𝔼 exp 𝑋 = exp 𝔼 𝑋 . exp
$
#
𝔼 (𝑋(𝑡)− < 𝑋 >&)#
¨ 𝔼 exp 𝑋 = exp 𝑀[𝑋(𝑡)] . exp
$
#
𝑉[𝑋(𝑡)]
¨ 𝔼 exp 𝑋 = exp[𝑀] . exp
$
#
𝑉
¨ 𝔼 𝑒: = 𝑒U. 𝑒
7
!
37

38. Luc_Faucheux_2021
Another useful relationship for a Gaussian process - X
¨ Note that in the Gaussian case we could also have explicitly derived the formula like we did
in the Langevin deck (couple of slides following)
38

39. Luc_Faucheux_2021
Some properties of the GBM - VIa
¨ 𝔼 exp(𝜎𝑊 𝑡 ) = exp
"!
#
𝑡
¨ 𝔼 𝑒"1 & = 𝑒
/!
!
&
¨ We can also derive this one explicitly from the the fact that 𝑊 𝑡 ~𝑁(0, 𝑡)
¨ 𝔼 exp(𝜎𝑊 𝑡 ) = ∫T8)E
T8FE
exp 𝜎𝑦 . 𝑝1 𝑦, 𝑡 . 𝑑𝑦
¨ And 𝑝1 𝑦, 𝑡 = ℎ 𝑦, 𝑡 =
$
#C&
. exp(
)T!
#&
)
¨ 𝔼 exp(𝜎𝑊 𝑡 ) = ∫T8)E
T8FE
exp 𝜎𝑦 .
$
#C&
. exp(
)T!
#&
) . 𝑑𝑦
¨ 𝔼 exp(𝜎𝑊 𝑡 ) = ∫T8)E
T8FE $
#C&
. exp 𝜎𝑦 . exp(
)T!
#&
) . 𝑑𝑦
39

40. Luc_Faucheux_2021
Some properties of the GBM - VIb
¨ 𝔼 exp(𝜎𝑊 𝑡 ) = ∫T8)E
T8FE $
#C&
. exp 𝜎𝑦 . exp(
)T!
#&
) . 𝑑𝑦
¨ 𝔼 exp(𝜎𝑊 𝑡 ) = ∫T8)E
T8FE $
#C&
. exp(
)T!
#&
+
#"T&
#&
) . 𝑑𝑦
¨ 𝔼 exp(𝜎𝑊 𝑡 ) = ∫T8)E
T8FE $
#C&
. exp(
)(T)"&)!
#&
+
("&)!
#&
) . 𝑑𝑦
¨ 𝔼 exp(𝜎𝑊 𝑡 ) = exp
"!
#
𝑡 . ∫T8)E
T8FE $
#C&
. exp(
)(T)"&)!
#&
) . 𝑑𝑦
¨ We do the change of variable: 𝜉 = 𝑦 − 𝜎𝑡
¨ 𝔼 exp(𝜎𝑊 𝑡 ) = exp
"!
#
𝑡 . ∫V8)E
V8FE $
#C&
. exp
)V!
#&
. 𝑑𝜉 = exp
"!
#
𝑡
¨ 𝔼 exp(𝜎𝑊 𝑡 ) = exp
"!
#
𝑡
40

41. Luc_Faucheux_2021
Another useful relationship for a Gaussian process - XI
¨ Before we leave this chapter, there is something else we need to point out, as we will use it
when looking at the Radon-Nikodym derivative.
¨ 𝔼 exp 𝑋 = exp 𝔼 𝑋 . exp
$
#
𝔼 (𝑋(𝑡)− < 𝑋 >&)#
¨ This also has to do with the Moment Generating function.
¨ The moment generating function of a variable 𝑋 is the function of the variable 𝜑
¨ 𝕄: 𝜑 = 𝔼 exp 𝜑𝑋
¨ 𝕄: 𝜑 = 𝔼 exp[𝜑𝑋] = ∑L85
L8E $
L!
. 𝔼 [𝜑𝑋]L = ∑L85
L8E W6
L!
. 𝔼 [𝑋]L
¨ The cool thing about the Moment Generating function (if it exists, unless the distribution is
pathological) is that the n-th derivative is the n-th moment of the distribution.
¨
=6
=W6 𝕄: 𝜑 |W85 = 𝔼 [𝑋]L
41

42. Luc_Faucheux_2021
Another useful relationship for a Gaussian process - XII
¨ 𝕄: 𝜑 = 𝔼 exp[𝜑𝑋] = ∑L85
L8E $
L!
. 𝔼 [𝜑𝑋]L = ∑L85
L8E W6
L!
. 𝔼 [𝑋]L
¨ 𝔼 exp 𝑋 = exp 𝔼 𝑋 . exp
$
#
𝔼 (𝑋(𝑡)− < 𝑋 >&)#
¨ If we define 𝑌 = 𝜑𝑋
¨ 𝔼 exp 𝑌 = exp 𝔼 𝑌 . exp
$
#
𝔼 (𝑌(𝑡)− < 𝑌 >&)#
¨ 𝔼 exp 𝑌 = 𝔼 exp 𝜑𝑋 = 𝕄: 𝜑
¨ 𝔼 (𝑌(𝑡)− < 𝑌 >&)# = 𝔼 (𝜑𝑋(𝑡)− < 𝜑𝑋 >&)#
¨ 𝔼 (𝑌(𝑡)− < 𝑌 >&)# = 𝔼 (𝜑𝑋(𝑡) − 𝜑. < 𝑋 >&)#
¨ 𝔼 (𝑌(𝑡)− < 𝑌 >&)# = 𝔼 𝜑#(𝑋(𝑡)− < 𝑋 >&)# = 𝜑#. 𝔼 (𝑋(𝑡)− < 𝑋 >&)#
¨ 𝔼 𝑌 = 𝔼 𝜑𝑋 = 𝜑. 𝔼 𝑋
42

43. Luc_Faucheux_2021
Another useful relationship for a Gaussian process - XIII
¨ 𝔼 exp 𝑌 = exp 𝔼 𝑌 . exp
$
#
𝔼 (𝑌(𝑡)− < 𝑌 >&)#
¨ 𝔼 𝑌 = 𝔼 𝜑𝑋 = 𝜑. 𝔼 𝑋
¨ 𝔼 (𝑌(𝑡)− < 𝑌 >&)# = 𝜑#. 𝔼 (𝑋(𝑡)− < 𝑋 >&)#
¨ 𝔼 exp 𝜑𝑋 = exp 𝜑. 𝔼 𝑋 . exp
$
#
𝜑# 𝔼 (𝑋(𝑡)− < 𝑋 >&)#
¨ For sake of notation, if 𝑋(𝑡) is Normal 𝑁 𝔼 𝑋 , 𝔼 𝑋 𝑡 − < 𝑋 >&
# = 𝑁(𝜇𝑡, 𝜎# 𝑡)
¨ 𝔼 exp 𝜑𝑋 = exp 𝜑. 𝜇𝑡 . exp
$
#
𝜑# 𝜎# 𝑡
43

44. Luc_Faucheux_2021
Another useful relationship for a Gaussian process - XIV
¨ 𝔼 exp 𝜑𝑋 = exp 𝜑. 𝜇𝑡 . exp
$
#
𝜑# 𝜎# 𝑡
¨ Now let’s get a taste of the Radon-Nykodim theorem.
¨ 𝔼 is the expectation associated to the random variable 𝑋
¨ Let’s call it 𝔼ℙ
¨ We now define a ℚ measure equivalent to the ℙ-measure, defined by :
¨ 𝔼ℚ 𝑋 𝑡 𝔉 0 = 𝔼ℙ{
=ℚ
=ℙ
𝑋(𝑡)|𝔉 0 }
¨ Suppose now that “out of nowhere) (Baxter p.71), we set the quantity
=ℚ
=ℙ
to be equal to:
¨
=ℚ
=ℙ
= exp(−𝜉. 𝑊 𝑡 −
$
#
𝜉#. 𝑡)
¨ Where 𝑊 𝑡 is the regular Brownian motion under the ℙ-measure (also sometimes called a
ℙ-Brownian motion)
44

45. Luc_Faucheux_2021
Another useful relationship for a Gaussian process - XV
¨ 𝔼ℙ exp 𝜑𝑋 = exp 𝜑. 𝜇𝑡 . exp
$
#
𝜑# 𝜎# 𝑡
¨
=ℚ
=ℙ
= exp(−𝜉. 𝑊 𝑡 −
$
#
𝜉#. 𝑡)
¨ 𝔼ℚ 𝑋 𝑡 𝔉 0 = 𝔼ℙ{
=ℚ
=ℙ
𝑋(𝑡)|𝔉 0 }
¨ Let’s try to calculate 𝔼ℚ exp 𝜑. 𝑊(𝑡) |𝔉 0
¨ 𝔼ℚ exp 𝜑. 𝑊(𝑡) |𝔉 0 = 𝔼ℙ{
=ℚ
=ℙ
. exp 𝜑. 𝑊(𝑡) |𝔉 0 }
¨ 𝔼ℚ exp 𝜑. 𝑊(𝑡) |𝔉 0 = 𝔼ℙ{(exp(−𝜉. 𝑊 𝑡 −
$
#
𝜉#. 𝑡)). exp 𝜑. 𝑊(𝑡) |𝔉 0 }
¨ 𝔼ℚ exp 𝜑. 𝑊(𝑡) |𝔉 0 = 𝔼ℙ{(exp(−𝜉. 𝑊 𝑡 + 𝜑. 𝑊(𝑡) −
$
#
𝜉#. 𝑡))|𝔉 0 }
45

46. Luc_Faucheux_2021
Another useful relationship for a Gaussian process - XVI
¨ 𝔼ℚ exp 𝜑. 𝑊(𝑡) |𝔉 0 = 𝔼ℙ{(exp(−𝜉. 𝑊 𝑡 + 𝜑. 𝑊(𝑡) −
$
#
𝜉#. 𝑡))|𝔉 0 }
¨ Just dropping the “|𝔉 0 ” for sake of simplicity
¨ 𝔼ℚ exp 𝜑. 𝑊(𝑡) = 𝔼ℙ{(exp(−𝜉. 𝑊 𝑡 + 𝜑. 𝑊(𝑡) −
$
#
𝜉#. 𝑡))}
¨ 𝔼ℚ exp 𝜑. 𝑊(𝑡) = exp −
$
#
𝜉#. 𝑡 . 𝔼ℙ{exp[ 𝜑 − 𝜉 . 𝑊 𝑡 ]}
¨ NOW, since 𝑊 𝑡 is a ℙ-Brownian motion, or is a Normal 𝑁(0, 𝑡)
¨ 𝔼ℙ exp 𝜑𝑋 = exp 𝜑. 𝜇𝑡 . exp
$
#
𝜑# 𝜎# 𝑡
¨ 𝔼ℙ exp 𝜑𝑊 = exp
$
#
𝜑# 𝑡
¨ 𝔼ℙ exp 𝜑 − 𝜉 . 𝑊 𝑡 = exp
$
#
𝜑 − 𝜉 # 𝑡
46

47. Luc_Faucheux_2021
Another useful relationship for a Gaussian process - XVII
¨ 𝔼ℚ exp 𝜑. 𝑊(𝑡) = exp −
$
#
𝜉#. 𝑡 . 𝔼ℙ{exp[ 𝜑 − 𝜉 . 𝑊 𝑡 ]}
¨ 𝔼ℙ exp 𝜑 − 𝜉 . 𝑊 𝑡 = exp
$
#
𝜑 − 𝜉 # 𝑡
¨ So combining the two equations above we have:
¨ 𝔼ℚ exp 𝜑. 𝑊(𝑡) = exp −
$
#
𝜉#. 𝑡 . exp
$
#
𝜑 − 𝜉 # 𝑡
¨ 𝔼ℚ exp 𝜑. 𝑊(𝑡) = exp −
$
#
𝜉#. 𝑡 +
$
#
𝜑 − 𝜉 # 𝑡
¨ 𝔼ℚ exp 𝜑. 𝑊(𝑡) = exp −
$
#
𝜉#. 𝑡 +
$
#
𝜉#. 𝑡 +
$
#
𝜑#. 𝑡 − 𝜑𝜉𝑡
¨ 𝔼ℚ exp 𝜑. 𝑊(𝑡) = exp −𝜑𝜉𝑡 +
$
#
𝜑# 𝑡
47

48. Luc_Faucheux_2021
Another useful relationship for a Gaussian process - XVIII
¨ So we get now:
¨ 𝔼ℚ exp 𝜑. 𝑊(𝑡) = exp −𝜑𝜉𝑡 +
$
#
𝜑# 𝑡
¨ But wait a second!
¨ We had started with:
¨ 𝔼ℙ exp 𝜑𝑋 = exp 𝜑. 𝜇𝑡 . exp
$
#
𝜑# 𝜎# 𝑡
¨ 𝔼ℙ exp 𝜑𝑊 = exp
$
#
𝜑# 𝑡
¨ So what the equation 𝔼ℚ exp 𝜑. 𝑊(𝑡) = exp −𝜑𝜉𝑡 +
$
#
𝜑# 𝑡 tells us is the following:
¨ The distribution of 𝑊(𝑡) under the ℚ-measure is ALSO a Normal distribution with mean
equal to −𝜉𝑡 and variance equal to (𝑡)
48

49. Luc_Faucheux_2021
Another useful relationship for a Gaussian process - XIX
¨ That is an awesome result that we need to ponder a little, and keep in the back of our mind
when doing the proper measure change through the CMG (Cameron-Martin-Girsanov)
theorem using the Radon-Nykodim derivative (not super rigorous at this point, but trying to
just get the jist of it)
¨ 𝑊 𝑡 is a ℙ-Brownian motion, with a Normal distribution 𝑁(0, 𝑡)
¨ 𝑊 𝑡 is ALSO a ℚ-Brownian motion, with a Normal distribution 𝑁(−𝜉𝑡, 𝑡)
¨ Let’s take a leap here and assume that what is true at time 𝑡 is also true for all prior times,
by defining a drifted process:
¨ h𝑊 𝑡 = 𝑊 𝑡 + 𝜉𝑡
49

50. Luc_Faucheux_2021
Another useful relationship for a Gaussian process - XX
¨ 𝑊 𝑡 is a ℙ-Brownian motion, with a Normal distribution 𝑁(0, 𝑡)
¨ 𝑊 𝑡 is ALSO a ℚ-Brownian motion, with a Normal distribution 𝑁(−𝜉𝑡, 𝑡)
¨ h𝑊 𝑡 = 𝑊 𝑡 + 𝜉𝑡, is a ℚ-Brownian motion, with a Normal distribution 𝑁(0, 𝑡)
¨ 𝔼ℙ exp 𝜑𝑊 (𝑡) = exp
$
#
𝜑# 𝑡
¨ 𝔼ℙ exp 𝜑 h𝑊 𝑡 = exp 𝜑𝜉𝑡 . exp
$
#
𝜑# 𝑡
¨ 𝔼ℚ exp 𝜑. 𝑊(𝑡) = exp −𝜑𝜉𝑡 +
$
#
𝜑# 𝑡
¨ 𝔼ℚ exp 𝜑. h𝑊(𝑡) = exp
$
#
𝜑# 𝑡
¨ Note that all those are on the marginal distribution with “|𝔉 0 ”
50

51. Luc_Faucheux_2021
Another useful relationship for a Gaussian process - XXI
¨ Where you need a little leap of faith is to also assume that what we did is also valid for times
prior to the terminal time.
¨ 𝔼ℚ exp 𝜑. h𝑊(𝑡) = exp
$
#
𝜑# 𝑡
¨ Which was really:
¨ 𝔼ℚ exp 𝜑. h𝑊(𝑡) |𝔉 0 = exp
$
#
𝜑# 𝑡
¨ By assuming: h𝑊 𝑡 = 𝑊 𝑡 + 𝜉𝑡, we are also implicitly assuming:
¨ 𝔼ℚ exp 𝜑. ( h𝑊 𝑡 − h𝑊 𝑠 |𝔉 𝑠 = exp
$
#
𝜑#(𝑡 − 𝑠)
¨
=ℚ
=ℙ
= exp(−𝜉. 𝑊 𝑡 −
$
#
𝜉#. 𝑡)
51

52. Luc_Faucheux_2021
Another useful relationship for a Gaussian process - XXII
¨ Sometimes to make the notation easier to understand, we can use: 𝔼&
1
¨ To note the expected value at time 𝑡 in the probability measure associated to the Brownian
motion 𝑊(𝑡)
¨ h𝑊 𝑡 = 𝑊 𝑡 + 𝜉𝑡
¨ 𝔼&
1 exp 𝜑𝑊 (𝑡)|𝔉 0 = exp
$
#
𝜑# 𝑡
¨ 𝔼&
1 exp 𝜑 h𝑊 𝑡 |𝔉 0 = exp 𝜑𝜉𝑡 +
$
#
𝜑# 𝑡
¨ 𝔼&
X1 exp 𝜑𝑊(𝑡) |𝔉 0 = exp −𝜑𝜉𝑡 +
$
#
𝜑# 𝑡
¨ 𝔼&
X1
exp 𝜑 h𝑊(𝑡) |𝔉 0 = exp
$
#
𝜑# 𝑡
¨
=ℚ
=ℙ
= exp(−𝜉. 𝑊 𝑡 −
$
#
𝜉#. 𝑡)
52

53. Luc_Faucheux_2021
Another useful relationship for a Gaussian process - XXIII
¨ Subject to some condition (most notably the Novikov condition), the results can be
extended from 𝜉 → 𝜉(𝑡) (Baxter p.74).
¨ “If 𝑊(𝑡) is a ℙ-Brownian motion and 𝜉(𝑡) is a 𝔉-previsible process satisfying the
boundedness condition 𝔼ℙ exp
$
#
∫A85
A8&
𝜉 𝑠 #. 𝑑𝑠 |𝔉 0 < ∞, then there exists a measure
ℚ such that:
¨ ℚ is equivalent to ℙ
¨
=ℚ
=ℙ
= exp(−𝜉. 𝑊 𝑡 −
$
#
𝜉#. 𝑡)
¨ Becomes:
¨
=ℚ
=ℙ
= exp[− ∫A85
A8&
𝜉(𝑠). 𝑑𝑊(𝑠) −
$
#
∫A85
A8&
𝜉 𝑠 #. 𝑑𝑠]
¨ h𝑊 𝑡 = 𝑊 𝑡 + ∫A85
A8&
𝜉(𝑠). 𝑑𝑠
¨ h𝑊 𝑡 is a ℚ-Brownian motion
53

54. Luc_Faucheux_2021
Another useful relationship for a Gaussian process - XXIV
¨ To quote Baxter, “within limits, drift is measure and measure is drift”
¨ To quote Gilles Franchini “the only thing we can really do in stochastic calculus is to calculate
expectations, so it would make sense that the only tools at our disposal are related to
changing the drift”
¨ Note that here we showed that if we define a new Brownian motion as the original one plus
a drift, we recover an equivalent measure.
¨ It is a little more complicated to convince yourself that if you have a measure, ANY other
equivalent measure is such that the two Brownian motions associated to each measures are
only different by a drift:
¨ ∫A85
A8&
𝜉(𝑠). 𝑑𝑠 = h𝑊 𝑡 − 𝑊(𝑡)
¨ and that the Radon-Nykodim derivative is given by:
¨
=ℚ
=ℙ
= exp[− ∫A85
A8&
𝜉(𝑠). 𝑑𝑊(𝑠) −
$
#
∫A85
A8&
𝜉 𝑠 #. 𝑑𝑠]
54

55. Luc_Faucheux_2021
Another useful relationship for a Gaussian process - XXV
¨ Another cool thing, the Radon-Nykodim derivative is a martingale under the ℙ-measure
¨
=ℚ
=ℙ
= exp[− ∫A85
A8&
𝜉 𝑠 . 𝑑𝑊 𝑠 −
$
#
∫A85
A8&
𝜉 𝑠 #. 𝑑𝑠]
¨ And we know from the first useful relationship that:
¨ 𝔼 exp ∫A85
A8&
𝑓 𝑠 . 𝑑𝑊 𝑠 = exp[∫A85
A8& $
#
𝑓 𝑠 #. 𝑑𝑠], where 𝔼 = 𝔼&
ℙ = 𝔼&
1
¨ 𝔼&
ℙ exp ∫A85
A8&
𝑓 𝑠 . 𝑑𝑊 𝑠 |𝔉 0 = exp[∫A85
A8& $
#
𝑓 𝑠 #. 𝑑𝑠]
¨ 𝔼&
ℙ exp ∫A85
A8&
𝑓 𝑠 . 𝑑𝑊 𝑠 −
$
#
∫A85
A8&
𝑓 𝑠 #. 𝑑𝑠 |𝔉 0 = 1
¨ Just switching 𝜉 𝑠 = −𝑓(𝑠)
¨ 𝔼&
ℙ exp ∫A85
A8&
𝜉 𝑠 . 𝑑𝑊 𝑠 −
$
#
∫A85
A8&
𝜉 𝑠 #. 𝑑𝑠 |𝔉 0 = 1
55

56. Luc_Faucheux_2021
Another useful relationship for a Gaussian process - XXVI
¨ 𝔼&
ℙ exp ∫A85
A8&
𝜉 𝑠 . 𝑑𝑊 𝑠 −
$
#
∫A85
A8&
𝜉 𝑠 #. 𝑑𝑠 |𝔉 0 = 1
¨
=ℚ
=ℙ
= exp[− ∫A85
A8&
𝜉 𝑠 . 𝑑𝑊 𝑠 −
$
#
∫A85
A8&
𝜉 𝑠 #. 𝑑𝑠]
¨ 𝔼&
ℙ =ℚ
=ℙ
|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 }
¨ We can replace 𝑋 𝑡 = 1 in the above definition and we will get:
¨ 𝔼ℚ 1 𝔉 0 = 1 = 𝔼ℙ{
=ℚ
=ℙ
|𝔉 0 }
¨ So we get: 𝔼&
ℙ =ℚ
=ℙ
|𝔉 0 = 1
56

57. Luc_Faucheux_2021
A quick note on martingale and driftless
processes
57

58. Luc_Faucheux_2021
Quick side note
¨
=ℚ
=ℙ
= exp[− ∫A85
A8&
𝜉 𝑠 . 𝑑𝑊 𝑠 −
$
#
∫A85
A8&
𝜉 𝑠 #. 𝑑𝑠]
¨ 𝔼&
ℙ =ℚ
=ℙ
|𝔉 0 = 1
¨ We also have if we define :
¨ 𝑌 𝑡 = 𝑌 𝑡 = 0 . exp(∫A85
A8&
𝜉 𝑠 . 𝑑𝑊 𝑠 −
$
#
∫A85
A8&
𝜉 𝑠 #. 𝑑𝑠)
¨ 𝔼&
ℙ
𝑌 𝑡 |𝔉 0 = 𝑌(0)
¨ So the process
¨ 𝑌 𝑡 = 𝑌 0 . exp(∫A85
A8&
𝜉 𝑠 . 𝑑𝑊 𝑠 −
$
#
∫A85
A8&
𝜉 𝑠 #. 𝑑𝑠)
¨ Is a martingale under the ℙ-measure associated with the Brownian motion 𝑊
58

59. Luc_Faucheux_2021
Quick side note - II
¨ 𝑌 𝑡 = 𝑌 0 . exp(∫A85
A8&
𝜉 𝑠 . 𝑑𝑊 𝑠 −
$
#
∫A85
A8&
𝜉 𝑠 #. 𝑑𝑠)
¨ Is a martingale under the ℙ-measure associated with the Brownian motion 𝑊
¨ So 𝑌 𝑡 is driftless and can be written (maybe) as the solution of an SDE that could look like:
¨ 𝑑𝑌 𝑡 = 0. 𝑑𝑡 + 𝑏 𝑌 𝑡 , 𝑡 . ([). 𝑑𝑊(𝑡)
¨ Let’s use ITO lemma on:
¨ 𝑌 𝑡 = 𝑌 0 . exp(∫A85
A8&
𝜉 𝑠 . 𝑑𝑊 𝑠 −
$
#
∫A85
A8&
𝜉 𝑠 #. 𝑑𝑠)
¨ 𝑋 𝑡 = ∫A85
A8&
𝜉 𝑠 . 𝑑𝑊 𝑠 −
$
#
∫A85
A8&
𝜉 𝑠 #. 𝑑𝑠
¨ We apply ITO lemma to 𝑌 𝑡 = 𝑌 0 . exp(𝑋 𝑡 )
59

60. Luc_Faucheux_2021
Quick side note - III
¨ Applying Ito lemma:
¨ 𝑓 𝑋 𝑡6 − 𝑓 𝑋 𝑡7 = ∫&8&7
&8&6 -9
-;
. ([). 𝑑𝑋(𝑡) + ∫&8&7
&8&6 $
#
.
-!9
-;! . ([). (𝛿𝑋)#
¨ In the ”limit” of small [me increments, this can be wrien formally as the Ito lemma:
¨ 𝛿𝑓 =
-9
-;
. ([). 𝛿𝑋 +
$
#
.
-!9
-;! . (𝛿𝑋)#
¨ For a function of the Brownian motion 𝑊(𝑡):
¨ 𝑓 𝑊 𝑡6 − 𝑓 𝑊 𝑡7 = ∫&8&7
&8&6 -9
-Y
. ([). 𝑑𝑊(𝑡) + ∫&8&7
&8&6 $
#
.
-!9
-Y! . ([). 𝑑𝑡
60

61. Luc_Faucheux_2021
Quick side note - IV
¨ 𝑓 𝑋 𝑡6 − 𝑓 𝑋 𝑡7 = ∫&8&7
&8&6 -9
-;
. ([). 𝑑𝑋(𝑡) + ∫&8&7
&8&6 $
#
.
-!9
-;! . ([). (𝑑𝑋)#
¨ 𝛿𝑓 =
-9
-;
. ([). 𝛿𝑋 +
$
#
.
-!9
-;! . (𝛿𝑋)#
¨ 𝑋 𝑡 = ∫A85
A8&
𝜉 𝑠 . 𝑑𝑊 𝑠 −
$
#
∫A85
A8&
𝜉 𝑠 #. 𝑑𝑠
¨ 𝑑𝑋 𝑡 = 𝜉 𝑡 . 𝑑𝑊 𝑡 −
$
#
𝜉 𝑡 #. 𝑑𝑡
¨ 𝑑𝑋 # 𝑡 = 𝜉 𝑡 #. 𝑑𝑡
¨ 𝑌 𝑡 = 𝑌 0 . exp(𝑋 𝑡 )
¨
-9
-;
= 𝑌 0 . exp(𝑋 𝑡 )
¨
-!9
-;! = 𝑌 0 . exp(𝑋 𝑡 )
61

62. Luc_Faucheux_2021
Quick side note - V
¨ 𝛿𝑓 =
-9
-;
. ([). 𝛿𝑋 +
$
#
.
-!9
-;! . (𝛿𝑋)#
¨ 𝑑𝑌(𝑡) = 𝑌(𝑡). ([). {𝜉 𝑡 . 𝑑𝑊 𝑡 −
$
#
𝜉 𝑡 #. 𝑑𝑡} +
$
#
. 𝑌 𝑡 . {𝜉 𝑡 #. 𝑑𝑡}
¨ 𝑑𝑌(𝑡) = 𝑌(𝑡). ([). {𝜉 𝑡 . 𝑑𝑊 𝑡 }
¨ 𝑑𝑌 𝑡 = 𝜉 𝑡 . 𝑌(𝑡). 𝑑𝑊 𝑡
¨ So we showed that the stochastic process:
¨ 𝑌 𝑡 = 𝑌 0 . exp(∫A85
A8&
𝜉 𝑠 . 𝑑𝑊 𝑠 −
$
#
∫A85
A8&
𝜉 𝑠 #. 𝑑𝑠)
¨ Is a solution (we leave to pure math people the rigorous work of showing unicity, stability,
well-behaved and all that good stuff)
¨ Is a solution of the SDE:
¨ 𝑑𝑌 𝑡 = 𝜉 𝑡 . 𝑌(𝑡). 𝑑𝑊 𝑡
62

63. Luc_Faucheux_2021
Quick side note - VI
¨ 𝑌 𝑡 = 𝑌 0 . exp(∫A85
A8&
𝜉 𝑠 . 𝑑𝑊 𝑠 −
$
#
∫A85
A8&
𝜉 𝑠 #. 𝑑𝑠)
¨ SDE: 𝑑𝑌 𝑡 = 𝜉 𝑡 . 𝑌(𝑡). 𝑑𝑊 𝑡
¨ SIE: 𝑌 𝑡6 − 𝑌 𝑡7 = ∫&8&7
&8&6
𝜉 𝑡 . 𝑌(𝑡). 𝑑𝑊 𝑡
¨ In the regular (Newtonian) calculus,
¨ 𝑑𝑌 𝑡 = 𝜉 𝑡 . 𝑌(𝑡). 𝑑𝑡
¨ Would yield:
¨ 𝑌 𝑡 = 𝑌 0 . exp(∫A85
A8&
𝜉 𝑠 . 𝑑𝑠)
¨ Which is the regular exponential function
63

64. Luc_Faucheux_2021
Quick side note - VII
¨ In the stochastic calculus (ITO), the solution of the SDE:
¨ 𝑑𝑌 𝑡 = 𝜉 𝑡 . 𝑌(𝑡). 𝑑𝑊 𝑡
¨ Is NOT the regular exponential that we are used to, but instead:
¨ 𝑌 𝑡 = 𝑌 0 . exp(∫A85
A8&
𝜉 𝑠 . 𝑑𝑊 𝑠 −
$
#
∫A85
A8&
𝜉 𝑠 #. 𝑑𝑠)
¨ Sometimes the above function is referred to the Doleans-Dade exponential in memory of
Catherine Doleans-Dade, and because is it so useful and used
¨ ℰ ∫5
&
𝜉 𝑠 . 𝑑𝑊 𝑠 = exp(∫5
&
𝜉 𝑠 . 𝑑𝑊(𝑠) − ∫5
& $
#
𝜉 𝑠 #. 𝑑𝑠)
¨ 𝑌 𝑡 = 𝑌 0 . exp(∫A85
A8&
𝜉 𝑠 . 𝑑𝑊 𝑠 −
$
#
∫A85
A8&
𝜉 𝑠 #. 𝑑𝑠)
¨ 𝑌 𝑡 = 𝑌 0 . ℰ ∫5
&
𝜉 𝑠 . 𝑑𝑊 𝑠
64

65. Luc_Faucheux_2021
Quick side note - VIII
¨ Note the formal analogy:
¨ REGULAR CALCULUS (Newtonian)
¨ 𝑑𝑌 𝑡 = 𝜉 𝑡 . 𝑌(𝑡). 𝑑𝑡
¨ 𝑌 𝑡 = 𝑌 0 . exp(∫A85
A8&
𝜉 𝑠 . 𝑑𝑠)
¨ STOCHASTIC CALCULUS (Brownian) in the ITO convention
¨ 𝑑𝑌 𝑡 = 𝜉 𝑡 . 𝑌 𝑡 . [ . 𝑑𝑊 𝑡
¨ 𝑌 𝑡 = 𝑌 0 . ℰ ∫5
&
𝜉 𝑠 . [ . 𝑑𝑊 𝑠
65

66. Luc_Faucheux_2021
Quick side note - IX
¨ The interesting thing is that:
¨ 𝑑𝑌 𝑡 = 𝜉 𝑡 . 𝑌 𝑡 . [ . 𝑑𝑊 𝑡
¨ Is driftless, and the solution of it is:
¨ 𝑌 𝑡 = 𝑌 0 . exp(∫A85
A8&
𝜉 𝑠 . [ . 𝑑𝑊 𝑠 −
$
#
∫A85
A8&
𝜉 𝑠 #. 𝑑𝑠)
¨ Such that it is a martingale:
¨ 𝔼&
ℙ 𝑌 𝑡 |𝔉 0 = 𝑌(0)
¨ That would be another way to recover the useful relationship, is to use the property that a
driftless process is a martingale.
¨ This is the end of this quick note, but I wanted to point out the nice connection between a
process that is driftless and the fact that it is a martingale, in the case where we can have an
explicit solution of the SDE
66

67. Luc_Faucheux_2021
Quick side note - X
¨ There is an awful lot of complicated math to prove the equivalence, but very roughly, if the
Novikov condition is respected:
¨ 𝔼&
ℙ exp(
$
#
∫A85
A8&
𝜉 𝑠 #. 𝑑𝑠)|𝔉 0 < ∞
¨ Then you have equivalence between driftless and martingale.
¨ Just like in Mario Kart, with the evil Wario, if the Novikov condition is not respected, then
the process becomes a wartingale
67

68. Luc_Faucheux_2021
Quick side note - XV
¨ In Finance you want to remove the drift (find the martingale)
¨ In Mario Kart, you want to control the drift especially around the corners
¨ I want to thank Gilles Franchini for pointing out how crucial the drift was in both situations
68

69. Luc_Faucheux_2021
Using the useful relationship
to re-derive Ho-Lee
𝔼 𝑒!
= 𝑒 𝔼{!}
. 𝑒
'[!]
&
69

70. Luc_Faucheux_2021
Re-deriving Ho-Lee using the useful relationship - I
¨ We put ourselves in the formalism where we have not yet calibrated the Ho-Lee model
¨ We start with:
¨ 𝑑𝑅 𝑡, 𝑡, 𝑡 = 𝜃 𝑡 . 𝑑𝑡 − 𝜎. ([). 𝑑𝑊(𝑡)
¨ 𝑅 𝑠, 𝑠, 𝑠 = 𝑅 0,0,0 + ∫Z85
Z8A
𝜃 𝑢 . 𝑑𝑢 − ∫Z85
Z8A
𝜎. ([). 𝑑𝑊(𝑢)
70

71. Luc_Faucheux_2021
Re-deriving Ho-Lee using the useful relationship - II
¨ We also have:
¨
[ 5,$$,&,&
2(5)
= 𝔼&
ℚ [ &,$$,&,&
2(&)
|𝔉(0) = 𝔼&
ℚ [ &,$$,&,&
]^_(∫89:
89;
a A,A,A .=A)
|𝔉(0)
¨
[ 5,$$,&,&
2(5)
=
[ 5,$$,&,&
$
= 𝑍𝐶 0,0, 𝑡 = 𝔼&
ℚ [ &,$$,&,&
]^_(∫89:
89;
a A,A,A .=A)
|𝔉(0)
¨ 𝑍𝐶 0,0, 𝑡 = 𝔼&
ℚ [ &,$$,&,&
]^_(∫89:
89;
a A,A,A .=A)
|𝔉(0) = 𝔼&
ℚ $
]^_(∫89:
89;
a A,A,A .=A)
|𝔉(0)
¨ 𝑍𝐶 0,0, 𝑡 = 𝔼&
ℚ
exp(− ∫A85
A8&
𝑅 𝑠, 𝑠, 𝑠 . 𝑑𝑠)|𝔉(0)
¨ 𝑑𝑅 𝑡, 𝑡, 𝑡 = 𝜃 𝑡 . 𝑑𝑡 − 𝜎. ([). 𝑑𝑊(𝑡)
¨ 𝑅 𝑠, 𝑠, 𝑠 = 𝑅 0,0,0 + ∫Z85
Z8A
𝜃 𝑢 . 𝑑𝑢 − ∫Z85
Z8A
𝜎. ([). 𝑑𝑊(𝑢)
71

72. Luc_Faucheux_2021
Re-deriving Ho-Lee using the useful relationship - III
¨ So the order is the following:
¨ 1) Solve the equation for the dynamics of the short rate.
¨ 2) integrate that solution 𝑅 𝑠, 𝑠, 𝑠 over the maturity of the zero coupon bond
¨ 3) Plug that integral ∫A85
A8&
𝑅 𝑠, 𝑠, 𝑠 . 𝑑𝑠 into: exp(− ∫A85
A8&
𝑅 𝑠, 𝑠, 𝑠 . 𝑑𝑠)
¨ 4) Finally evaluate the expected value: 𝑍𝐶 0,0, 𝑡 = 𝔼&
ℚ
exp(− ∫A85
A8&
𝑅 𝑠, 𝑠, 𝑠 . 𝑑𝑠)|𝔉(0)
¨ 𝑑𝑅 𝑡, 𝑡, 𝑡 = 𝜃 𝑡 . 𝑑𝑡 − 𝜎. ([). 𝑑𝑊(𝑡)
¨ Yields:
¨ 𝑅 𝑠, 𝑠, 𝑠 = 𝑅 0,0,0 + ∫Z85
Z8A
𝜃 𝑢 . 𝑑𝑢 − ∫Z85
Z8A
𝜎. ([). 𝑑𝑊(𝑢)
¨ Ok, we are done (formally) with step 1)
72

73. Luc_Faucheux_2021
Re-deriving Ho-Lee using the useful relationship - IV
¨ ∫A85
A8&
𝑅 𝑠, 𝑠, 𝑠 . 𝑑𝑠 = ∫A85
A8&
{𝑅 0,0,0 + ∫Z85
Z8A
𝜃 𝑢 . 𝑑𝑢 − ∫Z85
Z8A
𝜎. ([). 𝑑𝑊(𝑢)}. 𝑑𝑠
¨ ∫A85
A8&
𝑅 𝑠, 𝑠, 𝑠 . 𝑑𝑠 = ∫A85
A8&
𝑅 0,0,0 . 𝑑𝑠 + ∫A85
A8&
∫Z85
Z8A
𝜃 𝑢 . 𝑑𝑢 . 𝑑𝑠 − ∫A85
A8&
∫Z85
Z8A
𝜎. ([). 𝑑𝑊(𝑢) . 𝑑𝑠
¨ ∫A85
A8&
𝑅 𝑠, 𝑠, 𝑠 . 𝑑𝑠 = 𝑅 0,0,0 . 𝑡 + ∫A85
A8&
∫Z85
Z8A
𝜃 𝑢 . 𝑑𝑢 . 𝑑𝑠 − 𝜎. ∫A85
A8&
∫Z85
Z8A
1. ([). 𝑑𝑊(𝑢) . 𝑑𝑠
¨ In the third term we use our friend Fubini like we did in the first part:
73

74. Luc_Faucheux_2021
Re-deriving Ho-Lee using the useful relationship - V
¨ 𝑋 = ∫A85
A8&
{∫Z85
Z8A
𝜎. ([). 𝑑𝑊(𝑢)}. 𝑑𝑠
74
s
s=t
u
s
s=t
u

75. Luc_Faucheux_2021
Re-deriving Ho-Lee using the useful relationship - VI
¨ 𝑋 = ∫A85
A8&
{∫Z85
Z8A
𝜎. ([). 𝑑𝑊(𝑢)}. 𝑑𝑠
75
s
s = t
u
s
s = t
u
𝑋 = n
A85
A8&
𝑑𝑠 n
Z85
Z8A
𝜎. ([). 𝑑𝑊(𝑢) 𝑋 = n
Z85
Z8&
𝑑𝑊(𝑢) n
A8Z
A8&
𝜎. 𝑑𝑠

76. Luc_Faucheux_2021
Re-deriving Ho-Lee using the useful relationship - VII
¨ 𝑋 = 𝜎. ∫A85
A8&
∫Z85
Z8A
1. ([). 𝑑𝑊(𝑢) . 𝑑𝑠 = 𝜎. ∫Z85
Z8&
𝑑𝑊(𝑢) ∫A8Z
A8&
1. 𝑑𝑠
¨ 𝑋 = 𝜎. ∫Z85
Z8&
𝑑𝑊(𝑢) ∫A8Z
A8&
1. 𝑑𝑠 = 𝜎. ∫Z85
Z8&
𝑑𝑊 𝑢 . (𝑡 − 𝑢)
¨ 𝑋 = 𝜎. ∫Z85
Z8&
𝑑𝑊 𝑢 . (𝑡 − 𝑢) = 𝜎. ∫Z85
Z8&
𝑡 − 𝑢 . [ . 𝑑𝑊 𝑢
¨ ∫A85
A8&
𝑅 𝑠, 𝑠, 𝑠 . 𝑑𝑠 = 𝑅 0,0,0 . 𝑡 + ∫A85
A8&
∫Z85
Z8A
𝜃 𝑢 . 𝑑𝑢 . 𝑑𝑠 − 𝜎. ∫A85
A8&
∫Z85
Z8A
1. ([). 𝑑𝑊(𝑢) . 𝑑𝑠
¨ ∫A85
A8&
𝑅 𝑠, 𝑠, 𝑠 . 𝑑𝑠 = 𝑅 0,0,0 . 𝑡 + ∫A85
A8&
∫Z85
Z8A
𝜃 𝑢 . 𝑑𝑢 . 𝑑𝑠 − 𝜎. ∫Z85
Z8&
𝑡 − 𝑢 . [ . 𝑑𝑊 𝑢
76

77. Luc_Faucheux_2021
Re-deriving Ho-Lee using the useful relationship - VIII
¨ We now make use of the useful relationship to derive the mean and the variance of the
quantity: 𝑋 𝑡 = ∫A85
A8&
𝑅 𝑠, 𝑠, 𝑠 . 𝑑𝑠
¨ Note that in the first part we did it explicitly, this is a little more general as a derivation
¨ Since we are after:
¨ 𝑍𝐶 0,0, 𝑡 = 𝔼&
ℚ
exp(− ∫A85
A8&
𝑅 𝑠, 𝑠, 𝑠 . 𝑑𝑠)|𝔉(0)
¨ And we know that:
¨ 𝔼 exp 𝑋 = exp 𝔼 𝑋 . exp
$
#
𝔼 (𝑋(𝑡)− < 𝑋 >&)#
¨ 𝔼 exp 𝑋 = exp 𝑀[𝑋(𝑡)] . exp
$
#
𝑉[𝑋(𝑡)]
¨ 𝔼 exp 𝑋 = exp[𝑀] . exp
$
#
𝑉
77

78. Luc_Faucheux_2021
Re-deriving Ho-Lee using the useful relationship - IX
¨ 𝑋 𝑡 = ∫A85
A8&
𝑅 𝑠, 𝑠, 𝑠 . 𝑑𝑠
¨ 𝑋(𝑡) = 𝑅 0,0,0 . 𝑡 + ∫A85
A8&
∫Z85
Z8A
𝜃 𝑢 . 𝑑𝑢 . 𝑑𝑠 − 𝜎. ∫Z85
Z8&
𝑡 − 𝑢 . [ . 𝑑𝑊 𝑢
¨ 𝔼 exp 𝑋 = exp 𝑀[𝑋(𝑡)] . exp
$
#
𝑉[𝑋(𝑡)]
¨ Let’s first look at the mean (expected value of 𝑋 𝑡 )
¨ 𝑀 𝑋 𝑡 = 𝔼 𝑋(𝑡) = 𝔼&
ℚ
𝑋(𝑡)|𝔉(0) to be fully explicit
¨ Note that we are operating in the risk neutral measure that is associated with our Brownian
motion 𝑊(𝑢)
78

79. Luc_Faucheux_2021
Re-deriving Ho-Lee using the useful relationship - X
¨ 𝑋(𝑡) = 𝑅 0,0,0 . 𝑡 + ∫A85
A8&
∫Z85
Z8A
𝜃 𝑢 . 𝑑𝑢 . 𝑑𝑠 − 𝜎. ∫Z85
Z8&
𝑡 − 𝑢 . [ . 𝑑𝑊 𝑢
¨ The first two terms are deterministic:
¨ 𝔼 𝑅 0,0,0 . 𝑡 = 𝔼5
ℚ
𝑅 0,0,0 . 𝑡|𝔉(0) = 𝑅 0,0,0 . 𝑡
¨ 𝔼 ∫A85
A8&
∫Z85
Z8A
𝜃 𝑢 . 𝑑𝑢 . 𝑑𝑠 = 𝔼&
ℚ
∫A85
A8&
∫Z85
Z8A
𝜃 𝑢 . 𝑑𝑢 . 𝑑𝑠 |𝔉(0) = ∫A85
A8&
∫Z85
Z8A
𝜃 𝑢 . 𝑑𝑢 . 𝑑𝑠
¨ The third term is stochastic:
¨ 𝔼 𝜎. ∫Z85
Z8&
𝑡 − 𝑢 . [ . 𝑑𝑊 𝑢 = 𝔼5
ℚ
𝜎. ∫Z85
Z8&
𝑡 − 𝑢 . [ . 𝑑𝑊 𝑢 |𝔉(0)
¨ Here we use the fact that for any trading strategy (function of time only), the ITO integral is a
martingale:
¨ 𝔼&
ℚ
∫Z85
Z8&
𝑓(𝑢). [ . 𝑑𝑊 𝑢 |𝔉(0) = 0 for any function 𝑓(𝑢)
79

80. Luc_Faucheux_2021
Re-deriving Ho-Lee using the useful relationship - XI
¨ Here we use the fact that for any trading strategy (function of time only), the ITO integral is a
martingale:
¨ 𝔼&
ℚ
∫Z85
Z8&
𝑓(𝑢). [ . 𝑑𝑊 𝑢 |𝔉(0) = 0 for any function 𝑓(𝑢)
¨ Let’s make sure that we are firmly convinced of that fact.
¨ We saw that in the stochastic calculus deck, but always worth looking at it again.
¨ As always, replace the integral by a limit of a sum (with the proper convention, LHS for ITO,
Middle for Strato,..) so that you can switch the Expectation operator and the sum operator
¨ ∫Z85
Z8&
𝑓(𝑢). [ . 𝑑𝑊 𝑢 = lim
G→E
∑!85
!8G
𝑓 𝑠! . {𝑊 𝑠!F$ − 𝑊(𝑠!)}
¨ Let’s note 𝔼&
ℚ
∫Z85
Z8&
𝑓(𝑢). [ . 𝑑𝑊 𝑢 |𝔉(0) by 𝔼 ∫Z85
Z8&
𝑓(𝑢). [ . 𝑑𝑊 𝑢 for sake of
simplicity of notation
80

81. Luc_Faucheux_2021
Re-deriving Ho-Lee using the useful relationship - XII
¨ ∫Z85
Z8&
𝑓(𝑢). [ . 𝑑𝑊 𝑢 = lim
G→E
∑!85
!8G
𝑓 𝑠! . {𝑊 𝑠!F$ − 𝑊(𝑠!)}
¨ 𝔼 ∫Z85
Z8&
𝑓(𝑢). [ . 𝑑𝑊 𝑢 = 𝔼 lim
G→E
∑!85
!8G
𝑓 𝑠! . {𝑊 𝑠!F$ − 𝑊(𝑠!)}
¨ 𝔼 ∫Z85
Z8&
𝑓(𝑢). [ . 𝑑𝑊 𝑢 = lim
G→E
∑!85
!8G
𝔼{𝑓 𝑠! . {𝑊 𝑠!F$ − 𝑊(𝑠!)}}
¨ 𝔼 ∫Z85
Z8&
𝑓(𝑢). [ . 𝑑𝑊 𝑢 = lim
G→E
∑!85
!8G
𝑓 𝑠! . 𝔼{{𝑊 𝑠!F$ − 𝑊(𝑠!)}}
¨ 𝔼 ∫Z85
Z8&
𝑓(𝑢). [ . 𝑑𝑊 𝑢 = lim
G→E
∑!85
!8G
𝑓 𝑠! . 0
¨ 𝔼 ∫Z85
Z8&
𝑓(𝑢). [ . 𝑑𝑊 𝑢 = 0
¨ Note that this would ALSO be true in the Stratonovitch calculus, because it is a function of
time only
¨ ∫Z85
Z8&
𝑓(𝑢). ∘ . 𝑑𝑊 𝑢 = lim
G→E
∑!85
!8G
𝑓
A"FA"<3
#
. {𝑊 𝑠!F$ − 𝑊(𝑠!)}
81

82. Luc_Faucheux_2021
Re-deriving Ho-Lee using the useful relationship - XIII
¨ 𝔼 ∫Z85
Z8&
𝑓(𝑢). [ . 𝑑𝑊 𝑢 = 0
¨ 𝔼 ∫Z85
Z8&
𝑓(𝑢). ∘ . 𝑑𝑊 𝑢 = 0
¨ This would be different for a function of the stochastic driver (also a self replicating strategy)
¨ ∫Z8&7
Z8&6
𝑓 𝑊(𝑢) . (∘). 𝑑𝑊(𝑢) = ∫Z8&7
Z8&6
𝑓 𝑊(𝑢) . ([). 𝑑𝑊(𝑢) +
$
#
∫Z8&7
Z8&6
𝑓′ 𝑊(𝑢) . 𝑑𝑢
¨ 𝔼 ∫Z85
Z8&
𝑓 𝑊 𝑢 . ([). 𝑑𝑊(𝑢) = 0
¨ 𝔼 ∫Z85
Z8&
𝑓 𝑊(𝑢) . (∘). 𝑑𝑊(𝑢) ≠ 0
¨ 𝔼 ∫Z85
Z8&
𝑓 𝑊(𝑢) . (∘). 𝑑𝑊(𝑢) = 𝔼
$
#
∫Z85
Z8&
𝑓′ 𝑊(𝑢) . 𝑑𝑢
82

83. Luc_Faucheux_2021
Re-deriving Ho-Lee using the useful relationship - XIV
¨ 𝔼 ∫Z85
Z8&
𝑓 𝑊 𝑢 . ([). 𝑑𝑊(𝑢) = 0
¨ 𝔼 ∫Z85
Z8&
𝑓 𝑊(𝑢) . (∘). 𝑑𝑊(𝑢) ≠ 0
¨ 𝔼 ∫Z85
Z8&
𝑓 𝑊(𝑢) . (∘). 𝑑𝑊(𝑢) = 𝔼
$
#
∫Z85
Z8&
𝑓′ 𝑊(𝑢) . 𝑑𝑢
¨ Suppose that we take the simple function 𝑓 𝑤 = 𝑤
¨ Remember we try to stick to the notation where we take lower case for regular calculus
variable and upper case for stochastic variable
¨ We just write 𝑓 𝑊(𝑢) and 𝑓′ 𝑊(𝑢) for sake of simplicity
¨ But following Baxter, we should really write more precisely:
¨ 𝑓 𝑤 = 𝑊(𝑢) and
-9(Y)
-Y
|Y81(Z)
83

84. Luc_Faucheux_2021
Re-deriving Ho-Lee using the useful relationship - XV
¨ 𝑓 𝑤 = 𝑤 𝑓 𝑤 |Y81(Z) = 𝑊(𝑢)
¨ 𝑓′ 𝑤 = 1
-9(Y)
-Y
|Y81(Z) = 1
¨ 𝔼 ∫Z85
Z8&
𝑊(𝑢). ([). 𝑑𝑊(𝑢) = 0
¨ 𝔼 ∫Z85
Z8&
𝑊(𝑢). (∘). 𝑑𝑊(𝑢) ≠ 0
¨ 𝔼 ∫Z85
Z8&
𝑊(𝑢). (∘). 𝑑𝑊(𝑢) = 𝔼
$
#
∫Z85
Z8&
1. 𝑑𝑢 =
$
#
𝑡
84

85. Luc_Faucheux_2021
Re-deriving Ho-Lee using the useful relationship - XVI
¨ If you recall what we had from the stochastic calculus deck:
¨ Can you integrate l’𝑋 ?
85

86. Luc_Faucheux_2021
Re-deriving Ho-Lee using the useful relationship - XVII
¨ We had derived then
¨ Within the ITO convention
¨ ∫&85
&84
𝑋. ([). 𝑑𝑋 =
:(4)!
#
−
$
#
∫&85
&84
1. 𝑑𝑡 or ∫&85
&84
𝑋. 𝑑𝑋 =
:(4)!
#
−
$
#
𝑇
¨ Within the STRATANOVITCH convention
¨ ∫&85
&84
𝑋. ∘ . 𝑑𝑋 = ∫&85
&84
𝑋. ([). 𝑑𝑋 +
$
#
𝑇 =
:(4)!
#
−
$
#
𝑇 +
$
#
𝑇 =
:(4)!
#
¨ STRATANOVITCH as expected follows in a formal manner the usual rules of calculus
¨ With our current notations
¨ ∫Z85
Z8&
𝑊(𝑢). ([). 𝑑𝑊(𝑢) =
1(&)!
#
−
$
#
𝑡
¨ ∫Z85
Z8&
𝑊(𝑢). (∘). 𝑑𝑊(𝑢) =
1(&)!
#
86

87. Luc_Faucheux_2021
Re-deriving Ho-Lee using the useful relationship - XVIII
¨ ∫Z85
Z8&
𝑊(𝑢). ([). 𝑑𝑊(𝑢) =
1(&)!
#
−
$
#
𝑡
¨ ∫Z85
Z8&
𝑊(𝑢). (∘). 𝑑𝑊(𝑢) =
1(&)!
#
¨ So now taking the expectations and knowing that for the Brownian motion we have the
usual expectation:
¨ 𝔼 𝑊(𝑡) = 0
¨ 𝔼 𝑊(𝑡)# = 𝑡
¨ We then recover in a very consistent manner:
¨ 𝔼 ∫Z85
Z8&
𝑊(𝑢). ([). 𝑑𝑊(𝑢) = 𝔼
1(&)!
#
−
$
#
𝑡 = 𝔼
1(&)!
#
−
$
#
𝑡 =
$
#
𝑡 −
$
#
𝑡 = 0
¨ 𝔼 ∫Z85
Z8&
𝑊(𝑢). (∘). 𝑑𝑊(𝑢) = 𝔼
1(&)!
#
=
$
#
𝑡
87

88. Luc_Faucheux_2021
Re-deriving Ho-Lee using the useful relationship - XIX
¨ This is sometimes a very useful trick that you can use
¨ The ITO integral is a martingale for the measure associated with the Brownian motion
¨ The expectation is then 0
¨ 𝔼 ∫Z85
Z8&
[𝑆𝑜𝑚𝑒𝑡ℎ𝑖𝑛𝑔 𝑜𝑓 𝑊(𝑢)]. ([). 𝑑𝑊(𝑢)
¨ If the [𝑆𝑜𝑚𝑒𝑡ℎ𝑖𝑛𝑔 𝑜𝑓 𝑊(𝑢)] is a function 𝑓 𝑊(𝑢) with a well behaved first derivative, let’s
call it by the notation
-9(Y)
-Y
|Y81(Z) = 𝑓′(𝑊 𝑢 )
¨ You know the relationship between the ITO and STRATANOVITCH integral:
¨ ∫Z85
Z8&
𝑓 𝑊(𝑢) . (∘). 𝑑𝑊(𝑢) = ∫Z85
Z8&
𝑓 𝑊(𝑢) . ([). 𝑑𝑊(𝑢) +
$
#
∫Z85
Z8&
𝑓′ 𝑊(𝑢) . 𝑑𝑢
¨ You can take the expectations on both sides
¨ 𝔼 ∫=>?
=>'
𝑓 𝑊(𝑢) . (∘). 𝑑𝑊(𝑢) = 𝔼 ∫=>?
=>'
𝑓 𝑊(𝑢) . ([). 𝑑𝑊(𝑢) + 𝔼
.
-
∫=>?
=>'
𝑓′ 𝑊(𝑢) . 𝑑𝑢
88

89. Luc_Faucheux_2021
Re-deriving Ho-Lee using the useful relationship - XX
¨ 𝔼 ∫=>?
=>'
𝑓 𝑊(𝑢) . (∘). 𝑑𝑊(𝑢) = 𝔼 ∫=>?
=>'
𝑓 𝑊(𝑢) . ([). 𝑑𝑊(𝑢) + 𝔼
.
-
∫=>?
=>'
𝑓′ 𝑊(𝑢) . 𝑑𝑢
¨ 𝔼 ∫Z85
Z8&
𝑓 𝑊(𝑢) . (∘). 𝑑𝑊(𝑢) = 0 + 𝔼
$
#
∫Z85
Z8&
𝑓′ 𝑊(𝑢) . 𝑑𝑢
¨ 𝔼 ∫Z85
Z8&
𝑓 𝑊(𝑢) . (∘). 𝑑𝑊(𝑢) = 𝔼
$
#
∫Z85
Z8&
𝑓′ 𝑊(𝑢) . 𝑑𝑢
¨ And you are left with somewhat easier expressions to deal with
¨ IN PARTICULAR, you can rely on the fact that you can use the regular rules of ”Newtonian”
calculus (remember, only in a formal manner) within the Startanovitch calculus
¨ So that makes computing ∫Z85
Z8&
𝑓 𝑊(𝑢) . (∘). 𝑑𝑊(𝑢) easier
¨ Which makes it easier to also computes:
¨ ∫Z85
Z8&
𝑓 𝑊(𝑢) . ([). 𝑑𝑊(𝑢) = ∫Z85
Z8&
𝑓 𝑊(𝑢) . (∘). 𝑑𝑊(𝑢) −
$
#
∫Z85
Z8&
𝑓′ 𝑊(𝑢) . 𝑑𝑢
89

90. Luc_Faucheux_2021
Re-deriving Ho-Lee using the useful relationship - XXI
¨ OK, back to the problem at hand here:
¨ 𝑑𝑅 𝑡, 𝑡, 𝑡 = 𝜃 𝑡 . 𝑑𝑡 − 𝜎. ([). 𝑑𝑊(𝑡)
¨ 𝑅 𝑡, 𝑡, 𝑡 = 𝑅 0,0,0 + ∫Z85
Z8&
𝜃 𝑢 . 𝑑𝑢 − ∫Z85
Z8&
𝜎. ([). 𝑑𝑊(𝑢)
¨ 𝑀 𝑅 𝑡, 𝑡, 𝑡 = 𝔼 𝑅 𝑡, 𝑡, 𝑡 = 𝔼&
ℚ
𝑅 𝑡, 𝑡, 𝑡 |𝔉(0) to be fully explicit
¨ 𝔼 𝑅 𝑡, 𝑡, 𝑡 = 𝔼 𝑅 0,0,0 + ∫Z85
Z8&
𝜃 𝑢 . 𝑑𝑢 − ∫Z85
Z8&
𝜎. ([). 𝑑𝑊(𝑢)
¨ 𝔼 𝑅 𝑡, 𝑡, 𝑡 = 𝑅 0,0,0 + ∫Z85
Z8&
𝜃 𝑢 . 𝑑𝑢 − 𝔼 ∫Z85
Z8&
𝜎. ([). 𝑑𝑊(𝑢)
¨ You can use the fact that the ITO integral is a martingale or explicitly write
¨ ∫Z85
Z8&
𝜎. ([). 𝑑𝑊(𝑢) = 𝜎. 𝑊(𝑡)
90

91. Luc_Faucheux_2021
Re-deriving Ho-Lee using the useful relationship - XXII
¨ 𝔼 ∫Z85
Z8&
𝜎. ([). 𝑑𝑊(𝑢) = 0 because it is a ITO integral of a self-financing trading strategy
¨ Or:
¨ 𝔼 ∫Z85
Z8&
𝜎. ([). 𝑑𝑊(𝑢) = 𝔼 𝜎. 𝑊(𝑡) = 0
¨ In the specific case of the Brownian motion
¨ 𝔼 𝑅 𝑡, 𝑡, 𝑡 = 𝑀[𝑅 𝑡, 𝑡, 𝑡 ] = 𝑅 0,0,0 + ∫Z85
Z8&
𝜃 𝑢 . 𝑑𝑢
¨ For the variance:
¨ 𝔼 (𝑅(𝑡)− < 𝑅 >&)# = 𝑉[𝑅(𝑡)] for the Variance
¨ 𝑉 𝑅 𝑡, 𝑡, 𝑡 = 𝔼 (𝑅(𝑡, 𝑡, 𝑡) − 𝑀[𝑅 𝑡, 𝑡, 𝑡 ])#
91

92. Luc_Faucheux_2021
Re-deriving Ho-Lee using the useful relationship - XXIII
¨ 𝑉 𝑅 𝑡, 𝑡, 𝑡 = 𝔼 (𝑅(𝑡, 𝑡, 𝑡) − 𝑀[𝑅 𝑡, 𝑡, 𝑡 ])#
¨ 𝑅 𝑡, 𝑡, 𝑡 = 𝑅 0,0,0 + ∫Z85
Z8&
𝜃 𝑢 . 𝑑𝑢 − ∫Z85
Z8&
𝜎. ([). 𝑑𝑊(𝑢)
¨ 𝔼 𝑅 𝑡, 𝑡, 𝑡 = 𝑀[𝑅 𝑡, 𝑡, 𝑡 ] = 𝑅 0,0,0 + ∫Z85
Z8&
𝜃 𝑢 . 𝑑𝑢
¨ 𝑅 𝑡, 𝑡, 𝑡 − 𝑀 𝑅 𝑡, 𝑡, 𝑡 = − ∫Z85
Z8&
𝜎. ([). 𝑑𝑊(𝑢)
¨ (𝑅(𝑡, 𝑡, 𝑡) − 𝑀[𝑅 𝑡, 𝑡, 𝑡 ])#= (∫Z85
Z8&
𝜎. ([). 𝑑𝑊(𝑢))#
¨ (𝑅(𝑡, 𝑡, 𝑡) − 𝑀[𝑅 𝑡, 𝑡, 𝑡 ])#= (𝜎. 𝑊(𝑡))#
¨ 𝔼 (𝑅(𝑡, 𝑡, 𝑡) − 𝑀[𝑅 𝑡, 𝑡, 𝑡 ])# = 𝔼 (𝜎. 𝑊(𝑡))# = 𝜎#. 𝔼 (𝑊(𝑡))# = 𝜎# 𝑡
92

93. Luc_Faucheux_2021
Re-deriving Ho-Lee using the useful relationship - XXIX
¨ 𝑑𝑅 𝑡, 𝑡, 𝑡 = 𝜃 𝑡 . 𝑑𝑡 − 𝜎. ([). 𝑑𝑊(𝑡)
¨ 𝑅 𝑡, 𝑡, 𝑡 = 𝑅 0,0,0 + ∫Z85
Z8&
𝜃 𝑢 . 𝑑𝑢 − ∫Z85
Z8&
𝜎. ([). 𝑑𝑊(𝑢)
¨ 𝑀 𝑅 𝑡, 𝑡, 𝑡 = 𝔼 𝑅 𝑡, 𝑡, 𝑡 = 𝑅 0,0,0 + ∫Z85
Z8&
𝜃 𝑢 . 𝑑𝑢
¨ 𝑉 𝑅 𝑡, 𝑡, 𝑡 = 𝔼 (𝑅(𝑡, 𝑡, 𝑡) − 𝑀[𝑅 𝑡, 𝑡, 𝑡 ])# = 𝜎# 𝑡
¨ So far nothing special
¨ We then concerned ourselves with the new variable:
¨ 𝑋 𝑡 = ∫A85
A8&
𝑅 𝑠, 𝑠, 𝑠 . 𝑑𝑠
¨ 𝑋(𝑡) = 𝑅 0,0,0 . 𝑡 + ∫A85
A8&
∫Z85
Z8A
𝜃 𝑢 . 𝑑𝑢 . 𝑑𝑠 − 𝜎. ∫Z85
Z8&
𝑡 − 𝑢 . [ . 𝑑𝑊 𝑢
¨
93

94. Luc_Faucheux_2021
Re-deriving Ho-Lee using the useful relationship - XXX
¨ We had so far got the mean:
¨ 𝑋(𝑡) = 𝑅 0,0,0 . 𝑡 + ∫A85
A8&
∫Z85
Z8A
𝜃 𝑢 . 𝑑𝑢 . 𝑑𝑠 − 𝜎. ∫Z85
Z8&
𝑡 − 𝑢 . [ . 𝑑𝑊 𝑢
¨ The first two terms are deterministic:
¨ 𝔼 𝑅 0,0,0 . 𝑡 = 𝔼5
ℚ
𝑅 0,0,0 . 𝑡|𝔉(0) = 𝑅 0,0,0 . 𝑡
¨ 𝔼 ∫A85
A8&
∫Z85
Z8A
𝜃 𝑢 . 𝑑𝑢 . 𝑑𝑠 = 𝔼5
ℚ
∫A85
A8&
∫Z85
Z8A
𝜃 𝑢 . 𝑑𝑢 . 𝑑𝑠 |𝔉(0) = ∫A85
A8&
∫Z85
Z8A
𝜃 𝑢 . 𝑑𝑢 . 𝑑𝑠
¨ The third term is stochastic:
¨ 𝔼 𝜎. ∫Z85
Z8&
𝑡 − 𝑢 . [ . 𝑑𝑊 𝑢 = 𝔼5
ℚ
𝜎. ∫Z85
Z8&
𝑡 − 𝑢 . [ . 𝑑𝑊 𝑢 |𝔉(0)
¨ Here we use the fact that for any trading strategy (function of time only), the ITO integral is a
martingale:
¨ 𝔼&
ℚ
∫Z85
Z8&
𝑓(𝑢). [ . 𝑑𝑊 𝑢 |𝔉(0) = 0 for any function 𝑓(𝑢)
94

95. Luc_Faucheux_2021
Re-deriving Ho-Lee using the useful relationship - XXXI
¨ 𝑀 𝑋 𝑡 = 𝔼 𝑋 𝑡
¨ 𝑀 𝑋 𝑡 = 𝔼 𝑅 0,0,0 . 𝑡 + ∫A85
A8&
∫Z85
Z8A
𝜃 𝑢 . 𝑑𝑢 . 𝑑𝑠 − 𝜎. ∫Z85
Z8&
𝑡 − 𝑢 . [ . 𝑑𝑊 𝑢
¨ 𝑀 𝑋 𝑡 = 𝑅 0,0,0 . 𝑡 + ∫A85
A8&
∫Z85
Z8A
𝜃 𝑢 . 𝑑𝑢 . 𝑑𝑠 − 𝔼 𝜎. ∫Z85
Z8&
𝑡 − 𝑢 . [ . 𝑑𝑊 𝑢
¨ 𝑀 𝑋 𝑡 = 𝑅 0,0,0 . 𝑡 + ∫A85
A8&
∫Z85
Z8A
𝜃 𝑢 . 𝑑𝑢 . 𝑑𝑠
¨ OK, almost there, now we need to look at the variance of 𝑋 𝑡
¨ Because remember we are doing all of this because we are ultimately interested in the
expectation of 𝑒:using the useful relationship 𝔼 𝑒: = 𝑒 𝔼{:}. 𝑒
7[A]
! = 𝑒U[:]. 𝑒
7[A]
!
95

96. Luc_Faucheux_2021
Re-deriving Ho-Lee using the useful relationship - XXXII
¨ 𝑉 𝑋(𝑡) = 𝔼 (𝑋(𝑡) − 𝑀[𝑋(𝑡)])#
¨ 𝑋(𝑡) = 𝑅 0,0,0 . 𝑡 + ∫A85
A8&
∫Z85
Z8A
𝜃 𝑢 . 𝑑𝑢 . 𝑑𝑠 − 𝜎. ∫Z85
Z8&
𝑡 − 𝑢 . [ . 𝑑𝑊 𝑢
¨ 𝑋(𝑡) = 𝑀[𝑋(𝑡)] − 𝜎. ∫Z85
Z8&
𝑡 − 𝑢 . [ . 𝑑𝑊 𝑢
¨ 𝑋 𝑡 − 𝑀 𝑋 𝑡 = −𝜎. ∫Z85
Z8&
𝑡 − 𝑢 . [ . 𝑑𝑊 𝑢
¨ 𝑉 𝑋(𝑡) = 𝔼 (𝑋(𝑡) − 𝑀[𝑋(𝑡)])#
¨ 𝑉 𝑋(𝑡) = 𝔼 (𝜎. ∫Z85
Z8&
𝑡 − 𝑢 . [ . 𝑑𝑊 𝑢 )#
¨ So we can be flashy and use directly the isometry property of the ITO integral:
¨ (∫Z85
Z8&
𝑡 − 𝑢 . [ . 𝑑𝑊 𝑢 )#= ∫Z85
Z8&
𝑡 − 𝑢 #. 𝑑𝑢 = [−
&)Z 4
O
]Z85
Z8& =
&4
O
96

97. Luc_Faucheux_2021
Re-deriving Ho-Lee using the useful relationship - XXXIII
¨ Or we can be more pedestrian and recheck once again our hopefully firmly grounded
understanding of ITO integrals by going back once again to the definition of the ITO integral
as a limit of a sum, using the LHS (Left Hand Side) convention for where the function to be
integrated is evaluated
¨ ∫Z85
Z8&
𝑓 𝑢 . ([). 𝑑𝑊 𝑢 = lim
G→E
∑!85
!8G
𝑓 𝑠! . {𝑊 𝑠!F$ − 𝑊(𝑠!)}
¨ We will leave to the pure math guys the job of coming up with all the pathological cases
where a regular well behaved mesh does not work for regular well behaved functions
¨ 𝑉 𝑋(𝑡) = 𝔼 (𝜎. ∫Z85
Z8&
𝑡 − 𝑢 . [ . 𝑑𝑊 𝑢 )#
¨ 𝑉 𝑋(𝑡) = 𝜎#. 𝔼 (∫Z85
Z8&
𝑓(𝑢). [ . 𝑑𝑊 𝑢 )# with 𝑓 𝑢 = (𝑡 − 𝑢)
97

98. Luc_Faucheux_2021
Re-deriving Ho-Lee using the useful relationship - XXXIV
¨ 𝑉 𝑋(𝑡) = 𝜎#. 𝔼 (∫Z85
Z8&
𝑓(𝑢). [ . 𝑑𝑊 𝑢 )# with 𝑓 𝑢 = (𝑡 − 𝑢)
¨ [∫Z85
Z8&
𝑓 𝑢 . [ . 𝑑𝑊 𝑢 ]#= [lim
G→E
∑!85
!8G
𝑓 𝑢! . {𝑊 𝑢!F$ − 𝑊(𝑢!)} ]#
¨ [∫=>?
=>'
𝑓 𝑢 . [ . 𝑑𝑊 𝑢 ]-
= lim
C→E
∑$>?
$>C
𝑓 𝑢$ . 𝑊 𝑢$F. − 𝑊 𝑢$ . [lim
G→E
∑%>?
%>G
𝑓 𝑢% . {𝑊 𝑢%F. − 𝑊(𝑢%)} ]
¨ [∫=>?
=>'
𝑓 𝑢 . [ . 𝑑𝑊 𝑢 ]-
= lim
G→E
∑%>?
%>G
lim
C→E
∑$>?
$>C
𝑓 𝑢$ . 𝑓 𝑢% . 𝑊 𝑢$F. − 𝑊 𝑢$ . {𝑊 𝑢%F. − 𝑊(𝑢%)}
¨ 𝔼 𝑊 𝑢!F$ − 𝑊 𝑢! = 0
¨ 𝔼 𝑊 𝑢!F$ − 𝑊 𝑢! . [𝑊 𝑢(F$ − 𝑊(𝑢()] = 𝛿!,(. [𝑢(F$ − 𝑢(]
¨ Where we are using the usual Kronecker notation:
¨ 𝛿!,( = 1 if 𝑖 = 𝑗
¨ 𝛿!,( = 0 if 𝑖 ≠ 𝑗
98

99. Luc_Faucheux_2021
Re-deriving Ho-Lee using the useful relationship - XXXV
¨ 𝔼 𝑊 𝑢!F$ − 𝑊 𝑢! . [𝑊 𝑢(F$ − 𝑊(𝑢()] = 𝛿!,(. [𝑢(F$ − 𝑢(]
¨ [∫=>?
=>'
𝑓 𝑢 . [ . 𝑑𝑊 𝑢 ]-
= lim
G→E
∑%>?
%>G
lim
C→E
∑$>?
$>C
𝑓 𝑢$ . 𝑓 𝑢% . 𝑊 𝑢$F. − 𝑊 𝑢$ . {𝑊 𝑢%F. − 𝑊(𝑢%)}
¨ 𝔼{ ∫!"#
!"$
𝑓 𝑢 . [ . 𝑑𝑊 𝑢
%
} = 𝔼{ lim
&→(
∑)"#
)"&
lim
*→(
∑+"#
+"*
𝑓 𝑢+ . 𝑓 𝑢) . 𝑊 𝑢+,- − 𝑊 𝑢+ . {𝑊 𝑢),- − 𝑊(𝑢))} }
¨ 𝔼{ ∫!"#
!"$
𝑓 𝑢 . [ . 𝑑𝑊 𝑢
%
} = lim
&→(
∑)"#
)"&
lim
*→(
∑+"#
+"*
𝔼{𝑓 𝑢+ . 𝑓 𝑢) . 𝑊 𝑢+,- − 𝑊 𝑢+ . {𝑊 𝑢),- − 𝑊(𝑢))}}
¨ 𝔼{ ∫!"#
!"$
𝑓 𝑢 . [ . 𝑑𝑊 𝑢
%
} = lim
&→(
∑)"#
)"&
lim
*→(
∑+"#
+"*
𝑓 𝑢+ . 𝑓 𝑢) . 𝔼{ 𝑊 𝑢+,- − 𝑊 𝑢+ . {𝑊 𝑢),- − 𝑊(𝑢))}}
¨ 𝔼{ ∫!"#
!"$
𝑓 𝑢 . [ . 𝑑𝑊 𝑢
%
} = lim
&→(
∑)"#
)"&
lim
*→(
∑+"#
+"*
𝑓 𝑢+ . 𝑓 𝑢) . 𝛿+,). [𝑢),- − 𝑢)]
¨ 𝔼{ ∫=>?
=>'
𝑓 𝑢 . [ . 𝑑𝑊 𝑢
-
} = lim
G→E
∑%>?
%>G
𝑓 𝑢% . 𝑓 𝑢% . [𝑢%F. − 𝑢%]
¨ 𝔼{ ∫=>?
=>'
𝑓 𝑢 . [ . 𝑑𝑊 𝑢
-
} = ∫=>?
=>'
𝑓 𝑢 -
. 𝑑𝑠
99

100. Luc_Faucheux_2021
Re-deriving Ho-Lee using the useful relationship - XXXVI
¨ with 𝑓 𝑢 = (𝑡 − 𝑢)
¨ ∫Z85
Z8&
𝑓 𝑢 #. 𝑑𝑠 = ∫Z85
Z8&
𝑡 − 𝑢 #. 𝑑𝑢 = [−
&)Z 4
O
]Z85
Z8&
=
&4
O
¨ So we end up with:
¨ 𝑉 𝑋(𝑡) = 𝔼 (𝜎. ∫Z85
Z8&
𝑡 − 𝑢 . [ . 𝑑𝑊 𝑢 )#
¨ 𝑉 𝑋(𝑡) = 𝜎#. 𝔼 ∫Z85
Z8&
𝑡 − 𝑢 #. 𝑑𝑢 = 𝜎#. 𝔼
&4
O
= 𝜎#.
&4
O
100

101. Luc_Faucheux_2021
Re-deriving Ho-Lee using the useful relationship - XXXVII
¨ A LITTLE SUMMARY so far:
¨ 𝑑𝑅 𝑡, 𝑡, 𝑡 = 𝜃 𝑡 . 𝑑𝑡 − 𝜎. ([). 𝑑𝑊(𝑡)
¨ 𝑅 𝑡, 𝑡, 𝑡 = 𝑅 0,0,0 + ∫Z85
Z8&
𝜃 𝑢 . 𝑑𝑢 − ∫Z85
Z8&
𝜎. ([). 𝑑𝑊(𝑢)
¨ 𝑀 𝑅 𝑡, 𝑡, 𝑡 = 𝔼 𝑅 𝑡, 𝑡, 𝑡 = 𝑅 0,0,0 + ∫Z85
Z8&
𝜃 𝑢 . 𝑑𝑢
¨ 𝑉 𝑅 𝑡, 𝑡, 𝑡 = 𝔼 (𝑅(𝑡, 𝑡, 𝑡) − 𝑀[𝑅 𝑡, 𝑡, 𝑡 ])# = 𝜎# 𝑡
¨ 𝑋 𝑡 = ∫A85
A8&
𝑅 𝑠, 𝑠, 𝑠 . 𝑑𝑠
¨ 𝑀 𝑋 𝑡 = 𝑅 0,0,0 . 𝑡 + ∫A85
A8&
∫Z85
Z8A
𝜃 𝑢 . 𝑑𝑢 . 𝑑𝑠
¨ 𝑉 𝑋(𝑡) = 𝜎# &4
O
101

102. Luc_Faucheux_2021
Re-deriving Ho-Lee using the useful relationship - XXXVIII
¨ All right, almost there !
¨ We are after:
¨ 𝑍𝐶 0,0, 𝑡 = 𝔼&
ℚ
exp(− ∫A85
A8&
𝑅 𝑠, 𝑠, 𝑠 . 𝑑𝑠)|𝔉(0) = 𝔼 exp(− ∫A85
A8&
𝑅 𝑠, 𝑠, 𝑠 . 𝑑𝑠)
¨ 𝑍𝐶 0,0, 𝑡 = 𝔼 exp(−𝑋 𝑡
¨ 𝑍𝐶 0,0, 𝑡 is the current bond prices (also referred to as the current term structure)
¨ We now can use our useful relationship:
¨ 𝔼 𝑒: = 𝑒 𝔼{:}. 𝑒
7[A]
! = 𝑒U[:]. 𝑒
7[A]
!
¨ A last little twist because of the minus sign:
¨ 𝔼 𝑒): = 𝑒 𝔼{):}. 𝑒
7[2A]
! = 𝑒)U[:]. 𝑒
7[A]
!
¨ (you can convince yourself of it by doing 𝑋 → −𝑋)
102

103. Luc_Faucheux_2021
Re-deriving Ho-Lee using the useful relationship - XXXIX
¨ 𝑍𝐶 0,0, 𝑡 = 𝔼 exp(−𝑋 𝑡
¨ 𝔼 𝑒): = 𝑒 𝔼{):}. 𝑒
7[2A]
! = 𝑒)U[:]. 𝑒
7[A]
!
¨ 𝑀 𝑋 𝑡 = 𝑅 0,0,0 . 𝑡 + ∫A85
A8&
∫Z85
Z8A
𝜃 𝑢 . 𝑑𝑢 . 𝑑𝑠
¨ 𝑉 𝑋(𝑡) = 𝜎# &4
O
¨ 𝑍𝐶 0,0, 𝑡 = 𝔼 exp(−𝑋 𝑡 = 𝑒/0[2]. 𝑒
![#]
% = exp −𝑅 0,0,0 . 𝑡 − ∫4"#
4"$
∫!"#
!"4
𝜃 𝑢 . 𝑑𝑢 . 𝑑𝑠 . exp[𝜎% $&
5
]
¨ We have now recovered the expression for the Zeros using our useful relationship:
¨ 𝑍𝐶 0,0, 𝑡 = exp −𝑅 0,0,0 . 𝑡 − ∫A85
A8&
∫Z85
Z8A
𝜃 𝑢 . 𝑑𝑢 . 𝑑𝑠 . exp[𝜎# &4
~
]
103

104. Luc_Faucheux_2021
Re-deriving Ho-Lee using the useful relationship - XXXX
¨ 𝑍𝐶 0,0, 𝑡 = exp −𝑅 0,0,0 . 𝑡 − ∫A85
A8&
∫Z85
Z8A
𝜃 𝑢 . 𝑑𝑢 . 𝑑𝑠 . exp[𝜎# &4
~
]
¨ We can now calibrate the Ho-Lee model by deriving what 𝜃 𝑢 should be to fit the current
term structure of bond prices
¨ We did that in the previous deck
¨ But super quickly let’s just redo it
¨ 𝑍𝐶 0,0, 𝑡 = exp −𝑅 0,0,0 . 𝑡 − ∫A85
A8&
∫Z85
Z8A
𝜃 𝑢 . 𝑑𝑢 . 𝑑𝑠 . exp[𝜎# &4
~
]
¨ 𝑙𝑛𝑍𝐶 0,0, 𝑡 = −𝑅 0,0,0 . 𝑡 − ∫A85
A8&
∫Z85
Z8A
𝜃 𝑢 . 𝑑𝑢 . 𝑑𝑠 + 𝜎# &4
~
¨ ∫A85
A8&
∫Z85
Z8A
𝜃 𝑢 . 𝑑𝑢 . 𝑑𝑠 = −𝑙𝑛𝑍𝐶 0,0, 𝑡 − 𝑅 0,0,0 . 𝑡 + 𝜎# &4
~
104

105. Luc_Faucheux_2021
Re-deriving Ho-Lee using the useful relationship - XXXXI
¨ ∫A85
A8&
∫Z85
Z8A
𝜃 𝑢 . 𝑑𝑢 . 𝑑𝑠 = −𝑙𝑛𝑍𝐶 0,0, 𝑡 − 𝑅 0,0,0 . 𝑡 + 𝜎# &4
~
¨
-
-&
∫A85
A8&
∫Z85
Z8A
𝜃 𝑢 . 𝑑𝑢 . 𝑑𝑠 = ∫Z85
Z8&
𝜃 𝑢 . 𝑑𝑢
¨
-
-&
∫A85
A8&
∫Z85
Z8A
𝜃 𝑢 . 𝑑𝑢 . 𝑑𝑠 =
-
-&
−𝑙𝑛𝑍𝐶 0,0, 𝑡 − 𝑅 0,0,0 . 𝑡 + 𝜎# &4
~
¨
-
-&
∫A85
A8&
∫Z85
Z8A
𝜃 𝑢 . 𝑑𝑢 . 𝑑𝑠 =
-
-&
−𝑙𝑛𝑍𝐶 0,0, 𝑡 − 𝑅 0,0,0 + 𝜎# &!
#
¨ ∫Z85
Z8&
𝜃 𝑢 . 𝑑𝑢 =
-
-&
−𝑙𝑛𝑍𝐶 0,0, 𝑡 − 𝑅 0,0,0 + 𝜎# &!
#
¨ We recognize our old friend the instantaneous forward:
¨ 𝑅 𝑡, 𝑡!, 𝑡! = −
-./(*+ &,&,&"
-&"
¨ 𝑅 0, 𝑡, 𝑡 = −
-./(*+ 5,5,&
-&
105

106. Luc_Faucheux_2021
Re-deriving Ho-Lee using the useful relationship - XXXXII
¨ ∫Z85
Z8&
𝜃 𝑢 . 𝑑𝑢 =
-
-&
−𝑙𝑛𝑍𝐶 0,0, 𝑡 − 𝑅 0,0,0 + 𝜎# &!
#
¨ 𝑅 0, 𝑡, 𝑡 = −
-./(*+ 5,5,&
-&
¨ ∫Z85
Z8&
𝜃 𝑢 . 𝑑𝑢 = 𝑅 0, 𝑡, 𝑡 − 𝑅 0,0,0 + 𝜎# &!
#
¨ Almost there:
¨
-
-&
∫Z85
Z8&
𝜃 𝑢 . 𝑑𝑢 = 𝜃 𝑡 =
-
-&
𝑅 0, 𝑡, 𝑡 − 𝑅 0,0,0 + 𝜎# &!
#
=
-
-&
𝑅 0, 𝑡, 𝑡 + 𝜎#. 𝑡
¨ So we calibrated once again the Ho-Lee model to :
¨ 𝜃 𝑡 =
-
-&
𝑅 0, 𝑡, 𝑡 + 𝜎#. 𝑡
106

107. Luc_Faucheux_2021
Re-deriving Ho-Lee using the useful relationship - XXXXIII
¨ Redoing the summary so far:
¨ 𝑑𝑅 𝑡, 𝑡, 𝑡 = 𝜃 𝑡 . 𝑑𝑡 − 𝜎. ([). 𝑑𝑊(𝑡)
¨ 𝑅 𝑡, 𝑡, 𝑡 = 𝑅 0,0,0 + ∫Z85
Z8&
𝜃 𝑢 . 𝑑𝑢 − ∫Z85
Z8&
𝜎. ([). 𝑑𝑊(𝑢)
¨ 𝑀 𝑅 𝑡, 𝑡, 𝑡 = 𝔼 𝑅 𝑡, 𝑡, 𝑡 = 𝑅 0,0,0 + ∫Z85
Z8&
𝜃 𝑢 . 𝑑𝑢
¨ 𝑉 𝑅 𝑡, 𝑡, 𝑡 = 𝔼 (𝑅(𝑡, 𝑡, 𝑡) − 𝑀[𝑅 𝑡, 𝑡, 𝑡 ])# = 𝜎# 𝑡
¨ With the results of the calibration being:
¨ 𝜃 𝑡 =
-
-&
𝑅 0, 𝑡, 𝑡 + 𝜎#. 𝑡
¨ ∫Z85
Z8&
𝜃 𝑢 . 𝑑𝑢 = 𝑅 0, 𝑡, 𝑡 − 𝑅 0,0,0 + 𝜎# &!
#
¨ 𝑅 𝑡, 𝑡, 𝑡 = 𝑅 0, 𝑡, 𝑡 + 𝜎# &!
#
− ∫Z85
Z8&
𝜎. ([). 𝑑𝑊(𝑢)
107

108. Luc_Faucheux_2021
Re-deriving Ho-Lee using the useful relationship - XXXXIV
¨ 𝑅 𝑡, 𝑡, 𝑡 = 𝑅 0, 𝑡, 𝑡 + 𝜎# &!
#
− ∫Z85
Z8&
𝜎. ([). 𝑑𝑊(𝑢)
¨ 𝑅 𝑡, 𝑡, 𝑡 = 𝑅 0, 𝑡, 𝑡 + 𝜎# &!
#
− 𝜎. 𝑊(𝑡)
¨ 𝑀 𝑅 𝑡, 𝑡, 𝑡 = 𝔼 𝑅 𝑡, 𝑡, 𝑡 = 𝑅 0, 𝑡, 𝑡 + 𝜎# &!
#
¨ 𝑉 𝑅 𝑡, 𝑡, 𝑡 = 𝔼 (𝑅(𝑡, 𝑡, 𝑡) − 𝑀[𝑅 𝑡, 𝑡, 𝑡 ])# = 𝜎# 𝑡
¨ 𝑋 𝑡 = ∫A85
A8&
𝑅 𝑠, 𝑠, 𝑠 . 𝑑𝑠
¨ 𝑀 𝑋 𝑡 = 𝑅 0,0,0 . 𝑡 + ∫A85
A8&
∫Z85
Z8A
𝜃 𝑢 . 𝑑𝑢 . 𝑑𝑠
¨ 𝑉 𝑋(𝑡) = 𝜎# &4
O
¨ And from the calibration we have:
¨ ∫A85
A8&
∫Z85
Z8A
𝜃 𝑢 . 𝑑𝑢 . 𝑑𝑠 = −𝑙𝑛𝑍𝐶 0,0, 𝑡 − 𝑅 0,0,0 . 𝑡 + 𝜎# &4
~
108

109. Luc_Faucheux_2021
Re-deriving Ho-Lee using the useful relationship - XXXXV
¨ 𝑋 𝑡 = ∫A85
A8&
𝑅 𝑠, 𝑠, 𝑠 . 𝑑𝑠
¨ 𝑀 𝑋 𝑡 = 𝑅 0,0,0 . 𝑡 + ∫A85
A8&
∫Z85
Z8A
𝜃 𝑢 . 𝑑𝑢 . 𝑑𝑠
¨ 𝑉 𝑋(𝑡) = 𝜎# &4
O
¨ ∫A85
A8&
∫Z85
Z8A
𝜃 𝑢 . 𝑑𝑢 . 𝑑𝑠 = −𝑙𝑛𝑍𝐶 0,0, 𝑡 − 𝑅 0,0,0 . 𝑡 + 𝜎# &4
~
¨ 𝑀 𝑋 𝑡 = 𝑅 0,0,0 . 𝑡 − 𝑙𝑛𝑍𝐶 0,0, 𝑡 − 𝑅 0,0,0 . 𝑡 + 𝜎# &4
~
= −𝑙𝑛𝑍𝐶 0,0, 𝑡 + 𝜎# &4
~
¨ 𝑉 𝑋(𝑡) = 𝜎# &4
O
109

110. Luc_Faucheux_2021
Re-deriving Ho-Lee using the useful relationship - XXXXVI
¨ 𝑋 𝑡 = ∫A85
A8&
𝑅 𝑠, 𝑠, 𝑠 . 𝑑𝑠
¨ 𝑀 𝑋 𝑡 = −𝑙𝑛𝑍𝐶 0,0, 𝑡 + 𝜎# &4
~
¨ 𝑉 𝑋(𝑡) = 𝜎# &4
O
¨ 𝑍𝐶 0,0, 𝑡 = 𝔼 exp(−𝑋 𝑡
¨ 𝔼 𝑒): = 𝑒 𝔼{):}. 𝑒
7[2A]
! = 𝑒)U[:]. 𝑒
7[A]
!
¨ 𝑍𝐶 0,0, 𝑡 = exp 𝑙𝑛𝑍𝐶 0,0, 𝑡 − 𝜎# &4
~
. exp
$
#
𝜎# &4
O
= exp 𝑙𝑛𝑍𝐶 0,0, 𝑡 = 𝑍𝐶(0,0, 𝑡)
¨ 𝑍𝐶 0,0, 𝑡 = 𝑍𝐶(0,0, 𝑡)
¨ YEAH!! We did not drop terms or goofed up in our derivation
110

111. Luc_Faucheux_2021
Applying the useful relationship to recover the
bond prices dynamics
111

112. Luc_Faucheux_2021
Bond prices dynamics using mean and variance
¨ We can apply the same trick to recover the dynamics of Bond prices.
¨ The math is a little more complicated because instead of integrating from 0 to 𝑡, we will be
now integrating from 𝑡 to 𝑡!
¨ But in essence it will be the same
¨ We will calculate something that is the exponential of a stochastic process
¨ We will then compute the expectation of the exponential by using the useful relationship
¨ Let’s have a couple of slides to remind us about the expectations and how we are going to
use it
112

113. Luc_Faucheux_2021
Bond prices dynamics using mean and variance - II
¨ Note that in the Instantaneous and expectations section we had:
¨ 𝑍𝐶 0,0, 𝑡 = exp − ∫A85
A8&
𝑖𝑓𝑤𝑟 0, 𝑠 . 𝑑𝑢 = exp − ∫A85
A8&
𝑅 0, 𝑠, 𝑠 . 𝑑𝑢
¨ 𝑍𝐶 0,0, 𝑡 = exp −𝜏 0,0, 𝑡 . 𝑅 0,0, 𝑡
¨ 𝔼&"
*+
𝑉 𝑡!, $𝐼𝑆ℎ𝑅 𝑡! , 𝑡!, 𝑡! |𝔉(𝑡) = 𝑖𝑓𝑤𝑟 𝑡, 𝑡!
¨ 𝔼A
*+ 𝑉 𝑠, $𝐼𝑆ℎ𝑅 𝑠 , 𝑠, 𝑠 |𝔉(𝑡) = 𝑖𝑓𝑤𝑟 𝑡, 𝑠
¨ 𝔼A
*+ 𝑉 𝑠, $𝑅 𝑠, 𝑠, 𝑠 , 𝑠, 𝑠 |𝔉(𝑡) = 𝑅 𝑡, 𝑠, 𝑠
¨ 𝔼A
*+ 𝑉 𝑠, $𝑅 𝑠, 𝑠, 𝑠 , 𝑠, 𝑠 |𝔉(0) = 𝑅 0, 𝑠, 𝑠
¨ 𝑍𝐶 0,0, 𝑡 = exp − ∫A85
A8&
𝑅 0, 𝑠, 𝑠 . 𝑑𝑢
¨ 𝑍𝐶 0,0, 𝑡 = exp − ∫A85
A8&
𝔼A
*+ 𝑉 𝑠, $𝑅 𝑠, 𝑠, 𝑠 , 𝑠, 𝑠 |𝔉(0) . 𝑑𝑢
113

114. Luc_Faucheux_2021
Bond prices dynamics using mean and variance - III
¨ 𝑍𝐶 0,0, 𝑡 = 𝔼&
ℚ
exp(− ∫A85
A8&
𝑅 𝑠, 𝑠, 𝑠 . 𝑑𝑠)|𝔉(0)
¨ 𝑍𝐶 0,0, 𝑡 = 𝑉 0, $1, 𝑡, 𝑡 =
[ 5,$$,&,&
2(5)
¨
[ 5,$$,&,&
2(5)
= 𝔼&
ℚ [ &,$$,&,&
2(&)
|𝔉(0) = 𝔼&
ℚ [ &,$$,&,&
]^_(∫89:
89;
a A,A,A .=A)
|𝔉(0)
¨
[ 5,$$,&,&
2(5)
=
[ 5,$$,&,&
$
= 𝑍𝐶 0,0, 𝑡 = 𝔼&
ℚ [ &,$$,&,&
]^_(∫89:
89;
a A,A,A .=A)
|𝔉(0)
¨ 𝑍𝐶 0,0, 𝑡 = 𝔼&
ℚ [ &,$$,&,&
]^_(∫89:
89;
a A,A,A .=A)
|𝔉(0) = 𝔼&
ℚ $
]^_(∫89:
89;
a A,A,A .=A)
|𝔉(0)
¨ 𝑍𝐶 0,0, 𝑡 = 𝔼&
ℚ
exp(− ∫A85
A8&
𝑅 𝑠, 𝑠, 𝑠 . 𝑑𝑠)|𝔉(0)
114

115. Luc_Faucheux_2021
Bond prices dynamics using mean and variance - IV
¨ We have for ease of notation the rolling numeraire:
¨ 𝐵 𝑡 = exp(∫A85
A8&
𝑅 𝑠, 𝑠, 𝑠 . 𝑑𝑠
¨ We can rewrite the previous section as:
¨ 𝑍𝐶 0,0, 𝑡 = 𝔼&
ℚ
exp(− ∫A85
A8&
𝑅 𝑠, 𝑠, 𝑠 . 𝑑𝑠)|𝔉(0)
¨ 𝑍𝐶 0,0, 𝑡 = 𝔼&
ℚ $
2(&)
|𝔉(0)
¨ 𝑍𝐶 0,0, 𝑡! = 𝔼&"
ℚ $
2(&")
|𝔉(0) and remember that 𝐵 0 = 1
¨ Let’s convince ourselves that we also have the following relation:
¨ 𝑍𝐶 𝑡, 𝑡, 𝑡! = 𝔼&"
ℚ 2(&)
2(&")
|𝔉(𝑡) = 𝔼&"
ℚ
exp(− ∫A8&
A8&"
𝑅 𝑠, 𝑠, 𝑠 . 𝑑𝑠)|𝔉(𝑡)
¨ If we do, then we are in business.
115

116. Luc_Faucheux_2021
Bond prices dynamics using mean and variance - V
¨ 𝑑𝑅 𝑡, 𝑡, 𝑡 = 𝜃 𝑡 . 𝑑𝑡 − 𝜎. ([). 𝑑𝑊(𝑡)
¨ 𝑅 𝑡, 𝑡, 𝑡 = 𝑅 0,0,0 + ∫Z85
Z8&
𝜃 𝑢 . 𝑑𝑢 − ∫Z85
Z8&
𝜎. ([). 𝑑𝑊(𝑢)
¨ 𝑀 𝑅 𝑡, 𝑡, 𝑡 = 𝔼 𝑅 𝑡, 𝑡, 𝑡 = 𝑅 0,0,0 + ∫Z85
Z8&
𝜃 𝑢 . 𝑑𝑢
¨ 𝑉 𝑅 𝑡, 𝑡, 𝑡 = 𝔼 (𝑅(𝑡, 𝑡, 𝑡) − 𝑀[𝑅 𝑡, 𝑡, 𝑡 ])# = 𝜎# 𝑡
¨ We already know that we calibrated the model to the 𝑍𝐶 0,0, 𝑡 = 𝔼&
ℚ $
2(&)
|𝔉(0)
¨ 𝜃 𝑡 =
-
-&
𝑅 0, 𝑡, 𝑡 + 𝜎#. 𝑡
¨ We integrated from 0 to 𝑡, we will be now integrating from 𝑡 to 𝑡!
¨ We had : 𝑋 𝑡 = ∫A85
A8&
𝑅 𝑠, 𝑠, 𝑠 . 𝑑𝑠
¨ We now will calculate 𝑋 𝑡, 𝑡! = ∫A8&
A8&"
𝑅 𝑠, 𝑠, 𝑠 . 𝑑𝑠
116

117. Luc_Faucheux_2021
Bond prices dynamics using mean and variance - VI
¨ 𝑅 𝑠, 𝑠, 𝑠 = 𝑅 0,0,0 + ∫Z85
Z8A
𝜃 𝑢 . 𝑑𝑢 − ∫Z85
Z8A
𝜎. ([). 𝑑𝑊(𝑢)
¨ 𝑋 𝑡, 𝑡! = ∫A8&
A8&"
𝑅 𝑠, 𝑠, 𝑠 . 𝑑𝑠
¨ 𝜃 𝑡 =
-
-&
𝑅 0, 𝑡, 𝑡 + 𝜎#. 𝑡
¨ 𝜃 𝑢 =
-
-Z
𝑅 0, 𝑢, 𝑢 + 𝜎#. 𝑢
¨ 𝑅 𝑠, 𝑠, 𝑠 = 𝑅 0,0,0 + ∫Z85
Z8A
𝜃 𝑢 . 𝑑𝑢 − ∫Z85
Z8A
𝜎. ([). 𝑑𝑊(𝑢)
¨ 𝑅 𝑠, 𝑠, 𝑠 = 𝑅 𝑡, 𝑡, 𝑡 + ∫Z8&
Z8A
𝜃 𝑢 . 𝑑𝑢 − ∫Z8&
Z8A
𝜎. ([). 𝑑𝑊(𝑢) for 𝑠 > 𝑡
¨ 𝑋 𝑡, 𝑡! = ∫A8&
A8&"
𝑅 𝑠, 𝑠, 𝑠 . 𝑑𝑠
¨ 𝑋 𝑡, 𝑡! = ∫A8&
A8&"
{𝑅 𝑡, 𝑡, 𝑡 + ∫Z8&
Z8A
𝜃 𝑢 . 𝑑𝑢 − ∫Z8&
Z8A
𝜎. ([). 𝑑𝑊(𝑢)}. 𝑑𝑠
117

118. Luc_Faucheux_2021
Bond prices dynamics using mean and variance - VII
¨ 𝑋 𝑡, 𝑡! = ∫A8&
A8&"
{𝑅 𝑡, 𝑡, 𝑡 + ∫Z8&
Z8A
𝜃 𝑢 . 𝑑𝑢 − ∫Z8&
Z8A
𝜎. ([). 𝑑𝑊(𝑢)}. 𝑑𝑠
¨ 𝑅 𝑠, 𝑠, 𝑠 = 𝑅 0,0,0 + ∫Z85
Z8A
𝜃 𝑢 . 𝑑𝑢 − ∫Z85
Z8A
𝜎. ([). 𝑑𝑊(𝑢)
¨ 𝜃 𝑢 =
-
-Z
𝑅 0, 𝑢, 𝑢 + 𝜎#. 𝑢
¨ ∫Z8&
Z8A
𝜃 𝑢 . 𝑑𝑢 = ∫Z8&
Z8A
{
-
-Z
𝑅 0, 𝑢, 𝑢 + 𝜎#. 𝑢 }. 𝑑𝑢
¨ ∫Z8&
Z8A
𝜃 𝑢 . 𝑑𝑢 = ∫Z8&
Z8A -
-Z
𝑅 0, 𝑢, 𝑢 . 𝑑𝑢 + ∫Z8&
Z8A
{𝜎#. 𝑢 }. 𝑑𝑢
¨ ∫Z8&
Z8A
𝜃 𝑢 . 𝑑𝑢 = [𝑅 0, 𝑢, 𝑢 ]Z8&
Z8A +[
"!Z!
#
]Z8&
Z8A
¨ ∫Z8&
Z8A
𝜃 𝑢 . 𝑑𝑢 = [𝑅 0, 𝑠, 𝑠 − 𝑅(0, 𝑡, 𝑡)] + [
"!(A!)&!)
#
]
¨ ∫Z8&
Z8A
𝜃 𝑢 . 𝑑𝑢 = 𝑅 0, 𝑠, 𝑠 − 𝑅(0, 𝑡, 𝑡) +
"!
#
. (𝑠# −𝑡#)
118

119. Luc_Faucheux_2021
Bond prices dynamics using mean and variance - VIII
¨ 𝑑𝑅 𝑡, 𝑡, 𝑡 = 𝜃 𝑡 . 𝑑𝑡 − 𝜎. ([). 𝑑𝑊(𝑡)
¨ 𝑑𝑅 𝑢, 𝑢, 𝑢 = 𝜃 𝑢 . 𝑑𝑢 − 𝜎. ([). 𝑑𝑊(𝑢)
¨ Integrating between 𝑡 and 𝑠 yields:
¨ ∫Z8&
Z8A
𝑑𝑅 𝑢, 𝑢, 𝑢 = 𝑅 𝑠, 𝑠, 𝑠 − 𝑅(𝑡, 𝑡, 𝑡) = ∫Z8&
Z8A
{𝜃 𝑢 . 𝑑𝑢 − 𝜎. ([). 𝑑𝑊(𝑢) }
¨ 𝑅 𝑠, 𝑠, 𝑠 = 𝑅 𝑡, 𝑡, 𝑡 + ∫Z8&
Z8A
𝜃 𝑢 . 𝑑𝑢 − ∫Z8&
Z8A
𝜎. ([). 𝑑𝑊(𝑢)
¨ ∫Z8&
Z8A
𝜃 𝑢 . 𝑑𝑢 = 𝑅 0, 𝑠, 𝑠 − 𝑅(0, 𝑡, 𝑡) +
"!
#
. (𝑠# −𝑡#)
¨ 𝑅 𝑠, 𝑠, 𝑠 = 𝑅 𝑡, 𝑡, 𝑡 + 𝑅 0, 𝑠, 𝑠 − 𝑅(0, 𝑡, 𝑡) +
"!
#
. (𝑠# −𝑡#) − ∫Z8&
Z8A
𝜎. ([). 𝑑𝑊(𝑢)
119

120. Luc_Faucheux_2021
Bond prices dynamics using mean and variance - IX
¨ 𝑅 𝑠, 𝑠, 𝑠 = 𝑅 𝑡, 𝑡, 𝑡 + 𝑅 0, 𝑠, 𝑠 − 𝑅(0, 𝑡, 𝑡) +
"!
#
. (𝑠# −𝑡#) − ∫Z8&
Z8A
𝜎. ([). 𝑑𝑊(𝑢)
¨ We now want to compute: 𝑋 𝑡, 𝑡! = ∫A8&
A8&"
𝑅 𝑠, 𝑠, 𝑠 . 𝑑𝑠
¨ 𝑋 𝑡, 𝑡! = ∫A8&
A8&"
𝑅 𝑠, 𝑠, 𝑠 . 𝑑𝑠
¨ 𝑋 𝑡, 𝑡! = ∫A8&
A8&"
{𝑅 𝑡, 𝑡, 𝑡 + 𝑅 0, 𝑠, 𝑠 − 𝑅(0, 𝑡, 𝑡) +
"!
#
. (𝑠# −𝑡#) − ∫Z8&
Z8A
𝜎. ([). 𝑑𝑊(𝑢) }. 𝑑𝑠
¨ ∫A8&
A8&"
{𝑅 𝑡, 𝑡, 𝑡 }. 𝑑𝑠 = 𝑅 𝑡, 𝑡, 𝑡 . (𝑡! − 𝑡)
¨ ∫A8&
A8&"
{𝑅 0, 𝑡, 𝑡 }. 𝑑𝑠 = 𝑅 0, 𝑡, 𝑡 . (𝑡! − 𝑡)
¨ ∫A8&
A8&"
{
"!
#
. (𝑠# −𝑡#)}. 𝑑𝑠 =
"!
#
. ∫A8&
A8&"
(𝑠#−𝑡#). 𝑑𝑠 =
"!
#
. [
A4
O
− 𝑡#. 𝑠]A8&
A8&"
¨ ∫A8&
A8&"
{
"!
#
. (𝑠# −𝑡#)}. 𝑑𝑠 =
"!
#
. {
&"
4
O
−
&4
O
− 𝑡#. 𝑡! + 𝑡#. 𝑡}
120

121. Luc_Faucheux_2021
Bond prices dynamics using mean and variance - X
¨ ∫A8&
A8&"
{
"!
#
. (𝑠# −𝑡#)}. 𝑑𝑠 =
"!
#
. {
&"
4
O
−
&4
O
− 𝑡#. 𝑡! + 𝑡#. 𝑡}
¨ ∫A8&
A8&"
{
"!
#
. (𝑠# −𝑡#)}. 𝑑𝑠 =
"!
~
. {𝑡!
O − 𝑡O − 3. 𝑡# . 𝑡! + 3. 𝑡#. 𝑡}
¨ 𝑡!
O − 𝑡O = 𝑡! − 𝑡 . (𝑡!
# + 𝑡# + 𝑡!. 𝑡)
¨ 𝑡!
O − 𝑡O − 3. 𝑡# . 𝑡! + 3. 𝑡#. 𝑡 = 𝑡! − 𝑡 . (𝑡!
# + 𝑡# + 𝑡!. 𝑡) − 3. 𝑡#. 𝑡! + 3. 𝑡#. 𝑡
¨ 𝑡!
O − 𝑡O − 3. 𝑡# . 𝑡! + 3. 𝑡#. 𝑡 = 𝑡! − 𝑡 . 𝑡!
# + 𝑡# + 𝑡!. 𝑡 − 3. 𝑡! − 𝑡 . 𝑡#
¨ 𝑡!
O − 𝑡O − 3. 𝑡# . 𝑡! + 3. 𝑡#. 𝑡 = 𝑡! − 𝑡 . 𝑡!
# + 𝑡# + 𝑡!. 𝑡 − 3. 𝑡#
¨ 𝑡!
O − 𝑡O − 3. 𝑡# . 𝑡! + 3. 𝑡#. 𝑡 = 𝑡! − 𝑡 . 𝑡!
# + 𝑡!. 𝑡 − 2. 𝑡#
¨ 𝑡!
# + 𝑡!. 𝑡 − 2. 𝑡# = 𝑡! − 𝑡 . (𝑡! + 2𝑡)
¨ 𝑡!
O − 𝑡O − 3. 𝑡# . 𝑡! + 3. 𝑡#. 𝑡 = 𝑡! − 𝑡 . 𝑡! − 𝑡 . 𝑡! + 2𝑡 = 𝑡! − 𝑡 #. (𝑡! + 2𝑡)
121

122. Luc_Faucheux_2021
Bond prices dynamics using mean and variance - XI
¨ 𝑋 𝑡, 𝑡! = ∫A8&
A8&"
{𝑅 𝑡, 𝑡, 𝑡 + 𝑅 0, 𝑠, 𝑠 − 𝑅(0, 𝑡, 𝑡) +
"!
#
. (𝑠# −𝑡#) − ∫Z8&
Z8A
𝜎. ([). 𝑑𝑊(𝑢) }. 𝑑𝑠
¨ ∫A8&
A8&"
{𝑅 𝑡, 𝑡, 𝑡 }. 𝑑𝑠 = 𝑅 𝑡, 𝑡, 𝑡 . (𝑡! − 𝑡)
¨ ∫A8&
A8&"
{𝑅 0, 𝑡, 𝑡 }. 𝑑𝑠 = 𝑅 0, 𝑡, 𝑡 . (𝑡! − 𝑡)
¨ ∫A8&
A8&"
{
"!
#
. (𝑠# −𝑡#)}. 𝑑𝑠 =
"!
~
. 𝑡! − 𝑡 #. (𝑡! + 2𝑡)
¨ ∫A8&
A8&"
{𝑅 0, 𝑠, 𝑠 }. 𝑑𝑠 = ∫A8&
A8&"
{𝑅 0, 𝑠, 𝑠 }. 𝑑𝑠, we leave this one as is for now
¨ And we use our good old friend Guido Fubini on the last term:
¨ ∫A8&
A8&"
{∫Z8&
Z8A
𝜎. ([). 𝑑𝑊(𝑢)}. 𝑑𝑠
122

123. Luc_Faucheux_2021
Bond prices dynamics using mean and variance - XII
¨ ∫A8&
A8&"
{∫Z8&
Z8A
𝜎. ([). 𝑑𝑊(𝑢)}. 𝑑𝑠
¨ We just have to make sure that we are careful about the variables because they are not the
same ones we had on our previous graph
¨ Before we were starting from 0, we now start the integral at 𝑡
123

124. Luc_Faucheux_2021
Bond prices dynamics using mean and variance - XIII
¨ 𝑋 = ∫A8&
A8&"
{∫Z8&
Z8A
𝜎. ([). 𝑑𝑊(𝑢)}. 𝑑𝑠
124
s
𝑠 = 𝑡!
u
s
u
𝑠 = 𝑡 𝑠 = 𝑡!𝑠 = 𝑡

125. Luc_Faucheux_2021
Bond prices dynamics using mean and variance - XIV
¨ 𝑋 = ∫A8&
A8&"
{∫Z8&
Z8A
𝜎. ([). 𝑑𝑊(𝑢)}. 𝑑𝑠
125
s
𝑠 = 𝑡!
u
s
u
𝑠 = 𝑡 𝑠 = 𝑡!𝑠 = 𝑡
𝑋 = n
A8&
A8&"
{ n
Z8&
Z8A
𝜎. ([). 𝑑𝑊(𝑢)}. 𝑑𝑠 𝑋 = n
Z8&
Z8&"
{ n
A8Z
A8&"
𝜎. 𝑑𝑠}. ([). 𝑑𝑊(𝑢)

126. Luc_Faucheux_2021
Bond prices dynamics using mean and variance - XV
¨ 𝑋 = ∫A8&
A8&"
{∫Z8&
Z8A
𝜎. ([). 𝑑𝑊(𝑢)}. 𝑑𝑠
¨ 𝑋 = ∫Z8&
Z8&"
{∫A8Z
A8&"
𝜎. 𝑑𝑠}. ([). 𝑑𝑊(𝑢)
¨ 𝑋 = ∫Z8&
Z8&"
{𝜎. [𝑠]A8Z
A8&"
}. ([). 𝑑𝑊(𝑢)
¨ 𝑋 = ∫Z8&
Z8&"
{𝜎. (𝑡! − 𝑢)}. ([). 𝑑𝑊(𝑢)
¨ 𝑋 = ∫Z8&
Z8&"
𝜎. (𝑡! − 𝑢). ([). 𝑑𝑊(𝑢)
126

127. Luc_Faucheux_2021
Bond prices dynamics using mean and variance - XVI
¨ All right, we now have all the terms that we need:
¨ 𝑋 𝑡, 𝑡! = ∫A8&
A8&"
{𝑅 𝑡, 𝑡, 𝑡 + 𝑅 0, 𝑠, 𝑠 − 𝑅(0, 𝑡, 𝑡) +
"!
#
. (𝑠# −𝑡#) − ∫Z8&
Z8A
𝜎. ([). 𝑑𝑊(𝑢) }. 𝑑𝑠
¨ ∫A8&
A8&"
{𝑅 𝑡, 𝑡, 𝑡 }. 𝑑𝑠 = 𝑅 𝑡, 𝑡, 𝑡 . (𝑡! − 𝑡)
¨ ∫A8&
A8&"
{𝑅 0, 𝑡, 𝑡 }. 𝑑𝑠 = 𝑅 0, 𝑡, 𝑡 . (𝑡! − 𝑡)
¨ ∫A8&
A8&"
{
"!
#
. (𝑠# −𝑡#)}. 𝑑𝑠 =
"!
~
. 𝑡! − 𝑡 #. (𝑡! + 2𝑡)
¨ ∫A8&
A8&"
{𝑅 0, 𝑠, 𝑠 }. 𝑑𝑠 = ∫A8&
A8&"
{𝑅 0, 𝑠, 𝑠 }. 𝑑𝑠
¨ ∫A8&
A8&"
{∫Z8&
Z8A
𝜎. ([). 𝑑𝑊(𝑢)}. 𝑑𝑠 = ∫Z8&
Z8&"
𝜎. (𝑡! − 𝑢). ([). 𝑑𝑊(𝑢)
¨ We can now calculate the mean and the variance at time 𝑡! of the quantity 𝑋 𝑡, 𝑡! conditional
to 𝔉(𝑡)
127

128. Luc_Faucheux_2021
Bond prices dynamics using mean and variance - XVII
¨ 𝑋 𝑡, 𝑡! = ∫A8&
A8&"
{𝑅 𝑡, 𝑡, 𝑡 + 𝑅 0, 𝑠, 𝑠 − 𝑅(0, 𝑡, 𝑡) +
"!
#
. (𝑠# −𝑡#) − ∫Z8&
Z8A
𝜎. ([). 𝑑𝑊(𝑢) }. 𝑑𝑠
¨ 𝑋 𝑡, 𝑡+ = 𝑅 𝑡, 𝑡, 𝑡 . 𝑡+ − 𝑡 + ∫4"$
4"$'
{𝑅 0, 𝑠, 𝑠 }. 𝑑𝑠 − 𝑅 0, 𝑡, 𝑡 . 𝑡+ − 𝑡 +
6%
5
. 𝑡+ − 𝑡 %
. 𝑡+ + 2𝑡 − ∫!"$
!"$'
𝜎. (𝑡+ − 𝑢). ([). 𝑑𝑊(𝑢)
¨ 𝑋 𝑡, 𝑡+ = (𝑅 𝑡, 𝑡, 𝑡 − 𝑅 0, 𝑡, 𝑡 ). 𝑡+ − 𝑡 + ∫4"$
4"$'
{𝑅 0, 𝑠, 𝑠 }. 𝑑𝑠 +
6%
5
. 𝑡+ − 𝑡 %. 𝑡+ + 2𝑡 − ∫!"$
!"$'
𝜎. (𝑡+ − 𝑢). ([). 𝑑𝑊(𝑢)
¨ 𝑍𝐶 𝑡, 𝑡, 𝑡! = 𝔼&"
ℚ 2(&)
2(&")
|𝔉(𝑡) = 𝔼&"
ℚ
exp(− ∫A8&
A8&"
𝑅 𝑠, 𝑠, 𝑠 . 𝑑𝑠)|𝔉(𝑡)
¨ 𝑋 𝑡, 𝑡! = ∫A8&
A8&"
𝑅 𝑠, 𝑠, 𝑠 . 𝑑𝑠
¨ 𝑍𝐶 𝑡, 𝑡, 𝑡! = 𝔼&"
ℚ 2(&)
2(&")
|𝔉(𝑡) = 𝔼&"
ℚ
exp(−𝑋 𝑡, 𝑡! )|𝔉(𝑡)
¨ And we can use once again the nice relationship:
¨ 𝔼&"
ℚ
exp(−𝑋 𝑡, 𝑡! )|𝔉(𝑡) = 𝔼 exp 𝑋 = exp 𝑀[𝑋(𝑡)] . exp
$
#
𝑉[𝑋(𝑡)]
128

129. Luc_Faucheux_2021
Bond prices dynamics using mean and variance - XVIII
¨ 𝔼&"
ℚ
exp(−𝑋 𝑡, 𝑡! )|𝔉(𝑡) = 𝔼 exp 𝑋 = exp 𝑀[𝑋(𝑡)] . exp
$
#
𝑉[𝑋(𝑡)]
¨ 𝔼&"
ℚ
exp(−𝑋 𝑡, 𝑡! )|𝔉(𝑡) = exp 𝑀[𝑋 𝑡, 𝑡! ] . exp
$
#
𝑉[𝑋 𝑡, 𝑡! ]
¨ 𝑀[𝑋 𝑡, 𝑡! ] = 𝔼&"
ℚ
𝑋 𝑡, 𝑡! |𝔉(𝑡)
¨ 𝑉 𝑋 𝑡, 𝑡! = 𝔼&"
ℚ
(𝑋 𝑡, 𝑡! − 𝑀[𝑋 𝑡, 𝑡! ])#|𝔉(𝑡)
¨ So we just need to compute the mean and variance of 𝑋 𝑡, 𝑡! in order compute the bond
prices:
¨ 𝑍𝐶 𝑡, 𝑡, 𝑡! = 𝔼&"
ℚ 2(&)
2(&")
|𝔉(𝑡) = 𝔼&"
ℚ
exp(−𝑋 𝑡, 𝑡! )|𝔉(𝑡)
¨ 𝑍𝐶 𝑡, 𝑡, 𝑡! = exp −𝑀[𝑋 𝑡, 𝑡! ] . exp
$
#
𝑉[𝑋 𝑡, 𝑡! ]
129

130. Luc_Faucheux_2021
Bond prices dynamics using mean and variance - XIX
¨ 𝑋 𝑡, 𝑡+ = (𝑅 𝑡, 𝑡, 𝑡 − 𝑅 0, 𝑡, 𝑡 ). 𝑡+ − 𝑡 + ∫4"$
4"$'
{𝑅 0, 𝑠, 𝑠 }. 𝑑𝑠 +
6%
5
. 𝑡+ − 𝑡 %. 𝑡+ + 2𝑡 − ∫!"$
!"$'
𝜎. (𝑡+ − 𝑢). ([). 𝑑𝑊(𝑢)
¨ 𝑀[𝑋 𝑡, 𝑡! ] = 𝔼&"
ℚ
𝑋 𝑡, 𝑡! |𝔉(𝑡)
¨ 𝑀 𝑋 𝑡, 𝑡$ = (𝑅 𝑡, 𝑡, 𝑡 − 𝑅 0, 𝑡, 𝑡 ). 𝑡$ − 𝑡 + ∫H>'
H>'"
{𝑅 0, 𝑠, 𝑠 }. 𝑑𝑠 +
,#
I
. 𝑡$ − 𝑡 -
. 𝑡$ + 2𝑡
¨ We can further express the term ∫A8&
A8&"
{𝑅 0, 𝑠, 𝑠 }. 𝑑𝑠 since we recognize the expression for
our good old friend the Instantaneous Forward Rate
¨ 𝑅 𝑡, 𝑡!, 𝑡! = −
-./(*+ &,&,&"
-&"
¨ 𝑅 0, 𝑡, 𝑡 = −
-./(*+ 5,5,&
-&
¨ 𝑅 0, 𝑠, 𝑠 = −
-./(*+ 5,5,A
-A
130

131. Luc_Faucheux_2021
Bond prices dynamics using mean and variance - XX
¨ ∫A8&
A8&"
𝑅 0, 𝑠, 𝑠 . 𝑑𝑠 = ∫A8&
A8&"
−
-./(*+ 5,5,A
-A
. 𝑑𝑠 = [−ln(𝑍𝐶 0,0, 𝑠 ]A8&
A8&"
¨ ∫A8&
A8&"
𝑅 0, 𝑠, 𝑠 . 𝑑𝑠 = [ln(𝑍𝐶 0,0, 𝑡 − ln 𝑍𝐶 0,0, 𝑡! = ln(
*+ 5,5,&
*+ 5,5,&"
)
¨ 𝑀 𝑋 𝑡, 𝑡$ = (𝑅 𝑡, 𝑡, 𝑡 − 𝑅 0, 𝑡, 𝑡 ). 𝑡$ − 𝑡 + ∫H>'
H>'"
{𝑅 0, 𝑠, 𝑠 }. 𝑑𝑠 +
,#
I
. 𝑡$ − 𝑡 -
. 𝑡$ + 2𝑡
¨ 𝑀 𝑋 𝑡, 𝑡! = (𝑅 𝑡, 𝑡, 𝑡 − 𝑅 0, 𝑡, 𝑡 ). 𝑡! − 𝑡 + ln(
*+ 5,5,&
*+ 5,5,&"
) +
"!
~
. 𝑡! − 𝑡 #. 𝑡! + 2𝑡
¨ Now onto the variance.
131

132. Luc_Faucheux_2021
Bond prices dynamics using mean and variance - XXI
¨ 𝑉 𝑋 𝑡, 𝑡! = 𝔼&"
ℚ
(𝑋 𝑡, 𝑡! − 𝑀[𝑋 𝑡, 𝑡! ])#|𝔉(𝑡)
¨ 𝑀 𝑋 𝑡, 𝑡! = (𝑅 𝑡, 𝑡, 𝑡 − 𝑅 0, 𝑡, 𝑡 ). 𝑡! − 𝑡 + ln(
*+ 5,5,&
*+ 5,5,&"
) +
"!
~
. 𝑡! − 𝑡 #. 𝑡! + 2𝑡
¨ 𝑋 𝑡, 𝑡+ = (𝑅 𝑡, 𝑡, 𝑡 − 𝑅 0, 𝑡, 𝑡 ). 𝑡+ − 𝑡 + ln(
78 #,#,$
78 #,#,$'
) +
6%
5
. 𝑡+ − 𝑡 %. 𝑡+ + 2𝑡 − ∫!"$
!"$'
𝜎. (𝑡+ − 𝑢). ([). 𝑑𝑊(𝑢)
¨ 𝑋 𝑡, 𝑡! − 𝑀 𝑋 𝑡, 𝑡! = − ∫Z8&
Z8&"
𝜎. (𝑡! − 𝑢). ([). 𝑑𝑊(𝑢)
¨ (𝑋 𝑡, 𝑡! − 𝑀[𝑋 𝑡, 𝑡! ])#= (∫Z8&
Z8&"
𝜎. (𝑡! − 𝑢). ([). 𝑑𝑊(𝑢))#
¨ That is where we can use the good old property of isometry of the ITO integral (again remember
we are assuming that we are using ITO calculus throughout the rates deck)
¨ (𝑋 𝑡, 𝑡! − 𝑀[𝑋 𝑡, 𝑡! ])#= (∫Z8&
Z8&"
𝜎. (𝑡! − 𝑢). ([). 𝑑𝑊(𝑢))#= ∫Z8&
Z8&"
(𝜎. (𝑡! − 𝑢))#. 𝑑𝑢
132

133. Luc_Faucheux_2021
Bond prices dynamics using mean and variance - XXII
¨ ∫Z8&
Z8&"
(𝜎. (𝑡! − 𝑢))#. 𝑑𝑢 = 𝜎#. ∫Z8&
Z8&"
(𝑡! − 𝑢)#. 𝑑𝑢 = 𝜎#. [−
(&")Z)4
O
]Z8&
Z8&"
= 𝜎#. [−
&")&"
4
O
+
(&")&)4
O
]
¨ ∫Z8&
Z8&"
(𝜎. (𝑡! − 𝑢))#. 𝑑𝑢 = 𝜎#. −
&")&"
4
O
+
&")& 4
O
= 𝜎#.
&")& 4
O
¨ 𝑉 𝑋 𝑡, 𝑡! = 𝔼&"
ℚ
(𝑋 𝑡, 𝑡! − 𝑀[𝑋 𝑡, 𝑡! ])#|𝔉(𝑡)
¨ (𝑋 𝑡, 𝑡! − 𝑀[𝑋 𝑡, 𝑡! ])#= (∫Z8&
Z8&"
𝜎. (𝑡! − 𝑢). ([). 𝑑𝑊(𝑢))#= ∫Z8&
Z8&"
(𝜎. (𝑡! − 𝑢))#. 𝑑𝑢
¨ (𝑋 𝑡, 𝑡! − 𝑀[𝑋 𝑡, 𝑡! ])#= 𝜎#.
&")& 4
O
¨ 𝑉 𝑋 𝑡, 𝑡! = 𝔼&"
ℚ
(𝑋 𝑡, 𝑡! − 𝑀[𝑋 𝑡, 𝑡! ])#|𝔉(𝑡)
¨ 𝑉 𝑋 𝑡, 𝑡! = 𝔼&"
ℚ
𝜎#.
&")& 4
O
|𝔉(𝑡) = 𝜎#.
&")& 4
O
133

134. Luc_Faucheux_2021
Bond prices dynamics using mean and variance - XXIII
¨ So to recap we have for the Instantaneous Short Rate:
¨ 𝑑𝑅 𝑡, 𝑡, 𝑡 = 𝜃 𝑡 . 𝑑𝑡 − 𝜎. ([). 𝑑𝑊(𝑡)
¨ 𝑅 𝑡, 𝑡, 𝑡 = 𝑅 0,0,0 + ∫Z85
Z8&
𝜃 𝑢 . 𝑑𝑢 − ∫Z85
Z8&
𝜎. ([). 𝑑𝑊(𝑢)
¨ 𝑀 𝑅 𝑡, 𝑡, 𝑡 = 𝔼 𝑅 𝑡, 𝑡, 𝑡 = 𝑅 0,0,0 + ∫Z85
Z8&
𝜃 𝑢 . 𝑑𝑢
¨ 𝑉 𝑅 𝑡, 𝑡, 𝑡 = 𝔼 (𝑅(𝑡, 𝑡, 𝑡) − 𝑀[𝑅 𝑡, 𝑡, 𝑡 ])# = 𝜎# 𝑡
¨ We already know that we calibrated the model to the 𝑍𝐶 0,0, 𝑡 = 𝔼&
ℚ $
2(&)
|𝔉(0)
¨ 𝜃 𝑡 =
-
-&
𝑅 0, 𝑡, 𝑡 + 𝜎#. 𝑡
¨ ∫Z85
Z8&
𝜃 𝑢 . 𝑑𝑢 = 𝑅 0, 𝑡, 𝑡 − 𝑅 0,0,0 +
"!
#
. 𝑡#
134

135. Luc_Faucheux_2021
Bond prices dynamics using mean and variance - XXIV
¨ 𝑑𝑅 𝑡, 𝑡, 𝑡 = {
-
-&
𝑅 0, 𝑡, 𝑡 + 𝜎#. 𝑡}. 𝑑𝑡 − 𝜎. ([). 𝑑𝑊(𝑡)
¨ 𝑅 𝑡, 𝑡, 𝑡 = 𝑅 0,0,0 + 𝑅 0, 𝑡, 𝑡 − 𝑅 0,0,0 +
"!
#
. 𝑡# − ∫Z85
Z8&
𝜎. ([). 𝑑𝑊(𝑢)
¨ 𝑅 𝑡, 𝑡, 𝑡 = 𝑅 0, 𝑡, 𝑡 +
"!
#
. 𝑡# − ∫Z85
Z8&
𝜎. ([). 𝑑𝑊(𝑢)
¨ 𝑀 𝑅 𝑡, 𝑡, 𝑡 = 𝔼&
ℚ
𝑅 𝑡, 𝑡, 𝑡 |𝔉(0)
¨ 𝑀 𝑅 𝑡, 𝑡, 𝑡 = 𝑅 0, 𝑡, 𝑡 +
"!
#
. 𝑡#
¨ 𝑉 𝑋 𝑡, 𝑡! = 𝔼&
ℚ
(𝑅 𝑡, 𝑡, 𝑡 − 𝑀[𝑅 𝑡, 𝑡, 𝑡 ])#|𝔉(0)
¨ 𝑉 𝑅 𝑡, 𝑡, 𝑡 = 𝔼 (𝑅(𝑡, 𝑡, 𝑡) − 𝑀[𝑅 𝑡, 𝑡, 𝑡 ])# = 𝜎# 𝑡
135

136. Luc_Faucheux_2021
Bond prices dynamics using mean and variance - XXV
¨ We then construct the quantity: 𝑋 𝑡, 𝑡! = ∫A8&
A8&"
𝑅 𝑠, 𝑠, 𝑠 . 𝑑𝑠
¨ 𝑀[𝑋 𝑡, 𝑡! ] = 𝔼&"
ℚ
𝑋 𝑡, 𝑡! |𝔉(𝑡)
¨ 𝑀 𝑋 𝑡, 𝑡! = (𝑅 𝑡, 𝑡, 𝑡 − 𝑅 0, 𝑡, 𝑡 ). 𝑡! − 𝑡 + ln(
*+ 5,5,&
*+ 5,5,&"
) +
"!
~
. 𝑡! − 𝑡 #. 𝑡! + 2𝑡
¨ 𝑉 𝑋 𝑡, 𝑡! = 𝔼&"
ℚ
(𝑋 𝑡, 𝑡! − 𝑀[𝑋 𝑡, 𝑡! ])#|𝔉(𝑡)
¨ 𝑉 𝑋 𝑡, 𝑡! = 𝜎#.
&")& 4
O
136

137. Luc_Faucheux_2021
Bond prices dynamics using mean and variance - XXVI
¨ We then want to compute the expectation of the quantity: exp(−𝑋 𝑡, 𝑡! )
¨ 𝑋 𝑡, 𝑡! = ∫A8&
A8&"
𝑅 𝑠, 𝑠, 𝑠 . 𝑑𝑠
¨ 𝑍𝐶 𝑡, 𝑡, 𝑡! = 𝔼&"
ℚ 2(&)
2(&")
|𝔉(𝑡) = 𝔼&"
ℚ
exp(− ∫A8&
A8&"
𝑅 𝑠, 𝑠, 𝑠 . 𝑑𝑠)|𝔉(𝑡)
¨ 𝑍𝐶 𝑡, 𝑡, 𝑡! = 𝔼&"
ℚ
exp(−𝑋 𝑡, 𝑡! ) |𝔉(𝑡)
¨ 𝔼&"
ℚ
exp(−𝑋 𝑡, 𝑡! ) |𝔉(𝑡) = exp −𝑀[𝑋 𝑡, 𝑡! ] . exp
$
#
𝑉[𝑋 𝑡, 𝑡! ]
¨ 𝑍𝐶 𝑡, 𝑡, 𝑡! = exp −𝑀[𝑋 𝑡, 𝑡! ] . exp
$
#
𝑉[𝑋 𝑡, 𝑡! ]
¨ 𝑀 𝑋 𝑡, 𝑡! = (𝑅 𝑡, 𝑡, 𝑡 − 𝑅 0, 𝑡, 𝑡 ). 𝑡! − 𝑡 + ln(
*+ 5,5,&
*+ 5,5,&"
) +
"!
~
. 𝑡! − 𝑡 #. 𝑡! + 2𝑡
¨ 𝑉 𝑋 𝑡, 𝑡! = 𝜎#.
&")& 4
O
137

138. Luc_Faucheux_2021
Bond prices dynamics using mean and variance - XXVII
¨ 𝑍𝐶 𝑡, 𝑡, 𝑡! = exp 𝑀[𝑋 𝑡, 𝑡! ] . exp
$
#
𝑉[𝑋 𝑡, 𝑡! ]
¨ 𝑀 𝑋 𝑡, 𝑡! = (𝑅 𝑡, 𝑡, 𝑡 − 𝑅 0, 𝑡, 𝑡 ). 𝑡! − 𝑡 + ln(
*+ 5,5,&
*+ 5,5,&"
) +
"!
~
. 𝑡! − 𝑡 #. 𝑡! + 2𝑡
¨ 𝑉 𝑋 𝑡, 𝑡! = 𝜎#.
&")& 4
O
¨ 𝑍𝐶 𝑡, 𝑡, 𝑡$ =
JK ?,?,'"
JK ?,?,'
. exp{−(𝑅 𝑡, 𝑡, 𝑡 − 𝑅 0, 𝑡, 𝑡 ). 𝑡$ − 𝑡 −
,#
I
. 𝑡$ − 𝑡 -
. 𝑡$ + 2𝑡 +
.
-
. 𝜎-
.
'"(' $
L
}
¨ 𝑌 = −
"!
~
. 𝑡! − 𝑡 #. 𝑡! + 2𝑡 +
$
#
. 𝜎#.
&")& 4
O
= −
"!
~
. 𝑡! − 𝑡 #. 𝑡! + 2𝑡 − 𝑡! − 𝑡
¨ 𝑌 = −
"!
~
. 𝑡! − 𝑡 #. 𝑡! + 2𝑡 − 𝑡! − 𝑡 = −
"!
~
. 𝑡! − 𝑡 #. 3𝑡 = −
"!
#
. 𝑡! − 𝑡 #. 𝑡
¨ 𝑍𝐶 𝑡, 𝑡, 𝑡! =
*+ 5,5,&"
*+ 5,5,&
. exp{−(𝑅 𝑡, 𝑡, 𝑡 − 𝑅 0, 𝑡, 𝑡 ). 𝑡! − 𝑡 −
"!
#
. 𝑡! − 𝑡 #. 𝑡}
138

139. Luc_Faucheux_2021
Bond prices dynamics using mean and variance - XXVIII
¨ We have derived:
¨ 𝑍𝐶 𝑡, 𝑡, 𝑡! =
*+ 5,5,&"
*+ 5,5,&
. exp{−(𝑅 𝑡, 𝑡, 𝑡 − 𝑅 0, 𝑡, 𝑡 ). 𝑡! − 𝑡 −
"!
#
. 𝑡! − 𝑡 #. 𝑡}
¨ We can compare to what we derived in the first part of the deck:
¨ 𝑍𝐶 𝑡, 𝑡, 𝑡! = 𝑍𝐶 0, 𝑡, 𝑡! . exp{−[𝑅 𝑡, 𝑡, 𝑡 − 𝑅 0, 𝑡, 𝑡 ]. 𝑡! − 𝑡 −
"!
#
𝑡 𝑡! − 𝑡 #}
¨ Yep, we still got it…we ended up mot messing up too much in the derivation !!
¨ We can once again put in evidence the affine property of the Ho-Lee model
¨ This is a neat way to derive the dynamics of Bond prices using:
¨ Fubini theorem
¨ Isometry property of the ITO integral
¨ Useful relationship
139

140. Luc_Faucheux_2021
The deflated Zeros
140

141. Luc_Faucheux_2021
The deflated Zeros
¨ In some textbooks (Bjork for example), a very useful quantity is defined:
¨ The deflated Zeros
¨ The Zeros are the usual Zero Coupon Bonds: 𝑍𝐶 𝑡, 𝑡, 𝑡!
¨ By the way when I redo all those slides I will just use one letter 𝑍 𝑡, 𝑡, 𝑡! , not quite sure why
I started using 𝑍𝐶 instead of 𝑍
¨ I might even start now to start getting used to it.
¨ The other quantity is the rolling numeraire, or money market account:
¨ 𝐵 𝑡 = exp(∫A85
A8&
𝑅 𝑠, 𝑠, 𝑠 . 𝑑𝑠)
¨ The deflated zeros are defined as:
¨ {𝑍𝐶 𝑡, 𝑡, 𝑡! = |𝑍 𝑡, 𝑡, 𝑡! =
*+ &,&,&"
2(&)
=
* &,&,&"
2(&)
141

142. Luc_Faucheux_2021
The deflated Zeros - II
¨ {𝑍𝐶 𝑡, 𝑡, 𝑡! = |𝑍 𝑡, 𝑡, 𝑡! =
*+ &,&,&"
2(&)
=
* &,&,&"
2(&)
¨ The cool thing about the deflated Zeros is that in the Risk neutral measure they are
martingales, and so their SDE is driftless:
¨ 𝑑 |𝑍 𝑡, 𝑡, 𝑡! = 0. 𝑑𝑡 + 𝜎 ‚* 𝑡, 𝑡! . [ . 𝑑𝑊(𝑡)
¨ Let’s re-derive that just for sake of consistency.
¨ 𝐵 𝑡 = exp(∫A85
A8&
𝑅 𝑠, 𝑠, 𝑠 . 𝑑𝑠)
¨ 𝑑𝐵 𝑡 = 𝑅 𝑡, 𝑡, 𝑡 . 𝐵 𝑡 . 𝑑𝑡
¨ In the risk neutral measure we have for the Zeros:
¨
=*+ &,&,&"
*+ &,&,&"
= 𝑅 𝑡, 𝑡, 𝑡 . 𝑑𝑡 + 𝑉 𝑡, 𝑡!, 𝑡! . ([). 𝑑𝑊 𝑡
¨ With for Ho-Lee: 𝑉 𝑡, 𝑡!, 𝑡! = 𝜎. (𝑡! − 𝑡)
142