Lf 2020 bachelier

Luc_Faucheux_2020
Notes on Bachelier
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Luc_Faucheux_2020
A quick summary
¨ We are mostly using the book “Louis Bachelier’s Theory of Speculation”, Mark Davis and
Alison Etheridge
¨ We use it as a starting point to explore some properties of Brownian motion, Gaussian
processes and option pricing concepts
¨ It is trying to be as rigorous as possible without losing track of being pragmatic
¨ Those notes are trying to offer you an overview of some of the concepts around option
mathematics, and allow you to be a reference , and an introduction to some of the methods
and sometimes “tricks” that end up being useful
¨ Those notes tend to also be “non-linear”, meaning I will sometimes use a specific page from
the Bachelier book as a starting point to muse around Gaussian processes and option pricing
theory, hence the rather disorganized structure. I found out usually that at least for me this
is how I learn, by using a starting point and checking “what-ifs” and “what-nots” around it.
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Many disclaimers and apologies
¨ Apologies for the lack of, or incomplete references. This is still a work in progress, and I
would welcome any comment, or kind readers pointing out omissions and glaring mistakes
¨ I have tried to keep consistent notations throughout those notes. It is somewhat
impossible, because keeping with Bachelier’s original notations is not possible with any
conventional notations we see in recent textbooks, so again many apologies
¨ The structure of those notes is somewhat “free-flowing”, as they originated from reading
the original thesis, and going off on a tangent, writing down some notes or derivations, then
putting those down in Powerpoint
¨ Apologies for the Powerpoint format, it is a result of working in Finance for too long, even
though I have to say that I have learnt to appreciate the PowerPoint Equation Editor
¨ In many ways, those notes are nothing more than a rather pedestrian derivations in many
pages of what Bachelier did in a line or two, but they also present what I hope is a rather
extensive “bag of tricks” that one need to have handy when dealing with option theories.
¨ Pages numbers usually refer the ones in the Davis & Etheridge book, but I am not to the
point where I can produce a rigorous index or references list
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When in doubt, go to the book
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Bachelier’s thesis : March 29th, 1900
¨ Louis Bachelier defended his Ph.D. thesis in front of Paul Appell, Henri Poincarre and Joseph
Boussinesq, a formidable trio of “mousquetaires”.
¨ It is also quite remarkable that the “second oral presentation” that Bachelier had to do was
on the matter of the “Resistance of a sphere in a liquid” under Boussinesq supervision, 5
years before the seminal paper by Einstein that relates the thermal fluctuations to the
viscous dissipation (a precursor of the fluctuation-dissipation theorem) through the diffusion
constant: 𝐷 =
!!.#
$%&'
, where 𝑇is here the temperature, 𝑅 the radius of the sphere, the fluid
viscosity 𝜂 and 𝑘( is the famous Boltzmann constant.
¨ So the first part of Bachelier thesis dealt with stochastic processes in Finance
¨ The second part dealt with stochastic processes in Physics
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A few humbling examples
¨ I am using the pages of the Davis and Etheridge book
¨ In page 16, Bachelier fully describes contango and backwardation
¨ On page 34, Bachelier works out the continuous limit of a binomial process through the
Stirling formula
¨ On page 40, Bachelier derives the heat equation (Fourier equation) through a Taylor
expansion of the probability flow
¨ On page 44, Bachelier essentially derives the now celebrated Dupire formula (1994)
¨ On page 45, Bachelier derives the now common proxy for at-the-money options
¨ On page 66, Bachelier uses the Reflection principle to recover one of the most intriguing and
beautiful property of a Brownian motion: The probability that a price will be attained or
exceeded at time t is half the probability that the price will be attained or exceeded during
the interval of time up to t.
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A few humbling examples - II
¨ On page 70, Bachelier looks at some properties around first passage time, and shows that
the expected value is infinite (a version of the Doob paradox of 1948)
¨ On page 73, Bachelier uses the method of images from Lord Kelvin, essentially the backbone
for valuing simple barrier options (Carr, Reiner, Rubinstein, Haug) from the 1990
¨ The examples are too numerous, and I would not do justice to the way it is presented in the
David and Etheridge book, especially Chapter III
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Oil prices went negative
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Rates have been negative for a while
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Also spread have also been negative for a while
¨ Spread option pricing models allow for spreads (difference between two indices) to be
negative.
¨ One of the most infamous was the curve inversion in June 2008 in Europe between the 2
year swap and the 30 year swap (units are in % on the right scale)
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Black-Sholes does not allow for negative prices
¨ The lognormal distribution only allows for positive asset prices.
¨ The Normal distribution allows for negative prices, hence when Black-Sholes was developed
in the context of stocks and bonds, the geometric Brownian motion (lognormal) was
favored.
¨ Also it offered the advantage to lend itself nicely to change of numeraires, as the inverse of a
geometric process, powers, ratio and products will also be geometric. It does however,
because it is a non-linear function of a Brownian motion, require the full fledged Ito calculus
and Ito lemma (1951)
¨ In the 1990s mostly out of Japan in the rate space, people started hitting the limits of
Lognormal models. The easy way out of it was shifted-lognormal implementations, which
essentially translate the variable
¨ Just for completeness, I have included in the next slides the closed forms for Lognormal,
Normal and shifted lognormal (setting to 0 the rates, cost of carry, dividends,..) in order to
preserve the essence of the formula. Also of interest are the incremental PL in a Taylor
expansion and the Vega scaling (Greeks normalized to Vega), quite a crucial point when
trying to capture the impact of stochastic volatilities
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12
Some Equations (Lognormal Black-Scholes)
ò¥-
-
=
-=
+=
-=
x
dexN
Tdd
T
T
KFLn
d
dNKdNFTKFC
x
p
s
s
s
s
x
..
2
1
)(
2
1)(
)(.)(.),,,(
)
2
1
(
12
1
21
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Luc_Faucheux_2020
13
Greeks and Scaling in the Lognormal Model
Greeks Definition Black formula Units Incremental P/L Vega scaling
Delta
F
C
¶
¶
=D )( 1dN ($/bp) )( FdD
Gamma
2
2
F
C
¶
¶
=g )('
1
1dN
TFs
($/bp/bp) 2
)(
2
1
Fdg
TF s2
1
Theta
T
C
¶
¶
=Q )('
2
2dN
T
TKs ($/day) )( TdQ
T2
s
Vega
s¶
¶
=
C
Vega )(' 2dNTK ($/%) )(dsVega 1
Vanna
s¶¶
¶
=
F
C
Vanna
2
)('' 2dN
F
K
s
($/%/bp) ))(( dsdFVanna
TF
d
s
2-
Volga
2
2
s¶
¶
=
C
Volga )('' 2
1
dNTK
d
s
-
($/%/%) 2
)(
2
1
dsVolga
s
21dd

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14
Normal Model
{ }
T
KF
d
dNddNTTKFC
N
NN
s
ss
)(
)(.)('),,,(
-
=
+=

Luc_Faucheux_2020
15
Greeks and Scaling in the Normal Model
Delta
F
C
¶
¶
=D )(dN ($/bp) )( FdD
Gamma
2
2
F
C
¶
¶
=g )('
1
dN
TNs
($/bp/bp) 2
)(
2
1
Fdg
TNs
1
Theta
T
C
¶
¶
=Q )('
2
dN
T
TNs ($/day) )( TdQ
T
N
2
s
Vega
N
C
Vega
s¶
¶
= )(' dNT ($/%) )( NVega ds 1
Vanna
NF
C
Vanna
s¶¶
¶
=
2
)(''
1
dN
Ns
($/%/bp) ))(( NFVanna dsd
T
d
Ns
-
Volga
2
2
N
C
Volga
s¶
¶
= )('' dNT
d
Ns
-
($/%/%) 2
)(
2
1
NVolga ds
N
d
s
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Luc_Faucheux_2020
Shifted Lognormal Model
¨ Shifted Lognormal model with shift 𝛽:
¨ 𝐶 𝐹, 𝐾, 𝑇, 𝜎)* = 𝐹. 𝑁 𝑑+ − 𝐾. 𝑁(𝑑,)
¨ 𝑑+ =
+
-"# #
𝐿𝑛(
./0
1/0
) +
+
,
𝜎)* 𝑇
¨ 𝑑, = 𝑑+ − 𝜎)* 𝑇
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Luc_Faucheux_2020
Greeks and Scaling in the shifted Lognormal Model
17
Delta ($/bp)
Gamma ($/bp/bp)
Theta ($/day)
Vega ($/%) 1
Vanna ($/%/bp)
Volga ($/%/%)
F
C
¶
¶
=D )( 1dN )( FdD
2
2
F
C
¶
¶
=g
TF
dN
Ssb )(
)(' 1
+
2
)(
2
1
Fdg 2
)(
11
bs +FTS
T
C
¶
¶
=Q )('
2
)(
2dN
T
TK Ssb+
)( TdQ
T
S
2
s
S
C
Vega
s¶
¶
= )(')( 2dNTK b+ )( SVega ds
SF
C
Vanna
s¶¶
¶
=
2
)(''
1
)(
)(
2dN
F
K
Ssb
b
+
+
))(( SFVanna dsd
TF
d
Ssb
1
)(
2
+
-
2
2
S
C
Volga
s¶
¶
= )('')( 2
1
dNTK
d
S
b
s
+
- 2
)(
2
1
SVolga ds
S
dd
s
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The Ph.D. thesis of Louis Bachelier
¨ Reading the original thesis (both in French if you can and the excellent translation by Mark
Davis and Alison Etheridge) is humbling.
¨ Without a strong well-developed theory of stochastic calculus (Ito lemma) that only came
about in the 1960s or so
¨ Without a strong theoretical footing of what is a numeraire and how to price a derivative in
the risk-neutral probability associated to that numeraire (Pliska 1980 or so)
¨ Without yet the strong connection between PDE (Partial Differential Equations) and SDE
(Stochastic Differential Equations) that really came about from the Feynman-Kac formula
(1950 roughly)
¨ Louis Bachelier managed to not only built a theory of option pricing that is nowadays
coming back in fashion with a vengeance, but perusing through the rather short thesis, one
cannot but be amazed at the breadth of his genius, but also at his attention to details.
Bachelier at times go through numerical examples with the same precision and clarity of
thoughts that he displays in the other more theoretical parts of his thesis.
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De l’equation de Kolmogorov a une
solution Gaussienne
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Kolmogorov equation: Bachelier thesis (page 35)
¨ 𝑃 𝑥, 𝑡 . 𝑑𝑥 is the probability that the price is in 𝑥, 𝑥 + 𝑑𝑥 at time 𝑡
¨ 𝑃 𝑥, 𝑡+ . 𝑃 𝑧 − 𝑥, 𝑡, − 𝑡+ . 𝑑𝑥. 𝑑𝑧 is the probability that the price is (𝑥, 𝑡+) and (𝑧, 𝑡,)
¨ 𝑃 𝑧, 𝑡,|𝑥, 𝑡+ = 𝑃 𝑧 − 𝑥, 𝑡, − 𝑡+
¨ Note that just writing something like the above implies lot of things:
¨ Strong Markov property
¨ The price process is memoryless
¨ The price process is homogeneous in time and space
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Kolmogorov equation: Bachelier thesis (page 35) - b
¨ What it is saying if you break it down is:
¨ There must be a function 𝑃 𝑥, 𝑡 that is the probability density to find the price (particle,
random walker, stochastic process) at 𝑥 at time 𝑡
¨ This assumes that it is a function, that we can find, and one which we can perform usual
calculus (not completely obvious)
¨ It is then saying that before reaching the point (𝑥, 𝑡) the price might have reached another
level at some time before (rather obvious, but again has some mathematical consequences)
¨ Bachelier for some typographical reasons used 𝑥 and 𝑧, which we will stick to in some of the
following slides, but for ease of notations here:
¨ The probability density to reach 𝑥, 𝑡 is 𝑃 𝑥, 𝑡
¨ The probability density to reach 𝑥′, 𝑡′ is ALSO the same function 𝑃 𝑥′, 𝑡′
¨ The conditional probability density to go from 𝑥′, 𝑡′ to 𝑥, 𝑡 is ALSO assumed to be some
function that we will note 𝑃𝑅 𝑥2, 𝑡2, 𝑥, 𝑡
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Kolmogorov equation: Bachelier thesis (page 35) - c
¨ In order to recover 𝑃 𝑥, 𝑡 , we can sum over all the possible in-between states
¨ 𝑃 𝑥, 𝑡 = ∫32456
324/6
∫7$48
7$47
𝑃 𝑥′, 𝑡′ . 𝑃𝑅 𝑥2, 𝑡2, 𝑥, 𝑡 . 𝑑𝑥2. 𝑑𝑡′
¨ Graphically this creates somewhat of a Feynman diagram
¨ NOW (and again, either this is painfully obvious or rather deep and we need to pay attention
to), we can actually SEPARATE the space and the time variable (because we are dealing with
a well defined process 𝑋(𝑡)) (see next slide)
¨ So for a given time 𝑡′ we suppose that we can write something like
¨ 𝑃 𝑥, 𝑡 = ∫32456
324/6
𝑃 𝑥′, 𝑡′ . 𝑃𝑅 𝑥2, 𝑡2, 𝑥, 𝑡 . 𝑑𝑥’
¨ NOW is the big one, we assume that we can write: 𝑃𝑅 𝑥2, 𝑡2, 𝑥, 𝑡 = 𝑃(𝑥 − 𝑥2, 𝑡 − 𝑡2)
¨ This is actually again either obvious or not at all
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Kolmogorov equation: Bachelier thesis (page 35) - c-1
¨ There is no traveling back in time, so
¨ 𝑃𝑅 𝑥2, 𝑡2, 𝑥, 𝑡 = 0 if 𝑡2> 𝑡
¨ Also no disappearing and “re-apparate”
¨ So for process from 0 to 𝑡, this process WILL have to go through every intermediate time 𝑡′
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Kolmogorov equation: Bachelier thesis (page 35) - d
¨ 𝑃𝑅 𝑥2, 𝑡2, 𝑥, 𝑡 = 𝑃(𝑥 − 𝑥2, 𝑡 − 𝑡2)
¨ First of all this is assuming that the conditional probability is the same as the probability
density:
¨ It does not matter where you are starting from, and at what time, the probability to end up
at a different level is only a function of the distance to the original starting point, and the
time lapsed
¨ FURTHERMORE, that conditional probability is exactly the probability density we are looking
for
¨ Strong Markov property
¨ The price process is memoryless
¨ The price process is homogeneous in time and space
¨ No smile, no mean reversion, no time dependent volatility, and all functions are
mathematically well behaved
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Kolmogorov equation: Bachelier thesis (page 35) II
¨ NOW Bachelier writes what is now known as the Chapman-Kolmogorov equation:
¨ 𝑃 𝑥, 𝑡+ . 𝑃 𝑧 − 𝑥, 𝑡, − 𝑡+ . 𝑑𝑥. 𝑑𝑧 is the probability that the price is (𝑥, 𝑡+) and (𝑧, 𝑡,)
¨ 𝑃 𝑧, 𝑡, = ∫3456
34/6
𝑃 𝑥, 𝑡+ . 𝑃 𝑧 − 𝑥, 𝑡, − 𝑡+ . 𝑑𝑥
¨ Bachelier actually changes the notation a little and writes it as
¨ 𝑃 𝑧, 𝑡+ + 𝑡, = ∫3456
34/6
𝑃 𝑥, 𝑡+ . 𝑃 𝑧 − 𝑥, 𝑡, . 𝑑𝑥
¨ We can take a lucky guess like Louis did and write 𝑃 𝑥, 𝑡 = 𝐴. 𝑒𝑥𝑝(−𝐵,. 𝑥,)
¨ To be more exact, 𝑃 𝑥, 𝑡 = 𝐴(𝑡). 𝑒𝑥𝑝(−𝐵(𝑡),. 𝑥,)
¨ Note that this does not guarantee the unicity of a solution, only the existence
¨ Kolmogorov expressed 30 years later or so some Germanic displeasure with what he
perceived to be a lack of mathematical rigor from Louis
¨ “Dass die Bachelierschen Betrachtungen jeder mathematischen Strenge ganzlich entbehren”
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Kolmogorov equation: Bachelier thesis (page 35) III
¨ A couple of notes on the function 𝑃 𝑥, 𝑡 = 𝐴. 𝑒𝑥𝑝(−𝐵,. 𝑥,)
¨ 𝐼 = (∫56
/6
𝑒593%
. 𝑑𝑥), = ∫56
/6
𝑒593%
. 𝑑𝑥 . ∫56
/6
𝑒59:%
. 𝑑𝑦 = ∫56
/6
∫56
/6
𝑑𝑥𝑑𝑦 𝑒59(3%/:%)
¨ 𝐼 = ∫=48
=4,%
𝑑𝜃 ∫>48
>46
𝜌. 𝑑𝜌 . 𝑒59>%
= 2𝜋. ∫>48
>46
𝜌. 𝑑𝜌 . 𝑒59>%
= 2𝜋.
5?&'(%
,9
=
,%
,9
=
%
9
¨ 𝛼 = 𝐵,
¨ 𝐼 = (∫56
/6
𝑒593%
. 𝑑𝑥), so ∫56
/6
𝑒593%
. 𝑑𝑥 =
%
9
¨ We want ∫56
/6
𝑃 𝑥, 𝑡 . 𝑑𝑥 = 1, or ∫56
/6
𝐴. 𝑒𝑥𝑝(−𝐵,. 𝑥,) . 𝑑𝑥 = 1, or 𝐴.
%
@% = 1
¨ Sooo… 𝐵 = 𝐴. 𝜋
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Luc_Faucheux_2020
Kolmogorov equation: Bachelier thesis (page 35) IV
¨ 𝑃 𝑥, 𝑡 = 𝐴. 𝑒𝑥 𝑝 −𝐵,. 𝑥, = 𝐴. 𝑒𝑥 𝑝 −𝜋𝐴,. 𝑥, because ∫56
/6
𝑃 𝑥, 𝑡 . 𝑑𝑥 = 1
¨ For simplicity of notation, Bachelier writes 𝑝+ = 𝑃(𝑥 = 0, 𝑡+)
¨ 𝑃 𝑥 = 0, 𝑡 = 𝐴 = 𝐴 𝑡
¨ 𝑃 𝑥, 𝑡 = 𝑃 𝑥 = 0, 𝑡 . exp{−𝜋. 𝑃 𝑥 = 0, 𝑡 , . 𝑥,}
¨ This is true and quite elegant, and our friend 𝜋 appears somehow mysteriously
¨ Usual Gaussian : ℎ 𝑥, 𝑡 =
+
,%-%7
. exp(
53%
,-%7
)
¨ ℎ 𝑥, 𝑡 = ℎ 0, 𝑡 . exp[−𝜋. ℎ 0, 𝑡 ,. 𝑥,]
¨ This is actually quite powerful, the Kolmogorov equation looks fairly general and obvious,
and we just proved that at least one solution of it HAS to verify:
¨ 𝑃 𝑥, 𝑡 = 𝑃 𝑥 = 0, 𝑡 . exp{−𝜋. 𝑃 𝑥 = 0, 𝑡 , . 𝑥,}
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Kolmogorov equation: Bachelier thesis (page 35) V
¨ 𝑃 𝑥, 𝑡 = 𝑃 𝑥 = 0, 𝑡 . exp{−𝜋. 𝑃 𝑥 = 0, 𝑡 , . 𝑥,}
¨ 𝑝+ = 𝑃(𝑥 = 0, 𝑡+)
¨ 𝑝, = 𝑃 𝑥 = 0, 𝑡,
¨ 𝑃 𝑧, 𝑡+ + 𝑡, = ∫3456
34/6
𝑃 𝑥, 𝑡+ . 𝑃 𝑧 − 𝑥, 𝑡, . 𝑑𝑥
¨ 𝑃 𝑧, 𝑡+ + 𝑡, = ∫3456
34/6
𝑝+. exp −𝜋. 𝑝+
, . 𝑥, . 𝑝,. exp −𝜋. 𝑝,
, . (𝑧 − 𝑥), . 𝑑𝑥
¨ 𝑃 𝑧, 𝑡+ + 𝑡, = 𝑝+ 𝑝, exp −𝜋. 𝑝,
, . 𝑧,
∫3456
34/6
exp −𝜋. 𝑝+
, + 𝑝,
, . 𝑥, + 2𝜋𝑝,
, 𝑧𝑥 . 𝑑𝑥
¨ 𝑃 𝑧, 𝑡+ + 𝑡, = 𝑝+ 𝑝, exp −𝜋𝑝,
,
𝑧, . exp
%A%
).B%
A*
%/A%
% ∫56
/6
exp[−𝜋(𝑥 𝑝+
, + 𝑝,
, −
A%
%B
A*
%/A%
%
),]. 𝑑𝑥
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Kolmogorov equation: Bachelier thesis (page 35) VI
¨ 𝑃 𝑧, 𝑡+ + 𝑡, = 𝑝+ 𝑝, exp −𝜋𝑝,
, 𝑧, . exp
%A%
).B%
A*
%/A%
% ∫56
/6
exp[−𝜋(𝑥 𝑝+
, + 𝑝,
, −
A%
%B
A*
%/A%
%
),]. 𝑑𝑥
¨ We do the change of variable: 𝑢 = 𝑥 𝑝+
, + 𝑝,
, −
A%
%B
A*
%/A%
%
, 𝑑𝑢 = 𝑑𝑥. 𝑝+
, + 𝑝,
,
¨ 𝑃 𝑧, 𝑡+ + 𝑡, =
A*A%
A*
%/A%
%
exp −𝜋𝑝,
,
𝑧, . exp
%A%
).B%
A*
%/A%
% ∫C456
C4/6
exp[−𝜋𝑢,]. 𝑑𝑢
¨ And −𝜋𝑝,
,
𝑧, +
%A%
).B%
A*
%/A%
% =
5%A%
)B%5%A%
%A*
%B%/%A%
).B%
A*
%/A%
% =
5%A%
%A*
%B%
A*
%/A%
%
¨ 𝑃 𝑧, 𝑡+ + 𝑡, =
A*A%
A*
%/A%
%
exp
5%A%
%A*
%B%
A*
%/A%
% ∫C456
C4/6
exp[−𝜋𝑢,]. 𝑑𝑢
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Luc_Faucheux_2020
Kolmogorov equation: Bachelier thesis (page 35) VII
¨ ∫C456
C4/6
exp[−𝜋𝑢,]. 𝑑𝑢 =
%
9
with 𝛼 = 𝜋
¨ 𝑃 𝑧, 𝑡+ + 𝑡, =
A*A%
A*
%/A%
%
exp
5%A%
%A*
%B%
A*
%/A%
%
¨ And we know that
¨ 𝑃 𝑥, 𝑡 = 𝑃 𝑥 = 0, 𝑡 . exp{−𝜋. 𝑃 𝑥 = 0, 𝑡 , . 𝑥,}
¨ 𝑃 𝑧, 𝑡 = 𝑃 𝑧 = 0, 𝑡+ + 𝑡, . exp{−𝜋. 𝑃 𝑧 = 0, 𝑡+ + 𝑡,
, . 𝑧,}
¨ 𝑝+ = 𝑃(𝑥 = 0, 𝑡+) and 𝑝, = 𝑃 𝑥 = 0, 𝑡,
¨ 𝑃 𝑧 = 0, 𝑡+ + 𝑡,
, =
A%
%A%
%
A*
%/A%
% or keeping the notation: 𝑝+/,
, =
A%
%A%
%
A*
%/A%
%
¨ Also quite an elegant formulation for the relationship between the peaks (maximum of
probability) for different times
30

Luc_Faucheux_2020
Kolmogorov equation: Bachelier thesis (page 35) VIII
¨ 𝑝+/,
, =
A%
%A%
%
A*
%/A%
%, where 𝑝+ = 𝑃(𝑥 = 0, 𝑡+), 𝑝, = 𝑃 𝑥 = 0, 𝑡,
¨ So to make it simpler let’s use the notation 𝑝 = 𝑝(𝑡)
¨ 𝑝(𝑡+ + 𝑡,), = 𝑝+/,
,
=
A(7*)%A(7%)%
A(7*)%/A(7%)%
¨ Method #1 : let’s be lucky and guess that 𝑝 𝑡 = 𝐻. 𝑡 ⁄&*
% =
E
7
¨ 𝑝(𝑡), =
E%
7
¨
A(7*)%A(7%)%
A(7*)%/A(7%)% =
(
+%
,*
)(
+%
,%
)
+%
,*
/(
+%
,%
)
= 𝐻,.
+
7*7%
.
+
*
,*
/
*
,%
= 𝐻,.
+
7*7%
.
7*7%
7*/7%
= 𝐻,.
+
7*/7%
= 𝑝(𝑡+ + 𝑡,),
¨ It works !
31

Luc_Faucheux_2020
Kolmogorov equation: Bachelier thesis (page 35) VIII
¨ 𝑝 𝑡 = 𝐻. 𝑡 ⁄&*
% =
E
7
= 𝑃(𝑥 = 0, 𝑡)
¨ 𝑃 𝑥, 𝑡 = 𝑃 𝑥 = 0, 𝑡 . exp{−𝜋. 𝑃 𝑥 = 0, 𝑡 , . 𝑥,}
¨ So we finally have what we are looking for:
¨ 𝑃 𝑥, 𝑡 =
E
7
. exp{−
%E%3%
7
}
¨ We just need to normalize one more time: ∫56
/6
𝑃 𝑥, 𝑡 . 𝑑𝑥 = 1
¨ We already know that : 𝐼 = (∫56
/6
𝑒593%
. 𝑑𝑥), =
%
9
so ∫56
/6
𝑒593%
. 𝑑𝑥 =
%
9
¨ 𝛼 =
%E%
7
, so ∫56
/6
𝑃 𝑥, 𝑡 . 𝑑𝑥 = ∫56
/6 E
7
. exp{−
%E%3%
7
} . 𝑑𝑥 =
E
7
.
%
-+%
,
= 1
32

Luc_Faucheux_2020
Kolmogorov equation: Bachelier thesis (page 35) IX
¨ 𝑃 𝑥, 𝑡 =
E
7
. exp{−
%E%3%
7
} is already normalized
¨ We still need to solve for the value of 𝐻
¨ A couple of side notes first on 𝐼9 = ∫56
/6
𝑒593%
. 𝑑𝑥 =
%
9
¨ 𝑃9 𝑥 =
+
F'
. 𝑒593%
is the normalized probability distribution
¨ < 𝑥 > = ∫56
/6
𝑥. 𝑃9 𝑥 . 𝑑𝑥 = 0 because 𝑥. 𝑃9 𝑥 is an odd function
¨ < 𝑥! > = ∫56
/6
𝑥!. 𝑃9 𝑥 . 𝑑𝑥 = 0 because (𝑥!. 𝑃9 𝑥 ) is an odd function if 𝑘 is odd
33

Luc_Faucheux_2020
Kolmogorov equation: Bachelier thesis (page 35) X
¨ < 𝑥, > = ∫56
/6
𝑥,. 𝑃9 𝑥 . 𝑑𝑥 =
%
9
. ∫56
/6
𝑥,. 𝑒593%
. 𝑑𝑥
¨ Now:
G
G9
𝑒593%
= −𝑥, 𝑒593%
¨ So: < 𝑥, > =
%
9
. ∫56
/6
𝑥,. 𝑒593%
. 𝑑𝑥 =
%
9
. ∫56
/6 5G
G9
𝑒593%
. 𝑑𝑥 =
%
9
.
5G
G9
[∫56
/6
𝑒593%
. 𝑑𝑥]
¨ A little more formally:
¨ < 𝑥, > =
5+
F'
.
GF'
G9
¨ Replacing 𝐼9 =
%
9
, we get < 𝑥, > =
5+
-
'
.
G
-
'
G9
= − 𝛼.
G
G9
𝛼 ⁄&*
% =
+
,
. 𝛼 ⁄*
%. 𝛼 ⁄&.
% =
+
,9
34

Luc_Faucheux_2020
Kolmogorov equation: Bachelier thesis (page 35) XI
¨ < 𝑥, > =
+
,9
¨ < 𝑥,! > =
+
F'
. ∫56
/6
𝑥,!. 𝑒593%
. 𝑑𝑥 and again 𝐼9 = ∫56
/6
𝑒593%
. 𝑑𝑥 =
%
9
¨ It is easy to see that (𝑥,!. 𝑒593%
) =
G/
G9/ 𝑒593%
. (−1)!
¨ < 𝑥,! > =
+
F'
. ∫56
/6 G/
G9/ 𝑒593%
. (−1)! . 𝑑𝑥 =
+
F'
.
H/
H9/ 𝐼9 . (−1)!
¨ < 𝑥,! > =
+
F'
.
H/
H9/ 𝐼9 . (−1)!=
+
F'
.
H/
H9/ 𝐼9 . (−1)!= 𝛼 ⁄*
%.
H/
H9/ 𝛼 ⁄&*
% . (−1)!
¨
H/
H9/ 𝛼 ⁄&*
% = 𝛼 ⁄&*
%. 𝛼5!. ∏I4+
I4!
(
+
,
+ 𝑗 − 1) . (−1)!
¨ < 𝑥,! > = 𝛼5!. ∏I4+
I4!
(
+
,
+ 𝑗 − 1), with 𝑘 = 1, we recover indeed < 𝑥, > =
+
,9
35

Luc_Faucheux_2020
Kolmogorov equation: Bachelier thesis (page 35) XII
¨ 𝐼9 = ∫56
/6
𝑒593%
. 𝑑𝑥 =
%
9
¨ 𝑃9 𝑥 =
+
F'
. 𝑒593%
is the normalized probability distribution
¨ < 𝑥! > = ∫56
/6
𝑥!. 𝑃9 𝑥 . 𝑑𝑥
¨ < 𝑥,! > = 𝛼5!. ∏I4+
I4!
(
+
,
+ 𝑗 − 1)
¨ < 𝑥,!/+ > = 0
¨ We will also look like Bachelier did at the positive part of the price distribution
¨ < (𝑥!|𝑥 > 0) > = ∫8
/6
𝑥!. 𝑃9 𝑥 . 𝑑𝑥
36

Luc_Faucheux_2020
Kolmogorov equation: Bachelier thesis (page 35) XIII
¨ < 𝑥,! > = 𝛼5!. ∏I4+
I4!
(
+
,
+ 𝑗 − 1)
¨ For the regular Gaussian ℎ 𝑥, 𝑡 =
+
,%-%7
. exp(
53%
,-%7
) we have 𝛼 =
+
,-%7
¨ A somewhat useful notation:
¨ 𝑘! = ∏I4+
I4!
𝑗 is the usual factorial
¨ 𝑘!! = ∏I4+
I4!
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 ∏I4+
I4!
(
+
,
+ 𝑗 − 1) as:
¨ ∏I4+
I4!
(
+
,
+ 𝑗 − 1) = ∏I4+
I4!
(
,I5+
,
) = 25! ∏I4+
I4!
(2𝑗 − 1) = 25!. 2𝑘 − 1 ‼
¨ < 𝑥,! > = 𝛼5!. 25!. 2𝑘 − 1 ‼
37

Luc_Faucheux_2020
Kolmogorov equation: Bachelier thesis (page 35) XIV
¨ < 𝑥,! > = 𝛼5!. 25!. 2𝑘 − 1 ‼
¨ In the case of the Gaussian, 𝛼 =
+
,-%7
, so < 𝑥,! > = (2𝜎, 𝑡)!. 25!. 2𝑘 − 1 ‼
¨ So : < 𝑥,! > = (𝜎, 𝑡)!. 2𝑘 − 1 ‼ and < 𝑥,!/+ > = 0
¨ Another cute way to express it is the following:
¨ < 𝑥J > = (𝜎 𝑡)J. 𝑛 − 1 ‼ if 𝑛 is even, 0 otherwise
¨ This is quite compact and beautiful
¨ < 𝑥J|𝑥 > 0 > =
+
,
(𝜎 𝑡)J. 𝑛 − 1 ‼ if 𝑛 is even, “something else” otherwise
¨ Let’s try to calculate that “something else”
38

Luc_Faucheux_2020
Kolmogorov equation: Bachelier thesis (page 35) XV
¨ < 𝑥|𝑥 > 0 > = ∫8
/6
𝑥. 𝑃9 𝑥 . 𝑑𝑥 =
+
F'
. ∫8
/6
𝑥. 𝑒593%
. 𝑑𝑥
¨
G
G9
𝑒593%
= −𝑥,. 𝑒593%
¨
G
G3
𝑒593%
= −2𝑥𝛼. 𝑒593%
¨ < 𝑥|𝑥 > 0 > =
+
F'
. ∫8
/6
𝑥. 𝑒593%
. 𝑑𝑥 =
+
F'
. ∫8
/6 +
,9
5G
G3
𝑒593%
. 𝑑𝑥 =
+
F'
.
+
,9
[−𝑒593%
]348
346
¨ < 𝑥|𝑥 > 0 > =
+
F'
.
+
,9
−𝑒593%
348
346
=
+
,9F'
and 𝐼9 =
%
9
¨ < 𝑥|𝑥 > 0 > =
+
, %
. 𝛼 ⁄&*
% =
+
K%9
¨ In the Gaussian case, 𝛼 =
+
,-%7
and so : < 𝑥|𝑥 > 0 > =
-%7
,%
(Bachelier page 38)
39

Luc_Faucheux_2020
Kolmogorov equation: Bachelier thesis (page 35) XVI
¨ For the regular Gaussian ℎ 𝑥, 𝑡 =
+
,%-%7
. exp(
53%
,-%7
) we have 𝛼 =
+
,-%7
¨ Bachelier likes to use exp(
53%
K%!%7
) so 𝜎, = 2𝜋𝑘,
¨ So ℎ 𝑥, 𝑡 =
+
,%! 7
. exp(
53%
K%!%7
), a little more concise
¨ Because now for example:
¨ < 𝑥|𝑥 > 0 > =
-%7
,%
= 𝑘 𝑡, quite concise and beautiful !
¨ Note also that < 𝑥 > =< 𝑥 𝑥 > 0 > +< −𝑥 𝑥 < 0 > = 2. < 𝑥|𝑥 > 0 >
¨ And so: < 𝑥 > = 2. 𝑘 𝑡 = 2.
-%7
,%
= 𝜎, 𝑡.
,
%
=
+
%9
40

Luc_Faucheux_2020
Kolmogorov equation: Bachelier thesis (page 35) XVII
¨ More generally:
¨ < 𝑥,!| 𝑥 > 0 > = ∫8
/6
𝑥,!. 𝑃9 𝑥 . 𝑑𝑥 =
+
,
∫56
/6
𝑥,!. 𝑃9 𝑥 . 𝑑𝑥
¨ Because 𝑥,!. 𝑃9 𝑥 is an even function
¨ If you are not convinced and want to do it the hard way:
¨ < 𝑥,!| 𝑥 > 0 > = ∫8
/6
𝑥,!. 𝑃9 𝑥 . 𝑑𝑥 =
+
F'
. ∫8
/6
𝑥,!. 𝑒593%
. 𝑑𝑥
¨
G
G9
𝑒593%
= −𝑥,. 𝑒593%
¨
G/
G9/ 𝑒593%
= (−1)!. 𝑥,!. 𝑒593%
¨ < 𝑥,!| 𝑥 > 0 > =
+
F'
. ∫8
/6
𝑥,!. 𝑒593%
. 𝑑𝑥 =
(5+)/
F'
. ∫8
/6 G/
G9/ 𝑒593%
. 𝑑𝑥
41

Luc_Faucheux_2020
Kolmogorov equation: Bachelier thesis (page 35) XVIII
¨ < 𝑥,!| 𝑥 > 0 > =
+
F'
. ∫8
/6
𝑥,!. 𝑒593%
. 𝑑𝑥 =
(5+)/
F'
. ∫8
/6 G/
G9/ 𝑒593%
. 𝑑𝑥
¨ < 𝑥,!| 𝑥 > 0 > =
(5+)/
F'
.
H/
H9/ ∫8
/6
𝑒593%
. 𝑑𝑥
¨ And 𝐼9 = ∫56
/6
𝑒593%
. 𝑑𝑥 =
%
9
¨ Let’s call 𝐼9
/ = ∫8
/6
𝑒593%
. 𝑑𝑥 =
+
,
.
%
9
trivially, or 𝐼9
/ =
+
,
. 𝐼9
¨ < 𝑥,!| 𝑥 > 0 > =
(5+)/
F'
.
H/
H9/ ∫8
/6
𝑒593%
. 𝑑𝑥 =
(5+)/
F'
.
H/
H9/ 𝐼9
/ =
+
,
.
(5+)/
F'
.
H/
H9/ 𝐼9
¨ So < (𝑥,! 𝑥 > 0 > =
+
,
. < 𝑥,! >
42

Luc_Faucheux_2020
Kolmogorov equation: Bachelier thesis (page 35) XIX
¨ < 𝑥,!/+| 𝑥 > 0 > =
+
F'
. ∫8
/6
𝑥,!/+. 𝑒593%
. 𝑑𝑥 =
+
F'
. ∫8
/6
𝑥,!. 𝑥. 𝑒593%
. 𝑑𝑥
¨
G
G3
𝑒593%
= −2𝑥𝛼. 𝑒593%
¨ < 𝑥,!/+| 𝑥 > 0 > =
+
F'
. ∫8
/6
𝑥,!. 𝑥. 𝑒593%
. 𝑑𝑥 =
+
F'
. ∫8
/6
𝑥,!.
5+
,9
G
G3
𝑒593%
. 𝑑𝑥
¨ After integration by parts
¨ < 𝑥,!/+| 𝑥 > 0 > =
5+
,9F'
. 𝑥,!. 𝑒593%
8
6
+
+
,9F'
. ∫8
/6
2𝑘 . 𝑥,!5+. 𝑒593%
. 𝑑𝑥
¨ < 𝑥,!/+| 𝑥 > 0 > =
!
9F'
. ∫8
/6
𝑥,!5+. 𝑒593%
. 𝑑𝑥
¨ < 𝑥,!/+| 𝑥 > 0 > =
!
9
. < 𝑥,!5+| 𝑥 > 0 > and < 𝑥|𝑥 > 0 > =
+
, %
. 𝛼 ⁄&*
% =
+
K%9
43

Luc_Faucheux_2020
Kolmogorov equation: Bachelier thesis (page 35) XX
¨ < 𝑥,!/+ 𝑥 > 0 > =
!
9
. < 𝑥,!5+ 𝑥 > 0 > =
,!
,9
. < 𝑥,!5+| 𝑥 > 0 >
¨ For sake of notation we write 𝐴 2𝑘 + 1 =< 𝑥,!/+| 𝑥 > 0 >
¨ 𝐴 2𝑘 + 1 =
!
9
. 𝐴 2𝑘 − 1
¨ 𝐴 2𝑘 − 1 =
!5+
9
. 𝐴 2𝑘 − 3
¨ 𝐴 2𝑘 − 3 =
!5,
9
. 𝐴 2𝑘 − 5 ….
¨ 𝐴 3 =
+
9
. 𝐴 1 and 𝐴 1 =< 𝑥|𝑥 > 0 > =
+
, %
. 𝛼 ⁄&*
% =
+
K%9
44

Luc_Faucheux_2020
Kolmogorov equation: Bachelier thesis (page 35) XXI
¨ < 𝑥,!/+ 𝑥 > 0 > =
!
9
. < 𝑥,!5+ 𝑥 > 0 > =
,!
,9
. < 𝑥,!5+| 𝑥 > 0 >
¨ For sake of notation we write 𝐴 2𝑘 + 1 =< 𝑥,!/+| 𝑥 > 0 >
¨ Easier to see that:
¨ 𝐴 2𝑘 + 1 =
,!
,9
. 𝐴 2𝑘 − 1
¨ 𝐴 2𝑘 − 1 =
,!5,
,9
. 𝐴 2𝑘 − 3
¨ 𝐴 2𝑘 − 3 =
,!5K
,9
. 𝐴 2𝑘 − 5 ….
¨ 𝐴 3 =
,
,9
. 𝐴 1 and 𝐴 1 =< 𝑥|𝑥 > 0 > =
+
, %
. 𝛼 ⁄&*
% =
+
K%9
¨ 𝐴 2𝑘 + 1 =
,! ‼
(,9)/ . 𝐴(1) where 2𝑘 ‼ is the double factorial
45

Luc_Faucheux_2020
Kolmogorov equation: Bachelier thesis (page 35) XXII
¨ 𝐴 2𝑘 + 1 =
,! ‼
(,9)/ . 𝐴(1) where 2𝑘 ‼ is the double factorial
¨ Rewriting it as : 𝐴 𝑝 =< 𝑥A| 𝑥 > 0 > and 𝐴 1 =< 𝑥|𝑥 > 0 > =
+
, %
. 𝛼 ⁄&*
% =
+
K%9
¨ 𝐴 𝑝 =
A5+ ‼
(,9)/ . 𝐴(1) where 𝑝 − 1 ‼ is the double factorial AND 𝑝 = 2𝑘 + 1
¨ In the Gaussian case : 𝛼 =
+
,-%7
¨ Using Bachelier’s convention, 𝛼 =
+
K%!%7
, or 𝜎, = 2𝜋𝑘, (careful Bachelier 𝑘 is NOT our
integer)
¨ Recall that we had : < 𝑥J > = (𝜎 𝑡)J. 𝑛 − 1 ‼ if 𝑛 is even, 0 otherwise
¨ So we are trying to express < 𝑥J| 𝑥 > 0 > in a similar fashion
46

Luc_Faucheux_2020
Kolmogorov equation: Bachelier thesis (page 35) XXIII
¨ 𝐴 𝑛 =
J5+ ‼
(,9)/ . 𝐴(1) where 𝑝 − 1 ‼ is the double factorial AND 𝑛 = 2𝑘 + 1
¨ In the Gaussian case : 𝛼 =
+
,-%7
and 𝐴 1 =
+
K%9
=
+
,%
. (𝜎 𝑡)
¨ < 𝑥J| 𝑥 > 0 > = 𝑛 − 1 ‼ 2𝛼 5!.
+
,%
. (𝜎 𝑡)
¨ < 𝑥J| 𝑥 > 0 > = 𝑛 − 1 ‼ (𝜎, 𝑡)
0
%
5+
.
+
,%
. 𝜎 𝑡 = 𝑛 − 1 ‼ (𝜎 𝑡)J5
*
%.
+
,%
. (𝜎 𝑡)
¨ < 𝑥J| 𝑥 > 0 > = 𝑛 − 1 ‼ 𝜎 𝑡
J
.
+
,%
=
+
,%
. < 𝑥J > (in the case of 𝑛 being odd)
¨ ALSO < |𝑥J > = 2. < 𝑥J 𝑥 > 0 > =
,
%
. 𝑛 − 1 ‼ 𝜎 𝑡
J
(in the case of 𝑛 being odd)
47

Luc_Faucheux_2020
Kolmogorov equation: Bachelier thesis (page 35) XXIV
¨ So… to recap...
¨ < 𝑥J > = (𝜎 𝑡)J. 𝑛 − 1 ‼ if 𝑛 is even, 0 otherwise
¨ < |𝑥|J > = (𝜎 𝑡)J. 𝑛 − 1 ‼ if 𝑛 is even, (𝜎 𝑡)J. 𝑛 − 1 ‼ (
,
%
) otherwise
¨ < 𝑥J| 𝑥 > 0 > =
+
,
(𝜎 𝑡)J. 𝑛 − 1 ‼ if 𝑛 is even, (𝜎 𝑡)J. 𝑛 − 1 ‼ (
+
,%
) otherwise
¨ Super useful and quite elegant, once again…and using Bachelier 𝜎, = 2𝜋𝑘,
¨ < 𝑥J > = (𝑘 2𝜋𝑡)J. 𝑛 − 1 ‼ if 𝑛 is even, 0 otherwise
¨ < 𝑥J| 𝑥 > 0 > =
+
,
(𝑘 2𝜋𝑡)J. 𝑛 − 1 ‼ if 𝑛 is even,
+
,
(𝑘 2𝜋𝑡)J. 𝑛 − 1 ‼
,
%
otherwise
¨ In particular for 𝑛 = 1, < 𝑥| 𝑥 > 0 > = 𝑘 𝑡
48

Luc_Faucheux_2020
Kolmogorov equation: Bachelier thesis (page 35) (VIII)-XXV
¨ 𝑝+/,
, =
A%
%A%
%
A*
%/A%
%, where 𝑝+ = 𝑃(𝑥 = 0, 𝑡+), 𝑝, = 𝑃 𝑥 = 0, 𝑡,
¨ So to make it simpler let’s use the notation 𝑝 = 𝑝(𝑡)
¨ 𝑝(𝑡+ + 𝑡,), = 𝑝+/,
,
=
A(7*)%A(7%)%
A(7*)%/A(7%)%
¨ Method #1 : let’s be lucky and guess that 𝑝 𝑡 = 𝐻. 𝑡 ⁄&*
% =
E
7
¨ 𝑝(𝑡), =
E%
7
¨
A(7*)%A(7%)%
A(7*)%/A(7%)% =
(
+%
,*
)(
+%
,%
)
+%
,*
/(
+%
,%
)
= 𝐻,.
+
7*7%
.
+
*
,*
/
*
,%
= 𝐻,.
+
7*7%
.
7*7%
7*/7%
= 𝐻,.
+
7*/7%
= 𝑝(𝑡+ + 𝑡,),
¨ It works !
49

Luc_Faucheux_2020
Kolmogorov equation: Bachelier thesis (page 35) XXVI
¨ Let’s redo it the way Bachelier did it, using ODE (Ordinary Differential Equations)
¨ 𝑝(𝑡+ + 𝑡,), = 𝑝+/,
,
=
A(7*)%A(7%)%
A(7*)%/A(7%)% and let’s take the partial derivatives to 𝑡+and 𝑡,
¨
G
G7*
yields:
¨ 2𝑝2 𝑡+ + 𝑡, 𝑝 𝑡+ + 𝑡, = 𝑝 𝑡,
,{
,A$ 7* A 7*
A 7*
%/A 7%
% +
5+ .A 7*
%.,A$ 7* A 7*
{A 7*
%/A 7%
%}% }
¨
G
G7%
yields:
¨ 2𝑝2 𝑡+ + 𝑡, 𝑝 𝑡+ + 𝑡, = 𝑝 𝑡+
,{
,A$ 7% A 7%
A 7*
%/A 7%
% +
5+ .A 7%
%.,A$ 7% A 7%
{A 7*
%/A 7%
%}% }
50

Luc_Faucheux_2020
Kolmogorov equation: Bachelier thesis (page 35) XXVII
¨
G
G7*
yields:
¨ 2𝑝2 𝑡+ + 𝑡, 𝑝 𝑡+ + 𝑡, = 𝑝 𝑡,
,{
,A$ 7* A 7*
A 7*
%/A 7%
% +
5+ .A 7*
%.,A$ 7* A 7*
{A 7*
%/A 7%
%}% }
¨ 2𝑝2 𝑡+ + 𝑡, 𝑝 𝑡+ + 𝑡, = 𝑝 𝑡,
,{
,A$ 7* A 7* .A 7*
%/,A$ 7* A 7* .A 7%
%
{A 7*
%/A 7%
%}% +
5+ .A 7*
%.,A$ 7* A 7*
{A 7*
%/A 7%
%}% }
¨ 2𝑝2 𝑡+ + 𝑡, 𝑝 𝑡+ + 𝑡, = 𝑝 𝑡,
, ,A$ 7* A 7* .A 7%
%
{A 7*
%/A 7%
%}%
¨
G
G7%
yields: 2𝑝2 𝑡+ + 𝑡, 𝑝 𝑡+ + 𝑡, = 𝑝 𝑡+
, ,A$ 7% A 7% .A 7*
%
{A 7*
%/A 7%
%}%
¨ And so :
¨ 𝑝 𝑡,
, ,A$ 7* A 7* .A 7%
%
{A 7*
%/A 7%
%}% = 𝑝 𝑡+
, ,A$ 7% A 7% .A 7*
%
{A 7*
%/A 7%
%}%
51

Luc_Faucheux_2020
Kolmogorov equation: Bachelier thesis (page 35) XXVIII
¨ 𝑝 𝑡,
, ,A$ 7* A 7* .A 7%
%
{A 7*
%/A 7%
%}% = 𝑝 𝑡+
, ,A$ 7% A 7% .A 7*
%
{A 7*
%/A 7%
%}%
¨ 𝑝2 𝑡+ 𝑝 𝑡+ . 𝑝 𝑡,
K = 𝑝2 𝑡, 𝑝 𝑡, . 𝑝 𝑡+
K
¨
A$ 7*
A 7*
. =
A$ 7%
A 7%
. for all values of 𝑡+ and 𝑡,
¨ So
A$ 7
A 7 . = 𝑐𝑡𝑒 and
A$ 7
A 7 . = (
5+
,
)
H
H7
[
+
A 7 %]
¨ So
H
H7
+
A 7 % = 𝑐𝑡𝑒 = 𝛼 and
+
A 7 % = 𝛼𝑡 + 𝛽
¨ We can choose 𝛽 = 0 and rewrite
+
A 7 % = 𝛼𝑡 as 𝑝 𝑡 = zE
7
¨ And so we are back to the expression for the distribution with the explicit peak value
52

Luc_Faucheux_2020
Kolmogorov equation: Bachelier thesis (page 35) XXIX
¨ 𝑝 𝑡 = 𝐻. 𝑡 ⁄&*
% =
E
7
= 𝑃(𝑥 = 0, 𝑡)
¨ 𝑃 𝑥, 𝑡 = 𝑃 𝑥 = 0, 𝑡 . exp{−𝜋. 𝑃 𝑥 = 0, 𝑡 , . 𝑥,}
¨ So we finally have what we are looking for:
¨ 𝑃 𝑥, 𝑡 =
E
7
. exp{−
%E%3%
7
}
¨ Note how general the assumptions we made seem to be
¨ Note also that we have the existence of a solution, but we have said nothing about unicity
53

Luc_Faucheux_2020
Discrete to continuous – Radiation of probability
Le Rayonement de Probabilities
(pages 39 – 40)
54

Luc_Faucheux_2020
From the coin toss to the random walker
¨ Let us limit ourselves to a one-dimensional random walk
¨ A random walker will jump to the right or the left by one unit 𝜀 at equal time intervals 𝜏
¨ We assume equal probability (1/2) to jump to the right or the left
¨ 𝑋(𝑡) will be in bin [𝑖], 𝑋(𝑡 + 𝜏) will be in bin [𝑖 − 1] or [𝑖 + 1] with equal probability 50%
¨ Analogous to the coin toss
¨ The random walk can be mapped to the coin toss for money, the position on the X axis is the
current amount of money that the player has while playing a simple strategy where one
amount of currency ($1) is won or lost if the coin lands on heads or tail
55
Xi i+1i-1

Luc_Faucheux_2020
The random walk properties – Markov
¨ Markov property: The value of 𝑋(𝑡 + 𝜏) only depends on the value 𝑋(𝑡)
¨ The distribution of the value of the random variable 𝑋(𝑡 + 𝜏) conditional upon all the past
events only depends on the previous value 𝑋(𝑡)
¨ “The random walk has no memory beyond where it is now”
¨ Note: this does not mean that the expected value of 𝑋(𝑡 + 𝜏) is 𝑋(𝑡)
56

Luc_Faucheux_2020
The random walk properties - Martingale
¨ Martingale: the expected value of 𝑋(𝑡 + 𝜏) is 𝑋(𝑡)
¨ In terms of game:
– You know how much money you have (your current winnings)
– Your expected winnings after one more coin toss is your current winning
– By recurrence, your expected winnings after any number of coin toss is the value of your
current winnings (somewhat akin to the Tower property)
¨ Note: this does not mean that your winnings are stuck at their current value, it is only the
expected value of your winnings that is equal to the current value
57

Luc_Faucheux_2020
Searching for the PDE for the PDF
¨ 𝑃(𝑖, 𝑞) is the probability to find our random walker in the bin (𝑖) at the time (𝑞. 𝜏)
¨ Remember that time and position are discrete and NOT continuous
¨ Position is indexed by 𝑖 , and the size of the jump is 𝜀, at every 𝜏 in time.
¨ Another way to think about it is to have a large number of random walkers, and so at time
𝑡 = 𝑞. 𝜏 the number of walkers in a specific bin indexed by (𝑖) is 𝑁. 𝑃(𝑖, 𝑞)
¨ 𝑃 𝑖, 𝑞 + 1 =
+
,
. {𝑃 𝑖 − 1, 𝑞 + 𝑃 𝑖 + 1, 𝑞 }
¨ Because the random walker has no choice but to jump to one of the adjacent bins, the
probability after the jump to be in the bin [i] is half of the probability before the jump in the
left bin, and half of the probability before the jump in the right bin
¨ This is sometime called Master Equation, or Fokker-Planck equation
58

Luc_Faucheux_2020
Taylor expansion
¨ Even though we are in the discrete description, we are somewhat assuming that we can use
tools of continuous calculus like Taylor expansion on the function 𝑃 𝑖, 𝑞
¨ More rigorously, 𝑃 𝑖, 𝑞 is NOT a continuous function (just like BINOM.DIST was not either),
but we are looking for a continuous function •𝑃 𝑥, 𝑡 , that would match the discrete values
of 𝑃 𝑖, 𝑞 , or is not “too far” from it.
¨ Say it another way, we assuming that there is a limit for 𝑃 𝑖, 𝑞 that would be a regular
continuous function •𝑃 𝑥 𝑖 , 𝑞. 𝜏 , and because we are not that rigorous, we just use the
same notation 𝑃 𝑖, 𝑞 and 𝑃 𝑥, 𝑡
59

Luc_Faucheux_2020
Taylor expansion II
¨ So really we should have written
¨ 𝑃 𝑖, 𝑞 + 1 is the discrete probability to find the random walker in the bin 𝑖 after (𝑞 + 1)
jumps of size 𝜀 every time interval 𝜏
¨ We have a strong feeling that there might be a continuous function •𝑃 𝑥, 𝑡 which is a
function of the two continuous variables position and time, which is such that the discrete
function ”converges” to the continuous function under some limits
¨ By “converge”, what we mean is that there is a manner in which you can calculate the
“distance” between the continuous function and the discrete one, and we would like to say
something along the lines of “as the size of the jump 𝜀 goes to 0 and the period of the jump
𝜏 also goes to zero”
¨ Note that we have not defined the “distance”
¨ Note that we have not defined ”how we get to 0”
¨ We are trying to be pragmatic without butchering the actual math too much, so really get to
the essence but alert you that there are a couple of trees in the forest that you should pay
attention to, and some others that are not that important
60

Luc_Faucheux_2020
Taylor expansion III
¨ 𝑃 𝑖, 𝑞 + 1 =
+
,
. {𝑃 𝑖 − 1, 𝑞 + 𝑃 𝑖 + 1, 𝑞 }
¨ 𝑃 𝑖, 𝑞 + 1 = 𝑃 𝑖, 𝑞 + 𝜏.
G OP(3,7)
G7
+ 𝒪(𝜏,)
¨ 𝑃 𝑖 − 1, 𝑞 = 𝑃 𝑖, 𝑞 − 𝜀.
G OP(3,7)
G3
+
+
,
. 𝜀,.
G% OP(3,7)
G3% + 𝒪(𝜀R)
¨ 𝑃 𝑖 + 1, 𝑞 = 𝑃 𝑖, 𝑞 + 𝜀.
G OP(3,7)
G3
+
+
,
. 𝜀,.
G% OP(3,7)
G3% + 𝒪(𝜀R)
¨ 𝒪(. . ) means ”something of the order of”, meaning all the higher orders that we are
neglecting in the Taylor expansion
¨ You have to be careful to which order you go to, and also if the higher orders are indeed
negligible for what you are trying to achieve
61

Luc_Faucheux_2020
Taylor expansion IV
¨ 𝑃 𝑖, 𝑞 + 1 =
+
,
. {𝑃 𝑖 − 1, 𝑞 + 𝑃 𝑖 + 1, 𝑞 }
¨ 𝑃 𝑖, 𝑞 + 1 = 𝑃 𝑖, 𝑞 + 𝜏.
G OP(3,7)
G7
+ 𝒪(𝜏,)
¨ 𝑃 𝑖 − 1, 𝑞 = 𝑃 𝑖, 𝑞 − 𝜀.
G OP(3,7)
G3
+
+
,
. 𝜀,.
G% OP(3,7)
G3% + 𝒪(𝜀R)
¨ 𝑃 𝑖 + 1, 𝑞 = 𝑃 𝑖, 𝑞 + 𝜀.
G OP(3,7)
G3
+
+
,
. 𝜀,.
G% OP(3,7)
G3% + 𝒪(𝜀R)
¨ 𝑃 𝑖, 𝑞 + 𝜏.
G OP(3,7)
G7
+ 𝒪 𝜏, =
+
,
. {𝑃 𝑖, 𝑞 − 𝜀.
G OP(3,7)
G3
+
+
,
. 𝜀,.
G% OP(3,7)
G3% + 𝒪 𝜀R + 𝑃 𝑖, 𝑞 +
𝜀.
G OP(3,7)
G3
+
+
,
. 𝜀,.
G% OP(3,7)
G3% + 𝒪(𝜀R)}
¨ 𝜏.
G OP(3,7)
G7
+ 𝒪 𝜏, =
+
,
𝜀,.
G% OP(3,7)
G3% + 𝒪 𝜀R
62

Luc_Faucheux_2020
Taylor expansion V
¨ 𝜏.
G OP(3,7)
G7
+ 𝒪 𝜏, =
+
,
𝜀,.
G% OP(3,7)
G3% + 𝒪 𝜀R
¨
G OP(3,7)
G7
=
S%
,T
.
G% OP(3,7)
G3%
¨ Equation above is usually referred to as a “heat equation” or “diffusion equation”
¨ The diffusion coefficient is defined as 𝐷 =
S%
,T
¨ This is the PDE (Partial Differential Equation) for the PDF (Probability Distribution Function)
¨ Note that we were looking for a continuous limit when 𝜀 ”goes to zero” and 𝜏 “goes to zero”.
Obviously since we are dividing one by the other we are going to have to be a little careful
here.
63

Luc_Faucheux_2020
Conservation Equation
¨
G OP(3,7)
G7
= 𝐷
G% OP(3,7)
G3%
¨ We can rewrite the above as
¨
G OP(3,7)
G7
= 𝐷
G% OP(3,7)
G3% =
G
G3
[𝐷
G OP(3,7)
G3
]
¨ This is also known as a conservation equation, because it verifies the conservation of overall
probability (we do not lose any random walkers)
¨ The overall probability is the integral over the position axis of the function •𝑃(𝑥, 𝑡)
¨
H
H7
. ∫ •𝑃 𝑥, 𝑡 = ∫
G
G7
•𝑃 𝑥, 𝑡 = ∫
G
G3
[𝐷
G OP(3,7)
G3
] = 0
¨ So the overall probability is ”conserved”.
¨ Please note that we were quite liberal in taking the diffusion coefficient 𝐷 inside the partial
derivative, which can can only do if it has no dependence on the position. When it depends
on the position, this opens up the whole Ito-Stratonovitch can of worms
64

Luc_Faucheux_2020
Gradient and Diffusion current
¨
G OP(3,7)
G7
= 𝐷
G% OP(3,7)
G3% =
G
G3
[𝐷
G OP(3,7)
G3
]
¨ The diffusion current is sometimes defined as: 𝐽 𝑥, 𝑡 = −𝐷
G OP(3,7)
G3
¨ The above equation is sometimes called the Ficks’s law.
¨
G
G7
•𝑃 = −
G
G3
𝐽
¨
G OP(3,7)
G7
= 𝐷
G% OP(3,7)
G3%
¨ We know a solution of this equation : the Normal Distribution Function, or Gausssian.
¨ 𝐺 𝑥, 𝑡 =
+
K%U7
. 𝑒𝑥𝑝(−
3%
KU7
)
65

Luc_Faucheux_2020
Propagator and Green function
¨ 𝐺 𝑥, 𝑡 =
+
K%U7
. 𝑒𝑥𝑝(−
3%
KU7
) is one solution of the diffusion equation.
¨ Because the diffusion equation is linear, a linear combination of solutions is ALSO a solution
¨ The Gaussian function is self-similar, if you plot {𝐺 𝑥, 𝑡 . 4𝜋𝐷𝑡} as a function of the
rescaled variable 𝑦 =
3%
KU7
, you always get the same function 𝑒𝑥𝑝(−𝑦,)
¨ This is what we did in the spreadsheet with the BINOM.DIST function
¨ When t=0, the Gaussian function above converges to the Dirac function. It is a function
equal to 0 everywhere except at x=0, where it goes to infinity but in such a way that the
integral of the Gaussian over the x-axis is always conserved and equal to 1 (conservation of
probability)
66

Luc_Faucheux_2020
Propagator and Green functions II
¨ Take any arbitrary initial probability distribution function •𝑃(𝑥, 𝑡 = 0)
¨ This can be written formally as
¨ •𝑃 𝑥, 𝑡 = 0 = ∫ •𝑃 𝑥2, 𝑡 = 0 . 𝛿 𝑥 − 𝑥2 . 𝑑𝑥′
¨ The initial “peak” •𝑃 𝑥2, 𝑡 = 0 . 𝛿(𝑥 − 𝑥2) is centered around 𝑥 with integral •𝑃 𝑥, 𝑡 = 0
¨ This peak will diffuse with the Gaussian 𝐺 𝑥, 𝑥′, 𝑡 =
+
K%U7
. 𝑒𝑥𝑝(−
(353$)%
KU7
)
¨ and so for any time t, the solution of the diffusion equation that satisfies •𝑃(𝑥, 𝑡 = 0) will
be:
¨ •𝑃 𝑥, 𝑡 = ∫ •𝑃 𝑥2, 𝑡 = 0 . 𝐺 𝑥, 𝑥′, 𝑡 . 𝑑𝑥′
¨ 𝐺 𝑥, 𝑥′, 𝑡 is called the Green function, or the propagator
67

Luc_Faucheux_2020
Propagators and Green functions III
¨ The propagator technique is hugely helpful when discounting payoff
¨ The Black Sholes equation is a diffusion equation
¨ Note : the probability distribution function for the random variable “diffuses forward in
time”
¨ Note : the option value as a function of the random variable ”diffuses backward in time”
from the terminal payoff.
¨ The terminal payoff is sometimes called the ”boundary condition” for the diffusion equation
followed by the option value
68

Luc_Faucheux_2020
Diffusion and convexity
¨
G
G7
𝑃(𝑥, 𝑡) = 𝐷
G%
G3% 𝑃(𝑥, 𝑡)
¨ If the density probability has a “sharp peak”,
G%
G3% 𝑃(𝑥, 𝑡) is a large negative number, and so
G
G7
𝑃(𝑥, 𝑡) is also a large negative number, and so the density probability at that spot will
decrease rapidly in time.
¨ If the density probability has a “sharp trough”,
G%
G3% 𝑃(𝑥, 𝑡) is a large positive number, and so
G
G7
𝑃(𝑥, 𝑡) is also a large positive number, and so the density probability at that spot will
increase rapidly in time.
¨ The diffusion equation tends to “smooth out” any irregularity of the density probability
(forward in time), any sharp “kinks” diffuses over time
¨ Note: in regions of large convexity (Gamma), the time dependence (time decay) is also
maximum
69

Luc_Faucheux_2020
Diffusion and convexity II
¨ The steady-state solution (also called equilibrium solution) of the diffusion equation is a
solution where there is no dependence in time.
¨ In our simple case, it means
G%
G3% 𝑃 𝑥, 𝑡 = ∞ =
G%
G3% 𝑃?VCWXW(YWCZ 𝑥 = 0
¨ That is a straight line
¨ Any “kink” (places where the second spatial derivative was non-zero) got smoothed out
70

Luc_Faucheux_2020
Another scaling argument (Bachelier page 69)
¨ For a given 𝑥, the probability density function at a given time 𝑡 is given by:
¨ 𝑃 𝑥, 𝑡 =
+
K%U7
. 𝑒𝑥𝑝(−
3%
KU7
)
¨ 𝑃 𝑥, 𝑡 = 0 = 0 and lim
7→6
𝑃 𝑥, 𝑡 = 0
¨ For a given 𝑥, the function 𝑃 𝑥, 𝑡 will exhibit a positive maximum for a given time 𝑡
¨ 𝑃 𝑥, 𝑡 =
+
K%U7
. 𝑒𝑥𝑝(−
3%
KU7
)
¨
GP 3,7
G3
=
5+
,
.
+
K%U7
.
+
7
. 𝑒𝑥 𝑝 −
3%
KU7
+
+
K%U7
. 𝑒𝑥 𝑝 −
3%
KU7
. (
3%
KU
.
+
7%)
¨
GP 3,7
G3
= 0 at the maximum 𝑡 = 𝑡 implies 𝑡 =
3%
,U
¨ Again we see the neat scaling of the square of the distance to the first order in time appears
71

Luc_Faucheux_2020
A neat thing about the diffusion equation (Bachelier)
¨ 𝑃 𝑥, 𝑡 =
+
K%U7
. 𝑒𝑥𝑝(−
3%
KU7
) is a solution of
G
G7
𝑃(𝑥, 𝑡) = 𝐷
G%
G3% 𝑃(𝑥, 𝑡)
¨ We define 𝑁 𝑥, 𝑡 = ∫3
6
𝑃 𝑥′, 𝑡 . 𝑑𝑥′ as the probability to find the random variable at time 𝑡
at a distance greater than 𝑥
¨
G] 3,7
G7
= ∫3
6 GP 32,7
G7
. 𝑑𝑥′ = ∫3
6
𝐷
G%
G32% 𝑃 𝑥′, 𝑡 . 𝑑𝑥′ = 𝐷.
GP 3$,7
G3$
6
3$43 = −𝐷.
GP 3,7
G3
¨
G] 3,7
G7
= −𝐷.
GP 3,7
G3
¨ 𝑁 𝑥, 𝑡 = ∫3
6
𝑃 𝑥′, 𝑡 . 𝑑𝑥′ and so
G] 3,7
G3
= −𝑃(𝑥, 𝑡)
¨ And so the function 𝑁 𝑥, 𝑡 = ∫3
6
𝑃 𝑥′, 𝑡 . 𝑑𝑥′ ALSO follows the same equation diffusion as
𝑃(𝑥, 𝑡)
¨
G
G7
𝑁(𝑥, 𝑡) = 𝐷
G%
G3% 𝑁 𝑥, 𝑡 NOTE that 𝑁(𝑥, 𝑡) is NOT a Gaussian (unicity of solution)
72

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Bachelier on the rayonement de probabilite
¨ Let us limit ourselves to a one-dimensional random walk
¨ A random walker will jump to the right or the left by one unit 𝜀 at equal time intervals 𝜏
¨ We assume equal probability (1/2) to jump to the right or the left
¨ 𝑋(𝑡) will be in bin [𝑖], 𝑋(𝑡 + 𝜏) will be in bin [𝑖 − 1] or [𝑖 + 1] with equal probability 50%
¨ Analogous to the coin toss
¨ The random walk can be mapped to the coin toss for money, the position on the X axis is the
current amount of money that the player has while playing a simple strategy where one
amount of currency ($1) is won or lost if the coin lands on heads or tail
73
Xi i+1i-1

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Bachelier’s argument is slighty different
¨ 𝑃(𝑖, 𝑞) is the probability to ﬁnd our random walker in the bin (𝑖) at the tme (𝑞. 𝜏)
¨ We define 𝑁 𝑖, 𝑞 = ∑I4W
6
𝑃(𝑗, 𝑞)
¨ 𝑁 𝑖, 𝑞 is the probability to find the random walker on the right of the bin (𝑖) at the tme
(𝑞. 𝜏)
¨ Bachelier somehow was more interested in 𝑁 𝑖, 𝑞 than 𝑃(𝑖, 𝑞), because he was more
interested in pricing an option
¨ 𝑁 𝑖, 𝑞 = ∑I4W
6
𝑃(𝑗, 𝑞)
¨ 𝑁 𝑖 + 1, 𝑞 = ∑I4W/+
6
𝑃(𝑗, 𝑞)
¨ And so 𝑃 𝑖, 𝑞 = 𝑁 𝑖, 𝑞 − 𝑁 𝑖 + 1, 𝑞
¨ 𝑁 𝑖 + 1, 𝑞 = 𝑁 𝑖, 𝑞 +
G]
G3
. 𝜀 + 𝒪(𝜀,)
¨ 𝑃 𝑖, 𝑞 = −
G]
G3
. 𝜀 to the second order in 𝜀
74

Luc_Faucheux_2020
Rayonement de probability (Bachelier page)
¨ 𝑃 𝑖, 𝑞 = −
G]
G3
. 𝜀
¨ We also know that the random walker follows the jump dynamic of equal probability to the
right and the left at every discrete time increment
¨ 𝑁 𝑖, 𝑞 + 1 = 𝑁 𝑖, 𝑞 −
+
,
. 𝑃 𝑖, 𝑞 +
+
,
. 𝑃 𝑖 − 1, 𝑞
¨ 𝑁 𝑖, 𝑞 + 1 − 𝑁 𝑖, 𝑞 =
+
,
. {𝑃 𝑖 − 1, 𝑞 − 𝑃(𝑖, 𝑞)}
¨
G]
G7
. 𝜏 =
+
,
. −
GP
G3
. 𝜀 =
+
,
. 𝜀,.
G%]
G3% or
G]
G7
=
S%
,T
.
G%]
G3%
¨ This is the same diffusion equation or heat equation that we had for 𝑃.
¨ Note that the two functions are quite different (there is no unicity of the diffusion equation)
¨ Different Boundary conditions:
¨ Note that we were a little liberal mixing 𝑃 𝑖, 𝑞 and •𝑃 𝑥 𝑖 , 𝑞. 𝜏 for clarity sake
75

Luc_Faucheux_2020
Some concepts around time
(first passage,..)
76

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Some concepts around time
¨ A typical Brownian motion would look something like that (thanks you Excel):
77

Luc_Faucheux_2020
Some concepts around time II
¨ We can define for a given path a number of variables
¨ 𝑋 𝑡 is the Brownian variable, 𝑇 is the last time, 0 ≤ 𝑡 ≤ 𝑇
¨ We can define the Maximum value of the path : 𝑀𝐴𝑋 𝑇 = MAX(𝑋 𝑡 , 0 ≤ 𝑡 ≤ 𝑇)
¨ We can define the “First passage time”, the first time that the Brownian motion reaches the
value 𝑚, as 𝜏 𝑚, 𝑇 = min(𝑡 ≥ 0, 𝑋 𝑡 = 𝑚)
¨ It would be useful to be able to know the distribution probability for 𝜏 𝑚, 𝑇
¨ Bachelier devotes the last few pages of his thesis on this, and comes up with a number of
very useful “rules of thumb”
¨ Let’s introduce now the reflection principle or symmetry principle.
¨ Not only it is neat, but also it is used widely for example in reducing the CPU and time for
Monte Carlo simulations.
78

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Some concepts around time III
¨ Let’s do the following trick: As soon as 𝑚 gets reached at time 𝜏 𝑚, 𝑇 by the Brownian
motion 𝑋 𝑡 (we then have 𝑋 𝜏 𝑚, 𝑇 = 𝑚 ), we create a symmetrical Brownian motion,
where starting at time 𝜏 𝑚, 𝑇 , every time 𝑋 𝑡 goes up or down, 𝑋)^ 𝑡 does exactly the
opposite, example below 𝑚 = 2, 𝜏 𝑚, 𝑇 = 12, 𝑋 𝑡 solid orange line, 𝑋)^ 𝑡 is the
dashed blue line
79

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Some concepts around time III-b
80
Maximum to date 𝑀𝐴𝑋(𝑇)
End point 𝑋 𝑇 = 𝑋#
End point reflected:
𝑋)^ 𝑇 = 2𝑚 − 𝑋#
Level of first passage 𝑚

Luc_Faucheux_2020
Some concepts around time III-c
81
𝑋)^ 𝑇 = 2𝑚 − 𝑋#

Luc_Faucheux_2020
Some concepts around time III-d
82
-2
0
2
4
6
8
10
𝑋)^ 𝑇 = 2𝑚 − 𝑋#

Luc_Faucheux_2020
Some concepts around time IV
¨ If during the interval [0, 𝑇], 𝑋 𝑡 reaches 𝑚, then we have a reflected path 𝑋)^ 𝑡
¨ We have by construction: ABS(𝑋)^ 𝑇 − 𝑚) = ABS(𝑋 𝑡 − 𝑚)
¨ So for all time 𝑡 ≥ 𝜏 𝑚, 𝑇 , 𝑋 𝑡 and 𝑋)^ 𝑡 are symmetrical around 𝑚
¨ Let’s now define in a more general fashion a new terminal variable 𝑋#
¨ We now the probability that at time 𝑡 = 𝑇, the Brownian motion will end with a value such
that 𝑋 𝑡 ≥ 𝑋#
¨ This is the usual Gaussian distribution : ℎ 𝑥, 𝑡 =
+
,%-%7
. exp(
53%
,-%7
)
¨ So 𝑃 𝑋 𝑇 ≥ 𝑋# = ∫34_1
346
1. ℎ 𝑥, 𝑇 . 𝑑𝑥
¨ Let’s try to figure out the cumulative probability distribution for 𝜏 𝑚, 𝑇 , or more exactly:
¨ 𝑃 𝜏 𝑚, 𝑇 ≤ 𝑇
83

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Some concepts around time V
¨ We can write what is known as the “reflection formula”
¨ 𝑃 𝜏 𝑚, 𝑇 ≤ 𝑇, 𝑋 𝑇 ≤ 𝑋# = 𝑃 𝜏 𝑚, 𝑇 ≤ 𝑇, 𝑋 𝑇 ≥ 2𝑚 − 𝑋#
¨ Because if 𝑋 𝑡 did reach 𝑚 at some point, for every path 𝑋 𝑡 after 𝑡 = 𝜏 𝑚, 𝑇 , there exist
a symmetrical path 𝑋)^ 𝑡 around 𝑚.
¨ So the number of paths 𝑋 𝑡 that did reach 𝑚 at some point, and are now at a terminal
value 𝑋 𝑇 ≤ 𝑋#, is the same number of paths 𝑋)^ 𝑡 that are now at a terminal value
𝑋)^ 𝑇 ≥ 2𝑚 − 𝑋#
¨ Those paths 𝑋)^ 𝑡 are “valid” paths 𝑋 𝑡 , meaning that they are a specific realization of
the Brownian motion 𝑋 𝑡
¨ So re-stating again the above, the number of paths 𝑋 𝑡 that did reach 𝑚 at some point, and
are now at a terminal value 𝑋 𝑇 ≤ 𝑋#, is the same number of paths 𝑋 𝑡 that are now at a
terminal value 𝑋 𝑇 ≥ 2𝑚 − 𝑋#
¨ This is the reflection principle (Desiree Andre, 1840-1917) or also sometimes called the
ballot problem (Joseph Louis Bertrand, 1887)
84

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Some concepts around time VI
¨ Now let’s use the specific example of 𝑋# = 𝑚
¨ 𝑃 𝜏 𝑚, 𝑇 ≤ 𝑇, 𝑋 𝑇 ≤ 𝑚 = 𝑃 𝜏 𝑚, 𝑇 ≤ 𝑇, 𝑋 𝑇 ≥ 𝑚
¨ But we also have
¨ 𝑃 𝜏 𝑚, 𝑇 ≤ 𝑇 = 𝑃 𝜏 𝑚, 𝑇 ≤ 𝑇, 𝑋 𝑇 ≤ 𝑚 + 𝑃 𝜏 𝑚, 𝑇 ≤ 𝑇, 𝑋 𝑇 ≥ 𝑚
¨ 𝑃 𝜏 𝑚, 𝑇 ≤ 𝑇 = 2. 𝑃 𝜏 𝑚, 𝑇 ≤ 𝑇, 𝑋 𝑇 ≤ 𝑚 = 2. 𝑃 𝜏 𝑚, 𝑇 ≤ 𝑇, 𝑋 𝑇 ≥ 𝑚
¨ 𝑃 𝜏 𝑚, 𝑇 ≤ 𝑇 = 2. 𝑃 𝜏 𝑚, 𝑇 ≤ 𝑇, 𝑋 𝑇 ≥ 𝑚
¨ And if 𝑋 𝑇 ≥ 𝑚, then obviously 𝑋 𝑡 did reach 𝑚 before 𝑡 = 𝑇
¨ So : 𝑃 𝜏 𝑚, 𝑇 ≤ 𝑇 = 2. 𝑃 𝑋 𝑇 ≥ 𝑚 the usual Gaussian
¨ The probability that the Brownian motion will be greater than a given level at maturity is
half the probability that this given level will be reached or exceeded during the time interval
up to maturity
85

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Some concepts around time VI-a
¨ Note: Shrieve (p.112) looks at each Brownian motion path that reaches level 𝑚 prior to time
𝑇 but is at a level 𝑋# below 𝑚 at time 𝑇
¨ In that case, since 𝑋# ≤ 𝑚, we have automatically: (2𝑚 − 𝑋#) ≥ 𝑋#
¨ And so he writes from the start the reflection equality as:
¨ 𝑃 𝜏 𝑚, 𝑇 ≤ 𝑇, 𝑋 𝑇 ≤ 𝑋# = 𝑃 𝑋 𝑇 ≥ 2𝑚 − 𝑋# and stating 𝑋# ≤ 𝑚, 𝑚 > 0
¨ We only did it when equating 𝑋# = 𝑚, and then of course:
¨ 𝑃 𝜏 𝑚, 𝑇 ≤ 𝑇, 𝑋 𝑇 ≥ 2𝑚 − 𝑋# = 𝑃 𝜏 𝑚, 𝑇 ≤ 𝑇, 𝑋 𝑇 ≥ 𝑚 = 𝑃 𝑋 𝑇 ≥ 𝑚
¨ Do we have :
¨ Even in the cases where 𝑋# ≥ 𝑚, 𝑚 > 0 ?
86

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Some concepts around time VII
¨ 𝑃 𝜏 𝑚, 𝑇 ≤ 𝑇 = 2. 𝑃 𝑋 𝑇 ≥ 𝑚
¨ 𝑃 𝜏 𝑚, 𝑇 ≤ 𝑇 = 2. ∫34Z
346
1. ℎ 𝑥, 𝑇 . 𝑑𝑥
¨ 𝑃 𝜏 𝑚, 𝑇 ≤ 𝑇 = 2. ∫34Z
346
1.
+
,%-%#
. exp(
53%
,-%#
). 𝑑𝑥
¨ We can now compute things such as the average first passage time:
¨ Note that 𝑃 𝜏 𝑚, 𝑇 ≤ 𝑇 is the CUMULATIVE distribution function
¨ 𝑃 𝜏 𝑚, 𝑇 ≤ 𝑇 is the probability that 𝑋 𝑡 will exceed 𝑚 over the time interval [0, 𝑇]
¨ Between 𝑇 and (𝑇 + 𝑑𝑇), the density function is 𝑝 𝜏 𝑚, 𝑇 ≤ 𝑇 =
GP T Z,# `#
G#
¨ 𝑃 𝜏 𝑚, 𝑇 ≤ 𝑇 = ∫a48
a4#
𝑝 𝜏 𝑚, 𝜉 ≤ 𝜉 ). 𝑑𝜉
87

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Some concepts around time VIII
¨ 𝑃 𝜏 𝑚, 𝑇 ≤ 𝑇 is the probability that 𝑋 𝑡 will exceed 𝑚 over the time interval [0, 𝑇]
¨ 𝑃 𝜏 𝑚, 𝑇 + 𝑑𝑇 ≤ 𝑇 + 𝑑𝑇 is the probability that 𝑋 𝑡 will exceed 𝑚 over the time interval
[0, 𝑇 + 𝑑𝑇]
¨ 𝑃 𝜏 𝑚, 𝑇 + 𝑑𝑇 ≤ 𝑇 + 𝑑𝑇 − 𝑃 𝜏 𝑚, 𝑇 ≤ 𝑇 is thus the probability that 𝑋 𝑡 will exceed
𝑚 INSIDE the time interval [𝑇, 𝑇 + 𝑑𝑇]
¨ 𝑃 𝜏 𝑚, 𝑇 + 𝑑𝑇 ≤ 𝑇 + 𝑑𝑇 − 𝑃 𝜏 𝑚, 𝑇 ≤ 𝑇 is the probability that 𝜏 𝑚, 𝑇 is within the
interval [𝑇, 𝑇 + 𝑑𝑇]
¨ 𝑃 𝜏 𝑚, 𝑇 + 𝑑𝑇 ≤ 𝑇 + 𝑑𝑇 − 𝑃 𝜏 𝑚, 𝑇 ≤ 𝑇 =
GP T Z,# `#
G#
. 𝑑𝑇
¨ 𝑝 𝜏 𝑚, 𝑇 ≤ 𝑇 =
GP T Z,# `#
G#
is the probability density to have 𝜏 𝑚, 𝑇 at time 𝑇
¨ So less confusing to rewrite it as 𝑝 𝑚, 𝑇 , probability that the Brownian motion 𝑋 𝑡 will
exceed 𝑚 at time 𝑇
¨ It also makes things easier to grasp when you realize that 𝑃 𝜏 𝑚, 𝑇 ≤ 𝑇 can only increase
with 𝑇 (if 𝑚 was reached, it is obviously still reached)
88

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Some concepts around time IX
¨ Let’s rewrite a little the cumulative probability:
¨ 𝑃 𝜏 𝑚, 𝑇 ≤ 𝑇 = 2. ∫34Z
346
1.
+
,%-%#
. exp(
53%
,-%#
). 𝑑𝑥
¨ We rescale using the change of variable: 𝑦, =
3%
-%#
¨ 𝑥 = 𝑚 corresponds to 𝑦 =
Z
-%#
and 𝑑𝑥 = 𝑑𝑦. 𝜎, 𝑇
¨ 𝑃 𝜏 𝑚, 𝑇 ≤ 𝑇 = 2. ∫:4
2
3%1
:46 +
,%
. exp(
5:%
,
). 𝑑𝑦
¨ 𝑃 𝜏 𝑚, 𝑇 ≤ 𝑇 =
,
,%
. ∫:4
2
3%1
:46
exp(
5:%
,
). 𝑑𝑦
89

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Some concepts around time X
¨ 𝑃 𝜏 𝑚, 𝑇 ≤ 𝑇 =
,
,%
. ∫:4
2
3%1
:46
exp(
5:%
,
). 𝑑𝑦 and writing 𝑦Z =
Z
-%#
¨ 𝑃 𝜏 𝑚, 𝑇 ≤ 𝑇 =
,
,%
. ∫:4:2
:46
exp(
5:%
,
). 𝑑𝑦
¨ 𝑝 𝜏 𝑚, 𝑇 ≤ 𝑇 =
GP T Z,# `#
G#
=
GP T Z,# `#
G:2
.
G:2
G#
¨ 𝑝 𝜏 𝑚, 𝑇 ≤ 𝑇 =
,
,%
. −1 . exp
5:2
%
,
. −
+
,
.
+
#
.
Z
-%#
¨ 𝑝 𝜏 𝑚, 𝑇 ≤ 𝑇 =
Z
# ,%-%#
. exp
5:2
%
,
¨ 𝑝 𝜏 𝑚, 𝑇 ≤ 𝑇 =
Z
# ,%-%#
. exp
5Z%
,-%#
= 𝑝(𝑚, 𝑇)
90

Luc_Faucheux_2020
Some concepts around time X-a
¨ 𝑃 𝜏 𝑚, 𝑇 ≤ 𝑇 = 𝑃(𝑚, 𝑇) =
,
,%
. ∫:4:2
:46
exp(
5:%
,
). 𝑑𝑦 with 𝑦Z =
Z
-%#
¨ If 𝑇 → ∞, 𝑦Z → 0 and so 𝑃(𝑚, 𝑇 = ∞) =
,
,%
. ∫:48
:46
exp(
5:%
,
). 𝑑𝑦
¨ We always go back to : 𝐼9 = ∫56
/6
𝑒593%
. 𝑑𝑥 =
%
9
¨ 𝑃 𝑚, 𝑇 = ∞ =
,
,%
. ∫:48
:46
exp
5:%
,
. 𝑑𝑦 =
,
,%
.
+
,
.
%
⁄*
%
= 1
¨ 𝑃 𝑚, 𝑇 = ∞ = 1 for all 𝑚. So whatever the value of 𝑚, it will be reached at some point in
time by the stochastic process 𝑋 𝑡 with probability 1
91

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A little paradox (Doob)
¨ 𝑃 𝑚, 𝑇 = ∞ = 1 for all 𝑚. So whatever the value of 𝑚, it will be reached at some point in
time by the stochastic process 𝑋 𝑡 with probability 1
¨ A little detour through notations and martingale
¨ We are looking at a stochastic process 𝑆(𝑡) or in its discrete implementation 𝑆! = 𝑆(𝑡!)
¨ (Because this is usually about stocks, so we are using the letter 𝑆)
¨ You see sometimes the notation ℱ! which is sometimes referred to as a “filtration”
¨ Essentially it is the current set of information available on the world at time 𝑡!
¨ It is a collection of stuff
¨ You also sometimes see something that looks like this :
¨ ℱ! = ℱ(𝑡!) ⊂ ℱb = ℱ(𝑡b) for all 0 < 𝑘 < 𝑠
¨ That means that the set of information increases with time
92

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A little paradox (Doob) - II
¨ ℱ! = ℱ(𝑡!) ⊂ ℱb = ℱ(𝑡b) for all 0 < 𝑘 < 𝑠
¨ That means that the set of information increases with time
¨ Any information available at time 𝑡! is still available at time 𝑡!/+
¨ ℱ! is sometimes called a 𝜎-field on Ω (nothing to do with variance, it is just a name)
¨ Now, just to be super-formal, a sequence of filtrations (collection of 𝜎-fields), is also called a
filtration if the stream of information is increasing
¨ The collection (ℱ!, 𝑘 > 0) of 𝜎-fields on Ω is called a filtration if ℱ! ⊂ ℱb for all 0 < 𝑘 < 𝑠
¨ If (ℱ!, 𝑘 = 0,1, … ) is a sequence of 𝜎-fields on Ω and ℱ! ⊂ ℱ!/+ for all 𝑘, we call (ℱ!) a
filtration as well
93

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A little paradox (Doob) - III
¨ A stochastic process 𝑆! is said to be “adapted to the filtration” (ℱ!, 𝑘 > 0) if the value of 𝑆!
is completely determined by the information in ℱ!, which is to say that :
¨ 𝑆! = 𝐸[𝑆!|ℱ!]
¨ 𝐸[𝑆!|ℱ!] is the conditional expectation of 𝑆!
¨ Conditional expectation is not the conditional probability
¨ The conditional expectation is a weighted average of conditional probabilities
¨ A filtration 𝒮ℱ! is said to be “generated” by the stochastic process (𝑆8, 𝑆+, 𝑆,,… 𝑆!) if it
contains all the information, and only the information (𝑆8, 𝑆+, 𝑆,,… 𝑆!), also sometimes
referred to as 𝜎(𝑆I, 𝑗 ≤ 𝑘)
¨ The stochastic process is said to be adapted to the filtration” (ℱ!, 𝑘 > 0) if:
¨ 𝜎(𝑆!) ⊂ ℱ! = ℱ(𝑡!) for all 𝑘
¨ It essentially means that the stochastic process does not carry more information than the
filtration, or 𝒮ℱ! ⊂ ℱ! for all 𝑘
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A little paradox (Doob) - IV
¨ For example suppose that 𝒮ℱ! is said to be “generated” by the stochastic process
(𝑆8, 𝑆+, 𝑆,,… 𝑆!)
¨ (𝑆!) is adapted to 𝒮ℱ!
¨ (𝑆!/𝑆!5+) is adapted to 𝒮ℱ!
¨ (𝑀𝐴𝑋(𝑆I; 𝑗 < +𝑘)) is adapted to 𝒮ℱ!
¨ (𝑆!/+) is NOT adapted to 𝒮ℱ!, because it is an additional piece of information that was NOT
included in 𝒮ℱ!, but will be included in 𝒮ℱ!/+
¨ Any “trading strategy” is adapted to 𝒮ℱ!
¨ A “trading strategy” on the stock 𝑆! is a sequence of positions 𝑅!on the stock 𝑆!
¨ At time 𝑡!, the investor places a bet of size 𝑅!on the stock 𝑆!
¨ The trading strategy is adapted to 𝒮ℱ! means that the 𝑅!are being computed (decided) only
based on the information 𝒮ℱ! = 𝜎(𝑆I, 𝑗 ≤ 𝑘), or (𝑆8, 𝑆+, 𝑆,,… 𝑆!)
95

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A little paradox (Doob) - V
¨ The trading strategy is said to be “self-financing” if the only gain or losses result from the
movements in the stochastic variable 𝑆! (no one adds money or subtract money to the
account)
¨ “self-financing” is not the same as “replicating”
¨ The total winnings up to time 𝑆! are: 𝑌! = 𝑌!5+ + 𝑅!5+. (𝑆! − 𝑆!5+)
¨ A stochastic process 𝑆! is called a martingale with respect to ℱ! in the following fashion:
¨ 1) 𝐸 𝑎𝑏𝑠 𝑆! ℱ!] < ∞ for all 𝑘
¨ 2) 𝑆! is adapted to ℱ!
¨ 3) 𝐸[𝑆!| ℱI] = 𝑆I for all 0 ≤ 𝑗 < 𝑘, meaning that 𝑆I is the best predictor of 𝑆! given ℱI
¨ In particular 𝐸[𝑆!/+| ℱ!] = 𝑆!
¨ A martingale has the remarkable property that its expectation function is constant (and we
sometimes omit pointing out which exact filtration is being used)
96

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A little paradox (Doob) - VI
¨ Any self-financing trading strategy of a martingale is also a martingale
¨ The total winnings up to time 𝑆! are: 𝑌! = 𝑌!5+ + 𝑅!5+. (𝑆! − 𝑆!5+)
¨ 𝐸[𝑆!/+| ℱ!] = 𝑆!
¨ 𝐸[𝑌!/+| ℱ!] = 𝐸[𝑌! + 𝑅!. (𝑆!/+ − 𝑆!)| ℱ!]
¨ 𝐸[𝑌!/+| ℱ!] = 𝑌! + 𝑅!. (𝐸[𝑆!/+| ℱ!] − 𝑆!) = 𝑌!
¨ Martingales is also referred to as “fair game”
¨ Originally, martingale is a French word referring something you put on a horse to drive
him/her
97

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A little paradox (Doob) - VII
¨ Almost getting to the paradox, but we had to spend a little time on definitions first.
¨ Suppose that a stochastic process (a stock) is a martingale, and that 𝑆8 = 0
¨ We define the following trading strategy: 𝑅! = 1 if 𝑆! < 𝑚, 0 otherwise
¨ Recall that: 𝑃 𝜏 𝑚, 𝑇 ≤ 𝑇 = 𝑃(𝑚, 𝑇)
¨ 𝜏 𝑚, 𝑇 is here referred as the “stopping time”. As soon as 𝑆! reached 𝑚, 𝑆!stays stuck on
that value
¨ The trading strategy is 𝑌! = 𝑌!5+ + 𝑅!5+. (𝑆! − 𝑆!5+) with 𝑅! = 1 if 𝑆! < 𝑚, 0 otherwise
¨ So 𝑌! = 𝑆! if 𝑡! < 𝜏 𝑚, 𝑇 and 𝑌! = 𝑚 for all 𝑡! ≥ 𝜏 𝑚, 𝑇
¨ Now here is the paradox: from what is sometimes referred to as Doob’s theorem (you
cannot make an expected non-zero profit with a trading strategy on a martingale), we know
that 𝐸 𝑌! = 0
¨ HOWEVER we also know that 𝑃 𝑚, 𝑇 = ∞ = 1 for all 𝑚. So whatever the value of 𝑚, it
will be reached at some point in time by the stochastic process 𝑋 𝑡 with probability 1
98

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A little paradox (Doob) - VIII
¨ So if 𝑃 𝑚, 𝑇 = ∞ = 1 for all 𝑚, AND
¨ 𝑌! = 𝑆! if 𝑡! < 𝜏 𝑚, 𝑇 and 𝑌! = 𝑚 for all 𝑡! ≥ 𝜏 𝑚, 𝑇
¨ We deduce that 𝑌! = 𝑚 will be equal to 𝑚 with probability 1
¨ And so we would like to say that 𝐸[𝑌!] will be equal to 𝑚 with probability 1
¨ This obviously seems like a paradox.
¨ The fact of the matter is that for any given time 𝑇 that is large enough, 𝑌# is very likely to be
equal to 𝑚, however there is enough probability that it has very large negative value that
the expected value is still 0
¨ Also we will see in the following slides that the average first passage time is infinite, which
again seems somewhat confusing.
99

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A couple more example of martingales
¨ A Brownian bridge is NOT a martingale
¨ Ito integrals are martingale
¨ Stratonovitch integrals are NOT martingale
100

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Some concepts around time XI
¨ For the Gaussian case:
¨ ℎ 𝑥, 𝑡 =
+
,%-%7
. exp(
53%
,-%7
)
¨ 𝑝 𝑚, 𝑇 =
Z
# ,%-%#
. exp
5Z%
,-%#
=
Z
#
. ℎ(𝑚, 𝑇)
¨ Be careful that those two probabilities are not equal, and that can be confusing.
¨ ℎ 𝑥, 𝑡 is the probability density for 𝑋 𝑡 , i.e. for a given time 𝑡, what is the probability to
find 𝑋 𝑡 in the interval [𝑥, 𝑥 + 𝑑𝑥]
¨ 𝑝 𝑚, 𝑇 is, for a given 𝑚, the probability that the Brownian motion 𝑋 𝑡 will exceed 𝑚 in
the interval [𝑇, 𝑇 + 𝑑𝑇] for the first time
¨ 𝑝 𝑚, 𝑇 is, for a given 𝑚, the probability that the first passage time defined as :
𝜏 𝑚, 𝑇 = min(𝑡 ≥ 0, 𝑋 𝑡 = 𝑚) can be found in the interval [𝑇, 𝑇 + 𝑑𝑇]
101

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Some concepts around time XI-b
¨ 𝑝 𝑚, 𝑇 =
Z
# ,%-%#
. exp
5Z%
,-%#
=
Z
#
. ℎ(𝑚, 𝑇)
¨ 𝑇. 𝑝 𝑚, 𝑇 = 𝑚. ℎ(𝑚, 𝑇)
¨ This is quite elegant and somewhat intuitive
¨ ℎ(𝑚, 𝑇) is the probability density for a given time 𝑇 to find the stochastic variable 𝑋 𝑡 at
the position 𝑋 𝑡 = 𝑚
¨ ℎ 𝑚, 𝑇 . 𝑑𝑚 is the probability for a given time 𝑇 to find the stochastic variable 𝑋 𝑡 in the
interval [𝑚, 𝑚 + 𝑑𝑚]
¨ ℎ(𝑚, 𝑇) is normalized so that: ∫Z456
Z4/6
ℎ 𝑚, 𝑇 . 𝑑𝑚 = 1
¨ So ℎ(𝑚, 𝑇) has units of
[+]
[_]
102

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Some concepts around time XI-c
¨ 𝑇. 𝑝 𝑚, 𝑇 = 𝑚. ℎ(𝑚, 𝑇)
¨ 𝑝(𝑚, 𝑇) is the probability density for a given time level 𝑚 to find the first passage time
𝜏 𝑚, 𝑇 = min(𝑡 ≥ 0, 𝑋 𝑡 = 𝑚) at time 𝑇
¨ 𝑝 𝑚, 𝑇 . 𝑑𝑇 is the probability for a given level 𝑚 to find the the first passage time 𝜏 𝑚, 𝑇 =
min(𝑡 ≥ 0, 𝑋 𝑡 = 𝑚) in the interval 𝑇, 𝑇 + 𝑑𝑇
¨ Is 𝑝(𝑚, 𝑇) is normalized ? ∫#48
#4/6
𝑝 𝑚, 𝑇 . 𝑑𝑇 =?
¨ So 𝑝(𝑚, 𝑇) has units of
[+]
[#]
103

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Some concepts around time XI-d
¨ 𝑝 𝑚, 𝑇 =
Z
# ,%-%#
. exp
5Z%
,-%#
=
Z
#
. ℎ(𝑚, 𝑇)
¨ The average time is then < 𝜏 𝑚, 𝑇 > = ∫#48
#46
𝑇. 𝑝 𝑚, 𝑇 . 𝑑𝑇
¨ < 𝜏 𝑚, 𝑇 > = ∫#48
#46
𝑇.
Z
# ,%-%#
. exp
5Z%
,-%#
. 𝑑𝑇
¨ < 𝜏 𝑚, 𝑇 > = ∫#48
#46 Z
,%-%#
. exp
5Z%
,-%#
. 𝑑𝑇
¨ When 𝑇 → ∞, exp
5Z%
,-%#
→ 1, and so the large 𝑇 integral looks like ∫#48
#46 +
#
. 𝑑𝑇 → ∞
¨ We will explore this in more details, but the average first passage time is infinite.
¨ The stochastic process will reach any level with probability 1, but will take on average an
infinite amount of time to reach that level. This is a little weird.
104

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Some concepts around time XI-e
¨ Because of the fact that the average first passage time is infinite, in the literature you will
find what is called the typical time
¨ The typical time is the maximum of the function 𝑝 𝑚, 𝑇 as a function of time 𝑇
¨ 𝑝 𝑚, 𝑇 =
Z
# ,%-%#
. exp
5Z%
,-%#
=
Z
#
. ℎ(𝑚, 𝑇)
¨
GA(Z,#)
G#
=
5R
,
Z
## ,%-%#
. exp
5Z%
,-%#
+
Z%
,-%##
.
Z
# ,%-%#
. exp
5Z%
,-%#
¨
GA(Z,#)
G#
= 0 implies:
R
,
Z
## ,%-%#
=
Z%
,-%##
.
Z
# ,%-%#
¨
R
,
=
Z%
,-%#
, or again 𝑇 =
Z%
R-%, or using the diffusion notation 𝐷 =
-%
,
we have 𝑇 =
Z%
$U
¨ The typical time scales as the square of the level for the first passage time
105

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Another scaling argument - redux
¨
G OP(3,7)
G7
=
G% OP(3,7)
G3% , where we set (𝐷 = 1) for simplicity sake
¨ 𝐺 𝑥, 𝑡, 𝐷 = 1 =
+
K%7
. 𝑒𝑥𝑝(−
3%
K7
) is a solution
¨ You can check that the following function is ALSO a solution of the heat equation
¨ 𝑈 𝑥, 𝑡, 𝐷 = 1 = ∫56
( e4
,
)
exp(−
a%
K
). 𝑑𝜉 = 𝑈(
3
7
)
¨ This is another self-similarity argument, where again the position has to scale with the
square root of the time, but also where the integral of a solution to the diffusion equation
ALSO is a solution of the same diffusion equation.
¨ Sometimes those kind of “mapping” are useful because it is easier to work in a given
(function, variable) space and then “map” the results onto the final (function, variable)
space that you need to work in
106

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Another scaling argument – redux I-a
¨ 𝑈 𝑥, 𝑡, 𝐷 = 1 = ∫56
( e4
,
)
exp(−
a%
K
). 𝑑𝜉 = 𝑈(
3
7
)
¨
Gf(3,7)
G7
= exp −
3%
K7
. −
+
,
𝑥𝑡 ⁄&.
%
¨
Gf(3,7)
G3
= exp −
3%
K7
. 𝑡 ⁄&*
%
¨
G%f (3,7)
G3% = exp −
3%
K7
. 𝑡 ⁄&*
%. −
,3
K7
= exp −
3%
K7
. −
+
,
𝑥𝑡 ⁄&.
%
¨ So we do have indeed 𝑈 𝑥, 𝑡, 𝐷 = 1 solution of :
G%f (3,7)
G3% =
Gf(3,7)
G7
107

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Another scaling argument – redux II
¨ 𝑈 𝑥, 𝑡, 𝐷 = 1 = ∫56
( e4
,
)
exp(−
a%
K
). 𝑑𝜉 = 𝑈(
3
7
)
¨ It is also a solution of the Heat equation
Gf (3,7)
G7
=
G%f(3,7)
G3%
¨ 𝑃 𝜏 𝑚, 𝑇 ≤ 𝑇 =
,
,%
. ∫:4
2
3%1
:46
exp(
5:%
,
). 𝑑𝑦
¨ So 𝑃 𝜏 𝑚, 𝑇 ≤ 𝑇 looks like it could also a solution of a Heat equation, let’s see what we
can do
¨
GP(Z,#)
G#
= 𝑝 𝜏 𝑚, 𝑇 ≤ 𝑇 = 𝑝 𝑚, 𝑇 =
Z
#
. ℎ 𝑚, 𝑇 =
Z
# ,%-%#
. exp
5Z%
,-%#
¨
G%P (Z,#)
GZ% =
G%
GZ% {
,
,%
. ∫:4
2
3%1
:46
exp(
5:%
,
). 𝑑𝑦} =
,
,%
.
G
GZ
{
G:2
GZ
G
G:2
[∫:4
2
3%1
:46
exp(
5:%
,
). 𝑑𝑦]}
¨ With 𝑦Z =
Z
-%#
and
G:2
GZ
=
+
-%#
108

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Another scaling argument – redux III
¨
G%P (Z,#)
GZ% =
G%
GZ% {
,
,%
. ∫:4
2
3%1
:46
exp(
5:%
,
). 𝑑𝑦} =
,
,%
.
G
GZ
{
G:2
GZ
G
G:2
[∫:4:2
:46
exp(
5:%
,
). 𝑑𝑦]}
¨
G%P (Z,#)
GZ% =
,
,%
.
G
GZ
{
+
-%#
. −1 . exp
5:2
%
,
}
¨
G%P (Z,#)
GZ% = −
,
,%
.
+
-%#
.
G
GZ
exp
5:2
%
,
= −
,
,%
.
+
-%#
.
G:2
GZ
G
G:2
{exp
5:2
%
,
}
¨
G%P (Z,#)
GZ% = −
,
,%
.
+
-%#
.
+
-%#
. −
,:2
,
. exp
5:2
%
,
¨
G%P (Z,#)
GZ% = −
,
,%
.
+
-%#
.
+
-%#
. −
Z
-%#
. exp
5:2
%
,
=
:2
# ,%
. exp
5:2
%
,
. (
,
-%)
¨
GP(Z,#)
G#
=
Z
# ,%-%#
. exp
5Z%
,-%#
=
:2
# ,%
. exp
5:2
%
,
109

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Another scaling argument – redux IV
¨ We then have :
¨
GP(Z,#)
G#
=
-%
,
G%P (Z,#)
GZ% with 𝑃 𝜏 𝑚, 𝑇 ≤ 𝑇 =
,
,%
. ∫:4
2
3%1
:46
exp(
5:%
,
). 𝑑𝑦
¨ Recall that this was for a usual Gaussian: ℎ 𝑥, 𝑡 =
+
,%-%7
. exp(
53%
,-%7
)
¨ Which is the solution of the heat equation:
G
G7
ℎ(𝑥, 𝑡) =
-%
,
G%
G3% ℎ(𝑥, 𝑡)
¨ In terms of the Diffusion equation, one has the notation:
¨
G
G7
𝑃(𝑥, 𝑡) = 𝐷
G%
G3% 𝑃(𝑥, 𝑡)
¨ 𝑃 𝑥, 𝑡 =
+
K%U7
. 𝑒𝑥𝑝(−
3%
KU7
) with 𝐷 = (𝜎,/2)
110

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Another scaling argument – redux V
¨ This is kind of a neat result.
¨ The cumulative distribution function for the random variable which is the first passage time:
𝜏 𝑚, 𝑇 = min(𝑡 ≥ 0, 𝑋 𝑡 = 𝑚) is 𝑃 𝑚, 𝑇 =
,
,%
. ∫:4
2
3%1
:46
exp(
5:%
,
). 𝑑𝑦
¨ The underlying Brownian motion 𝑋 𝑡 follows ℎ 𝑥, 𝑡 =
+
,%-%7
. exp(
53%
,-%7
)
¨ ℎ 𝑥, 𝑡 follows the diffusion equation:
G
G7
ℎ(𝑥, 𝑡) =
-%
,
G%
G3% ℎ(𝑥, 𝑡)
¨ 𝑃 𝑚, 𝑇 follows the diffusion equation:
GP(Z,#)
G#
=
-%
,
G%P (Z,#)
GZ%
¨ 𝑃 𝑚, 𝑇 diffuses in the space (𝑚, 𝑇) with the SAME diffusion as ℎ 𝑥, 𝑡 in (𝑥, 𝑡)
¨ Note again that it can get confusing to compare those two probability functions, one is a
cumulative, the other one is a density function. Compare that to the next slide where the
density and the cumulative for 𝑥 follows the SAME diffusion equation.
111

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Another scaling argument – redux VI
¨ This is reminiscent of a Dupire like equation:
¨
GP(Z,#)
G#
=
-%
,
G%P (Z,#)
GZ%
¨ 𝑝 𝑚, 𝑇 =
Z
# ,%-%#
. exp
5Z%
,-%#
=
Z
#
. ℎ 𝑚, 𝑇 =
GP(Z,#)
G#
¨ So:
¨ ℎ 𝑚, 𝑇 = (
#
Z
)
-%
,
G%P (Z,#)
GZ%
¨ So if we know 𝑃(𝑚, 𝑇), we can deduce the probability density : ℎ 𝑚, 𝑇
¨ Note: Dupire equation: if we know the Call options prices 𝐶(𝑚, 𝑇) we can deduce the
probability density (also Bachelier p. 51 of his thesis)
¨ ℎ 𝑚, 𝑇 =
G%g (Z,#)
GZ%
112

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A neat thing about the diffusion equation (Bachelier) -redux
¨ 𝑃 𝑥, 𝑡 =
+
K%U7
. 𝑒𝑥𝑝(−
3%
KU7
) is a solution of
G
G7
𝑃(𝑥, 𝑡) = 𝐷
G%
G3% 𝑃(𝑥, 𝑡)
¨ We define 𝑁 𝑥, 𝑡 = ∫3
6
𝑃 𝑥′, 𝑡 . 𝑑𝑥′ as the probability to find the random variable at time 𝑡
at a distance greater than 𝑥
¨
G] 3,7
G7
= ∫3
6 GP 32,7
G7
. 𝑑𝑥′ = ∫3
6
𝐷
G%
G32% 𝑃 𝑥′, 𝑡 . 𝑑𝑥′ = 𝐷.
GP 3$,7
G3$
6
3$43 = −𝐷.
GP 3,7
G3
¨
G] 3,7
G7
= −𝐷.
GP 3,7
G3
¨ 𝑁 𝑥, 𝑡 = ∫3
6
𝑃 𝑥′, 𝑡 . 𝑑𝑥′ and so
G] 3,7
G3
= −𝑃(𝑥, 𝑡)
¨ And so the function 𝑁 𝑥, 𝑡 = ∫3
6
𝑃 𝑥′, 𝑡 . 𝑑𝑥′ ALSO follows the same equation diffusion as
𝑃(𝑥, 𝑡)
¨
G
G7
𝑁(𝑥, 𝑡) = 𝐷
G%
G3% 𝑁 𝑥, 𝑡 NOTE that 𝑁(𝑥, 𝑡) is NOT a Gaussian (unicity of solution)
113

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Another scaling argument – redux VI
¨
GP(Z,#)
G#
=
-%
,
G%P (Z,#)
GZ% with 𝑃 𝜏 𝑚, 𝑇 ≤ 𝑇 =
,
,%
. ∫:4
2
3%1
:46
exp(
5:%
,
). 𝑑𝑦
¨ 𝑝 𝜏 𝑚, 𝑇 ≤ 𝑇 =
GP T Z,# `#
G#
=
Z
# ,%-%#
. exp
5Z%
,-%#
=
Z
#
. ℎ(𝑚, 𝑇)
¨ Does 𝑝 𝜏 𝑚, 𝑇 ≤ 𝑇 also follows a diffusion equation?
114

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Some more about the Maximum
¨ Let’s look again at the reflection principle
115
𝑋)^ 𝑇 = 2𝑚 − 𝑋#

Luc_Faucheux_2020
Some more about the Maximum - a
116
-2
0
2
4
6
8
10
𝑋)^ 𝑇 = 2𝑚 − 𝑋#

Luc_Faucheux_2020
Some more about the Maximum II
¨ We can define the Maximum value of the path : 𝑀𝐴𝑋 𝑇 = MAX(𝑋 𝑡 , 0 ≤ 𝑡 ≤ 𝑇)
¨ For positive 𝑚, we have 𝑀𝐴𝑋 𝑇 ≥ 𝑚 if and only if 𝜏 𝑚, 𝑇 = min(𝑡 ≥ 0, 𝑋 𝑡 = 𝑚) is
such that 𝜏 𝑚, 𝑇 ≤ 𝑇
¨ (You cannot reach a maximum higher than the level 𝑚 if you have not reached that level yet)
¨ The reflection equality was:
¨ Now :
¨ 𝑃 𝜏 𝑚, 𝑇 ≤ 𝑇, 𝑋 𝑇 ≤ 𝑋# = 𝑃 𝑀𝐴𝑋 𝑇 ≥ 𝑚, 𝑋 𝑇 ≤ 𝑋#
¨ So we have expressed something in terms of probabilities of 𝑀𝐴𝑋 𝑇 and 𝑋 𝑇 being above
some levels. This indicates that we should be able to define and maybe calculate a joint
probability for {𝑀𝐴𝑋 𝑇 , 𝑋 𝑇 } that we define as 𝑓 𝑀𝐴𝑋 𝑇 = 𝑚, 𝑋 𝑇 = 𝑋# = 𝑓(𝑀, 𝑋)
117

Luc_Faucheux_2020
Some more about the Maximum III
¨ Here we use the following trick: we do not try to explicitly derive 𝑓(𝑀, 𝑋), but write
equations that 𝑓(𝑀, 𝑋) verifies, and from those try to derive 𝑓(𝑀, 𝑋)
¨ 𝑃 𝜏 𝑚, 𝑇 ≤ 𝑇, 𝑋 𝑇 ≤ 𝑋# = 𝑃 𝑀𝐴𝑋 𝑇 ≥ 𝑚, 𝑋 𝑇 ≤ 𝑋#
¨ 𝑃 𝑀𝐴𝑋 𝑇 ≥ 𝑚, 𝑋 𝑇 ≤ 𝑋# = ∫4Z
46
𝑑𝑀 ∫_456
_4_1
𝑑𝑋 . 𝑓(𝑀, 𝑋)
¨ 𝑃 𝜏 𝑚, 𝑇 ≤ 𝑇, 𝑋 𝑇 ≥ 2𝑚 − 𝑋# = ∫34,Z5_1
346
1. ℎ(𝑥, 𝑡). 𝑑𝑥
¨ 𝑃 𝜏 𝑚, 𝑇 ≤ 𝑇, 𝑋 𝑇 ≥ 2𝑚 − 𝑋# = ∫34,Z5_1
346
1.
+
,%-%#
. exp(
53%
,-%#
). 𝑑𝑥
¨ Because if 𝑋 𝑇 ≥ 2𝑚 − 𝑋#, then by construction 𝑋 𝑇 has reached the level 𝑚 before 𝑇
¨ Note that we follow here p. 114 of Shrieve (so 𝑋# < 𝑚)
¨ Will try to give a shot later at a more general formula
118

Luc_Faucheux_2020
Some more about the Maximum IV
¨ So we have the following:
¨ ∫4Z
46
𝑑𝑀 ∫_456
_4_1
𝑑𝑋 . 𝑓 𝑀, 𝑋 = ∫34,Z5_1
346
1.
+
,%-%#
. exp(
53%
,-%#
). 𝑑𝑥
¨ Now bear in mind that we do not still know 𝑓 𝑀, 𝑋
¨ But we will take the derivative of the above equation with respect to 𝑋# and 𝑚
¨
G
GZ
∫4Z
46
𝑑𝑀 ∫_456
_4_1
𝑑𝑋 . 𝑓 𝑀, 𝑋 = − ∫_456
_4_1
𝑑𝑋 . 𝑓(𝑚, 𝑋)
¨
G
GZ
∫34,Z5_1
346
1.
+
,%-%#
. exp(
53%
,-%#
). 𝑑𝑥 = −
+
,%-%#
. exp(
5(,Z5_1)%
,-%#
)
¨ So we now have:
¨ ∫_456
_4_1
𝑑𝑋 . 𝑓 𝑚, 𝑋 =
+
,%-%#
. exp(
5(,Z5_1)%
,-%#
)
119

Luc_Faucheux_2020
Some more about the Maximum V
¨ ∫_456
_4_1
𝑑𝑋 . 𝑓 𝑚, 𝑋 =
+
,%-%#
. exp(
5(,Z5_1)%
,-%#
)
¨ We now take the derivative with respect with 𝑋#
¨
G
G_1
∫_456
_4_1
𝑑𝑋 . 𝑓 𝑚, 𝑋 = 𝑓(𝑚, 𝑋#)
¨
G
G_1
+
,%-%#
. exp(
5(,Z5_1)%
,-%#
) =
+
,%-%#
. exp
5 ,Z5_1
%
,-%#
. 2. 2𝑚 − 𝑋# .
+
,-%#
¨ We have now determined for 𝑋# < 𝑚
¨ 𝑓 𝑚, 𝑋# =
,Z5_1
-%#
.
+
,%-%#
. exp
5 ,Z5_1
%
,-%#
=
,Z5_1
-%#
. ℎ(2𝑚 − 𝑋#, 𝑇)
¨ ℎ(𝑥, 𝑡) is the regular Gaussian
¨ 𝑓 𝑚, 𝑋# is the joint probability density at time 𝑇 to have a maximum 𝑚 and terminal value
𝑋#
120

Luc_Faucheux_2020
Some more about the Maximum VI
¨ The joint probability density to reach within the interval [0, 𝑇] a maximum value 𝑀and
having a terminal value for the Brownian motion 𝑋# is:
¨ 𝑓 𝑀, 𝑋# =
,5_1
-%#
. ℎ(2𝑀 − 𝑋#, 𝑇)
¨ 𝑝 𝑚, 𝑇 is the probability density function for the reaching the level 𝑚 for the first time at
time 𝑇
¨ 𝑝 𝑚, 𝑇 . 𝑑𝑇 is the probability for a given level 𝑚 to find the the first passage time 𝜏 𝑚, 𝑇 =
min(𝑡 ≥ 0, 𝑋 𝑡 = 𝑚) in the interval 𝑇, 𝑇 + 𝑑𝑇
¨ 𝑝 𝑚, 𝑇 =
Z
#
. ℎ 𝑚, 𝑇
121

Luc_Faucheux_2020
Some more about the Maximum VII
¨ ℎ 𝑚, 𝑇 has unit of
[+]
[_]
, ℎ 𝑚, 𝑡 =
+
,%-%7
. exp(
5Z%
,-%7
)
¨ 𝑝 𝑚, 𝑇 has unit of
[+]
[#]
𝑝 𝑚, 𝑇 =
Z
#
. ℎ 𝑚, 𝑇
¨ 𝜎, 𝑇 has units of [𝑋,]
¨ 𝑓 𝑀, 𝑋# has units of
[+]
[_%]
𝑓 𝑀, 𝑋# =
,5_1
-%#
. ℎ(2𝑀 − 𝑋#, 𝑇) with (𝑋#< 𝑀)
¨ What does that mean to set 𝑋# = 𝑀 ?
¨ 𝑓 𝑀, 𝑀 =

-%#
. ℎ(𝑀, 𝑇)
¨ 𝑓 𝑚, 𝑚 =
Z
-%#
. ℎ 𝑚, 𝑇 =
+
-% . 𝑝(𝑚, 𝑇)
122

Luc_Faucheux_2020
Some more about the Maximum VII-b
¨ 𝑓 𝑚, 𝑚 =
Z
-%#
. ℎ 𝑚, 𝑇 =
+
-% . 𝑝(𝑚, 𝑇)
¨ Units are still correct
¨ 𝑓 𝑚, 𝑚 has to be integrated of 𝑋 then again over 𝑋 to return a dimensionless number
¨ 𝑝(𝑚, 𝑇) has to be integrated over time to return a dimensionless number
¨ The probability density to end up at time 𝑇 at a terminal value 𝑚, with the time 𝑇 being the
first time that this value 𝑚 is reached (since it is the maximum, so was never reached
before), is equal to the probability density (in time) to have the first passage time for the
level 𝑚 at the terminal time 𝑇, scaled by the square of the volatility
¨ 𝑓 𝑚, 𝑚 . 𝜎, 𝑇 = 𝑇. 𝑝(𝑚, 𝑇)
123

Luc_Faucheux_2020
Some more about the Maximum VIII
¨ Joint density and conditional density
¨ 𝑓 𝑀, 𝑋# is the joint density to find the maximum within [𝑀, 𝑀 + 𝑑𝑀] and the terminal
value within [𝑋#, 𝑋# + 𝑑𝑋#] (with for now the condition (𝑋#< 𝑀)
¨ Sometimes it is easier from a numerical point of view to simulate the Brownian motion
(process 𝑋(𝑇)) and THEN simulate another process for the maximum 𝑀. This second step
requires a slightly different probability, we need to know in this case the distribution of the
maximum 𝑀 between [0, 𝑇], conditioned on the value of 𝑋# = 𝑋(𝑇)
¨ (Shrieve p.114)
¨ The conditional density is the joint density divided by the marginal density of the
conditioning random variable.
¨ We are looking for the conditional density 𝑓 𝑀|𝑋#
¨ 𝑃𝑟𝑜𝑏 𝑀|𝑋# = 𝑃𝑟𝑜𝑏 𝑀, 𝑋# /𝑃𝑟𝑜𝑏 𝑋#
¨ 𝑓 𝑀|𝑋# = 𝑓 𝑀, 𝑋# /ℎ 𝑋#
124

Luc_Faucheux_2020
Some more about the Maximum IX
¨ 𝑓 𝑀|𝑋# = 𝑓 𝑀, 𝑋# /ℎ 𝑋# and has unit of
[+]
[_]
¨ ℎ 𝑋# = ℎ 𝑋#, 𝑇 =
+
,%-%#
. exp
5_1
%
,-%#
¨ 𝑓 𝑀, 𝑋# =
,5_1
-%#
. ℎ(2𝑀 − 𝑋#, 𝑇)
¨ And so we get:
¨ 𝑓 𝑀|𝑋# =
,5_1
-%#
.
h ,5_1,#
h _1,#
=
,5_1
-%#
. exp
/_1
%
,-%#
. exp
5(,5_1)%
,-%#
¨ We also have: −(2𝑀 − 𝑋#), + 𝑋#
,
= −4𝑀, + 4𝑀𝑋# = −4𝑀(𝑋# − 𝑀)
¨ 𝑓 𝑀|𝑋# =
,5_1
-%#
. exp
5K(_15)
,-%#
¨ That is kind of it, not sure if I can find any insightful thing to say about this
125

Luc_Faucheux_2020
Some more concepts about time – first return
¨ We have looked at the first passage, now let’s gain some intuition on the first return (and
also last return), first return in blue dot, successive returns in grey, last return in red for the
return to the origin
126

Luc_Faucheux_2020
Some more concepts about time – first return II
¨ Following some of the convention we had before, let us define as 𝑝𝑓𝑟(0, 𝑇)the probability
density distribution for the first return time to the origin at time 𝑇
¨ Note that we can always shift later the distribution around a “new” origin, one of the nice
properties of a Kolmogorov-like process
¨ 𝑝𝑓𝑟(0, 𝑇) is the probability density for the stochastic process 𝑋 𝑡 to return for the first
time back to the origin a time 𝑇
¨ The cumulative function of 𝑝𝑓𝑟(0, 𝑇) is:
¨ 𝑃𝑅𝐹 0, 𝑇 = ∫748
74#
𝑝𝑓𝑟 0, 𝑡 ). 𝑑𝑡
¨ 𝑝𝑓𝑟 0, 𝑇 =
GP.' 8,#
G#
¨ 𝑃𝑅𝐹 0, 𝑇 is the cumulative probability that the stochastic process 𝑋 𝑡 has returned to the
origin (at least once) by the time 𝑇
127

Luc_Faucheux_2020
Some more concepts about time – first return III
¨ 𝑃𝑅𝐹 0, 𝑇 is the cumulative probability that the stochastic process 𝑋 𝑡 has returned to the
origin (at least once) by the time 𝑇
¨ {1 − 𝑃𝑅𝐹 0, 𝑇 } is the cumulative survival probability noted 𝑆 0, 𝑇 that the stochastic
process 𝑋 𝑡 has NOT returned to the origin by the time 𝑇
¨ 𝑝𝑓𝑟 0, 𝑇 =
GP.' 8,#
G#
= −
G) 8,#
G#
¨ In order to evaluate 𝑆 0, 𝑇 , we need to enumerate all the paths that never returned to the
origin after time 𝑇 (or after 𝑁 steps where the time interval is 𝛿𝑡 = 𝑇/𝑁)
¨ We need to calculate the probability that a path never returns to the origin.
¨ This is a variant of the “ballot theorem”: in a ballot where candidates A and B have a and b
total votes respectively, what is the probability that when counting the votes, the tally for A
is always higher than B (A always leads the vote tally and there is no tie, and no time when B
is leading in the vote).
¨ Desire Andre and Joseph Louis Francois Bertrand (1887)
128

Luc_Faucheux_2020
Some more concepts about time – first return IV
¨ The remarkably simple result is that the probability of such a path is
(i5()
(i/()
¨ There are a couple of ways we can convince ourselves of this result.
¨ Proof by reflection, we suppose a>b (A is the winner, so the path always stays above 0)
¨ Any sequence that starts with a B must reach a tie at some point because A wins.
¨ Also any sequence that starts with B has B leading, so has to be excluded
¨ So we are left with sequences that start with A
¨ Some of those will never reach a tie, and some will
¨ For those who do reach a tie, we will use the reflection trick again, by reflecting the votes up
to the point of first tie (𝑋 𝑡 crossing the origin again). The reflected new sequence will
start with a B
129

Luc_Faucheux_2020
Some more concepts about time – first return V
¨ A sequence in space and its corresponding voting tally in AB
130

Luc_Faucheux_2020
Some more concepts about time – first return VI
¨ A reflected sequence up to the first point of tie, reflected path in blue
131
-3
-2
-1
0
1
2
3
4

Luc_Faucheux_2020
Some more concepts about time – first return VII
¨ Note: this is why it is sometime so helpful in gaining intuition to run simulations. It was very
hard to find a “nice” looking graphs.
¨ A lot of graphs either has the first return very close to the origin, or never returned
¨ This is somewhat counter-intuitive because a lot of people would expect that for a fair game,
each player would be on the winning side for about half the time, and that the lead will pass
not infrequently from one player to the other.
¨ It is actually not the case, and we will show that actually first returns and last returns are
actually much more likely to occur either very early or very late in the random walk.
¨ It is highly probable to remain on one side of the origin for nearly the entire walk, leading to
long waiting time before the tie
132

Luc_Faucheux_2020
Some more concepts about time – first return VIII
¨ A couple of F9
133
-8
-6
-4
-2
0
2
4
6
-3
-2
-1
0
1
2
3
4
5
6
7
8
-4
-3
-2
-1
0
1
2
3
4
5
6
-6
-4
-2
0
2
4
6
8
0
2
4
6
8
10
12
14

Luc_Faucheux_2020
Some more concepts about time – first return IX
¨ So to recap, looking at the survival probability for path that always stay above the origin
¨ Every sequence that starts with B is excluded
¨ Every sequence that starts with A, either ties or does not
¨ If it does, we built the reflected path in blue, reflecting up to the point of the first tie. This
reflected sequence will start with B and will cross the origin (will tie)
¨ Over 𝑁 votes, we have 𝑎 votes for A (or jump up in space) and 𝑏 votes for B (or jump down
in space)
¨ Because of the reflection, the number of sequences that start with A and tie is equal to the
number of sequences starting with B and do also tie
¨ Looking at the outcome with A being the winner (𝑎 > 𝑏), any sequence starting with B will
automatically tie at some point
¨ So we are counting twice the number of sequences starting with B
¨ The probability that a sequence starts with B is: ⁄𝑏 (𝑎 + 𝑏)
134

Luc_Faucheux_2020
Some more concepts about time – first return X
¨ So the survival probability that we are after is :
¨ 1 − 2. z𝑏 𝑎 + 𝑏 =
i5(
i/(
¨ Another way to look at it is by induction: suppose that the formula is true for (𝑁 − 1) steps,
can we extend it to 𝑁 steps?
¨ So for (𝑁 − 1) , we had either (𝑎 − 1, 𝑏) or (𝑎, 𝑏 − 1) votes for A and B
¨ The probability of no tie in the case (𝑁 − 1, 𝑎 − 1, 𝑏) is
i5+5(
i5+/(
¨ The probability of no tie in the case (𝑁 − 1, 𝑎, 𝑏 − 1) is
i5(/+
i/(5+
¨ Going into (𝑁) we need to count the last vote, the probability of a vote for A is
i
i/(
and the
probability of a vote for B is
(
i/(
(reverse the order and treat the last vote as the first one,
and read the sequence backward)
135

Luc_Faucheux_2020
Some more concepts about time – first return XI
¨ The survival probability at the level (𝑁)is then:
¨
i
i/(
.
i5+5(
i5+/(
+
(
i/(
.
i5(/+
i/(5+
=
ii5i5i(/(i5((/(
(i/()(i/(5+)
=
(i/(5+)(i5()
(i/()(i/(5+)
=
(i5()
(i/()
¨ Another notation would be 𝑁/ = 𝑎 and 𝑁5 = 𝑏, 𝑁/ + 𝑁5 = 𝑁
¨ The total number of possible paths is
]!
]5!]&!
for a given set (𝑁, 𝑁/, 𝑁5)
¨ The paths that end with positive end value are such that 𝑁/ > 𝑁5
¨ The number of paths never crossing (never tie-ing) is the total number of paths multiplied by
the probability that a path will not tie
¨ We will then have to sum all those numbers over the possible end points (because we have
it currently fixed by having a given set (𝑁, 𝑁/, 𝑁5), with those being above the origin, or
ensuring that (𝑁/ > 𝑁5)
¨ Then multiply by the probability for one path which is (25]) in the binomial discrete model
136

Luc_Faucheux_2020
Some more concepts about time – first return XII
¨ We then have for the survival probability:
¨ 𝑆 0, 𝑇 = 𝑆 0, 𝑁 = (25]). ∑]&48
]&k]5 ]!
]5!]&!
.
]55]&
]5/]&
and we have: 𝑁/+𝑁5= 𝑁
¨ So 𝑁5 < 𝑁/ is equivalent to 𝑁5 < 𝑁 − 𝑁5 or 𝑁5 < 𝑁/2
¨ 𝑆 0, 𝑁 = (25]). ∑]&48
]&k
6
% ] !
(]5]&)!(]&)!
. (
]5,]&
]
)
¨ (2]). 𝑆 0, 𝑁 = ∑]&48
]&k
6
%
[
] !
(]5]&)!(]&)!
− 2
]&
]
.
] !
(]5]&)!(]&)!
]
¨ (2]). 𝑆 0, 𝑁 = ∑]&48
]&k
6
%
[
] !
(]5]&)!(]&)!
− 2 .
]5+ !
]5]& !(]&5+)!
]
137

Luc_Faucheux_2020
Some more concepts about time – first return XIII
¨ We also use the result from the Pascal triangle
¨
] !
(]5]&)!(]&)!
=
]5+ !
(]5+5]&)!(]&)!
+
]5+ !
]5+5(]&5+) !(]&5+)!
or: 𝐶J
A
= 𝐶J5+
A
+ 𝐶J5+
A5+
¨ 𝐶J5+
A
+ 𝐶J5+
A5+
=
J5+ !
J5+5A !A!
+
J5+ !
J5+5 A5+ !(A5+)!
=
J5+ !
J5+5A !A!
+
J5+ !
J5A !(A5+)!
¨ 𝐶J5+
A
+ 𝐶J5+
A5+
=
J5+ !
J5+5A !A!
+
J5+ !
J5A !(A5+)!
=
J5+ !(J5A)
J5A !A!
+
J5+ !A
J5A !(A)!
=
J5+ ! J5A / J5+ !A
J5A !A!
¨ 𝐶J5+
A
+ 𝐶J5+
A5+
=
J5+ ! J5A / J5+ !A
J5A !A!
=
J5+ ! J5A/A
J5A !A!
=
J5+ ! J
J5A !A!
=
J !
J5A !A!
= 𝐶J
A
¨ So we can rewrite:
¨ (2]). 𝑆 0, 𝑁 = ∑]&48
]&k
6
%
[
]5+ !
(]5+5]&)!(]&)!
+
]5+ !
]5+5(]&5+) !(]&5+)!
− 2 .
]5+ !
]5]& !(]&5+)!
]
138

Luc_Faucheux_2020
Some more concepts about time – first return XIV
¨ (2]). 𝑆 0, 𝑁 = ∑]&48
]&k
6
%
[
]5+ !
(]5+5]&)!(]&)!
+
]5+ !
]5]&) !(]&5+)!
− 2 .
]5+ !
]5]& !(]&5+)!
]
¨ (2]). 𝑆 0, 𝑁 = ∑]&48
]&k
6
%
[𝐶]5+
]&
+ 𝐶]5+
]&5+
− 2 . 𝐶]5+
]&5+
]
¨ (2]). 𝑆 0, 𝑁 = ∑]&48
]&k
6
%
[𝐶]5+
]&
+𝐶]5+
]&5+
]
¨ In the above sum, terms cancel each other out up until the last one
¨ (2]). 𝑆 0, 𝑁 = 𝐶]5+
6
%
5+
¨ We now make use of the Stirling approximation : 𝑁! ~ 2𝜋𝑁( ⁄]
?)]~ 2𝜋𝑁exp(𝑁𝑙𝑛𝑁 − 𝑁)
139

Luc_Faucheux_2020
Some more concepts about time – first return XV
¨ (2]). 𝑆 0, 𝑁 = 1 + ∑]&4+
]&k
6
%
[
]5+ !
(]5+5]&)!(]&)!
−
]5+ !
]5]& !(]&5+)!
]
140

Luc_Faucheux_2020
Another look at first passage time
¨ Let’s look at the problem in a slightly different fashion
+
,%-%7
. exp(
5(35l7)%
,-%7
)
¨ The normalization is: ∫3456
34/6
ℎ 𝑥, 𝑡 . 𝑑𝑥 = 1
G
G7
ℎ 𝑥, 𝑡 =
-%
,
G%
G3% ℎ 𝑥, 𝑡 − 𝜇.
G
G3
ℎ 𝑥, 𝑡
¨ The corresponding SDE is : 𝑑𝑋 = 𝜇. 𝑑𝑡 + 𝜎. 𝑑𝑊
¨ In general, we will look at SDE <-> PDE but the simple mapping is:
¨ 𝑑𝑋 = 𝑎. 𝑑𝑡 + 𝑏. 𝑑𝑊
¨
G
G7
ℎ 𝑥, 𝑡 = −
G
G3
[𝐴. ℎ 𝑥, 𝑡 −
G
G3
(𝐵. ℎ 𝑥, 𝑡 )] with 𝐴 = 𝑎 and 𝐵 =
+
,
. 𝑏,
141

Luc_Faucheux_2020
Another look at first passage time - II
¨ Without any drift for now, this reads:
+
,%-%7
. exp(
53%
,-%7
)
34/6
ℎ 𝑥, 𝑡 . 𝑑𝑥 = 1
G
G7
ℎ 𝑥, 𝑡 =
-%
,
G%
G3% ℎ 𝑥, 𝑡
¨ The corresponding SDE is : 𝑑𝑋 = 𝜎. 𝑑𝑊
¨ In general, we will look at SDE <-> PDE but the simple mapping is:
¨ 𝑑𝑋 = 𝑏. 𝑑𝑊
¨
G
G7
ℎ 𝑥, 𝑡 =
G
G3
[
G
G3
(𝐵. ℎ 𝑥, 𝑡 )] with 𝐵 =
+
,
. 𝑏,
142

Luc_Faucheux_2020
Another look at first passage time - III
¨ The diffusion equation is a linear equation
¨ Any linear combination of solutions will itself be a solution
¨ We can identify a spanning set of solutions:
¨ The underlying Brownian motion 𝑋 𝑡 follows ℎ 𝑥, 𝑥8, 𝑡 =
+
,%-%7
. exp(
5(3537)%
,-%7
)
34/6
ℎ 𝑥, 𝑥8, 𝑡 . 𝑑𝑥 = 1
¨ And the initial starting point: ℎ 𝑥, 𝑥8, 𝑡 = 0 = 𝛿(𝑥 − 𝑥8)
¨ Let’s consider what is sometimes referred to as the “cliff” problem: a random walker
diffuses from its initial position 𝑥8 up until it meets the “cliff” at position 𝑥 = 𝑚, and then
“falls off the cliff” and disappears.
¨ So we are looking for a solution •ℎ 𝑥, 𝑥8, 𝑡 that obeys the diffusion equation in the interval
] − ∞, 𝑚]
¨ For all time we need to verify: •ℎ 𝑥 = 𝑚, 𝑥8, 𝑡 = 0
143

Luc_Faucheux_2020
Another look at first passage time - IV
¨ Note that the conservation of probability ∫3456
34/6 •ℎ 𝑥, 𝑥8, 𝑡 . 𝑑𝑥 = 1 obviously will not apply
¨ If anything we will define the survival probability, which is the probability that the random
walker did not yet fall off the cliff
¨ 𝑆𝑅 𝑡 = ∫3456
34Z •ℎ 𝑥, 𝑥8, 𝑡 . 𝑑𝑥
¨ This probability is the probability that the random walker did not reach the level 𝑥 = 𝑚 up
until the time 𝑡
¨ And so the probability that the random walker would have met the level 𝑥 = 𝑚 within the
time interval [0, 𝑡] is 1 − 𝑆𝑅 𝑡 = 𝑃 𝜏 𝑚, 𝑡 ≤ 𝑡 in the previous notation
¨ So now, either we know 𝑃 𝜏 𝑚, 𝑡 ≤ 𝑡 and we can calculate the Survival Probability 𝑆𝑅 𝑡
¨ Conversely, if we can find an easy way to calculate 𝑆𝑅 𝑡 we know 𝑃 𝜏 𝑚, 𝑡 ≤ 𝑡
144

Luc_Faucheux_2020
Another look at first passage time - V
¨ Note that the two processes are NOT the same.
¨ In the case of the cliff problem, the density is not conserved and the random walker is
“taken out” as soon as it hits the level 𝑥 = 𝑚
¨ In the case of the regular diffusion we looked at, the process is not impacted when crossing
the level 𝑥 = 𝑚
¨ However, for the purpose of calculating the First Passage Probability 𝑃 𝜏 𝑚, 𝑡 ≤ 𝑡 we can
use either
¨ So let’s see if we can find an easy solution to the cliff problem
¨ Remember that the diffusion equation is linear, so if we could find a linear combination of
Gaussians that matches •ℎ 𝑥 = 𝑚, 𝑥8, 𝑡 = 0, we would have at least one solution to work
with (maybe not unique, but at least something we could use)
145

Luc_Faucheux_2020
Another look at first passage time – VI – Image method
¨ Let’s look at:
¨ •ℎ 𝑥, 𝑥8, 𝑡 =
+
,%-%7
. exp
5 3537
%
,-%7
−
+
,%-%7
. exp(
5(35(,Z537))%
,-%7
)
¨ •ℎ 𝑥, 𝑥8, 𝑡 = ℎ 𝑥, 𝑥8, 𝑡 − ℎ 𝑥, (2𝑚 − 𝑥8), 𝑡
¨ Note that we see the beautiful symmetry principle at work again here
¨ •ℎ 𝑥, 𝑥8, 𝑡 follows the diffusion equation
¨ •ℎ 𝑥 = 𝑚, 𝑥8, 𝑡 = 0 for all time 𝑡
¨ We are in business
¨ 𝑆𝑅 𝑡 = ∫3456
34Z •ℎ 𝑥, 𝑥8, 𝑡 . 𝑑𝑥 = ∫3456
34Z
ℎ 𝑥, 𝑥8, 𝑡 . 𝑑𝑥 − ∫3456
34Z
ℎ 𝑥, (2𝑚 − 𝑥8), 𝑡 . 𝑑𝑥
¨ We also have: ∫3456
34Z
ℎ 𝑥, 𝑥8, 𝑡 . 𝑑𝑥 = ∫3456
34Z537
ℎ 𝑥, 𝑥8 = 0, 𝑡 . 𝑑𝑥
146

Luc_Faucheux_2020
Another look at first passage time - VII
¨ 𝑆𝑅 𝑡 = ∫3456
34Z
ℎ 𝑥, 𝑥8, 𝑡 . 𝑑𝑥 − ∫3456
34Z
ℎ 𝑥, (2𝑚 − 𝑥8), 𝑡 . 𝑑𝑥
¨ 𝑆𝑅 𝑡 = ∫3456
34Z537
ℎ 𝑥, 0, 𝑡 . 𝑑𝑥 − ∫3456
34Z5(,Z537)
ℎ 𝑥, 0, 𝑡 . 𝑑𝑥
¨ 𝑆𝑅 𝑡 = ∫3456
34Z537
ℎ 𝑥, 0, 𝑡 . 𝑑𝑥 − ∫3456
34375Z
ℎ 𝑥, 0, 𝑡 . 𝑑𝑥
¨ 𝑆𝑅 𝑡 = ∫34375Z
34Z537
ℎ 𝑥, 0, 𝑡 . 𝑑𝑥 = ∫34375Z
34Z537 +
,%-%7
. exp
53%
,-%7
. 𝑑𝑥
¨ 𝑆𝑅 𝑡 = ∫
34
47&2
%3%,
34
2&47
%3%, +
%
. exp −𝜉, . 𝑑𝜉 with 𝜉 =
3
,-%7
and 𝑑𝑥 = 2𝜎, 𝑡. 𝑑𝜉
¨ 𝑆𝑅 𝑡 = erf
Z537
,-%7
= erf(
Z537
KU7
)
¨ Using the diffusion notation, 𝐷 =
-%
,
147

Lf 2020 bachelier

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