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Transcript of Stochastic Processes for Finance
Patrick Roger
Stochastic Processes forFinance
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Stochastic Processes for Finance
Patrick Roger
Strasbourg University, EM Strasbourg Business School
June 2010
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Stochastic Processes for Finance© 2010 Patrick Roger & Ventus Publishing ApSISBN 978-87-7681-666-7
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Stochastic Processes for Finance
4
Contents
Contents Introduction 7
1 Discrete-time stochastic processes 91.1 Introduction 91.2 The general framework 101.3 Information revelation over time 121.3.1 Filtration on a probability space 121.3.2 Adapted and predictable processes 141.4 Markov chains 171.4.1 Introduction 171.4.2 Definition and transition probabilities 191.4.3 Chapman-Kolmogorov equations 191.4.4 Classification of states 211.4.5 Stationary distribution of a Markov chain 241.5 Martingales 251.5.1 Doob decomposition of an adapted process 291.5.2 Martingales and self-financing strategies 301.5.3 Investment strategies and stopping times 341.5.4 Stopping times and American options 39
2 Continuous-time stochastic processes 432.1 Introduction 43
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Stochastic Processes for Finance
5
Contents
2.2 General framework 442.2.1 Filtrations, adapted and predictable processes 482.2.2 Markov and diffusion processes 512.2.3 Martingales 532.3 The Brownian motion 552.3.1 Intuitive presentation 552.3.2 The assumptions 572.3.3 Definition and general properties 612.3.4 Usual transformations of the Wiener process 642.3.5 The general Wiener process 672.3.6 Stopping times 692.3.7 Properties of the Brownian motion paths 71
3 Stochastic integral and Itô’s lemma 733.1 Introduction 733.2 The stochastic integral 753.2.1 An intuitive approach 753.2.2 Counter-example 783.2.3 Definition and properties of the stochastic integral 803.2.4 Calculation rules 833.3 Itô’s lemma 853.3.1 Taylor’s formula, an intuitive approach to Itô’s lemma 863.3.2 Itô’s lemma 883.3.3 Applications 883.4 The Girsanov theorem 91
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Stochastic Processes for Finance
6
Contents
3.4.1 Preliminaries 913.4.2 Girsanov theorem 933.4.3 Application 933.5 Stochastic differential equations 953.5.1 Existence and unicity of solutions 953.5.2 A specific case: linear equations 97
Bibliography 100
Index 103
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Stochastic Processes for Finance
7
Introduction
0, 1, .., T. [0;T ] .
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Stochastic Processes for Finance
8
Introduction
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Stochastic Processes for Finance
9
Discrete-time stochastic processes
T
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Stochastic Processes for Finance
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Discrete-time stochastic processes
T = {0, 1, ..., T} T < +∞. T− = {0, 1, ..., T − 1} ; T − 1 T. T
(,A, P ) A P A.
X = (X0, ...,XT ) (,A) R.
Xt n
R BR
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Stochastic Processes for Finance
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Discrete-time stochastic processes
Xt X
Xt = Xt−1 × Yt
Yt u d p 1 − p. X0 X Xt Xt R+
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Maersk.com/Mitas
�e Graduate Programme for Engineers and Geoscientists
Month 16I was a construction
supervisor in the North Sea
advising and helping foremen
solve problems
I was a
hes
Real work International opportunities
�ree work placementsal Internationaor�ree wo
I wanted real responsibili� I joined MITAS because
Maersk.com/Mitas
�e Graduate Programme for Engineers and Geoscientists
Month 16I was a construction
supervisor in the North Sea
advising and helping foremen
solve problems
I was a
hes
Real work International opportunities
�ree work placementsal Internationaor�ree wo
I wanted real responsibili� I joined MITAS because
Maersk.com/Mitas
�e Graduate Programme for Engineers and Geoscientists
Month 16I was a construction
supervisor in the North Sea
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solve problems
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Real work International opportunities
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I wanted real responsibili� I joined MITAS because
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Stochastic Processes for Finance
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Discrete-time stochastic processes
X1(ω) X2(ω)ω1 ω2 ω3 ω4
X1 X2
Xt X1
X2 X1 X2 .
= {ωi, i = 1, ..., 4} A = P() X1 X2
X1 X1 =100 {ω1, ω2} X1 =105 {ω3, ω4} . X2 X1 {ω1, ω2} X1 T = 2 X2
X1 X2
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Stochastic Processes for Finance
13
Discrete-time stochastic processes
X1 X2 σ X1
σ (X1, X2) .
t t
s t
(,A, P ) F = {F0,F1, ...,FT} A. (,A, P,F)
Ft
Ft−1 ⊆ Ft
t ≥ 1. T,
T. FT =A.
F0 = {∅ }
F0 = {∅,}F1 = {∅, {ω1, ω2} , {ω3, ω4} ,}F2 = P()
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Stochastic Processes for Finance
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Discrete-time stochastic processes
t = 1 {ω1, ω2} {ω3, ω4} X1
X Xt
t. t, Xt Xt t+ 1 Xt+1 Xt t t.
F = {F0,F1, ...,FT} Ft t {Xt ≤ 100} Xt Xt Ft t).
X = (X0, ..., XT ) F t, Xt Ft
X FX A FX
t Xs, s ≤ t.
Xt =Xt−1Yt Yt u d p 1− p. T = 2, X t = 0 T = 2. 4 = {uu, ud, du, dd} X1 up down . {uu, ud} {du, dd}
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Stochastic Processes for Finance
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Discrete-time stochastic processes
F1 = {∅, {uu, ud} , {du, dd} ,} . X1
X1(uu) = X1(ud) = uX0
X1(du) = X1(dd) = dX0
F1 F = FX .
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Stochastic Processes for Finance
16
Discrete-time stochastic processes
X0 X1(ω) X2(ω)ω1 1 u u2
ω2 1 u udω3 1 d duω4 1 d d2
X
Card() = n n L2 (,A, P ) Rn. L2 (,Ft, P ) , Ft L2 (,A, P )
t. L2 (,F1, P )
L2 (,F1, P ) =(x, y, z, t) ∈ R4 x = y z = t
F1 X0 = 1
K Xk, k = 1, ...,K.
t t+1. t t + 1 θ′
t =θ1t , ...θ
Kt
θ′
t+1 =θ1t+1, ...θ
Kt+1
t t+ 1.
θt+1 t
X = (X1..., XT ) t ≥ 1, Xt Ft−1
t = 1
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Stochastic Processes for Finance
17
Discrete-time stochastic processes
X1 F0 θ θ0 θ0 θ1.
rt [t; t+ 1] Bt t
rt t r F , B
Bt =t−1
s=0
(1 + rs)
Bt t− 1 Bt Ft−1
p,
p+1. p p+1.
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Stochastic Processes for Finance
18
Discrete-time stochastic processes
p p+1.
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Stochastic Processes for Finance
19
Discrete-time stochastic processes
X (,A, P,F) (B1, ..., Bn) ∈ Bn
R (t1, ..., tn) ∈ T n t1 < .... < tn
PXt ∈ Bn
Xt ∈ Bj, j = 1, .., n− 1= P
Xt ∈ Bn
Xt− ∈ Bn−1
tn−1
tn−1 tn. tn−1.
(Xp, p ∈ N) (x1, ..., xn)
(Xp, p ∈ N) (x1, ..., xn) ;
π(xi, p− 1, p, xj) = P (Xp = xj |Xp−1 = xi )
Πp−1 = (π(xi, p− 1, p, xj), i, j = 1, ..., n) p− 1.
p, π(xi, p− 1, p, xj) = πij
π(xi, p − 1, p, xj) X xj
p xi p − 1.
m
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Stochastic Processes for Finance
20
Discrete-time stochastic processes
m
X = (Xp, p ∈ N) N π = (πij, i, j = 1, ...,N) .
π(m+n)ij = P (Xm+n = xj |X0 = xi ) m + n
i j.
π(m+n)ij =
n
k=1π
(m)ik π
(n)kj
m = n = 1 N = 3. π(2)12
x2 x1.
πր x1
πց
x1π→ x2
π→ x2
ցπ x3
րπ
π(2)12 = π11π12 + π12π22 + π13π32
xi xj m + n π
(m+n)ij i m
π
(m)ij , i, j = 1, ..., n
j n
π
(n)ij , i, j = 1, ..., n
.
X = (Xp, p ∈ N) N π = (πij, i, j = 1, ...,N) .
π(m)ij = P (Xm = xj |X0 = xi ) m
i j π(m) m π
(m)ij .
π(m) = πm
πm m π.
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Stochastic Processes for Finance
21
Discrete-time stochastic processes
xj X xi k > 0
π(k)ij > 0
xi xj k k′
π(k)ij > 0 π
(k′)ji > 0
π.
π =
0.6 0.4 0 00.3 0.7 0 00 0 0.5 0.50 0 0.7 0.3
x1 x2, x3
x4 x1
x2 x3 x4 x1 x2 {x1, x2} {x3, x4} .
X = (Xp, p ∈ N) N π = (πij , i, j = 1, ..., N) . (C1, C2, ...CM) N Ck
R, x R x x x
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Stochastic Processes for Finance
22
Discrete-time stochastic processes
S t+ 1
St+1 =
uSt pdSt 1− p
d = 1/u. S0
X = (Xp, p ∈ N) N π = (πij , i, j = 1, ..., N) .
xi, t(i) m π(m)(i, i) > 0. t(i) = 0 m, π(m)(i, i) = 0.
X t(i) = 1 i.
X = (Xp, p ∈ N) π.
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Stochastic Processes for Finance
23
Discrete-time stochastic processes
pi(n) i n i +∞
i=1pi(n) = 1
i
limm→+∞
π(m)ii = qi > 0
i i n
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Stochastic Processes for Finance
24
Discrete-time stochastic processes
X = (Xp, p ∈ N) .
qi =+∞
j=1qjπij
+∞
i=1qi = 1. qi
∀i, qi ≥ 0 +∞
i=1qi = 1
qi =+∞
i=1qjπij
qi
πm m πm (qi, i = 1, ..., n).
π =
0.6 0.40.4 0.6
π
π2 =
0.52 0.480.48 0.52
π3 =
0.504 0.4960.496 0.504
lim
m→+∞πm =
0.5 0.50.5 0.5
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Stochastic Processes for Finance
25
Discrete-time stochastic processes
(Y,Z) L2 (,A, P ) B,B′
A B ⊂ B′
Z c ∈ R, E (Z |B ) = c
∀(a, b) ∈ R2, E (aZ + bY |B ) = aE (Z |B ) + bE (Y |B )
Z ≤ Y, E (Z |B ) ≤ E (Y |B )
E (E (Z |B′ ) |B ) = E (Z |B )
Z B E (ZY |B ) = Z E (Y |B )
Z B, E (Z |B ) = E(Z)
(,A,F , P ) X = (X0, ..., XT ) (F , P )
X F ∀t ∈ T , Xt ∈ L1 (,A, P ) ∀t ∈ T ∗, Xt−1 = E [Xt |Ft−1 ]X (F , P ) Xt−1 ≥ E [Xt |Ft−1 ]X (F , P ) Xt−1 ≤ E [Xt |Ft−1 ]
(F , P )
P
E [Xt −Xt−1 |Ft−1 ] = 0 X F Xt−1 Ft−1
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Stochastic Processes for Finance
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Discrete-time stochastic processes
X (s, t), s ≤ t
E [Xt |Fs ] = (≤,≥) Xs
E [Xt+1 |Ft−1 ] . Xt−1 =E [Xt |Ft−1 ] Xt = E [Xt+1 |Ft ] .
Xt−1 = E [E [Xt+1 |Ft ] |Ft−1 ]
Xt−1 = E [Xt+1 |Ft−1 ]
Ft−1 ⊂ Ft+1.
X E [Xt |Ft−1 ] Ft−1 Xt X Xt−1 Xt t− 1. Xt−1 Xt Ft−1
E(Xt) .
X X0 = c ∈R
Xt = Xt−1 + Yt
Yt
E (E [X |F− ]) = E [X] = E [X−]
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Stochastic Processes for Finance
27
Discrete-time stochastic processes
X0 t Xt
Xt = Xt−1 + Yt
Yt t Ys
t
Xt = X0 +t
s=1
Ys
F Y, Fs Yu, u ≤ s. Ys s Xs s X F
E [Xt |Ft−1 ] = E [Xt−1 + Yt |Ft−1 ]
= E [Xt−1 |Ft−1 ] + E [Yt |Ft−1 ]
X F, Xt−1 Yt Yt Ft−1. E [Yt |Ft−1 ] = E [Yt] = 0⇒ E [Xt |Ft−1 ] = Xt−1, X .
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Stochastic Processes for Finance
28
Discrete-time stochastic processes
Y ∈ L1 (,A, P ) X
Xt = E [Y |Ft ]
X
E [Xt |Ft−1 ] = E [E [Y |Ft ] |Ft−1 ]
= E [Y |Ft−1 ] = Xt−1
E [Xt] t)
T t ≤ T Y ( Z) Y Z).
Xt = Yt − Zt t X0 = 0
Xt = Xt−1 + δt
δt = 1 t δt = −1
E [Xt |Ft−1 ] = Xt−1 + E [δt |Ft−1 ]
δt Ft−1. Yt = s Zt = t−s
P (δt+1 = 1 |Yt = s) =T2− s
T − t
12 s = t
2,
E [δt+1 |Yt = s ] = 0.
X t, E(Yt) = E(Zt) =t2 E [Xt] = 0. X
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Stochastic Processes for Finance
29
Discrete-time stochastic processes
0 t t
2
Y Z YT = ZT = T
2 XT = 0.
F0
X F , Xt X
∀t ∈ T , Xt = X0 +Mt +At
M M0 = 0 A A0 = 0. X A (At ≤ At+1)
Xt = X0 +Mt +At,
E [Xt −Xt−1 |Ft−1 ] = E [Mt −Mt−1 |Ft−1 ] + E [At − At−1 |Ft−1 ]
M A At − At−1.
At =t
s=1
E [Xs −Xs−1 |Fs−1 ]
X A M
Mt = Xt −X0 −t
s=1
E [Xs −Xs−1 |Fs−1 ]
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Stochastic Processes for Finance
30
Discrete-time stochastic processes
Xt 0 t [s; s+ 1]
Xt −Xt−1 = Mt −Mt−1 +At − At−1
E [Xt −Xt−1 |Ft−1 ] = E [At − At−1 |Ft−1 ] = At −At−1
At −At−1
K t Xt = (X1
t , ....,XKt )
Xk θt = (θ1t , ...., θ
Kt )
t− 1 t. t
Vt(θ) =K
k=1
θktXkt
Vt(θ) V0(θ) s, 0 < s < t 5× $100 + 10× $60 = $1100.
5×$130+10×$65 = $1300.
T
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Stochastic Processes for Finance
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Discrete-time stochastic processes
5 10 100 60 6 8 130 65 4 11 120 80
S
6×$130+8×$65 = $1300,
.
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Stochastic Processes for Finance
32
Discrete-time stochastic processes
θ θ t
K
k=1
θktXkt =
K
k=1
θkt+1Xkt
t θt. θt+1 t
X =X1, ..., XK
RK
θ =θ1, ..., θK
Y
Y0 =K
k=1
θk1Xk0
Yt =K
k=1
θktXkt
Yt t θ, Y0
E [Yt − Yt−1 |Ft−1 ] = E
K
k=1
θktX
kt − θkt−1X
kt−1
|Ft−1
E [Yt − Yt−1 |Ft−1 ] = E
K
k=1
θktXk
t −Xkt−1
|Ft−1
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Stochastic Processes for Finance
33
Discrete-time stochastic processes
K
k=1 θktX
kt =
Kk=1 θ
kt+1X
kt .
θ θt
E [Yt − Yt−1 |Ft−1 ] =K
k=1
θktE
Xkt −Xk
t−1
|Ft−1
X
E [Yt − Yt−1 |Ft−1 ] = 0
E [Yt |Ft−1 ] = Yt−1
X =X1, ..., XK
RK
θ =θ1, ..., θK
Z
Z0 = 0
Zt =K
k=1
t
s=1
θksXk
s −Xks−1
E [Zt − Zt−1 |Ft−1 ] = E
K
k=1
θktXk
t −Xkt−1
|Ft−1
E [Zt − Zt−1 |Ft−1 ] =K
k=1
θktE
Xkt −Xk
t−1
|Ft−1
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Stochastic Processes for Finance
34
Discrete-time stochastic processes
X
Z Y fY FY
E(Y ) =
+∞
−∞y fY (y) dy =
+∞
−∞y dFY (y)
E(Y ) = E(Y ) = Y,L .
Y y1 <y2, ... < yn;
E(Y ) =
yi (FY (yi)− FY (yi−1)) =
+∞
−∞y dFY (y)
Z
Zt =K
k=1
t
0
θksdXks
t
0θksdX
ks θ k
t. E(X) Zt; E(X) Zt Y Xk
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Stochastic Processes for Finance
35
Discrete-time stochastic processes
Xt t;
Xt = X0 +t
s=1
Ys
Ys X0
t Xt = X0 + 1 X0 + 1 T
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Stochastic Processes for Finance
36
Discrete-time stochastic processes
v (,A, P ) T
∀t ∈ T {v = t} = {ω ∈ v(ω) = t} ∈ Ft
ν {v = t} ∈Ft t
FT Ft
, {v = t} {v ≤ t} , {v > t} . Ft−1 ⊂ Ft, {v = t− 1} ∈ Ft−1 {v = t− 1} ∈ Ft. t−k k < t. Ft {v ≤ t} ∈Ft ⇒ {v ≤ t}c = {v > t} ∈ Ft.
v X Xv E(Xv) = E (X0) .
v X K t ∈ T
ω ∈ , |Xt(ω)| ≤ K.
Xt
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Stochastic Processes for Finance
37
Discrete-time stochastic processes
t , ν
Y
Yt =
Xt t < vXv t ≥ v
Y X, Xv v .
ν X F , Y F X Y.
ξs = {v≥s}; {v ≥ s} Y
Yt = X0 +t
s=1
ξs (Xs −Xs−1)
t < v, Yt = Xt. t > v 1 s ≤ v Yt = Xv. {v ≥ t} = {v ≤ t− 1}c ξ Y
X Y ξ
E [Yt − Yt−1 |Ft ] = E [ξt (Xt −Xt−1) |Ft ]
= ξtE [(Xt −Xt−1) |Ft ]
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Stochastic Processes for Finance
38
Discrete-time stochastic processes
X
E [(Xt −Xt−1) |Ft ] = 0
X0 = 0
t
Xt = −t
s=1
2s−1 = −t−1
s=0
2s
t
Xt = −2t − 1
2− 1= −(2t − 1)
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Stochastic Processes for Finance
39
Discrete-time stochastic processes
t+ 1 2t. Xt+1 = 1 = X0 + 1.
v = inf {t / Xt = 1}
E(v) =+∞
s=1
1
2s
s
+∞
s=1
1
2s
s =
+∞
s=1
+∞
u=s
1
2u
+∞
u=s
12
= 1
2−
E(v) =+∞
s=1
1
2s= 2
v X
T K K, T T ).
Yt t. Yt = max(K − St; 0), St t t K ≤ St. K
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Stochastic Processes for Finance
40
Discrete-time stochastic processes
K − St
Y F X
XT = YT
Xt = max (Yt;E (Xt+1 |Ft )) t < T
Y.
X Y Y, Xt ≥ Yt t.
Xt ≥ E (Xt+1 |Ft ) ; X Z Y ∀t, Zt ≥ Xt . t = T X. Zs ≥ Xs s ≥ t0 Zt−1 ≥ Xt−1.
Z
Zt−1 ≥ E (Zt |Ft−1 ) ≥ E (Xt |Ft−1 )
Z Y ;
Zt−1 ≥ Yt−1
Zt−1 ≥ max (Yt−1;E (Xt |Ft−1 )) = Xt−1
t.
t Xt = Yt.
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Stochastic Processes for Finance
41
Discrete-time stochastic processes
v
v = inf (t / Xt = Yt)
v Xv
X v ≤ T
{v = s} =s−1
u=1
{Xu > Yu}{Xs = Ys}
Fs Xu, Yu, u ≤ s Fs {v = s} ∈ Fs v
Xvt ξs = {v≥s},
Xvt = X0 +
t
s=1
ξs (Xs −Xs−1)
Xs = max (Ys;E (Xs+1 |Fs )) Xvs (ω) = Xs(ω) ω ∈
{v ≥ s} ,
Xs(ω) = E (Xs+1 |Fs ) (ω) ω ∈ {v ≥ s} Xv
s − Xvs−1 = ξs (Xs −E (Xs |Fs−1 )) .
{v ≥ s} , Xs−Xs−1 Xs. {v < s} , Xv
s = Xvs−1 = Yv
{v < s}
EXv
s −Xvs−1 |Fs−1
= E (ξs (Xs −E (Xs |Fs−1 )) |Fs−1 )
ξ ξs
EXv
s −Xvs−1 |Fs−1
= 0
Xv
v
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Stochastic Processes for Finance
42
Discrete-time stochastic processes
T T, v
v = inf {t / Xt = Yt}
E [Yv] = supu∈T
E [Yu]
Yt = max(K−St; 0) t Xt − Yt
T T, v
v = inf {t / Xt = Yt}
E [Yv] = supu∈T
E [Yu]
Yt = max(K−St; 0) t Xt − Yt
- ©
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Stochastic Processes for Finance
43
Continuous-time stochastic processes
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Stochastic Processes for Finance
44
Continuous-time stochastic processes
(,A, P ) T T = [0;T ] , T < +∞
X =(Xt, t ∈ T ) (R,BR) .
T
t → Xt(ω) ω ∈ X.
[0;T ]
•
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Stochastic Processes for Finance
45
Continuous-time stochastic processes
t→ Xt(ω)
•
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Stochastic Processes for Finance
46
Continuous-time stochastic processes
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Stochastic Processes for Finance
47
Continuous-time stochastic processes
• Xt(ω) Xs(ω), s < t
T .
Y X t ∈ T t = {Xt = Yt} Y X.
X Y
∃∗ ∈ A P (∗) = 1 ∀ω ∈ ∗, ∀t ∈ T , Xt(ω) = Yt(ω)
= [0; 1] , T =[0; 1] X Xt = 0 t ω. Y
Yt(ω) = 1 t = ω
= 0
P ({Xt = Yt}) = 1 P Y X. X Y
t t = ∅.
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Stochastic Processes for Finance
48
Continuous-time stochastic processes
F = {Ft, t ∈ T } A. (,A,F , P )
F t < T, Ft =
s>tFs
F Ft
X F t, Xt Ft
X, FX , X FX
t Xs, s ≤ t.
P (ω) > 0 ω. B ∈ Ft P (B) = 0 A B Ft. P (A) ≤ P (B) A ⊂ B P (A) A
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Stochastic Processes for Finance
49
Continuous-time stochastic processes
t t − 1 Xt Ft−1 Xt Ft−1
X F
f : R→R f(x) x < x0. f(x0)?
f
limx→xx<x
f(x) = f(x0)
x → x0 x < x0 x →x−
0 .
t < t0, t
X (,A, P ) ;
X t1 ≤ t2 ≤ ... ≤ tn, Xt −Xt−
X t ∈ T h > 0 t + h ∈ T , Xt+h −Xt h t
B∗ ×T B∗
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Stochastic Processes for Finance
50
Continuous-time stochastic processes
X
X0 = 0
Xt = Xt−1 + Yt
Yt X Yt.
Xt S) Xt − Xs [s; t] . Xt = ln(St)
Xt −Xs = ln(St)− ln(Ss) = ln
St
Ss
X Xt −Xs
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Stochastic Processes for Finance
51
Continuous-time stochastic processes
X λ X0 = 0 X (s, t) ∈ R+×R+ s ≤ t, Xt−Xs
λ(t− s).
∀k ∈ N, P (Xt −Xs = k) =(λ(t− s))k
k!exp (− (λ(t− s)))
Xt N .
Y t,
Claimt =X
i=1
Yi
Xt t.
h h.
P (Xt+h −Xt = 2) =(λh)2
2exp (−λh) ≃ λ2
2h2
h h λh.
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Stochastic Processes for Finance
52
Continuous-time stochastic processes
X (,A, P,F) (B1, ..., Bn) ∈ Bn
R (t1, ..., tn) ∈ T n t1 < .... < tn
PXt ∈ Bn
Xt ∈ Bj , j = 1, .., n− 1= P
Xt ∈ Bn
Xt− ∈ Bn−1
Xs s ≤ tn−1 Xt− .
Bn−1 tn−1 Xt −Xt−. tn−1
X X σ R×T R R+
(x, t) = limh→0
E [Xt+h −Xt |Xt = x ]
h
σ2(x, t) = limh→0
V [Xt+h −Xt |Xt = x ]
h
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Stochastic Processes for Finance
53
Continuous-time stochastic processes
σ Xt t Xt+h −Xt [t; t+ h] . (x, t) t X x. h
σ2 σ σ.
(,A,F , P )
(F , P ) X F ,
∀(s, t) ∈ T 2, s ≤ t⇒ E [Xt |Fs ] = Xs
X (F , P ) E [Xt |Fs ] ≤Xs.
X (F , P ) E [Xt |Fs ] ≥Xs.
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Stochastic Processes for Finance
54
Continuous-time stochastic processes
(F , P ) X X X
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Stochastic Processes for Finance
55
Continuous-time stochastic processes
X F X
∀t ∈ T , Xt = X0 +Mt +At
M M0 = 0 A A0 = 0. X A
X
X0 = a
Xn = Xn−1 + Yn
a Yn T [0;T ] N
h = T/N N T
T = {0, 1, ..., N} .
X Yn σ −σ p = 1/2
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Stochastic Processes for Finance
56
Continuous-time stochastic processes
X
Cov(Xn, Xm) = σ2 min(n,m)
Yn. m < n
Cov(Xn, Xm) = Cov(Xm +n
k=m+1
Yk,Xm)
= Cov(Xm,Xm) +n
k=m+1
cov(Yk, Xm)
= V (Xm) +n
k=m+1
Cov (Yk,Xm)
Cov(Xm, Xm) = V (Xm). Cov (Yk, Xm) k > m Yk Xm = X0 +
k≤m Yk
V (Yk) = σ2 V (Xm) =m
k=1 V (Yk) = mσ2. n < m, Cov(Xn,Xm) = V (Xn) = nσ2.
Xn t t = nh X0 = 0 Xn [0; t] . T = {0, 1, ..., N} [0;T ]
N → +∞ [0;T ] ΛT , ΛT = limN→+∞ XN .
σ N Nσ2 [0;T ] T N = 13 N = 91
T
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Stochastic Processes for Finance
57
Continuous-time stochastic processes
σ N u d
Y Nn , n = 1, ...,N
T = {0, 1, ...,N} σh Y Nn
T XN = X0 +Nn=1 Y
Nn .
N, XN −X0 [0;T ] , ΛT . N h
N
A1 > 0, N,
∀n ∈ T , V (Xn) ≥ A1
A2 > 0, N, :
V (XN ) ≤ A2
A3 > 0, N, n ∈ T
V (Yn)
maxNj=1 V (Yj)≥ A3
• 0 t
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Stochastic Processes for Finance
58
Continuous-time stochastic processes
• [0;T ] u d
•
•
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Stochastic Processes for Finance
59
Continuous-time stochastic processes
A3 = 1.
O o.
f g R R. f(h) = O(g(h)) limh→0
f(h)g(h)
< +∞ f(h) = o(g(h))
limh→0
f(h)g(h)
= 0.
f(h) = O(g(h)) f g h f(h) = o(g(h)) f g h
σ2h
σ2h = O(h) σ2
h = o(h)
O(h) o(h)
V (XN) =N
n=1
V (Y Nn ) = Nσ2
h
Y Nn .
Nσ2h ≤ A2 N T
h
σ2h ≤
A2
Th
σ2h = O(h).
Nσ2h ≥ A1
σ2h ≥ A
Th. σ2
h = o(h).
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Stochastic Processes for Finance
60
Continuous-time stochastic processes
Yn O(h), Yn σh = c
√h −c
√h c
O(√h) h.
c = 1 X
Xn = Xn−1 + δn√h
δn +1 −1
h = 1
h = 1 h = 0.2
±
√hh, ± 1√
h.
h h h
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Stochastic Processes for Finance
61
Continuous-time stochastic processes
h = 0.2
ΛT = limN→+∞
TN
Nn=1 δn δn
. ΛT
√T .
T T ′ , T < T ′, ΛT ′−ΛT
√T ′ − T ΛT ′ −ΛT
ΛT − Λ0.
,A, P ) Z
Z0 = 0 P.a.s
Z
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Stochastic Processes for Finance
62
Continuous-time stochastic processes
∀(s, t) ∈ R+ ×R+, s < t, Zt − Zs ∼ N (0,√t− s)
Z
Zt
√t.
Z t t + h
√h
h
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Stochastic Processes for Finance
63
Continuous-time stochastic processes
Z
Z
cov(Zt, Zs) = min(t, s)
(1)
Zt =Zs + (Zt − Zs) s < t) Zs (Zt − Zs) (Zt − Zs) FZ
s , Z.
(2), s t s < t.
EZt
FZs
= E
Zs + (Zt − Zs)
FZs
= EZs
FZs
+ E
Zt − Zs
FZs
Zt−Zs FZ
s EZt − Zs
FZs
= E [Zt − Zs] = 0
Z FZ , Zs FZs
EZs
FZs
= Zs
EZt
FZs
= Zs
(3), s < t)
cov(Zt, Zs) = cov(Zt − Zs + Zs, Zs)
= cov(Zt − Zs, Zs) + V (Zs)
= cov(Zt − Zs, Zs − Z0) + s
= s
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Stochastic Processes for Finance
64
Continuous-time stochastic processes
Zt − Zs Zs − Z0
t < s, Zs Zs − Zt + Zt .
(x, t) = limh→0
E [Xt+h −Xt |Xt = x ]
h
σ2(x, t) = limh→0
V [Xt+h −Xt |Xt = x ]
h
((x, t) = 0) σ(x, t) 1
X Y
Xt = Z2t − t
Yt = exp
γZt −
γ2t
2
γ X Y FZ .
Z Xt Zt Yt
Zt = Zs+(Zt−Zs) s < t. E
Xt
FZs
EXt
FZs
= E
Z2
t
FZs
− t
= E(Zt − Zs)
2 + 2ZtZs − Z2s
FZs
− t
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Stochastic Processes for Finance
65
Continuous-time stochastic processes
Zs FZ
s − Z Zt − Zs FZ
s .
EXt
FZs
= E
(Zt − Zs)
2FZ
s
+ 2E
ZtZs
FZs
− E
Z2
s
FZs
− t
= E(Zt − Zs)
2FZ
s
+ 2E
ZtZs
FZs
− Z2
s − t
= E(Zt − Zs)
2FZ
s
+ 2ZsE
Zt
FZs
− Z2
s − t
= E(Zt − Zs)
2FZ
s
+ 2Z2
s − Z2s − t
E(Zt − Zs)
2+ Z2
s − t
(t− s) + Z2s − t
= Z2s − s = Xs
Y
EYt
FZs
= exp
−γ2t
2
Eexp (γZt)
FZs
= exp
−γ2t
2
Eexp (γ(Zt − Zs)) exp(γZs)
FZs
= exp
−γ2t
2
E (exp (γ(Zt − Zs))) exp(γZs)
exp (γ(Zt − Zs)) 0 γ
√t− s. E (exp (γ(Zt − Zs))) = γ(t−s)
2.
EYt
FZs
= exp
γZs −
γ2s
2
= Ys
X Y
(x, t) → f(x, t) = x2 − t
(x, t) → g(x, t) = exp
γx− γ2t
2
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Stochastic Processes for Finance
66
Continuous-time stochastic processes
z(x, t) Zt
z(x, t) =1√2πt
exp
−x2
2t
f g
h(0, s) =
+∞
−∞h(u, t+ s)z(u, t)du
f, f(0, s) = −s
+∞
−∞f(u, t+ s)z(u, t)du =
1√2πt
+∞
−∞(u2 − (t+ s)) exp
−u2
2t
du
=1√2πt
+∞
−∞u2 exp
−u2
2t
du− (t+ s)
1√2πt
+∞−∞ u2 exp
−u
2t
du Zt, t. g,
+∞
−∞g(u, t+ s)z(u, t)du =
1√2πt
+∞
−∞exp
γu− γ2(t+ s)
2− u2
2t
du
= exp
−γ2(t+ s)
2
1√2πt
+∞
−∞exp
γu− u2
2t
du
exp
γu− u2
2t
= exp
− 1
2t(u− γt)2 +
γ2t
2
+∞
−∞g(u, t+ s)z(u, t)du = exp
−γ2s
2
1√2πt
+∞
−∞exp
− 1
2t(u− γt)2
du
= exp
−γ2s
2
= g(0, s)
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Stochastic Processes for Finance
67
Continuous-time stochastic processes
E(Zt) = 0 t. V (Zt) = t
σ σ2
W σ Wt
W0 = 0
Wt = t+ σZt
Z
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Stochastic Processes for Finance
68
Continuous-time stochastic processes
W Z.
W
s < t, Wt −Ws ∼ N ((t− s), σ
t− s)
cov(Wt,Ws) = σ2(t− s)
W (<)0.
W
1. Wt [0; t] , 0 r, r.
h W [t; t+ h]
Wt+h −Wt = h+ σ(Zt+h − Zt)
Z, Zt+h−Zt
O(√h), h h
W , σ(Zt+h − Zt) h h .
= 0, [a; b])
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Stochastic Processes for Finance
69
Continuous-time stochastic processes
=10% σ = 20%)
= 10% σ = 20%, . t = 0.75),
(,A,F , P ) τ T {+∞} {τ ≤ t} ∈Ft t ∈ T .
{τ ≤ t} {τ = t} . P ({τ = t}) = 0 t
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Stochastic Processes for Finance
70
Continuous-time stochastic processes
+∞ τ = +∞.
T (C1, ..., CT ) t Ft < Ct τ τ = inf {t / Ft < Ct} τ = +∞ T.
τ τ ′ F τ + τ ′, min(τ , τ ′), max(τ , τ ′)
min(τ , τ ′) τ ∧ τ ′ τ ∨ τ ′ max(τ , τ ′).
τ ∧ τ ′ τ ∨ τ ′ τ τ ′ min(τ , τ ′)) max(τ , τ ′)) |τ − τ ′|
t t ∧ τ
Fτ τ A {τ ≤ t} A ∈ A
τ X
τ Fτ
τ X F Xτ Fτ
τ τ ′ τ ≤ τ ′, Fτ ⊆ Fτ ′
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Stochastic Processes for Finance
71
Continuous-time stochastic processes
W (0, σ) a b a < 0 < b τ
τ = inft ∈ R+ / Wt = a Wt = b
P (Wτ = a) =b
b− a
E(τ) =−ab
σ2
τ a b. a τ b σ, τ a b σ P (Wτ = a) =P (Zτ = a).
X,
Xt = Z2t − t
τ ∗ = τ ∧ s s Xτ∗
t EXτ∗
t
= E(Xτ∗
0 ) = 0
EZ2
t∧τ∗
= E(t ∧ τ ∗) ≤ (b− a)2
τ Z a b. s,
E(τ ) = lims→+∞
E(τ ∗) ≤ (b− a)2
τ E(Zτ ) = E(Z0) = 0
E(Zτ ) = aP (Zτ = a) + b(1− P (Zτ = a)) = 0
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Stochastic Processes for Finance
72
Continuous-time stochastic processes
P (Zτ = a) =b
b− a
X
E (Xτt ) = 0 = E
Z2
τ − τ
E(Z2τ ) = a2 b
b− a+ b2
−a
b− a= −ab
Wt = σZt,
Z
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Stochastic Processes for Finance
73
Stochastic integral and Itô’s lemma
Yn.
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Stochastic Processes for Finance
74
Stochastic integral and Itô’s lemma
(x, t) σ(x, t)
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Stochastic Processes for Finance
75
Stochastic integral and Itô’s lemma
[0;T ] N h h = T/N. 0, h, 2h, ...., Nh
X
X0 = 0
Xn = Xn−1 + Yn
n ∈ T = {1, ...,N} . Xn(ω) X nh ω h Yn h.
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Stochastic Processes for Finance
76
Stochastic integral and Itô’s lemma
Yn
Yn = n−1h+ σn−1
Znh − Z(n−1)h
Z n−1 σn−1 F(n−1)h F Z n−1 σn−1
n−1 X [(n− 1)h;nh] σn−1 σ− F(n−1)h. σn−1 Znh−Z(n−1)h V
Znh − Z(n−1)h
= h.
X Y
ΛT [0;T ] ΛT
XN N
XN =N
n=1
Yn = hN
n=1
n−1 +N
n=1
σn−1
Znh − Z(n−1)h
h TN,
XN =N
n=1
Yn = T
1
N
N
n=1
n−1
+N
n=1
σn−1
Znh − Z(n−1)h
= AN +BN
AN = T
1N
Nn=1 n−1
BN =
Nn=1 σn−1
Znh − Z(n−1)h
A = limN→+∞ AN B = limN→+∞ BN .
αN
αNs = n−1 s ∈ [(n− 1)h;nh[
αNs (ω)
[(n− 1)h;nh[ . αN(ω)
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Stochastic Processes for Finance
77
Stochastic integral and Itô’s lemma
ω,
AN (ω) =
T
0
αNs (ω)ds
N → +∞ ω, A(ω)
A(ω) =
T
0
αs(ω)ds
αs(ω) = limN→+∞ αNs (ω). α
BN ,
Znh − Z(n−1)h
√h.
σ Z
limN→+∞
BN =
T
0
σsdZs
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Stochastic Processes for Finance
78
Stochastic integral and Itô’s lemma
CN DN
CN =N
n=1
Znh
Znh − Z(n−1)h
DN =N
n=1
Z(n−1)h
Znh − Z(n−1)h
Z Z
limN→+∞
CN = limN→+∞
DN =
T
0
ZsdZs =Z2
T − Z20
2=
Z2T
2
CN DN N Zt = Zs + (Zt−Zs)
E(CN ) = E
N
n=1
Znh
Znh − Z(n−1)h
=N
n=1
EZnh
Znh − Z(n−1)h
Znh = Z(n−1)h + (Znh − Z(n−1)h),
E(CN) =N
n=1
E
Z(n−1)h + (Znh − Z(n−1)h)
Znh − Z(n−1)h
= E
N
n=1
Z(n−1)h
Znh − Z(n−1)h
+E
N
n=1
Znh − Z(n−1)h
2
=N
n=1
E
Znh − Z(n−1)h
2+
N
n=1
EZ(n−1)h
Znh − Z(n−1)h
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Stochastic Processes for Finance
79
Stochastic integral and Itô’s lemma
Znh − Z(n−1)h
E
Znh − Z(n−1)h
2= V
Znh − Z(n−1)h
= nh− (n− 1)h = h
Nh = T.
N
n=1
EZ(n−1)h
Znh − Z(n−1)h
=
N
n=1
EZ(n−1)h
EZnh − Z(n−1)h
= 0
E(CN ) = T
[0;T ] N.
E(DN) CN DN
DN
gn =
n
s=1
θnhXnh −X(n−1)h
θnh F(n−1)h θ
σn−1 Yn F(n−1)h
BN
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Stochastic Processes for Finance
80
Stochastic integral and Itô’s lemma
E (B2N)
EB2
N
= E
(BN −BN−1 +BN−1)
2
= E(BN −BN−1)
2+ EB2
N−1
+ 2E (BN−1 (BN −BN−1))
= Eσ2N−1
ZNh − Z(N−1)h
2+ E
B2
N−1
+ 2E (BN−1)E (BN −BN−1)
= hEσ2N−1
+ E
B2
N−1
B.
ZNh − Z(N−1)h
σ2
N−1 E
ZNh − Z(N−1)h
2=
h.
EB2
N
= T
1
N
N
n=1
E(σ2n−1)
βN [0;T ]
βNs = σn−1 s ∈ [(n− 1)h;nh[
EB2
N
= E
T
0
βNs
2ds
βN σ.
E T
0(σs)
2 ds
< +∞ BN
Z ,A,FZ , P
σ FZ σ
E
T
0
σ2sds
< +∞
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Stochastic Processes for Finance
81
Stochastic integral and Itô’s lemma
σ [0;T ] Z T
0σsdZs
T
0
σsdZs = limN→+∞
N
n=1
σn−1
Znh − Z(n−1)h
n [0;T ] N [0;T ] [ti; ti+1[ tN = T
T
0
σsdZs = limN→+∞
N
i=1
σt−
Zt − Zt−
max(ti− ti−1) N → +∞. T.
u
0σsdZs, u ∈ T
(Iu(σ), u ∈ T ) . σ σn−1
σn θ,
T
0θsdZs
Z.
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Stochastic Processes for Finance
82
Stochastic integral and Itô’s lemma
X Y t
0XsdZs
t
0YsdZs t ≤ T
∀(a, b) ∈ R2, t
0(aXs + bYs)dZs = a
t
0XsdZs + b
t
0bYsdZs
∀(t, u) ∈ T × T , t < u, u
0XsdZs =
t
0XsdZs +
u
tXsdZs
∀(t, u) ∈ T × T , t < u, E u
0XsdZs |Ft
= t
0XsdZs
∀t ∈ T , E
t
0XsdZs
2= E
t
0X2
sds
X
u
0XsdZs, u ∈ T
t
0XsdZs L2
X
X0 = 0
Xn = Xn−1 + Yn = Xn−1 + n−1h+ σn−1
Znh − Z(n−1)h
X
Xt = X0 +
t
0
sds+
t
0
σsdZs
X s = (Xs, s) σs = σ(Xs, s)
dXt = (Xt, t)dt+ σ(Xt, t)dZt
X dZt
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Stochastic Processes for Finance
83
Stochastic integral and Itô’s lemma
X W σ ,
dWt = dt+ σdZt
W0 = 0)
t
0
dWs = Wt =
t
0
ds+
t
0
σdZs
=
t
0
ds+ σ
t
0
dZs
= t+ σZt
dZt). dZt Zt+dt − Zt dt > dt dt.
E(dZt) = 0 V (dZt) = dt
V (dZ2t ) = o(dt)
dZt.dt = o(dt)
E (dZtdZt) = 0 t1 = t2
Z Z∗
E (dZtdZ∗t ) = rtdt
V (dZtdZ∗t ) = o(dt)
dZt Zt+dt−Zt V (Zt − Zs) = t− s t > s.
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Stochastic Processes for Finance
84
Stochastic integral and Itô’s lemma
(dZt)2
Z
√dt −
√dt (dZt)
2
dt. dZt = O(
√dt) dZt.dt = O(dt
) = o(dt).
t1 = t2, dZt dZt
rt O(
√dt). O(dt).
O(dt2), o(dt).
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Stochastic Processes for Finance
85
Stochastic integral and Itô’s lemma
T K.
CT = max (XT −K; 0) .
C = (Ct, t ∈ T ) C0 . Ct = g(Xt, t) g R+×T R+. X C.
X Yt = exp(Xt)? Y Xt = ln(Yt)?
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Stochastic Processes for Finance
86
Stochastic integral and Itô’s lemma
f R2 R f (x0, t0)
f(x, t) = f(x0, t0) +∂f
∂x(x0, t0)(x− x0) +
∂f
∂t(x0, t0)(t− t0)
+1
2
∂2f
∂x2(x0, t0)(x− x0)
2 +1
2
∂2f
∂t2(x0, t0)(t− t0)
2
+∂2f
∂x∂t(x0, t0)(x− x0)(t− t0) + ε(x0, t0)
ε(x0, t0) ∼ o ((x− x0)2 + (t− t0)
2) . ε
x t− t0
√t− t0.
∂f∂x
(x0, t0)(x − x0)2
O(t− t0). ∂f∂t(x0, t0)(t− t0).
df(x0, t0) = f(x, t)−f(x0, t0) ; t−t0 = dt x−x0 =dx;
df(x0, t0) =∂f
∂x(x0, t0)dx+
∂f
∂t(x0, t0)dt+
1
2
∂2f
∂x2(x0, t0)(dx)
2
+1
2
∂2f
∂t2(x0, t0)(dt)
2 +∂2f
∂x∂t(x0, t0)dxdt+ ε(x0, t0)
x Xt X
dXt = (Xt, t)dt+ σ(Xt, t)dZt
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Stochastic Processes for Finance
87
Stochastic integral and Itô’s lemma
(Xt, t)
df(Xt, t) =∂f
∂x[(Xt, t)dt+ σ(Xt, t)dZt] +
∂f
∂tdt
+1
2
∂2f
∂x2[(Xt, t)dt+ σ(Xt, t)dZt]
2
+1
2
∂2f
∂t2(dt)2 +
∂2f
∂x∂t[(Xt, t)dt+ σ(Xt, t)dZt] dt+ ε(Xt, t)
∂f
∂t ∂f
∂x∂t o(dt)).
df(Xt, t) =∂f
∂x[(Xt, t)dt+ σ(Xt, t)dZt] +
∂f
∂tdt
+1
2
∂2f
∂x2[(Xt, t)dt+ σ(Xt, t)dZt]
2 + ε′(Xt, t)
ε′ = o(dt).
[(Xt, t)dt+ σ(Xt, t)dZt]2 ,
O(dt2)) dZtdt O(dt ))
dZ2t O(dt)
df(Xt, t) =∂f
∂x((Xt, t)dt+ σ(Xt, t)dZt) +
∂f
∂tdt
+1
2
∂2f
∂x2σ2(Xt, t)dt+ ε′′(Xt, t)
ε′′ = o(dt). f(Xt, t)
dt dZt
df(Xt, t) =
∂f
∂x(Xt, t) +
∂f
∂t+
1
2
∂2f
∂x2σ2(Xt, t)
dt+σ(Xt, t)
∂f
∂xdZt+ε′′(Xt, t)
ε(Xt, t) ε dt
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Stochastic Processes for Finance
88
Stochastic integral and Itô’s lemma
X
dXt = (Xt, t)dt+ σ(Xt, t)dZt
f : R2 → R Y Yt = f(Xt, t)
dYt =
∂f
∂x(Xt, t) +
∂f
∂t+
1
2
∂2f
∂x2σ2(Xt, t)
dt+ σ(Xt, t)
∂f
∂xdZt
Y
dYt = Y (Yt, t) dt+ σY (Yt, t) dZt
Y (Yt, t) =∂f
∂x(Xt, t) +
∂f
∂t+
1
2
∂2f
∂x2σ2(Xt, t)
σY (Yt, t) = σ(Xt, t)∂f
∂x
Y (Yt, t) f(Xt, t) σY (Yt, t)
W σ W Y Yt = f(Wt) = exp (Wt) . Y t ∂f
∂t= 0
Y
Y (Yt, t) =∂f
∂x+
∂f
∂t+
1
2
∂2f
∂x2σ2 = exp (Wt)
+
σ2
2
= Yt
+
σ2
2
σY (Yt, t) = σ∂f
∂x= σYt
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Stochastic Processes for Finance
89
Stochastic integral and Itô’s lemma
dYt
Yt=
+
σ2
2
dt+ σdZt
Y
Y
Y0 = 1
dYt = Ytdt+ σYtdZt
X Xt = g(Yt) = ln(Yt).
∂g
∂x(Yt, t) +
∂g
∂t+
1
2
∂2g
∂x2σ2(Yt, t) =
∂g
∂xYt +
1
2
∂2g
∂x2σ2Yt
= − σ2
2
σYt∂g
∂x= σ
dXt =
− σ2
2
dt+ σdZt
X
dXt = α(β −Xt)dt+ σdZt
Xt
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Stochastic Processes for Finance
90
Stochastic integral and Itô’s lemma
β) Xt
β. α
σ Xt. Z dt
√dt,
Z
dXt = α(β −Xt)dt+ σ
XtdZt
αβdt
β
β) Xt
β. α
σ Xt. Z dt
√dt,
Z
dXt = α(β −Xt)dt+ σ
XtdZt
αβdt
β
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Stochastic Processes for Finance
91
Stochastic integral and Itô’s lemma
W (, σ) , (,A,F , P ) F W. Z, Wt = t + σZt. Wt = ln(St) St Wt−Ws [s; t] . r Q W r.
P Q EP EQ P Q.
r = .
X ∼ N (0, 1) (,A, P ) Q
∀A ∈ A, Q(A) = EP
A exp
αX − α2
2
Q P X ∼ N (α, 1) Q.
EP
exp
αX − α
2
= exp
−α
2
EP [exp (αX)] exp (αX)
(0, α) ,
EP [exp (αX)] = exp
α
2
.
Q() = EP
exp
αX − α
2
= 1.
P Q
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Stochastic Processes for Finance
92
Stochastic integral and Itô’s lemma
EQ [X] =
XdQ =
X exp
αX − α2
2
dP
=
+∞
−∞x exp
αx− α2
2
fX(x)dx
=1√2π
+∞
−∞x exp
αx− α2
2
exp
−x2
2
dx
=1√2π
+∞
−∞x exp
−1
2(x− α)2
dx = α
fX X P. X Q, α exp
αX − α
2
Q P, dQ
dP.
Xt ∼ N (t, σ2t) t = EP (Xt) σ2t = VP (Xt) Xt rt =EQ(Xt) σ2t = VQ (Xt) .
X Xt = t+ σZt = t+ σ
√tYt
Z Yt ∼ N (0, 1) . Yt N (α, 1) α Q
EQ [Xt] = t+ σ√tα = rt
α
α =(r − )
√t
σ
dQ
dP= exp
(r − )
√t
σYt −
1
2
(r − )
√t
σ
2
= exp
−− r
σZt −
t
2
− r
σ
2
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Stochastic Processes for Finance
93
Stochastic integral and Itô’s lemma
λ = (λt, t ∈ [0;T ]) F L = (Lt, t ∈ [0;T ])
Lt = exp
− t
0
λsdZs −1
2
t
0
λ2sds
λ
EP
exp
1
2
T
0
λ2sds
< +∞
λ
L P− Z∗
Z∗t = Zt +
t
0
λsds
(,A,F , Q) Q
dQ
dP= LT
λ L
Lt = exp
−λZt −
λ2
2t
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Stochastic Processes for Finance
94
Stochastic integral and Itô’s lemma
dSt = Stdt+ σStdZt
St = S0 exp
− σ2
2
t+ σZt
exp (−rt)St = S0 exp
− r − σ2
2
t+ σZt
St = S0 exp
−σ2
2t+ σZ∗
t
Z∗t = Zt +
−rσ
t.
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Stochastic Processes for Finance
95
Stochastic integral and Itô’s lemma
σ
F (,A, P ) Z.
X0 = c
dXt = (Xt, t) dt+ σ (Xt, t) dZt
c c
X [0;T ]
X F σ
T
0
| (Xt, t)| dt < +∞ T
0
σ2 (Xt, t) dt < +∞
X
Xt = X0 +
t
0
(Xs, s) ds+
t
0
σ (Xs, s) dZs
σ
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Stochastic Processes for Finance
96
Stochastic integral and Itô’s lemma
P X
F , EP
T
0X2
t dt
< +∞.
m > 0 ∀t ∈ [0;T ] , ∀ (x, y) ∈ R2
max (|(x, t)− (y, t)| ; |σ(x, t)− σ(y, t)|) ≤ m |x− y|(x, t)2 + σ(x, t)2 ≤ m
1 + x2
(x, t)
X0 Ft t.
σ t. σ X (1 + x2)
.
(x, t) = exp(x).
Z (X,Z) σ
Xt
c.
σ t, X σ.
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Stochastic Processes for Finance
97
Stochastic integral and Itô’s lemma
X0 = c
dXt = aXtdt+ σtdZt
a
Xt = c exp(at) +
t
0
exp [a (t− s)] σsdZs
Xt exp(−at) = c+
t
0
exp (−as) σsdZs
Y,
Y0 = c
dYt = exp(−at)σtdZt
exp(at)Yt = f (Yt, t) , f
∂f
∂t= a exp(at)Yt
∂f
∂Yt
= exp(at)
∂2f
∂Y 2t
= 0
df (Yt, t) = a exp(at)Ytdt+ exp(at) exp(−at)σtdZt
= a exp(at)Ytdt+ σtdZt
Yt exp(−at)Xt
dXt = aXtdt+ σtdZt
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Stochastic Processes for Finance
98
Stochastic integral and Itô’s lemma
X0 = c
dXt = a(t)Xtdt+ σtdZt
X
Xt = γt
c+
t
0
γ−1s σsdZs
γ
f′(t) = a(t)f(t)
Y
Y0 = y0
dYt = α (β − Yt) dt+ σdZt
Xt = (Yt − β) exp(αt) = f(Yt, t); f
∂f
∂t= α exp(αt) (Yt − β)
∂f
∂Yt= exp(αt)
∂2f
∂Y 2t
= 0
dXt = [α (Yt − β) exp(αt) + exp(αt)α (β − Yt)] dt+ σ exp(αt)dZt
= σ exp(αt)dZt
X0 = y0 − β,
Xt = y0 − β + σ
t
0
exp(αs)dZs
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Stochastic Processes for Finance
99
Stochastic integral and Itô’s lemma
X Y
Yt = β +Xt exp(−αt)
Yt = β + exp(−αt)
y0 − β + σ
t
0
exp(αs)dZs
= β (1− exp(−αt)) + y0 exp(−αt) + σ
t
0
exp(−α(t− s))dZs
Yt
EP [Yt |Y0 = y0 ] = β (1− exp(−αt)) + y0 exp(−αt)
VP [Yt |Y0 = y0 ] = σ2
t
0
exp(−2α(t− s))ds
=σ2
2α[1− exp (−2αt)]
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I wanted real responsibili� I joined MITAS because
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Stochastic Processes for Finance
100
Bibliography
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Stochastic Processes for Finance
101
Bibliography
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Stochastic Processes for Finance
102
Bibliography
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Stochastic Processes for Finance
103
Index
Index
Brownian motion, 57 general, 63 geometric, 85 Markov process, 59 martingale, 59 path simulation, 65 stopping time, 67 transformation, 60
Doob decomposition, 25, 51 martingale, 24
Filtration, 9, 44 complete, 44 filtered probability space, 44 natural, 10, 44 right-continuous, 44 Girsanov theorem, 89 application, 89 Novikov condition, 89
Itô lemma, 84 application, 84 Taylor series expansion, 82 process, 48 diffusion coefficient, 49 drift, 49
Landau notations, 55
Markov chain, 15 accessible, 17 aperiodic, 18 Chapman-Kolmogorov equations, 16 communicating class, 17 communication, 17 homogeneous, 15 irreducible, 18 periodicity, 18
positively recurrent, 19 recurrent, 19 stationary distribution, 20 transient, 19 no memory process, 48 process, 15, 48 Brownian motion, 59 transition matrix, 15Martingale, 21, 49 Brownian motion, 59 Doob, 24 submartingale, 21, 49 super-martingale, 21, 49Modi cation stochastic process, 43
Novikov condition, 89
Path, 40 càdlàg, 41 càglàd, 43 continuous, 40 Brownian motion, 58 LCRL, 43 nowhere differentiable, 59 RCLL, 41Poisson distribution, 47 process, 47Probability transition, 15
Snell envelope, 36Stochastic differential equation, 91 linear, 93 solution conditions, 92 de nition, 91 Markov, 92Stochastic integral, 71 calculation rules, 79 definition, 76 properties, 78 stochastic differential, 78
Stochastic Processes for Finance
104
Index
Stochastic process adapted, 10, 44 Brownian motion, 57 continuous-time, 40 diffusion, 48 diffusion coefficient, 49 discrete-time, 6 drift, 49 increments independent, 45 stationary, 45 indistinguishable, 43 Itô, 48 Markov, 15, 48 modification, 43 Ornstein-Uhlenbeck, 85 path, 40 Poisson, 47 predictable, 12, 45 random walk, 22, 46, 51 square root, 86 stopped process, 33, 37 trajectory, 40 Wiener, 57Stopping time, 32, 37, 65 American option, 35 optional stopping theorem, 32Strategy doubling, 34 portfolio, 28 self- nancing, 28
Transition matrix, 15 probability, 15
Wiener process, 57 general, 63