Stochastic Calculus Cheatsheet
Standard Brownian Motion / Wiener process
E[dX 2 ] = dt
E[dX] = 0
limdt→0 dX 2 = dt
√
Discrete approx: dX = φ dt where φ ∼ N (0, 1)
dX is O(dt1/2 )
dtdX is O(dt3/2 )
Itô Product Rule
Characterization:
1.
2.
3.
4.
X(0) = 0
Continuous everywhere, differentiable nowhere
X(t) − X(s) ∼ N (0, |t − s|)
X(t + s) − X(t) is independent of X(t)
Levy’s characterization:
3. Xt is a martingale w.r.t. the filtration Ft
4. |X|2 − t is a martingale w.r.t. the filtration Ft
If dXt = αdt + βdWt and dYt = γdt + λdWt ,
d(Xt Yt ) = Xt dYt + Yt dXt + dXdY
1
= Xt dYt + Yt dXt + βλdt
2
Stochastic Differential Equations (General Form)
dS
= f (t, S) dt + g(t, S) dXi
dSi
= fi (t, S0 , . . . , Sn ) dt + gi (t, S0 , . . . , Sn ) dXi
where f is the drift, g is the diffusion
Itô’s Lemma and Basic Stochastic Integration
For F (Xt )
dF
1 d2 F
dF =
dXt +
dt
dX
2 dX 2
Z
F (Xt ) = F (X0 ) +
0
t
dF
1
dXτ +
dX
2
Z
0
t
d2 F
dτ
dX 2
For F (Xt , t)
dF =
∂F
dXt +
∂X
∂F
1 ∂2F
+
∂t
2 ∂X 2
t
Z
dt
F (Xt , t) = F (X0 , 0) +
0
∂F
dXτ +
∂X
Z t
0
1 ∂2F
∂F
+
∂t
2 ∂X 2
Functions of Stochastic Functions
1-dimensional: V (t, S)
dV
=
=
1. Apply Taylor expansion on V
2. Apply Itô’s Lemma:
∂V
∂V
1 ∂2V
dt +
dS + g 2 2 dt
∂t
∂S
2 ∂S
∂V
∂V
1 ∂2V
∂V
+f
+ g 2 2 dt + g
dX
∂t
∂S
2 ∂S
∂S
• dXi2 → dt
• dXi dXj → ρij dt
3. Regroup the terms in dt and dXi
4. Sto.integ.: integrate the resulting DE
2-dimensional: V (t, S1 , S2 )
dV =
∂V
∂V
∂V
1 ∂2V
∂2V
1 ∂2V
+ f1
+ f2
+ g12 2 + ρg1 g2
+ g22 2
∂t
∂S1
∂S2
2 ∂S1
∂S1 ∂S2
2 ∂S2
dt + g1
∂V
∂V
dX1 + g2
dX2
∂S1
∂S2
n-dimensional: V (t, S1 , . . . , Sn )

dV = 
n
X
n
1X
∂V
∂V
+
fi
+
∂t
∂S
2
i
i=1
i=1
gi2
n
X

n
X
∂ V
∂ V 
∂V
+
ρ
g
g
dt
+
gi
dXi
ij i j
∂Si2 i=1,j>1
∂Si ∂Sj
∂S
i
i=1
2
2
dτ
Transition Density Functions
Solution
Forward Kolmogorov
1 ∂2
∂p
∂
=
B(y 0 , t0 )2 p − 0 (A(y 0 , t0 )p)
0
02
∂t
2 ∂y
∂y
log
p(S, t; S 0 , t0 ) =
1
σS 0
p
2π(t0 − t)
−
S
S0
e
2
+ µ − 21 σ 2 (t0 − t)
2σ 2 (t0 − t)
Common Processes/Dynamics
Geometric Brownian Motion (Lognormal)
Brownian Motion with Drift
dS = µS dt + σS dX
dS = µ dt + σ dX
dS
= µ dt + σ dX
S
Cox, Ingersoll, Ross
Vasiček (1977)
dS = γ(r̄ − r) dt + σ dX
FIXME TODO add others, Ho Lee and company...
1
dS = (υ − σS) dt + σS 2 dX
All you need to know about Sto.Calc
(FIXME integrate these words of wisdom from Antoine.)
• If Xt → N (µ, σ) then E(xXt ) = eµ+
σ2
2
.
• Itô: d(f (Xt ))
• Itô: d(Xt Yt ) = Xt dYt + Yt dXt + 12 βλdt where dXt = αdt + βdWt and dYt = γdt + λdWt
R
• E[ Xt dWt ] = 0
R
R
• V ar[ Xt dWt ] = Xt2 dt
• Girsanov’s theorem.
• Generating correlated X and Y .
Martingales
Probability Spaces
Unconditional Expectation
“Let (Ω, F, P) be a probability space. . . ”
Expected value under a prob. measure (Lebesgue integral):
Z
Z
Z
E[h(X)] =
h(x)p(x)dx =
h(x)d(P(x)) =
h(x)d P
Ω
ZΩ
ZΩ
E[1{X∈A} ] =
1{X∈A} d P = d P = P(A)
• Ω: sample space
• F: filtration (information set),
(Note that Ft1 ⊆ Ft2 ⊆ FT ≡ F)
• P: probability measure
Ω
A
Conditional Expectation
Martingales (Definition)
(Use these to prove that a process is a Martingale; use the definition.)
E[Mt ] < ∞
E[Mt+1 |Ft ] = Mt
∀0 ≤ s ≤ t
E[Mt+1 |Ft ] ≤ Mt
(supermartingale)
E[Mt+1 |Ft ] ≥ Mt
(submartingale)
1. Linearity: E[aX + bY |F] = aE[X|F] + bE[Y |F]
2. Tower Property: if F ⊂ G,
E[E[X|G] |F] = E[X|F]
Wiener ∈ Martingale (driftless) ⊂ Markov (memoryless) ⊂ nonMarkov
E[E[X|F]] = E[X]
3. Taking out what is known:
Equivalent Measures
E[X|F] = X
Absolute continuity: if P (A) = 0 → Q(A) = 0
Q is “absolutely continuous” w.r.t. P, and Q << P.
∀A.
“It is allright to tinker with the probabilities as
long as we do not tinker with the
(im)possibilities.”
Equivalent measures: if Q << P and P << Q.
Radon-Nikodým Theorem
Q(A) =
Z
A
ΛdP
E[XY |F] = XE[Y |F]
4. Independence: if X is independent from F,
E[X|F] = E[X]
5. Positivity: if X ≥ 0 then E[X|F] ≥ 0
6. Jensen’ Inequality: if f is a convex function, then
f (E[X|F]) ≤ E[f (X)|F]
Exponential Martingale
dQ
is the R.N. derivative.
dP
where Λ =
(if X is F-measurable but not Y :)
M (t) = exp(St + f (t))
where f (t) = −(µ +
1 2
σ )t
2
Itô Integrals & Martingales
Properties of Itô Integrals
Itô integrals are Martingales:
Z
1. Linearity:
T
E[
g(t, Xt )dXt ] = 0
0
Z
T
Z
(αf (t)+βg(t))dXt =
Martingale Representation Theorem
If M is a Martingale, there exists g(t, X) such that
Z
T
MT = M0 +
g(t, X)dXt
0
0
T
Z
αf (t)dXt +
0
βg(t)dXt
0
2. Isometry:

2 
"Z
#
Z T
T
E 
f (t)dXt  = E
|f (t)|2 dt
0
0
The rightmost term is an Itô integral (and thus also a Martingale).
3. Martingale:
"Z
Fubini’s Theorem
"Z
E
T
#
Z
f (Xt )dt =
0
E
T
E [f (Xt )] dt
0
T
0
T
# Z
s
f (t)dXt Fs =
f (t)dXt
0
Application of Martingales to Asset Pricing
Warning: I still need to complete and arrange this page of notes.
Fundamental Asset Pricing Formula
Value = E Meas. [P V (expected cash flows)]
Novikov Condition
h 1 RT 2 i
E e 2 0 θs ds < ∞
Risk-free Asset
dBt = rBt dt,
Mtθ = e(−
B(0) = B0
B(t) = B0 ert
0
θs dXs− 21
Rt
θs2 ds)
0
is a Martingale
Girsanov’s Theorem
Rt
Rt 2
1
dQ
= e(− 0 θs dXs− 2 0 θs ds)
dP
Z t
XtQ = XtP +
θ(s)ds
Underlying S
dSt = µSt dt + σSt dX,
S(t) = S0 e
Rt
S(0) = S0
µt− 12 σ 2 +σXt
0
• Provides an expression for the Radon-Nikodým
derivative.
Removing the TVM
• Gives an explicit correspondence btw P and Q in
terms of their Brownian motion.
S(T )
S (T ) = rt
e
1 2
∗
S (t) = S0∗ e(µ−r− 2 σ )t+σXt
∗
dS ∗ = (µ − r)S ∗ dt + σS ∗ dX
Self-financing Portfolios
. . . but does not tell you what θ is. We assume θ and
check that it satisfies the Novikov condition. Then we
have the RN derivative, and we can change measures!
Doléans/Stochastic Exponential
Trading Strategy:
t
Z
φt = (φSt , φB
t ) processes
E
θs dXs
θs dXs −
0
Value :
Vt (φ) =
φSt St
+
Z t
Vt (φ) = V0 (φ) +
Z
∀t ∈ [0, T ]
Z t
φSu dSu +
φB
u dBu
Z
t
θs2 ds
0
t
θ(s)ds
Feynman-Kač Equivalence
0
Arbitrage opportunity:
PDE:
V0 (φ) = 0
and
1
2
0
φB
t Bt
0
with P (VT (φ) > 0) > 0
0
XtQ = XtP −
Self-financing portfolio: no in/out flows.
t
Z
= exp
P (VT (φ) < 0) = 0
∂V
∂V
1 ∂2V
+µ
+ σ 2 2 − rV = 0,
∂t
∂S
2 ∂S
dSt = µ(t, St )dt + σ(t, St )dXt
V (T, S) = G(S)
m
Expectation: V (t, St ) = e−r(T −t) E [G(ST )|Ft ]