Stochastic Calculus Cheatsheet

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 )

Itˆo 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

Itˆo’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 Itˆo’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

Vasiˇcek (1977) dS = γ(¯ r − 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

.

• Itˆo: d(f (Xt )) • Itˆo: 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