Stochastic Calculus Formulae
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Formula Page • Normal cdf : Φ(x) := R x -∞ e - 1 2 x 2 dx √ 2π . • Moments of Normals : If Z is a normal r.v. with mean 0 and variance 1, then the m.g.f. is E[e aZ ]= e 1 2 a 2 and the fourth moment is E[Z 4 ] = 3. • Ito’s Lemma : If W t is a standard Brownian motion and X t satisfies the SDE: dX t = μ(t, X t ) dt + σ(t, X t ) dW t , and U t = f (t, X t ), where f (t, x) is twice differentiable in x and once differentiable in t, then dU t = ∂ x f (t, X t ) dX t + ( ∂ t + 1 2 σ 2 (t, X t ) ∂ xx ) f (t, X t ) dt = ( ∂ t + μ(t, X t )∂ x + 1 2 σ 2 (t, X t ) ∂ xx ) f (t, X t ) dt + σ(t, X t ) ∂ x f (t, X t ) dW t • Ito’s Isometry : If W t is a standard Brownian motion, then E Z t 0 g(s, W s ) dW s Z t 0 h(s, W s ) dW s = E Z t 0 g(s, W s )h(s, W s ) ds . • Feynman-Kac : Suppose a function f (y,t) satisfies the PDE: ( ∂ t + a(t, y)∂ y + 1 2 b 2 (t, y)∂ yy ) f (t, y) = c(y,t) f (t, y), f (T,y) = ϕ(y). Then f (y,t) admits the unique solution f (y,t)= E P * [e - R T t c(u,Yu)du ϕ(Y T ) | Y t = y] where, dY t = a(t, Y t ) dt + b(t, Y t ) dW t and W t is a P * -standard Brownian motion. • Black-Scholes : In the Black-Scholes model, dSt St = r dt + σdW t where W t is a risk-neutral Brownian motion. Moreover, the price of put and call options with strike K and maturity T are given by: V call t = S t Φ(d + ) - Ke -r(T -t) Φ(d - ) V put t = Ke -r(T -t) Φ(-d - ) - S t Φ(-d + ) where d ± = ln S t K +(r± 1 2 σ 2 )(T -t) σ √ T -t . 2
description
Some important formulae in stochastic calculus.Useful as reference for undergraduate mathematics / engineering / physics students.
Transcript of Stochastic Calculus Formulae
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Formula Page
Normal cdf: (x) := x e 12x2 dx2pi . Moments of Normals: If Z is a normal r.v. with mean 0 and variance 1, then the m.g.f. isE[eaZ ] = e 12a2 and the fourth moment is E[Z4] = 3.
Itos Lemma: If Wt is a standard Brownian motion and Xt satisfies the SDE:
dXt = (t,Xt) dt+ (t,Xt) dWt ,
and Ut = f(t,Xt), where f(t, x) is twice differentiable in x and once differentiable in t, then
dUt = xf(t,Xt) dXt +(t +
122(t,Xt) xx
)f(t,Xt) dt
=(t + (t,Xt)x +
122(t,Xt) xx
)f(t,Xt) dt+ (t,Xt) xf(t,Xt) dWt
Itos Isometry: If Wt is a standard Brownian motion, then
E
[( t0
g(s,Ws) dWs
)( t0
h(s,Ws) dWs
)]= E
[ t0
g(s,Ws)h(s,Ws) ds
].
Feynman-Kac: Suppose a function f(y, t) satisfies the PDE:(t + a(t, y)y +
12b2(t, y)yy
)f(t, y) = c(y, t) f(t, y),
f(T, y) = (y).
Then f(y, t) admits the unique solution
f(y, t) = EP [e Tt c(u,Yu)du (YT ) | Yt = y] where, dYt = a(t, Yt) dt+ b(t, Yt) dWt
and Wt is a P-standard Brownian motion.
Black-Scholes: In the Black-Scholes model, dStSt
= r dt+ dWt where Wt is a risk-neutral Brownian
motion. Moreover, the price of put and call options with strike K and maturity T are given by:
V callt = St(d+)K er(Tt)(d) V putt = K er(Tt)(d) St(d+)
where d =ln
StK
+(r 122)(Tt)
Tt .
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![Pathwise stochastic calculus with local timesmdavis/docs/AIHP792.pdf · similar results for a large class of Dirichlet processes; see also Coutin, Nualart and Tudor [5] (who consider](https://static.fdocument.pub/doc/165x107/5f3f5e4a2a630a76653785fd/pathwise-stochastic-calculus-with-local-mdavisdocsaihp792pdf-similar-results.jpg)
