Page 357 - Numerical Methods for Chemical Engineering

346 7 Probability theory and stochastic simulation

and apply (7.162),
1 2

dG = − 1 + (2)[1] dt + 2 Y − W t k (1)dW t = Y − W t k dW t (7.168)
2
to yield the integrand of (7.166). Using (7.166) in (7.165), we obtain the Mil’shtein rule

∂b 1 2
)( W t ) + ) [( W t ) − t]
X t k+1 ≈ X t k + a(t k , X t k )( t) + b(t k , X t k b(t k , X t k 2
∂ X
(t k ,X t k )
(7.169)
Further higher-order methods are discussed in Kloeden & Platen (2000).
Example. Stochastic calculus in quantitative finance

Stochasticcalculusisusedheavilyinquantitativefinance,asignificantemployerofnumerate
engineers. In Problem 6.B.5, we solved the Black–Scholes equation for the fair value of an
option. Here, we show how this equation is obtained, through stochastic calculus.
Consider the spot (market) price S(t) of some financial asset as a function of time. We
sample the price at uniform time periods t k = k( t) and let S k = S(t k ). A reasonable model
of the behavior of many assets is the lognormal random walk, which assumes that the return
between successive time periods
S k+1 − S k
R k = (7.170)
S k
is normally distributed so that the spot price is governed by the SDE
(7.171)
dS = µSdt + σ SdW t
µ is the drift rate, and is computed from a sequence of returns by
N s
1
µ = R k (7.172)
N s ( t)
k=1
σ is the volatility of the asset, and is estimated from
1 N s
2 2
σ = (R k − R ) R = µ( t) (7.173)
(N s − 1)( t)
k=1
In a simple type of derivative, a European option, we purchase at time t the right to either
buy (a call option) or sell (a put option) the underlying asset at some time T > t in the future
at an exercise price E. Thus, at time T, if we purchase the option, we will have a payoff

max(S(T ) − E, 0), call option
payoff(S(T )) = (7.174)
max(E − S(T ), 0), put option
What is the fair price V (S, t) of this option at t < T when the spot price of the underlying
asset is S?
To compute this value, assume that we purchase an option and at the same time short
(i.e., sell assets we don’t actually have – this is often possible and legal) a quantity of the
underlying asset. The value of this portfolio is = V (S, t) − S. Applying (7.162), the