Solution of Stochastic Differential Equations in Finance

Monte-Carlo sampling of SDE paths for option pricing

As an illustrative example of the use of SDE solvers for option pricing, consider the European call, whose value at expiration time T is max{X(T ) − K, 0}, where X(t) is the price of the underlying stock, K is the strike price. The no- arbitrage assumptions of Black-Scholes theory imply that the present value of such an option is

C(X₀, T ) = e^−rT E(max{X(T ) − K, 0}) (20)
where r is the fixed prevailing interest rate during the time interval [0, T ], and where the underlying stock price X(t) satisfies the stochastic differential equation
dX = rX dt + σX dW_t.
The value of the call option can be determined by calculating the expected value (20) explicitly. Using the Euler-Maruyama method for following solu- tions to the Black-Scholes SDE, the value X(T ) at the expiration time T can be determined for each path, or realization of the stochastic process. For a given n realizations, the average (max{X(T ) − K, 0}) can be used as an ap- proximation to the expected value in (20). Carrying this out and comparing with the exact solution from the Black-Scholes formula
C(X, T ) = XN (d₁) − Ke^−rT N (d₂) (21)

where
log(X/K) + (r + ¹ σ²)T



2
log(X/K) + (r − ¹ σ²)T

d₁ =
2
σ^√T
, d₂ =
σ^√T ^,

yields the errors plotted as circles in Fig. 5.

0
10

error
−1
10
−2
10 _{2 3}
10 10
number of realizations n