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 dWt.
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
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