Solution of Stochastic Differential Equations in Finance

Fig. 6. Pricing error for barrier down-and-out call option. Error is propor- tional to the square root of the number of Monte-Carlo realizations.
The quasi-random approach can become too cumbersome if the number of steps m along each SDE trajectory becomes large. As an example, consider a barrier option, whose value is a function of the entire trajectory. For a down- and-out barrier call, the payout is canceled if the underlying stock drops belong a certain level during the life of the option. Therefore, at time T the payoff is max(X(T ) − K, 0) if X(t) > L for 0 < t < T , and 0 otherwise. For such an option, accurate pricing is dependent on using a relatively large number of steps m per trajectory. Results of a Monte-Carlo simulation of this modified call option are shown in Fig. 6, where the error was computed by comparison with the exact price

V (X, T ) = C(X, T ) −
_{X 1−} σ₂

2r
L


C(L²/X, T )

where C(X, t) is the standard European call value with strike price K. The tra- jectories were generated with Euler-Maruyama approximations with pseudo- random number increments, where m = 1000 steps were used.

Other approaches to making Monte-Carlo sampling of trajectories more efficient fall under the umbrella of variance reduction. The idea is to calculate the expected value more accurately with fewer calls to the random number generator. The concept of antithetic variates is to follow SDE solutions in pairs, using the Wiener increment in one solutions and its negative in the other solution at each step. Due to the symmetry of the Wiener process,
the solutions are equally likely. For the same number of random numbers generated, the standard error is decreased by a factor of ^√2.
A stronger version of variance reduction in computing averages from SDE trajectories can be achieved with control variates. We outline one such ap- proach, known as variance reduction by delta-hedging. In this method the quantity that is being estimated by Monte-Carlo is replaced with an equiva- lent quantity of smaller variance. For example, instead of approximating the expected value of (20), the cash portion of the replicating portfolio of the Eu- ropean call can be targeted, since it must equal the option price at expiration.

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10

error
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10 10 10
number of realizations n