Solution of Stochastic Differential Equations in Finance

Fig. 5. Option pricing comparison between pseudo-random and quasi- random numbers. Circles (squares) represent error in Monte-Carlo estimation of European call by following SDE paths using pseudo-random (quasi-random) num- bers to generate increments. Settings were X(0) = 10, K = 12, r = 0.05, σ = 0.5, expiration time T = 0.5. The number of Wiener increments per trajectory was m = 8.
The results above were attained using pseudo-random numbers to generate the Wiener increments ∆W in the Euler-Maruyama method. An improvement in accuracy can be achieved by using quasi-random numbers instead.
By definition, standard normal pseudo-random numbers are created to be independent and identically-distributed, where the distribution is the stan- dard normal distribution. For many Monte-Carlo sampling problems, the in- dependence is not crucial to the computation. If that assumption can be dis- carded, then there are more efficient ways to sample, using what are called low- discrepancy sequences. Such quasi-random sequences are identically-distributed but not independent. Their advantage is that they are better at self-avoidance than pseudo-random numbers, and by essentially reducing redundancy they can deliver Monte-Carlo approximations of significantly reduced variance with the same number of realizations.
Consider the problem of estimating an expected value like (20) by calculat- ing many realizations. By Property 2 of the Wiener process, the m increments
∆W₁, . . . , ∆W_m of each realization must be independent. Therefore along the trajectories, independence must be preserved. This is accomplished by using m different low-discepancy sequences along the trajectory. For example, the base-p low discrepancy sequences due to Halton (1960) for m different prime numbers p can be used along the trajectory, while the sequences themselves run across different realizations.

Fig. 5 shows a comparison of errors for the Monte-Carlo pricing of the European call, using this approach to create quasi-random numbers. The low-discrepancy sequences produce nonindependent uniform random numbers, and must be run through the Box-Muller method (Box and Muller, 1958) to produce normal quasi-random numbers. The pseudo-random sequences show error proportional to n^−0.5, while the quasi-random appear to follow approx- imately n^−0.7.

More sophisticated low-discrepancy sequences, due to Faure, Niederreiter, Xing, and others, have been developed and can be shown to be more effi- cient than the Halton sequences. The chapter in this volume by Niederreiter (Niederreiter, 2010) describes the state of the art in generating such sequences.
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