How about estimating the standard deviation, as opposed to the vari- ance? We can use s as an estimator of the standard deviation. We can derive an expression for the standard deviation of the standard deviation estimator. For the case of a Gaussian parent distribution, it can be shown that the standard error in the estimation of the standard deviation is (Lupton, 1993)
(5.18) Assessment of this error may become important in value-at-risk com- putations.
where n is the number of cycles needed to achieve the accuracy we want, and is the computational work per cycle. Replacing for n,
(5.20) Assume now that two estimators with variance and and work per Monte Carlo cycle 1 and 2, respectively. Define , the ratio of computational work per replication. We can now define an acceleration factor that results by using estimator 2 instead of estimator 1:
(5.21)
We would like y to be as large as possible. The next section discusses the main strategies available to accomplish this. We will then go into some of those strategies in greater detail.
Increasing Simulation Efficiency
If we do nothing about efficiency, the number of Monte Carlo replications we need to achieve acceptable pricing accuracy may be surprisingly large.
It may in fact be so large that it renders a naive implementation of the Monte Carlo method all but useless given the practical requirements of trading and risk management. For example, to price a typical European call down to one cent per $10 we may need over 1 million replications. In other instruments, such as index amortizing swaps, the computational time with naive Monte Carlo may be impractically long. As a result, in many cases variance reduction is not just a mathematical nicety, but a practical requirement.
The most commonly used strategies for variance reduction are the following.
■ Antithetic variates: We construct the estimator by using two Brownian motion trajectories that are mirror images of each other. This causes cancellation of dispersion. This method tends to reduce the variance modestly, but it is extremely easy to implement and as a result, very commonly used.
■ Control variates: The estimator includes a problem highly correlated with the one we want to solve. We must know the expectation of the correlated problem either analytically or numerically. The combined problem has less variance. The correlated problem is called the control variate. In either case, we must know the expectation of the control
Work
2 Error ---
∝
12 22
1
2
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=
y 12 22 ---
=
variate very well, because any uncertainty in the control variate will contaminate our desired results. This is the methodology of choice in many pricing problems. If carried out properly, we can accomplish extremely high improvements in efficiency. Significant insight and understanding of the problem is required, however.
■ Importance sampling: We take expectations using a different probabil- ity density than that of the problem we are solving. This is effectively an application of the Girsanov theorem, where the measure is distorted in a manner that the variance is reduced. It can be very effective in problems involving jumps.
■ Stratification: We arrange the Monte Carlo replications within prede- termined regions of the distribution, thus covering the space spanned by the random variables more evenly. Stratification can be particularly effective when important events that need to be captured occur in the tails of distributions.
■ Moment matching: This consists in ensuring that the moments of the sample of the underlying processes are matched to the moments of the population. It follows the intuitive but rather vague notion that if we want to get the correct expectation, we should have the correct underlying process. It is straightforward to implement, but it is not guaranteed to work.
■ Low-discrepancy sequences: We discussed this topic to some extent in the previous chapter. Also known as quasi–Monte Carlo methods, these approaches use sequences that cover the space of underlying random vari- ables more evenly than regular Monte Carlo. Essentially, low-discrepancy sequences get around the problem known as clustering, which happens with regular Monte Carlo. Clustering means that random points gener- ated by Monte Carlo in a multidimensional space will not be spread out in a manner that could be considered necessarily optimal for a given num- ber of dimensions. Low-discrepancy sequences get around this problem by controlling the way sampling points are arranged in a multidimen- sional space. When the number of dimensions is small, the idea of evening out the arrangement of sampling points may appear intuitive and attrac- tive. As the number of dimensions increases, however, attempts to even out the distribution of sampling points ends up translating into regions of space (you can visualize these as holes) that don’t get sampled. Quasi–
Monte Carlo methods were popularized in the early 1990s, but the fact that many of the interesting and useful properties of simulation depend on the use of (pseudo-) random numbers has affected the growth of the popularity of low-discrepancy methods.
Next, we will concentrate on the four of these strategies that have proven to be the most fruitful in pricing applications.