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In Peter Jaeckel, Monte Carlo in Finance book, I read the following sentence:

Whenever the first realised moment of the underlying variate draws $\{z_i\}$ has a strong impact on the result of the overall simulation, antithetic sampling is an easy way of improving the accuracy of the calculation since it corrects the first moment. Of course, if we use Monte Carlo simulation to calculate the value of derivatives contracts that mainly (or only) depend on higher moments, antithetic sampling will not help.

From what I understood from Antithetic sampling, it reduces the variance of the mean estimator if:

$$ Cov(v(z_i), v(-z_i)) < 0$$ Since no proofs were shown in the book, my questions then are:

  • How does it correct the first moment?
  • Why it does not help for derivatives contract that depend on higher moments?
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    $\begingroup$ This is indeed a confused explanation. Have a look at the more thorough analysis in Glasserman Chapter 4.2 $\endgroup$ – g g Jun 24 at 21:37
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1) Antithetical sampling reduces the variance in that for each path generated by random numbers in the interval [0,1] (representing probabilities), it generates another path that is correlated to that path (for example by taking the 1 - random number from the first path). As such, by construction, it forces there to be another correlated path in the simulation and therefore a reduction in variance when compared to that of a series of only random numbers.

To illustrate by example:

Say one ascribes to the "Brownian Motion with Drift model" for equity returns, and that this model accurately describes the distribution of returns for the stock you are modeling.

Your equity returns can then be modeled using the following equation:

$$E(r)_{t} = μt + \sigma\epsilon*t^{0.5}$$

where ϵ is a random number drawn from a standard normal.

Say one is generating 2 possible paths of returns for one period, using 2 methods.

Method 1: Monte Carlo simulation using antithetic sampling; using random samples for path 1 and antithetic technique (1 - random number) for the second path.

Method 2: Monte Carlo simulation; using random samples for both paths

In order to generate random paths, say one draws from uniform distribution [0,1] to generate a random probability and transforms this probability to using the standard normal. Say that, this random probability is 0.84 resulting in a z-score of 1 for the first path under both Method 1 and Method 2.

Under Method 1, using antithetic sampling, one would then generate the second path using (1 - random prob from the first path) to generate a probability of 0.14 resulting in z-score of -1. It is easy to see that if one does this for a number of paths, what will result is a distribution of returns that is normally distributed around μ, arriving at an accurate first moment, since each of the z-scores derived from the simulation will have an equal but opposite z-score from the antithetical technique. This is consistent with lognormality assumption of the distribution of equities.

Under Method 2, the second random draw could result in any z-score for the second path and could then result in a volatility greater than that of Method 1, the antithetic sampling method. For example, if the second random number was 0.9 resulting in z score of 1.28, one would have ϵ's of 1 and 1.28 to generate the distribution of returns using the equation above (as opposed to 1 and -1 from using antithetical sampling). In this case, the pure randomness of your sampling technique would introduce volatility, whereas in Method 1, you are forcing the distribution to adhere to the correct first moment by construction. Of course if one had a truly random number generator and with enough paths, these two would converge; but in general, result in a volatility higher than that of antithetic sampling and would require a larger number of paths to achieve a distribution of normally distributed returns.

2) Antithetical sampling does not help in pricing derivatives that depend on higher moments in that you are forcing the distribution of returns to be correlated through your sampling technique. If the distribution of returns is distorted as a result of using antithetical techniques, or not generating random paths and a simulation that incorporates the volatility of the higher order underlying of your derivative, one will get an inaccurate valuation of derivatives that are based on higher moments.

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