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+ ßxMXxiX;) + ß2MX2(Xi) + • • • + ßkMXk(Xi)]

What still remains to be specified is how to choose the weights /J,. Notice that it is not strictly necessary to determine these weights, which could be all set equal to one. If the 'simpler' option is indeed very similar to the option we want to evaluate, this choice would produce almost as good a result as the more accurate least-square procedure described in the following. In general, however, the factors are designed to account for all sources of imperfections in the rebalancing of the hedge. To see how this can be accomplished, let us consider the vector P,

made up of the n final realisations VQii(T)) + ßiMX\(T) + ß2MXi2(T), the matrix MX, where MXj (T) denotes the realisation of the ¿th martingale variate at expiration time T for the y'th simulation,

and the vector /J:

Since one can clearly write r l l

a least-square estimate of fi is given by

where, as usual, the superscript T indicates transpose, and the superscript —1 inverse. Equation (10.28) therefore gives the 'best' combination of the parameters {/J} which can afford the maximum achievable variance reduction for the estimate of the option price.

Little more can be said about variance-reduction techniques based on the idea of taking advantage in a skilful way of the high degree of correlation between the target and a known option without examining the specific details of each individual problem. A second class of techniques, however, exists which rely on the strong negative correlation between two estimates in order to achieve the desired reduction in variance. The simplest of these antithetic methods consists of running two simulations in parallel, the first with a series of Gaussian random draws {z;}, and the second with the series of draws {— z;}. Given that, if z is a zero-mean, unit-variance random variable, then also —z is a zero-mean, unit-variance random variable, also the second series of draws constitutes a feasible realisation for the Brownian process to be simulated. Therefore the antithetic estimate

where Vest({z}) is the naive estimate obtained using Equation (10.11) and the random draws {z}, is also an unbiased estimator of the true option value, of variance

Antithetic Paths

Antithetic Variable

Figure 10.1 A log-normal path with its mirror image, obtained using the same random draws with opposite signs

Time Step

Figure 10.1 A log-normal path with its mirror image, obtained using the same random draws with opposite signs

As long as the covariance between Vest({z}) and Vest({-z}) is negative, as it will be for the choice of antithetic variable here suggested, the overall variance will have been substantially reduced. Intuitively, as shown in Figure 10.1, the reduction in variance stems from associating to each 'outlier' the opposite and mirror-image 'outlier', thereby cancelling to some extent the biasing effects of both. This technique is particularly effective when the integration is carried over a symmetric range, and the value function is approximately linear, as in the case of the evaluation of the present value of a swap; in this case, in fact, to each 'abnormally' high and positive realisation there corresponds an 'abnormally' high and negative value. In the case of options this anti-correlation is less pronounced, more and more so as the option is out-of-the-money, and an increasing fraction of zero outcomes occur.

10.4 handling american options

American options are notoriously difficult to evaluate using forward-induction procedures such as Monte Carlo. As is well known, the problem stems from the fact that, along a given path at time t, the expectation of future payoffs is not available. Backward-induction methods such as tree methodologies, for which expectations of future payoffs are readily available, suffer from the 'symmetric' problem of poor handling of path-dependent options; one method knows only about the past, the other about the future.

In theory, expectations could be evaluated using MC: from each time/state point reached during the course of a simulation where the expectation is needed a 'new' MC simulation could in principle be started, until the next exercise time is reached (see Figure 10.2). At this point, and from each of the sub-paths, yet another MC simulation would have to be started, and so on. Needless to say, the dimensionality of the underlying integration, which requires a number of paths growing exponentially with the number of exercise possibilities, is such that the procedure in this naive form is totally impractical.

The problem, however, can be made considerably simpler by combining forward and backward induction. Let us consider first the case of a Markovian one-dimensional model, and, for the sake of concreteness, let us assume that the underlying variables evolved through the simulation are forward rates; let us also consider the case when the option to be valued is an American swaption. Despite the fact that the following treatment would apply to the case of any interest-rate model implemented using Monte Carlo, the discussion will be cast with the HJM approach in mind, since it constitutes the most popular and important model to require MC for a general implementation. This section could therefore be profitably re-read after completing Chapter 20.

Under these assumptions, the 'normal' MC procedure can first be followed in order to obtain the sampled future values of a European option expiring at the last exercise opportunity (time T). For the specific case here examined these

Sub-Paths originating from a MC path

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