Approximated Moment-Matching Dynamics for Basket-Options Simulation
39 Pages Posted: 29 Apr 2001
Date Written: November 14, 2001
In this paper we start by introducing the standard moment-matching procedure that one can apply to simulate the average price of a basket of basic assets. The basic idea is that of approximating the actual process of the basket value by a sufficiently simple stochastic process. The expression "sufficiently simple" should be interpreted as "simple enough to allow for analytic solutions to the pricing problem at hand".
The approximation happens on the basis of a moment matching principle, which can be stated as follows: set the parameters of the approximating process so that as many moments of the actual basket-price process as possible are exactly reproduced. With the usual lack of fantasy, the market choice of an approximating process seems to have fallen onto the lognormal one. The distinctive parameters of such a process being only two (the average return and the return's standard deviation over the time horizon set by the option to price) the moment matching procedure can only match the first two moments of the original distribution. The lengthy calculations can be performed so as to take into account the effect of dividends, either continuous or discrete (but in any case deterministic, both in payment dates and in amounts). A more compact formulation of this method is obtained by resorting to forward prices, which incorporate interest rates and dividends. We describe this basic framework in detail and then move to the three moments matching procedure, obtained by shifting the approximating Lognormal basket by a deterministic constant parameter. This new parameter allows to fit the first three moments without losing analytical tractability, in that we can immediately characterize the distributional properties of the resulting process trivially.
We then move to an empirical analysis of the two and three moments matching approximations, where we study the case of a basket of two equities in the Italian stock exchange and compare results by resorting to a Monte Carlo simulation to obtain the "true" distribution and statistics of the basket.
We subsequently move to analyze specifically the implications of the three moments method as far as a call option pricing is concerned.
The second part of the paper address the problem of computing a synthetic but at the same time rigorous measure of the deviation of the approximated baskets distributions from the real basket distribution. To characterize rigorously this distributional discrepancy, we introduce both the Kullback-Leibler information and the Hellinger distances in suitable spaces of probability densities, and explain how this can help us in our investigation. We compute the distances of the real basket from the parametric families of densities being used in the two and three moments approximations through Monte Carlo simulation. The two families are respectively the lognormal and shifted lognormal families. Finally, we try and isolate the variables and the situations causing this distance to increase drastically via a case study, thus characterizing the case where the two and three moments approximations can fail.
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