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This paper has four goals: (a) relate ladder height distributions to option values; (b) show how Laguerre expansions may be used in the computation of densities, distribution functions, and option prices; (c) derive some new results on the integral of geometric Brownian motion over a finite interval; and (d) apply the preceding results to the determination of the distribution of the integral of geometric Brownian motion and the computation of Asian option values. The usual fixed-strike options on the average are treated, as well as options with payoffs expressed in terms of one over the average of the underlying security, which this author calls "reciprocal Asian options." In all cases the underlying asset is represented by geometric Brownian motion, the averages are performed continuously, and the options are of European type. Copyright Blackwell Publishers, Inc..

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This paper is about the probability law of the integral of geometric Brownian motion over a finite time interval. A partial differential equation is derived for the Laplace transform of the law of the reciprocal integral, and is shown to yield an expression for the density of the distribution. This expression has some advantages over the ones obtained previously, at least when the normalized drift of the Brownian motion is a non-negative integer. Bougerol's identity and a relationship between Brownian motions with opposite drifts may also be seen to be special cases of these results.

Two techniques are described for approximating distributions on the positive half-line by combinations of exponentials. One is based on Jacobi polynomial expansions, and the other on the logbeta distribution. The techniques are applied to some well-known distributions (degenerate, uniform, Pareto, lognormal and others). In theory, the techniques yield sequences of combination of exponentials that always converge to the true distribution, but their numerical performance depends on the particular distribution being approximated. An error bound is given in the case the logbeta approximations. class we are considering is then a subset of the rational family of distributions (also called the matrix-exponential family, or Cox family), which comprises those distributions with a Laplace transform which is a rational function (a ratio of two polynomials). Another subset of the rational family is the class of phase-type distributions [1], which includes the hyper-exponentials but not all combinations of exponentials. There is an expanding literature on these classes of distributions, see Reference [2] for more details and a list of references. For our purposes, the important property of combinations of exponentials is that they are dense in the set of probability distributions on ½0; 1Þ:The problem of fitting distributions from one of the above classes to a given distribution is far from new; in particular, the fitting of phase-type distributions has attracted attention recently, see Reference [3]. However, this author came upon the question in particular contexts (see below) where combinations of exponentials are the ideal tool, and where there is no reason to restrict the search to hyper-exponentials or phase-type distributions. Known methods rely on least-squares, moment matching, and so on. The advantage of the methods described in this paper is that they apply to all distributions, though they perform better for some distributions than for others. For a different fitting technique and other references, see the paper by Feldmann and Whitt [4]; they fit hyper-exponentials to a certain class of distributions (a subset of the class of distributions with a tail fatter than the exponential).The references cited above are related to queueing theory, where there is a lot of interest in phase-type distributions. However, this author has come across approximations by combinations of exponentials in relation to the following three problems.Risk theory. It has been known for some time that the probability of ruin is simpler to compute if the distribution of the claims, or that of the inter-arrival times of claims, is rational, see Reference [5] for details and references. The simplifications which occur in risk theory have been well-known in the literature on random walks (see, for instance, the comments in Reference [6]) and are also related to queueing theory.Convolutions. The distribution of the sum of independent random variables with a lognormal or Pareto distribution is not known in simple form. Therefore, a possibility is...