Economics - Research Publications
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ItemBessel processes and a functional of Brownian motion( 2004-08)The goal of this paper is to give a concise account of the connection between Bessel processes and the integral of geometric Brownian motion. The latter appears in the pricing of Asian options. Bessel processes are defined and some of their properties are given. The known expressions for the probability density function of the integral of geometric Brownian motion are stated, and other related results are given, in particular the Geman &Y or (1993) Laplace transform for Asian option prices.
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ItemFitting combinations of exponentials to probability distributions( 2005-11)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 log-beta 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 log-beta approximations.
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ItemFourier inversion formulas in option pricingand insurance( 2005-12)Several authors have used Fourier inversion to compute optionprices. In insurance, the expected value of max(S − K, 0) also arisesin excess-of-loss or stop-loss insurance, and similar techniques maybe used. Lewis (2001) used Parseval’s theorem to find formulas foroption prices in terms of the characteristic function of the log-price.This paper aims at taking the same idea further: (1) formulas requiringweaker assumptions; (2) relationship with classical inversiontheorems; (3) formulas for payoffs which occur in insurance.
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ItemBessel Processes and a Functional of Brownian Motion( 2004)The goal of this paper is to give a concise account of the connection between Besselprocesses and the integral of geometric Brownian motion. The latter appears in thepricing of Asian options. Bessel processes are defined and some of their propertiesare given. The known expressions for the probability density function of the integralof geometric Brownian motion are stated, and other related results are given, inparticular the Geman &Y or (1993) Laplace transform for Asian option prices.
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ItemA general class of risk models( 2002-01)We consider the actuarial risk model when the waiting times or claims have a Laplace transform which is a rational function. This generalizes the classical model, where the waiting times are expotenial, and give more flexibility in the modelling of a risk business. Ruin is seen as a random walk crossing a barrier; the summands of the random walk are expressed as the difference of the waiting time and the claim. The class R of distributions which have finite Laplace transforms includes the so-called phase-type distributions. For waiting times in R , the Laplace transform of the ruin probability is obtained explicitly; if the calims are in R , then the probability of ruin is a combination of exponentials times polynomials, which can be found in closed-form.
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ItemThe integrated square-root process( 2001-11)The square-root process has been used to model interest rates and stochastic volatility. This paper studies some of its properties, particularly those of the integral of the process over time. After summarizing the properties of the square-root process, the Laplace transform of the integral of the square-root process is derived. Three methods for the computation of the moments of this integral are given, as well as some properties of the density of the integral. The last section studies the relationship between the Laplace transforms of a variable and of its reciprocal, a topic which arises in the previous analysis and elsewhere. An application to the generalized inverse Gaussian distribution is given
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ItemAsian and Basket Asymptotics( 2002-07)The pricing of Asian or basket options is directly related to finding the distributions of sums of lognormal random variables. There is no general explicit formula for those distributions. This paper looks at the limit distributions of sums of lognormal variables when volatility, or maturity, tends to either 0 or to infinity. The limits obtained are either normal or lognormal, depending on the normalization chosen. This justifies the lognormal approximation, much used in practice, and also gives an asymptotically exact distribution for averages of lognormals with a relatively small volatility; it has been noted that all the analytical pricing formulas for Asian options perform poorly for small volatilities. Asymptotic formulas are also found for the moments of the sums of lognormals. Results are given for both discrete and continuous averages.