Economics - Research Publications

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Start date: 2001 × End date: 2002 × Author: Dufresne, Daniel ×

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    A general class of risk models
    DUFRESNE, DANIEL ( 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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    The integrated square-root process
    DUFRESNE, DANIEL ( 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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    Asian and Basket Asymptotics
    DUFRESNE, DANIEL ( 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.