Economics - Research Publications

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Author: Dufresne, Daniel × End date: 2004 × Start date: 2001 × Subject: exponential functional of Brownian motion ×

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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.