Optimal Continuous-Time Hedging with Leptokurtic Returns

Imperial College Business School Discussion Paper No. 04/33

Mathematical Finance, 2007, 17(2), 175-203

34 Pages Posted: 4 May 2005 Last revised: 22 Jun 2020

See all articles by Aleš Černý

Aleš Černý

The Business School (formerly Cass), City, University of London

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Date Written: April 1, 2005

Abstract

We examine the behaviour of optimal mean-variance hedging strategies at high rebalancing frequencies in a model where stock prices follow a discretely sampled exponential Levy process and one hedges a European call option to maturity. Using elementary methods we show that all the attributes of a discretely rebalanced optimal hedge, i.e. the mean value, the hedge ratio and the expected squared hedging error, converge pointwise in the state space as the rebalancing interval goes to zero. The limiting formulae represent 1-D and 2-D generalized Fourier transforms which can be evaluated much faster than backward recursion schemes, with the same degree of accuracy.

In the special case of a compound Poisson process we demonstrate that the convergence results hold true if instead of using an infinitely divisible distribution from the outset one models log returns by multinomial approximations thereof. This result represents an important extension of Cox, Ross and Rubinstein (1979) to markets with leptokurtic returns.

Keywords: Hedging error, Fourier transform, mean-variance hedging, exponential Levy process, incomplete market, option pricing

JEL Classification: G11, C61

Suggested Citation

Černý, Aleš, Optimal Continuous-Time Hedging with Leptokurtic Returns (April 1, 2005). Imperial College Business School Discussion Paper No. 04/33, Mathematical Finance, 2007, 17(2), 175-203, Available at SSRN: https://ssrn.com/abstract=713361 or http://dx.doi.org/10.2139/ssrn.713361

Aleš Černý (Contact Author)

The Business School (formerly Cass), City, University of London ( email )

106 Bunhill Row
London, EC1Y 8TZ
United Kingdom

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