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Optimal continuous-time hedging with leptokurtic returns

Černý, A. (2007). Optimal continuous-time hedging with leptokurtic returns. Mathematical Finance, 17(2), pp. 175-203. doi: 10.1111/j.1467-9965.2007.00299.x


We examine the behavior of optimal mean–variance hedging strategies at high re-balancing frequencies in a model where stock prices follow a discretely sampled exponential Lévy 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 to markets with leptokurtic returns.

Publication Type: Article
Publisher Keywords: hedging error, Fourier transform, mean–variance hedging, locally optimal strategy, exponential Lévy process, incomplete market, option pricing
Subjects: H Social Sciences > HG Finance
Departments: Bayes Business School > Finance
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