Abstract:

It is well known that stock returns on short time horizons are highly non-normal, contrary to the assumptions in the
Black--Scholes model. The present paper shows that non-normality of stock returns introduces a sizeable hedging error, even if one
hedges optimally, continuously and in the absence of transaction costs. Our finding is in sharp contrast with the standard textbook
knowledge claiming that continuous hedging is risk-free.

This paper gives a theoretical description of optimal continuous-time mean--variance hedging strategies in a world with leptokurtic stock returns. We find closed form expressions for the optimal delta, the unconditional variance of the optimal hedging error and the dynamic Sharpe ratio of the entire hedging strategy, and suggest an efficient scheme for their evaluation using the fast Fourier transform. The analysis presented here extends the work of Cox, Ross and Rubinstein (1979) to the world of fat-tailed IID returns, and at the same time it adds an important time dimension to the optimal portfolio framework of Markowitz (1952) and Sharpe (1966).

In much of the asset pricing literature (with the notable exception of Duffie and Richardson 1991, Toft 1996, Cochrane and Saa-Requejo 2000, and Khanna and Madan 2004) the hedging error is either not modelled by assuming market completeness, or it is effectively ignored by considering the so-called representative agent price, corresponding to the price at which a trader would not wish to buy or sell an option given her risk preferences. In practice traders sell large amounts of option contracts and, in the presence of hedging error, by doing so the traders enter into a risky position. To make the trading activity worthwhile the option price must therefore include a risk premium proportional to the hedging error, implying that option price can move in a bound around the Black-Scholes price. The width of the bound increases with the Sharpe ratio of the optimal hedging strategy. We find that a calibrated model of high frequency FT100 returns yields robust and non-trivial option price bounds.