Options and Discontinuity: An Asymptotic Decomposition for Trading Algorithms

Options and Discontinuity: An Asymptotic
Decomposition for Trading Algorithms

Feb 19, 2001

REC 123

Seongjoo Song, University of Chicago

Abstract: The problem of hedging contingent claims is well understood in a complete financial market. In such a market, any contingent claim can be replicated exactly by trading available securities with large enough initial capital. On the other hand, the risk of any option cannot be hedged away completely when the market is incomplete. There are many different causes of incompleteness. Among them, discontinuity of the underlying asset price process is a very important cause. this is because the discontinuous model fits the data better than any continuous model, and in particular because it incorporates such very real phenomena as crashes and devaluations, which can upset any trading strategy.

This paper studies the problem of option pricing and hedging in the presence of such discontinuities by adopting an asymptotic approach, letting securities prices converge to continuous processes. We then study the first order error in this convergence.

The first order error term after we hedge an option with the classical Black-Scholes strategy is decomposed into a part which can be traded away and a part which is purely unreplicable. First, I modify the Black-Scholes hedging strategy by adding the replicable part of the first order error and secondly, I adopt the mean-variance hedging method by Duffie and Richardson (1991) and Schweizer (1992) for the nonreplicable part. Under some regularity conditions, the closed form solution is obtained for the hedging, in this setting, I also study the properties of hedging at intervals, as the length of such intervals goes to zero.

Some results of simulation and real market data application are also provided. In simulation, we see that the new hedging strategy improves the classical Black-Scholes hedging strategy up to 30% in terms of the mean square of hedging error, when the distribution of log stock price is skewed.