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When You Cannot Hedge Continuously, The Corrections of Black-Scholes

The insight behind the Black-Scholes formula for options valuation is the recognition that, if you know the future volatility of a stock, you can replicate an option payoff exactly by a continuous rebalancing of a portfolio consisting of the underlying stock and a risk-free bond. If no arbitrage is possible, then the value of the option should be the cost of the replication strategy.

It is widely recognized that the Black-Scholes model is imperfect in many respects. It makes the idealized assumption that future stock evolution is lognormal with a known volatility, it ignores transaction costs and market impact, and it assumes that trading can be carried out continuously. Nevertheless the model is rational, and therefore amenable to rational modification of its assumptions and results; the Black-Scholes theory can be used to investigate its own shortcomings. This is an active area of research for the Quantitative Strategies Group at Goldman Sachs.

In this note we examine the effect of dropping only one of the key Black-Scholes assumptions, that of the possibility of continuous hedging. We examine the error that arises when you replicate a single option throughout its lifetime according to the Black-Scholes replication strategy, with the constraint that only a discrete number of rebalancing trades at regular intervals are allowed. To study the replication error, we carry out Monte Carlo simulations of the Black-Scholes hedging strategy for a single European-style option and analyze the resulting uncertainty (that is, the error) in the replication value. We will show that that these errors follow a simple rule-of-thumb that is related to statistical uncertainty that arises in estimating volatility from a discrete number of observations: the typical error in the replication value is proportional to the vega of the option (its volatility sensitivity, sometimes also called kappa) multiplied by the uncertainty in its observed volatility.

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