Optimal no-arbitrage bounds on S&P 500 index options and the volatility smile

This article shows that the volatility smile is not necessarily inconsisten t with the Black-Scholes analysis. specifically, when transaction costs are present, the absence of arbitrage opportunities does not dictate that ther e exists a unique price for an option. Rather, there exists a range of pric es within which the option's price may fall and still be consistent with th e Black-Scholes arbitrage pricing argument. This article uses a linear prog ram (LP) cast in a binomial framework to determine the smallest possible ra nge of prices for Standard & Poor's 500 Index options that are consistent w ith no arbitrage in the presence of transaction costs. The LP method employ s dynamic trading in the underlying and risk-free assets as well as fixed p ositions in other options that trade on the same underlying security. One-w ay transaction-cost levels on the index, inclusive of the bid-ask spread, w ould have to be below six basis points for deviations from Black-Scholes pr icing to present an arbitrage opportunity. Monte Carlo simulations are empl oyed to assess the hedging error induced with a 12-period binomial model to approximate a continuous-time geometric Brownian motion. Once the risk cau sed by the hedging error is accounted for, transaction costs have to be wel l below three basis points for the arbitrage opportunity to be profitable t wo times out of five. This analysis indicates that market prices that devia te from those given by a constant-volatility option model, such as the Blac k-Scholes model, can be consistent with the absence of arbitrage in the pre sence of transaction costs. (C) 2001 John Wiley & Sons, Inc.