Optimal No-Arbitrage Bounds on S&P 500 Index Options and the Volatility Smile

✍ Scribed by Patrick J. Dennis

• Publisher: John Wiley and Sons
• Year: 2001
• Tongue: English
• Weight: 176 KB
• Volume: 21
• Category: Article
• ISSN: 0270-7314
• DOI: 10.1002/fut.2203

✦ Synopsis

Abstract

This article shows that the volatility smile is not necessarily inconsistent with the Black–Scholes
analysis. Specifically, when transaction costs are present, the absence of arbitrage opportunities does not
dictate that there exists a unique price for an option. Rather, there exists a range of prices within which the
option's price may fall and still be consistent with the Black–Scholes arbitrage pricing argument.
This article uses a linear program (LP) cast in a binomial framework to determine the smallest
possible range of prices for Standard & Poor's 500 Index options that are consistent with no arbitrage
in the presence of transaction costs. The LP method employs dynamic trading in the underlying and
risk‐free assets as well as fixed positions in other options that trade on the same underlying security.
One‐way transaction‐cost levels on the index, inclusive of the bid–ask spread, would have to
be below six basis points for deviations from Black–Scholes pricing to present an arbitrage opportunity.
Monte Carlo simulations are employed to assess the hedging error induced with a 12‐period binomial model
to approximate a continuous‐time geometric Brownian motion. Once the risk caused by the hedging error is
accounted for, transaction costs have to be well below three basis points for the arbitrage opportunity to be
profitable two times out of five. This analysis indicates that market prices that deviate from those given by a
constant‐volatility option model, such as the Black–Scholes model, can be consistent with the
absence of arbitrage in the presence of transaction costs. © 2001 John Wiley & Sons, Inc. Jrl Fut Mark
21:1151–1179, 2001