Why are those options smiling?

Abstract

This paper tests whether the true smile in implied volatilities is flat. The smile in observed Black-Scholes implied volatilities has often been attributed to deficiencies in the B-S model, such as the assumption of constant volatility, which cause the implied volatilities calculated using the B-S formula to differ from the true volatilities. If such deficiencies are the sole cause, then if the implied volatilities were calculated correctly (i.e., using the true though possibly unknown model), the smile should disappear or become flat. Using stock index options data, we test and reject the hypothesis that the true smile in stock index option prices is flat. If the true smile is flat, then a trading strategy in which one buys options at the bottom of the incorrect Black-Scholes smile and sells options at the top(s) should not be profitable even on a pre-transaction-cost basis. However, we find that such a delta-gamma neutral strategy yields substantial pre-transaction-cost profits. Moreover, the profits are large when the B-S model predicts large profits and small when small profits are predicted. Our results indicate that while part of the observed Black-Scholes smile appears due to deficiencies in the Black-Scholes model, a substantial part reflects a smile in the true implied volatilities. We argue that the true smile persists despite these substantial pretransaction-cost profits, because maintaining the trading portfolio’s original low risk profile requires frequent re-balancing which quickly eats away the profits.