Trading TP₂ option violations

Call option prices in the Black-Scholes model are totally positive of order 2 ((Formula presented.)), meaning that the ratio of the price of a higher-strike call to a lower-strike call increases with time-to-expiry, with adjustments for dividends and interest. This property, which strengthens the absence of calendar-spread arbitrage, holds in many, but not all standard theoretical models. We investigate whether it holds empirically in the market for call options on the S&P 500 index, and whether a closely related property holds for puts. We find that violations of (Formula presented.) are rare and usually reverse quickly. We examine the combinations of strikes and expiries most likely to produce violations, and we investigate the impact of market conditions on violation rates. We propose long-short option trading strategies designed to profit from violations. In our preferred implementation, these strategies substantially outperform the index on both an absolute and risk-adjusted basis. Individual trades based on individual (Formula presented.) violations have very high hit rates. These findings suggest that deviations from (Formula presented.) in market prices are anomalies and can be exploited profitably when they occur.