Black and Scholes (1973) implied volatilities tend to be systematically related to the option's exercise price and time to expiration. Derman and Kani (1994), Dupire (1994), and Rubinstein (1994) attribute this behavior to the fact that the Black-Scholes constant volatility assumption is...

Stocks that hedge against sustained market downturns — periods from peak to trough in S&P500 levels at the business cycle frequency — should have low expected returns, but they do not. We use ex-ante firm characteristics and covariances to construct a tradeable Safe Minus Risky (SMR)...