When Are Option Prices TP2$\text{TP}_2$?

Call option prices in the Black–Scholes model, viewed as functions of strike and maturity, are totally positive of order two (TP2$\text{TP}_2$), meaning that the price ratio of a higher‐strike call to a lower‐strike call increases with maturity, with adjustments for dividends and interest. We develop conditions for this property in other models and contrast it with full total positivity, which holds only for out‐of‐the‐money strikes in the Black–Scholes model. Related properties apply to puts. We give a simple sufficient condition for TP2$\text{TP}_2$ based on the unimodality of ratios of densities of the underlying asset at different dates. We show that the TP2$\text{TP}_2$ property entails a strengthening of monotonicity of the underlying asset in the convex order and thus a strengthening of the absence of static arbitrage. We construct examples illustrating the gaps between these properties. We develop connections between TP2$\text{TP}_2$ and the shape of the implied volatility surface—in particular, connections with supermodularity of implied variance, a condition implying that lines of implied variance for different maturities fan out at high strikes. An examination of S&P 500 options market data indicates that TP2$\text{TP}_2$ violations are infrequent and typically reverse quickly.