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 ((Formula presented.)), 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 (Formula presented.) based on the unimodality of ratios of densities of the underlying asset at different dates. We show that the (Formula presented.) 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 (Formula presented.) 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 (Formula presented.) violations are infrequent and typically reverse quickly.