Total Positivity and Relative Convexity of Option Prices

Paul Glasserman and Dan Pirjol

Chapter 12 in Peter Carr Gedenkschrift:Research Advances in Mathematical Finance, 2023, pp 393-443 from World Scientific Publishing Co. Pte. Ltd.

Abstract: This chapter studies total positivity and relative convexity properties in option pricing models. We introduce these properties in the Black–Scholes setting by showing the following: out-of-the-money calls are totally positive in strike and volatility, out-of-the-money puts have a reverse sign rule property, calls and puts are convex with respect to at-the-money prices and relative convexity of option prices implies a convexity-in-time property of the underlying. We then extend these properties to other models, including scalar diffusions, mixture models and certain Lévy processes. We show that relative convexity typically holds in time-homogeneous local volatility models through the Dupire equation. We develop implications of these ideas for empirical option prices, including constraints on the at-the-money skew. We illustrate connections with models studied by Peter Carr, including the variance-gamma, CGMY, Dagum and logistic density models.

Keywords: Mathematical Finance; Quantitative Finance; Option Pricing; Derivatives; No Arbitrage; Asset Price Bubbles; Asset Pricing; Equilibrium; Volatility; Diffusion Processes; Jump Processes; Stochastic Integration; Trading Strategies; Portfolio Theory; Optimization; Securities; Bonds; Commodities; Futures (search for similar items in EconPapers)
JEL-codes: C02 C6 (search for similar items in EconPapers)
Date: 2023
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