TOTAL POSITIVITY AND RELATIVE CONVEXITY OF OPTION PRICES

Paul Glasserman, Dan Pirjol

Research output: Contribution to journalArticlepeer-review

Abstract

This paper 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.

Original languageEnglish
Pages (from-to)1-32
Number of pages32
JournalFrontiers of Mathematical Finance
Volume2
Issue number1
DOIs
StatePublished - Mar 2023

Keywords

  • Lévy processes
  • Total positivity
  • implied volatility skew
  • option prices
  • relative convexity

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