Total Positivity and Relative Convexity of Option Prices

Paul Glasserman, Dan Pirjol

Research output: Chapter in Book/Report/Conference proceedingChapterpeer-review

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 variancegamma, CGMY, Dagum and logistic density models.

Original languageEnglish
Title of host publicationPeter Carr Gedenkschrift
Subtitle of host publicationResearch Advances in Mathematical Finance
Pages393-443
Number of pages51
ISBN (Electronic)9789811280306
DOIs
StatePublished - 1 Jan 2023

Keywords

  • Implied volatility skew
  • Levy processes
  • Option prices
  • Relative convexity
  • Total positivity

Fingerprint

Dive into the research topics of 'Total Positivity and Relative Convexity of Option Prices'. Together they form a unique fingerprint.

Cite this