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 language | English |
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Title of host publication | Peter Carr Gedenkschrift |
Subtitle of host publication | Research Advances in Mathematical Finance |
Pages | 393-443 |
Number of pages | 51 |
ISBN (Electronic) | 9789811280306 |
DOIs | |
State | Published - 1 Jan 2023 |
Keywords
- Implied volatility skew
- Levy processes
- Option prices
- Relative convexity
- Total positivity