TY - JOUR
T1 - TOTAL POSITIVITY AND RELATIVE CONVEXITY OF OPTION PRICES
AU - Glasserman, Paul
AU - Pirjol, Dan
N1 - Publisher Copyright:
© 2023, American Institute of Mathematical Sciences. All rights reserved.
PY - 2023/3
Y1 - 2023/3
N2 - 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.
AB - 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.
KW - Lévy processes
KW - Total positivity
KW - implied volatility skew
KW - option prices
KW - relative convexity
UR - http://www.scopus.com/inward/record.url?scp=85184264086&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=85184264086&partnerID=8YFLogxK
U2 - 10.3934/fmf.2023001
DO - 10.3934/fmf.2023001
M3 - Article
AN - SCOPUS:85184264086
VL - 2
SP - 1
EP - 32
JO - Frontiers of Mathematical Finance
JF - Frontiers of Mathematical Finance
IS - 1
ER -