Total positivity and relative convexity of option prices

Paul Glasserman; Dan Pirjol

doi:10.3934/fmf.2023001

Article Contents

2023, Volume 2, Issue 1: 1-32. Doi: 10.3934/fmf.2023001 open access

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Total positivity and relative convexity of option prices

Paul Glasserman^1, , and
Dan Pirjol^1,2,

1.
Business School, Columbia University, USA
2.
Business School, Stevens Institute of Technology, USA

^*Corresponding author: Paul Glasserman
^*Corresponding author: Paul Glasserman

Received: September 2022

Revised: December 2022

Early access: February 2023

Published: March 2023

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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.

Keywords:

Mathematics Subject Classification: Primary: 60E15, 60E07; Secondary: 60G42.

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Figure 1. Implications among total positivity and relative convexity properties for out-of-the-money calls. The properties in the figure are stated for $ x\ge 1 $ and $ K\ge 1 $. An alternative to the implication (d) is discussed in Remark 2.2

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Figure 2. Implications among RR and relative convexity properties for out-of-the-money puts. The properties are stated for $ x,K\in (0,1] $

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Figure 3. The ratio $ f(x,\sigma_1)/f(x,\sigma_2) $, $ \sigma_1 = 0.5<\sigma_2 = 1 $, is increasing for $ x\in(0,1] $ and decreasing for $ x\ge 1 $

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Figure 4. Points in the plane $ (K_i, K_j) $ with $ \delta_{ij}< 0 $. They correspond to exceptions from TP$ _2 $ property for OTM call options (left) and from RR$ _2 $ property for OTM put options (right). Upper: $ T_1 = 10/15/21, T_2 = 11/19/21 $, middle: $ T_1 = 10/15/21, T_2 = 12/17/21 $ and lower: $ T_1 = 11/19/21, T_2 = 12/17/21 $. The vertical/horizontal red lines are at $ K = S_0 $. Pricing date 10/4/21

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Table 1. The first few expiries of the SPX index options, the forward price $ F(T) $ and the number of strikes for which option pricing data is available. The pricing date is 4-Oct-2021, with closing price $ S_0 = 4,300.46 $

Expiry $ F(T) $ $ n_K $

15-Oct-2021 4,300.86 407

19-Nov-2021 4,295.84 367

17-Dec-2021 4,290.59 399

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