Additive Logistic Processes in Option Pricing

38 Pages Posted: 16 Sep 2020 Last revised: 20 Jul 2021

See all articles by Peter Carr

Peter Carr

New York University (NYU) - Finance and Risk Engineering Department

Lorenzo Torricelli

University of Parma, Department of Economics and Management

Date Written: August 3, 2020

Abstract

In option pricing it is customary to first specify a stochastic underlying model and then extract valuation equations from it. Still, it is possible to reverse this paradigm: starting from an arbitrage-free option valuation formula, one could derive a family of risk-neutral probabilities and a corresponding risk-neutral underlying asset process. In this paper we start from two simple arbitrage-free valuation equations, inspired by the log-sum-exponential function and an $l^p$ vector norm. Such expressions lead respectively to logistic and Dagum (or ``log-skew-logistic'') risk-neutral distributions for the underlying security price. We proceed to exhibit supporting martingale processes of additive type for underlying securities having as time marginals two such distributions. By construction, these processes produce closed-form valuation equations which are even simpler than those of the Bachelier and Samuelson-Black-Scholes models.
Additive logistic processes provide parsimonious and simple option pricing models capturing various important stylized facts at the minimum price of a single market observable input.

Keywords: Logistic distribution, additive processes, derivative pricing, Dagum distribution, generalized-z distributions

JEL Classification: 91G20, 60H99

Suggested Citation

Carr, Peter and Torricelli, Lorenzo, Additive Logistic Processes in Option Pricing (August 3, 2020). Available at SSRN: https://ssrn.com/abstract=3647466 or http://dx.doi.org/10.2139/ssrn.3647466

Peter Carr

New York University (NYU) - Finance and Risk Engineering Department ( email )

6 Metrotech Center
New York, NY 11201
United States

Lorenzo Torricelli (Contact Author)

University of Parma, Department of Economics and Management ( email )

Viale Kennedy 6
Parma
Italy

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