Dynamic programming for valuing American options under a variance‐gamma process

• Author: Rim Chérif,Hatem Ben‐Ameur,Bruno Rémillard
• DOI: http://doi.org/10.1002/fut.22148
• Published date: 01 October 2020
• Date: 01 October 2020

J Futures Markets. 2020;40:1548–1561.wileyonlinelibrary.com/journal/fut1548

|

© 2020 Wiley Periodicals LLC

Received: 28 June 2019

|

Accepted: 3 June 2020

DOI: 10.1002/fut.22148

RESEARCH ARTICLE

Dynamic programming for valuing American options

under a variance‐gamma process

Hatem Ben‐Ameur |Rim Chérif |Bruno Rémillard

HEC Montréal, 3000 chemin de la Côte

Sainte‐Catherine, Montréal,

Québec, Canada

Correspondence

Rim Chérif, HEC Montréal, 3000 chemin

de la Côte Sainte‐Catherine, Montréal,

QC H3T 2A7, Canada.

Email: rim.cherif@hec.ca

Funding information

Natural Sciences and Engineering

Research Council of Canada; Fonds pour

la formation de Chercheurs et l'Aide à la

Recherche

Abstract

Lévy processes provide a solution to overcome the shortcomings of the log-

normal hypothesis. A growing literature proposes the use of pure‐jump Lévy

processes, such as the variance‐gamma (VG) model. In this setting, explicit

solutions for derivative prices are unavailable, for instance, for the valuation of

American options. We propose a dynamic programming approach coupled

with finite elements for valuing American‐style options under an extended VG

model. Our numerical experiments confirm the convergence and show the

efficiency of the proposed methodology. We also conduct a numerical in-

vestigation that focuses on American options on S&P 500 futures contracts.

KEYWORDS

primary 60F05, secondary 62E20, American options, calibration, dynamic programming, finite

elements, jump‐diffusion process, maximum likelihood,

1|INTRODUCTION

A special class of diffusion processes, specifically, the generalized hyperbolic distributions, have straightforward par-

allels with Lévy processes. They have become extremely relevant in mathematical finance as they provide tools to

accurately consider enough of the desired properties of asset returns, both in the real and the risk‐neutral worlds. One

member of the generalized hyperbolic family, and one of the most popular Lévy processes used in financial modeling, is

the variance‐gamma (VG) process that, in itself, synthesizes several desired movement types for the asset price. On the

one hand, it allows an infinite number of low‐amplitude jumps that behave like diffusion, and, on the other hand,

permits a finite number of high‐amplitude jumps whose intensity decreases as amplitude increases.

We evaluate options under an extended VG model in line with Ben‐Ameur, Chérif, and Rémillard (2016), which can

improve the model estimation step. A special case is the VG model of Madan, Carr, and Chang (1998). Our approach is

based on dynamic programming (DP) coupled with finite elements. The value function under consideration is ap-

proximated by a piecewise polynomial at each decision date. High‐order polynomials are accurate but are time con-

suming. We compare piecewise‐constant, ‐linear, and ‐quadratic approximations, and conclude this last one to be the

most efficient. We show that our methodology is an acceptable compromise between tractability and efficiency. We also

discuss its potential expandability to higher‐dimensional state spaces.

The VG model is unique in that Brownian motion varies according to a stochastic time scale given by the gamma

process. With this connection, price fluctuations are expressed according to a business time scale rather than calendar time.

Thus, a stochastic time change can have two effects: it can speed up calendar time and subject the market to turbulence, or

slow down calendar time and maintain an unperturbed market. The VG process enables accurate financial applications in,

for example, modeling oil price dynamics (Askari & Krichene, 2008), credit risk (Fiorani, Luciano, & Semeraro, 2010), and

options on stocks, energy, and currency prices (Daal & Madan, 2005; Pinho & Madaleno, 2011). The VG process' flexibility is