The Binomial CEV Model and the Greeks

• Published date: 01 January 2017
• DOI: http://doi.org/10.1002/fut.21791
• Date: 01 January 2017

The Binomial CEV Model and the Greeks

Aricson Cruz and Jos´

e Carlos Dias*

This article compares alternative binomial approximation schemes for computing the option

hedge ratios studied by Chung and Shackleton (2002), Chung, Hung, Lee, and Shih (2011),

and Pelsser and Vorst(1994) under the lognormal assumption, but now considering the con-

stant elasticity of variance (CEV) process proposed by Cox (1975) and using the continuous-

time analytical Greeks recently offered by Larguinho, Dias, and Braumann (2013) as the

benchmarks. Among all the binomial models considered in this study, we conclude that an

extended tree binomial CEV model with the smooth and monotonic convergence property is

the most efficient method for computing Greeks under the CEV diffusion process because

one can apply the two-point extrapolation formula suggested by Chung et al. (2011). ©2016

Wiley Periodicals, Inc. Jrl Fut Mark 37:90–104, 2017

1. INTRODUCTION

Option traders need to repeatedly and accurately calculate options sensitivity measures (usu-

ally known as Greeks) to successfully implement hedging strategies in their risk management

activities, especially in the case of naked short options positions. This is so mainly because

the option’s risk characteristics change dynamically as the underlying stock price and the

remaining time to maturity change.

Given the absence of closed-form solutions for pricing and hedging many financial

option contracts possessing early exercise features and/or exotic payoffs, binomial models—

such as the one initially proposed by Cox, Ross, and Rubinstein (1979)—are commonly

used by both academics and practitioners to value and hedge such derivative products. The

computation of the required Greek measures is then often performed through a numerical

differentiation procedure. However, it is well known that the use of such scheme for comput-

ing Greeks (and prices) may be flawed by the nature of the binomial discretization behavior

observed in tree methods. See, for instance, Chung and Shackleton (2002, 2005), Chung

et al. (2011), and Pelsser and Vorst (1994) for details under the geometric Brownian motion

(henceforth, GBM) setup.

The main purpose of this article is to revisit the analysis performed by Chung and

Shackleton (2002), Chung et al. (2011), and Pelsser and Vorst (1994) for choosing appro-

priate methods when calculating option hedge ratios under the GBM assumption, but now

using the constant elasticity of variance (henceforth, CEV) diffusion process proposed by

Aricson Cruz and Jos´

e Carlos Dias are at the Instituto Universit´

ario de Lisboa (ISCTE-IUL), Lisboa, Portu-

gal. Unidade de Investigac¸ ˜

ao em Desenvolvimento Empresarial (UNIDE-IUL), Lisboa, Portugal. We thank

the comments of the editor Bob Webb. We are particularly grateful for the suggestions and comments of

an anonymous referee. Aricson Cruz gratefully acknowledges the financial support provided by Fundac¸ ˜

ao

Millenium BCP.

*Correspondence author,Instituto Universit ´

ario de Lisboa (ISCTE-IUL), Edif´

ıcio II, Av.Prof. An´

ıbal Bettencourt,

1600-189 Lisboa, Portugal. Tel:+351 21 7903977, Fax: +351 21 7964710, e-mail: jose.carlos.dias@iscte.pt

Received October 2015; Accepted March 2016

The Journal of Futures Markets, Vol. 37, No.1, 90–104 (2017)

©2016 Wiley Periodicals, Inc.

Published online 6 May 2016 in Wiley Online Library (wileyonlinelibrary.com).

DOI: 10.1002/fut.21791