Repeated Richardson extrapolation and static hedging of barrier options under the CEV model

• Author: Jia‐Hau Guo,Lung‐Fu Chang
• Date: 01 June 2020
• Published date: 01 June 2020
• DOI: http://doi.org/10.1002/fut.22100

J Futures Markets. 2020;40:974–988.wileyonlinelibrary.com/journal/fut974

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© 2020 Wiley Periodicals, Inc.

Received: 25 November 2019

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Accepted: 13 January 2020

DOI: 10.1002/fut.22100

RESEARCH ARTICLE

Repeated Richardson extrapolation and static hedging of

barrier options under the CEV model

Jia‐Hau Guo

1

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Lung‐Fu Chang

2

1

Department of Information Management

and Finance, College of Management,

National Chiao Tung University, Hsinchu,

Taiwan

2

Department of Finance, College of

Business, National Taipei University of

Business, Taipei, Taiwan

Correspondence

Lung‐Fu Chang, Department of Finance,

College of Business, National Taipei

University of Business, No. 321, Sec. 1,

Jinan Road, Zhongzheng District, Taipei

100, Taiwan.

Email: lfchang@ntub.edu.tw

Abstract

This paper proposes an accelerated static replication approach for continuous

European‐style barrier options by employing the repeated Richardson extra-

polation technique with the Romberg sequence. This approach is developed

under the constant elasticity of variance (CEV) model of Cox (1975) and Cox

and Ross (1976) using the framework offered by Derman, Ergener, and Kani

(1995; DEK) and its modified method of Chung et al. (2010, 2013a, 2013b) and

Tsai (2014). The numerical results indicate that our method could significantly

reduce replication errors for European knock‐out call options and may be

superior to the imposition of the theta‐matching condition on the DEK method.

KEYWORDS

barrier options, constant elasticity of variance, Richardson extrapolation, static hedging portfolio,

theta‐matching condition

JEL CLASSIFICATION

G13

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INTRODUCTION

Hedging exotic options, such as barrier options, is important to capital markets for financial institutions to manage certain types

of risk. Traditionally, static hedging portfolio (SHP) and dynamic delta‐hedging approaches are two main hedging categories in

the financial literature. The SHP approach could be more affordable than the dynamic hedging approach because it does not

require weights to be frequently rebalanced in the hedging portfolio over passes or as the underlying price moves, especially

when the transaction cost is high. Chung, Shih, and Tsai (2013a, 2013b) further show that the hedging effectiveness of a

bimonthly SHP is far less risky than that of a delta‐hedging portfolio with daily rebalancing. Two subcategories of SHP have

been developed. The first approach, proposed by Bowie and Carr (1994), Carr and Chou (1997), and Carr, Ellis, and Gupta

(1998), is to construct SHP in a continuum of standard European options with the same maturity as the exotic option but

different strike prices. The second approach, developed by Derman, Ergener, and Kani (1995; DEK), utilizes a standard

European option to match the boundary at the maturity of the exotic option and a continuum of standard European options

with the same strike price but different maturities to match the boundary before the maturity of the exotic option. The

replication error of DEK results from the nonzero value of SHP on the barrier except at discrete time points. Increasing the

number of time points of value‐matching can enhance the accuracy of static replication but at the expense of being

computationally time consuming. Therefore, the DEK method is modified by Chung, Shih, and Tsai (2010), Chung et al.

(2013a), and Tsai (2014) with the imposition of the theta‐matching condition on discrete time points to enhance its efficiency.

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Nunes, Ruas, and Dias (2015) also impose the theta‐matching condition in their SHP for American knock‐in options on defaultable stocks.