Cev Model

Constant Elasticity of Variance (CEV) Option Pricing Model:Integration and Detailed Derivation

Ying-Lin Hsu

Department of Applied Mathematics National Chung Hsing University Co-authors: T. I. Lin and C. F. Lee

Oct. 21, 2008

Ying-Lin Hsu (NCHU)

Oct. 21, 2008

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Outline

Introduction Transition Probability Density Function Noncentral Chi-Square Distribution The Noncentral Chi-Square Approach to Option Pricing Model Detailed Derivations of C1 and C2 β Approaches to 2 Some Computational Considerations Special Cases Concluding Remarks

Ying-Lin Hsu (NCHU)

Oct. 21, 2008

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Introduction

CEV model

The CEV option pricing model is defined as dS = µSdt + σS β/2 dZ, β < 2,

where dZ is a Wiener process and σ is a positive constant.

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Oct. 21, 2008

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Introduction

The elasticity is β − 2 since the return variance υ(S, t) = σ 2 S β−2 with respect to price S has the following relationship dυ(S, t)/dS = β − 2, υ(S, t)/S which implies that dυ(S, t)/υ(S, t) = (β − 2)dS/S.

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Oct. 21, 2008

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Introduction

If β = 2, then the elasticity is zero and the stock prices are lognormally distributed as in the Black and Scholes model (1973). If β = 1, then the elasticity is -1. The model proposed by Cox and Ross (1976).

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Oct. 21, 2008

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Introduction

We will focus on the case of β < 2 since many empirical evidences (see Campbell (1987), Glosten et al. (1993), Brandt and Kang (2004)) have shown that the relationship between the stock price and its return volatility is negative. The transition density for β > 2 is given by Emanuel and Macbeth (1982)

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Oct. 21, 2008

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Transition Probability Density Function

Consider the constant elasticity of variance diffusion, dS = µ(S, t) + σ(S, t)dZ, where µ(S, t) = rS − aS, and σ(S, t) = σS β/2 , 0 ≤ β < 2. Then

dS = (r − a)Sdt + σS β/2 dZ.

Ying-Lin Hsu (NCHU)...