# Asymptotic option pricing under the CEV diffusion

Sang Hyeon Park, Jeong Hoon Kim

Research output: Contribution to journalArticle

15 Citations (Scopus)

### Abstract

In finance, many option pricing models generalizing the Black-Scholes model do not have closed form, analytic solutions so that it is hard to compute the solutions or at least it requires much time to compute the solutions. Therefore, asymptotic representation of options prices of various type has important practical implications in finance. This paper presents asymptotic expansions of option prices in the constant elasticity of variance model as the parameter appearing in the exponent of the diffusion coefficient tends to 2 which corresponds to the well-known Black-Scholes model. We use perturbation theory for partial differential equations to obtain the relevant results for European vanilla, barrier, and lookback options. We make our application of perturbation theory mathematically rigorous by supplying error bounds.

Original language English 490-501 12 Journal of Mathematical Analysis and Applications 375 2 https://doi.org/10.1016/j.jmaa.2010.09.060 Published - 2011 Mar 15

### Fingerprint

Black-Scholes Model
Option Pricing
Finance
Perturbation Theory
Lookback Options
Barrier Options
European Options
Asymptotic Representation
Analytic Solution
Diffusion Coefficient
Error Bounds
Asymptotic Expansion
Elasticity
Costs
Closed-form
Partial differential equation
Exponent
Tend
Model
Partial differential equations

### All Science Journal Classification (ASJC) codes

• Analysis
• Applied Mathematics

### Cite this

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abstract = "In finance, many option pricing models generalizing the Black-Scholes model do not have closed form, analytic solutions so that it is hard to compute the solutions or at least it requires much time to compute the solutions. Therefore, asymptotic representation of options prices of various type has important practical implications in finance. This paper presents asymptotic expansions of option prices in the constant elasticity of variance model as the parameter appearing in the exponent of the diffusion coefficient tends to 2 which corresponds to the well-known Black-Scholes model. We use perturbation theory for partial differential equations to obtain the relevant results for European vanilla, barrier, and lookback options. We make our application of perturbation theory mathematically rigorous by supplying error bounds.",
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Asymptotic option pricing under the CEV diffusion. / Park, Sang Hyeon; Kim, Jeong Hoon.

In: Journal of Mathematical Analysis and Applications, Vol. 375, No. 2, 15.03.2011, p. 490-501.

Research output: Contribution to journalArticle

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AU - Kim, Jeong Hoon

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AB - In finance, many option pricing models generalizing the Black-Scholes model do not have closed form, analytic solutions so that it is hard to compute the solutions or at least it requires much time to compute the solutions. Therefore, asymptotic representation of options prices of various type has important practical implications in finance. This paper presents asymptotic expansions of option prices in the constant elasticity of variance model as the parameter appearing in the exponent of the diffusion coefficient tends to 2 which corresponds to the well-known Black-Scholes model. We use perturbation theory for partial differential equations to obtain the relevant results for European vanilla, barrier, and lookback options. We make our application of perturbation theory mathematically rigorous by supplying error bounds.

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