Valuing American options under the CEV model by Laplace–Carson transforms

doi:10.1016/j.orl.2010.07.006

Operations Research Letters

Volume 38, Issue 5, September 2010, Pages 474-481

https://doi.org/10.1016/j.orl.2010.07.006 Get rights and content

Abstract

This paper investigates American option pricing under the constant elasticity of variance (CEV) model. Taking the Laplace–Carson transform (LCT) to the corresponding free-boundary problem enables the determination of the optimal early exercise boundary to be separated from the valuation procedure. Specifically, a functional equation for the LCT of the early exercise boundary is obtained. By applying Gaussian quadrature formulas, an efficient method is developed to compute the early exercise boundary, American option price and Greeks under the CEV model.

Introduction

The American feature in an option contract represents the right for the option holders to exercise their option prematurely. The early exercise strategy is not unknown in advance, and must be determined within the pricing mechanism. Under the classical Black–Scholes dynamics, there are different ways to handle the valuation of American options. For instance, Kim [10] finds that the American option price can be expressed as a European option plus an early exercise premium. McKean [13], Merton [14] formulate the problem as a free boundary problem for a partial differential equation (PDE). Carr [1] develops the randomization approach, which actually considers the Laplace–Carson transform (LCT) of a fixed maturity barrier put, maximized over barriers. The pioneer work of Carr [1] also stimulates [11] to value finite-lived Russian options by the LCT approach.

Empirical evidence, however, suggests that the Black–Scholes model is inadequate to describe asset returns and the behavior of option markets. In particular, asset return distributions exhibit excess kurtosis and fat tails, whereas the volatility implied by the option prices shows a smile or skew shape across strike prices. This leads to a practical need to establish a more flexible model to cope with the empirical facts. One popular alternative is the constant elasticity of variance (CEV) model suggested by Cox [4]. Cox [4], and Emanuel and MacBeth [8] derive the closed-form solutions for European options under different elasticity factors. Davydov and Linetsky [5] study the pricing and hedging of barrier and lookback options under the CEV model. For the valuation of American options, Detemple and Tian [7] derive a recursive integral equation for the exercise boundary. Wong and Zhao [19] generalize the artificial boundary finite difference method to the CEV model. Zhao and Wong [20] investigate closed-form solutions in the sense of homotopy expansion for American option pricing under general diffusion models, which nest the CEV model as special cases. Nunes [16] proposes an alternative characterization of the early exercise premium that is valid for the CEV model. Erhan [9] proves the convexity of the perpetual American put option price of the level dependent volatility model with compound Poisson jumps and the validity of the smooth pasting condition. When the underlying asset price jumps, Chockalingam and Muthuraman [3] extend the moving-boundary approach to price American options. Liu and Hong [12] revisit the stochastic mesh method for pricing American options from a conditioning viewpoint.

However, the analytical valuation of American options under the CEV model is yet to be solved. By the Laplace–Carson transform, we show that the determination of the optimal early exercise boundary can be separated from the American option valuation procedure, enabling the option holders to know the early exercise strategy in advance. The LCTs of the Greeks are derived in explicit forms. This may help managing market risks through hedging. Although the paper focuses on American put options, the method is generally applicable to other exotic derivatives with continuous earlier exercise rights.

The remainder of the paper is organized as follows. Section 2 formulates the corresponding free-boundary problem for American options under the CEV model. Section 3 proposes the valuation procedures with Laplace–Carson transforms. Numerical examples are given in Section 4.

Section snippets

The free-boundary problem

The CEV model assumes that the risk-neutral process of the underlying asset price evolves according to the stochastic differential equation: $d S_{t} = (r - q) S_{t} d t + δ S_{t}^{β + 1} d Z_{t},$ where $r$ is the risk-free interest rate, $q$ is the dividend yield, and $Z_{t}$ is the Wiener process. It belongs to the class of local volatility models with $σ (S) = δ S^{β}$ . Thus, $β$ can be interpreted as the elasticity of the local volatility function because $\frac{d σ / σ}{d S / S} = β$ , and $δ$ is the scale parameter fixing the initial instantaneous volatility

Valuation with Laplace–Carson transforms

For $λ > 0$ , define the Laplace–Carson transform (LCT) of the American put option price $P_{A} (S, τ)$ as ${\hat{P}}_{A} (S, λ) = \int_{0}^{\infty} P_{A} (S, τ) λ e^{- λ τ} d τ ≔ LC [P_{A} (S, τ)] (λ) .$ Similarly, we denote the LCTs of $S_{f} (τ)$ by ${\hat{S}}_{f} (λ)$ . There is no essential difference between the LCT and the Laplace Transform (LT) except that the use of LCT simplifies notation in the later analysis of this paper. Under the Black–Scholes assumption, LCT has been adopted by Carr [1] to value American options and by Kimura [11] to value finite-lived Russian

Numerical examples

This section provides numerical examples to illustrate the American option valuation under the CEV model using the Laplace–Carson transform. In fact, the implementation of our solution is straightforward given the analytical solutions. The first step is to numerically calculate the early exercise boundary. To this end, the secant method (or any other root-finding procedure) is applied to determine the unique root of the functional equation in Proposition 1 over the interval $(0, K)$ . The Gaussian

Acknowledgements

Research grant HKSAR-RGC403608, provided by the Research Grant Council of Hong Kong, is gratefully acknowledged.

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Valuing American options under the CEV model by Laplace–Carson transforms

Abstract

Introduction

Section snippets

The free-boundary problem

Valuation with Laplace–Carson transforms

Numerical examples

Acknowledgements

European Journal of Operational Research

Operations Research Letters

Randomization and the American put

Review of Financial Studies

Alternative characterizations of American put options

Mathematical Finance

Pricing American options when asset prices jump

Operations Research Letters

Pricing and hedging path dependent options under the CEV process

Management Science

American options: symmetry property

The valuation of American options for a class of diffusion processes

Management Science

Further results on the constant elasticity of variance call option pricing model

Journal of Financial and Quantitative Analysis