Pricing Step Options under the CEV and other Solvable Diffusion Models

TL;DR

This paper develops new closed-form spectral expansion formulas for pricing proportional step options under nonlinear diffusion models, including the CEV model, providing efficient and exact solutions for complex occupation-time derivatives.

Contribution

It introduces analytically exact spectral expansion formulas for pricing step options under a class of nonlinear diffusion processes, including the CEV model, with practical applications.

Findings

Derived closed-form spectral expansion formulas for transition densities.

Applied formulas to price step options in four solvable models.

Formulas are rapidly convergent and easy to implement.

Abstract

We consider a special family of occupation-time derivatives, namely proportional step options introduced by Linetsky in [Math. Finance, 9, 55--96 (1999)]. We develop new closed-form spectral expansions for pricing such options under a class of nonlinear volatility diffusion processes which includes the constant-elasticity-of-variance (CEV) model as an example. In particular, we derive a general analytically exact expression for the resolvent kernel (i.e. Green's function) of such processes with killing at an exponential stopping time (independent of the process) of occupation above or below a fixed level. Moreover, we succeed in Laplace inverting the resolvent kernel and thereby derive newly closed-form spectral expansion formulae for the transition probability density of such processes with killing. The spectral expansion formulae are rapidly convergent and easy-to-implement as they are based simply on knowledge of a pair of fundamental solutions for an underlying solvable diffusion process. We apply the spectral expansion formulae to the pricing of proportional step options for four specific families of solvable nonlinear diffusion asset price models that include the CEV diffusion model and three other multi-parameter state-dependent local volatility confluent hypergeometric diffusion processes.