Pricing and Hedging Path-Dependent Options Under the CEV Process

Abstract

M uch of the work on path-dependent options assumes that the underlying asset price follows geometric Brownian motion with constant volatility. This paper uses a more general assumption for the asset price process that provides a better fit to the empirical observations. We use the so-called constant elasticity of variance (CEV) diffusion model where the volatility is a function of the underlying asset price. We derive analytical formulae for the prices of important types of path-dependent options under this assumption. We demonstrate that the prices of options, which depend on extrema, such as barrier and lookback options, can be much more sensitive to the specification of the underlying price process than standard call and put options and show that a financial institution that uses the standard geometric Brownian motion assumption is exposed to significant pricing and hedging errors when dealing in path-dependent options.

Chapter 8 Discrete Barrier and Lookback Options

“…Most models assume the continuous time version mainly because this leads to analytical solutions; see, for example, Gatto et al (1979), Goldman et al (1979), and Conze and Viswanathan (1991), for continuous lookback options; and see, for example, Merton (1973), Kat (1994a, 1994b), Rubinstein and Reiner (1991), Chance (1994), and Kunitomo and Ikeda (1992) for various formulae for continuously monitored barrier options under the classical Brownian motion framework. Recently, Boyle and Tian (1999) and Davydov and Linetsky (2001) have priced continuously monitored barrier and lookback options under the CEV model using lattice and Laplace transform methods, respectively; see Kou andWang (2003, 2004) for continuously monitored barrier options under a jump-diffusion framework.…”

Laplace transform identities for diffusions, with applications to rebates and barrier options

“…We note that (6) was obtained by Davidov and Linetsky [7,Prop. 2] by an application of Fubini's theorem.…”

“…We also present some ancillary identities for the Laplace transforms of expressions involving first-passage times. A few of these originate from Davidov and Linetsky [7], but the arguments used there are different. However, two of the identities are new, and are used later to obtain expressions for rebate prices.…”

“…The approach there employed contour integration to perform analytic inversion, yielding series expansions for the prices of the instruments considered. Finally, Davidov and Linetsky [7] priced rebates, lookback options and barrier options on a security with CEV dynamics, by numerical Laplace transform inversion. We acknowledge the influence of the latter article, in particular, on our work.…”

“…Although there are no guaranteed error bounds, the same method has been used successfully before, e.g. by Davidov and Linetsky [7,8] and Fu et al [10], to invert Laplace transforms associated with derivative valuation problems. We also draw the reader's attention to the comparative study of numerical schemes for Laplace transform inversion in Craddock et al [6], which focused specifically on applications to derivative pricing.…”