Displaced Diffusion as an Approximation of the Constant Elasticity of Variance

Abstract

The CEV (constant elasticity of variance) and displaced diffusion processes have been posited as suitable alternatives to a lognormal process in modelling the dynamics of market variables such as stock prices and interest rates. Marris (1999) noted that, for a certain parameterization, option prices produced by the two processes display close correspondence across a range of strikes and maturities. This parametrization is a simple linearization of the CEV dynamics around the initial value of the underlying and we quantify the observed agreement in option prices by performing a small time expansion of the option prices around the forward-at-the-money value of the underlying. We show further results regarding the comparability of the conditional probability density functions of the two processes and hence the associated moments.

Key Words:

• Constant elasticity of variance (CEV)
• displaced diffusion
• option pricing
• asymptotic expansions

Acknowledgements

Many thanks to Ben Hambly, Sam Howison and William Shaw for invaluable assistance and Mark Greenwood for useful comments and discussions.

Notes

¹See Howison (2005) for details.

²See Howison (2005), footnote 2.

³ Extending the approach of Howison (2005) it is easy to show that if and , a particular solution takes the form .