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 (Citation1999) 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.
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 (Citation2005) for details.
2See Howison (Citation2005), footnote 2.
3 Extending the approach of Howison (Citation2005) it is easy to show that if and
, a particular solution
takes
the
form
.