S\&P type indices and call option values under a CEV diffusion process
The focus of this study is on estimating the diffusion characteristics of stock index prices, primarily the S&P 500 index, and on the corresponding evaluation of some call option pricing models. Here we are concerned with only European style index options, that is those index options that can be exercised only at expiration.^ An important aspect about the value of an option is that it depends critically on the assumption made about the distributional characteristics of the price dynamics of the asset on which the option is written. The basic Black-Scholes (1973) option valuation model assumes a lognormal distribution for the underlying asset. An important assumption of this model is that the variance of returns for the underlying asset is a constant.^ We attempt to answer the question: Is there empirical evidence to justify a CEV (constant elasticity of variance) specification for describing the dynamics of stock index prices? We then simulate a comparison of the basic Black-Scholes model with our estimated version of the CEV model for the S&P 500. We find that for the period under consideration there is sufficient empirical evidence to support a CEV specification for the dynamics of index prices. This also suggests an inverse association between the variance of returns on the index and the level of the index. The simulation results suggest that in general the estimated version of the CEV model does correct for the observed bias in the Black-Scholes model.^ We conclude by suggesting some extensions of this study as it applies to implied volatility. We also suggest a few references where currently, applications of the general method of contingent claims pricing are being utilized. ^
Samanta, Prodyot, "S\&P type indices and call option values under a CEV diffusion process" (1995). ETD Collection for Fordham University. AAI9520614.