In this paper, we examine the pricing of European call options on stocks that have variance rates that change randomly. We study continuous time diffusion processes for the stock return and the standard deviation parameter, and we find that one must use the stock and two options to form a riskless hedge. The riskless hedge does not lead to a unique option pricing function because the random standard deviation is not a traded security. One must appeal to an equilibrium asset pricing model to derive a unique option pricing function. In general, the option price depends on the risk premium associated with the random standard deviation. We find that the problem can be simplified by assuming that volatility risk can be diversified away and that changes in volatility are uncorrelated with the stock return. The resulting solution is an integral of the Black-Scholes formula and the distribution function for the variance of the stock price. We show that accurate option prices can be computed via Monte Carlo simulations and we apply the model to a set of actual prices.

Louis O. Scott, The Stationarity of the Conditional Mean of Real Rates of Return on Common Stocks: An Empirical Investigation, The Journal of Financial and Quantitative Analysis, Vol. 19, No. 2 (Jun., 1984), pp. 217-230

A method for estimating the marginal rate of substitution in the intertemporal capital asset pricing model is presented. The marginal rate of substitution is treated as an unobservable and one-period returns are used to develop a method of moments estimator that is consistent. Consistency depends on both a large number of time observations and a large number of securities. In the last section of the paper, the estimates of the marginal rate of substitution are used to test whether stock prices are unbiased predictors of ex post market fundamentals. Copyright 1989 by MIT Press.

The variance bounds tests of the present value model of stock prices are re-examined in this paper.A direct test of the model based on ordinary least squares estimation of a simple regression equation is proposed as an alternative and it is shown that this regression approach has several advantages over the variance bounds tests.This test is easily adapted to the important case in which the percentage changes in real dividends and real stock prices are stationary pro- cesses.The t*ests are applied to quarterly data for the Standard & Poor's Index of 500 Common Stocks and tJie results are much more > conclusive than those obtained by previous tests.

[Solutions are presented for prices on interest rate options in a two-factor version of the Cox-Ingersoll-Ross model of the term structure. Specific solutions are developed for caps on floating interest rates and for European options on discount bonds, coupon bonds, coupon bond futures, and Euro-dollar futures. The solutions for the options are expressed as multivariate integrals, and we show how to reduce the calculations to univariate numerical integrations, which can be calculated very quickly. The two-factor model provides more flexibility in fitting observed term structures, and the fixed parameters of the model can be set to capture the variability of the term structure over time.]

Marc Chesney, Louis Scott, Pricing European Currency Options: A Comparison of the Modified Black- Scholes Model and a Random Variance Model, The Journal of Financial and Quantitative Analysis, Vol. 24, No. 3 (Sep., 1989), pp. 267-284

Solutions are presented for prices on interest rate options in a two-factor version of the Cox–Ingersoll–Ross model of the term structure. Specific solutions are developed for caps on floating interest rates and for European options on discount bonds, coupon bonds, coupon bond futures, and Euro-dollar futures. The solutions for the options are expressed as multivariate integrals, and we show how to reduce the calculations to univariate numerical integrations, which can be calculated very quickly. The two-factor model provides more flexibility in fitting observed term structures, and the fixed parameters of the model can be set to capture the variability of the term structure over time.