This paper examines the pricing of European call options on stocks with randomly changing variance. It studies continuous-time diffusion processes for stock returns and standard deviation, finding that a riskless hedge requires the stock and two options. However, the random standard deviation is not a traded security, so an equilibrium asset pricing model is needed to derive a unique option pricing function. The option price depends on the risk premium associated with the random standard deviation. The problem can be simplified by assuming volatility risk can be diversified and changes in volatility are uncorrelated with stock returns. The solution is an integral of the Black-Scholes formula and the distribution function for the stock price variance. Accurate option prices can be computed via Monte Carlo simulations and applied to actual prices. The paper also presents an application of the model to Digital Equipment Corporation (DEC) options, showing that the random variance model outperforms the Black-Scholes model in explaining actual option prices. The model incorporates random variation in the volatility parameter and uses Monte Carlo simulations to compute option prices. The paper also discusses parameter estimation for the variance process, using methods such as the method of moments and Kalman filters. The results show that the random variance model provides more accurate option prices than the Black-Scholes model, especially for out-of-the-money options. The paper concludes that the random variance model is a better estimate of the underlying variance rate and that the Black-Scholes model with a single variance estimate is clearly rejected by the data.This paper examines the pricing of European call options on stocks with randomly changing variance. It studies continuous-time diffusion processes for stock returns and standard deviation, finding that a riskless hedge requires the stock and two options. However, the random standard deviation is not a traded security, so an equilibrium asset pricing model is needed to derive a unique option pricing function. The option price depends on the risk premium associated with the random standard deviation. The problem can be simplified by assuming volatility risk can be diversified and changes in volatility are uncorrelated with stock returns. The solution is an integral of the Black-Scholes formula and the distribution function for the stock price variance. Accurate option prices can be computed via Monte Carlo simulations and applied to actual prices. The paper also presents an application of the model to Digital Equipment Corporation (DEC) options, showing that the random variance model outperforms the Black-Scholes model in explaining actual option prices. The model incorporates random variation in the volatility parameter and uses Monte Carlo simulations to compute option prices. The paper also discusses parameter estimation for the variance process, using methods such as the method of moments and Kalman filters. The results show that the random variance model provides more accurate option prices than the Black-Scholes model, especially for out-of-the-money options. The paper concludes that the random variance model is a better estimate of the underlying variance rate and that the Black-Scholes model with a single variance estimate is clearly rejected by the data.