This paper examines the pricing of European call options on stocks with randomly varying variance rates. The author studies continuous-time diffusion processes for stock returns and the standard deviation parameter, finding that a riskless hedge requires one stock and two options. The riskless hedge does not yield a unique option pricing function due to the non-tradability of random standard deviation. An equilibrium asset pricing model is used to derive a unique option pricing function, which depends on the risk premium associated with the random standard deviation. The problem is simplified by assuming that volatility risk can be diversified away and that changes in volatility are uncorrelated with stock returns. The resulting solution involves an integral of the Black-Scholes formula and the distribution function for the variance of the stock price. The paper also discusses the estimation of parameters for the variance process and applies the model to actual option prices for Digital Equipment Corporation (DEC) from July 1982 to June 1983. The random variance model outperforms the Black-Scholes model with daily variance rates that change, showing a marginally better fit to actual option prices.This paper examines the pricing of European call options on stocks with randomly varying variance rates. The author studies continuous-time diffusion processes for stock returns and the standard deviation parameter, finding that a riskless hedge requires one stock and two options. The riskless hedge does not yield a unique option pricing function due to the non-tradability of random standard deviation. An equilibrium asset pricing model is used to derive a unique option pricing function, which depends on the risk premium associated with the random standard deviation. The problem is simplified by assuming that volatility risk can be diversified away and that changes in volatility are uncorrelated with stock returns. The resulting solution involves an integral of the Black-Scholes formula and the distribution function for the variance of the stock price. The paper also discusses the estimation of parameters for the variance process and applies the model to actual option prices for Digital Equipment Corporation (DEC) from July 1982 to June 1983. The random variance model outperforms the Black-Scholes model with daily variance rates that change, showing a marginally better fit to actual option prices.