Discrete Hedging in the Mean/Variance Model for European Call Options

Abstract

We consider a portfolio with the call option and the relevant asset under the standard assumption that the market price is a random variable with a lognormal distribution. Minimizing the variance (hedging risk) of the portfolio on the maturity date of the option, we find the relative value of the asset per option unit. As a direct consequence, we obtain a statistically fair price of the call option explicitly. Unlike the well-known Black–Scholes theory, the portfolio cannot be risk-free, because no additional transactions within the contract are allowed, but the sequence of portfolios reduces the risk to zero asymptotically. This property is illustrated in the experimental section on the example of the daily stock prices of 18 leading Australian companies over a three year period.