Mean-Variance Hedging for Pricing European Options Under Assumption of Non-continuous Trading

TL;DR

This paper develops a mean-variance hedging approach for European options assuming non-continuous trading, deriving explicit prices and demonstrating risk reduction through independent portfolios with real stock data.

Contribution

It introduces a novel mean-variance hedging method under non-continuous trading assumptions and derives explicit lookback option prices, contrasting with Black-Scholes.

Findings

Explicit formula for lookback call option price.

Portfolio risk asymptotically approaches zero with independent trading.

Empirical validation using Australian stock data.

Abstract

We consider a portfolio with call option and the corresponding underlying asset under the standard assumption that stock-market price represents a random variable with lognormal distribution. Minimizing the variance (hedging risk) of the portfolio on the date of maturity of the call option we find a fraction of the asset per unit call option. As a direct consequence we derive the statistically fair lookback call option price in explicit form. In contrast to the famous Black-Scholes theory, any portfolio can not be regarded as risk-free because no additional transactions are supposed to be conducted over the life of the contract, but the sequence of independent portfolios will reduce risk to zero asymptotically. This property is illustrated in the experimental section using a dataset of daily stock prices of 18 leading Australian companies for the period of 3 years.