Valuing Asian and portfolio options by conditioning on the geometric mean price

The valuation of Asian, or average price, options and of European options on portfolios in a ‘Black-Scholes’ environment has given researchers trouble. The difficulty with these problems is that the probability distribution of the variable which determines the option payoff at expiration, a sum of correlated lognormal random variables, has no closed-form representation. For the Asian case the approach generally taken has been to approximate the distribution of the arithmetic average price, while for the portfolio option case, attempts have focused on discretizing the joint distribution of the terminal prices of the assets comprising the portfolio and approximating the expected risk-neutral option payoff with a discrete sum. These approaches are not entirely satisfactory. The distribution-approximating procedures for Asian options are not very accurate for some cases, while the computational requirements for obtaining a reasonably accurate estimate using the discretizing or multinomial approaches for portfolio options become excessive as the number of assets rises above four or five, because the computation time is exponential in the number of assets. This paper presents a method based on conditioning on the geometric mean price which results in a far more efficient technique for valuing these options.