The Algebra of Option Pricing: Theory and Application

Abstract

We consider a financial market with a risk-free money market account and a finite number of risky assets. We assume that the real-world prices of the risky assets are multivariate log-normally distributed with a non-singular valid correlation matrix. The mean and standard deviation per time unit of the return differences on these risky assets, as well as the volatility of the return, are chosen as (possibly time dependent) parameters. The obtained representation includes two of the most popular return models, namely the Black-Scholes and the Vasicek (Ornstein-Uhlenbeck) models. Solving the required martingale conditions, we construct an explicit state-price deflator, called Black-Scholes-Vasicek (BSV) deflator. Based on two multivariate normal integral identities of independent interest, we use the BSV deflator to derive in a unified and elementary probabilistic algebraic manner the price of several multiple risk options. We obtain general analytical pricing formulas for the multivariate maximum and minimum options. These are used to price related options including the best of assets or cash, the rainbow call on the maximum option, and the multivariate maximum spread option. Alternative formulas for some double triggered options, which are of importance in business applications, are also presented.