Option Pricing on Multiple Assets

Abstract

Since 1973, the Black–Scholes formula has been used in financial markets to price financial derivatives such as options. In the classical Black–Scholes model for the market, the following type of mix is assumed or postulated: in the simplest case, it consists of an essentially riskless bond and a single risky asset. Hence, certainty mixed with uncertainty: safe vs risky! Here we consider more complex products where each component in a portfolio entails several variables with constraints. This leads to elegant models based on multivariable stochastic integration, and describing several securities simultaneously [Etheridge, A Course in Financial Calculus, Cambridge University Press, UK (2002), Jiang, Mathematical Modeling and Methods of Option Pricing, Higher Education, Beijing, China (2003)] and [Broadie, Detemple, Math. Financ. 7:241–286 (1997)]. We derive a general asymptotic solution in a short time interval using the heat kernel expansion on a Riemannian metric. We then use our formula to predict the better price of options on multiple underlying assets. We then apply our method to the case known as the two-color rainbow option, i.e., the special case of the model with two underlying assets. This asymptotic solution is important, as it explains hidden effects in a class of financial models.