An Alternative Model for Contingent Claims

The fundamental valuation equation of Cox, Ingersoll and Ross was expressed in terms of the indirect utility of wealth function. As closed-form solution for the indirect utility is generally unobtainable when investment opportunities are stochastic, existing contingent claims models involving general asset price processes were almost all derived under the restrictive log utility assumption. An alternative valuation equation is proposed here that depends only on the direct utility function. This alternative valuation model is applied to derive closed-form solutions for bonds, bond options, individual stocks, and stock options under both power utility and exponential utility functions. Allowable processes for aggregate output, firms' dividends, and state variables are quite general and empirically plausible. The resulting interest rates and stock price dynamics have many empirically plausible properties. Our option pricing model with stochastic volatility and stochastic interest rates has most existing models nested in it. This means most existing models also hold for economies with power or exponential utility functions. The option pricing model is also shown to have the ability to reconcile certain puzzling empirical regularities such as the volatility smile.