Adding and Subtracting Black-Scholes: A New Approach to Approximating Derivative Prices in Continuous Time Models

This paper develops a new systematic approach to implement approximate solutions to asset pricing models within multi-factor diffusion environments. For any model lacking a closed-form solution, we provide a solution obtained by expanding the analytically intractable model around a known auxiliary pricing function. We derive power series expansions, which provide increasingly improved refinements to the initial mispricing arising from the use of the auxiliary model. In practice, the expansions can be truncated to include only a few terms to generate extremely accurate approximations. We illustrate our methodology in a variety of contexts, including option pricing with stochastic volatility, volatility contracts and the term-structure of interest rates.