Divisions of Research & Statistics and Monetary Affairs Federal Reserve Board , Washington , D . C . Expectations of functions of stochastic time with application to credit risk modeling

We develop two novel approaches to solving for the Laplace transform of a time-changed stochastic process. We discard the standard assumption that the background process (Xt) is Lévy. Maintaining the assumption that the business clock (Tt) and the background process are independent, we develop two different series solutions for the Laplace transform of the time-changed process X̃t = X(Tt). In fact, our methods apply not only to Laplace transforms, but more generically to expectations of smooth functions of random time. We apply the methods to introduce stochastic time change to the standard class of default intensity models of credit risk, and show that stochastic time-change has a very large effect on the pricing of deep out-of-the-money options on credit default swaps. JEL Codes: G12, G13