We present in this chapter several extensions to the asymptotic approach developed in the previous chapters. In Section 9.1, we extend the results of Chapter 4 to the case where the short rate is also varying, driven by the same factors driving the volatility, and the stock is paying dividends. In Sections 9.3 and 9.4, we derive the second-order corrections generated by the fast and the slow factors. These second terms produce the smile of implied volatilities as illustrated with the Heston model in Chapter 10. In Section 9.5, we show that a periodic daily component can easily be incorporated in the model without additional difficulty in the asymptotic analysis. In Section 9.6, we introduce jumps in the fast volatility factor. We indicate how to generalize the perturbation method to multidimensional models in Section 9.7.

Dividends and Varying Interest Rates

In this section, we present generalizations of the first-order approximation to the cases where dividends are paid and/or where the interest rate is varying as a function of the factors (Y, Z). We also present the probabilistic representations of these approximations.

Dividends

In this section we indicate how the perturbation theory explained above can easily be modified to incorporate dividend modeling into the stock price model. For simplicity we consider a continuous dividend yield D, which is the fraction of the stock price received by a stockholder per unit of time.