ABSTRACT
Abstract In this chapter we pursue the application of the heat kernel expansion for stochastic volatility models (SVM). In our geometric framework, a SVM corresponds to a complex curve, also called Riemann surfaces. By using the classification of conformal metrics on a Riemann surface, we show that SVMs fall into two classes. In particular, the SABR model corresponds to the Poincare´ hyperbolic surface. We derive the first-order asymptotics for implied volatility for any timehomogeneous SVM. This general formula, particularly useful for calibration purposes, reproduces and improves the well-known asymptotic implied volatility in the case of the SABR model. This expression only depends on the geometric objects (metric, connection) characterizing a specific SVM. We apply this formula to the SABR model with a mean-reverting drift and the Heston model. Finally, in order to show the strength of our geometrical framework, we give an exact solution to the Kolmogorov equation for the normal (resp. log-normal) SABR model. The solutions are connected to the Laplacian heat kernel on the two-dimensional (resp. three-dimensional) hyperbolic surface. Deterministic interest rates will be assumed throughout this chapter.
