4 - First-Order Perturbation Theory

Summary

In this chapter, we present the multiscale perturbation analysis in the case of European options. We take as a starting point the class of multiscale stochastic volatility models introduced in Chapter 3. Recall that these models have two stochastic volatility factors, one fast and one slow. In the Markovian framework that we consider here, option prices are obtained as solutions of partial differential equations. The two time scales translate into two small parameters in these equations and we use a combination of singular and regular perturbation techniques to derive approximations for the option prices. Here we consider only the first-order corrections, while the second-order corrections will be derived in Chapter 9. The approximations we derive for European call options are crucial for the calibration procedure that we discuss in the next chapter. There, we use the implied volatility skew to calibrate the few universal parameters which arise in our first-order perturbation theory. These are the only parameters needed for computing approximations of prices of more complicated path-dependent derivatives discussed in detail in the later chapters. The perturbation theory presented here in the context of equity markets will also be the basis for our analysis of the fixed income (Chapter 12) and credit markets (Chapters 13–14).

Option Pricing under Multiscale Stochastic Volatility

In this section, we recall the multiscale stochastic volatility models introduced in the previous chapters.