Numerical Methods for Pricing and Calibration
The models presented in this book have the advantage that they have the flexibility to provide prices for exotic options that are more consistent with the market. The downside is that the extra complexity means that closed-form solutions do not exist, in all but a few simple examples, and therefore we are forced to resort to numerical methods for pricing options with these models. This is an extensive subject, and in this chapter we can only scratch the surface. Further references are given for those who wish to follow up in more detail, but this is a practical subject and the best way to learn is to implement these techniques in code.
This chapter covers numerical techniques used for both calibration and pricing. Though it may seem counterintuitive we shall (a) start with simple numerical optimisation schemes that we find useful for calibration, then (b) consider pricing using techniques such as Monte Carlo and numerical methods for partial differential equations and, finally, (c) return to calibration, but this time considering the sometimes difficult issue of the calibration of the LSV model using numerical methods for the forward Fokker–Planck equation.
7.1 ONE-DIMENSIONAL ROOT FINDING – IMPLIED VOLATILITY CALCULATION
The simplest numerical method to begin with is an implied volatility calculator – a routine for a Black–Scholes model with known domestic and foreign interest rates rd and rf and initial spot S0. The requirement ...