This chapter takes a hands-on approach and dives into a market-based valuation without paying too much attention to the theoretical and numerical foundations. It addresses all main steps of such a valuation: market modeling, European call valuation via Fourier techniques, calibration of a market model to European call option quotes and simulation of the calibrated model.
The exposition might seem a bit bumpy. However, all aspects are addressed and are made somewhat more precise later in this part of the book. Those with some background knowledge will find in this first example and the accompanying Python scripts a kind of sandbox in which first steps in other directions can be taken.
Section 8.2 introduces the market model. Section 8.3 addresses valuation via Fourier-based approaches. Section 8.4 calibrates the model to real market data. Finally, section 8.5 simulates the calibrated model and values a European call option by simulation.
8.2 Market Model
We consider the jump-diffusion model of Merton (cf. Merton (1976), M76) as already sketched out in section 6.6. The plan is to completely implement this specific model numerically and technically. The time horizon T is fixed, 0 < T < ∞. In this continuous market model, the index level has risk-neutral dynamics of the form
The variables and parameters ...