Chapter 8 conducts a model calibration and market-based valuation with the jump-diffusion model of Merton (1976). The calibration effort reveals that a jump component alone is not capable of replicating a typical volatility surface. It is rather necessary to include at least a stochastic volatility component (as already pointed out in Chapter 3). In addition, we also need a stochastic short rate component to accommodate stylized facts of interest rate markets.
This chapter therefore introduces in section 9.2 the model framework of Bakshi, Cao and Chen (1997, BCC97, Bakshi et al. (1997)) that includes as special cases a number of popular financial models, like the Black-Scholes-Merton model. Section 9.3 briefly recaps the main statistical features a realistic market model should exhibit. That section also cites a number of empirical findings regarding the performance of the framework under different parametrizations. Section 9.5 then concerns itself with the valuation of European options in the general framework—a necessary prerequisite for an efficient calibration procedure.
9.2 The Framework
Given is a filtered probability space representing uncertainty in the model economy with final date T where 0 < T < ∞. Ω denotes the continuous ...