Pricing outside the standard Black and Scholes model
The standard Black and Scholes model relies on several assumptions which allow for explicit formulas and easy calculations in most cases. Unfortunately some of these hypotheses like the constant volatility, Gaussianity of the returns and the continuity of the paths of the geometric Brownian motion process are unlikely to hold for many observed prices. Indeed, prices often show jumps, changes in volatility (see e.g. Section 6.6.1) and the distribution of the returns is usually skewed and with high tails (see e.g. Section 5.4). Most of these stylized facts are indeed captured by Lévy processes as we have seen. In this chapter we mainly consider the problem of pricing under the assumption that the dynamic of the financial prices includes some kind of jump process and/or non-Gaussian behaviour. We start with the simpler and most studied Lévy process.
8.1 The Lévy Market Model
Consider again the simple exponential Lévy model analyzed in Sections 4.5.5 and 5.4
where Zt is a Lévy process with triplet (b, c, ν) and canonical decomposition
such that . We have seen that it is possible to estimate the infinitely distribution ...