Author(s): Boyarchenko SI
By now, the drawbacks of the Gaussian modelling in Financial Markets and Investment under Uncertainty are well-known. In particular, Gaussian models cannot produce so-called fat tails of observed probability densities, which leads to under-pricing of financial risks. One can hardly make a mistake by saying that the under-pricing of the risk was the main reason for the Long Term Capital Management disaster or recent failures of rating agencies to warn investors of a series of the defaults of the investment- graded firms. The purpose of the book is to introduce an analytically tractable and computationally effective class of non-Gaussian models for shocks (Regular Levy Processes of Exponential type (RLPE)), and related analytical meth- ods similar to the initial Merton-Black-Scholes approach, which we call the Merton-Black-Scholes theory (MBS-theory). The potential range of appli- cations of the non-Gaussian variant of the MBS-theory is huge, and the list of results we have obtained so far does not exhaust all the possibilities. As applications to Financial Mathematics, we solve pricing problems for several types of perpetual American options, barrier options, touch-and-out options and some other options, provide analogues of several approximate methods for pricing of American options in the finite horizon case, and deduce explicit analytical formulas for the locally risk-minimizing hedging. We suggest fast computational procedures for pricing of European options; they can be used for hedging and pricing of American and barrier options as well.