《Fourier based valuation methods》.pdf

Chapter 1 Fourier based valuation methods in mathematical finance Ernst Eberlein 1.1 Introduction A fundamental problem of mathematical finance is the explicit computation of ex- pectations which arise as prices of derivatives. What leads to simple formulas in the classical setting when the underlying random quantity is modeled by a geometric Brownian motion, turns out to be rather nontrivial in more sophisticated modeling approaches. There is overwhelming statistical evidence that Brownian motion as the driver of models in equity, fixed income, credit and foreign exchange markets produces distributions which are far from reality and can be considered as first ap- ´ proximations at best. Levy processes are a much more flexible class of drivers. They can be parametrized with a low-dimensional set and at the same time generate dis- ´ tributions which are more realistic from a statistical point of view. However in Levy models simple closed-form valuation formulas are typically not available even in the case of plain vanilla European options. The situation is worse for more complicated exotic options. Efficient methods to compute prices of derivatives are crucial in particular for calibration purposes. During a calibration procedure in each iteration step typi- cally a large number of model prices has to be computed and compared to mar- ket prices. Models which cannot be calibrated within reasonable time limits are useless for most applications. A method which almost always works to get expec- tations is Monte Carlo simulation. Its disadvantage is that it is computer intensive and therefore too slow for many purposes. Another classical approach is to repre- sent prices as solutions of partial differential equations (PDEs) which in the case of ´ Levy processes with jumps become partial integro