8.1 Introduction
The Black-Scholes formula, one of
the major breakthroughs of modern finance, allows for an easy and
fast computation of option prices. But some of its assumptions,
like constant volatility or log-normal distribution of asset
prices, do not find justification in the markets.
More complex models, which take into account the empirical facts,
often lead to more computations and this time burden can become
a severe problem when
computation of many option prices is required, e.g. in calibration of the implied
volatility surface. To overcome this problem
Carr and Madan (1999) developed a fast method to compute option
prices for a whole range of strikes. This method and its
application are the theme of this
chapter.
In Section 8.2, we briefly discuss the Merton,
Heston, and Bates models concentrating on aspects relevant
for the option pricing method. In the following section, we
present the
method of Carr and Madan which is based on the fast Fourier transform (FFT) and can be applied to
a variety of models. We also consider briefly some further developments
and give a short introduction to the FFT algorithm. In the last
section, we apply the method to the three analyzed models, check the results by Monte Carlo simulations and comment on some numerical issues.