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 intoaccount 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 1.3, we briefly discuss the Merton, Heston and Bates models concentrating on aspects relevant for the option pricing method. In the followingsection, we present the method of Carr and Madan which is based on the fastFourier transform (FFT) and can be applied to a variety of models. We also consider briefly some further developments and give a short introduction to theFFT algorithm. In the last section, we apply the method to the three analyzed models, check the results by Monte Carlo simulations and comment on somenumerical issues.
