Direct integration techniques have often been limited to the valuation of vanilla options, but their efficiency makes them particularly suitable for calibration purposes.
A large part of state of the art numerical integration techniques relies on a transformation to the Fourier domain, the probability density function f(y|x) appears in the integrand in the original pricing domain (for example the price or the log-price), but is not known analytically for many important pricing processes. The characteristic functions of these processes, on the other hand, can often be expressed analytically, where the characteristic function of a real valued random variable X is the Fourier transform of its distribution.
The probability density function and its corresponding characteristic function thus form a Fourier pair,
Many probabilistic properties of random variables correspond to analytical properties of their characteristic functions, making them a very useful concept for studying random variables.
The characteristic function of the Heston model is given by
In the next blog post on friday we will discuss in more detail, how the Heston parameters effect the form and properties of its characteristic functions.