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### Heston Model cont'd - Characteristic Functions

We start with a short survey of methods for option pricing. The conditional expectation of the value of a contract payoff function under the risk neutral measure can be linked to the solution of a partial (integro-) differential equation (PIDE). This PIDE can then be solved using discretisation schemes, such as Finite Differences (FD) and Finite Elements (FEM), or by Wavelet-based methods, together with appropriate boundary and terminal conditions. A direct discretization of the underly- ing stochastic differential equation, on the other hand, leads to (Quasi)Monte Carlo (QMC) methods. Both groups of numerical techniques – discretization of the P(I)DE as well as dis- cretisation of the SDE – are well known and widely used in quantitative finance. A third group of methods, directly applies numerical integration techniques to the the risk neutral valuation formula for European options.

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

where

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.