The Heston Model

For the analysis of many exotic financial derivatives, the Heston model, a stochastic volatility model, is widely used. Its specific parameters have to be identified from sets of options market data with different strike prices and maturities, leading to a minimization problem for the least square error between the model prices and the market prices. It is intrinsic to the Heston  model that this error functional typically exhibits a large number of local minima, therefore techniques from global optimization have to be applied or combined with local optimisation techniques to deliver a trustworthy optimum. 

In our discussion of models the Heston model will get a prominent place and we will examine its features in detail. We start with the model equations will discuss the properties and will also show some examples how reliable model parameters can be obtained.

The Heston stochastic volatility model [1] relaxes the constant volatility assumption in the classical Black Scholes model by incorporating an instantaneous short term variance process (CIR)

where r denotes the domestic yield curve, v(t) denotes the stock price variance and dW's are standard Brownian motions with correlation ρ, κ is the reversion speed (represents the degree of volatility clustering), Θ is the long term level of the variance process and σ is called the volatility of volatility (although technically a volatility of variance).
The variance process is always positive if

 (Feller condition).

[1] S. Heston: A closed-form solution for options with stochastic volatility with applications to bond and currency options, Review of Financial Studies 1993,6, 327-343.