Heston Model

In the Heston Model, the volatility of the asset is a sotchastic process. The price of the asset is given by

$$ dS_t = \mu S_t dt + \sqrt{V_t} S_t dW_t^S, $$

$$ dV_t = \alpha(b-V_t)dt + \sigma \sqrt{V_t} dW_t^V, $$

and the correlation between the two Brownian motions is given by

$$ dW_t^S dW_t^V = \rho dt. $$

Using the Feynman-Kac formula we can calculate the option price as an expectation value $$c_t = \mathbf{E}[e^{-r(T-t)} {\rm max}(0,S_T-K) \, \, | \, \, S_t] $$ $$p_t = \mathbf{E}[e^{-r(T-t)} {\rm max}(0,K-S_T) \, \, | \, \,S_t ] $$ where $c_t$ ($p_t$) corresponds to a call (put) option, $S_T$ is the stock price at maturity and $K$ is the strike price.

The Cox-Ingersoll-Ross model for interest rates has the same equation as the volatility in this model. The former is used when interest rates are expected to remain always positive

$$ dr = \alpha(b-r)dt + \sigma \sqrt{r} dW_t. $$

The expectation and variance for the interest rate $r(t)$ correspond to

$$ \mathbf{E}[r(t)] = e^{-\alpha t} r_0 + b(1-e^{-\alpha t}) $$

$$ {\rm Var}[r(t)] = \frac{\sigma^2}{\alpha} r_0 (e^{-\alpha t} - e^{-2\alpha t}) + \frac{b\sigma^2}{2\alpha} (1- e^{-\alpha t})^2 $$