A type of stochastic volatility model developed by associate finance professor Steven Heston in 1993 for analyzing bond and currency options. The Heston model is a closed-form solution for pricing options that seeks to overcome the shortcomings in the Black-Scholes option pricing model related to return skewness and strike-price bias.
The Heston model assumes that the underlying stock price,$S_t$ follows a Geometric Brownian Motion process but with a stochastic variance $v_t$ that follows a Cox, Ingersoll, and Ross (1985) process. Hence, the Heston model is represented by the bivariate system of stochastic differential equations $$ dS_t=\mu S_t dt+\sqrt{v_t}S_t dW_1(t)$$ $$dv_t=\kappa(\theta-v_t)dt+\sigma\sqrt{v_t}\,dW_2(t)$$ where $\mathbb{E}^P=[dW_1(t),dW_2(t)]=\rho dt$.The parameters of the model are
- $\mu$ the drift of the process for the stock.
- $\kappa>0$ the mean reversion speed for the variance.
- $\theta > 0$ the mean reversion level for the variance.
- $\sigma > 0$ the volatility of the variance.
- $\rho \in [−1, 1]$ the correlation between the two Brownian motions $W_1$ and $W_2$.
If the parameters obey the following condition (known as the Feller condition) then the process $v_t$ is strictly positive.
$$2\kappa\theta\geq\sigma^2$$
