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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

  1. $\mu$ the drift of the process for the stock.
  2. $\kappa>0$ the mean reversion speed for the variance.
  3. $\theta > 0$ the mean reversion level for the variance.
  4. $\sigma > 0$ the volatility of the variance.
  5. $\rho \in [−1, 1]$ 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.

$$2\kappa\theta\geq\sigma^2$$

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