Stochastic Process Equations of the Heston Model - 2023.2 English

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2023.2 English

The continuous form of Heston Model is:

\[\mathrm{d} S_t = \mu S_t \mathrm{d}t + \sqrt[2]{\nu} S_t \mathrm{d}W_t^S\]
\[\mathrm{d} \nu_t = \kappa (\theta - \nu_t) \mathrm{d}t + \sigma \sqrt[2]{\nu_t} \mathrm{d}W_t^\nu\]
\[Corr(W_t^S, W_t^\nu)) = \rho\]

Where \(S_t\) is random variable, represent a stock price at time \(t\). \(\mu\) is the stock’s expected rate of return. \(\sqrt[2]{\nu_t}\) is the volatility of stock price and \(\nu_t\) is also a random variable. \(\theta\) is the long term variance. \(\kappa\) is the rate at which \(\nu_t\) revert to \(\theta\). \(\sigma\) is the volatility of the volatility. \(\mathrm{d}W_t^S\) and \(\mathrm{d}W_t^\nu\) are Wiener processes with correlation \(\rho\).