I am trying to simulate the volatility process in the Heston model using the relation between the CIR Process and Ornstein–Uhlenbeck processes. In fact, giving $\mathbf{X}$ a $n$-dimensional vector valued OU process with \begin{equation} \mathrm{d}X_t^i = \alpha X_t^i \mathrm{d}t + \beta \mathrm{d}W_t^i, \end{equation} where $\mathbf{W}$ is a $n$-dimensional vector of independent Brownian motions.

Then, we know that the process \begin{equation} Y_t = \sum_{i = 1}^n \left( X_t^i \right)^2. \end{equation}

is a CIR process, such that

\begin{eqnarray} \mathrm{d}Y_t & = & \left( 2 \alpha Y_t + n \beta^2 \right) \mathrm{d}t + 2 \beta \sqrt{Y_t} \mathrm{d}\widetilde{W}_t\\ & = & \kappa \left( \theta - Y_t \right) \mathrm{d}t + \xi \sqrt{Y_t} \mathrm{d} \widetilde{W}_t, \end{eqnarray}

where $\kappa = -2 \alpha$, $\theta = -n \beta^2 / 2 \alpha$, $\xi = 2 \beta$ and \begin{equation} \widetilde{W}_t = \int_0^t \frac{1}{\sqrt{Y_u}} \sum_{i = 1}^n X_u^i \mathrm{d}W_u^i. \end{equation}

For n=1, the simulation is clear since basically in that case $\widetilde{W}_t=W_t$. However, it is not clear to me how to do the simulation for $n>1$ ($n$ integer) since, for the Heston model, I also need to construct the noise driving the asset price which is needed to be correlated to $\widetilde{W}_t$?! I know that I can just simulate $\widetilde{W}_t$ from its expression above but I thought there maybe a more efficient way than that! Any guidance for this issue? Many thanks!


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.