Representing a CIR Variance Process with Squared Ornstein–Uhlenbeck Processes

Summary

The document describes a representation of the Cox–Ingersoll–Ross (CIR) process as the sum of squares of independent Ornstein–Uhlenbeck processes. It derives how the drift and diffusion parameters of the resulting process relate to those of the underlying OU processes, and identifies the Brownian motion that drives the aggregate CIR process.

The practical question is how to simulate this construction for the Heston model when the sum includes multiple OU processes. In particular, the author needs the variance-driving noise to have a specified correlation with the asset-price noise, and asks whether there is a more efficient construction than computing the aggregate Brownian motion from its defining expression. The document supplies the representation and parameter mapping but no answer, simulation procedure, or performance comparison. It therefore raises a useful modeling question rather than establishing that the construction is more efficient or suitable for Heston simulation.

Key ideas

• A sum of squares of independent OU processes can be represented as a CIR process under the stated parameter mapping.
• The CIR-driving Brownian motion is formed from the OU processes and their underlying Brownian motions.
• In Heston simulation, the asset-price noise must also be coupled to the variance-driving noise with the desired correlation.
• The document asks how to construct that correlated noise efficiently for a sum involving multiple OU processes but does not resolve the question.

Tags

# 46212


# Simulating volatility process in the Heston model using the relation between the CIR Process and Ornstein–Uhlenbeck processes












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!

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