Simulating Correlated Brownian Drivers in a Multi-Asset Heston Model
Summary
The document addresses how to extend a single-asset Heston quadratic-exponential simulation to multiple correlated assets. Its central step is to generate independent standard normal draws at each time step, then transform them using a matrix square root of the instantaneous correlation matrix. Cholesky decomposition supplies that transformation when the matrix is suitable, and a constant correlation matrix allows the factorization to be reused.
This explains how to correlate Gaussian Brownian increments, but does not provide a complete multi-asset QE algorithm or resolve how to handle each asset’s correlation between its price and variance processes. The correlation matrix itself must be specified or estimated separately; the answer mentions historical estimates and calibration to correlation-sensitive instruments as possibilities. The method assumes a valid correlation matrix and does not discuss numerical treatment of near-singular or time-varying cases.
Key ideas
- Generate independent Gaussian draws for the Brownian drivers at each simulation step.
- Factor the instantaneous correlation matrix, for example with Cholesky decomposition.
- Multiply the independent draws by the factor to obtain correlated Gaussian increments.
- Correlation estimation or calibration is a separate modeling decision from the simulation transformation.
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# How to do QE scheme for n correlated assets?
# How to do QE scheme for n correlated assets?
I'm trying to simulate correlated assets under Heston model. I coded the QE scheme for a single asset but i dont understand the next step: How should i set the correlation matrix given my n-asset independent stock and variance paths ? Should i generate those paths first given my asset-variance correlation, then decorrelate my subsystem (i mean from the same asset-variance system) or what ? And if so, how ? My question is really what's the first step ? From what i read in the papers below ill have to then calibrate my asset-asset correlation given the historical one but that's an another issue. I read wadman, dimitroff and De Innocentis but I'm still struggling. Help would be appreciated. Many thanks
Wadman, 2010: an advanced Monte Carlo for the multi asset Heston model
Dimitroff: a parsimonious multi asset Heston model: calibration and derivative pricing
De Innocentis: Efficient simulation of the multi asset Heston model
## Answer by byouness (score 1)
https://quant.stackexchange.com/a/39826
I am not familiar with the QE scheme, but I think your question is more general: You want to do a multi-variate diffusion, for $n$ correlated processes.
You have your instantaneous correlations matrix $R = (\rho_{i,j})_{i,j}$ where $d \langle W^i, W^j \rangle_t = \rho_{i,j} dt$, and I am assuming here you know how to simulate brownian increments for a single asset.
At each time step $t$, simulating $n$ correlated brownian increments boils down to simulating $n$ correlation gaussian variables $Z = (z_i)_i$. To achieve this, you need to:
- Simulate n uncorrelated gaussian variables $\bar{Z} = (\bar{z_i})_i$;
- Apply Cholesky decomposition to your correlation matrix $R$ in order to get its so-called square root $L$ (if the correlation is constant then you will do this step only once).
- Multiply the obtained matrix by the vector of the uncorrelated gaussian to get your correlated gaussians: $L \bar{Z} = Z$.
As you said in the question, the correlation calculation or calibration is another issue. Depending on the context, it might be better to use the historical correlation or to calibrate it from correlation swaps, or other financial products.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.