How to do QE scheme for n correlated assets?

Question

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 1

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.