Simulating Heston Volatility with a Drift-Adjusted Correlated Brownian Motion
Summary
The document addresses how to simulate the variance process in a Heston-style model when its correlated Brownian driver includes a drift adjustment linked to the wealth process. The proposed approach is to substitute the adjusted Brownian differential into the correlation relation, then substitute that relation into the variance equation. This expresses the variance dynamics using the underlying Brownian increments and the current variance, making them usable in a simulation step.
The question gives the model equations and identifies the apparent circularity between wealth and volatility. The answer sketches an algebraic substitution rather than presenting a discretization scheme, simulation results, or a validation. It therefore offers a way to rewrite the dynamics, while leaving numerical choices and checks to the practitioner.
Key ideas
- The variance driver is correlated with a Brownian motion that includes a variance-dependent drift term.
- Substituting the adjusted Brownian increment into the correlation equation removes the apparent circular definition.
- The resulting variance dynamics can be written in terms of the current variance and the original Brownian increments.
- The answer does not specify a numerical discretization or test the simulation.
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# Simulation Heston Model, markovianity
# Simulation Heston Model, markovianity
I am trying to simulate the instanteneous volatility of a Heston process.
My equations are the following :
wealth process: $$dX_t = r_t X_t + \theta \sqrt {V_t} u_t dt + u_t dW_{1t}$$
Volatility: $$dV_t = (\kappa \phi - \lambda V_t) dt + \sigma \sqrt {V_t} dB_t $$
With, I start my simulations with a 2D brownian motion : $(W_1, W_2)$ and another "corrolated" Brownian motion $B_t = \rho d \tilde{W}_{1t} + \sqrt{1- \rho^2} dW_{2t} $
My problem lies in the $d \widetilde{W}_{1t}$. Its definition is :
$$ \widetilde{W}_{1t} = W_{1t} + 2 \theta \int_0^t \sqrt {V_s} ds $$.
So I know how to simulate the wealth process, it s a classical "flow".
The volatility follows the same pattern, iff the brownian motion $dB_t$ is a classical one. Here there is a drift movement which makes the whole simulation cyclic. I have no idea how to deal with it.
- Is it possible to simulate that ? Is my problem markovian ?
- How would one deal with that problem. I simply need a solution for $\widetilde{W}_{1t} $, I'll deal with the rest.
Thank you
## Answer by Valometrics.com (score 1, accepted)
https://quant.stackexchange.com/a/50770
You should replace the differential of the correlated process dBt with its value in the volatility equation, then replace dW~t in the same equation with: dW~t=dWt+ 2*theta*square_root(Vt)*dt you will get an formula with Vt,W1t and W2t. You can then simulate the volatility.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.