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.

  1. Is it possible to simulate that ? Is my problem markovian ?
  2. How would one deal with that problem. I simply need a solution for $\widetilde{W}_{1t} $, I'll deal with the rest.

Thank you

  • $\begingroup$ There are several undefined processes, e.g. $r_t$, $u_t$, and possibly also $\phi_t$. Could you specify which are parameters and otherwise the definitions of the other processes. $\endgroup$ – oliversm Jan 20 '20 at 11:49
  • $\begingroup$ @oliversm thanks for the comment. The parameters are constant, $ \phi_t = \phi $. What for $r_t, u_t$, I could define them ( $r_t$ is the interest rate, can be considered constant, $u_t$ would take an equation in order to be defined), however, my main problem was with $ \widetilde{W}_{1t} $ and those parameters do not intervene within the equation. Why do you need more information about it ? Knowing why you need that information would help me define and give you additional information that you'd need $\endgroup$ – Marine Galantin Jan 20 '20 at 13:36

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.

  • $\begingroup$ so you mean that $$ d \widetilde{W}_{1t} = d W_{1t} + 2 \theta \sqrt{V_t} dt $$ ? $\endgroup$ – Marine Galantin Jan 20 '20 at 21:40
  • $\begingroup$ square_root of Vt $\endgroup$ – Valometrics.com Jan 20 '20 at 21:45
  • $\begingroup$ mhm I believe this should indeed work, so my question was kinda easy $\endgroup$ – Marine Galantin Jan 20 '20 at 21:46
  • $\begingroup$ not easy at all! 3 days without answer! $\endgroup$ – Valometrics.com Jan 20 '20 at 21:48
  • $\begingroup$ you're right @user1987 but I should have thought about it myself ! Now that you say it, it is obvious that the answer you gave is the right one. I believe I thought that the expression is in fact : $$ d\widetilde{W}_{1t} = d W_{1t} + 2 \theta \int_0^t \sqrt {V_s} ds $$ This is what I must have read in my head. But you're right, you can apply the transformation you did. Thank you so much. $\endgroup$ – Marine Galantin Jan 20 '20 at 21:55

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