Heston ADI Time-Constraints when Vol-of-Vol > 1

[rmg]

I have been working on an explicit and implicit ADI (Douglas, Craig-Sneyd) finite difference solver to the Heston model and have been able to reproduce the closed-form solutions for the cases where the vol-of-vol, is < 1. In the ADI case, I am able to gain a speed-up over the explicit solver as well as second-order convergence. However, in the case where vol-of-vol is > 1, (such as in Nimalin Moodley's thesis and in many other cases of equity options) I have to take very small time-steps for the solution to remain stable. At first, I thought it could be resolved by using upwind differencing in the volatility dimension, but even when this is implemented, both my ADI and explicit solutions are unstable for larger delta-t.

I'm assuming that there is a Von-Neuman-like stability condition related to the coefficient of the vol-diffusion term that I'm missing; though I have been unable to derive the condition.

Has anyone else run into this? Any ideas would be greatly appreciated!

 

[Daniel Duffy]

I have been working on an explicit and implicit ADI (Douglas, Craig-Sneyd) finite difference solver to the Heston model and have been able to reproduce the closed-form solutions for the cases where the vol-of-vol, is < 1. In the ADI case, I am able to gain a speed-up over the explicit solver as well as second-order convergence. However, in the case where vol-of-vol is > 1, (such as in Nimalin Moodley's thesis and in many other cases of equity options) I have to take very small time-steps for the solution to remain stable. At first, I thought it could be resolved by using upwind differencing in the volatility dimension, but even when this is implemented, both my ADI and explicit solutions are unstable for larger delta-t.

I'm assuming that there is a Von-Neuman-like stability condition related to the coefficient of the vol-diffusion term that I'm missing; though I have been unable to derive the condition.

Has anyone else run into this? Any ideas would be greatly appreciated!

Is your scheme monotone? That's the acid test.

1. Does it still have problems with rh0 = 0?
2. Exponential fitting (see thesis by Roelof Sheppard).

BTW ADI is a bit long in the tooth.

 

[rmg]

Thanks for your response! I will take a look at the questions you asked and follow up with you. The scheme is monotone. Thanks for your note on ADI- if you have recommendations for a better method, I'd love to try to implement something new.

 

[Daniel Duffy]

The scheme is monotone.

Sure?

 

[rmg]

Thanks for your recommendations and insights. I ended up trying the same Heston parameters in Rouah & Heston's code from their excellent book on the topic, and it too crashed unless I took a large enough number of time steps (small enough dt). I knew there would be a stability condition determined by the equations, but now realize that when vol-of-vol > 1 (as is the case in Moodley's thesis and many equity options) it further restricts the minimum number of time steps required to keep the FD solution stable. When dt is sufficiently small for the code to converge and remain stable, the solutions are the same as Rouah's solvers.

Moral of the story: if your code blows up trying to replicate Moodley's thesis or a Heston model with sigma/xi/vol-of-vol > 1; test with small dt first. I took 3,000-10,000 steps. Hope that helps anyone who might run into the same problems.