2
$\begingroup$

I have the following non-linear PDE and I have no idea how to go about solving it using a finite difference scheme in Python. Can someone get me started and/or point me to an algorithm for doing this? It represents the price of a derivative in the Uncertain Volatility Model (where $\sigma \in [\sigma_{low}, \sigma_{high}]$).

$$\partial_t u(t,x) + H(x, D_x^2u(t,x)) = 0$$ $(t,x) \in [0,T) \times\mathbb{R}$

where

$$H(x,\Gamma) = \frac{1}{2}x^2\Sigma(\Gamma)^2\Gamma$$ $$\Sigma(\Gamma) = \sigma_{low}1_{\Gamma < 0} + \sigma_{high}1_{\Gamma \ge 0}$$

$\endgroup$
1
$\begingroup$

I don't know of any libraries for this. There is a pretty good literature on the problem you mention though. I suggest https://cs.uwaterloo.ca/~paforsyt/numuncert.pdf as a good paper to follow; they study numerical techniques, document pitfalls, and even prove something about convergence of their preferred approach.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.