Finite-Difference Methods for the Uncertain Volatility PDE

Summary

The document frames numerical solution of a nonlinear pricing PDE for a derivative under the uncertain volatility model. Volatility is selected from a lower or upper bound according to the sign of the solution’s second spatial derivative, so the PDE is nonlinear even though the volatility bounds are specified. The question asks how to implement a finite-difference scheme in Python.

The response offers no step-by-step discretization or code. Instead, it points to academic literature on numerical methods for this problem, describing the cited paper as covering techniques, implementation pitfalls, and convergence analysis for a preferred approach. This is a useful direction for further study, but the document itself does not identify the scheme, boundary conditions, time-stepping choices, or parameter requirements. Its practical guidance is therefore limited to locating a reference rather than providing an immediately implementable algorithm.

Key ideas

• The uncertain volatility model chooses volatility based on the sign of the option value’s curvature.
• This curvature-dependent choice makes the derivative pricing PDE nonlinear.
• Finite differences are posed as a possible numerical solution method.
• The response directs readers to research on numerical techniques, pitfalls, and convergence but gives no implementation details.

Tags

# Solving a Non-Linear PDE using a Finite Difference Scheme


# Solving a Non-Linear PDE using a Finite Difference Scheme












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

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

## Answer by q.t.f. (score 1)

https://quant.stackexchange.com/a/17675

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.

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