# A model-free backward and forward nonlinear PDEs for implied volatility

## TL;DR

This paper introduces new nonlinear PDEs for modeling implied volatility of convex payoff contingent claims, providing a model-free approach with numerical solutions.

## Contribution

It derives backward and forward nonlinear PDEs for implied volatility applicable to a broad class of contingent claims, including convex payoffs, without relying on specific models.

## Key findings

- Derived PDEs for implied volatility applicable to convex payoffs
- Proposed iterative finite-difference numerical method for solving the PDEs
- Discussed initial and boundary conditions for practical implementation

## Abstract

We derive a backward and forward nonlinear PDEs that govern the implied volatility of a contingent claim whenever the latter is well-defined. This would include at least any contingent claim written on a positive stock price whose payoff at a possibly random time is convex. We also discuss suitable initial and boundary conditions for those PDEs. Finally, we demonstrate how to solve them numerically by using an iterative finite-difference approach.