Model-Free Backward and Forward Nonlinear PDEs for Implied Volatility
The authors derive backward and forward nonlinear partial differential equations 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. The authors also discuss suitable initial and boundary conditions for those partial differential equations. Finally, we demonstrate how to solve them numerically by using an iterative finite-difference approach. TOPICS: Volatility measures, options Key Findings • In this article, we derive backward and forward quasilinear parabolic partial differential equations (PDEs) that govern the implied volatility of a contingent claim whenever the latter is well defined. Alternatively, we have derived a forward nonlinear hyperbolic PDE of the first order, which also governs evolution of the implied volatility in (K, T, Z) space. We discuss suitable initial and boundary conditions for those PDEs. • We develop an iterative numerical method to solve the PDEs by using a finite-difference approach. The method is of second order of approximation in both space and time, is unconditionally stable, and preserves positivity of the solution. • Using this method, we compute the PDE implied volatility and find that our intuition behind the main idea of the article is correct. In other words, performance of the finite-difference solver exceeds that of the traditional approach by factor of 40. However, this result is subject to some details, which are highlighted in the article.