Nonlinear Black–Scholes Equations in Finance: Associated Control Problems and Properties of Solutions

Abstract

We study properties of solutions to fully nonlinear versions of the standard Black–Scholes partial differential equation. These equations have been introduced in financial mathematics in order to deal with illiquid markets or with stochastic volatility. We show that typical nonlinear Black–Scholes equations can be viewed as dynamic programming equation of an associated control problem. We establish existence and comparison results and show that the equation induces a convex risk measure on the set of all continuous terminal value claims. Moreover, we study the asymptotic behavior of solutions as market frictions get “large.” Finally, the pricing of individual contracts relative to a book of derivatives is discussed.