A Stochastic Target Approach for P&L Matching Problems

Within a Brownian diffusion Markovian framework, we provide a direct PDE characterization of the minimal initial endowment required so that the terminal wealth of a financial agent (possibly diminished by the payoff of a random claim) can match a set of constraints in probability. Such constraints should be interpreted as a rough description of a targeted profit and loss (P&L) distribution. This allows us to give a price to options under a P&L constraint, or to provide a description of the discrete P&L profiles that can be achieved given an initial capital. This approach provides an alternative to the standard utility indifference (or marginal) pricing rules that is better adapted to market practices. From the mathematical point of view, this is an extension of the stochastic target problem under controlled loss, studied in Bouchard, Touzi, and Elie (2009) , to the case of multiple constraints. Although the associated Hamilton‐Jacobi‐Bellman operator is fully discontinuous, and the terminal condition is irregular, we are able to construct a numerical scheme that converges at any continuity points of the pricing function.