Backward Stochastic Differential Equations

We saw in Chap. 4 that the problem of pricing and hedging financial derivatives can be modeled in terms of (possibly reflected) backward stochastic differential equations (BSDEs) or, equivalently in the Markovian setup, by partial integro-differential equations or variational inequalities (PIDEs or PDEs for short). Also, Chaps. 10 and 11 just provided thorough illustrations of the abilities of simulation/regression numerical schemes for solving high-dimensional pricing equations: very large systems of partial differential equations in Chap. 10 and Markov chain related systems of ODEs in Chap. 11 . Now that we experimented the power of the theory, let’s dig into it. The next few chapters provides a thorough mathematical treatment of the BSDEs and PDEs that are of fundamental importance for our approach. More precisely, Chaps. 12 to 14 develop, within a rigorous mathematical framework, the connection between backward stochastic differential equations and partial differential equations. This is done in a jump-diffusion setting with regime switching, which covers all the models considered in the book. To start with, Chap. 12 establishes the well-posedness of a Markovian reflected BSDE in a rather generic jump-diffusion model with regime switching, denoted by (X,N), which covers all the models considered in this book. In standard applications, the main component of the model, in which the payoffs of a derivative are expressed, is X. The other model component N can be used to represent a pricing regime, which may also be viewed as a degenerate form of stochastic volatility. More standard diffusive forms of stochastic volatility may also be accounted for in X. The presence of jumps in X is motivated by the empirical evidence of the short-term volatility smile in the market. In credit and counterparty risk modeling, the main model component (the one which drives the cash flows) is the Markov-chain-like-component N, representing a vector of default status and/or credit ratings of reference obligors; a jump-diffusion-like-component X can be used to represent the evolution of economic variables modulating the dynamics of N. Frailty and default contagion are accounted for by the coupled interaction between N and X.