Martingale Modeling

In this chapter, we show how the task of pricing and hedging financial derivatives can be reduced to that of solving related backward stochastic differential equations, called stochastic pricing equations in this book, equivalent to the deterministic pricing equations that arise in Markovian setups. The deterministic pricing equations, starting with the celebrated Black–Scholes equation, are better known to practitioners. However, these deterministic partial-differential equations, also including integral terms in models with jumps, are more “model dependent” than the stochastic pricing equations. Moreover, the deterministic pricing equations are less general since they are only available in Markovian setups. In addition, the mathematics of pricing and hedging financial derivatives is simpler in terms of the stochastic pricing equations. Indeed, rigorous demonstrations based on the deterministic pricing equations involve technical notions of viscosity or Sobolev solutions (at least as soon as the problem is nonlinear, e.g. when early exercise clauses and related obstacles in the deterministic equations come into the picture).