Tree Methods

Abstract

Tree pricing schemes are natural in finance because of their Markov chain interpretation as discrete time pricing models. From a practical point of view, trees are often rather obsolete as compared with more sophisticated finite difference or finite element technologies. However, in a number of situations, they remain an adequate and simple alternative. Moreover, from the theoretical point of view, the Markov chain interpretation underlies interesting probabilistic convergence proofs of the related (deterministic) pricing schemes.

Note that there is no hermetic frontier between deterministic and stochastic pricing schemes. In essence, all these numerical schemes are based on the idea of propagating the solution, starting from a surface of the time-space domain on which it is known (the maturity of the derivative), along suitable (random) “characteristics” of the problem (here “characteristics” refers to Riemann’s method for solving hyperbolic first-order equations). From the point of view of control theory, all these numerical schemes can be viewed as variants of Bellman’s dynamic programming principle. Monte Carlo pricing schemes may thus be regarded as one-time-step multinomial trees, converging to a limiting jump diffusion when the number of space discretization points (tree branches) goes to infinity. The difference between a tree method in the usual sense and a Monte Carlo method is that a Monte Carlo computation mesh is stochastically generated and nonrecombining.