Effective binomial discretizations of bivariate diffusion processes

In this paper, we investigate the general conditions under which a bivariate continuous-time stochastic process can be approximated by a computationally tractable discrete-time bivariate binomial process. The main requirement is the explicit solvability of a specific system of partial differential equations associated with the norm of the volatility vectors. As a key application, we develop a simple recombining bivariate binomial tree for the stochastic volatility model introduced by Heston (1993). We then employ this discrete framework to compute no-arbitrage prices of European call and put options, obtaining results that are consistent with those generated by established numerical methods. Finally, we perform a detailed analysis of the two-dimensional free boundaries of American call and put options, examining their dependence on both the spot price and the spot volatility.