On the Complexity of Bivariate Lattice with Stochastic Interest Rate Models

Many complex financial instruments with multiple state variables have no analytical formulas and thus must be priced by numerical methods like lattice. The bivariate lattice is a numerical method that is widely used to work with correlated state variables. Some research has focused on bivariate lattices for models with two state variables: stochastic underlying asset prices (e.g., stock prices) and stochastic interest rates. However, when the interest rate model allows rates to grow superpolynomially, the said lattices generate invalid transition probabilities. With the trinomial lattice and the mean-tracking techniques, this paper presents the first bivariate lattice that guarantees valid probabilities. It also proves that any bivariate lattice for stock price and interest rate must grow superpolynomially if the interest rate model allows rates to grow superpolynomially.