On the construction and complexity of the bivariate lattice with stochastic interest rate models

Complex financial instruments with multiple state variables often have no analytical formulas and therefore must be priced by numerical methods, like lattice ones. For pricing convertible bonds and many other interest rate-sensitive products, research has focused on bivariate lattices for models wit...

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Bibliographic Details
Published in:Computers & mathematics with applications (1987) 2011-02, Vol.61 (4), p.1107-1121
Main Authors: Lyuu, Yuh-Dauh, Wang, Chuan-Ju
Format: Article
Language:English
Subjects:
Complexity
Interest rates
Lattice
Lattices
Mathematical analysis
Mathematical models
Optimization
Pricing
Raw materials
Stochastic interest rate model
Stochasticity
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Summary:Complex financial instruments with multiple state variables often have no analytical formulas and therefore must be priced by numerical methods, like lattice ones. For pricing convertible bonds and many other interest rate-sensitive products, research has focused on bivariate lattices for models with two state variables: stock price and interest rate. This paper shows that, unfortunately, when the interest rate component allows rates to grow in magnitude without bounds, those lattices generate invalid transition probabilities. As the overwhelming majority of stochastic interest rate models share this property, a solution to the problem becomes important. This paper presents the first bivariate lattice that guarantees valid probabilities. The proposed bivariate lattice grows (super)polynomially in size if the interest rate model allows rates to grow (super)polynomially. Furthermore, we show that any valid constant-degree bivariate lattice must grow superpolynomially in size with log-normal interest rate models, which form a very popular class of interest rate models. Therefore, our bivariate lattice can be said to be optimal.
ISSN:0898-1221
1873-7668
DOI:10.1016/j.camwa.2010.12.061