Abstract
The Lie-algebraic approach has been applied to solve the bond pricing problem in single-factor interest rate models. Four of the popular single-factor models, namely the Vasicek model, Cox-Ingersoll-Ross model, double square-root model, and Ahn-Gao model, are investigated. By exploiting the dynamical symmetry of their bond pricing equations, analytical closed-form pricing formulae can be derived in a straightfoward manner. Time-varying model parameters could also be incorporated into the derivation of the bond price formulae, and this has the added advantage of allowing yield curves to be fitted. Furthermore, the Lie-algebraic approach can be easily extended to formulate new analytically tractable single-factor interest rate models.
Highlights
In this paper we apply the Lie-algebraic method to tackle the bond pricing problem in single-factor interest rate models
(1) By choosing α = 0, γ = 1/2, and λ = 1/4 − κθ/σ2, the bond pricing equation of the Cox-Ingersoll-Ross model with constant model parameters can be expressed in terms of the differential operators realizing the subalgebra L as
Since all the four bond pricing equations exhibit the dynamical symmetry SU(1, 1) ⊕ h(1) or its subgroup, their solutions can be derived in a unified manner and have very similar mathematical structures
Summary
In this paper we apply the Lie-algebraic method to tackle the bond pricing problem in single-factor interest rate models. By exploiting the dynamical symmetry SU(1, 1) ⊕ h(1) of the bond pricing equation, we derive analytically tractable single-factor interest rate models in a unified manner and obtain their closed-form bond pricing formulae. The Lie-algebraic method was introduced by Lo and Hui [5,6,7] to the field of finance for the pricing of financial derivatives with time-dependent model parameters. This new method is very simple and consists of two basic ingredients: (1) identifying the dynamical symmetries of the given pricing partial differential equations and (2) applying the Wei-Norman theorem [8] to solve the equations and obtain analytical closed-form pricing formulae. A recent review of the applications of the Lie theory to problems in mathematical finance and economics can be found in [18]
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