Lie-Algebraic Approach for Pricing Zero-Coupon Bonds in Single-Factor Interest Rate Models

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.

1. Introduction

In this paper we apply the Lie-algebraic method to tackle the bond pricing problem in single-factor interest rate models. In particular, we investigate the bond pricing equation of the form 𝐻(𝑡)𝐵()≡1𝑟,𝑡2𝜎(𝑡)²𝑟^𝜈𝜕²𝜕𝑟²()𝜕+𝜇𝑟,𝑡𝐵()=()𝜕𝑟−𝑟𝑟,𝑡𝜕𝐵𝑟,𝑡,𝜕𝑡 (1) where 𝜈 is a real parameter, 𝜇(𝑟,𝑡) is a real function of both spot interest rate 𝑟 and time-to-maturity 𝑡, 𝜎(𝑡) is the time-varying volatility, and 𝐵(𝑟,𝑡) denotes the price of a zero-coupon bond of duration 𝑇 with a value of unity at maturity; that is, 𝐵(𝑟,0)=1. 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. It is found that not only four of the popular single-factor models, namely the Vasicek model [1], Cox-Ingersoll-Ross model [2], double square-root model [3], and Ahn-Gao model [4], can be derived in a straightforward manner, but also new analytically tractable models can be formulated systematically. Moreover, time-varying model parameters can be incorporated into the derivation of the bond pricing formulae without difficulty. This has the added advantage of allowing yield curves to be fitted, and thus a “no-arbitrage” yield curve model can be developed to match the current market data.

The Lie-algebraic method was introduced by Lo and Hui [5–7] 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. For demonstration, the Lie-algebraic approach has already been applied to price European options for the constant elasticity of variance processes and corporate discount bonds with default risk, multiasset financial derivatives, and so forth. It should be noted that the Lie-algebraic method is different from the Lie group analysis which was introduced by Ibragimov and his coauthors [9, 10] to tackle partial differential equations occurring in financial problems. The Lie group analysis is a mathematical theory developed by Sophus Lie and classifies partial differential equations in terms of their symmetry groups, thereby identifying the set of equations which could be integrated or reduced to low-order equations by group theoretic algorithms. (Details of the Lie group analysis and its application to partial differential equations can be found in, for example, Hydon (2000) and Bluman et al. (2010) [11, 12].) Further applications of the Lie group analysis in mathematical finance were subsequently explored by a number of papers, for example, Goard [13], Pooe et al. [14], Goard et al. [15], Leach et al. [16], and Sinkala et al. [17]. A recent review of the applications of the Lie theory to problems in mathematical finance and economics can be found in [18].

2. Lie-Algebraic Approach

We consider a possible set of differential operators realizing the Lie algebra 𝑆𝑈(1,1)⊕ℎ(1) [19]: 𝑊₁=𝑟^𝛾𝜕𝜕𝑟−𝜆𝑟^𝛼,𝑊₂=1𝑟1−𝛾^1−𝛾,𝑊₃𝐾=1,₋≡12𝑊²₁=12𝑟^2𝛾𝜕²𝜕𝑟²+12𝛾𝑟^2𝛾−1−𝜆𝑟^𝛼+𝛾𝜕+1𝜕𝑟2𝜆𝑟^𝛼𝜆𝑟^𝛼−𝛼𝑟^𝛾−1,𝐾₀≡14𝑊₁𝑊₂+𝑊₂𝑊₁=12𝑟𝜕1−𝛾+1𝜕𝑟4−𝜆2𝑟1−𝛾^{𝛼+1−𝛾},𝐾₊≡12𝑊²₂=121−𝛾²𝑟^2(1−𝛾), (2) where 𝛼, 𝛾, and 𝜆 are real adjustable parameters (see Appendix A). Then we try to look for appropriate linear combinations of these operators, namely, 𝐻(𝑡)≡𝐴₋(𝑡)𝐾₋+𝐴₀(𝑡)𝐾₀+𝐴₊(𝑡)𝐾₊+𝐴₁(𝑡)𝑊₁+𝐴₂(𝑡)𝑊₂+𝐴₃(𝑡)𝑊₃, where the coefficients are arbitrary scalar functions of 𝑡 only, to produce the bond pricing equation given in (1). Here are some illustrative examples.

(1)For 𝐴₋(𝑡)=𝜎(𝑡)², 𝐴₀(𝑡)=−4𝐴₃(𝑡)=−2𝜅(𝑡), 𝐴₊(𝑡)=0, 𝐴₁(𝑡)=𝜅(𝑡)𝜃(𝑡), 𝐴₂(𝑡)=−1, 𝛾=𝜆=0, and 𝛼 being arbitrary, we recover the bond pricing equation of the Vasicek model:12𝜎(𝑡)²𝜕²𝜕𝑟²(𝑡𝜃(𝑡)]𝜕+𝜅)[−𝑟𝐵()=()𝜕𝑟−𝑟𝑟,𝑡𝜕𝐵𝑟,𝑡.𝜕𝑡 (3) By applying the Wei-Norman theorem, we can derive the bond price 𝐵(𝑟,𝑡) as (see Appendix B) 𝐵(𝑟,𝑡)=𝑈₀(𝑡)𝑈_𝐼(𝑡)𝐵(𝑟,0),𝐵(𝑟,0)=1, (4) where 𝑈₀(𝑡)=exp𝑐₁(𝑡)𝐾₊exp𝑐₂(𝑡)𝐾₀exp𝑐₃(𝑡)𝐾₋,𝑈_𝐼(𝑡)=exp𝑔₁(𝑡)𝑊₁exp𝑔₂(𝑡)𝑊₂exp𝑔₃(𝑡)𝑊₃,𝑐₁(𝑡)𝑐=0,₂(𝑡)=−2^𝑡₀𝜅(𝜏)𝑐𝑑𝜏,₃(𝑡)=^𝑡₀𝜎(𝜏)²exp𝑐₂(𝜏𝑔)𝑑𝜏,₁(𝑡)=^𝑡₀𝜅(𝜏)𝜃(𝜏)1exp2𝑐₂(𝜏)+𝑐₃(𝜏)−1exp2𝑐₂(𝜏𝑔)𝑑𝜏,₂(𝑡)=−^𝑡₀−1exp2𝑐₂(𝜏𝑔)𝑑𝜏,₃(𝑡)=^𝑡₀12𝜅(𝜏)+𝑔₁(𝜏)−1exp2𝑐₂(𝜏)𝑑𝜏. (5) As a result, the bond price 𝐵(𝑟,𝑡) can be expressed as 𝐵()1𝑟,𝑡=exp4𝑐₂(𝑡)+𝑔₃(𝑡)+𝑔₂(𝑡)×𝑔₁(𝑡)+12𝑐₃(𝑡)𝑔₂(𝑡𝑔)×exp₂(𝑡)1exp2𝑐₂(𝑡)𝑟. (6) In the special case of constant model parameters, that is, 𝜎(𝑡)=𝜎₀, 𝜅(𝑡)=𝜅₀, and 𝜃(𝑡)=𝜃₀, the 𝑐_𝑖(𝑡) and 𝑔_𝑖(𝑡) can be analytically determined as 𝑐₁(𝑡)𝑐=0,₂(𝑡)=−2𝜅₀𝑐𝑡,₃(𝑡)=𝜎²₀21−exp−2𝜅₀𝑡𝜅₀,𝑔₁(𝑡)=𝜅₀𝜃₀−𝜎²₀2𝜅₀1−exp−2𝜅₀𝑡𝜅₀+𝜎²₀2𝜅₀exp𝜅₀𝑡−1𝜅₀,𝑔₂(𝑡)𝜅=−exp₀𝑡−1𝜅₀,𝑔₃(𝑡)=12𝜅₀𝑡−𝜃₀−𝜎²₀2𝜅²₀𝑡+𝜃₀−𝜎²₀2𝜅²₀×exp𝜅₀𝑡−1𝜅₀+𝜎²₀4𝜅₀exp𝜅₀𝑡−1𝜅₀², (7) and the bond price 𝐵(𝑟,𝑡) is reduced to the well-known closed-form expression [1]: 𝐵(𝑟,𝑡)=exp𝜃₀−𝜎²₀2𝜅²₀1−exp−𝜅₀𝑡𝜅₀−𝜎−𝑡₂₀4𝜅₀1−exp−𝜅₀𝑡𝜅₀²−1−exp−𝜅₀𝑡𝜅₀−𝑡𝑟. (8)

(2) For 𝐴₋(𝑡)=𝜎(𝑡)², 𝐴₀(𝑡)=−4𝐴₃(𝑡)=−2𝜆(𝑡), 𝐴₊(𝑡)=−1/2, 𝐴₁(𝑡)=−𝜅(𝑡), 𝐴₂(𝑡)=0, 𝛾=1/2, 𝜆=0, and 𝛼 being arbitrary, the bond pricing equation of the double square-root model is reproduced: 12𝜎(𝑡)²𝑟𝜕²𝜕𝑟²+14𝜎(𝑡)²−𝜅(𝑡)√𝜆(𝑡)𝑟𝜕𝑟−2()=()𝜕𝑟−𝑟×𝐵𝑟,𝑡𝜕𝐵𝑟,𝑡.𝜕𝑡 (9) As in the Vasicek model, we apply the Wei-Norman theorem to determine the bond price 𝐵(𝑟,𝑡) as (see Appendix B) 𝐵(𝑟,𝑡)=𝑈₀(𝑡)𝑈_𝐼(𝑡)𝐵(𝑟,0),𝐵(𝑟,0)=1, (10) where 𝑈₀(𝑡)=exp𝑐₁(𝑡)𝐾₊exp𝑐₂(𝑡)𝐾₀exp𝑐₃(𝑡)𝐾₋,𝑈_𝐼(𝑡)=exp𝑔₁(𝑡)𝑊₁exp𝑔₂(𝑡)𝑊₂exp𝑔₃(𝑡)𝑊₃,𝑑𝑐₁(𝑡)𝜆(𝑡)𝑐𝑑𝑡=−1−2₁(𝑡)+12𝜎(𝑡)²𝑐₁(𝑡)²,𝑐₁(0)𝑐=0,₂(𝑡)=^𝑡₀−2𝜆(𝜏)+𝜎(𝜏)²𝑐₁(𝜏𝑐)𝑑𝜏,₃(𝑡)=12^𝑡₀𝜎(𝜏)²exp𝑐₂(𝜏𝑔)𝑑𝜏,₁(𝑡)=1√2^𝑡0𝜅(𝜏)𝑐₁(𝜏)𝑐₃(𝜏)−1exp2𝑐₂(𝜏1)−exp2𝑐₂(𝜏𝑔)𝑑𝜏,₂(𝑡)1=−√2^𝑡0𝜅(𝜏)𝑐₁(𝜏)−1exp2𝑐₂(𝜏𝑔)𝑑𝜏,₃(𝑡)=^𝑡₀12𝜆(𝜏)+1√2𝜅(𝜏)𝑐₁(𝜏)−1exp2𝑐₂(𝜏)𝑔₁(𝜏)𝑑𝜏. (11) Accordingly, the bond price 𝐵(𝑟,𝑡) is given by 𝐵()𝑔𝑟,𝑡=exp₃(𝑡)+𝑔₁(𝑡)𝑔₂(𝑡)+12𝑐₃(𝑡)𝑔₂(𝑡)²+14𝑐₂(𝑡𝑔)×exp₂(𝑡)1exp2𝑐₂(𝑡)√2𝑟+𝑐₁(𝑡)𝑟. (12) In the special case of constant model parameters, that is, 𝜎(𝑡)=𝜎₀, 𝜅(𝑡)=𝜅₀, and 𝜆(𝑡)=𝜆₀, the 𝑐_𝑖(𝑡) and 𝑔_𝑖(𝑡) can be analytically determined as 𝑐₁(𝑡)2=−exp𝛾𝑡−1𝛾+2𝜆₀=2𝜆exp𝛾𝑡−1+2𝛾₀−𝛾𝜎²₀+2𝛾𝜎²₀1−𝐶₀,𝑐exp𝛾𝑡₂(𝑡)=−2𝜆₀𝑡)𝜆+2ln2𝛾exp(1/2𝛾+2₀𝑡𝛾+2𝜆₀exp𝛾𝑡−1+2𝛾=𝛾𝑡+2ln2𝛾𝛾−2𝜆₀1−𝐶₀,𝑐exp𝛾𝑡₃(𝑡)1=−2𝜎²₀𝑐₁(𝑡),𝑔₁(𝑡)=√2𝜅₀𝜆₀𝛾²−1exp21𝛾𝑡1−exp2𝛾𝑡²−𝜅₀√12𝛾exp2−1𝛾𝑡−exp2,𝑔𝛾𝑡₂(𝑡)=√2𝜅₀𝛾²−1exp21𝛾𝑡1−exp2𝛾𝑡²,𝑔₃(𝑡)=12𝜆₀−𝜅²₀𝛾²𝜅𝑡−²⁰𝜆₀𝛾⁴×1exp−𝛾𝑡1−exp2𝛾𝑡⁴+𝜅²₀2𝛾³exp𝛾𝑡−exp−𝛾𝑡 (13) with 𝛾=4𝜆²₀+2𝜎²₀ and 𝐶₀=(2𝜆₀+𝛾)/(2𝜆₀−𝛾). Consequently, we are able to recover the well-known closed-form expression of the bond price 𝐵(𝑟,𝑡) [3]: 𝐵(𝑟,𝑡)=Ψ(𝑡)expΩ(𝑡)𝑟+Γ(𝑡)√𝑟, (14) where Ψ(𝑡)=1−𝐶₀1−𝐶₀𝛼exp𝛾𝑡×exp₁+𝛼₂𝛼𝑡+³+𝛼₄exp(1/2)𝛾𝑡1−𝐶₀,Ω(𝑡)=2𝜆exp𝛾𝑡₀−𝛾𝜎²₀+2𝛾𝜎²₀1−𝐶₀,Γ(𝑡)=exp𝛾𝑡2𝜅₀2𝜆₀+𝛾1−exp(1/2)𝛾𝑡²𝛾𝜎²₀1−𝐶₀,exp𝛾𝑡 (15) with 𝛼₁𝜅=−²⁰𝛾³𝜎²₀4𝜆₀+𝛾2𝜆₀,𝛼−𝛾₂=2𝜆₀+𝛾4−𝜅²₀𝛾²,𝛼₃=4𝜅²₀𝛾³𝜎²₀2𝜆²₀−𝜎²₀,𝛼₄=−8𝜅²₀𝜆₀𝛾³𝜎²₀2𝜆₀+𝛾. (16)

(3) For 𝐴₋(𝑡)=𝜎(𝑡)², 𝐴₀(𝑡)=−4𝐴₃(𝑡)/3, 𝐴₊(𝑡)=𝐴₁(𝑡)=0, 𝐴₂(𝑡)=−1, 𝛾=0, and 𝜆=𝛼=−1, a bond pricing equation with a special time-dependent nonlinear drift term can be obtained: 12𝜎(𝑡)²𝜕²𝜕𝑟²+12𝐴₀(𝑡)𝜎(𝑡)𝑟+²𝑟𝜕𝐵()=()𝜕𝑟−𝑟𝑟,𝑡𝜕𝐵𝑟,𝑡𝜕𝑡 (17) which can be straightforwardly solved as in the Vasicek model. As a result, the bond price 𝐵(𝑟,𝑡) can be expressed as (see Appendix B) 𝐵()3𝑟,𝑡=exp4𝑐₂(𝑡)+𝑔₃(𝑡)+𝑔₂(𝑡)×𝑔₁(𝑡)+12𝑐₃(𝑡)𝑔₂(𝑡)×exp𝑔₂(𝑡)1exp2𝑐₂(𝑡)𝑟×1+𝑔₁(𝑡)+𝑐₃(𝑡)𝑔₂(𝑡1)exp2𝑐₂(𝑡)1𝑟, (18) where 𝑐₂(𝑡)=^𝑡₀𝐴₀(𝜏)𝑐𝑑𝜏,₃(𝑡)=^𝑡₀𝜎(𝜏)²exp𝑐₂(𝜏𝑔)𝑑𝜏,₁(𝑡)=^𝑡₀𝑐₃(𝜏)−1exp2𝑐₂(𝜏𝑔)𝑑𝜏,₂(𝑡)=−^𝑡₀−1exp2𝑐₂(𝜏𝑔)𝑑𝜏,₃(𝑡)=^𝑡₀−34𝐴₀(𝜏)+𝑔₁(𝜏)−1exp2𝑐₂(𝜏)𝑑𝜏. (19)

(4) For 𝐴₋(𝑡)=𝜎(𝑡)², 𝐴₀(𝑡)=−4𝐴₃(𝑡)/3, 𝐴₊(𝑡)=𝐴₂(𝑡)=0, 𝐴₁(𝑡)=1, 𝛾=2, and 𝜆=𝛼=1, we can derive the bond pricing equation: 12𝜎(𝑡)²𝑟⁴𝜕²𝜕𝑟²+𝑟²−12𝐴₀(𝑡)𝑟𝜕𝐵()=()𝜕𝑟−𝑟𝑟,𝑡𝜕𝐵𝑟,𝑡,𝜕𝑡 (20) which has both the 𝑟² dependence of volatility and a time-dependent nonlinear drift term. By performing the same analysis as in the other three cases, the bond price 𝐵(𝑟,𝑡) is found to be given by (see Appendix B) 𝐵(𝑟,𝑡)=1−𝑔₁(𝑡)−1exp2𝑐₂(𝑡)𝑟, (21) where 𝑐₂(𝑡)=^𝑡₀𝐴₀(𝜏)𝑔𝑑𝜏,₁(𝑡)=^𝑡₀1exp2𝑐₂(𝜏)𝑑𝜏. (22)

Next we apply the same analysis to derive the Cox-Ingersoll-Ross model and Ahn-Gao model from an alternative set of differential operators realizing the subalgebra 𝔏 of the Lie algebra 𝑆𝑈(1,1)⊕ℎ(1): 𝐾₋=12𝑟^2𝛾𝜕²𝜕𝑟²+12𝛾−𝜆𝑟^2𝛾−1𝜕𝜕𝑟+𝛼𝑟^2𝛾−2,𝐾₀=12𝑟𝜕1−𝛾+1𝜕𝑟4−𝜆2,𝐾1−𝛾₊=121−𝛾²𝑟^2(1−𝛾),𝑊₃=1, (23) where 𝛼, 𝛾, and 𝜆 are real adjustable parameters (see Appendix A). The subalgebra 𝔏 is actually the reductive Lie algebra 𝑈(1,1).

(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 𝜕𝐵(𝑟,𝑡)=1𝜕𝑡2𝜎²𝑟𝜕²𝜕𝑟²()𝜕+𝜅𝜃−𝑟𝐵()=𝜎𝜕𝑟−𝑟𝑟,𝑡₂𝐾₋−𝜅𝐾₀−12𝐾₊+𝜅²𝜃𝜎²𝑊₃𝐵(𝑟,𝑡). (24) (Strictly speaking, the model parameters 𝜅, 𝜃, and 𝜎 could be time-varying with the constraint that 𝜅𝜃/𝜎² is independent of time 𝑡.)

By the Wei-Norman theorem, the bond price 𝐵(𝑟,𝑡) can be easily determined as (see Appendix B) 𝐵(𝑟,𝑡)=exp2𝑐₁(𝑡)𝑟exp𝜅𝜃𝜎²𝜅𝑡+𝑐₂(𝑡), (25) where 𝑑𝑐₁(𝑡)1𝑑𝑡=−2−𝜅𝑐₁(𝑡)+𝜎²𝑐₁(𝑡)²,𝑐₁(0)𝑐=0,₂(𝑡)=−𝜅𝑡+𝜎^2𝑡₀𝑐₁(𝜏)𝑑𝜏. (26) The Riccati equation with constant coefficients in (26) can be straightforwardly solved to yield 𝑐₁(𝑡)=−exp𝛾𝑡−1𝛾+𝜅exp𝛾𝑡−1+2𝛾 (27) with 𝛾=√𝜅²+2𝜎². Once the 𝑐₁(𝑡) has been found, we are also able to obtain 𝑐₂(𝑡))𝑡=−𝜅𝑡+2ln2𝛾exp(1/2𝛾+𝜅𝛾+𝜅exp𝛾𝑡−1+2𝛾 (28) via analytical integrations. Beyond question, our finding is in agreement with the well-known closed-form result [2]: 𝐵()=)𝑡𝑟,𝑡2𝛾exp(1/2𝛾+𝜅𝛾+𝜅exp𝛾𝑡−1+2𝛾^{2𝜅𝜃/𝜎2}−2𝑟×expexp𝛾𝑡−1.𝛾+𝜅exp𝛾𝑡−1+2𝛾 (29)

(2) By setting 𝛼=−1/𝜎², 𝛾=3/2, and 𝜆=𝑞+3/4, we can cast the bond pricing equation of the Ahn-Gao model, which includes nonlinearity in the drift term and a realistic 𝑟^3/2 dependence in the volatility (not only the nonlinear drift term of the Ahn-Gao model is consistent with the empirical findings of Aït-Sahalia [20], but also the chosen 𝑟^3/2 dependence of volatility is the best fit power law for volatility [21, 22]), in terms of the differential operators realizing the subalgebra 𝔏 into the form 𝜕𝐵(𝑟,𝑡)=1𝜕𝑡2𝜎²𝑟³𝜕²𝜕𝑟²+𝜎²𝑎(𝑡)𝑟−𝑞𝑟²𝜕𝐵()=𝜎𝜕𝑟−𝑟𝑟,𝑡₂𝐾₋−𝜎²𝑎(𝑡)𝐾₀+𝜎²𝑞+1𝑎(𝑡)𝑊₃𝐵(𝑟,𝑡), (30) where 𝑎(𝑡) is a real function of 𝑡. Then applying the Wei-Norman theorem allows us to represent the bond price 𝐵(𝑟,𝑡) by (see Appendix B) 𝐵(𝑟,𝑡)=exp𝜎²𝑞+1^𝑡₀𝑎(𝜏)𝑑𝜏exp𝑐₂(𝑡)𝐾₀×exp𝑐₃(𝑡)𝐾₋𝐵(𝑟,0), (31) where 𝐵(𝑟,0)=1 and 𝑐₂(𝑡)=−𝜎^2𝑡₀𝑎(𝜏)𝑐𝑑𝜏,₃(𝑡)=𝜎^2𝑡₀exp𝑐₂(𝜏)𝑑𝜏. (32) Without loss of generality, we suppose that 𝐵(𝑟,0)=𝑥^−(2𝑞+1)𝑉(𝑥), where 𝑥=2/√𝑟, 𝑉(𝑥)=^∞₀𝑑𝜈𝜈𝐽_𝑝(𝑥𝜈)^∞₀𝑑𝑦𝑦𝐽_𝑝𝑦𝜈𝑉𝑦, (33) and 𝑝=(2𝑞+1)²+8/𝜎². It is not difficult to show that the bond price 𝐵(𝑟,𝑡) is given by 𝐵(𝑟,𝑡)=^∞₀𝑑𝑥^′𝐺𝑥,𝑡;𝑥^′,0𝐵𝑟^′,0, (34) where 𝑥^′=2/𝑟^′ and 𝐺𝑥,𝑡;𝑥^′,0=𝑥^′𝑥^′𝑥exp𝑐₂(𝑡)/2^2𝑞+1×^∞₀𝑑𝜈𝜈𝐽_𝑝𝑐𝑥𝜈exp²(𝑡)2×𝐽_𝑝𝑥^′𝜈−𝑐exp³(𝑡)2𝜈². (35) The function 𝐽_𝑝(𝜉) is the Bessel function of the first kind of order 𝑝. Here we have made use of the fact that 𝑥^−(2𝑞+1)𝐽_𝑝(𝑥𝜈) is an eigenfunction of the operator 𝐾₋ with the eigenvalue −𝜈²/2. The integral over 𝜈 can be analytically evaluated to give [23] 1𝑐₃(𝑡)−𝑥exp^′2+𝑥²exp𝑐₂(𝑡)2𝑐₃(𝑡)𝐼_𝑝𝑥^′𝑥exp𝑐₂(𝑡)/2𝑐₃(𝑡) (36) for 𝑝>−1, 𝑥^′>0, 𝑥exp{𝑐₂(𝑡)/2}>0, and |arg[𝑐₂(𝑡)/2]^1/2|<𝜋/4. The function 𝐼_𝑝(𝜉) is the modified Bessel function of the first kind of order 𝑝. As a result, 𝐺(𝑥,𝑡;𝑥^′,0) is found to be given by 𝐺𝑥,𝑡;𝑥^′=𝑥,0^′𝑐₃(𝑡)exp2𝑞+1𝑐₂(𝑡)𝑥/2𝑥^′2𝑞+1×𝐼_𝑝𝑥^′𝑥exp𝑐₂(𝑡)/2𝑐₃(𝑡)−𝑥×exp^′2+𝑥²exp𝑐₂(𝑡)2𝑐₃(𝑡). (37) Since 𝐵(𝑟,0)=1, we can readily derive the bond price 𝐵(𝑟,𝑡) as follows: 𝐵()=𝑥𝑟,𝑡^−(2𝑞+1)𝑐₃(𝑡)exp2𝑞+1𝑐₂(𝑡)−𝑥/2×exp²exp𝑐₂(𝑡)2𝑐₃(𝑡)×^∞₀𝑑𝑥^′𝑥^′2(𝑞+1)−𝑥exp^′22𝑐₃(𝑡)×𝐼_𝑝𝑥^′𝑥exp𝑐₂(𝑡)/2𝑐₃(𝑡)=Γ𝑝+1−𝜔Γ𝑀𝑐𝑝+1𝜔,𝑝+1,−2exp₂(𝑡)𝑐₃(𝑡)𝑟×2exp𝑐₂(𝑡)𝑐₃(𝑡)𝑟^𝜔, (38) where 𝜔=−(2𝑞+1−𝑝)/2, Γ(𝜉) denotes the Gamma function and 𝑀(𝜉,𝜒,𝜌) is the standard confluent hypergeometric function [23, 24]. Furthermore, (38) will reproduce the well-known closed-form result if the model parameter 𝑎(𝑡) is independent of time [4].

3. Conclusion

In this paper the Lie-algebraic method 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, and analytical closed-form pricing formulae are derived. 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. This interesting feature helps shed new light upon the systematic formulation of new analytically tractable single-factor interest rate models, as demonstrated in Section 2. Time-varying model parameters could also be incorporated into the derivation of the bond price formulae without difficulty. This has the added advantage of allowing yield curves to be fitted, and thus a “no-arbitrage” yield curve model can be developed to match the current market data. Hence, we believe that the Lie-algebraic method will provide an easy-to-use analytical tool for the bond pricing problem. Furthermore, the Lie-algebraic approach can be easily extended to the pricing of other standard European interest rate derivatives for they differ from the zero-coupon bonds in the final payoff conditions only [25].

Appendices

A. Generators of the Lie Algebra 𝑆𝑈(1,1)⊕ℎ(1) and Its Subalgebras

The generators {𝑊₁,𝑊₂,𝑊₃} of the Heisenberg-Weyl Lie algebra ℎ(1) obey the set of commutation relations [19]: 𝑊₁,𝑊₂=𝑊₃,𝑊₁,𝑊₃=𝑊₂,𝑊₃=0. (A.1) By direct substitution, a possible set of differential operators realizing the Lie algebra can be identified as 𝑊₁=𝑟^𝛾𝜕𝜕𝑟−𝜆𝑟^𝛼,𝑊₂=1𝑟1−𝛾^1−𝛾,𝑊₃=1, (A.2) where 𝛼, 𝛾, and 𝜆 are real adjustable parameters. Then, in terms of these generators one can construct the generators {𝐾₊,𝐾₀,𝐾₋} of the Lie algebra 𝑆𝑈(1,1) as follows: 𝐾₋≡12𝑊²₁=12𝑟^2𝛾𝜕²𝜕𝑟²+12𝛾𝑟^2𝛾−1−𝜆𝑟^𝛼+𝛾𝜕+1𝜕𝑟2𝜆𝑟^𝛼𝜆𝑟^𝛼−𝛼𝑟^𝛾−1,𝐾₀≡14𝑊₁𝑊₂+𝑊₂𝑊₁=12𝑟𝜕1−𝛾+1𝜕𝑟4−𝜆2𝑟1−𝛾^{𝛼+1−𝛾},𝐾₊≡12𝑊²₂=121−𝛾²𝑟^2(1−𝛾) (A.3) which satisfy the set of commutation relations [19]: 𝐾₊,𝐾₋=−2𝐾₀,𝐾₀,𝐾_±=±𝐾_±. (A.4) These six generators {𝑊₁,𝑊₂,𝑊₃,𝐾₊,𝐾₀,𝐾₋} in turn form the Lie algebra 𝑆𝑈(1,1)⊕ℎ(1), which is defined by the following set of commutation relations [19]: 𝑊₁,𝑊₂=𝑊₃,𝑊₁,𝑊₃=𝑊₂,𝑊₃𝐾=0,₊,𝐾₋=−2𝐾₀,𝐾₀,𝐾_±=±𝐾_±,𝑊₁,𝐾₊=𝑊₂,𝑊₁,𝐾₀=12𝑊₁,𝑊₂,𝐾₀1=−2𝑊₂,𝑊₂,𝐾₋=−𝑊₁,𝑊₁,𝐾₋=𝑊₂,𝐾₊=𝑊₃,𝐾₀=𝑊₃,𝐾_±=0. (A.5) In addition to the subalgebras 𝑆𝑈(1,1) and ℎ(1), we may also form another subalgebra 𝔏 in terms of the four generators {𝑊₃,𝐾₊,𝐾₀,𝐾₋} satisfying the commutation relations: 𝐾₊,𝐾₋=−2𝐾₀,𝐾₀,𝐾_±=±𝐾_±,𝑊₃,𝐾₀=𝑊₃,𝐾_±=0. (A.6) The subalgebra 𝔏 is actually the reductive Lie algebra 𝑈(1,1). Moreover, it is not difficult to show that an alternative realization of the set of generators of the Lie algebra 𝔏 is given by 𝐾₋=12𝑟^2𝛾𝜕²𝜕𝑟²+12𝛾−𝜆𝑟^2𝛾−1𝜕𝜕𝑟+𝛼𝑟^2𝛾−2,𝐾₀=12𝑟𝜕1−𝛾+1𝜕𝑟4−𝜆2,𝐾1−𝛾₊=121−𝛾²𝑟^2(1−𝛾),𝑊₃=1, (A.7) where 𝛼, 𝛾, and 𝜆 are real adjustable parameters.

B. Wei-Norman Theorem

Consider the linear operator differential equation of the first order 𝑑𝑈(𝑡)(𝑡)𝑈(𝑡)(0)𝑑𝑡=𝐻,𝑈=1, (B.1) where 𝐻 and 𝑈 are both time-dependent linear operators in a Banach space or a finite-dimensional space. According to the Wei-Norman theorem [8], if the operator 𝐻 can be expressed as 𝐻(𝑡)=^𝑁_𝑛=1𝑎_𝑛(𝑡)𝐿_𝑛, (B.2) where 𝑎_𝑛’s are scalar functions of time and 𝐿_𝑛 are the generators of an 𝑁-dimensional solvable Lie algebra or a real split 3-dimensional simple Lie algebra, then the operator 𝑈 can assume the following form: 𝑈(𝑡)=^𝑁_𝑛=1exp𝑔_𝑛(𝑡)𝐿_𝑛. (B.3) Here the 𝑔_𝑛’s are time-dependent scalar functions to be determined. To find the 𝑔_𝑛’s, we simply substitute (B.2) and (B.3) into (B.1) and compare the two sides term by term to obtain a set of coupled nonlinear differential equations 𝑑𝑔_𝑛(𝑡)=𝑑𝑡^𝑁_𝑚=1𝜂_𝑛𝑚𝑎_𝑚(𝑡),𝑔_𝑛(0)=0, (B.4) where 𝜂_𝑛𝑚 are nonlinear functions of 𝑔_𝑛’s. Thus, we have transformed the linear operator differential equation in (B.1) to a set of coupled nonlinear differential equations of scalar functions in (B.4).

Moreover, according to Levi’s Theorem, “If 𝐿 is a finite-dimensional Lie algebra with radical 𝑅 which is the maximal solvable ideal of the Lie algebra, then there exists a semisimple subalgebra 𝑆 of 𝐿 such that 𝐿 is the semidirect sum 𝐿=𝑆⊕𝑅,” in the equation 𝑑𝑈(𝑡)/𝑑𝑡=𝐻(𝑡)𝑈(𝑡), where 𝐻(𝑡) generates 𝐿, the decomposition 𝐿=𝑆⊕𝑅 gives rise to the corresponding decomposition 𝐻(𝑡)=𝐻_𝑆(𝑡)+𝐻_𝑅(𝑡), where 𝐻_𝑆(𝑡)∈𝑆 and 𝐻_𝑅(𝑡)∈𝑅. Then it is easy to verify that 𝑈(𝑡)=𝑈_𝑆(𝑡)𝑈_𝑅(𝑡) where 𝑈_𝑆(𝑡) and 𝑈_𝑅(𝑡) satisfy 𝜕𝑈_𝑆(𝑡)𝜕𝜏=𝐻_𝑆(𝑡)𝑈_𝑆(𝑡),𝜕𝑈_𝑅(𝑡)=𝑈𝜕𝑡^𝑆(𝑡)⁻¹𝐻_𝑅(𝑡)𝑈_𝑆(𝑡)𝑈_𝑅(𝑡). (B.5)(B.6) Since 𝑅 is an ideal in 𝐿, we can easily see that 𝑈_𝑆(𝑡)⁻¹𝐻_𝑅(𝑡)𝑈_𝑆(𝑡) is in 𝑅. The fact that 𝑅 is solvable makes it easy to find 𝑈_𝑅(𝑡) once 𝑈_𝑆(𝑡) has been found. More details can be found in [8].

For illustration, we apply the Wei-Norman theorem to the following cases.

B.1. Heisenberg-Weyl Lie Algebra ℎ(1)

The Heisenberg-Weyl Lie algebra ℎ(1) is defined by the set of commutation relations [17]: 𝑊₁,𝑊₂=𝑊₃,𝑊₁,𝑊₃=𝑊₂,𝑊₃=0, (B.7) of its generators {𝑊₁,𝑊₂,𝑊₃}. Given that 𝐻(𝑡)=𝑎₁(𝑡)𝑊₁+𝑎₂(𝑡)𝑊₂+𝑎₃(𝑡)𝑊₃, (B.8) the Wei-Norman theorem states that 𝑈(𝑡) can be expressed as 𝑈(𝑡)=exp𝑔₁(𝑡)𝑊₁exp𝑔₂(𝑡)𝑊₂exp𝑔₃(𝑡)𝑊₃, (B.9) where the time-dependent functions 𝑔_𝑛’s satisfy a set of three coupled nonlinear differential equations: 𝑑𝑔₁(𝑡)𝑑𝑡=𝑎₁(𝑡),𝑑𝑔₂(𝑡)𝑑𝑡=𝑎₂(𝑡),𝑑𝑔₃(𝑡)𝑑𝑡+𝑔₁(𝑡)𝑑𝑔₂(𝑡)𝑑𝑡=𝑎₃(𝑡). (B.10) It is obvious that the set of differential equations can be easily solved by quadrature: 𝑔₁(𝑡)=^𝑡₀𝑑𝜏𝑎₁(𝜏),𝑔₂(𝑡)=^𝑡₀𝑑𝜏𝑎₂(𝜏),𝑔₃(𝑡)=^𝑡₀𝑑𝜏𝑎₃(𝜏)−𝑎₂(𝜏)𝑔₁(𝜏). (B.11) As a result, the operator 𝑈(𝑡) is thus determined.

B.2. 𝑆𝑈(1,1) Lie Algebra

We consider the evolution equation of the operator 𝑈(𝑡): 𝑑𝑈(𝑡)=𝑎𝑑𝑡¹(𝑡)𝐾₊+𝑎₂(𝑡)𝐾₀+𝑎₃(𝑡)𝐾₋𝑈(𝑡),𝑈(0)=1, (B.12) where the operators {𝐾₊,𝐾₀,𝐾₋} form the 𝑆𝑈(1,1) Lie algebra defined by the commutation relations [19]: 𝐾₊,𝐾₋=−2𝐾₀,𝐾₀,𝐾_±=±𝐾_±. (B.13) According to the Wei-Norman theorem, the operator 𝑈(𝑡) can be expressed in the product form 𝑈(𝑡)=exp𝑐₁(𝑡)𝐾₊exp𝑐₂(𝑡)𝐾₀exp𝑐₃(𝑡)𝐾₋,𝑐_𝑖(0)=0, (B.14) where the time-dependent functions 𝑐_𝑛’s satisfy a set of three coupled nonlinear differential equations: 𝑑𝑐₁(𝑡)𝑑𝑡−𝑐₁(𝑡)𝑑𝑐₂(𝑡)𝑑𝑡+𝑐₁(𝑡)²exp−𝑐₂(𝑡)𝑑𝑐₃(𝑡)𝑑𝑡=𝑎₁(𝑡),𝑑𝑐₂(𝑡)𝑑𝑡−2𝑐₁(𝑡)exp−𝑐₂(𝑡)𝑑𝑐₃(𝑡)𝑑𝑡=𝑎₂(𝑡),exp−𝑐₂(𝑡)𝑑𝑐₃(𝑡)𝑑𝑡=𝑎₃(𝑡). (B.15) After further simplification, these three differential equations become 𝑑𝑐₁(𝑡)𝑑𝑡=𝑎₁(𝑡)+𝑎₂(𝑡)𝑐₁(𝑡)+𝑎₃(𝑡)𝑐₁(𝑡)²,𝑐₁(0)𝑐=0,₂(𝑡)=^𝑡₀𝑎₂(𝜏)+2𝑎₃(𝜏)𝑐₁(𝜏𝑐)𝑑𝜏,₃(𝑡)=^𝑡₀𝑎₃(𝜏)exp𝑐₂(𝜏)𝑑𝜏. (B.16) Hence, once the 𝑐₁(𝑡) is found by solving the Riccati equation, the 𝑐₂(𝑡) and 𝑐₃(𝑡) can be readily determined by quadrature.

B.3. 𝑆𝑈(1,1)⊕ℎ(1) Lie Algebra

If 𝐻(𝑡) is a linear combination of the six generators {𝑊₁,𝑊₂,𝑊₃,𝐾₊,𝐾₀,𝐾₋} of the Lie algebra 𝑆𝑈(1,1)⊕ℎ(1), then, according to Levi’s theorem, we may decompose the 𝐻(𝑡) into two parts 𝐻_𝑆(𝑡)=𝑎₁(𝑡)𝐾₊+𝑎₂(𝑡)𝐾₀+𝑎₃(𝑡)𝐾₋ and 𝐻_𝑅(𝑡)=𝑏₁(𝑡)𝑊₁+𝑏₂(𝑡)𝑊₂+𝑏₃(𝑡)𝑊₃, and the operator 𝑈(𝑡) assumes the product form 𝑈(𝑡)=𝑈_𝑆(𝑡)𝑈_𝑅(𝑡) where 𝑈_𝑆(𝑡) and 𝑈_𝑅(𝑡) satisfy (B.5) and (B.6), respectively. It is obvious that the operator 𝑈_𝑆(𝑡) is given by 𝑈_𝑆(𝑡)=exp𝑐₁(𝑡)𝐾₊exp𝑐₂(𝑡)𝐾₀exp𝑐₃(𝑡)𝐾₋, (B.17) where 𝑑𝑐₁(𝑡)𝑑𝑡=𝑎₁(𝑡)+𝑎₂(𝑡)𝑐₁(𝑡)+𝑎₃(𝑡)𝑐₁(𝑡)²,𝑐₁(0)𝑐=0,₂(𝑡)=^𝑡₀𝑎₂(𝜏)+2𝑎₃(𝜏)𝑐₁(𝜏𝑐)𝑑𝜏,₃(𝑡)=^𝑡₀𝑎₃(𝜏)exp𝑐₂(𝜏)𝑑𝜏. (B.18) Next, in order to determine 𝑈_𝑅(𝑡), we need to evaluate 𝐻_𝐼(𝑡)≡𝑈_𝑆(𝑡)⁻¹𝐻_𝑅(𝑡)𝑈_𝑆(𝑡). Using the explicit form of the operator 𝑈_𝑆(𝑡), we can apply the Baker-Hausdorff formula [26] to derive the operator 𝐻_𝐼(𝑡): 𝐻_𝐼(𝑡)=𝑎₁(𝑡)𝑊₁+𝑎₂(𝑡)𝑊₂+𝑎₃(𝑡)𝑊₃, (B.19) where 𝑎₁(𝑡)=𝑏₁(𝑡)1exp2𝑐₂(𝑡)−𝑏₁(𝑡)𝑐₁(𝑡)+𝑏₂(𝑡)×𝑐₃(𝑡)−1exp2𝑐₂(𝑡),𝑎₂(𝑡)=𝑏₁(𝑡)𝑐₁(𝑡)+𝑏₂(𝑡−1)exp2𝑐₂(𝑡),𝑎₃(𝑡)=𝑏₃(𝑡). (B.20) Then, the operator 𝑈_𝑅(𝑡) can be easily found to be given by 𝑈_𝑅(𝑡)=exp𝑔₁(𝑡)𝑊₁exp𝑔₂(𝑡)𝑊₂exp𝑔₃(𝑡)𝑊₃, (B.21) where 𝑔₁(𝑡)=^𝑡₀𝑑𝜏𝑎₁(𝜏),𝑔₂(𝑡)=^𝑡₀𝑑𝜏𝑎₂(𝜏),𝑔₃(𝑡)=^𝑡₀𝑑𝜏𝑎₃(𝜏)−𝑎₂(𝜏)𝑔₁(𝜏). (B.22)