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

Authors: C. F. Lo
Journal: Journal of Applied Mathematics
 (2013)

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## 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.


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## Body

## 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(1)H(t)B(r,t)≡{12σ(t)2rν∂2∂r2+μ(r,t)∂∂r-r}B(r,t)=∂B(r,t)∂t,
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 algebraSU(1,1)⊕h(1) [19]:
(2)W1=rγ∂∂r-λrα,W2=11-γr1-γ,W3=1,K-≡12W12=12r2γ∂2∂r2+(12γr2γ-1-λrα+γ)∂∂r+12λrα(λrα-αrγ-1),K0≡14(W1W2+W2W1)=12(1-γ)r∂∂r+14-λ2(1-γ)rα+1-γ,K+≡12W22=12(1-γ)2r2(1-γ),
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:(3){12σ(t)2∂2∂r2+κ(t)[θ(t)-r]∂∂r-r}B(r,t)=∂B(r,t)∂t.
By applying the Wei-Norman theorem, we can derive the bond price B(r,t) as (see Appendix B)
(4)B(r,t)=U0(t)UI(t)B(r,0),B(r,0)=1,
where
(5)U0(t)=exp{c1(t)K+}exp{c2(t)K0}exp{c3(t)K-},UI(t)=exp{g1(t)W1}exp{g2(t)W2}exp{g3(t)W3},c1(t)=0,c2(t)=-2∫0t‍κ(τ)dτ,c3(t)=∫0t‍σ(τ)2exp{c2(τ)}dτ,g1(t)=∫0t‍[κ(τ)θ(τ)exp{12c2(τ)}+c3(τ)exp{-12c2(τ)}]dτ,g2(t)=-∫0t‍exp{-12c2(τ)}dτ,g3(t)=∫0t‍[12κ(τ)+g1(τ)exp{-12c2(τ)}]dτ.
As a result, the bond price B(r,t) can be expressed as
(6)B(r,t)=exp{14c2(t)+g3(t)+g2(t)×[g1(t)+12c3(t)g2(t)]}×exp{g2(t)exp[12c2(t)]r}.
In the special case of constant model parameters, that is, σ(t)=σ0, κ(t)=κ0, and θ(t)=θ0, the ci(t) and gi(t) can be analytically determined as
(7)c1(t)=0,c2(t)=-2κ0t,c3(t)=σ022{1-exp(-2κ0t)κ0},g1(t)=(κ0θ0-σ022κ0){1-exp(-2κ0t)κ0}+σ022κ0{exp(κ0t)-1κ0},g2(t)=-{exp(κ0t)-1κ0},g3(t)=12κ0t-(θ0-σ022κ02)t+(θ0-σ022κ02)×{exp(κ0t)-1κ0}+σ024κ0{exp(κ0t)-1κ0}2,
and the bond price B(r,t) is reduced to the well-known closed-form expression [1]:
(8)B(r,t)=exp{(θ0-σ022κ02)[1-exp(-κ0t)κ0-t]-σ024κ0[1-exp(-κ0t)κ0]2-[1-exp(-κ0t)κ0-t]r}.(
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:
(9){12σ(t)2r∂2∂r2+[14σ(t)2-κ(t)r-2λ~(t)r]∂∂r-r}×B(r,t)=∂B(r,t)∂t.
As in the Vasicek model, we apply the Wei-Norman theorem to determine the bond price B(r,t) as (see Appendix B)
(10)B(r,t)=U0(t)UI(t)B(r,0),B(r,0)=1,
where
(11)U0(t)=exp{c1(t)K+}exp{c2(t)K0}exp{c3(t)K-},UI(t)=exp{g1(t)W1}exp{g2(t)W2}exp{g3(t)W3},dc1(t)dt=-1-2λ~(t)c1(t)+12σ(t)2c1(t)2,c1(0)=0,c2(t)=∫0t‍{-2λ~(τ)+σ(τ)2c1(τ)}dτ,c3(t)=12∫0t‍σ(τ)2exp{c2(τ)}dτ,g1(t)=12∫0t‍κ(τ)[c1(τ)c3(τ)exp{-12c2(τ)}-exp{12c2(τ)}]dτ,g2(t)=-12∫0t‍κ(τ)c1(τ)exp{-12c2(τ)}dτ,g3(t)=∫0t‍[12λ~(τ)+12κ(τ)c1(τ)exp{-12c2(τ)}g1(τ)]dτ.
Accordingly, the bond price B(r,t) is given by
(12)B(r,t)=exp{+12c3(t)g2(t)2+14c2(t)g3(t)+g1(t)g2(t)=+12c3(t)g2(t)2+14c2(t)}×exp{g2(t)exp[12c2(t)]2r+c1(t)r}.
In the special case of constant model parameters, that is, σ(t)=σ0, κ(t)=κ0, and λ~(t)=λ~0, the ci(t) and gi(t) can be analytically determined as
(13)c1(t)=-2(exp{γt}-1)(γ+2λ~0)(exp{γt}-1)+2γ=2λ~0-γσ02+2γσ02[1-C0exp{γt}],c2(t)=-2λ~0t+2ln{2γexp[(1/2)(γ+2λ~0)t](γ+2λ~0)(exp{γt}-1)+2γ}=γt+2ln{2γ(γ-2λ~0)[1-C0exp{γt}]},c3(t)=-12σ02c1(t),g1(t)=2κ0λ~0γ2exp{-12γt}(1-exp{12γt})2-κ02γ(exp{12γt}-exp{-12γt}),g2(t)=2κ0γ2exp{-12γt}(1-exp{12γt})2,g3(t)=(12λ~0-κ02γ2)t-κ02λ~0γ4exp{-γt}×(1-exp{12γt})4+κ022γ3(exp{γt}-exp{-γt})
with γ=4λ~02+2σ02 and C0=(2λ~0+γ)/(2λ~0-γ). Consequently, we are able to recover the well-known closed-form expression of the bond price B(r,t) [3]:
(14)B(r,t)=Ψ(t)exp{Ω(t)r+Γ(t)r},
where
(15)Ψ(t)=1-C01-C0exp{γt}×exp(α1+α2t+α3+α4exp{(1/2)γt}1-C0exp{γt}),Ω(t)=2λ~0-γσ02+2γσ02[1-C0exp{γt}],Γ(t)=2κ0(2λ~0+γ)(1-exp{(1/2)γt})2γσ02[1-C0exp{γt}],
with
(16)α1=-κ02γ3σ02(4λ~0+γ)(2λ~0-γ),α2=2λ~0+γ4-κ02γ2,α3=4κ02γ3σ02(2λ~02-σ02),α4=-8κ02λ~0γ3σ02(2λ~0+γ).(
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:
(17){12σ(t)2∂2∂r2+[12A0(t)r+σ(t)2r]∂∂r-r}B(r,t)=∂B(r,t)∂t
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)
(18)B(r,t)=exp{(g1(t)+12c3(t)g2(t))34c2(t)+g3(t)+g2(t)×(g1(t)+12c3(t)g2(t))}×exp{g2(t)exp(12c2(t))r}×{1+[g1(t)+c3(t)g2(t)]exp(12c2(t))1r},
where
(19)c2(t)=∫0t‍A0(τ)dτ,c3(t)=∫0t‍σ(τ)2exp{c2(τ)}dτ,g1(t)=∫0t‍c3(τ)exp{-12c2(τ)}dτ,g2(t)=-∫0t‍exp{-12c2(τ)}dτ,g3(t)=∫0t‍[-34A0(τ)+g1(τ)exp{-12c2(τ)}]dτ.(
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:
(20){12σ(t)2r4∂2∂r2+[r2-12A0(t)r]∂∂r-r}B(r,t)=∂B(r,t)∂t,
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)
(21)B(r,t)=1-g1(t)exp{-12c2(t)}r,
where
(22)c2(t)=∫0t‍A0(τ)dτ,g1(t)=∫0t‍exp{12c2(τ)}dτ.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 subalgebra𝔏 of the Lie algebra SU(1,1)⊕h(1):
(23)K-=12r2γ∂2∂r2+(12γ-λ)r2γ-1∂∂r+αr2γ-2,K0=12(1-γ)r∂∂r+14-λ2(1-γ),K+=12(1-γ)2r2(1-γ),W3=1,
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
(24)∂B(r,t)∂t={12σ2r∂2∂r2+κ(θ-r)∂∂r-r}B(r,t)={σ2K--κK0-12K++κ2θσ2W3}B(r,t).
(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 priceB(r,t) can be easily determined as (see Appendix B)
(25)B(r,t)=exp{2c1(t)r}exp{κθσ2[κt+c2(t)]},
where
(26)dc1(t)dt=-12-κc1(t)+σ2c1(t)2,c1(0)=0,c2(t)=-κt+σ2∫0t‍c1(τ)dτ.
The Riccati equation with constant coefficients in (26) can be straightforwardly solved to yield
(27)c1(t)=-exp{γt}-1(γ+κ)(exp{γt}-1)+2γ
with γ=κ2+2σ2. Once the c1(t) has been found, we are also able to obtain
(28)c2(t)=-κt+2ln(2γexp{(1/2)(γ+κ)t}(γ+κ)(exp{γt}-1)+2γ)
via analytical integrations. Beyond question, our finding is in agreement with the well-known closed-form result [2]:
(29)B(r,t)=(2γexp{(1/2)(γ+κ)t}(γ+κ)(exp{γt}-1)+2γ)2κθ/σ2×exp{-2(exp{γt}-1)r(γ+κ)(exp{γt}-1)+2γ}.(
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 r3/2 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 r3/2 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
(30)∂B(r,t)∂t={12σ2r3∂2∂r2+σ2[a(t)r-qr2]∂∂r-r}B(r,t)={σ2K--σ2a(t)K0+σ2(q+1)a(t)W3}B(r,t),
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)
(31)B(r,t)=exp{σ2(q+1)∫0t‍a(τ)dτ}exp{c2(t)K0}×exp{c3(t)K-}B(r,0),
where B(r,0)=1 and
(32)c2(t)=-σ2∫0t‍a(τ)dτ,c3(t)=σ2∫0t‍exp{c2(τ)}dτ.
Without loss of generality, we suppose that B(r,0)=x-(2q+1)V(x), where x=2/r,
(33)V(x)=∫0∞‍dννJp(xν)∫0∞‍dyyJp(yν)V(y),
and p=(2q+1)2+8/σ2. It is not difficult to show that the bond price B(r,t) is given by
(34)B(r,t)=∫0∞‍dx′G(x,t;x′,0)B(r′,0),
where x′=2/r′ and
(35)G(x,t;x′,0)=x′[x′xexp{c2(t)/2}]2q+1×∫0∞‍dννJp(xνexp{c2(t)2})×Jp(x′ν)exp{-c3(t)2ν2}.
The function Jp(ξ) is the Bessel function of the first kind of order p. Here we have made use of the fact that x-(2q+1)Jp(xν) is an eigenfunction of the operator K- with the eigenvalue -ν2/2. The integral over ν can be analytically evaluated to give [23]
(36)1c3(t)exp{-x′2+x2exp{c2(t)}2c3(t)}Ip(x′xexp{c2(t)/2}c3(t))
for p>-1, x′>0, xexp{c2(t)/2}>0, and |arg[c2(t)/2]1/2|<π/4. The function Ip(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
(37)G(x,t;x′,0)=x′c3(t)exp{(2q+1)c2(t)/2}(x′x)2q+1×Ip(x′xexp{c2(t)/2}c3(t))×exp{-x′2+x2exp{c2(t)}2c3(t)}.
Since B(r,0)=1, we can readily derive the bond price B(r,t) as follows:
(38)B(r,t)=x-(2q+1)c3(t)exp{(2q+1)c2(t)/2}×exp{-x2exp{c2(t)}2c3(t)}×∫0∞‍dx′x′2(q+1)exp{-x′22c3(t)}×Ip(x′xexp{c2(t)/2}c3(t))=Γ(p+1-ω)Γ(p+1)M(ω,p+1,-2exp{c2(t)}c3(t)r)×{2exp{c2(t)}c3(t)r}ω,
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 symmetrySU(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].



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