No. 96-01

An Interest Rate Model with Upper and Lower Bounds

by Freddy Delbaen^{†} and Hiroshi Shirakawa^{‡}

First Version, August 1996 Revised Version August 1997 This Version, August 2002

†Department of Mathematics, Eidgenössische Technische Hochschule Zürich ‡Department of Industrial and Systems Engineering, Tokyo Institute of Technology

An Interest Rate Model with Upper and Lower Bounds

Freddy Delbaen Department of Mathematics, Eidgenössische Technische Hochschule Zürich CH-8092 Zürich, Switzerland E-mail : delbaen@math.ethz.ch and Hiroshi Shirakawa^{†} Department of Industrial Engineering and Management, Tokyo Institute of Technology 2-12-1 Oh-Okayama, Meguro-ku, Tokyo 152, Japan E-mail : sirakawa@me.titech.ac.jp

Abstract: We propose a new interest rate dynamics model where the interest rates fluctuate in a bounded region. The model is characterised by five parameters which are sufficiently flexible to reflect the prediction of the future interest rates distribution. The interest rate converges in law to a Beta distribution and has transition probabilities which are represented by a series of Jacobi polynomials. We derive the moment evaluation formula of the interest rate. We also derive the arbitrage free pure discount bond price formula by a weighted series of Jacobi polynomials. Furthermore we give simple lower and upper bounds for the arbitrage free discount bond price which are tight for the narrow interest rates region case. Finally we show that the numerical evaluation procedure converges to the exact value in the limit and evaluate the accuracy of the approximation formulas for the discount bond prices.

Keywords: Bounded state space, Beta distribution, Jacobi polynomials, Discount bond price, Lower and Upper Bounds, Numerical computation.

1 Introduction

In this paper, we study a new interest rate dynamics model which is defined by the following Markov type stochastic differential equation :

dr_{t} = α(r_{μ} − r_{t})dt + β\sqrt{(r_{t} − r_{m})(r_{M} − r_{t})}dW_{t}.  (1.1)

W = {W_{t}; 0 ≤ t} is a standard Wiener process on the probability space (Ω, F, P) equiped with the natural filteration {F_{t}; 0 ≤ t} generated by W, i.e. {F_{t}; 0 ≤ t} satisfies the usual assumptions. We assume that α, β > 0 and r_{m} < r_{μ} < r_{M} which will garantee the existence of stationary distribution. This type of diffusions (called Jacobi processes) have been well studied in genetics (see Ethier-Kurtz [3] and Karlin-Taylor [5]). Also Warren-Yor [8] studied the law of functionals of Jacobi process through Girsanov and time-change techniques. Here we use this model to study the interest rate process {r_{t}} which fluctuates in the bounded region [r_{m}, r_{M}]. The same model was studied by Fujita, see [4]. The methods used are (of course) related but also at the same time different. Fujita essetially uses the perturbation method whereas we use a spectral method. In our model, the parameter α reflects the speed of reversion to the longrun mean r_{μ} and β represents the scale of variance caused by the random process W. Let σ(x) = β\sqrt{(x − r_{m})(r_{M} − x)}. Then for any x, y ∈ [r_{m}, r_{M}],

|σ^{2}(x) − σ^{2}(y)| = β|r_{m} + r_{M} − (x + y)||x − y| ≤ β(r_{M} − r_{m})|x − y|.

†Part of the research was done while the second auther was visiting the Vrije Universiteit Brussel, Departement Wiskunde.

He wishes to extend his deep thanks for their hospitality. Also we appreciate Marc Yor for his helpful comments. This research is partly supported by the Dai-ichi Life Insurance Co. and the Toyo Trust and Banking Co. Ltd.

This means that the diffusion coefficient function σ(x) is Hölder \frac{1}{2} continuous. Since the drift function μ(x) = α(r_{μ} − x) is Lipschitz continuous, the pathwise uniqueness of the stochastic differential equation (1.1) is guaranteed by the general uniqueness theorem ([6], Theorem 4.5, p.360). The object of this paper is to evaluate the arbitrage free discount bond price for the Jacobi-type interest rate processes. The use of such models could be questionable. In particular it is not clear how the upper and lower bounds, r_{M} and r_{m} can be fixed or derived from data. However in risk management of portfolios of interest related instruments, we might be interested in the influence of high and low interest rates. As an example we could ask whether the value (under a certain model) of the position is due to the probability that the interest rate reaches a high level. In such a case we might want to evaluate the value under a model that does not allow for such higher values of the interest rate. In risk management it is quite normal to analyse the value of a portfolio under alternative models. The model that we describe leads to tractable expressions and therefore might be of some use in practice. By carefully adapting the coefficient β we can see the CIR-model, [2] as a limit of the present model. More precisely one can easily show that if we put r_{m} = 0, if we take β_{M} so that β_{M}r_{M}^{1/2} = σ_{0} and if r_{M} → +∞, the law of the process (1.1) tends to the CIR-model.

Let’s consider the following variable transformation :

z_{t} = \frac{r_{t} − r_{m}}{r_{M} − r_{m}} .  (1.2)

Then z_{t} follows

dz_{t} = α(γ − z_{t})dt + β\sqrt{z_{t}(1 − z_{t})}dW_{t},  (1.3)

where γ = \frac{r_{μ}−r_{m}}{r_{M} −r_{m}}. Equation (1.3) is a special case of equation (1.1). However we can derive the properties of {r_{t}; t ≥ 0} from those of {z_{t}; t ≥ 0} through the inverse transformation r_{t} = r_{m} + (r_{M} − r_{m})z_{t}. Hence we shall consider (1.3) to study the equation (1.1).

This paper is organized as follows. In Section 2, we summarize the basic properties of the process {r_{t}} defined by (1.1), e.g. the hitting probability of boundaries, transition probability density and the moment evaluation. In Section 3, we study the arbitrage free pure discount bond price and give a spectral representation by a weighted series of Jacobi polynomials. Then we consider the approximate evaluation of the bond prices by the trancated sum of a series representation and we derive lower and upper bounds in Section 4. Finally in Section 5, we show that the numerical computation scheme which enables us to evaluate the exact bond prices, converges.

2 Basic Properties of Model

2.1 Hitting Probability

First we derive the hitting probability for the boundaries 0 or 1. Suppose that the process is stopped when it hits the boundaries 0 or 1. Let τ_{y} be the stopping time :

τ_{y} = inf{t ≥ 0; z_{t} = y}, 0 ≤ y ≤ 1.  (2.1)

Warren-Yor [8] have already derived the Laplace transform of τ_{y} using the Jacobi process representation by Bessel processes. Here we consider the standard argument to get the hitting probabilities in simple and easy way. Let ρ_{x,y} be the probability that z_{t} hits y in finite time when it starts from x,

ρ_{x,y} = P [τ_{y} < ∞|Z_{0} = x].  (2.2)

Then by the general result for one dimensional diffusions, ρ_{x,y} is given by (see e.g., Proposition 9.4, [1, p.419]) :

ρ_{x,0} = lim_{y→0,z→0} \frac{B_{x,z}(p, q)}{B_{y,z}(p, q)},  (2.3)

ρ_{x,1} = lim_{y→0,z→0} \frac{B_{y,x}(p, q)}{B_{y,z}(p, q)},  (2.4)

where

\{ \begin{matrix}p & = & 1 − \frac{2αγ}{β^{2}} \\ q & = & 1 − \frac{2α(1−γ)}{β^{2}} \\ B_{x,y}(u, v) & = & ∫_{x}^{y} z^{u−1}(1 − z)^{v−1}dz. \end{matrix}  (2.5)

For x ∈ (0, 1), lim_{y→0} B_{y,x} < ∞ ⇔ p > 0 and lim_{z→1} B_{x,z} < ∞ ⇔ q > 0. Hence from (2.3) and (2.4), we have the following conditions for the attainability of the boundary.

ρ_{x,0} > 0 ⇔ lim_{y→0} B_{y,x}(p, q) < ∞ ⇔ \frac{2αγ}{β^{2}} < 1,  (2.6)
ρ_{x,1} > 0 ⇔ lim_{z→1} B_{x,z}(p, q) < ∞ ⇔ \frac{2α(1 − γ)}{β^{2}} < 1.  (2.7)

From (2.6) and (2.7), the boundary \{0, 1\} is inaccessible if and only if \frac{β^{2}}{2α} ≤ γ ≤ 1 − \frac{β^{2}}{2α}.

2.2 Transition Probability

We can derive the transition probability for (1.1) or (1.3) by spectral methods. Let p(x, s; y, t) be the transition probability density for process {z_{t}; t ≥ 0},

p(x, s; y, t) = \frac{∂}{∂y} P [z_{t} ≤ y|z_{s} = x].  (2.8)

Then p(x, s; y, t) is given by [1, p.410] :

p(x, s; y, t) = ∑_{n=0}^{∞} ϕ_{n}(x)ϕ_{n}(y)π(y)e^{−λ_{n}(t−s)}.  (2.9)

Here π is the nonnegative function which is proportional to the stationary distribution of (1.3),

π(x) = \frac{2K}{β^{2}x(1 − x)} exp\{I(c, x)\}, K > 0,  (2.10)

where

I(c, x) = ∫_{c}^{x} \frac{2(γ − z)}{β^{2}z(1 − z)}dz, c ∈ (0, 1).  (2.11)

{(ϕ_{n}, λ_{n}); n ≥ 0} are the normalized orthogonal eigenfunctions and associated eigenvalues such that

∫_{0}^{1} ϕ_{m}(x)ϕ_{n}(x)π(x)dx = \{ \begin{matrix}1, & m = n, \\ 0, & m ≠ n, \end{matrix}  (2.12)

and

Lϕ_{n} + λ_{n}ϕ_{n} = 0, n ≥ 0,  (2.13)

where L is the differential operator :

Lf(x) = α(γ − x)\frac{∂f(x)}{∂x} + \frac{β^{2}x(1 − x)}{2} \frac{∂^{2}f(x)}{∂x^{2}}.  (2.14)

The complete orthogonal basis {(ϕ_{n}, λ_{n}); n ≥ 0} for the operator L is given by Jacobi polynomials. That is :

J_{n}(x; u, v) = ∑_{k=0}^{n}(−1)^{k} ( \begin{matrix}n \\ k \end{matrix} ) \frac{(u + n)_{k}}{(v)_{k}} x^{k}, n ≥ 0,  (2.15)

where (x)_{k} = \frac{Γ(x+k)}{Γ(x)}. The properties of Jacobi polynomials are summarized as follows [7].

Proposition 2.1 (1) For each n ≥ 0, J_{n}(x; u, v) is the solution of the differential equation,

x(1 − x)y′′ + [v − (u + 1)x]y′ + n(u + n)y = 0, y(0) = 1, y′(0) = −\frac{n(u + n)}{v}.  (2.16)

(2) Let u − v > −1 and v > 0. Then the following properties hold for each m, n ≥ 0.

(i) ∫_{0}^{1} x^{v+k−1}(1 − x)^{u−v}J_{n}(x; u, v)dx = 0, for k = 0, 1, ⋯, n − 1,  (2.17)
(ii) ∫_{0}^{1} x^{v−1}(1 − x)^{u−v}J_{m}(x; u, v)J_{n}(x; u, v)dx
= \{ \begin{matrix}\frac{n!Γ(n+u−v+1)Γ^{2}(v)}{(u+2n)Γ(u+n)Γ(v+n)}, & if m = n, \\ 0, & if m ≠ n.  □ \end{matrix}  (2.18)

From these we can easily check the following explicit representation for (2.9) : Theorem 2.2

p(x, s; y, t) = ∑_{n=0}^{∞} k_{n}ψ_{n}(x)ψ_{n}(y)w(y)e^{−λ_{n}(t−s)},  (2.19)

where

\{ \begin{matrix}a & = & \frac{2αγ}{β^{2}} > 0 \\ b & = & \frac{2α(1−γ)}{β^{2}} > 0 \\ λ_{n} & = & αn + \frac{β^{2}}{2} n(n − 1) \\ k_{n} & = & \frac{(a+b+2n−1)Γ(a+n)Γ(a+b+n−1)}{n!Γ(a)^{2}Γ(b+n)} \\ ψ_{n}(x) & = & ∑_{k=0}^{n}(−1)^{k} ( \begin{matrix}n \\ k \end{matrix} ) \frac{(a+b+n−1)_{k}}{(a)_{k}} x^{k} \\ w(x) & = & x^{a−1}(1 − x)^{b−1}. \end{matrix}  (2.20)

Especially, the stationary and limit distribution is given by the Beta distribution with density :

\frac{Γ(a + b)}{Γ(a)Γ(b)}w(y). □  (2.21)

2.3 Moment Evaluation

From the transition probability (2.19), we can deduce the moments of z_{t}. That is

E[z_{t}^{n}|z_{s} = z]

= ∫_{0}^{1} x^{n}p(z, s; x, t)dx
= ∑_{v=0}^{∞} k_{v}ψ_{v}(z)e^{−λ_{v}(t−s)} ∫_{0}^{1} x^{n}ψ_{v}(x)w(x)dx
= ∑_{v=0}^{n} k_{v}ψ_{v}(z)e^{−λ_{v}(t−s)} ∑_{k=0}^{v}(−1)^{k} ( \begin{matrix}v \\ k \end{matrix} ) \frac{Γ(a + b + v + k − 1)Γ(a)}{Γ(a + b + v − 1)Γ(a + k)} ∫_{0}^{1} x^{n+k}w(x)dx
= ∑_{v=0}^{n} \{∑_{k=0}^{v}(−1)^{k} ( \begin{matrix}v \\ k \end{matrix} ) \frac{(a + b + 2v − 1)(a)_{v}(a + k)_{n}}{v!(b)_{v}(a + b + v + k − 1)_{n−v+1}} \} ψ_{v}(z)e^{−λ_{v}(t−s)}.  (2.22)

The third equality follows from (2.15), (2.17) and the last equality follows from (2.20) and B_{0,1}(p, q) = \frac{Γ(p)Γ(q)}{Γ(p+q)}. From (2.15), (2.20) and (2.22), we can derive the moment formulas explicitly for n=1, 2.

Proposition 2.3 The first and second moments of z_{t} are given as follows.

E[z_{t}|z_{s} = z] = γ + (z − γ)e^{−α(t−s)},  (2.23)
E[z_{t}^{2}|z_{s} = z] = \begin{matrix}\frac{(2αγ+β^{2})γ}{2α+β^{2}} + \frac{(2αγ+β^{2})γ}{α+β^{2}} (z − γ)e^{−α(t−s)} \\ + (z^{2} − \frac{2αγ+β^{2}}{α+β^{2}} z + \frac{αγ(2αγ+β^{2})}{(2α+β^{2})(α+β^{2})}) e^{−(2α+β^{2})(t−s)}.  □ \end{matrix} (2.24)

Next we shall consider the moments of r_{t}. From (1.2) and the binomial theorem,

E[r_{t}^{n}|r_{s} = r]
= E[(r_{m} + (r_{M} − r_{m})z_{t})^{n}|z_{s} = z]
= r_{m}^{n} ∑_{k=0}^{n} ( \begin{matrix}n \\ k \end{matrix} ) (\frac{r_{M} − r_{m}}{r_{m}} )^{k} E[z_{t}^{k}|z_{s} = z],  (2.25)

where z = \frac{r−r_{m}}{r_{M} −r_{m}}. Substituting (2.23), (2.24) to (2.25), we get the following formulas for the moments of r_{t}. Corollary 2.4 The first and second moments of r_{t} are given as follows.

E[r_{t}|r_{s} = r] = r_{μ} + (r − r_{μ})e^{−α(t−s)},  (2.26)
E[r_{t}^{2}|r_{s} = r] = r_{μ}^{2} + \frac{β^{2}}{2α + β^{2}} (r_{μ} − r_{m})(r_{M} − r_{μ})
+\frac{2(r − r_{μ})}{α + β^{2}} (\frac{β^{2}(r_{m} + r_{M})}{2} + αr_{μ}) e^{−α(t−s)}  (2.27)

+\frac{1}{2α + β^{2}} (\frac{2α(r − r_{μ})}{α + β^{2}} (α(r − r_{μ}) − β^{2} (\frac{r_{m} + r_{M}}{2} − r))

−β^{2}(r − r_{m})(r_{M} − r))e^{−(2α+β^{2})(t−s)}. □

3 Arbitrage Free Pure Discount Bond Price

We assume that the probability measure P is already the risk neutral measure. Then the arbitrage free pure discount bond price at time t which pays $ 1 at maturity T, is given by:

B(r, t, T) = E [exp (− ∫_{t}^{T} r_{u}du)| r_{t} = r] .  (3.1)

In general, it is difficult to get the exact evaluation of (3.1). However using the transition probability (2.19), we can represent (3.1) by a weighted series of Jacobi polynomials. Theorem 3.1

B(r, t, T)
= e^{−r_{m}(T −t)} \{ \begin{matrix}1 + ∑_{n=1}^{∞}(r_{M} − r_{m})^{n} \\ ∑_{(v_{n},···,v_{1})∈V^{n}} ψ_{v_{n}} (\frac{r−r_{m}}{r_{M} −r_{m}} ) ∏_{j=n}^{1} k_{v_{j}}q(v_{j}, v_{j−1})I_{t,T}^{(n)}(λ_{v_{n}}, ⋯, λ_{v_{1}}) \end{matrix} \} , (3.2)

where

V^{n} = {(v_{n}, ⋯, v_{1}) ∈ Z_{+}^{n}; |v_{j} − v_{j−1}| ≤ 1, 1 ≤ j ≤ n, v_{0} = 0},  (3.3)

q(v_{j}, v_{j−1}) = \{ \begin{matrix}\frac{(2v(a+b+v−1)+a(a+b−2))Γ^{2}(a)v!Γ(b+v)}{(a+b+2v)(a+b+2v−1)(a+b+2v−2)Γ(a+v)Γ(a+b+v−1)}, & if v_{j} = v_{j−1}, \\ −\frac{v!Γ^{2}(a)Γ(b+v)}{(a+b+2v−1)(a+b+2v−2)(a+b+2v−3)Γ(a+v−1)Γ(a+b+v−2)}, & if |v_{j} − v_{j−1}| = 1, \end{matrix}

v = v_{j} ∨ v_{j−1},  (3.4)
I_{t,T}^{(n)}(λ_{n}, ⋯, λ_{1}) = ∫_{t}^{T} ∫_{s_{n}}^{T} ⋯ ∫_{s_{2}}^{T} exp{− ∑_{j=n}^{1} λ_{j}(s_{j} − s_{j+1})}ds_{1} ⋯ ds_{n}, s_{n+1} = t.  (3.5)

Proof. From (1.2),

B(r, t, T) = E [exp {− ∫_{t}^{T} r_{u}du}| r_{t} = r]
= e^{−r_{m}(T −t)}E_{z,t} [exp {−(r_{M} − r_{m}) ∫_{t}^{T} z_{u}du}]
= e^{−r_{m}(T −t)} \{1 + ∑_{n=1}^{∞} \frac{(−1)^{n}(r_{M} − r_{m})^{n}}{n!} E_{z,t} [(∫_{t}^{T} z_{u}du)^{n}]\} ,

where z = \frac{r−r_{m}}{r_{M} −r_{m}} and E_{z,t}[ · ] = E[ · |z_{t} = z]. Substituting the equality :

E_{z,t} [(∫_{t}^{T} z_{u}du)^{n}] = n! ∫_{t}^{T} ∫_{s_{n}}^{T} ⋯ ∫_{s_{2}}^{T} E_{z,t}[z_{s_{n}}z_{s_{n−1}} ⋯ z_{s_{1}}]ds_{1} ⋯ ds_{n−1}ds_{n},

we get

B(r, t, T) = e^{−r_{m}(T −t)} {1 + ∑_{n=1}^{∞}(−1)^{n}(r_{M} − r_{m})^{n} ∫_{t}^{T} ⋯ ∫_{s_{2}}^{T} E_{z,t}[z_{s_{n}} ⋯ z_{s_{1}}]ds_{1} ⋯ ds_{n}} .  (3.6)

For the given (s_{n}, ⋯, s_{1}),

E_{z,t}[z_{s_{n}} ⋯ z_{s_{1}}] = ∫_{0}^{1} ⋯ ∫_{0}^{1} z_{n} ⋯ z_{1} ∏_{j=n}^{1} p(z_{j+1}, s_{j+1}; z_{j}, s_{j})dz_{1} ⋯ dz_{n},  (3.7)

where s_{n+1} = t, z_{n+1} = z. Since the probability density in (3.7) is given by

∏_{j=n}^{1} p(z_{j+1}, s_{j+1}; z_{j}, s_{j})

= ∏_{j=n}^{1} ∑_{v_{j}=0}^{∞} k_{v_{j}}ψ_{v_{j}}(z_{j+1})ψ_{v_{j}}(z_{j})w(z_{j})e^{−λ_{v_{j}} (s_{j}−s_{j+1})}
= ∑_{(v_{n},···,v_{1})∈Z_{+}^{n}} ψ_{v_{n}}(z) ∏_{j=n}^{1} k_{v_{j}}ψ_{v_{j}}(z_{j})ψ_{v_{j−1}}(z_{j})w(z_{j})e^{−λ_{v_{j}} (s_{j}−s_{j+1})}, v_{0} = 0,

we have

E_{z,t}[z_{s_{n}} ⋯ z_{s_{1}}] = ∑_{(v_{n},···,v_{1})∈Z_{+}^{n}} ψ_{v_{n}}(z) ∏_{j=n}^{1} k_{v_{j}}e^{−λ_{v_{j}} (s_{j}−s_{j+1})} ∫_{0}^{1} z_{j}ψ_{v_{j}}(z_{j})ψ_{v_{j−1}}(z_{j})w(z_{j})dz_{j}. (3.8)

Furthermore as shown in Lemma A.1,

∫_{0}^{1} z_{j}ψ_{v_{j}}(z_{j})ψ_{v_{j−1}}(z_{j})w(z_{j})dz_{j} = \{ \begin{matrix}q(v_{j}, v_{j−1}), & if |v_{j} − v_{j−1}| ≤ 1, \\ 0, & if |v_{j} − v_{j−1}| ≥ 2. \end{matrix}  (3.9)

Then from (3.8) and (3.9), we get

E_{z,t}[z_{s_{n}} ⋯ z_{s_{1}}] = ∑_{(v_{n},···,v_{1})∈V^{n}} ψ_{v_{n}}(z) (∏_{j=n}^{1} k_{v_{j}}q(v_{j}, v_{j−1})) exp {− ∑_{j=n}^{1} λ_{v_{j}}(s_{j} − s_{j+1})} . (3.10)

Combining (3.6) and (3.10), we obtain (3.2). □

4 Approximate Evaluation

From Theorem 3.1, we can approximate the arbitrage free pure discount bond price by the trancated sum of series (3.2). Let

B^{(j)}(r, t, T)
= e^{−r_{m}(T −t)} \{ \begin{matrix}1 + ∑_{n=1}^{j}(r_{M} − r_{m})^{n} \\ ∑_{(v_{n},···,v_{1})∈V^{n}} ψ_{v_{n}} (\frac{r−r_{m}}{r_{M} −r_{m}} ) ∏_{j=n}^{1} k_{v_{j}}q(v_{j}, v_{j−1})I_{t,T}^{(n)}(λ_{v_{n}}, ⋯, λ_{v_{1}}) \end{matrix} \} , 0 ≤ j.

(4.1)

After the “tedious” but elementary calculation, we get the following results for 0 ≤ j ≤ 2. The proof is left to the reader.

Proposition 4.1 The truncated sums of (3.2) up to second order are given as follows.

B^{(0)}(r, t, T) = e^{−r_{m}(T −t)},  (4.2)
B^{(1)}(r, t, T) = e^{−r_{m}(T −t)} (1 − (r_{μ} − r_{m})(T − t) − (r − r_{μ})\frac{1 − e^{−α(T −t)}}{α} ) ,  (4.3)

B^{(2)}(r, t, T)

= e^{−r_{m}(T −t)} \{ \begin{matrix}1 − (r_{μ} − r_{m})(T − t) − (r − r_{μ})\frac{1−e^{−α(T −t)}}{α} + \frac{1}{2}(r_{μ} − r_{m})^{2}(T − t)^{2} \\ + (\frac{β^{2}}{2α+β^{2}} (r_{μ} − r_{m})(r_{M} − r_{μ}) + (r_{μ} − r_{m})(r − r_{μ})) \frac{T −t}{α} \\ + ( \begin{matrix}\frac{α}{α+β^{2}} (r − r_{μ})^{2} + 2 (\frac{β^{2}}{α+β^{2}} )^{2} (r − r_{μ}) (\frac{r_{m}+r_{M}}{2} − r_{μ}) \\ −\frac{β^{2}}{α+β^{2}} (r_{μ} − r_{m})(r_{M} − r_{μ}) \end{matrix} ) \frac{1−e^{−α(T −t)}}{α^{2}} \\ − (\frac{α}{α+β^{2}} (r_{μ} − r_{m})(r − r_{μ}) + \frac{β^{2}}{α+β^{2}} (r − r_{μ})(r_{M} − r_{μ})) \frac{(T −t)e^{−α(T −t)}}{α} \\ − ( \begin{matrix}\frac{α}{α+β^{2}} \frac{β^{2}}{2α+β^{2}} (r_{μ} − r_{m})(r_{M} − r_{μ}) \\ +\frac{α}{α+β^{2}} (r − r_{μ})^{2} − \frac{β^{2}}{α+β^{2}} (r − r_{m})(r_{M} − r) \end{matrix} ) \frac{1−e^{−(2α+β^{2})(T −t)}}{(α+β^{2})(2α+β^{2})} \end{matrix} \} .

□(4.4)

From (4.2) through (4.4), we see that the volatility coefficient β appeared only for j = 2. This means that we should use at least B^{(2)}(r, t, T) to take into account the volatility effect in the approximation. Also B^{(0)} is an obvious upper bound of the discount bond price for any initial interest rate. However we can derive more sophisticated upper bounds as will be shown in Theorem 4.2.

We proposed the approximation formula (4.1) assuming that the higher order terms are negligible. Hence we should consider the lower and upper bounds to evaluate the error magnitude. Since the interest rate distribution has bounded support [r_{m}, r_{M}], we can derive both the lower and upper bounds.

Theorem 4.2 (1) Lower bound :

exp \{−r_{μ}(T − t) − (r − r_{μ})\frac{1 − e^{−α(T −t)}}{α} \} ≤ B(r, t, T).  (4.5)

(2) Upper Bound :

B(r, t, T) ≤ \begin{matrix}(1 − γ − (z − γ)\frac{1−e^{−α(T −t)}}{α(T −t)} ) e^{−r_{m}(T −t)} \\ + (γ + (z − γ)\frac{1−e^{−α(T −t)}}{α(T −t)} ) e^{−r_{M} (T −t)}, \end{matrix}  (4.6)

where z = \frac{r−r_{m}}{r_{M} −r_{m}}.

Proof. Let X be a random variable with mean x_{μ} and with a distribution supported by [x_{m}, x_{M}]. Then for any convex function f, the following inequalities are obvious consequences of Jensen’s inequality.

f(x_{μ}) ≤ E[f(X)] ≤ \frac{x_{M} − x_{μ}}{x_{M} − x_{m}} f(x_{m}) + \frac{x_{μ} − x_{m}}{x_{M} − x_{m}} f(x_{M}).  (4.7)

The left hand side is attained when X is degenerated to the mean x_{μ} and the right hand side is attained when X follows two point distribution such that X ∈ {x_{m}, x_{M}} and E[X] = x_{μ}. Since f(x) = e^{−x} is a convex function, we can apply (4.7) for X = ∫_{t}^{T} r_{u}du. From (2.26),

E [∫_{t}^{T} r_{u}du| r_{t} = r] = ∫_{t}^{T} (r_{μ} + (r − r_{μ})e^{−α(u−t)})du
= r_{μ}(T − t) + (r − r_{μ})\frac{1 − e^{−α(T −t)}}{α} .  (4.8)

Furthermore since r_{u} ∈ [r_{m}, r_{M}] for all u, r_{m}(T − t) ≤ ∫_{t}^{T} r_{u}du ≤ r_{M}(T − t). This together with (4.7) and (4.8), we get (4.5) and (4.6). □

Remark 4.3 The lower bound (4.5) is valid for all the linear mean reversion type interest rate models, e.g. [2]. However (4.5) is expecially useful in our model since the lower bound seems to be close to the exact value for reasonable parameters. This comes from the boundedness of the fluctuation. We will study this effect numerically in the next section.

5 Numerical Computation

For the sophisticated approximate evaluation of discount bond price (3.1), we can use a numerical computation method. Let us define the discrete state space S^{(Δ,ϵ,c_{m},c_{M})} = {r_{n}^{(Δ,ϵ,c_{m},c_{M})}; n ∈ I^{(Δ,ϵ,c_{m},c_{M})}} and the indices set I^{(Δ,ϵ,c_{m},c_{M})} = {−N_{m}^{(Δ,ϵ,c_{m})}, −N_{m}^{(Δ,ϵ,c_{m})} + 1, ⋯, 0, ⋯, N_{M}^{(Δ,ϵ,c_{M})} − 1, N_{M}^{(Δ,ϵ,c_{M})}} for (Δ, ϵ, c_{m}, c_{M}) ∈ R_{+}^{4} by

r_{n}^{(Δ,ϵ,c_{m},c_{M})} = \{ \begin{matrix}r_{μ} + (r_{M} − r_{μ})(1 − αΔ)^{N_{M}^{(Δ,ϵ,c_{M})}−n}, & if 1 ≤ n ≤ N_{M}^{(Δ,ϵ,c_{M})}, \\ r_{μ}, & if n = 0, \\ r_{μ} − (r_{μ} − r_{m})(1 − αΔ)^{N_{m}^{(Δ,ϵ,c_{M})}+n}, & if − N_{m}^{(Δ,ϵ,c_{m})} ≤ n ≤ −1, \end{matrix}  (5.1)

where

N_{x}^{(Δ,ϵ,c)} = ⌈\frac{log\{(1 + ϵ)βcη_{x}\sqrt{Δ}\}}{log(1 − αΔ)}⌉ , x = m, M,  (5.2)

η_{x} = \{ \begin{matrix}\sqrt{\frac{γ}{1−γ}} , & if x = M, \\ \sqrt{\frac{1−γ}{γ}} , & if x = m, \end{matrix}  (5.3)
⌈x⌉ = inf{n ∈ Z; n ≥ x}.

We can easily check that

lim_{Δ→0} N_{x}^{(Δ,ϵ,c)} = ∞,  (5.4)
lim_{Δ→0} sup_{−N_{m}^{(Δ,ϵ,cm)}≤n≤N_{M}^{(Δ,ϵ,c_{M})}−1} |r_{n+1}^{(Δ,ϵ,c_{m},c_{M})} − r_{n}^{(Δ,ϵ,c_{m},c_{M})}| = 0.  (5.5)

Let us define the drift and diffusion term for this discrete time state space :

μ_{n}^{(Δ,ϵ,c_{m},c_{M})} = α(r_{μ} − r_{n}^{(Δ,ϵ,c_{m},c_{M})})Δ1{|n| ≥ 2}
= \{ \begin{matrix}−αΔ(1 − αΔ)^{N_{M}^{(Δ,ϵ,c_{M})}−n}(r_{M} − r_{μ}), & if n ≥ 2, \\ 0, & if |n| ≤ 1, \\ αΔ(1 − αΔ)^{N_{m}^{(Δ,ϵ,cm)}+n}(r_{μ} − r_{m}), & if n ≤ −2, \end{matrix}  (5.6)

and

σ_{n}^{(Δ,ϵ,c_{m},c_{M})}
= β\sqrt{(r_{n}^{(Δ,ϵ,c_{m},c_{M})} − r_{m})(r_{M} − r_{n}^{(Δ,ϵ,c_{m},c_{M})})Δ}

= \{ \begin{matrix}β(r_{M} − r_{m})\sqrt{\begin{matrix}(1 − γ)(γ + (1 − γ)(1 − αΔ)^{N_{M}^{(Δ,ϵ,c_{M})}−n}) \\ ×(1 − (1 − αΔ)^{N_{M}^{(Δ,ϵ,c_{M})}−n})Δ \end{matrix}} , & if n ≥ 1, \\ β(r_{M} − r_{m})\sqrt{γ(1 − γ)Δ}, & if n = 0, \\ β(r_{M} − r_{m})\sqrt{\begin{matrix}γ(1 − γ + γ(1 − αΔ)^{N_{m}^{(Δ,ϵ,cm)}+n}) \\ ×(1 − (1 − αΔ)^{N_{m}^{(Δ,ϵ,cm)}+n})Δ \end{matrix}} , & if n ≤ −1. \end{matrix}  (5.7)

(5.6) and (5.7) correspond to the discretized value of the drift and diffusion coefficient functions μ, σ for all n ∈ I^{(Δ,ϵ,c_{m},c_{M})} except for μ_{±1}. From the definition of state space, we can easily deduce

μ_{n}^{(Δ,ϵ,c_{m},c_{M})} = \{ \begin{matrix}r_{n−1}^{(Δ,ϵ,c_{m},c_{M})} − r_{n}^{(Δ,ϵ,c_{m},c_{M})}, & if n ≥ 2, \\ 0, & if |n| ≤ 1, \\ r_{n+1}^{(Δ,ϵ,c_{m},c_{M})} − r_{n}^{(Δ,ϵ,c_{m},c_{M})}, & if n ≤ −2. \end{matrix}  (5.8)

This together with the next lemma play a key role to construct a sequence of Markov chains which converge to (1.1).

Lemma 5.1 For sufficiently small Δ > 0, there exist (n_{1}, n_{2}) for n ∈ I^{(Δ,ϵ,c_{m},c_{M})} such that

Case I : n > 0; 0 ≤ n_{1} ≤ n ≤ n_{2} ≤ N_{M}^{(Δ,ϵ,c_{M})},
r_{n}^{(Δ,ϵ,c_{m},c_{M})} + \frac{\sqrt{α(1 − γ)}}{(1 + ϵ)β}σ_{n}^{(Δ,ϵ,c_{m},c_{M})} ≤ r_{n_{2}}^{(Δ,ϵ,c_{m},c_{M})},  (5.9)
r_{(n−1)∨1}^{(Δ,ϵ,c_{m},c_{M})} − c_{M}σ_{n}^{(Δ,ϵ,c_{m},c_{M})} ≥ r_{n_{1}}^{(Δ,ϵ,c_{m},c_{M})}.  (5.10)
Case II : n < 0; −N_{m}^{(Δ,ϵ,c_{m})} ≤ n_{2} ≤ n ≤ n_{1} ≤ 0,
r_{(n+1)∧−1}^{(Δ,ϵ,c_{m},c_{M})} + c_{m}σ_{n}^{(Δ,ϵ,c_{m},c_{M})} ≤ r_{n_{1}}^{(Δ,ϵ,c_{m},c_{M})},  (5.11)
r_{n}^{(Δ,ϵ,c_{m},c_{M})} − \frac{\sqrt{αγ}}{(1 + ϵ)β} σ_{n}^{(Δ,ϵ,c_{m},c_{M})} ≥ r_{n_{2}}^{(Δ,ϵ,c_{m},c_{M})}.  (5.12)
Case III : n = 0; n_{1} = −1, n_{2} = 1,
r_{n}^{(Δ,ϵ,c_{m},c_{M})} + c_{M}σ_{n}^{(Δ,ϵ,c_{m},c_{M})} ≤ r_{n_{2}}^{(Δ,ϵ,c_{m},c_{M})},  (5.13)
r_{n}^{(Δ,ϵ,c_{m},c_{M})} − c_{m}σ_{n}^{(Δ,ϵ,c_{m},c_{M})} ≥ r_{n_{1}}^{(Δ,ϵ,c_{m},c_{M})}.  (5.14)

Proof. Since the proof is similar, we prove only (5.9) and (5.10). If n = N_{M}^{(Δ,ϵ,c_{M})}, then σ_{n}^{(Δ,ϵ,c_{m},c_{M})} = 0 and hence there exist n_{1} = n − 1 and n_{2} = n which satisfy (5.9) and (5.10). Let us assume 0 < n < N_{M}^{(Δ,ϵ,c_{M})}. Then

lim_{Δ→0} sup_{0<n<N_{M}^{(Δ,ϵ,c_{M})}} \frac{\sqrt{α(1 − γ)}}{(1 + ϵ)β} \frac{σ_{n}^{(Δ,ϵ,c_{m},c_{M})}}{r_{M} − r_{n}^{(Δ,ϵ,c_{m},c_{M})}}

= lim_{Δ→0} sup_{0<n<N_{M}^{(Δ,ϵ,c_{M})}} \frac{1}{1 + ϵ}\sqrt{(\frac{1}{1 − (1 − αΔ)^{N_{M}^{(Δ,ϵ,c_{M})}}} − (1 − γ)) αΔ}
= lim_{Δ→0} \frac{1}{1 + ϵ}\sqrt{1 − (1 − γ)αΔ}
= \frac{1}{1 + ϵ} < 1.

Hence (5.9) holds. On the other hand,

lim_{Δ→0} sup_{0<n<N_{M}^{(Δ,ϵ,c_{M})}} \frac{c_{M}σ_{n}^{(Δ,ϵ,c_{m},c_{M})} + r_{n}^{(Δ,ϵ,c_{m},c_{M})} − r_{(n−1)∨1}^{(Δ,ϵ,c_{m},c_{M})}}{r_{n}^{(Δ,ϵ,c_{m},c_{M})} − r_{0}^{(Δ,ϵ,c_{m},c_{M})}}

= lim_{Δ→0} sup_{0<n<N_{M}^{(Δ,ϵ,c_{M})}} c_{M}β \sqrt{\begin{matrix}(_{\frac{γ}{(1−γ)(1−αΔ)^{N{(Δ,ϵ,c_{M}) \atop M}−n}}} + 1) \\ × (_{\frac{1}{(1−αΔ)^{N{(Δ,ϵ,c_{M}) \atop M}−n}}} − 1) Δ \end{matrix}} + αΔ1\{n > 1\}

= lim_{Δ→0} c_{M}β\sqrt{(\frac{γ}{(1 − γ)(1 − αΔ)^{N_{M}^{(Δ,ϵ,c_{M})}−1}} + 1) (\frac{1}{(1 − αΔ)^{N_{M}^{(Δ,ϵ,c_{M})}−1}} − 1) Δ}

= lim_{Δ→0} c_{M}β\sqrt{\frac{γ}{1 − γ} (\frac{(1 − αΔ)^{ϵ′}}{(1 + ϵ)βc_{M}η_{M} \sqrt{Δ}})^{2} Δ}
= \frac{1}{1 + ϵ} < 1.

In the third equality, ϵ′ ∈ [0, 1) is determined by the truncation operator ⌈ · ⌉. Hence (5.10) holds. □

Let us now for each given Δ > 0, define a Markov chain . Let ϵ > 0 and

c_{m}^{∗} = max \{1, \frac{(1 + ϵ)β}{\sqrt{αγ}} \} ,  (5.15)
c_{M}^{∗} = max \{1, \frac{(1 + ϵ)β}{\sqrt{α(1 − γ)}}\} .  (5.16)

Then we can construct the discrete time Markov chain X^{(Δ)} = {X_{u}^{(Δ)}; u = 0, Δ, 2Δ, ⋯} on the state space S^{(Δ,ϵ,c_{m}^{∗},c_{M}^{∗})} with the indices set I^{(Δ,ϵ,c_{m}^{∗},c_{M}^{∗})}. The transition probability is stationary and is defined by

p_{i,j}^{(Δ)} = P r[X_{t+Δ}^{(Δ)} = r_{j}^{(Δ,ϵ,c_{m}^{∗},c_{M}^{∗})}|X_{t}^{(Δ)} = r_{i}^{(Δ,ϵ,c_{m}^{∗},c_{M}^{∗})}]

= \{ \begin{matrix}\frac{r_{j_{2}(i)}−r_{j_{1}(i)}}{r_{j_{3}(i)}−r_{j_{1}(i)}} \frac{σ_{i}^{2}}{(r_{j_{2}(i)}−r_{j_{1}(i)})(r_{j_{3}(i)}−r_{j_{2}(i)})}, & if j = j_{3}(i), \\ 1 − \frac{σ_{i}^{2}}{(r_{j_{2}(i)}−r_{j_{1}(i)})(r_{j_{3}(i)}−r_{j_{2}(i)})}, & if j = j_{2}(i), \\ \frac{r_{j_{3}(i)}−r_{j_{2}(i)}}{r_{j_{3}(i)}−r_{j_{1}(i)}} \frac{σ_{i}^{2}}{(r_{j_{2}(i)}−r_{j_{1}(i)})(r_{j_{3}(i)}−r_{j_{2}(i)})}, & if j = j_{1}(i), \\ 0, & elsewhere, \end{matrix}  (5.17)

where

j_{1}(i) = \{ \begin{matrix}(N_{M}^{(Δ,ϵ,c_{M})} − ⌈^{\frac{log\{(1−αΔ)^{N{(Δ,ϵ,c{∗ \atop M}) \atop M}−i+1}− \frac{c{∗ \atop M}}{r_{M} −rμ} σ_{i}^{(Δ,ϵ,c^{∗}_{m},c{∗ \atop M})}\}}{log(1−αΔ)}} ⌉) ∨ 0, & if \begin{matrix}1 < i \\ ≤ N_{M}^{(Δ,ϵ,c_{M})}, \end{matrix} \\ −1\{i = 0\}, & if |i| ≤ 1, \\ (⌈^{\frac{log\{(1−αΔ)^{N{(Δ,ϵ,c{∗ \atop M}) \atop m}+i+1}− \frac{c^{∗}_{m}}{rμ−rm} σ_{i}^{(Δ,ϵ,c^{∗}_{m},c{∗ \atop M})}\}}{log(1−αΔ)}} ⌉ − N_{m}^{(Δ,ϵ,c_{m}^{∗})}) ∧ 0, & if \begin{matrix}−N_{m}^{(Δ,ϵ,c_{m}^{∗})} \\ ≤ i < −1, \end{matrix} \end{matrix}

j_{2}(i) = \{ \begin{matrix}i − 1, & if 1 < i ≤ N_{M}^{(Δ,ϵ,c_{M})}, \\ i, & if |i| ≤ 1, \\ i + 1, & if − N_{m}^{(Δ,ϵ,c_{m}^{∗})} ≤ i < −1, \end{matrix}

j_{3}(i) = \{ \begin{matrix}N_{M}^{(Δ,ϵ,c_{M})} − ⌊^{\frac{log\{(1−αΔ)^{N{(Δ,ϵ,c{∗ \atop M}) \atop M}−i}+^{\frac{\sqrt{α(1−γ)}}{(r_{M} −rμ)(1+ϵ)β}} σ_{i}^{(Δ,ϵ,c^{∗}_{m},c{∗ \atop M})}\}}{log(1−αΔ)}} ⌋ , & if \begin{matrix}1 ≤ i \\ ≤ N_{M}^{(Δ,ϵ,c_{M})}, \end{matrix} \\ 1, & if i = 0, \\ ⌊^{\frac{log\{(1−αΔ)^{N{(Δ,ϵ,c{∗ \atop M}) \atop m}+i}+ \frac{\sqrt{αγ}}{(rμ−rm)(1+ϵ)β} σ_{i}^{(Δ,ϵ,c^{∗}_{m},c{∗ \atop M})}\}}{log(1−αΔ)}} ⌋ − N_{m}^{(Δ,ϵ,c_{m}^{∗})}, & if \begin{matrix}−N_{m}^{(Δ,ϵ,c_{m}^{∗})} \\ ≤ i ≤ −1, \end{matrix} \end{matrix}

⌊x⌋ = sup{n ∈ Z; n ≤ x}.  (5.18)

From Lemma 5.1, (5.18) is well defined on I^{(Δ,ϵ,c_{m}^{∗},c_{M}^{∗})}. We can easily check that the following properties hold for the transition probabilities (5.17).

(1) ∑_{j∈I^{(Δ,ϵ,c^{∗}_{m},c{∗ \atop M})}} p_{i,j}^{(Δ)} = 1,  (5.19)
(2) ∑_{j∈I^{(Δ,ϵ,c^{∗}_{m},c{∗ \atop M})}}(r_{j}^{(Δ,ϵ,c_{m}^{∗},c_{M}^{∗})} − r_{i}^{(Δ,ϵ,c_{m}^{∗},c_{M}^{∗})})p_{i,j}^{(Δ)} = μ_{i}^{(Δ,ϵ,c_{m}^{∗},c_{M}^{∗})},  (5.20)
(3) ∑_{j∈I^{(Δ,ϵ,c^{∗}_{m},c{∗ \atop M})}}(r_{j}^{(Δ,ϵ,c_{m}^{∗},c_{M}^{∗})} − r_{i}^{(Δ,ϵ,c_{m}^{∗},c_{M}^{∗})})^{2}p_{i,j}^{(Δ)} = (μ_{i}^{(Δ,ϵ,c_{m}^{∗},c_{M}^{∗})})^{2} + (σ_{i}^{(Δ,ϵ,c_{m}^{∗},c_{M}^{∗})})^{2} .

(5.21)

Furthermore we can show that for sufficiently small Δ, the p_{i,j} are nonnegative.

Lemma 5.2 There exists δ > 0 such that

p_{i,j}^{(Δ)} ≥ 0, for all i, j ∈ I^{(Δ,ϵ,c_{m}^{∗},c_{M}^{∗})}, 0 ≤ Δ ≤ δ.  (5.22)

Proof. From (5.17),

p_{i,j}^{(Δ)} ≥ 0
⇔ (r_{j_{2}(i)}^{(Δ,ϵ,c_{m}^{∗},c_{M}^{∗})} − r_{j_{1}(i)}^{(Δ,ϵ,c_{m}^{∗},c_{M}^{∗})}) (r_{j_{3}(i)}^{(Δ,ϵ,c_{m}^{∗},c_{M}^{∗})} − r_{j_{2}(i)}^{(Δ,ϵ,c_{m}^{∗},c_{M}^{∗})}) ≥ (σ_{i}^{(Δ,ϵ,c_{m}^{∗},c_{M}^{∗})})^{2} . (5.23)

Suppose 1 < i ≤ N_{M}^{(Δ,ϵ,c_{M}^{∗})}. Then

(r_{j_{2}(i)}^{(Δ,ϵ,c_{m}^{∗},c_{M}^{∗})} − r_{j_{1}(i)}^{(Δ,ϵ,c_{m}^{∗},c_{M}^{∗})}) (r_{j_{3}(i)}^{(Δ,ϵ,c_{m}^{∗},c_{M}^{∗})} − r_{j_{2}(i)}^{(Δ,ϵ,c_{m}^{∗},c_{M}^{∗})})

= (r_{M} − r_{μ})^{2} ((1 − αΔ)^{N_{M}^{(Δ,ϵ,c{∗ \atop M})}−j_{2}(i)} − (1 − αΔ)^{N_{M}^{(Δ,ϵ,c{∗ \atop M})}−j_{1}})

((1 − αΔ)^{N_{M}^{(Δ,ϵ,c{∗ \atop M})}−j_{3}(i)} − (1 − αΔ)^{N_{M}^{(Δ,ϵ,c{∗ \atop M})}−j_{2}})

= (r_{M} − r_{μ})^{2}(1 − αΔ)^{−ϵ′′}

((1 − αΔ)^{N_{M}^{(Δ,ϵ,c{∗ \atop M})}−i+1} − ((1 − αΔ)^{N_{M}^{(Δ,ϵ,c{∗ \atop M})}−i+1} − \frac{c_{M}^{∗}}{r_{M} −r_{μ}} σ_{i}^{(Δ,ϵ,c_{m}^{∗},c_{M}^{∗})}) (1 − αΔ)^{ϵ′})
(((1 − αΔ)^{N_{M}^{(Δ,ϵ,c{∗ \atop M})}−i} + ^{\frac{\sqrt{α(1−γ)}}{(r_{M} −r_{μ})(1+ϵ)β}} σ_{i}^{(Δ,ϵ,c_{m}^{∗},c_{M}^{∗})}) − (1 − αΔ)^{N_{M}^{(Δ,ϵ,c{∗ \atop M})}−i+1+ϵ′′}) .

In the last equality, ϵ′, ϵ′′ ∈ [0, 1) are determined by the truncation operators ⌈ · ⌉ and ⌊ · ⌋. From Lemma 5.1, we deduce

r_{i−1}^{(Δ,ϵ,c_{m}^{∗},c_{M}^{∗})} − r_{μ} − c_{M}^{∗}σ_{i}^{(Δ,ϵ,c_{m}^{∗},c_{M}^{∗})} = (r_{M} − r_{μ}) (1 − αΔ)^{N_{M}^{(Δ,ϵ,c{∗ \atop M})}−i+1} − \frac{c_{M}^{∗}}{r_{M} − r_{μ}} σ_{i}^{(Δ,ϵ,c_{m}^{∗},c_{M}^{∗})}) ≥ 0.

Therefore

(r_{j_{2}(i)}^{(Δ,ϵ,c_{m}^{∗},c_{M}^{∗})} − r_{j_{1}(i)}^{(Δ,ϵ,c_{m}^{∗},c_{M}^{∗})}) (r_{j_{3}(i)}^{(Δ,ϵ,c_{m}^{∗},c_{M}^{∗})} − r_{j_{2}(i)}^{(Δ,ϵ,c_{m}^{∗},c_{M}^{∗})})

≥ c_{M} \frac{\sqrt{α(1 − γ)}}{(1 + ϵ)β} (σ_{i}^{(Δ,ϵ,c_{m}^{∗},c_{M}^{∗})})^{2}
≥ (σ_{i}^{(Δ,ϵ,c_{m}^{∗},c_{M}^{∗})})^{2} .

This together with (5.23) gives (5.22). We can handle the cases −N_{m}^{(Δ,ϵ,c_{m}^{∗})} ≤ i < −1 and |i| ≤ 1 in the same way. □

The Markov chain X̃^{(Δ)} converges to the process {r_{t}; 0 ≤ t} defined by (1.1).

Theorem 5.3 Consider the continuous time Markov chain X̃^{(Δ)} defined by

X̃_{t}^{(Δ)} = X_{n_{t}Δ}^{(Δ)}  (5.24)

where n_{t} = ⌊\frac{t}{Δ}⌋. Then

\{X̃_{t}^{(Δ)}; 0 ≤ t\} \underrightarrow{law} \{r_{t}; 0 ≤ t\} as Δ → 0.  (5.25)

Proof. From (5.20) and (5.21), we see that the mean and the quadratic variational process of X̃_{t}^{(Δ)} converge to those of {r_{t}}. Also the transition jump size goes to 0 as Δ → 0. Hence from the martingale central limit theorem ([3], Theorem5.1, p.354), (5.25) holds. □

A Appendix

Lemma A.1 For any m, n ∈ Z, 0 ≤ m ≤ n and u, v ∈ R, u > 0, u − v > −1,

∫_{0}^{1} x^{v}(1 − x)^{u−v}J_{m}(x; u, v)J_{n}(x; u, v)dx

= \{ \begin{matrix}0, & if 0 ≤ m ≤ n − 2, \\ −\frac{n!Γ(n+u−v+1)Γ^{2}(v)}{(u+2n)(u+2n−1)(u+2n−2)Γ(u+n−1)Γ(v+n−1)}, & if 0 ≤ m = n − 1, \\ \frac{(2n(u+n)+v(u−1))Γ^{2}(v)n!Γ(n+u−v+1)}{(u+2v+1)(u+2n)(u+2n−1)Γ(u+n)Γ(v+n)}, & if 0 ≤ m = n. \end{matrix}  (A.1)

Proof. From (2.15) and (2.17), we deduce

∫_{0}^{1} x^{v}(1 − x)^{u−v}J_{m}(x; u, v)J_{n}(x; u, v)dx
= ∑_{k=0}^{m}(−1)^{k} ( \begin{matrix}m \\ k \end{matrix} ) \frac{(u + m)_{k}}{(v)_{k}} ∫_{0}^{1} x^{v+k}(1 − x)^{u−v}J_{n}(x; u, v)dx
= ∑_{k=n−1}^{m}(−1)^{k} ( \begin{matrix}m \\ k \end{matrix} ) \frac{(u + m)_{k}}{(v)_{k}} ∫_{0}^{1} x^{v+k}(1 − x)^{u−v}J_{n}(x; u, v)dx,  (A.2)

where ∑_{k=n−1}^{m} = 0 for n ≥ m + 2. Case I: 0 ≤ m ≤ n − 2. (A.1) follows from (A.2) directly. Case II: 0 ≤ m = n − 1. First we prove the following equation.

∑_{k=0}^{n}(−1)^{k} ( \begin{matrix}n \\ k \end{matrix} ) \frac{(u + n)_{k}}{(v)_{k}} \frac{Γ(v + n + k)Γ(u − v + 1)}{Γ(u + n + k + 1)}  (A.3)
= \frac{(−1)^{n}n!Γ(u − v + n + 1)Γ(v)}{Γ(u + 2n + 1)}, v > 0, u − v > −1.

From (2.15), (2.17) and (2.18) for m = n,

∫_{0}^{1} x^{v−1}(1 − x)^{u−v}J_{n}^{2}(x; u, v)dx

= ∑_{k=0}^{n}(−1)^{k} ( \begin{matrix}n \\ k \end{matrix} ) \frac{(u + n)_{k}}{(v)_{k}} ∫_{0}^{1} x^{v+k−1}(1 − x)^{u−v}J_{n}(x; u, v)dx
= (−1)^{n} \frac{(u + n)_{n}}{(v)_{n}} ∫_{0}^{1} x^{v+n−1}(1 − x)^{u−v}J_{n}(x; u, v)dx
= (−1)^{n} \frac{(u + n)_{n}}{(v)_{n}} ∑_{k=0}^{n}(−1)^{k} ( \begin{matrix}n \\ k \end{matrix} ) \frac{(u + n)_{k}}{(v)_{k}} ∫_{0}^{1} x^{v+n+k−1}(1 − x)^{u−v}dx
= (−1)^{n} \frac{Γ(u + 2n)Γ(v)}{Γ(u + n)Γ(v + n)} ∑_{k=0}^{n}(−1)^{k} ( \begin{matrix}n \\ k \end{matrix} ) \frac{(u + n)_{k}}{(v)_{k}} \frac{Γ(v + n + k)Γ(u − v + 1)}{Γ(u + n + k + 1)}
= \frac{n!Γ(u − v + n + 1)Γ^{2}(v)}{(u + 2n)Γ(u + n)Γ(v + n)}.  (A.4)

Rearranging the last equality, we obtain (A.3). Next we shall show (A.1) for Case II. From (A.2),

∫_{0}^{1} x^{v}(1 − x)^{u−v}J_{n−1}(x; u, v)J_{n}(x; u, v)dx
= (−1)^{n−1} \frac{(u + n − 1)_{n−1}}{(v)_{n−1}} ∫_{0}^{1} x^{v+n−1}(1 − x)^{u−v}J_{n}(x; u, v)dx
= (−1)^{n−1} \frac{(u + n − 1)_{n−1}}{(v)_{n−1}} ∑_{k=0}^{n}(−1)^{k} ( \begin{matrix}n \\ k \end{matrix} ) \frac{(u + n)_{k}}{(v)_{k}} ∫_{0}^{1} x^{v+n+k−1}(1 − x)^{u−v}dx
= (−1)^{n−1} \frac{(u + n − 1)_{n−1}}{(v)_{n−1}} ∑_{k=0}^{n}(−1)^{k} ( \begin{matrix}n \\ k \end{matrix} ) \frac{(u + n)_{k}}{(v)_{k}} \frac{Γ(v + n + k)Γ(u − v + 1)}{Γ(u + n + k + 1)}
= (−1)^{n−1} \frac{(u + n − 1)_{n−1}}{(v)_{n−1}} \frac{(−1)^{n}n!Γ(u − v + n + 1)Γ(v)}{Γ(u + 2n + 1)}
= −\frac{n!Γ(u − v + n + 1)Γ^{2}(v)}{(u + 2n)(u + 2n − 1)(u + 2n − 2)Γ(u + n − 1)Γ(v + n − 1)},

where the fourth equality follows from (A.3). Case III : 0 ≤ m = n. First we prove the following equation by induction with respect to n.

∑_{k=0}^{n}(−1)^{k} ( \begin{matrix}n \\ k \end{matrix} ) \frac{(u + n)_{k}}{(v)_{k}} \frac{Γ(v + n + k + 1)Γ(u − v + 1)}{Γ(u + n + k + 2)}  (A.5)
= \frac{(−1)^{n}(n + 1)!(v + n)Γ(u − v + n + 1)Γ(v)}{Γ(u + 2n + 2)}, v > 0, u − v > −1.

We can easily check that (A.5) holds for n = 0. Suppose that (A.5) holds for n. Then

∑_{k=0}^{n+1}(−1)^{k} ( \begin{matrix}n + 1 \\ k \end{matrix} ) \frac{(u + n + 1)_{k}}{(v)_{k}} \frac{Γ(v + n + k + 2)Γ(u − v + 1)}{Γ(u + n + k + 3)}

= ∑_{k=0}^{n+1}(−1)^{k} ( \begin{matrix}n + 1 \\ k \end{matrix} ) \frac{(u + n + 1)_{k}}{(v)_{k}} \frac{Γ(v + n + k + 1)Γ(u − v + 1)}{Γ(u + n + k + 2)} (1 − \frac{u − v + 1}{u + n + k + 2})
= ∑_{k=0}^{n+1}(−1)^{k} ( \begin{matrix}n + 1 \\ k \end{matrix} ) \frac{(u + n + 1)_{k}}{(v)_{k}} \frac{Γ(v + n + k + 1)Γ(u − v + 1)}{Γ(u + n + k + 2)}
− ∑_{k=0}^{n+1}(−1)^{k} (( \begin{matrix}n \\ k \end{matrix} ) + ( \begin{matrix}n \\ k − 1 \end{matrix} )) \frac{(u + n + 1)_{k}}{(v)_{k}} \frac{Γ(v + n + k + 1)Γ(u − v + 2)}{Γ(u + n + k + 3)}.  (A.6)

From (A.3) and the assumption of induction for n,

(A.6) = \frac{(−1)^{n+1}(n + 1)!Γ(u − v + n + 2)Γ(v)}{Γ(u + 2n + 3)}

− ∑_{k=0}^{n}(−1)^{k} ( \begin{matrix}n \\ k \end{matrix} ) \frac{(u + n + 1)_{k}}{(v)_{k}} \frac{Γ(v + n + k + 1)Γ(u − v + 2)}{Γ(u + n + k + 3)}

+ ∑_{k=0}^{n}(−1)^{k} ( \begin{matrix}n \\ k \end{matrix} ) \frac{(u + n + 1)_{k+1}}{(v)_{k+1}} \frac{Γ(v + n + k + 2)Γ(u − v + 2)}{Γ(u + n + k + 4)}

= \frac{(−1)^{n+1}(n + 1)!Γ(u − v + n + 2)Γ(v)}{Γ(u + 2n + 3)}

+\frac{(−1)^{n+1}(n + 1)!(v + n)Γ(u − v + n + 2)Γ(v)}{Γ(u + 2n + 3)}

+\frac{u + n + 1}{v} ∑_{k=0}^{n}(−1)^{k} ( \begin{matrix}n \\ k \end{matrix} ) \frac{(u + n + 2)_{k}}{(v + 1)_{k}} \frac{Γ(n + k + v + 2)Γ(u − v + 2)}{Γ(u + n + k + 4)}

= \frac{(−1)^{n+1}(n + 1)!Γ(u − v + n + 2)Γ(v)(v + n + 1)}{Γ(u + 2n + 3)}

+\frac{u + n + 1}{v} \frac{(−1)^{n}(n + 1)!(v + n + 1)Γ(u − v + n + 2)Γ(v + 1)}{Γ(u + 2n + 4)}

= \frac{(−1)^{n+1}(n + 2)!(v + n + 1)Γ(u − v + n + 2)Γ(v)}{Γ(u + 2n + 4)}.

Hence (A.5) holds for n + 1. Next we shall show (A.1) for Case III. From (2.15), (A.2) and (A.4),

∫_{0}^{1} x^{v}(1 − x)^{u−v}J_{n}^{2}(x; u, v)dx
= (−1)^{n−1}n\frac{(u + n)_{n−1}}{(v)_{n−1}} ∫_{0}^{1} x^{v+n−1}(1 − x)^{u−v}J_{n}(x; u, v)dx

+(−1)^{n} \frac{(u + n)_{n}}{(v)_{n}} ∫_{0}^{1} x^{v+n}(1 − x)^{u−v}J_{n}(x; u, v)dx

= (−1)^{n−1}n\frac{(u + n)_{n−1}}{(v)_{n−1}} \frac{(−1)^{n}n!Γ(u − v + n + 1)Γ(v)}{Γ(u + 2n + 1)}

+(−1)^{n} \frac{Γ(u + 2n)Γ(v)}{Γ(u + n)Γ(v + n)} ∑_{k=0}^{n}(−1)^{k} ( \begin{matrix}n \\ k \end{matrix} ) \frac{(u + n)_{k}}{(v)_{k}} ∫_{0}^{1} x^{v+n+k}(1 − x)^{u−v}dx

= −\frac{nΓ(u − v + n + 1)Γ^{2}(v)n!}{Γ(u + n)Γ(v + n − 1)(u + 2n)(u + 2n − 1)}

+(−1)^{n} \frac{Γ(u + 2n)Γ(v)}{Γ(u + n)Γ(v + n)} ∑_{k=0}^{n}(−1)^{k} ( \begin{matrix}n \\ k \end{matrix} ) \frac{(u + n)_{k}}{(v)_{k}} \frac{Γ(v + n + k + 1)Γ(u − v + 1)}{Γ(u + n + k + 2)}dx

= −\frac{nΓ(u − v + n + 1)Γ^{2}(v)n!}{Γ(u + n)Γ(v + n − 1)(u + 2n)(u + 2n − 1)}

+(−1)^{n} \frac{Γ(u + 2n)Γ(v)}{Γ(u + n)Γ(v + n)} \frac{(−1)^{n}(n + 1)!(v + n)Γ(u − v + n + 1)Γ(v)}{Γ(u + 2n + 2)}

= \frac{(2n(u + n) + v(u − 1))Γ^{2}(v)n!Γ(n + u − v + 1)}{(u + 2v + 1)(u + 2n)(u + 2n − 1)Γ(u + n)Γ(v + n)},

where the fourth equality follows from (A.5). □

References

[1] Bhattacharya, R. N. and E. C. Waymire (1990), Stochastic Processes with Applications, John Wiley & Sons, Inc.

[2] Cox, J. C. , J. E. Ingersoll and S. A. Ross (1985), “A Theory of the Term Structure of Interest Rates,” Econometrica, 53, 385-407.

[3] Ethier, S. and T. G. Kurtz (1986), Markov Processes : Characterization and Convergence, John Wiley & Sons, Inc.

[4] Fujita, T (1999??), Pricing Derivatives in Zone Model, ??

[5] Karlin and Tayler (1985), Second Course of Stochastic Processes, 2nd vol. , John Wiley & Sons, Inc.

[6] Revuz, D. and M. Yor (1991), Continuous Martingales and Brownian Motion, Springer-Verlag, Berlin.

[7] Szegö, G. (1939), Orthogonal Polynomials, Amer. Math. Soc., Providence, Phod, Island.

[8] Warren, J. and M. Yor (1996), “Skew-products Involving Bessel and Jacobi Processes,” working paper.