A spectral method for bonds

Computers & Operations Research 35 (2008) 64–75www.elsevier.com/locate/cor

A spectral method for bonds�

Javier de Frutos∗

Departamento de Matemática Aplicada, Universidad de Valladolid, Spain

Available online 23 March 2006

Abstract

We present an spectral numerical method for the numerical valuation of bonds with embedded options. We use a CIR model forthe short-term interest rate. The method is based on a Galerkin formulation of the partial differential equation for the value of thebond, discretized by means of orthogonal Laguerre polynomials. The method is shown to be very efficient, with a high precision forthe type of problems treated here and is easy to use with more general models with nonconstant coefficients. As a consequence, itcan be a possible alternative to other approaches employed in practice, specially when a calibration of the parameters of the modelis needed to match the observed market data.� 2006 Elsevier Ltd. All rights reserved.

Keywords: Bonds; Embedded options; Spectral methods; Laguerre polynomials

1. Introduction

A bond is a contract paying a known fixed amount, the principal, at a given date in the future, called the maturity. Thecontract can pay smaller quantities, the coupons, periodically at specific dates up to the maturity date. When it givesno coupons it is called a zero-coupon bond. Many contracts incorporate one or several embedded options such as call,put or conversion features. A call option gives the issuer the right to purchase back the bond for a fixed quantity calledthe call price of the bond. A put option gives the holder the right to return the bond to the issuer for a specified amountknown as the put price of the bond. Convertible bonds add the possibility of exchanging the bond for a fixed numberof shares of the issuer. As it has been pointed out in [1], the embedded options are an integral part of the contract andcannot be traded independently of the bond.

The optional features of the contract, call, put or conversion option, can usually be exercised at given dates during aspecified period of the life of the bond, that is, the contract allows for early exercise so that the bond with its embeddedoptions are an American-style interest rate derivative. As it is well-known, approximate methods, fully numerical orsemi-analytical, are necessary for pricing American style contracts, as there are no known closed-form formulae evenin the simpler cases. A wide variety of numerical methods have been used in the literature for the numerical valuationof bonds with embedded options. Finite-difference methods were used in [2], whereas trinomial trees were later used in[3]. Finite elements coupled with the characteristic method for two factor convertible bonds were used in [4,5] wherethe author used the method of lines and an efficient implicit–explicit Runge–Kutta method for the time integration.

� This research has been supported by Spanish Minister of Education and Science Project MTM2004-02847 (cofinanced by FEDER funds).∗ Tel.: +34 983 184680; fax: +34 983 423013.

E-mail address: [email protected].

0305-0548/$ - see front matter � 2006 Elsevier Ltd. All rights reserved.doi:10.1016/j.cor.2006.02.014

http://www.elsevier.com/locate/cor

mailto:[email protected]

J. de Frutos / Computers & Operations Research 35 (2008) 64–75 65

In [6], a semi-analytical approach based on the Green function coupled with numerical integration were employed forthe valuation of callable bonds with notice period. The same problem was solved in [7] by means of a finite volumemethod with stabilization and in [8] by means of a finite element/characteristic method. Recently, in [1] the authorspropose a dynamic programing approach which is proved to be a very efficient way to numerically evaluate callablebonds with notice period.

In this paper, we present a spectral numerical method for the valuation of such bonds. The numerical procedurewe present here is based in a Galerkin formulation of the relevant model-dependent partial differential equation forthe value of the bond, discretized by means of orthogonal Laguerre polynomials. The method is proved to be veryefficient; it shows a high precision for the type of problems we treat here and it is easy to use with general modelswith nonconstant coefficients. Although, for brevity, we shall use a CIR model [9] for the dynamics of the short-terminterest rate, the methodology can be easily extended to other type of models, such as Vasicek [10] or Hull and White[3] models, for example. It is worth remarking that time-dependent models are needed in practice to calibrate theparameters of the model to match the observed yield curves, see [4]. Calibration of parameters is a difficult problem,specially for American derivatives. Mathematically, it is a inverse problem for a nonlinear equation, see [11]. This kindof problem is usually very sensitive to the computed values of the solution of the equation and therefore extremelyprecise numerical results are needed to correctly solve it. The method we present here can be a possible alternative toother approaches employed in practice such as finite differences or finite elements.

A common problem shared by all the numerical methods mentioned above is the necessity of introducing artificialboundaries in order to discretize the infinite interval in which the state variable, i.e. the interest rate, is a prioridefined. Although it is possible to show convergence for the approximations defined in increasing finite intervals, see,for example, [12,13], in practice this is the cause of difficulty in controlling numerical errors which, if not treatedappropriately, can ruin any attempt to calibrate the parameters of the model to match the observed market data. Inthis paper, we present a numerical method based on Laguerre polynomials. The method does not require any artificialboundary and it can be extremely efficient at least for a class of problems related to the valuation of bonds.

2. A model problem

Several diffusion models have been used to describe the dynamics of the short-term risk-free interest rate. In thispaper, we consider the Cox, Ingersoll and Ross (CIR) model [9] for the short-term interest rate described by thestochastic differential equation

drt = k(r − rt ) dt + �r1/2t dBt , 0� t �T , (1)

where {Bt , t �0} is a standard Brownian motion, k is the reverting rate, r is the reverting level and � is the volatility.The numerical method we present is easily adaptable to other alternatives models as Vasicek model [10] or modelswith time-dependent parameters such as Hull and White model [3].

Let P(r, t, T ), 0� t �T , be the value at time t of a zero coupon bond with maturity date T and principal scaled to 1.By standard hedging arguments

P(r, t, T ) = EQ

[exp

(−∫ T

t

r(s) ds

)∣∣∣∣ r(t) = r

], (2)

where EQ[·|r(t) = r] represent the expectation conditional on r(t) = r with respect to the risk-neutral probabilitymeasure Q. Equivalently, P(r, t, T ) represents the discount factor over the period [t, T ] when r(t) = r . A partialdifferential equation for the price of the bond under the CIR model is

�P

�t+ 1

2�2r

�2P

�r2+ (k(r − r) − qr)

�P

�r− rP = 0, (3)

where qr is the risk premium corresponding to the choice of q(r, t) = −qr1/2/� as market price of risk.Eq. (3) together with the final condition P(r, T , T ) = 1 is a well-posed problem. Note that the partial differential

equation (3) is singular at r =0. No boundary condition at r =0 is needed as long as the condition kr ��2/2 is satisfied.We refer to [7] for a study of the problem of assigning boundary conditions at the singular boundary.

66 J. de Frutos / Computers & Operations Research 35 (2008) 64–75

A closed-form formula is known for the value of the straight bond under the CIR model, see, for example, [14],

P(r, t, T ) = A(T − t) exp(−rB(T − t)), (4)

with

A(�) =[

2de(k+q+d)�/2

(k + q + d)(ed� − 1) + 2d

]2kr/�2

,

B(�) = 2(ed� − 1)

(k + q + d)(ed� − 1) + 2d,

d =√

(k + q)2 + 2�2.

We now provide a formulation of the model for the pricing of a bond with an embedded call option with noticeperiod. We shall follow closely the dynamic programming approach in [1].

The contract we shall price is a coupon bearing bond with an embedded call option. The bond, with principal scaledto 1, pays a coupon c at given dates ti , i = 1, . . . , n, with tn = T , T being the maturity date. We put t0 = 0. The bond iscallable, after a protection period tn∗ , at coupon dates tm, m�n∗. The exercise decision is made at times �m = tm − �m

where �m is the notice period. The coupon dates tm and notice dates �m are known in advance as well as the call priceCm. For convenience, we define �n+1 = tn = T .

Let V (t, r) represent the value of the bond. The ex-coupon holding value at time �m when the spot interest rate is ris defined by

V h(t, r) = EQ

[V (�m+1, r(�m+1)) exp

(−∫ �m+1

t

r(s) ds

)∣∣∣∣ r(t) = r

](5)

for �m � t ��m+1, n∗ �m�n, and by

V h(t, r) = EQ

[V (�n∗ , r(�n∗)) exp

(−∫ �n∗

t

r(s) ds

)∣∣∣∣ r(t) = r

](6)

for 0� t ��n∗ . Then, at notice dates �m, m=n∗, . . . , n, the value of the callable bond satisfies the dynamic programmingrecurrence equations, see [1],

V (T , r) = 1 + c, (7)

V (�m, r) = cP (r, �m, tm) + min(CmP (r, �m, tm), V h(�m, r)), n∗ �m�n, (8)

V (0, r) = V h(0, r) + c

n∗∑m=1

P(r, 0, tm). (9)

We assume, as is usual in practice, that Cn = 1. In this case at the last notification date �n, we have simply

V (�n, r) = V h(�n, r) = (1 + c)P (r, �n, T ).

Using the Feynman–Kac theorem [15], we see that, for �m � t ��m+1 the holding value function V h(t, r) is thesolution of the backward parabolic problem

�V h

�t+ 1

2�2r

�2V h

�r2+ (k(r − r) − qr)

�V h

�r− rV h = 0, (10)

V h(�m+1, r) = V (�m+1, r). (11)

For 0� t ��n∗ , V h(t, r) can be computed from Eq. (10) together with the final condition

V h(�n∗ , r) = V (�n∗ , r). (12)


3. Review on Laguerre polynomials

The Laguerre polynomial of degree n is defined as the solution of the singular Sturm–Liouville problem,

rd2Ln

dr2+ (1 − r)

dLn

dr+ nLn = 0, (13)

satisfying the normalizing condition

Ln(0) = 1.

Explicitly, the nth degree Laguerre polynomial has the expression

Ln(r) =n∑

k=0

(n

n − k

)(−1k)

k! rk .

As it is well known [16], Laguerre polynomials are orthogonal in (0, +∞) with respect to the weight e−r , that is∫ +∞

0Ln(r)Lm(r)e−r dr = �nm,

where �nm is delta function, �nm = 0 if n �= m, �nn = 1.Let H 0

� (R+), Hs� (R+), 0 < ��1, be the Hilbert spaces

H 0� (R+) =

{u measurable

∣∣∣∣∫ +∞

0u2(r)e−�r dr < + ∞

},

H s� (R+) =

{u ∈ H 0

� (R+)

∣∣∣∣dpu

dup∈ H 0

� (R+), 0�p�s

}.

The Laguerre polynomials are a complete orthogonal system in H 01 (R+), that is, each function u ∈ H 0

1 (R+) can bewritten as

u(r) =∞∑

n=0

unLn(r),

with

un =∫ +∞

0u(r)Ln(r)e

−r dr . (14)

If u ∈ Hs1−�(R+), 0 < � < 1, the following approximation property is satisfied [17]:

∫ +∞

0

∣∣∣∣∣u(r) −N∑

n=0

unLn(r)

∣∣∣∣∣2

e−r dr �CN−s‖u‖s,1−�, (15)

where ‖ · ‖s,1−� denotes the natural norm in u ∈ Hs1−�(R+).

We note that the convergence order in (15) only depends on the regularity of the function u. Then, a Laguerrepolynomial-based method is expected to be of arbitrarily high order of convergence for smooth functions. In general, inan spectral method, that is, in an orthogonal polynomial based method, the order is exponential for analytic functions,see the numerical experiments section. This is the so-called spectral order of convergence.

The Laguerre polynomials satisfy the following recurrence relation:

Ln(r) = 2n − 1 − r

nLn−1(r) − n − 1

nLn−2(r), n�2, (16)

with L0(r) = 1, L1(r) = 1 − r .


Another useful recurrence equation satisfied by the derivatives of the Laguerre polynomials is

Ln(r) = L′n(r) − L′

n+1(r), n�0. (17)

Formula (17) can be equivalently written as

L′n(r) = −

n−1∑k=0

Lk(r), n�0. (18)

Using (16) and (17) we get the following recurrence for the derivatives of the Laguerre polynomials:

L′n+1(r) = 2n − r

nL′

n(r) − L′n−1(r). (19)

Let us denote by (·, ·) the inner product in H 01 (R+), that is,

(u, v) =∫ +∞

0u(r)v(r)e−r dr .

Using (16)–(18) and the differential equation (13) we can obtain the following relations that we are going to use in thesequel:

(Ln, Lm) = �n,m,

(rL′n, L

′m) = n�n,m,

(L′n, Lm) = −

n−1∑k=0

�k,m,

(rL′n, Lm) = n�n,m − n�n,m+1,

(rLn, Lm) = (2n + 1)�n,m − n�n,m+1 − (n + 1)�n+1,m. (20)

The Laguerre coefficients of a smooth given function u are not, in general, easily computable. They are usuallyapproximated by means of the Gauss–Laguerre or Gauss–Radau–Laguerre quadrature. This last option is more conve-nient to approximate Eq. (10). The Gauss–Radau–Laguerre quadrature of order N is written as

N∑j=0

u(xj )wj ≈∫ +∞

0u(r)e−r dr , (21)

where the nodes rj , 1�j �N , are the zeros of the polynomial L′N+1(r), the derivative of the Laguerre polynomial of

degree N + 1, and r0 = 0. The weights wj are defined by

wj = 1

(N + 1)L2N+1(xj )

, 0�j �N .

Gauss–Radau–Laguerre quadrature is exact when the function u is a polynomial of degree less than or equal to 2N .Given a smooth function u we define the interpolation polynomial at the Gauss–Radau–Laguerre nodes as the only

polynomial of degree N, INu(r), that satisfies

uN(rj ) = u(rj ), 0�j �N .

We can write INu in the Laguerre basis as

INu(r) =N∑

n=0

unLn(r),


where the coefficients un, 0�n�N , are computed from

un =N∑

j=0

u(rj )Ln(rj )wj .

The approximation of the Laguerre coefficients un in (14) by means of the quadrature (21) is roughly equivalent tosubstituting the function u to its interpolation polynomial INu. We remark that interpolation at points different to thenodes rj can be efficiently performed by means of the recursion formula (16).

4. The numerical method

In this section we present a numerical procedure to solve the partial differential equation (10) with final condition(11). We remark that the state variable r is defined in the interval (0, +∞). Any numerical method needs some type oftruncation of the infinite state variable interval. This localization of the original problem introduces numerical errorsthat are difficult to quantify and can ruin the precision of the computed bond values. On the other hand, usually thevalue of the bond is only needed in a very small interval near r =0, corresponding to realistic values of the spot interestrate. Our objective is to present a numerical method that is able to deal simultaneously with these two requirements:high precision in a small interval near r = 0 and approximation over an infinite interval without introducing artificialboundary values.

Let us denote by HN the space of polynomials of degree N. Our objective is to approximate the holding value functionV h(t, r) by a polynomial V h

N(t, r) ∈ HN by imposing that the residual of (10) with respect to V hN(t, r) is orthogonal

to the set HN . To this end, we substitute V hN(t, r) in Eq. (10), multiply by a test function �N(r)e−r with �N ∈ HN and

integrate by parts to get the discrete variational equation

−(

�V hN

�t, �N

)= − �2

2

(r�V h

N

�r,��N

�r

)+(

kr − �2

2

)(�V h

N

�r, �N

)

−(

k + q − �2

2

)(r�V h

N

�r, �N

)−(rV h

N, �N

). (22)

The final conditions (11) and (12) are approximated by means of

V hN(�m+1, r) = VN(�m+1, r), (23)

V hN(�n∗ , r) = VN(�n∗ , r), (24)

where

VN(�m, r) = IN(cP (r, �m, tm) + min(CmP (r, �m, tm) + V hN(�m, r))). (25)

Alternatively, we can define the polynomial VN(�m, r) approximating V (�m, r) by the interpolation conditions:

VN(�m, rj ) = cP (rj , �m, tm) + min(CmP (rj , �m, tm) + V hN(�m, rj )),

where rj are the Gauss–Radau–Laguerre quadrature nodes.Writing

V hN(r, t) =

N∑n=0

vn(t)Ln(r),


we easily see that (22) is equivalent to the following system of ordinary differential equations:

−N∑

n=0

dvn

dt(Ln, Lm) = − �2

2

N∑n=0

vn(rL′n, L

′m) +

(kr − �2

2

) N∑n=0

vn(L′n, Lm)

−(

k + q − �2

2

) N∑n=0

vn(rL′n, Lm)

−N∑

n=0

vn(rLn, Lm), m = 0, 1, . . . , N . (26)

Using properties (20) of the Laguerre polynomials we can write system (26) in matrix form as

−dvdt

= −�2

2Kv +

(kr − �2

2

)C1v −

(k + q − �2

2

)C2v − Mv, (27)

where the vector v collects the coefficients of the Laguerre expansion of V hN(t),

v(t) = [v0, v1, . . . , vN ]T,

and the matrices K, C1, C2 and M are defined by

K = ((rL′

n, L′m))Nn,m=0,

C1 = ((L′

n, Lm))Nn,m=0,

C2 = ((rL′

n, Lm))Nn,m=0,

M = ((rLn, Lm))Nn,m=0.

We remark that the structure of the matrices involved in system (27) is very convenient for the task of its numericalresolution. More precisely, matrix K is a diagonal matrix, C1 is upper triangular, C2 is bidiagonal and M is a tridiagonalmatrix.

Eq. (27) is a stiff system of ordinary differential equations that must be solved by means of an implicit, stiff accurate,high-order numerical time integrator. High-order time integrations is needed because the spot rate being discretizationof spectral type, a low-order numerical scheme, such as a standard finite difference method, can ruin the high precisionthat is expected from a spectral method. On the other hand, a stiff accurate numerical method is needed. As it is wellknown, explicit time integrators are usually inefficient when applied to the time integration of semidiscretizations ofparabolic partial differential equations because of the severe time step restrictions that must be imposed for stabilityreasons when the spot rate discretization is refined by increasing the number of degrees of freedom, that is, in our case,increasing the degree of the polynomials used in the Laguerre discretization.

We remark that second-order finite differences in time have been widely used in the financial literature in conjunctionwith low-order finite differences in the state variable, the spot rate in our case, see [18], for example, or [7] where aCrank–Nicolson method is used in combination with finite volumes. However, when high-order state discretization isused it is usually more convenient to resort to high-order time discretization in order to attain high-accuracy in the fullydiscrete method.

Although for this problem there are several numerical methods and software that can be used with reasonable results,in this paper we shall use an implicit–explicit Runge–Kutta type time discretization that is accurate enough for ourpurposes and has been proved to be very efficient [5,19] in other financial problems.

We start by rewriting Eq. (27) in the form:

−dvdt

= L(v) + N(v),


with

L(v) =(

−�2

2K − M

)v,

N(v) =((

kr − �2

2

)C1 −

(k + q − �2

2

)C2

)v.

Let K be a positive integer and define �t = (�m+1 − �m)/K . We denote by vk = v(�m + k�t). The equations of themethod are

Y1 = vk+1,

Yi = vk+1 + �t

i∑j=2

�ij L(Yj ) + �t

i−1∑j=1

�ij N(Yj ), i = 2, 3, 4,

vk = vk+1 + �t

4∑j=2

�j

(L(Yj ) + N(Yj )

),

where aii = is the middle root of 6x3 − 18x2 + 9x − 1, and

�32 = (1 − )/2, �42 = − 322 + 4 − 1

4 , �43 = 322 − 5 + 5

4 ,

�21 = , �31 = 1 +

2− �32, �32 =

13 − 22 − 2�3�43

(1 − ),

�42 = 1 − �43, �43 = − 720 , �2 = �42,

�3 = �43, �4 = .

The implicit–explicit method has been designed in [20] to have order three and good stability properties when appliedto convection diffusion equations, a class of equations that includes the partial differential equation (10). We remarkthat, at each time step, three linear systems of equations with the same matrix I + �t (�2/2K +M) have to be solved.Thanks to the properties of the Laguerre discretization, the matrix is tridiagonal and positive definite. In consequence,the linear systems can be solved very efficiently resulting in a very cheap and easy to implement numerical procedure.

In practice the time integrator is implemented in variable step mode so that the time step is automatically chosenby the code depending on an estimation of the so-called local errors and a user supplied tolerance that is intended tobe the desired precision of the computed approximation. We refer to [19,20] for further details and discussion on theprocedure.

5. Numerical experiments

In this section we present some numerical experiments to demonstrate the qualities of the proposed method. In orderto compare the results with other numerical approaches we take the same parameters for the dynamics of the spotinterest rate as in [1,6,7]. They are, k = 0.54958046, � = 0.38757496, r = 0.0348468515 and q = −0.40663675.

We start with the problem of valuation of a zero-coupon bond with maturity T = 10. For this problem we have anexact closed form solution given by (4), so we can measure the errors committed in the discretization. In Table 1 wepresent the results obtained for several values of the time integrator tolerance ranging from 10−5 to 10−8 (columnTOL) and degrees of the polynomials in the Laguerre discretizations N = 10, 20, 40 and 80 (column N ). The errorsare reported in the column ERROR. We have reported also the number of time steps employed by the code (columnSteps) and the CPU time in seconds (column CPU). The code was implemented in MATLAB�. All runs were madeon a modest personal computer with a 3 GHz processor.

First, we shall comment on the behaviour of the time integrator. As we can see in Table 1, the number of time stepsonly depends on the time integrator tolerance, being nearly independent of the number of degrees of freedom N. This isthe expected behaviour of a stiff time integrator. Furthermore, the number of time steps, and in consequence the CPUtime, is roughly doubled each time the tolerance is divided by 10 for the same value of N. Once the error in the spot


Table 1Errors in the fully discrete numerical method

TOL N ERROR Steps CPU

10−5 10 1.6177 × 10−1 81 0.0310−6 10 1.6178 × 10−1 158 0.0510−7 10 1.6178 × 10−1 320 0.0810−8 10 1.6178 × 10−1 670 0.1510−5 20 1.1995 × 10−2 77 0.0310−6 20 1.1997 × 10−2 146 0.0510−7 20 1.1997 × 10−2 292 0.0910−8 20 1.1997 × 10−2 601 0.1710−5 40 3.7558 × 10−5 77 0.0610−6 40 3.8611 × 10−5 146 0.0810−7 40 3.8724 × 10−5 290 0.1610−8 40 3.8736 × 10−5 597 0.2810−5 80 1.7475 × 10−6 77 0.1210−6 80 1.7587 × 10−7 146 0.1910−7 80 1.7429 × 10−8 290 0.2310−8 80 1.8399 × 10−9 597 0.41

Table 2Errors in the Laguerre semidiscretization

TOL N ERROR

10−12 5 4.4825 × 10−1

10−12 10 1.6178 × 10−1

10−12 20 1.1997 × 10−2

10−12 40 3.8737 × 10−5

10−12 80 2.6399 × 10−10

interest rate discretization is small enough, N = 80 in Table 1, we can observe that the error due to the time integrationis proportional to the tolerance prescribed. Together, these considerations show the high order of convergence of thetime integrator and the robust behaviour of the automatic time step selection procedure.

Next, let us analyse the behaviour of the Laguerre spot rate discretization. We can observe in Table 1 that for N =10,20 and 40, the error is dominated by the errors in the Laguerre discretization. A reduction in the time integrator tolerancedoes not give any reduction in the error of the computed solution. The component of the error due to the Laguerrediscretization can be evaluated as being approximately 1.6×10−1 for N =10, 1.2×10−2 for N =20 and 3.9×10−5 forN = 40. For N = 80, the Laguerre discretization error is less than 1.84 × 10−9. We can see that the rate of convergenceof the spectral Laguerre method must be necessarily very high and surely higher than in any finite difference, finitevolume or finite element method.

In Table 2 we report the errors obtained for a very small value of the time integration tolerance (10−12) and severalvalues of the degree of the Laguerre polynomials. Then, the errors showed in the table are caused only by the spotrate discretization. We can observe that, once enough resolution of the solution has been obtained, the error decreasesdramatically with increasing values of N. Doubling the degree of the Laguerre polynomials, from N = 20 to 40, resultsin reduction in the error by three orders of magnitude; doubling again from N = 40 to 80 results in a reduction of fiveorders of magnitude. We remark that the final approximation with error of order of 10−10 is really difficult to obtainwith a standard finite difference or finite element method.

We can observe better high order of convergence in Fig. 1 where we have represented the number of degrees offreedom (degree of the Laguerre polynomials used in the computations) on the abscissa axis and the error in logarithmicscale on the ordinate axis. Each circle corresponds to a result in Table 2. We can see that the resulting relation is nearly


0 10 20 30 40 50 60 70 8010-10

10-8

10-6

10-4

10-2

100

Number of degrees of freedom

Err

or

Fig. 1. Exponential convergence of Laguerre semidiscretization.

Table 3Optimized values of tolerance and number of degrees of freedom

TOL N ERROR Steps CPU

10−3 30 6.5121 × 10−4 26 0.0210−4 40 3.0098 × 10−5 43 0.0410−5 50 2.5254 × 10−6 77 0.0710−6 60 2.1523 × 10−7 146 0.1010−7 70 1.9320 × 10−8 290 0.1910−8 80 1.8399 × 10−9 597 0.41

linear, corresponding to an exponential decrease of the error. In this experiment, the error E behaves as

E = Ce−N ,

or equally well as

log(E) = log(C) − N ,

with C ≈ 2.77 and ≈ 0.29. That is, the convergence is exponential with respect to the degree of the Laguerrepolynomials. This is what has been called spectral order of convergence in the numerical analysis literature, see [21]for example.

In a practical situation, an equilibration of temporal discretization and spot rate discretization errors is convenientin order to optimize the CPU time that is needed for a given level of error. This is specially true if a spectral method isemployed for the spot rate discretization. For example, in Table 1 we can see that for N = 20 all the runs with differentvalues of the tolerance of the time integrator give approximately the same error. This means that the error is saturatedby the spot discretization error and, in consequence, the same results could be obtained for a greater tolerance in lessCPU time.

In Table 3 we show the results for optimized values of the tolerance in the time integrator and the number of degreesof freedom in the spot rate discretization. As we can see, a really fast convergence can be obtained in a really smallCPU time. In this example, an error of 6.5121 × 10−4 is obtained in approximately 0.02 s. An extremely precise resultwith error 1.8399 × 10−9 is obtained in less than half of a second. This clearly shows the good performance of thefully discrete numerical method.

Finally, we examine the behaviour of the method in a more interesting problem. We apply the Laguerre spectralmethod to the valuation of a 4.25% callable bond with notice period. The parameters for the dynamics of the spotinterest rate are the same as in the preceding example. At t = 0, the pricing date, the time to maturity of the contract is


Table 4Callable bond results

r DFVL ABKE LAG

0.01 0.93926 0.93921 0.939220.02 0.91598 0.91595 0.915960.03 0.89333 0.89330 0.893310.04 0.87127 0.87125 0.871250.05 0.84980 0.84978 0.849790.06 0.82890 0.82888 0.828890.07 0.80855 0.80854 0.808540.08 0.78874 0.78873 0.788730.09 0.76945 0.76945 0.769450.10 0.75067 0.75067 0.75067

N = 200, TOL = 10−4, CPU time is 0.75 s.

T = tn = 20.172 with n = 21. The bond has a principal scaled to 1 and a coupon c = 0.0425 coming once a year. Thefirst coupon is delivered at t = 0.172. The notice period is 2 months, that is �m = tm − 0.1666. There is a protectionperiod of tn∗ = 10.172 years with n∗ = 11. The call prices are C11 = 1.025, C12 = 1.020, C13 = 1.015, C14 = 1.010,C15 = 1.005 and Cm = 1 for m = 16, . . . , 21. This contract has been previously used as a test in [1,6,7]. Here, we shalluse the results reported in [1,7] as a reference to our computations with the spectral Laguerre method.

The full algorithm is as follows: let VN(t, r) be the discrete value of the bond and V hN(t, r) the discrete holding

value. We put VN(T , r) = 1 + c, compute VN(�n, r) = IN((1 + c)P (r, �n, T )) and put �m+1 = �n.

1. Solve backwards the discrete bond equation (22) for V hN(�m, r), using the implicit–explicit time integrator, from

t = �m+1 to t = �m.2. Compute the bond value by means of Eq. (25). Put �m+1 = �m and repeat step 1 until �m = �n∗ .3. If �m = �n∗ , solve the discrete bond equation from �n∗ to t = 0 and add the coupon payments by means of

VN(0, r) = V hN(0, r) + c

n∗∑m=1

IN(P (r, 0, tm)).

We show the results in Table 4 (column LAG) where for comparison we have included the results reported in [7] inthe column denoted DFVL and the results reported in [1] in the column denoted ABKE. The numerical parameters inthis calculation were N = 200 for the spot rate discretization and T OL = 10−4 for the tolerance of the time integrator.As we can see the Laguerre spectral discretization gives nearly the same results as reported in [1,7], with a maximumdifference of 10−5, in approximately 0.75 s of CPU time. For the sake of comparison we note that the CPU time reportedin [1,7] was of 2 s for ABKE computations and 12 s for DFVL computations. Taking into account that the computationshave been done in different computers, this is not a measure of the relative efficiency of the methods but at least wecan say that the Laguerre spectral method can be competitive even when compared with an extremely efficient methodsuch as the dynamic programming, finite element method of [1] or when compared with the sophisticated finite volumemethod presented in [1].

We remark that the spectral Laguerre approximation VN(t, r) to the value function V (t, r) is a polynomial in thevariable r which can be used to efficiently compute, by means of recursion (16), a price for any value of the spot interestrate. Equally well some quantities of interest to the practitioners, such as the duration or the convexity for example,related to the derivatives of the bond value can be easily computed by means of recursion (19).

References

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A spectral method for bonds

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