THE VALUATION OF BONDS AND BOND OPTIONS: SOME EMPIRICAL TESTS

7/28/2019 THE VALUATION OF BONDS AND BOND OPTIONS: SOME EMPIRICAL TESTS

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Initial Version: April 1991Current Version: October 1991

THE VALUATION OF BONDS AND BOND OPTIONS: SOME EMPIRICAL TESTS

by E. Barone* and D. Cuoco**

*) I.M.I.

**) University of California at Berkeley

We thank Andrea Cividini for Marquardts algorithm and Gianluca Salsecci for his helpful sugges-

tions. This paper has been prepared for the International Workshop on Large-Scale Economic andFinancial Applications: New Tools and Methodologies held in Urbino (May 10th, 1991).


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SUMMARY

This paper presents a general method for valuing fixed rate bonds and options written on them. Inthe first part, of a theoretical nature, we present valuation formulae, derived within the framework ofthe Cox, Ingersoll and Ross (CIR) model, both for bonds and for European options written on bonds

(with and without coupons) and yields. We recall the theoretical parity of put and call options. In thesecond part we describe a procedure that can be used to estimate the CIR model using the prices ofriskless coupon bonds. Lastly, we describe the valuation of bonds denominated in three different cur-rencies and of the options implied in some Italian government securities.

CONTENTS

1. INTRODUCTION 1

2. THEORETICAL ASPECTS 1

2.1 Description of the CIR model 12.2 Pure discount bonds 22.3 Coupon bonds and the stochastic duration 32.4 Options on pure discount bonds 4

Put-call parity. 52.5 Options on coupon bonds 5

Put-call parity. 52.6 Options on yields 5

Put-call parity. 6A special case 6Options on continuously compounded interest rates 6

3. ESTIMATION METHODS 7

3.1 The one-stage approach 73.2 The score function: least squares 83.3 The weighting of the errors 83.4 Marquardts algorithm 83.5 The Gauss-Newton algorithm 103.6 The finite differences method 10

4. APPLICATIONS 11

4.1 Lira, ecu and dollar bonds 114.2 (Implied) options on coupon bonds 14

5. CONCLUSIONS 16

APPENDIX A 19

Estimates of the CIR Model Based on BTPs 19

APPENDIX B 24

Estimates of the Options Implied in CTOs 24

REFERENCES 27

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1. INTRODUCTION

The valuation of the different contracts referring to bonds -- spot contracts, forward or futures con-tracts, repos, the different types of options on prices and yields (call, put, etc.), swaps, caps and

floors, etc. -- cannot be based on a series of ad hoc methods. A general approach nonetheless re-quires a theoretical model that formalizes, with an appropriately small number of parameters, the in-fluence of the most important factors and the random components.

The model used in this paper is the one-factor version of that proposed by Cox, Ingersoll andRoss (CIR). According to this model, there exists a pivot rate that determines the whole termstructure of interest rates: the instantaneous rate, i.e. the rate on an investment of instantaneous dura-tion. The dynamics of the instantaneous rate are of the mean-reverting type: even if disturbed byshocks, the rate tends to return towards a long-run average level. The market equilibrium conditionis embodied in the local expectations hypothesis: the expected return in the next instant of time on aninvestment in bonds of any maturity is equal to the instantaneous rate plus a risk premium/discount(the amount of risk is equal to the elasticity of the price with respect to the instantaneous rate).

The model describing the dynamics of bond prices is more complex than that generally

adopted for share prices (geometric Brownian motion). This is because the prices of bonds, unlikethose of shares, depend on their maturity. It follows that the valuation of bond options is more com-plex than that of share options. In order to value bond options, it is necessary to estimate four pa-rameters instead of just one (volatility in the case of the Black and Scholes formula).

In this paper we describe a general method for valuing fixed interest bonds and options writtenon them. In the first part, of a theoretical nature, we present the valuation formulae derived withinthe framework of the CIR model for both bonds and European options written on bonds (with andwithout coupons) and yields (2). We also show the theoretical parity of call and put options. In thesecond part we describe a procedure that can be used to estimate the CIR model using the prices ofriskless coupon bonds (3). In the following section we examine the valuation of bonds denominatedin three different currencies and of the options implied in some Italian government securities (4).

2. THEORETICAL ASPECTS

2.1 Description of the CIR model

The one-factor version of the Cox, Ingersoll and Ross (CIR) model assumes that:1. the instantaneous interest rate (r) follows the process described by the stochastic differential

equation

( ) dzrdtrdr +=

(1)

where is the speed of adjustment of the interest rate rtowards its long-run average , r is thevolatility of the changes in rand dzs a standardized Wiener process; if 0 < < 1 the instantaneousrate converges towards its average value (a mean-reverting process);

2. the expected instantaneous return on bonds of any maturity is equal to the interest rate r plus a

risk premium (the local expectations hypothesis):

dtP

Prr

P

dPE r

+=

(2)

whereP=P(r, t, T) is the price of a pure discount bond with residual maturity = Tt, is themarket price of risk and rPr/Pis the elasticity of the bonds price with respect to r;

3. the usual perfect market assumptions apply.Under these assumptions it can be shown that if the valueH=H(r, t, T) of a financial asset dependsexclusively on the current term structure of interest rates, it will follow (in the interval between twosuccessive coupons) the stochastic differential process defined by (3):

( )[ ] 0 2 =++ rHHHrrrH trrr . (3)


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This equation can be solved once appropriate boundary conditions are imposed on H(r, t, T), accord-ing to the specific features of each type of financial asset.

2.2 Pure discount bonds

For the priceP(r, t, T) of a pure discount bond with residual maturity = Tt, the solution of equa-tion (3), subject to the boundary condition

( ) 1,, =TTrP (4)

is

( ) ( ) ( )rTtGeTtFTtrP ,,,, =

(5)

where

( )

( )

3

1

2

12

1

1,

+=

e

eTtF

and

( )( )

.1

1,

121

1

+

=

e

eTtG

Equation (5) gives the value of a pure discount bond with residual maturity as a function of thestate variable rand of the three parameters 1, 2 and 3, where:

( ) 221 2 ++= (6)

2

1

2

++=

(7)

.

223

=

(8)

Equation (5) can be used to derive the whole term structure of interest rates R(r, t, T) by using thefollowing relationship:

( )( )[ ]

.,,ln

,,tT

TtrPTtrR

=

(9)

In particular:

( ) rTtrRR tT == ,,lim0 (10)

( ) ( ) .,,lim 321 ==

TtrRRT

(11)

The parameterof equation (1) can be determined using the relationship:

( ).2 2221 = (12)

The economic significance of the three parameters 1, 2 and 3 is clear when the value ofR issubstituted for. In this case we have:

=1 (13)

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=2 (14)

.3

=

(15)

The derivatives ofPwith respect to r, 1, 2 and 3 are as follows:

PGr

P=

(16)

( )( )[ ] ( )

( )[ ]( )[ ] ( )( )

( )[ ]212212

2

12

2112

1

12

13

1

1

111

1

11

1

1

1111

1

1212

3

1

2

+

++

+

+

++

+=

e

eeeePr

ee

eeee

e

eP Gr

(17)

( )( )[ ] ( )

( )[ ]

( )

2

12

2

12

1121

1

12

13

2

1

1

1

11

1

1

1

1

1212

3

1

2

+

+

+

+

+

+=

e

ePr

ee

eeee

e

eP Gr

(18)

( ).

1ln

12

1

3 1

2Gre

e

eF

P

+=

(19)

2.3 Coupon bonds and the stochastic duration

The price of a coupon bond is obtained simply as the sum of the bonds payments, with each pay-ment multiplied by the price of the corresponding pure discount bond:

( ) ( )( ) ( )=

=

==m

j

jj

tsstrRm

j

j strPaeatrBjj

1

,,

1

,,;, as,

(20)

where:

B(r, t; s, a) is the price at time t of a coupon bond;

a is the vector of payments aj (j = 1, 2, ..., m);s is the vector of payment datessj (t


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( )

==

aP

aPG

aP

aP

B

B rr

(22)

This measure of the risk associated with coupon bonds can be expressed in units of time. In this case

Cox, Ingersoll and Ross (1979) used the term stochastic duration. The stochastic duration () of acoupon bond is equal to the maturity of the pure discount bond with the same degree of risk:

( )

+==

2

1

1

1

/1

/1ln

1/

BB

BBBBG

r

rr

(23)

2.4 Options on pure discount bonds

Let c(r, t, T;s, K) be the value at time tof a call option, with striking price Kand maturity T, writtenon a pure discount bond with maturity s (s > T> t). The price of the option, obtained by solvingequation (3), subject to the boundary condition

( ) ( )[ ]0,,,max,;,, KsTrPKsTTrc = (24)

is

( ) ( ) ( ) ( ) ( )2222

1112 ,,,,,,,,,;,, ncdfdTtrKPncdfdstrPKsTtrC =

(25)

where 2(d, df, nc) is the non-central chi-square distribution function,

1with dfdegrees of freedom

and non-centrality parameternc, valued at point d, and, respectively, according to Cox, Ingersoll andRoss (1985) and Jamshidian (1990):

( )[ ] ( )[ ]sTGvrdsTGrd ,2,2 11 +=++=

(26)

( ) vrdrd =+= 22 2

(27)

321321 22 ==== dfdfdfdf (28)

( )

( ) ( )sTGvvbr

ncsTG

renc

tT

,2,

21

2

1

1

+=

++=

(29)

( )brnc

renc

tT

=+

=

2

2

2

12

(30)

( )[ ] ( )TtGv

e tT ,

4

1

2

22

1

1

=

=

(31)

( )TtvGb ,'

222 ==

(32)

( )

( )

( )

( )sTGK

sTF

rsTG

K

sTF

r,

,ln

,

,ln

**

=

=

(33)

1

. For the calculation of the non-central chi-square distribution function, see Farebrother (1987) and Ashour and Ab-del-Samad (1990).

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The value rdefined by equation (33) is the critical interest rate below which the call option will beexercised and is obtained by putting P(r

*, T,s) =Kand solving with respect to r

*.

A call option, with striking priceKand maturity T, written on a pure discount bond with matur-ity s (s> T> t) can thus be replicated by a portfolio constructed by buying

2(d1, df1, nc1) pure dis-

count bonds with maturity s and selling K2(d2, df2, nc2) pure discount bonds with maturity T.

Put-call parity.

Letp(r, t, T;s,K) be the value at time t of a put option, with striking price Kand maturity T, writtenon a pure discount bond with maturity s. The value of the put option can be obtained by way of theput-call parity, which, in the case of options on pure discount bonds, is given by:

( ) ( ) ( ) ( ),,,,,;,,,;,, TtrKPstrPKsTtrpKsTtrc += . (34)

Such options are markedly different from those written on shares. In particular, the values of a calland a put can move in the same direction when the price of the underlying bond changes and theycan also be inversely correlated with the maturity.

2.5 Options on coupon bondsWith the CIR model, as with all one-factor models, the option on a coupon bond is equivalent to aportfolio of options on pure discount bonds [Jamshidian (1989)]

2

( ) ( )

( ) ( )[ ] ( ) ( 2222,1,1,12 ,,,,,,,,

0,,;,,,;,,

ncdfdTtrKPncdfdstrPa

KsTtrcaK,Ttrc

m

qj

jjjjj

m

qj

jjj

=

==

=

=

as

)

(35)

where aj is the bonds payment at timesj (TT>t) can thus be replicated by a portfolio constructed by buying aj

2(d1,j, df1,j, nc1,j) pure discountbonds with maturitysj (j = m, ..., n) and sellingKj

2(d2,j, df2,j, nc2,j) pure discount bonds with matur-

ity T.

Put-call parity.

The value at time tof a put option, with striking priceKand maturity T, written on a coupon bondwith payments a at dates s, can be obtained by way of the put-call parity, which, in the case of op-tions on coupon bonds, is given by:

( ) ( ) ( ) ( ) ( )TtrKP,trB,trBK,TtrpK,Ttrc ,,';,;,,;,,,;,, += asasasas (36)

where:


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( ) ( )[ ]{ }0,1,,/1max,;,, iii KsTrPKsTTrc = (37)

Equation (3) can be solved using numerical methods. It can be rewritten as a function of the parame-ters 1, 2 and 3, as well as of the state variable r, by using the definitions (6),(7) and (8). Refer-

ring to equation (12), we have:( )[ ] 02 12322 =++ rHHHrrH trrr (38)

The solution of the differential equation (38), subject to the boundary condition (37), can be ap-proximated using numerical algorithms.

Put-call parity.

The value at time tof a put option, with striking rateKi and maturity T, written on the yield relativeto the period (T,s) (s > T> t) can be obtained by way of the put-call parity, which, in the case of op-tions on yields, is given by:

( ) ( ) ( ) ( )

TtrPKsTtrcKsTtrpKsTtrc iiiiii ,,0,;,,,;,,,;,, += (39)

A special case

If the settlement of the final value of options on yields is not made at maturity, but deferred for a pe-riod equal to that to which the optionable interest rate refers, the price of the options can be obtainedexplicitly using (25) and (34).

In this case the final value at time Tof a call option, with striking rate Ki, maturity Tand set-tlement at times, written on the yield relative to the period (T,s) (s > T> t) is:

( ) ( )[ ]{ } ( )( ) ( ){ }0,,,max/1

,,0,1,,/1max,;,,

sTrPKK

sTrPKsTrPKsTTrc iii

=

==

(40)

whereK= 1/(1 +Ki Accordingly, we have:

( ) ( )

++=

i

iiiK

sTtrpKKsTtrc1

1,;,,1,;,,

(41)

Analogously:

( ) ( )

++=

i

iiiK

sTtrcKKsTtrp1

1,;,,1,;,,

(42)

The put-call parity for such options is:

( ) ( ) ( ) ( ) ( ).,,,,1,;,,,;,, TtrPstrPKKsTtrpKsTtrc iiiii ++= (43)

In addition, the value at time tof a forward contract, with forward rateKi, maturity Tand settlement

ats, written on the yield relative to the period (T,s) (s > T> t) is:( ) ( ) ( )

( ) ( ) ( ).,,1,,,;,,,;,,,;,,

strPKTtrP

KsTtrpKsTtrcKsTtrF

i

iiiii

+=

==

(44)

The value at time tof this contract is null if the forward rateKi is equal to that implied in the termstructure of interest rates,

( )( )

1,,

,,=

strP

TtrPKi

(45)

Options on continuously compounded interest rates

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In the case of options written on the yields R(r, t, T) defined by equation (9), ], Longstaff (1990b)obtained the following closed-form formula:

3

( ) ( ) ( ) ( )[ ] /,,,4,,,,;,, ++= ncdfdQKncdfdQRTtrPKsTtrc RiR (46)

where Q(d, df, nc) = 1

2

(d, df, nc) is the non-central chi-square complementary distribution func-tion and:

( )[ ]{ }( ) ( )TGG

FKd R

2

ln4 +=

(47)

32=df (48)

( ) ( )[ ]{ }

( )

Ge

RFTGenc

T

T

22

21

1

ln4

1

1

+=

(49)

( )2

2/1

111 1

=

T

eeTGT

(50)

( )[ ] ( ) ( ) ( ) ( )

( )[ ] ( ).,4,ln

,2,2

,,ln2

3

ncdfdQF

ncdfdQGTGncdfdQF

++

+++=

(51)

The analytical properties of these options are very different from those usually encountered. In par-ticular, call options can have a value that is higher than the underlying interest rate or less than theirintrinsic value. The use of the Black and Scholes formula to value options written on interest rates,

and hence also caps and floors, can therefore lead to erroneous results, since it is sometimes impos-sible to generate the correct theoretical value, no matter how large is the adjustment of the varianceparameter.

3. ESTIMATION METHODS

3.1 The one-stage approach

The Cox, Ingersoll and Ross model has been estimated using the one-stage approach suggested byBrown and Dybvig (1986). It is assumed, for each of the n coupon bonds observed at a certain date,that:

( ) ,,2,1;, niutrBB iiei K=+= as, .

(52)

whereBi is the actual (cum-coupon) price of the i-th security Bi theoretical price, s the vector ofthe m payment dates a the vector of the payments and ui, , the error term.

Using equation (20), equation (52) can be written as follows:

( ) .,,2,1,,1

niustPaB ij

m

j

jei K=+=

=

(53)

where Pis defined by equation (5) and = (r, 1, 2 e 3) is the vector of the parameters to be esti-mated. In more concise form:

3. Longstaff (1990b) also shows how to value options on interest rate "portfolios" (i.e. options on the average of in-

terest rates or on the spread between two interest rates) as well as options on the maximum (or the minimum) of tworates.

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( ) u,APBe +=

(54)

where Beis the vector (n x 1) ) of the actual(cum-coupon)prices, A is the payments matrix (n x

m) of which the generic element aij with i = 1, 2, ..., n ej=1, 2, ..., m indicates the payment made on

the i-th bond at the j-th date, P[; ] is the vector (m x 1) of the coefficients to be estimated and u isthe vector (n x 1) of the errors. It should be noted that the coefficients P are linked, by way of equa-tion (5), to the vector (known) of the maturities s and to the vector (unknown) of the parameters .

3.2 The score function: least squares

In order to estimate the parameters of model (54) it is necessary to define the score function that is tobe minimized, within the predetermined range.

The score function adopted in this paper is the one most commonly used: the sum of squarederrors. However, this function is particularly sensitive to the values of outliers, so that a material er-ror in the data (e.g. a decimal point in the wrong place) can seriously distort the results. Accordingly,we modified the estimation procedure by repeating the search for the minimum after automaticallyexcluding the outliers from the data set.

The method involves identifying the residuals that are greater in absolute terms than 2.57 timestheir standard deviation. On the assumption of normality, the probability of such residuals occurringis one per cent. The observation corresponding to the largest standardized error

4in absolute terms is

excluded and the search repeated. This procedure is iterated until all the outliers have been elimi-nated.

3.3 The weighting of the errors

Before carrying out the estimation, we adopted the hypothesis that the variance of the error term u in(54) increases with the duration of the securities. It is in fact reasonable to assume that operators findit more difficult to determine the value of long-term securities than that of short-term securities, sothat the order of magnitude of the errors in the prices of securities closer to maturity will be smaller.

Accordingly, we assumed that the variance of the error is directly proportional to the Hicksianduration (multiplied by the cum-coupon price and divided by the return) or, in other words, to thederivative of the price with respect to the yield to maturity with the sign changed. So that:

( )

+

= 1

22

ei

u

DB

(55)

( )

=++

=m

j

jj

j

aD

11

1

(56)

whereD is the Hicksian duration and is the yield to maturity.Hence, in order to make the error term homoschedastic, both the members of equation

(54) are divided by the square root ofDBie

, giving:

( ), XPY += . (57)

3.4 Marquardts algorithm

The parameters of model (57) were therefore estimated by minimizing the sum of the squared er-rors. Denoting this sum as S() we have:

( ) 'S MinMin = (58)

4. The standardized error is defined asu/ u2/(n 4).

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or, in other terms:

( ) ( )[ ] ( )[ ],',MinMin XPYXPYS = . (59)

First order conditions require the gradient, i.e. the vector of the four partial derivatives of S() with

respect to

, to be null:( ) ( )[ ] 0= ,XPYX'',z

(60)

where:

( )( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

=

=

=

321

3

2

2

2

1

22

3

1

2

1

1

11

,,,,

,,,,

,,,,

mmmm PPPr

P

PPP

r

P

PPP

r

P

LLLL

,P,z

(61)

and the derivatives P/rP/1P/2P/3 are defined by (16),(17),(18) and (19).In order to find the vectorP(; ) that satisfies equation (60) we used Marquardts algorithm,

which provides, for each stage of the iterative procedure, both the direction matrix and the optimalstep length. At the (k+ 1)- )-th iteration the vectork+1 is given by:

5

( )[ ] ( ) ( )[ ]kkkkkkk XPYX',zMdiagM1

,'1 ++=

+ (62)

where:

( ) ( )kkk ,XzX'',zM = (63)

diag (Mk) is a matrix with the same elements as Mkalong the main diagonal and null

elements elsewhere.(64)

For each iteration, if

( ) ( )kk SS +1 (66)

a sub-iteration is started in which the vector of the parameters k is modified on the basis of equation(62) using ankincreasingly large (k= 10k) until inequality (65) holds.

7

The iterative procedure is continued until the following convergence condition is satisfied:

( ) ( )( )

k

kk

THE VALUATION OF BONDS AND BOND OPTIONS: SOME EMPIRICAL TESTS

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