Computation of Japanese bonds and derivative securities

K. Ben Nowmana,*, Ghulam Sorwarb

a Department of Investment, Risk Management and Insurance and Centre for Mathematical Trading and Finance, City

University Business School, Frobisher Crescent, Barbican Centre, London EC2Y 8HB, UKb Department of Accounting, Finance and Law, Faculty of Management, University of Stirling, Scotland, FK9 4LA, UK

Received 16 April 1998; accepted 19 June 1998

Abstract

In this paper, we use the Box numerical method to compute implied bond and option prices starting from the general CKLS

interest rate model based on Japanese interbank data. In particular, we compute numerically implied prices from the CKLS,

Vasicek, Cox±Ingersoll±Ross and Brennan±Schwartz models. We also compare the prices with those obtained from the exact

analytical formulae of the Cox±Ingersoll±Ross model. We find that the implied bond and option prices vary across models for

Japan. # 1998 IMACS/Elsevier Science B.V.

Keywords: Term structure; Auctions; Numerical methods

1. Introduction

The seminal papers of Black±Scholes [1] and Merton [2] stimulated growth not only of equity andcommodity options1 but also of term structure interest rate models and the valuation of bonds andoptions based on these term structure of interest rate models. Today research into term structure modelsis important both to academics and commercial institutions alike. Unfortunately bond prices andoptions based on term structure interest rate models, with few exceptions cannot be valued analytically.To date a number of numerical methods have been used to solve this problem. In this paper, we use therecently introduced Box method into finance by Barone-Adesi et al. [3] from the engineering literatureto value bonds and options prices using the historical estimates obtained from Japanese interbank data.Our main results indicate that bond prices and options prices are sensitive to the underlying interest ratemodel used.

Mathematics and Computers in Simulation 47 (1998) 583±588

ÐÐÐÐ

* Corresponding author.1An option is a financial instrument that derives its characteristics from the underlying asset on which it is based. For

example an equity call and put options derive their value from the stock price. Similarly interest rate call and put options

derive their value from the prevalent bond price.

0378-4754/98/$19.00 # 1998 IMACS/Elsevier Science B.V. All rights reserved

PII S 0 3 7 8 - 4 7 5 4 ( 9 8 ) 0 0 1 4 0 - 2

The outline of the paper is as follows: Section 2 introduces the models and briefly discuses thenumerical methods for the valuation of bond and option prices. Section 3 presents the implied bond andoption prices. Section 4 contains some conclusions.

2. Interest rate models and the Box method

Historically, it has been observed that periods of high interest rates have led to periods of low interestrates and vice versa. This process is known as mean reversion. Furthermore, it has been observed that arandom element exists in the movements of interest rates which is generally dependent on the currentlevel of interest rates. Both these important historical characteristics are incorporated by the CKLSmodel [4].

The CKLS model giving the change in the instantaneous short term interest rate is represented by Eq.(1) and allows the conditional mean and variance to depend on the level of rates.

drt � k��ÿ �r � ���dt � �r dzt (1)

where dzt is the Wiener process driving the term structure movements, a parameter, rt the short terminterest rate at time t, k the speed of adjustment, � the long term interest rate and � the risk premium.

Setting �0 yields the Vasicek [5] model, �0.5 yields the Cox±Ingersoll±Ross [6] model and �1yields the Brennan±Schwartz [7] model. Eq. (1) is a stochastic differential equation. Applying Ito'slemma [5] to Eq. (1) for a general function u(rt, t) which may represent either a bond price or an optionprice yields.

du � k��ÿ �r � ��� @u

@r� @u

@t� 1

2�2r2 @

2u

@r2

� �dt � �r dzt (2)

Financial arguments based on those of Black±Scholes are then used to construct a portfolio, whicheliminates the random component of the above equation, yielding the following partial differentialequation.

1

2�2r2 @

2u

@r2� k��ÿ �r � ��� @u

@rÿ ru� @u

@t� 0 (3)

The above pricing equation is transformed such that either the bond or the contingent claim evolvesfor the options expiration date2 or the bonds maturity date3 to the present, i.e. we let ��Tÿt. The aboveequation then becomes

1

2�2r2 @

2u

@r2� k��ÿ �r � ��� @u

@rÿ ru � @u

@�(4)

In Eq. (4) u�rt; t� may represent either the bond price B�rt; t; T��, maturing at time T*, or the call and

put option price P�rt; t; T�;T�, expiring at time T. The bond and the option prices are subject to the

2Options have defined periods of time during which they can be exercised, i.e. exchanged for the underlying asset. There are

principally two types of options with regard to the exercise date. Options which can be exercised at the exercise date only are

called European; whereas options which can be exercised anytime until the exercise date is called American.3Bonds are, `I owe you', financial instruments issued by national governments or financial institutions to raise capital. In

return the borrowers pay interest at or until the date at which the bond expires ± called the maturity date.

584 K.B. Nowman, G. Sorwar / Mathematics and Computers in Simulation 47 (1998) 583±588

following boundary conditions:

B�0; t; T�� � 1

B�1; t;T�� � 0

with P�rt; t; T�;T� representing an American call option it is subject to the following boundary

conditions:

P�rt; t;T�; T� � max�B�rt;T

�; T� ÿ E�P�1; t;T�; T� � 0

P�rt; t;T�; T� � max�B�rt; t;T

�; T� ÿ E;P�rt; t; T�;T��

Finally with P�rt; t; T�;T� representing an American put option it is subject to the following boundary

conditions:

P�rt; t;T�; T� � max�E ÿ B�rt; T

�;T��P�1; t;T�; T� � E

P�rt; t;T�; T� � max�E ÿ B�rt; t; T

�; T�;P�rt; t; T�;T��

To solve the above partial differential equation numerically, a grid of size M�N is constructed forvalues of um

n � u�n�r;m�t� ± the value of u at time increment tm and interest rate increment rn where

�tm � t0 � m�t; m � 0; 1; . . . ;M� and �rn � r0 � n�t; n � 0; 1; . . . ;N�The values of individual elements of um

n are computed column by column from the left column to theright column. Also within each column, we solve from bottom to the top.

A number of researchers have used the finite difference method to solve the above partial differentialequation subject to the appropriate boundary conditions. However, Barone-Adesi et al. [3] find that fora certain combination of parameters, the finite difference solution does not converge to the analyticalsolution, where one is available. Barone-Adesi et al. [3] applied the Box method to solve the pricingequation for bond and option based on the CKLS interest rate model. Barone-Adesi et al. [3] solveEq. (4) using the Box method and derive the following iterative equation (Barone-Adesi et al. [3],Eq. (3.20)):

�numÿ1n � �num

nÿ1 � �numn � �num

n�1 (5)

where taking

ra � rn � rnÿ1

2and rb � rn�1 � rn

2

�n � dr1ÿ2 b

�t�1ÿ 2 � 1ÿ ra

rb

� �1ÿ2 !

if 6� 12

or 6� 1

� ÿ d

�tln

ra

rb

� �for � 1

2

� d

�t

1

ra

ÿ 1

rb

� �for � 1

K.B. Nowman, G. Sorwar / Mathematics and Computers in Simulation 47 (1998) 583±588 585

�n � ÿ 1

�r

�ra��rn�

�n � ÿ 1

�r

�rb��rn�

�n � 1

�r

�rb��rn� �

�ra��rn�

� �� X

where

X � cr2ÿ2 b

2ÿ 2 1ÿ ra

rb

� �1ÿ2 !

� dr1ÿ2 b

�t�1ÿ 2 � 1ÿ ra

rb

� �1ÿ2 !

provided 6� 12

or 6� 1

� c�rb ÿ ra� ÿ d

�tln

ra

rb

� �for � 1

2

� ÿc lnra

rb

� �� d

�t

1

ra

ÿ 1

rb

� �for � 1

�r� � expar1ÿ2

1ÿ 2 ÿ br2ÿ2

2ÿ 2

� �provided 6� 1

2or 6� 1

� exp�ÿbr�ra for � 12

� expÿa

r

� �rÿb for � 1

Barone-Adesi et al. [3] used the following SOR iteration process to determine bond and option prices:

zmn �

1

�n

�numÿ1n ÿ �num

nÿ1 ÿ �numÿ1n�1

ÿ �(6)

umn � !zm

n � �1ÿ !�umÿ1n (7)

for n � 1; . . . ;N ÿ 1, and !��1; 2�. To estimate the CKLS model historically we used the approach ofNowman [8] who estimated the CKLS model on the US and UK data using a discrete model (seeNowman [8] for full details). The short-term interest rate used in this study is the Japanese 1 monthinterbank obtained from Datastream. The data is monthly, covering the period January 1986±January1998.

3. Analysis of results

In this section we discuss the results in Table 1 which are organised such that in the first section ofthe table we analyse bond prices. Bond prices are calculated for maturities ranging from 5 to 15 yearsand across constant rates form 5% to 11%. Bond prices are calculated using the Box method for theVasicek model ( �0), Cox±Ingersoll±Ross (CIR) model ( �0.5), Brennan±Schwartz model ( �1) and

586 K.B. Nowman, G. Sorwar / Mathematics and Computers in Simulation 47 (1998) 583±588

the actual market . Further we also calculate analytical bond prices for the CIR model using theformula in the original CIR paper. In the second part of the table we calculate both American type calland put options based on the zero coupon bonds. Note that as the underlying instrument is a zerocoupon the value of American call option is the same as European call option. We exploit this feature tocheck the accuracy of our numerical CIR call price. We calculate analytical call prices using theformula provided by CIR in their original paper. We calculate both short dated and long dated calloptions. The short dated call options are based on a 5 year bond with a expiry date of 1 year and isduring the last year before the bond matures. Similarly long dated options are based on 10 year bondwith an expiry date of 5 years during the last 5 year's of the bond. Finally call and put option prices arecalculated across a wide range of exercise prices. The exercise prices are chosen so as to highlight thevariation of contingent claim prices across the standard models. We assume, the market price of risk is

Table 1

1 month Japan, ��0.0128, ��0.1872, ��0.1776, market �0.4700, �t�0.05, �r�0.5%: all options written on zero coupon

bonds with a face value of US$100.00

Maturity

of bond

Expiry

of option

r (%) Exercise

price

Asset/

option

�0 Analytic

�0.5

�0.5 �1 Market

5 5 Bond 85.4138 78.4195 78.5097 76.4128 78.8435

8 78.0791 68.9109 69.0234 66.2689 69.4274

11 71.0240 60.5553 60.6918 57.5499 61.1244

10 5 Bond 81.4017 65.8038 65.9367 56.6603 67.0771

8 73.6156 54.1756 54.3031 43.5633 55.5977

11 66.1507 44.6022 44.7260 33.7283 46.0419

15 5 Bond 78.5774 58.2901 58.4500 41.2758 60.1965

8 71.0123 46.9067 47.0360 28.9151 48.9338

11 63.7609 37.7463 37.8537 20.5827 39.7304

5 1 8 65 Call 21.9962 10.5910 10.6029 6.3901 11.1569

70 18.4601 7.2592 7.2390 2.4475 7.8008

75 15.0410 4.5287 4.4870 0.3506 5.0198

80 11.7246 2.4781 2.4417 0.0110 2.8860

5 1 8 65 Put 5.4319 2.2237 0.3280 2.4038

70 6.8835 4.0570 3.7312 4.2115

75 8.5144 6.8940 8.7312 6.9132

80 10.3216 10.9766 13.7312 10.7029

10 5 8 50 Call 35.3803 21.1394 21.0614 10.6316 22.2013

55 31.6418 18.2351 18.1145 7.6061 19.2251

60 27.9162 15.4571 15.2922 4.8862 16.3628

65 24.2039 12.8180 12.6094 2.6625 13.6255

10 5 8 50 Put 5.2426 5.3592 6.4368 5.4506

55 6.3480 7.3575 11.4368 7.3670

60 7.5684 9.8215 16.4368 9.6952

65 8.9156 12.8120 21.4368 12.4795

K.B. Nowman, G. Sorwar / Mathematics and Computers in Simulation 47 (1998) 583±588 587

zero as in Barone-Adesi et al. [3]. The analysis is based on annualised estimates in the table to make itconsistent with the grid.

The results for the unrestricted gamma for the one month rate is �0.4700. These results compare toTse's [9] estimate for 3 month market data of 0.6187 indicating volatility dependence has fallenrecently. In Table 1 we find that actual bond and option prices are very close to �0.5 bond andoption prices. For example, for a 10 year maturity bond at 8% interest, we find that the market price is55.5977 and �0.5 price is 54.3031. Furthermore, the analytical �0.5 is very close to the numerical �0.5 bond price at 54.1756. Similarly for a call option based on a 5 year bond with 1 year expiry dateat exercise price of $65 market call price is 11.1569 and �0.5 call price is 10.6029 (numerical) and10.5910 (analytical). Further, we find that for �0, bond prices are higher than actual bond pricesparticularly at longer maturities. The reverse is true for �1. This difference in bond prices leads tocorresponding difference in option prices.

4. Conclusions

In this paper, we used the Box method to value bonds and derivative securities based on the CKLSmodel outlined in Barone-Adesi et al. [3]. We compared both bond and option prices based on the 1month Japanese interbank rate. Our main conclusions are as follows: Firstly, both bond and optionprices vary widely across the standard interest rate models. Secondly the Box method leads tonumerical solution which are in excellent agreement with analytical solutions where available.

References

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[2] R.C. Merton, The theory of rational option pricing, Bell Journal of Economics and Management Science 4 (1973) 141±

183.

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contingent claims, Centre for Mathematical Trading and Finance Technical Report, City University Business School,

1997.

[4] K.C. Chan, G.A. Karolyi, F.A. Longstaff, A.B. Sanders, An empirical comparison of alternative models of the short-term

interest rate, Journal of Finance 47 (1992) 1209±1227.

[5] O. Vasicek, An equilibrium characterization of the term structure, Journal of Financial Economics 5 (1977) 177±188.

[6] J.C. Cox, J.E. Ingersoll, S.A. Ross, A theory of the term structure of interest rates, Econometrica 53 (1985) 385±407.

[7] M.J. Brennan, E.S. Schwartz, Analyzing convertible bonds, Journal of Financial and Quantitative Analysis 15 (1980)

907±929.

[8] K.B. Nowman, Gaussian estimation of single-factor continuous time models of the term structure of interest rates,

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[9] Y.K. Tse, Some international evidence on the stochastic behavior of interest rates, Journal of International Money and

Finance 14 (1995) 721±738.

588 K.B. Nowman, G. Sorwar / Mathematics and Computers in Simulation 47 (1998) 583±588