Consider the European call option written on a zero

coupon bond. Suppose the call option has maturity T and

strike price K while the bond has maturity S  T . We propose a numerical method for evaluating the call option

price under the Chan, Karolyi, Longstaff and Sanders (CKLS)

model in which the increment of the short rate over a time

interval of length dt , apart from being independent and

stationary, is having the quadratic-normal distribution with

mean zero and variance dt. The key steps in the numerical

procedure include (i) the discretization of the CKLS model;

(ii) the quadratic approximation of the time-T bond price as a function of the short rate rT  at time T; and (iii) the

application of recursive formulas to find the moments of

r(t+dt) given the value of r(t). The numerical results thus

found show that the option price decreases as the parameter

 in the CKLS model increases, and the variation of the

option price is slight when the underlying distribution of the increment departs from the normal distribution

Khor, C. Y.

Ng, Kok Haur

Pooi, Ah Hin *

Sunway Institutional Repository

Proceedings of 2nd Regional Conference on Applied and Engineering Mathematics (RCAEM-II) 2012  179  BOND OPTION PRICING UNDER THE CKLS MODEL   C. Y. Khor1, A. H. Pooi2 and K. H. Ng3  1Faculty of Computing and Informatics, Multimedia University  Jalan Multimedia, 63100 Cyberjaya, Selangor, Malaysia.  2Sunway University Business School  No.5, Jalan Universiti, Bandar Sunway, 46150 Petaling Jaya, Selangor, Malaysia.  3Institute of Mathematical Sciences, Faculty of Science Building University of Malaya, 50603 Kuala Lumpur, Malaysia.  e-mail: 1cykhor@mmu.edu.my, 2ahhinp@sunway.edu.my, 3kokhaur@um.edu.my   Abstract—Consider the European call option written on a zero coupon bond. Suppose the call option has maturity T and strike price K while the bond has maturity TS  . We propose a numerical method for evaluating the call option price under the Chan, Karolyi, Longstaff and Sanders (CKLS) model in which the increment of the short rate over a time interval of length dt , apart from being independent and stationary, is having the quadratic-normal distribution with mean zero and variance dt. The key steps in the numerical procedure include (i) the discretization of the CKLS model; (ii) the quadratic approximation of the time-T bond price as a function of the short rate  Tr  at time T; and (iii) the application of recursive formulas to find the moments of  dttr   given the value of  tr . The numerical results thus found show that the option price decreases as the parameter   in the CKLS model increases, and the variation of the option price is slight when the underlying distribution of the increment departs from the normal distribution. Keywords: zero coupon bond, CKLS model, option price. I. INTRODUCTION One factor interest rate models are a well-known class of interest rate models in the pricing of interest rate derivatives. There are many types of one factor interest rate models, such as the Vasicek model, Cox, Ingersoll and Ross (CIR) model, Hull-White model, Chan, Karolyi, Longstaff and Sanders (CKLS) model and others (see Vasicek(1977)[1], CIR(1985)[2], Hull-white(1990)[3] and CKLS(1992)[4]). The Vasicek model is considered as the most fundamental one factor interest rate model which exhibits mean reversion. The Vasicek model follows the following stochastic differential equation:       tdwdttrmatdr   . where  tr  is the interest rate at time 0t , a the drift factor with 0a , m  the long run mean around which mean reversion occurs,  the volatility factor and  tw  a Brownian motion. In mean reversion, if the interest rate  tr  is above the long run mean m, then the coefficient of a performs a negative drift such that the interest rate is pulled down back to the long run mean m. Likewise, if the interest rate  tr  is below the long run mean m, then the coefficient of a performs a positive drift such that the interest rate is pulled up back to the long run mean m. Thus, the coefficient of a is a speed of adjustment of interest rate towards m when it wanders away. However, there are some drawbacks of the Vasicek model. One of the main drawbacks is that the interest rate is theoretically possible to become negative which obviously does not make sense in market. This main shortcoming was then fixed in the CIR model. The CIR model follows the following stochastic differential equation:         tdwtrdttrmatdr   . In the CIR model, the term  tr  is imposed and its standard deviation factor becomes  tr . This standard deviation factor guarantees a nonnegative interest rate and hence eliminates the main drawback of the Vasicek model. In 1992, Chan, Karolyi, Longstaff and Sanders introduced a model which covers both Vasicek and CIR models. The CKLS model follows the following stochastic differential equation:             ,0; 0       rrtdwtrdttrmatdr     (1) where a is the drift factor, m the mean around which mean reversion occurs,  the volatility factor,   a positive constant, 0r  the initial interest rate at 0t  and  tw  a Brownian motion. Analytical solutions have been found in many of the one factor interest rate models. Unfortunately, there are no analytical solutions for the bond and option prices based on the CKLS model. Therefore, numerical approaches are required. In 2008, Pooi et al. and Ng et al. as in [5] and [6] proposed numerical methods to approximate the distributions Proceedings of 2nd Regional Conference on Applied and Engineering Mathematics (RCAEM-II) 2012  180 of the short term interest rates in one factor models. Barone-Adesi et al. (1997, 1999) as in [7] and [8] provided a method call the Box method for valuing the prices of the zero coupon bond and the call option written on the zero coupon bond in the CKLS model featured by a Brownian motion. The discretized version of (1) featured by a Levy process with quadratic-normal increments can be expressed as   ,1,111      ktwrtrmarr kkkkk    (2) where ,,,),( 21 wwtkttrrk        are independent, kw  has a quadratic-normal distribution (see Pooi (2003) [9]) with parameters 0 and λ , [  λ,0~ QNwk ] and λ  is such that   ,2,1,1  kwVar k   . Let STB denotes the bond price at time Tt   of a zero coupon bond that matures at time St  , ST 0 . The bond price STB  can be expressed as   .  STdssrST eEB            (3) In 2009, Khor and Pooi as in [10] proposed a numerical method for evaluating the bond price under the CIR model featured by a Levy process. While in 2010, Khor, Ng and Pooi as in [11] proposed a numerical method for finding bond price of a zero coupon bond with maturity T under the CKLS model featured by a Levy process.  The price of the call option with maturity T and strike price K written on the zero coupon bond that matures at time S is given by     ,exp0    KBdssrECSTT       (4) where     0,max    KBKB STST  is the payoff function.  In this paper, we present an alternative method based on numerical integration for finding the price of European call option written on a zero coupon bond when the interest rate follows the CKLS model featured by a Levy process. The rest of this paper is structured as follows. In Section II, we explain two methods and an analytical formula of evaluating the price of the European call option written on a zero coupon bond. Section III shows some results found from the methods in Section II. The analysis of the results is discussed as well. We conclude our work in Section IV. II. METHODS OF EVALUATING THE PRICE OF THE EUROPEAN CALL OPTION WRITTEN ON A ZERO COUPON BOND We present two methods of finding the price of the call option written on a zero coupon bond. These methods are respectively the simulation method and the method based on numerical integration.                Figure 1.  Short rate  tr and times of the maturity of Bond and Option Suppose the interval  T,0  is divided into cN  intervals each of length t  and the interval  ST ,  into bN  intervals each also of length t (see fig. 1). The two methods for finding the call option price written on a zero coupon bond are described below. 2.1 Simulation Method We first use (2) to generate cM  values of Vc =   tuuucN,...,, 21  where iu  is the short rate at time tit  . For each generated value of cV , we again use (2) to generate bM  values of   trrr bNb  ,...,, 21V  where jr is the short rate at time . tjTt  Denote the thi  generated value of cV by   tuuu ci iNiic  ,...,, 21V  and the thj  generated of bV  by   trrr bj jNjjb  ,...,, 21V . For the thi  generated valueicV , we compute the following bond price,  ip  and payoff,  iq .    bjMjbbi VMp1exp1 and      Kpq ii , where   trrrVbj jNjjb ...21 . The price C of the European call option written on a zero coupon bond can be estimated using the following expression:    ciMiiccqVMC1exp1, where   tuuuVci iNiic ...21 . 2.2  Numerical Integration Method In Khor, Ng and Pooi (2010) as in [11], the polynomial approximations of the first four moments of kr  (see (2)) conditioned on the given value of 1kr  are given as  Bond matures T S 2u  1cNu  cNu  1r  2r  1bNr  bNr  Option matures  tr  t 1u   Proceedings of 2nd Regional Conference on Applied and Engineering Mathematics (RCAEM-II) 2012  181   ,| 111101   kkk rccrrE                     (5)   ,| 2 1221212012   kkkk rcrccrrE                (6)   ,| 3 1332 1321313013   kkkkk rcrcrccrrE       (7)   ,| 4 1443 1432 1421414014   kkkkkk rcrcrcrccrrE (8) where 441110 ,...,, ccc  are some constants. We now use the polynomial approximations for the first four moments of kr  conditioned on the value of 1kr  to find the bond price STB . Let   trrrVbNb ...21 . Applying (5) with 1,...,1,  bb NNk , we can get  bVE  in terms of a linear function  of 10c  and 11c . Applying (5) and (6) with 1,...,1,  bb NNk , we can get  2bVE  in terms of a quadratic function of 10c , 11c , 20c , 21c  and 22c . Applying (5), (6) and (7),  3bVE  is found in terms of a cubic function of 10c , 11c , 20c , 21c , 22c , 30c , 31c , 32c  and 33c . Applying (5), (6), (7) and (8), we get  4bVE  in terms of a quartic function of 10c , 11c , 20c , 21c , 22c , 30c , 31c , 32c , 33c , 40c , 41c , 42c , 43c  and 44c . By using (3), the following bond price can then can be obtained for a given value of 0r :       24621432bbbbSTVEVEVEVEB  . After computing STB  for a number of selected values of 0r  in  1,0 , we then apply the regression technique to find the following quadratic approximation for STB . 202010 rgrggBST             (9) We note that the value of 0r  coincides with the value cNu which appears in the vector   tuuucc NN ,...,, 21V . Thus from (4) and (9), we get  EC                   (10)  1231201 ||||...cNcNuuuuuuuu EEEE , where         ,24621exp221043222100       KuguggVVVVKuguggdssrccccNNccccNNTwith   tuuuVcNc ...21  Next, we let  11 | cNcNcNuuu EF .        (11)  1212 || cNcNcNcNcNuuuuu EEF .     (12)                         123121 |||...cNcNuuuuuuu EEEF .     (13) To find 1cNuF , we first use numerical integration to compute , KcNugcNuggcNucNucNuEcNuE 22101|1,,4,3,2,1,0 and express each 1, cNuE as a low degree polynomial function of 1cNu : .0,2 12110, 1           cccN NNuuuE      (14) .2,1,3 13212110, 1                                               ccccNNNNuuuuE     (15) .4,3,4 14313212110, 1                            cccccNNNNNuuuuuE       (16) By using (14), (15) and (16), we can find 1cNuF  approximately (see (11)) and express 1cNuF  as a low degree polynomial function 1cNuF  of 1cNu . We next use (5) – (8) to compute 2cNuF  given by (12): 1212 | cNcNcNcNuuuu FEF . Similarly by using the iterative formulas in (5) – (8), we can also find ,3cNuF …, 2uF , 1uF and C. 2.3  Analytical Formula When the increment )()( twdttw   in the CIR model is having the normal distribution with mean zero and variance dt, the analytical result for the price of the European call option written on a zero coupon bond was derived in [2]. The explicit expression for the price is given below.           STBtThtrSTBrStPKSTtZBC,exp2,4;,2,,,, 222              tThtrrTtKPexp2,4;2,222, where Proceedings of 2nd Regional Conference on Applied and Engineering Mathematics (RCAEM-II) 2012  182      0,exp),, rUtBUtAUtP   , TU  or S ,         221exp221exp2,tUhhhtUhhUtA ,           1exp21exp2,tUhhhhtUUtB,    STBKSTATSrr,,ln ,     1exp22 tThhtT , 2h , 22 2 h . The function   ,;2 x  is a non-central chi-squared cumulative distribution function with   degree of freedom and non-centrality parameter  . In Section III, the numerical results based on the above formula will be compared with those obtained by using the simulation method and the proposed numerical method. III.    NUMERICAL RESULTS In this section, we present some numerical values for the prices of the European call option written on a zero coupon bond when the interest rate follows the CKLS model. We denote  3kwE  and  4kwE  as 3m  and 4m  respectively (see (2)). When the random variable kw  in (2) has a standard normal distribution, we have 03 m  and 0.34 m .The following table shows the values of the parameters used. TABLE 1.  VALUES OF PARAMETERS Parameter Value a  1.0 m  1.0   1.0 t  3651  0r   K  0.4  Table 2 shows the option prices obtained by using the methods in Sections 2.1 to 2.3 when 5.0 , 03 m  and 0.34 m  TABLE 2.  OPTION PRICE WHEN 5.0 , 0.03 m  AND 0.34 m   tkT   k Option Price Simulation Numerical Integration Analytical Formula 50 0.21827092935 0.2221749346 0.2202023774 100 0.1806177512 0.1863042828 0.1831610025 200 0.1272962859 0.1344418227 0.1307168901 300 0.0959408320 0.1003385409 0.0976330740 365 0.0809268886 0.0832110177 0.0822883281 This table shows that the option prices found from the methods of simulation and numerical integration (Section 2.1 and Section 2.2) agree fairly well with those found from the analytical method (Section 2.3). When the random variable kw  is quadratic-normally distributed, the option prices obtained from the methods of simulation and numerical integration for the CIR model are shown in Table 3. TABLE 3.  OPTION PRICES WHEN 5.0 AND tT  365  FOR VARIOUS VALUES OF  43 ,mm    Option Price Simulation Numerical Integration 0.0 2.6 0.0822254808 0.0832112631 0.0 8.0 0.0788519675 0.0832079504 0.5 6.2 0.0810694488 0.0830742762 3.0 20.0 0.0820817169 0.0823858031 From Table 3, we observe that when kw  has the quadratic-normal distribution, the option prices do not differ much from the result obtained by the analytical formula for the case when 5.0  and    3,0, 43  mm . At the same time, we produce option prices when the   is varied from 0.1 to 0.9. Table 4 shows the option prices when 0.03 m  and 4m  is varied whereas Table 5 gives the option prices when 3m  is nonzero. TABLE 4.  OPTION PRICES FOR DIFFERENT VALUES OF  FOR 0.03 m  AND tT  365   003 .m  624 .m  034 .m  084 .m  0.1 0.1003085479 0.100308491 0.100307757 0.2 0.0920748876 0.0920748095 0.092073829 0.3 0.0881931422 0.0881300059 0.088191230 0.4 0.0855136680 0.0855134764 0.0855110817 0.5 0.0832112631 0.0832110177 0.0832079504 0.6 0.0810360881 0.0810357640 0.0810317101 0.7 0.0789214506 0.0789209927 0.0789152552 0.8 0.0768586515 0.0768579490 0.0768491154 0.9 0.0748481852 0.0748470100 0.0748321504   0.1 4m  3m  Proceedings of 2nd Regional Conference on Applied and Engineering Mathematics (RCAEM-II) 2012  183 TABLE 5.  OPTION PRICES FOR DIFFERENT VALUES OF  FOR 0.03 m AND tT  365    503 .m  264 .m  033 .m  0204 .m  0.1 0.1000572229 0.0987054007 0.2 0.0918911949 0.0909350768 0.3 0.0880423613 0.0872776490 0.4 0.0853754988 0.0846821386 0.5 0.0830742762 0.0823858031 0.6 0.0808936705 0.0801695653 0.7 0.0787690562 0.0779761557 0.8 0.0766911151 0.0757819327 0.9 0.0746554390 0.0735216583 From Tables 4 and 5, we see that the option prices decrease when   increases, but the variation of 3m  and 4m  has not much influence on the prices. IV.    CONCLUDING REMARKS As the parameter   in the CKLS model has a significant effect on the option price, it is important that   should be determined carefully. Although the values of 3m  and 4m  have only slight effects on the option price, it is also important to determine these values accurately. The reason is that when the total number of units of option traded is large, a small difference in option price might still have a significant effect on the total value of the large number of units of option.     ACKNOWLEDGMENT We thank to the RCAEM-II 2012 for giving us a chance to participate in the conference.   REFERENCES [1] Vasicek, O.A., “An equilibrium characterization of the term structure,” Journal of Financial Economics, vol. 5, pp. 177-188, 1977. [2] John C Cox, Jonathan E Ingersoll, and Stephen A Ross, “A theory of the term structure of interest rates,” Econometrica, vol. 53, issue 2, March 1985. [3] Hull, J., and A. White, “Pricing interest rate derivatives securities,” The Review  of Financial Studies, vol. 3, num. 4, pp. 573-592, 1990. [4] Chan, K. C., G. A. Karolyi, F. A. Longstaff, and A. B. Sanders., “An empirical comparison of alternative models of the short-term interest rate,” The Journal of Finance, vol. 47, no. 3, pp. 1209-1227, July 1992. [5] A.H,Pooi, K.H,Ng and Y.C,Wong, “Distribution of short-term rate in one-factor models,” Proceedings of 16th annual conference on Pacific Basin Finance Economics Accounting Management (PBFEAM2008), 2-4 July 2008. [6] K.H,Ng, A.H,Pooi, Y.C,Wong and W.L,Beh, “Effects of non-normality on the distribution of short-term interest rate in the CIR model,” Proceedings of 16th annual conference on Pacific Basin Finance Economics Accounting Management (PBFEAM2008), 2-4 July 2008. [7] Barone-Adesi, G., Allergretto, W., Dinenis, W. and Sorwar G., “New Numerical Methods for the Evaluation of Interest Rate Contingent Claims,” technical report, Center for Mathematical Trading and Finance, City University Business School, London, 1997. [8] Barone-Adesi, G., Allergretto, W., Dinenis, W. and Sorwar G., “Valuation of Derivatives Based on CKLS Interest Rate Models,” technical note no. 99-06, Center for Mathematical Trading and Finance, City University Business School, London, 1999. [9] A. H. Pooi, “Effects of Non-Normality on Confidence Intervals in Linear Models,” technical report no. 6/2003, Institute of Mathematical Sciences, University of Malaya, 2003. [10] C.Y.Khor, A.H.Pooi, “A method for evaluating bond price under the CIR model,” Proceedings of 5th Asian Mathematical Conference, vol. 1, pp. 182-187, June 2009. [11] C.Y.Khor, K.H.Ng, A.H.Pooi, “Bond price under the CKLS model featured by a Levy process,” conference paper of PBFEAM 2010, 23-25 July 2010. 

Bond option pricing under the CKLS model

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