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Applied Mathematics, 2010, 1, 37-43 doi:10.4236/am.2010.11006 Published Online May 2010 (http://www.SciRP.org/journal/am) Copyright © 2010 SciRes. AM Implied Bond and Derivative Prices Based on Non-Linear Stochastic Interest Rate Models Ghulam Sorwar1, Sharif Mozumder2 1Nottingham University Business School, Jubilee Campus, Nottingham, UK 2Department of Mathematics, University of Dhaka, Dhaka, Bangladesh E-mail: ghulam.sorwar@nottingham.ac.uk, sharif_math2000 @yahoo.com Received March 8, 2010; revised April 2, 2010; accepted April 30, 2010 Abstract In this paper we expand the Box Method of Sorwar et al. (2007) to value both default free bonds and interest rate contingent claims based on one factor non-linear interest rate models. Further we propose a one-factor non-linear interest rate model that incorporates features suggested by recent research. An example shows the extended Box Method works well in practice . Keywords: Stochastic, Interest Rates, Derivatives, Box Method 1. Introduction Stochastic differential equations are the foundations on which modern option pricing methodology is based. However, non-linear stochastic differential equations for interest rate models have been proposed that captures the non-linear dynamics of the spot interest rates. There are two aspects to the modeling of interest rate term structure models and interest rate contingent claims. The first concerns the econometric aspects (see for example, [1]) and the second the numerical implementation of the re- sulting models. With regard to the numerical aspects of interest rate modeling, there exist three different ap- proaches. The first is the lattice approach introduced by Cox-Ross-Rubinstein (1979) [2]. However, as Ba- rone -Adesi, Dinenis and Sorwar (1997) [3] have demon- strated the lattice approach does not always lead to mea- ningful bond and hence contingent claim prices. The second approach is the Monte-Carlo simulation approach introduced by Boyle (1977) is mainly used to value path dependent European type contingent claims. To date no single accepted Monte-Carlo simulation scheme has been put forward for the valuation of American type contin- gent claims. The third approach is the partial differential equation (PDE) approach. With this approach, the par- tial first and second order derivatives are discretized to produce a system of equations which are then solved iteratively to obtain the bond and contingent claim prices. However, Sorwar et al. (2007) have shown that the usual finite difference approach used to discretize the PDE does not always lead to bond and contingent claim prices that correspond with analytical prices where these prices are available. Sorwar et al. (2007) intro d uc ed the Box Method from engineering to improve on the standard finite difference approach. Sorwar et al. (2007) focused on the CKLS (1992) model. Sorwar et al. (2007) did not attempt to value bonds and contingent claims based on non-linear interest rate models. Ait-Sahalia (1996) [4] non-and Conley et al. (1997) [5] propose parametric linear one-factor which allows non-linear parameterisation. Our main objective in this paper is to expand the Box Method of Sorwar et al. (1997) to price bonds and contingent claims based on both linear and non-linear interest rate models. The outline of the paper is as follows: Section 2 the general non-linear parametric model and the resulting partial differential equation for default free bonds and contingent claims is outlined. We then derive the Ex- panded Box Method (EBM) for the valuation of default free bonds and contingent claims. Using US estimates we compute implied bond and contingent claims prices in Section 3. Section 4 contains a summary and conclusion. 2. Expanded Box Method (EBM) In this section we discuss the valuation of the bond and contingent claim prices based on the extended Ait-Sahalia (1996) [4] and Conley et al. (1997) [5] framework. Following Sorwar et al. (2007) we let: ( ) * t Br ,t,T : price of a discount bond at time t whi ch G. SORWAR ET AL. Copyright © 2010 SciRes. AM 38 Table 1. lternative Parametric Specifications of the Spot Interest Rate Process ( )( ) 2 t ttt drrdtr dW µσ = + . Drift function ( ) r µ Diffusion function ( ) 2 r σ Refer en ce 01 r αα + 0 β Vasicek (1977) [6] 01 r αα + 1 r β Cox-Ingersoll-Ross(1985) [7] Br o wn-Dybvig(1986) [8] Gibbons-Ramaswamy(1993) [9] 01 r αα + 2 2 r β Courtadon (1982) [10] 01 r αα + 3 2r β β Chen et al. (1992) 23 01 2 rr r α αα α ++ + 3 01 2 rr β ββ β ++ Ait-Sahalia (1996) [4] 3 5 4 01 2 rrr α α α αα α +++ 3 01 2 rr β ββ β ++ matures at time * T with the generated spot rate t r . ( ) * P t,T,T : price of a contingent claim at time t which expires at time T based on a discount bond which matures at time * T subject to suitable boundary conditions. In a risk-neutral world, the drift rate is adjusted by the market price of risk r λ 1 so that the short-term interest process beco mes: ( ) ( ) 35 3 012 4 01 2 t t drrrr dt rr dW αα β α αλαα ββ β − = +++++ ++ (1) The resulting partial differential equation is: ( ) 3 35 2 01 22 012 4 1 2 0 U rr r UU r rrrU rt β αα ∂ ββ β∂ ∂∂ α αλαα∂∂ − ++ + ++++−+= (2) In equation (2) ( ) t Ur ,t may represent either ( ) * t Br ,t,T or ( ) * P t,T,T subject to the appropriate boundary conditions (see [10] for more details). Follow- ing Sorwar et al. (2007) we transform the above pricing equation such that either the bond or the contingent claims evolves from the options expiration date or the bonds maturity date to the present, i.e. we let Tt τ = − . The above equation then becomes: ( ) 35 3 2 012 4 2 01 2 2rr r UU r rrr αα β α αλαα ∂∂ ∂ ∂ββ β − ++ ++ +− ++ 33 01 201 2 22rU U rr rr ββ ∂ ∂τ ββ βββ β = ++++ (3) We now choose a general function ( ) R r,, αβ such that: ( ) 35 3 2 2 012 4 01 2 1 2 UU R Rr rr rrr U r rr αα β ∂∂ ∂ ∂∂ ∂ α αλαα∂ ∂ ββ β − = + ++ ++ ++ (4) The above expression simplifies to yield: ( ) 35 3 012 4 01 2 12rr r R Rr rr αα β α αλαα ∂ ∂ββ β − ++ ++ = ++ (5) We now integrate from the general value r ( ) 11nn r rr −+ << to the lower limit of integration 0r= to obtain: ( ) ( ) 35 3 1 012 4 01 2 2 n r r R r,, rr r exp dr rr αα β αβ α αλαα φββ β − − = ++ ++ + ++ ∫ where ( ) 0ln R,, φ αβ = . We further note that: 11UU RQ Rrr Qrr ∂∂ ∂∂ ∂∂ ∂∂ = where: 1Risk premium is treated differently by researchers. Vasicek (1977) [6] takes ( ) r λλ = , Chan et al. (1992) [1] take ( ) 0r λ = , ox et al. (1985) , we take ( ) rr λλ = . G. SORWAR ET AL. Copyright © 2010 SciRes. AM 39 ( ) ( ) 35 3 012 4 001 2 2 r Q r,, rr r exp dr rr αα β αβ α αλαα ββ β − = ++ ++ ++ ∫ So equation (3) becomes: 3 01 2 12Ur QU Qr rrr β ∂∂ ∂∂ββ β −= ++ We now transform the interest rate as: 3 01 2 2U rr β ∂ ∂τ ββ β ++ (6) cr cr s+ =1 where c is a constant. (7) This leads to the transformation of equation (6) as: ( )( )( ) ( )() ( ) ( )() 33 32 2 11 02 02 1 221 11 11 11 Us UU s Qs ssc scs ss ss cs cscs cs ββ τ ββ ββ ββ ∂∂ ∂ Ψ− = ∂∂ ∂ −− ++ ++ −− −− (8) where: ( )()( ) ( )( ) ( ) ( )( )( ) ( )( ) 35 3 2 1 02 4 2 01 02 1 11 1 2 1 11 s scs Qs s ss cs cscs Q sexpdr cs ss cs cs αα β αλ αα α β ββ − Ψ=− + ++ + −− − = − ++ −− ∫ Following the set-up of Sorwar et al. (2007) a grid of size MN× is constructed for values of ( ) m n UU n r,m t= ∆∆ - the value of U at time increment m t and interest rate increment n s , for each method, where: 0m ttmt =+∆ 01m, ,....,M= 2 1 nn s sa + ∆ =∆+ 1n ,....,N= where a is an arbitrary constant. Using the Euler backward difference for the time de- rivative gives: 0 UU U t ∂ ∂τ − =∆ , where 0 U and U refers to bond or contingent claims prices at time step m-1 and m respectively. Integrating equation (8) from the point 1 1 2 2 nn n ss s − − + = to point 1 1 2 2 nn n ss s + + + = , we have: ( )() ( )()() ( )() () ( )() 11 1 22 2 11 1 22 2 1 2 1 2 32 2 0 2 1 22 11 1 2 1 nn n nn n n n ss s ss s s s Us tsdstQsfs UdsQsfs Uds ss c scs Qsfs U ds cs ++ + −− − + − ∂∂ −∆Ψ+ ∆+ ∂∂ −− =− ∫∫ ∫ ∫ (9) Discretizing each of the above integrals, and rearrang- ing gives us the following matrix equation: 1 11 mm mm nnnnnn nn UU UU αχ ηβ − −+ = ++ (10) where: ( ) 1 2 1 2 n n s s U ts ds ss + − ∂∂ −∆ Ψ+ ∂∂ ∫ () ( )( ) 1 2 1 2 3 2 2 1 n n s s s tQsfs Uds cs + − ∆+ − ∫ 2Where a and 0 s∆ are arbitrary constants. A derivation of this expres- sion can be found in Settari and Aziz (1972) [11]. G. SORWAR ET AL. Copyright © 2010 SciRes. AM 40 () ( )( ) 1 2 1 2 2 1 2 1 n n s s Qsfs Uds cs + −− ∫ () ( )( ) 1 2 1 2 0 2 1 2 1 n n s s Qsfs U ds cs + − =− ∫ (9) Discretizing each of the above integrals, and rearrang- ing gives us the following matrix equation: 1 11 mm mm nnnnnn nn UU UU αχ ηβ − −+ = ++ (10) where: () ( ) () ( ) () ( ) () ( ) ( ) ( ) () ( ) ( ) () 1 2 1 2 11 22 11 22 11 22 1 1 1 01 11 03 2 12 2 22 1 1 n n n nn n n n nn n nn n nnnn nn nn nn n n nn n I s t s sQs s t ss Qs ss tt tI I ssQsss Qs sf s I ss cs fs I ss cs α χ β η − − + + −+ −+ +− +− = Ψ −∆ =− Ψ −∆ =− ΨΨ ∆∆ =++∆+ −− = − − = − − The matrix equation linking all bond prices or contingent claim prices between two successive time steps m-1 and m is: 1 01 1 01 1 11 01 22 201 33 3 333 22 2 11 1 00 00 00 0 0 00 0 0 0 0 00 m m m NN m m NNN NN N m NN NN U U U U U U α α α ηβ α χηβ α χηβ χχβ χηβ χη α − − − −−− −−− −− − = Sorwar et al. [12] used the following SOR iteration process to determine bond and contingent claims prices: ( ) 11 11 1 mmmm nnnn nnn n zU UU α χβ η −− −+ = −− (11) In particular they evaluated bond using the following expression: ( ) 1 1 mmm nn n Uz U ωω − = +− (12) Contingent claims were calculated using: ( ) 1 1 m mm n nn UmaxZ ,zU ωω − = +− (13) where Z is the intrinsic value of the contingent claim and for n=1,......,N-1, and ( ] 12, ω ∈ 3. 3. Analysis of Results In this section we apply the EBM using recent estimates of the non-linear model of Ait-Sahalia (1996) [4] on 7-day Eurodollar deposit spot rate over 1973-1995 to demonstrate the method. Ait -Sahalia (1996, Table 4) [4] obtained the following estimates: 3 01 21 23 4 64310 4 333101 143102 ., ., .,, αα αα − −− =−× = ×=−× = 4 45 1 304101., αα − =×= . 4 01 33 23 1 10810 1 883109 681102 073 ., .,. ,. ββ ββ − −− =×= −×= ×=. Table 2 reports the bond prices for maturities ranging from 6 months to 30 years and across interest rates of 2% to 16%. Table III reports both the value of call and put options across a wide range of interest rates. We consider both short and long dated call and put options. The short dated call and put options are based on a 5-year bond with an expiry date of 1 year and is during the last year before the bond matures. Similarly long dated options are based on 10-year bond with an expiry date of 5 years during the last 5 years of the bond. Finally both call and put option prices are calculated across a wide range of exercise prices. The exercise prices are chosen so as to highlight variation of prices for both in-the-money and out-of-the-money options. We assume λ , the market price of risk is zero. Turning to Table 2, we find that at lower interest rate bond prices decay slowly as the term to maturity in- creases. For example, at 2% interest rate a 1 year matur- ity bond is valued at 98.1119, whilst a 30 year bond is valued at 74.8290. At high interest rates, the bond price decay is more rapid for example at 16% interest rate, a 1 year maturity bond is valued at 85.2915, whist a 30 year maturity bond is valued at 1.1770. Turning to Table 3, we observe the following features. Short expiry call op 3 ω is determined by numerical experimentation. For all our calcula- tions we took 1 85. ω = G. SORWAR ET AL. Copyright © 2010 SciRes. AM 41 Table 2. All options written on zero coupon bonds with a face value of $100.00. Interest Rate Ma t ur ity of Bond 2% 4% 6% 8% 10% 12% 14% 16% 0.5 99.028 6 98.037 0 96.985 5 96.088 5 95.131 5 94.184 4 93.250 6 92.340 3 1 98.111 9 96.143 4 94.080 5 92.340 6 90.505 0 88.705 9 86.956 6 85.291 5 5 92.240 0 83.303 5 74.341 3 67.468 5 60.862 3 54.901 0 49.732 4 45.621 2 10 87.043 1 71.953 5 56.701 7 46.171 7 37.275 0 30.119 3 24.783 4 21.303 8 15 83.108 9 64.153 8 44.665 1 32.180 0 22.949 1 16.526 7 12.393 3 10.131 7 20 79.922 8 58.647 3 36.472 3 22.964 4 14.317 8 9.0809 6.2237 4.8889 25 77.215 6 54.633 8 30.873 1 16.883 2 9.0110 5.0032 3.1400 2.3870 30 74.8290 51.602 1 27.0075 12. 8582 5.7491 2.7 679 1.5 921 1.1770 Table 3. All options written on zero coupon bonds with a face value of $100.00. r(%) Exercise- Price 5 year ma- turity 1 year ex- piry 5 year ma- turity 1 year ex- piry Exercise- Price 10 year maturity 5 year ex- piry 10 year maturity 5 year ex- piry (83.3035) call put (71.9535) call put 4 70 16.003 1 0.0000 60 21.971 3 0.0007 75 11.195 9 0.0000 65 17.806 2 0.0493 80 6.3895 0.0050 70 13.641 8 0.6489 85 1.9369 1.6966 75 9.5270 3.1894 90 0.1421 6.6966 80 5.7979 8.0466 (67.4685) (46.1717) 8 55 16.681 1 0.0000 35 22.557 8 0.0000 60 12.064 1 0.0000 40 19.184 3 0.0000 65 7.4471 0.0000 45 15.810 9 0.0058 70 2.8302 2.5315 50 12.437 5 3.8283 75 0.0203 7.5315 55 9.0641 8.8283 12 (54.9010) (30.1193) 45 14.934 1 0.0000 20 19.139 5 0.0000 50 10.4996 0 .0000 25 16.3942 0 .0000 55 6.0652 0.1561 30 13.649 2 0.0183 60 1.6310 5.1561 35 10.904 2 4.8804 65 0.0000 10.156 1 40 8.1591 9.8804 16 (45.6212) (21.3038) 35 15.7692 0 .0000 10 16.7416 0 .0000 40 11.504 6 0.0000 15 14.460 6 0.0000 45 7.2400 0.0005 20 12.179 5 0.0001 50 2.9755 4.3788 25 9.8985 3.6962 55 0.0129 9.3782 30 7.6174 8.6962 tions decay faster than longer expiry call options; for example at r = 4%; the price of a call option decreases from 16.0031 to 11.1959 when the exercise price in creases from 70 to 75. For a similar 5 year call option the price decreases from 21.9713 to 17.8062, when the exer- cise price increases from 60 to 65. Furthermore, the call option prices decrease at a slower rate at high interests. This feature becomes more pronounced for longer expiry call options. With regard to put options we find, the prices are very close to zero, when the options are at-the- money or out-of-the -money. Finally, we find that the value of in-the-money put options is dominated by the intrinsic-va lue . 4. Conclusions The introduction of non-linear stochastic interest rate models has led to the possibility of valuing interest con- tingent claims that reflects the characteristics of the yield curve more accurately. In this paper we have expanded the Box Method to value both bond and American type interest rate contingent claims based on single factor non-linear interest rate models. We have found that the G. SORWAR ET AL. Copyright © 2010 SciRes. AM 42 Expanded Box Method works well with the example considered. 5. References [1] K. C. Chan, G. A. Karolyi, F. A. Longstaff and A. B. Sanders, “An Empirical Comparison of Alternative Mod- els of the Short-Term Interest Rate,” Journal of Finance, Vol. 47, No. 3, 1992, pp. 1209-1227. [2] J. C. Cox and S. A. Ross, “Option Pricing: A Simplified Approach,” Journal of Financial Economics, Vol. 7, No. 3, 1979, pp. 229-264. [3] G. Barone-Adesi, E. Dinenis and G. Sorwar, “A Note on the Convergence of Binomial Approximations for Interest Rate Models,” Journal of Financial Engineering, Vol. 6, No. 1, 1997, pp. 71-78. [4] Y. Ait-Sahalia and Y. Testing “Continuous-Time Models of the Spot Interest Rate,” Review of Financial Studies, Vol. 9, No. 2, 1996, pp. 385-426. [5] T. G. Conley, L. P. Hansen, E. G. J. Luttmer and J. A. Scheinkman, “Short-Term Interest Rates as Subordinated Diffusions,” Review of Financial Studies, Vol. 10, No. 3, 1997, pp. 525-577. [6] O. A. Vasicek, “An Equilibrium Characterization of the Term Structure,” Journal of Financial Economics, Vol. 5, No. 2, 1977, pp. 177-188. [7] J. C. Cox, J. E. Ingersoll and S. A. Ross, “A Theory of the Term Structure of interest Rates,” Econometrica, Vol. 53, No. 2, 1985, pp. 385-407. [8] S. J. Brown, P. H. Dybvig, “The Empirical Implications of the Cox, Ingersoll, Ross Theory of the Term Structure of Interest Rates,” Journal of Finance, Vol. 41, No. 3, 1986, pp. 617-630. [9] M. R. Gibbons and K. Ramaswamy, “A Test of the Cox, In- gersoll, and Ross Model of the Term Structure,” Review of Financial Studies, Vol. 6, No. 3, 1993, pp. 619-658. [10] G. Courtadon, “The Pricing of Options on Default-Free Bonds,” Journal of Financial and Quantitative Analysis, Vol. 17, No. 1, 1982, pp. 75-100. [11] A. Settari and K. Aziz, “Use of Irregular grid in Reservoir Simulation,” Society of Petroleum Engineering Journal, Vol. 12, No. 2, 1972, pp. 103-114. [12] G. Sorwar and G. Barone-Adesi, W. Allegretto, “Valua- tion of Derivatives Based on Single-Factor Interest Rate Models,” Global Finance Journal, Vol. 18, No. 2, 2007, pp. 251-269. G. SORWAR ET AL. Copyright © 2010 SciRes. AM 43 Appendix ( ) ()() 1 2 11 22 11 1 22 2 n nn n s nn ss s U UU sds ss sss s + +− − +− ∂∂∂ ∂ Ψ ≈Ψ−Ψ ∂∂∂ ∂ ∫ Furt he r : 1 2 1 2 1 1 1 1 n n mm nn s nn mm nn snn UU U s ss UU U s ss + − + + − − − ∂≈ ∂− − ∂≈ ∂− Substitution of the above approximation yields: ( ) () ()()() 1 21 2 1 2 11 1 22 2 1 1 1 11 1 n n s nm n nn s nn n mm nn nn nnnn s U s dsU ss ss ss s UU ss ssss + − + + + +− − − +− − Ψ ∂∂ Ψ≈ − ∂∂ − ΨΨΨ ++ −− − ∫ () ( )() 1 2 1 2 3 2 2 1 n n s s s tfs QsUds cs + − ∆≈ − ∫ ( ) () ( ) 1 2 1 2 3 2 2 1 n n s m nn s s tQsUfs ds cs + − ∆− ∫ We further take: () ( ) () ( ) () 1 2 11 22 1 2 33 22 11 n n s n nnn sn s sfsdsfs ss cs cs + − +− ≈− −− ∫ Similar approximation yields: () ( )() ( )() ( ) () () ( )() ( )() ( ) () 1 2 1 2 11 22 1 2 1 2 11 22 2 2 0 2 1 2 1 2 1 1 2 1 1 2 1 1 2 1 n n n n s s m nnnnn n s s m nnn nn n fs QsUds cs QsUfsss cs fs QsUds cs QsUf sss cs + − + − +− − +− ≈ − − − ≈ − − − ∫ ∫ |