1 Academic Journal of Computational and Applied Mathematics /August 2013/ UISA Pricing European and American bond option under the Hull White extended Vasicek model Eva Maria Rapoo 1, Mukendi Mpanda 2 Department of Mathematics and Applied Mathematics, University of South Africa, South Africa Abstract In this paper, we consider the Hull-White term structure problem with the boundary value condition given as the payoff of a European and American option. We restrict ourselves to the case where parameters of the model are constants and we first derive simple closed form expression for pricing European bond option in the Hull-White extended Vasicek model framework in term of forward price. The analytic representation of American bond option being very hard to handle, we are forced to resort numerical experiments. We transform the Hull-White term structure equation into the diffusion equation and we first solve it through implicit, explicit and Crank-icolson (C) difference methods. As these finite difference methods require truncation of the domain from infinite to finite one, which may deteriorate the computational efficiency for American bond option, we build a C method over the unbounded domain. We introduce an exact artificial boundary condition in the pricing boundary value problem to reduce the original problem to an initial boundary problem. Then the C method is used to solve the reduced problem. The results through illustration show that our method is more efficient and accurate than standard FDMs. Keywords: Term Structure Equation, Hull-White extended Vasicek model, Coupon bearing and zero coupon bonds, European and American bond option, Diffusion equation, Finite Difference Methods and Artificial Boundary method. 1 Dr. Rapoo, Senior lecture, Dept. of Mathematical Sciences, UISA (Contact: rapooe@unisa.ac.za) 2 Mr. Mpanda, Master student, Dept. of Mathematical Sciences, UISA (Contact: 48301914@mylife.unisa.ac.za)
2 Academic Journal of Computational and Applied Mathematics /August 2013/ UISA 1. Introduction Mathematical models that describe the future evolution of interest rates by describing the stochastic movement of the instantaneous short-term interest rate, the short rate models play an important role in fixed-income security pricing. Among them, the Hull-White model [12-16]. As an extension of Vasicek model [12], the Hull-White model assumes that the short rate follows the mean-reverting stochastic differential equation and present special features which are analytical tractability on liquidly traded derivatives [12], super calibration ability to the initial term structure [13] and elegant tree-building procedure [15]. These make the model very attractive as a practical tool. On another hand, if we want to price interest rate derivatives such as bond option, interest rate swap, interest rate cap and interest rate swaption we need to perform options on these derivatives. One attractive and simple option that gives us nice analytic results, is the European option [2] through which the option is exercised only on the expiration date. For the case where there is early exercise of the option, we talk in term of American option [19]. Thus, an American option is a European one with the additional right to exercise it any time prior to expiration. In the arbitrage free framework, pricing interest rate derivatives under the short rate model lead us to the parabolic partial differential equation called term structure equation [2] with the boundary condition given as the payoff function. The main problem for pricing options written on interest rate derivatives under interest rate models is how to solve these kinds of stochastic PDE associated to a given payoff option. The bond option being considered as a standard interest rate derivative 3, we turn our intention to bond option. Many papers have addressed the solution to the problem stated above, including Amin and Madsen [1], Brace and Musiela [6], and Madsen [21], who all worked within the Gaussian Heath-Jarrow-Morton framework under different short rate models. Jamshidian [18], as for him, derives a simple closed-expression for pricing European bond option under the Vasicek model where the resulting pricing formula resembles the Black-Scholes formula [5] and has a similar interpretation. In this paper, by referring mainly to [18], we also derive a formula for pricing European bond option under the Hull-White extended Vasicek model. Due to the complexity of American bond option, we rely on numerical experiments. We first reduce the Hull-White term structure equation which is a parabolic PDE to the diffusion equation with the help of some transformations and define a pricing boundary value problem under the diffusion equation which shall be discussed. As finite difference methods (FDMs) are straightforward to implement and the resulting uniform rectangular grids are comfortable, we then first use these methods, especially explicit, implicit and Crank-icolson methods to solve the obtained pricing boundary value problem. 3 Simply because from which we may derive other interest rate derivatives without any difficulties.
3 Academic Journal of Computational and Applied Mathematics /August 2013/ UISA It is well-known that the explicit Finite Difference Method requires the condition of the type where and represent respectively the small time step and the step width of the scheme for stability (see for e.g [22]). In practice, it is sometimes desirable to change the length of time step. In contrast, both the implicit FDM and Crank-icolson method can achieve unconditional stability [22]. Unfortunately, implicit schemes including both implicit FDM and C method are constructed for a PDE with a bounded domain. Therefore, the implementation requires the truncation of the infinite domain to the finite one which may deteriorate the computational efficiently of American bond option. To circumvent the issue stated above, Kangro and icolaides [20] study the boundary condition of the PDE of the Black Scholes type. In their performance, they stipulate that an alternative method to solve problems with unbounded domains is to impose an artificial boundary condition and then an exact boundary condition is derived on the artificial boundary based on the original problem. In the field of interest rate derivatives, Hun and Wu [17] extend the Kangro and icolaides results and propose an artificial eumann boundary condition for pricing American bond option under Black-Scholes dynamics. Wong and Zhao [25] generalize the artificial boundary condition to the CEV model 4 and show that the proposed artificial boundary condition is exact and the corresponding implicit scheme is unconditionally stable, efficient and accurate. In contrast, Tangman et al. [24] develop a high-order optimal compact scheme for pricing American options under the Black-Scholes dynamics without considering artificial boundary conditions as in [20]. To make more consistent these approaches listed above, Wong and Zhao [26] propose recently an artificial boundary method based on the PDEs to price interest rate derivatives with early exercise feature. This approach is accurate, efficient and robust to the truncation. On the debit side of the balance sheet, the obtained result is very complex and very difficult to implement numerically. The second interesting feature of this paper is the extension of Wong and Zhao [25, 26] studies. We study a C method over an unbounded domain into which we perform the C method on the initial boundary value problem obtained from an exact artificial boundary condition. We then compare our performance with Explicit, Implicit and standard C methods. The rest of the paper is structured as follows. Section 2 presents the statement of the problem. Section 3 derives simple closed form expression for pricing European bond option under the Hull-White extended Vasicek model, an illustration and results are provided. Section 4 deals more with numerical methods for pricing American bond option. We transform the Hull-White term structure problem into the diffusion equation, we apply standard FDMs and we derive the C over an 4 CEV is an acronym of Constant Elasticity of Variance widely used in stochastic volatility model and resembles Cox Ingersool Roll short interest rate model.
4 Academic Journal of Computational and Applied Mathematics /August 2013/ UISA unbounded domain. An illustration and results are also provided. Section 5 concludes. 2. Statement of the problem Let consider the price of the European option denoted by which is function of the time where the option has been made, the one-factor short interest rate and the maturity date Then the price is the solution of the boundary problem given by ( ) ( ) (2.1) (2.2) where the equation (2.1) is called the Hull-White term structure equation [2]. We note that the process represents the one-factor Hull-White extended Vacisek model [12-13] defined by the following Ornstein-Uhlenbeck stochastic differential equation ( ) (2.3) In equation (2.1) as well as (2.3), the deterministic time functions given by, and are respectively called the time independent drift, the speed of reversion and the volatility term of the stochastic process is the standard Brownian motion with respect to the risk neutral probability measure Q. The boundary condition (2.2) is called the payoff function of the European call option on a coupon bearing bond with strike price maturing at the time T. The process represents the price of a coupon-bearing and it is defined by (2.4) where are amounts paid at the maturity dates. If, then we get the zero-coupon bond. We further note that and the dynamics of under the Hull-White extended Vasicek model is given by the following Geometric Brownian motion, - (2.5) where is the Brownian motion with respect to the risk neutral probability measure Q and where the function is given by ( ) The purpose of this study is to solve the boundary problem (2.1)-(2.2) under the Hull-White model (2.3). Initially, from the risk neutral valuation formula [11], the solution to the boundary value problem (2.1) - (2.2) is given by 64 8 9 57 (2.6)
5 Academic Journal of Computational and Applied Mathematics /August 2013/ UISA where, - is the natural filtration generated by the standard Brownian motion W. The next section provides a simple closed form expression for (2.6) in our framework. 3. Pricing European bond option under the Hull-White extended Vasicek model. 3.1. Derivation of an analytic formula. In this section, our idea is mainly drawn from [1], [6], [21] and [18]. Let consider the boundary condition (2.2) given by 4 5 (3.1) ow we introduce the forward price defined by with By virtue of Ito calculus, the dynamics of noted by is given by (3.2) where And where From (3.2) we get ( ) 4 5 Writing shortly, we have where and where The boundary condition (3.1) becomes (3.3) 6 7 Where the set is defined by
6 Academic Journal of Computational and Applied Mathematics /August 2013/ UISA 8 9 and is the indicator function. From the risk neutral valuation formula and by referring to [18, formula (8)], we have ( ) Let say that 4 ( ) 5 ( ) 4 ( ) 5 (3.4) and then the price ( ( ) (3.5) ) can be written as The process is, under the probability measure Q, a Gaussian process, independent of the field with expected value zero and variance and the Gaussian law is given by. Hence ( ) 4 5 ( ) ( ) Therefore we should express as follows ( ) where ( ) is given by 4 5 To evaluate the expression (3.4), we introduce an auxiliary probability measure by setting 4 5 Here is the Likelihood process. Then by Girsanov Theorem, the process
7 Academic Journal of Computational and Applied Mathematics /August 2013/ UISA, - is the standard Brownian motion under. ote that the process admits the following representation under 4 5 We can write the equality above as where The random variable ( ) is independent of, with Gaussian law under. In addition, [ ] [ ] [ ]. Then we get the following result Furthermore Finally we have where.( ) / ( ) ( ) ( ) ( ) Therefore the price of a European call option on a coupon bearing bond is given by the following result (3.8) where processes and are given by ( ) (3.9) ( ) (3.10) and where are random variable whose distribution under Q is Gaussian, with zero expected value and covariance matrix given by
8 Academic Journal of Computational and Applied Mathematics /August 2013/ UISA. / [ ] [ ] (3.11) and the variance given by (. / ) (3.12) If we set and, then and is given by ( ( ) ) Therefore the price is given by (3.13) where parameters and are given by ( ) ( ) ( ) ( ) ( ) (3.13 ) ote that these results are equivalent to Jamshidian formula [18, formula (9)] simply because both Hull-White and Vasicek models have the same volatility term of the bond price process. Consequently, the price formulas (3.8) and (3.13) resemble Black-Scholes formula [5] and have the same interpretations. 3.2. Illustration and results Let consider four yield curves named the flat yield curve, upward yield curve, downward yield curve and the humped yield curve defined by: (3.13 ) Let consider again parameters of the Hull-White model given by Speed reversion and Volatility term with strike price given by 0.8. By applying the formula (3.13 ), then the price of a two year maturity bond in one year s time call option for these four yield curves are given by Table 3.1 Results of European bond option. Yield curves Prices (call option) 0.175131 0.172118 0.178177 0.167961 Prices (put option) 0.009723 0.010018 0.009434 0.010445
Prices 9 Academic Journal of Computational and Applied Mathematics /August 2013/ UISA Figure 3.2. Prices of European call/put option for, 0.18 0.16 0.14 0.12 0.1 0.08 Call option Put option 0.06 0.04 0.02 0 0.026 0.028 0.03 0.032 0.034 0.036 0.038 0.04 Yield Curves In the figure above, we have four values of the yield curves computed by using the formula (3.13 ) where and, we have:,, and We observe that for the same fixed parameters both for the Hull-White model and Vasicek model, the downward yield curve has the highest price and the humped yield curve has the lowest price for an European call bond option. In contrast, for a European put option, the humped yield curve has the highest price and the downward yield curve has the lowest price. 4. Pricing American bond option under the Hull White extended Vasicek model. In the previous section we have examined the pricing of European bond option where the option can be exercised only on the expiration date. So in this section, we examine the case of early exercise opportunity commonly called American option. To make our analysis easier, we transform the Hull-White term structure problem into the diffusion problem. Here we refer to [12, 13, 14, 16, 25 and 26]. 4.1. From the Hull White TSE to the diffusion equation In this subsection we make transformations of the Hull-White term structure equation (2.1) until we get the simplest diffusion or heat equation. As in the Hull- White model the drift term and the volatility term are given or determined statistically, then the main purposes of these transformations are to eliminate the time-independent drift which is unknown and to deal with only the obtained diffusion equation for pricing methodology. Let reconsider the Hull-White term structure equation (2.1) ( ) (4.1)
10 Academic Journal of Computational and Applied Mathematics /August 2013/ UISA In order to eliminate the unknown function, we introduce a deterministic variable such that ( ) and we define a new variable given by It follows that (4.2) We then adapt the governing equation (4.1) to the stochastic process (4.2) by replacing respectively the expression and the short interest rate process by and. We find ( ) For simplicity, let assume that which are strictly positive constants, we thus rewrite the governing equation (4.1) as ( ) (4.3) ext, we reverse the time by in the governing equation (4.3), we get (4.4) with Let assume that gets the following form where with and are defined by It is follow that ( ) 0 1 64 5 7 6 7 (4.5) where Let again 6 7, -
11 Academic Journal of Computational and Applied Mathematics /August 2013/ UISA then partial differential operators 4 (4.6) 5 By making substitution of partial differential operators (4.5) and (4.6) into the governing partial differential equation (4.4), we arrive to Finally by introducing two other variables; the variable and z such that ( ) with and ( ) And by setting we find the so called diffusion or heat equation given by: (4.7) with the overall changes summarized as follows ( ) ( ) (4.8) where ( ) ( ) (4.9) 0 1 4.2. A new formulation of a boundary value problem. We reconsider a boundary value condition (3.1) given by 4 5 (4.10) As discussed in [19], a European option can have a value smaller than the payoff but it cannot happen with American options. Thus the boundary condition (4.10) under the American bond option is then given by
12 Academic Journal of Computational and Applied Mathematics /August 2013/ UISA ( ) 4 5 (4.10 ) This can be written as where ( ) 4 5 (4.10 ) or equivalently 4 5 4 5 (4.10 ) With the initial conditions given by So that Therefore, the Hull-White term structure problem is reduced to the following boundary value problem: (4.11) The problem (4.11) above can be considered as the free boundary value problem of American bond option under the Hull-White extended Vasicek model. Due to the complexity of the problem which is very hard to resolve analytically, we rely to numerical experiments in our last subsection. Our choice falls to finite difference method (FDM) because of their easy implementations. 4.3. Solution of the obtained diffusion problem trough FDM In this section we develop three cases of FDM which are explicit, implicit and Crank- icolson (C) difference scheme (See [22]) and to the end we derive another special case the Crank-icolson method over an unbounded domain. Explicit, Implicit and Crank-icolson schemes For all these three types of schemes, we need first transform the domain of the continuous problem * + into a discretized domain and which must be approximated by a finite truncated interval [ ] where to achieve a given level of accuracy requires M to be large enough. We begin to build finite difference schemes by defining a grid of points in the plane. For any arbitrary integer n and m, we denote the value of at the grid point that can be shortly written as. The grid is then constructed for considering values of when the time is equal to
13 Academic Journal of Computational and Applied Mathematics /August 2013/ UISA and when the variable is equal to The general finite difference scheme is given by the following approximations These approximations inserted into the heat equation (4.7) give (4.11a ) where and a constant taking values in the set }. According to whether get value 0, 1 or we have respectively explicit, implicit and Crank icolson method. (4.12) is equivalent to By letting, - we get the following result Let us define the following vectors as (4.11b) (4.11c) [ ] [ ] [ ] Where and where approximate the value of at the grid point ( The initial and terminal conditions in the vector ) are given by We may formulate the American boundary value problem as
14 Academic Journal of Computational and Applied Mathematics /August 2013/ UISA (4.11d) where is a square tridiagonal matrix with We note that the explicit FDM is stable for In contrast, both Implicit and C methods are unconditionally stable which means that their stability holds for all time step. Moreover the C method has the highest order of convergence among FDMs. For more details, see [23, pages 117, 118, and 121]. Crank icolson method over an unbounded domain The three cases of finite difference method discussed above are constructed for a partial differential equation with a bounded domain. Therefore, their implementation requires the truncation of the infinite domain into the finite one (See for e.g. [22, 23]) which may deteriorate the computation efficiently. As the C method is the well-known highest order of convergence and efficiently method among standard finite difference methods, we thus derive the C method over an unbounded domain in order to increase the degree of accuracy. The main idea behind is to perform the C method on the initial boundary value problem obtained from an exact artificial boundary condition. As the solution of the problem (4.11) exists, then in the derivation of the exact boundary value problem we subdivide the domain into two: the interior domain containing the initial condition and exterior domain. These two domains are separated by the so called artificial boundary The initial condition being defined in the interior domain, then obviously it will be zero in the exterior domain. ow let consider the problem (4.11) defined on an unbounded domain given by where * + ( ) Let us define the artificial boundary as * + which divides the unbounded domain into the interior domain and exterior domain defined respectively by and * +
15 Academic Journal of Computational and Applied Mathematics /August 2013/ UISA where * + Then the derivation of the exact artificial boundary condition will be based on the interior problem defined by: (4.12) By using Laplace Transform and by the Duhamel Theorem (see [7, pp. 31]), we may find: (4.13) Proposition 4.1 The solution of the original problem (4.11) over an unbounded domain satisfies the following partial differential equation over a bounded domain (4.14) where is defined in (4.10 ) or (4.10 ). Moreover the problem above admits a unique solution. Proof: See Appendix A. ow let us first approximate the third boundary condition of the Problem (4.14). We know from the theory of approximation that the integral in that boundary condition can be approximated as Where (4.15) 4 5 ( ) From the Crank-icolson scheme, we have (4.16) Approximations (4.15) and (4.16) into the third boundary condition of the Problem (4.14) lead to
16 Academic Journal of Computational and Applied Mathematics /August 2013/ UISA which can be rewritten as (4.17) The Crank-icolson method over an unbounded domain is then given by (4.18) (4.19) (4.20) (4.21) We observe that terms and in equation (4.21) are unknown, so we need to eliminate them. In order to do so, we combine the equation (4.18) for with the equation (4.21) and we obtain the following result: As we have done in Section 4.3.1, let [ ] [ ] [ ] where with the initial and terminal conditions respectively given by and Finally, we arrive to the following problem
17 Academic Journal of Computational and Applied Mathematics /August 2013/ UISA (4.22) where is a square tridiagonal matrix with The solution to problems (4.15) and (4.22) above is done iteratively. To find their numerical solutions, we prefer to use the Successive over Relaxation (SOR) method because of its high speed of convergence. Since our problems are more complex and the standard SOR method cannot support this kind of problem, we provide in Appendix B. an adapted SOR method for our problems (4.15) and (4.22) which is a slight modification of the standard SOR method. ow let us give an illustration. 4.4. Illustration and results. We consider a one-year call option on a zero coupon bond of strike price 0.8 with early exercise feature on a two-year with face value equals to unity. The model parameters are given as and. Comparisons are made by using the explicit FDM, Crank-icolson method and the Crank-icolson method over an unbounded domain. We are doing our essay with six numbers of steps: and we regard the results of the explicit FDM with as the true value. 5 With the help of the Matlab 7.1 codes, we may arrive to the following summarized results: Table 4.1: Call option on American bond option under the C method over an unbounded domain. C Method over an unbounded domain Flat Upward Downward Humped 120 0.098817 0.09182 0.106204 0.078117 240 0.098863 0.091898 0.106275 0.078222 360 0.098889 0.091935 0.10631 0.078285 480 0.098896 0.091942 0.106314 0.078296 600 0.098898 0.091944 0.106316 0.078298 True(=1200) 0.098901 0.091948 0.106323 0.078299 5 We acknowledge that in doing this, a discretization or programming error could affect what we take to be a true value
18 Academic Journal of Computational and Applied Mathematics /August 2013/ UISA Table 4.2. Relative errors in percentage for the C method over an unbounded domain. Relative errors of C Method over an unbounded domain Flat Upward Downward Humped 120-0.084933418-0.139209118-0.1119231-0.2324423 240-0.038422261-0.054378562-0.04514545-0.09834097 360-0.012133345-0.014138426-0.01222689-0.01788018 480-0.005055561-0.006525427-0.00846477-0.00383147 600-0.003033336-0.004350285-0.00658371-0.00127716 1200 0 0 0 0 Table 4.3. Call option on American coupon bearing bond option under the C method. C method Flat Upward Downward Humped 120 0.098775715 0.09190874 0.10617924 0.07814658 240 0.098936218 0.09188911 0.106262956 0.078275906 360 0.098868091 0.091980332 0.106324155 0.078340569 480 0.098896381 0.091935298 0.106306257 0.078306506 600 0.098909083 0.091931257 0.106308566 0.078292649 1200 0.098901 0.091948 0.106323 0.078299 Table 4.4 Relative errors of explicit FDM Relative errors of C method Flat Upward Downward Humped 120-0.126677132-0.042697829-0.135210773-0.194664555 240 0.035609701-0.064046744-0.056473576-0.02949463 360-0.033274638 0.035162918 0.00108603 0.053090333 480-0.004670125-0.013814004-0.015747439 0.009585755 600 0.008172718-0.018209368-0.013575379-0.008111023 1200 0 0 0 0
Prices x 100 19 Academic Journal of Computational and Applied Mathematics /August 2013/ UISA Table 4.5. Call option on American coupon bearing bond option under the explicit FDM. Explicit FDM Flat Upward Downward Humped 120 0.099034 0.091888 0.106453 0.078381 240 0.098802 0.092 0.106379 0.078262 360 0.098946 0.091879 0.106308 0.078213 480 0.098904 0.091964 0.106343 0.078283 600 0.098884 0.091973 0.106341 0.078309 True(=1200) 0.098901 0.091948 0.106323 0.078299 Table 4.6. Relative errors in percentage for modified Trinomial Lattice tree Relative errors of Explicit FDM Flat Upward Downward Humped 120 0.134477912-0.065254274 0.122268935 0.104726753 240-0.1001001 0.056553704 0.052669695-0.04725475 360 0.045500046-0.075042415-0.01410795-0.10983537 480 0.003033336 0.01740114 0.018810605-0.02043449 600-0.017188906 0.027189281 0.016929545 0.012771555 1200 0 0 0 0 Figure 4.1: Results of the Explicit FDM, C method and C method over an unbounded domain (UD) for Flat yield Curve 9.905 9.9 9.895 True Value Explicit FDM C method with UD C method 9.89 9.885 9.88 9.875 100 150 200 250 300 350 400 450 500 550 600
Prices x 100 Prices x 100 20 Academic Journal of Computational and Applied Mathematics /August 2013/ UISA Figure 4.2: Results of the Explicit FDM, C method and C method over an unbounded domain (UD) for Upward yield Curve 9.21 9.205 9.2 True Value Explicit FDM C method with UD C method 9.195 9.19 9.185 9.18 100 150 200 250 300 350 400 450 500 550 600 Figure 4.3: Results of the Explicit FDM, C method and C method over an unbounded domain (UD) for Downward yield Curve 10.655 10.65 10.645 10.64 True Value Explicit FDM C method with UD C method 10.635 10.63 10.625 10.62 10.615 100 150 200 250 300 350 400 450 500 550 600
Relative errors in % Relative errors in % Prices x 100 21 Academic Journal of Computational and Applied Mathematics /August 2013/ UISA Figure 4.4: Results of the Explicit FDM, C method and C method with an unbounded domain (UD) for humped yield Curve 7.86 7.85 True Value Explicit FDM C method with UD C method 7.84 7.83 7.82 7.81 100 150 200 250 300 350 400 450 500 550 600 Figure 4.5: Relative Errors estimation of the Explicit FDM, C method and C method over an unbounded domain for Flat yield Curve 0.15 0.1 0.05 True Value Explicit FDM C method with UD C method 0-0.05-0.1-0.15-0.2 100 150 200 250 300 350 400 450 500 550 600 Figure 4.6: Relative Errors estimation of the Explicit FDM, C method and C method over an unbounded domain for Upward yield Curve 0.15 0.1 0.05 0-0.05-0.1-0.15-0.2 True Value Explicit FDM Modified C method C method -0.25 100 150 200 250 300 350 400 450 500 550 600
Relative errors in % Relative errors in % 22 Academic Journal of Computational and Applied Mathematics /August 2013/ UISA Figure 4.7: Relative Errors estimation of the Explicit FDM, C method and C method over an unbounded domain for Downpward yield Curve 0.1 0.05 0-0.05-0.1 True Value Explicit FDM C method with UD C method -0.15 100 150 200 250 300 350 400 450 500 550 600 Figure 4.8: Relative Errors estimation of the Explicit FDM, C method and C method over an unbounded domain for Humped yield Curve 0.15 0.1 0.05 0-0.05-0.1-0.15-0.2 True Value Explicit FDM C method with UD C method -0.25 100 150 200 250 300 350 400 450 500 550 600 5. Conclusion In this work, by introducing forward price and by applying the risk neutral valuation formula and referring to Jamshidian Work [17], we have derived a simple closedform expression for pricing European option on the zero-coupon and couponbearing bonds under the Hull-White extended Vasicek model. We draw two important findings: o The price of a European bond option under the Hull-White extended Vasicek model is equivalent to Jamshidian formula [18], consequently, the result resembles the Black-Scholes formula [5] and has the same interpretation. o The price of the call option is greater than the price of put option for any yield curve to maturity. As there is not analytical solution for American option, we have used numerical methods. After transformation from the Hull-White term structure equation to the
23 Academic Journal of Computational and Applied Mathematics /August 2013/ UISA diffusion equation, we have applied the finite difference method especially explicit, implicit and Cranck-icolson methods. As FDMs require truncation of interval from infinite to finite one, we have built one method which remedies to that; the Crank- icolson method over an unbounded domain into which we have derived the corresponding exact artificial boundary condition and the Crank-icolson scheme has been used in order to find the numerical solution. We find out that the C method with an unbounded domain outperforms FDMs in term of both efficiently and accuracy when we price American Bond option. Acknowledgement This work has been supported by the Department of Mathematical Sciences, College of Science, Engineering and Technology, University of South Africa. Appendix A: Proof of Proposition 4.1. Assume that and are two solutions to problem (4.64). We define their difference to be. In satisfies: By multiplying, we obtain: (A.1) by both sides of the PDE (4.14) and performing integrations over (A.2) We then consider the following problem on the unbounded domain : Given, the problem above has a unique solution. Moreover (A.3) (A.4) By multiplying by both sides of (A.4) and integrating over, we obtain (A.5) Then it follows that
24 Academic Journal of Computational and Applied Mathematics /August 2013/ UISA (A.6) Finally, combining (A.4), (A.5) and (A.6)we find. This means that Appendix B: SOR method to Problems (4.15) and (4.22). Letting and, we have So the problem becomes where and. From the standard SOR method, we have (B.1) As in our problem A is the tridiagonal matrix with and, then it is follow that o For the case where. / [ ] (B.2) o For the case where [ ] o For the case where [ ] Since then we may write. / [ ]} By setting. / and by adapting for to or equivalently, we arrive to the following algorithm for the adapted SOR model.
25 Academic Journal of Computational and Applied Mathematics /August 2013/ UISA Algorithm B.1: Adapted SOR model.. / 2 0 13 ote that the test allow us to get off the loop and the algorithm above is also valid for Crank icolson over an unbounded domain by replacing by which lead us to. / We note furthermore, the above algorithm can be particularized to the European option by replacing the line 2 0 13 by 0 1 References [1] Amin, K., Jarrow, R., Pricing options on risky assets in a stochastic interest rates economy. Journal of financial mathematics Vol. 2(1992) pp. 217-237. [2] Bjork, T., Arbitrage theory in continuous time, Oxford University Press, Oxford (2004). [3] Black,F., Karasinki,P., Bond and option pricing when short rates are lognormal, Financial Analysts Journal (1991) pp. 52 59. [4] Black, F. Derman, E., Toy, W., A one-factor model of interest rates and its application to treasury bond options, Financial analysts Journal (1990) pp. 33-39.
26 Academic Journal of Computational and Applied Mathematics /August 2013/ UISA [5] Black, F. Sholes, M., The pricing of options and corporate liabilities, Journal of Political Economics, vol. (81) (1973), pp. 637 654. [6] Brace, A., Musiela, M. (1997)., A Multifactor Gauss Markov implementation of Heath, Jarrow and Merton Journal of mathematics finance, Vol. (4)(1994), pp. 336-368. [7] Carslaw, H.S., Jaeger, J.C., Conduction of Heat in solids, Second Edition, Oxford University Press, ew York (1959). [8] El Karoui.,., Rochet, J.C. A pricing formula for options on coupon bonds. Working paper, SDEES, (1989). [9] Han H., Huang Z., A class of Artificial Boundary Conditions for Heat Equation in Unbounded Domains, International Journal of computers and Mathematics with applications, Vol.43(2002), pp 889 900. [10] Heath, D., Jarrow, R., Morton, A., Bond pricing and term structure of interest rates : A new methodology for contingent claims valuation, Econometrica, vol. 60(1992) pp. 77 105. [11] Hull, J.C., Options, Futures and Other derivatives, 7 th edition, Pearson Prentice Hall, Saddle River (2009c). [12] Hull, J., White, A., Pricing interest rate derivative securities, Review of Financial studies, vol.3 (1990) pp. 573 592. [13] Hull, J., White, A., The general Hull White model and supper calibration, Journal of finance, vol.57 (2001) pp. 875 903. [14] Hull, J., White, A., Efficient procedures for valuing European and American path dependent options, Journal of derivatives, (1993b) pp. 21 31. [15] Hull, J., White, A., umerical procedures for implementing term structure models I: Single Factor models, Journal of derivatives, (1994) pp. 7 16. [16] Hull, J., White, A., One factor interest rate derivative securities, Journal of financial quantitative, vol.28 (1995) pp. 97 102. [17] Hun, H., Wu, A fast numerical method for the Black Scholes equation of American options, SIAM Journal of umerical Analysis, vol. 41 (2003) pp 2081-2095. [18] Jamshidian, F., An exact bond option formula, Journal of financial, Vol.44(1989) pp.1 14. [19] Jamshidian, F., An analysis of American options, Review of futures Markets, vol.11(1)(1992) pp. 72 82. [20] Kangro, R., icolaides, Far field boundary conditions for Black- Scholes equations, SIAM Journal of umerical Analysis, vol. 38 (2000) pp 1357-1368. [21] Madsen, C., The pricing of options on coupon bonds. Working paper, Realkredit Danmark, Copenhagen (1994a). [22] Scott, L.R., umerical analysis, Princeton (2011). [23] Seydel, R., Tools for Computational Finance, Springer, ew York (2003). [24] Tangman DY., Gopaul A., Bhuruth M., A fast high order finite difference algorithm for pricing American options, Journal of computational and applied mathematics, Vol 222 (2008) pp. 17-29.
27 Academic Journal of Computational and Applied Mathematics /August 2013/ UISA [25] Wong H.Y., Zhao, J., An artificial boundary method for American option pricing under the CEV model, SIAM Journal of numerical analysis, Vol. 46 (2008) pp. 2183 2209. [26] Wong H.Y., Zhao, J., An artificial boundary method for the Hull White model of American interest rate derivatives, Journal of Applied mathematics and computation, Vol. 217(2011) pp. 4627 4643.