# A Robust Spline Collocation Method for Pricing American Put Options

Authors: Zhongdi Cen; Anbo Le; Aimin Xu
Journal: Discrete Dynamics in Nature and Society
 (2019)

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## Abstract

In this paper a robust numerical method is proposed for pricing American put options. The Black-Scholes differential operator in the original form is discretized by using a quadratic spline collocation method on a piecewise uniform mesh for the spatial discretization and the implicit Euler scheme for the time discretization. The position of collocation points is chosen so that the spline difference operator satisfies the discrete maximum principle, which guarantees that the scheme is maximum-norm stable. The error estimation is derived by applying the maximum principle to the discrete linear complementarity problem in two mesh sets. It is proved that the scheme is second-order convergent with respect to the spatial variable and first-order convergent with respect to the time variable. Numerical results demonstrate that the scheme is stable and accurate.


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## Body

## 1. Introduction

The American option is an important financial instrument that gives the holder the right, but not obligation, to buy (call option) or sell (put option) an asset at any time prior to its maturity date. One way to price American options is to solve a linear complementarity problem involving the Black-Scholes differential operator [1]. Since this complementarity problem is, in general, not analytically solvable, numerical methods are required to obtain the approximate solution.The spline approximation methods have become interesting and very promising in solving differential equations due to their flexibility in practical applications. The spline solution has its own advantages, for example, once the solution has been computed, the information required for the spline interpolation between mesh points is easy to obtain. A few papers have used the spline approximation methods to solve option pricing problems. Khabir and Patidar [2] applied a B-spline collocation method to solve the heat equation which is obtained from the Black-Scholes equation by an Euler transformation. Kadalbajoo et al. [3, 4] used cubic B-spline collocation methods for the Euler transformed generalized Black-Scholes equation. Mohammadi [5] developed a quintic B-spline collocation method for solving the generalized Black-Scholes equation governing option pricing. Christara and Leung [6] derived a quadratic spline collocation method on a uniform mesh in the pricing problem under a finite activity jump-diffusion model. Rashidinia and Jamalzadeh [7] proposed a modified B-spline collocation approach for pricing American style Asian options.In this paper we adopt a quadratic spline collocation method on a piecewise uniform mesh to solve the continuous linear complementarity problem arising from American put option pricing. The Black-Scholes differential operator in the original form is discretized. By applying the technique of Surla et al. [8, 9], the position of collocation points is chosen so that the spline difference operator on a piecewise uniform mesh satisfies the discrete maximum principle, which guarantees that the discretization scheme is maximum-norm stable. The error estimation is derived by applying the maximum principle to the discrete linear complementarity problem in two mesh sets. It is proved that the convergence order of the scheme is second-order with respect to the spatial variable and first-order with respect to the time variable. The scheme examined in the present paper overcomes the difficulty in the derivation of the error estimation for the linear complementarity problem under the difference schemes. Numerical results are presented to support these theoretical results.The outline of the paper is as follows. In Section2 the continuous linear complementarity problem for pricing American put options is given. In Section 3 the spline difference scheme is derived. In Section 4 the stability and error analysis for the spline difference scheme are proved. In Section 5 numerical experiments are carried out. Finally conclusion and discussion are indicated in Section 6.


## 2. The Continuous Problem

It is well known that the value of an American put optionv(x,t) satisfies the following continuous linear complementarity problem [1, 10]:(1)Lvx,t≥0,x>0andt∈0,T,(2)vx,t-gx≥0,x>0andt∈0,T,(3)Lvx,t·vx,t-gx=0,x>0andt∈0,T,(4)vx,T=gx,x≥0andt=T,(5)v0,t=E,x=0andt∈0,T,(6)vx,t→0,x→+∞andt∈0,T, where L represents the Black-Scholes differential operator defined by (7)Lvx,t≡-∂v∂t-12σ2x2∂2v∂x2-r-dx∂v∂x+rv,x is the underlying asset price, t is the time, σ is the volatility of the underlying asset price, r is the risk-free interest rate, d is the continuous dividend rate, E is the strike price, T is the maturity date, and g(x) is the payoff function given by(8)gx=max⁡E-x,0. Here we assume that r>d.The infinite domain(0,+∞) needs to be truncated into a bounded domain (0,X) for solving the problem by the numerical method, where X is taken as X=4E based on Willmott et al.’s estimate [1]. The boundary condition at x=X is set to be v(X,t)=0. A detailed discussion of the choice of the linear boundary conditions can be found in [11]. Normally, the error in the computed option price due to the domain truncation is negligible [12]. Therefore, in the remaining of this paper we will focus on solving the following linear complementarity problem(9)Lvx,t≥0,x,t∈0,X×0,T,(10)vx,t-gx≥0,x,t∈0,X×0,T,(11)Lvx,t·vx,t-gx=0,x,t∈0,X×0,T,(12)vx,T=gx,x∈0,X,(13)v0,t=E,vX,t=0,t∈0,T.


## 3. Discretization

Since the Black-Scholes differential operator atx=0 is degenerative, the Euler transformation is usually used to remove the singularity of the differential operator at x=0 (corresponding to -∞ in the transformed domain); see, e.g., Schwartz [13], Tangman et al. [14], and Zhao et al. [15]. However, this will result in additional computational errors due to the truncation on the left-hand side of the domain. Furthermore, the originally mesh points will concentrate around x=0 inappropriately by using the uniform mesh on the transformed interval. In this paper we will discretize the Black-Scholes equation in the original form, but with a quadratic spline collocation method for the spatial discretization and the implicit Euler scheme for the temporal discretization. It should be remarked that if the commonly quadratic spline collocation method is adopted on a uniform mesh for the original Black-Scholes equation, the stability of the scheme could not be guaranteed. The analogous problems have been discussed in [16, 17].For the spatial discretization, a piecewise uniform meshΩN is constructed to guarantee the stability of the discretization scheme:(14)xi=hi=1,h1+σ2r-di-1i=2,⋯,N,where(15)h=X1+σ2/r-dN-1.For the temporal discretization, a uniform mesh ΩK on [0,T] with K mesh elements is used. Then the domain Ω≡(0,X)×(0,T) is divided into the piecewise uniform mesh ΩN×K≡ΩN×ΩK. The mesh sizes hi=xi-xi-1 and Δt=tj-tj-1, respectively, satisfy(16)hi=hi=1,σ2r-dhi=2,⋯,N,and (17)Δt=TK,j=1,⋯,K.It will be shown that the matrix associated with the spline difference operator on the above piecewise uniform mesh is an M-matrix.The implicit Euler scheme onΩK is used for the time discretization to get the following linear complementarity problem:(18)L~vjx≥vj+1xΔt,x∈0,X,(19)vjx-gx≥0,x∈0,X,(20)L~vjx-vj+1xΔt·vjx-gx=0,x∈0,X,(21)vKx=gx,x∈0,X,(22)vj0=E,vjX=0, where (23)L~vjx=-12σ2x2d2vjdx2-r-dxdvjdx+r+1Δtvjx,and vj(x) denotes the approximation of the exact solution v(x,t) at j-th time level.The solutionvj(x) of the problem (18)-(22) is approximated by a quadratic spline function u(x)∈C1[0,X] which on each subinterval [xi,xi+1] has the following form:(24)ux=ui+x-xiui′+12x-xi2ui′′,x∈xi,xi+1.The collocation points are chosen in a nonstandard way as that in [8, 9]: (25)ξi=α1,ixi-1+1-α1,ixionxi-1,xi,ηi=α2,ixi+1-α2,ixi+1onxi,xi+1for 1≤i<N, where 0<α1,i,α2,i<1. Then the collocation equations are defined by(26)-12σ2ξi2u′′ξi-r-dξiu′ξi+r+1Δtuξi=L~uξi,ξi∈xi-1,xi,(27)-12σ2ηi2u′′ηi-r-dηiu′ηi+r+1Δtuηi=L~uηi,ηi∈xi,xi+1.We can obtainu′′(ξi)=ui-1′′, u′′(ηi)=ui′′ from the quadratic spline function (24). Since ξi-xi-1=hi1-α1,i, ηi-xi=hi+11-α2,i, we have (28)uξi=ui-1+hi1-α1,iui-1′+12hi21-α1,i2ui-1′′,uηi=ui+hi+11-α2,iui′+12hi+121-α2,i2ui′′,u′ξi=ui-1′+hi1-α1,iui-1′′,u′ηi=ui′+hi+11-α2,iui′′.Substituting the above expressions for u(ηi),u′(ηi), and u′′(ηi) into (27) on the interval xi,xi+1 with (29)ui+1=ui+hi+1ui′+12hi+12ui′′,ui+1′=ui′+hi+1ui′′,we have(30)ui′=ui+1-uiQi+r+1/Δthi+12ui-hi+12L~uηihi+1Pi,(31)ui+1′=2ui+1-uihi+1-ui+1-uiQi+r+1/Δthi+12ui-hi+12L~uηihi+1Pi,where(32)Qi=-σ2ηi2-2r-dηihi+11-α2,i+r+1Δthi+121-α2,i2,Pi=-σ2ηi2+r-dηihi+1-1+2α2,i-r+1Δthi+12α2,i1-α2,i.Using the same technique on the interval xi-1,xi with the collocation equation (26) and replacing i by i-1 in (31), we also can obtain(33)ui′=2ui-ui-1hi-ui-ui-1Wi+r+1/Δthi2ui-1-hi2L~uξihiMi,where (34)Wi=-σ2ξi2-2r-dξihi1-α1,i+r+1Δthi21-α1,i2Mi=-σ2ξi2+r-dξihi-1+2α1,i-r+1Δthi2α1,i1-α1,i.Combining (30) with (33), we have the following spline difference scheme(35)ai-ui-1+aicui+ai+ui+1=qi-L~uξi+qi+L~uηi,where (36)ai-=-ZihiMi,ai+=-Qihi+1Pi,aic=2hi-r+1Δthi+1Pi+Qihi+1Pi-WihiMi,qi-=-hiMi,qi+=-hi+1Pi,Zi=-σ2ξi2+2r-dξihiα1,i+r+1Δthi2α1,i2.Two degrees of freedom are provided by parameters α1,i and α2,i, which can be used to guarantee that the matrix associated with the discrete operator is an M-matrix. Here we choose α1,i=Δt and α2,i=1-Δt for 1≤i<N. In the next section we will prove that ai-<0,ai+<0,qi->0,qi+>0 and ai-+aic+ai+-qi-/Δt-qi+/Δt>0.Combining (18)-(22) with (35) we can obtain the following fully discretization scheme(37)LN,KVij≥0,i,j∈Ω~h,(38)Vij-gi≥0,i,j∈Ω~h,(39)LN,KVij·Vij-gi=0,i,j∈Ω~h,(40)ViK=gi,i=0,1,⋯,N,(41)V0j=E,VNj=0,j=K-1,⋯,1,0,where(42)LN,KVij≡ai-Vi-1j+aicVij+ai+Vi+1j-qi-Δtα1,iVi-1j+1+1-α1,iVij+1-qi+Δtα2,iVij+1+1-α2,iVi+1j+1.Here we have used the linear interpolation to get the approximated solutions Vj+1 at mesh points ξi and ηi since we only know the numerical solution on mesh points at j+1-th time level.There exists a unique solutionV for the above linear complementarity problem (37)-(41). A detailed discussion of existence and uniqueness of the solution for problem (37)-(41) can be found in [18]. Then from the above solution V we can obtain the optimal stopping price which is the maximum asset price such that Vij=gi for each tj.Remark 1.
In general the pointsxi+1/2 are used as the collocation points [19, 20], i.e., α1,i=α2,i=1/2. But for the problem (18)-(22) the discrete operator with α1,i=α2,i=1/2 does not satisfy the discrete maximum principle. Hence, in order to apply the discrete maximum principle and barrier function technique to analyze the stability and error estimate we use collocation points ξi on [xi-1,xi] and ηi on [xi,xi+1] for 1≤i<N.


## 4. Analysis of the Method

The discrete maximum principle, truncation error analysis techniques, and barrier function techniques are used for analysis of the scheme.Lemma 2 (discrete maximum principle).
Whenα1,i=Δt and α2,i=1-Δt for 1≤i<N, the operator LN,K on ΩN×K satisfies a discrete maximum principle; i.e., if wij is a mesh function satisfying w0j≥0, wNj≥0(0≤j<K), wiK≥0(0≤i≤N) and LN,Kwij≥0(1≤i<N, 0≤j<K), then wij≥0 for all i,j.Proof.
Whenα1,i=Δt and α2,i=1-Δt, it is easy to check that (43)Zi<0,Mi<0,andQi<0 for sufficiently small Δt and h. We also have (44)Pi<-σ2ηi2+r-dηihi+1=ηi-σ2ηi+r-dhi+1<ηi-σ2xi+r-dhi+1<ηi-σ2h+r-dσ2r-dh=0. Hence we can conclude that (45)ai-<0,ai+<0,qi->0,qi+>0, and (46)aic=-ai--ai++r+1Δtqi-+r+1Δtqi+>0,ai-+aic+ai+-1Δtqi-+qi+=rqi-+qi+>0for 1≤i<N and 0≤j<K. Hence the matrix associated with LN,K is an M-matrix, which reveals that the operator LN,K satisfies the discrete maximum principle [21].By using Taylor expansions we can get the following bound of the truncation error(47)τijv=LN,Kvij-qi-Lvξi,tj-qi+Lvηi,tj=p0vxi,tj+p1∂v∂xxi,tj+p2∂2v∂x2xi,tj+p3∂v∂txi,tj+∑n=112πi,njv, where p0=p1=p2=p3=0 and (48)πi,1jv=ai+R2xi,xi+1,vjx,πi,2jv=-ai-R2xi,xi-1,vjx,πi,3jv=-12σ2xi2qi-R0xi,ξi,d2vjdx2,πi,4jv=-12σ2xi2qi+R0xi,ηi,d2vjdx2,πi,5jv=-r-dxiqi-R1xi,ξi,dvjdx,πi,6jv=-r-dxiqi+R1xi,ηi,dvjdx,πi,7jv=r+1Δtqi-R2xi,ξi,vjx,πi,8jv=r+1Δtqi+R2xi,ηi,vjx,πi,9jv=-qi-Δtα1,iR1ξi,xi-1,vx,tj+1+1-α1,iR1ξi,xi,vx,tj+1,πi,10jv=-qi-Δtα2,iR1ηi,xi,vx,tj+1+1-α2,iR1ηi,xi+1,vx,tj+1,πi,11jv=-qi-ΔtR1xi,ξi,vx,tj-qi+ΔtR1xi,ηi,vx,tjπi,12jv=-qi-ΔtR1tj,tj+1,vξi,t-qi+ΔtR1tj,tj+1,vηi,t,Rnx,z,u=z-xn+1n+1!un+1s,s∈x,z.Thus, when α1,i=Δt and α2,i=1-Δt for 1≤i<N, from (47) we have(49)τij≤Cqi-+qi+h2+Δt1≤i<N,0≤j<K,where C is a positive constant independent of the mesh.The following theorem is our main result for the quadratic spline collocation scheme.Theorem 3.
Letv(x,t) be the solution of problem (9)-(13) and Vij the solution of problem (37)-(41). Then if we set α1,i=Δt and α2,i=1-Δt for 1≤i<N, there exists a positive constant C independent of the mesh such that (50)vxi,tj-Vij≤Ch2+Δt,0≤i≤N,0≤j≤K.Proof.
The error estimate will be derived by applying the maximum principle to the discrete linear complementarity problem in two mesh sets as that in [22]. Set (51)Ω¯h=i,j∣0≤i≤N,0≤j≤K,Ω~h=i,j∣1≤i≤N-1,0≤j≤K-1,Ω1=i,j∈Ω~h∣vxi,tj=gi,Ω2=Ω~h\Ω1,∂Ωh=Ω¯h\Ω~h. From (9)-(13) we have (52)Lvxi,tj≥0,i,j∈Ω1,Lvxi,tj=0,i,j∈Ω2. Let (53)Ωh1=i,j∈Ω~h∣Vij=gi,Ωh2=Ω~h\Ωh1. It is easy to check that(54)LN,KVij=0,i,j∈Ωh2. Define the barrier function(55)Φij=Ch2+Δt>0,i,j∈Ω~h,where C is a sufficiently large constant independent of h and Δt.
For(i,j)∈Ωh(2) we have (56)LN,Kvxi,tj-Vij+Φij=LN,Kvxi,tj+LN,KΦij=LN,Kvxi,tj-qi-Lvξi,tj-qi+Lvηi,tj+LN,KΦij+qi-Lvξi,tj+qi+Lvηi,tj≥0 by using Lv(ξi,tj)≥0,Lv(ηi,tj)≥0 and (49), (54), and (55). On the “boundary” of the Ωh(2) we have (57)vxi,tj-Vij+Φij=vxi,tj-gi+Φij≥0fori,j∈Ωh1, and (58)vxi,tj-Vij+Φij=Φij≥0fori,j∈∂Ωh. Then we have (59)vxi,tj-Vij+Φij≥0,i,j∈Ωh2by using the maximum principle (Lemma 2) to Ωh(2). Hence, we have(60)vxi,tj-Vij+Φij≥0,i,j∈Ω¯h.
For(i,j)∈Ω(2),Lv(xi,tj)=0, but LN,KVij≥0; thus for the sufficiently large constant C we have (61)LN,Kvxi,tj-Vij-Φij=LN,Kvxi,tj-qi-Lvξi,tj-qi+Lvηi,tj-LN,KΦij+qi-Lvξi,tj+qi+Lvηi,tj-LN,KVij≤LN,Kvxi,tj-qi-Lvξi,tj-qi+Lvηi,tj-LN,KΦij+qi-α1,i1-α1,i+qi+α2,i1-α2,iOh2+qi-α1,iLvxi-1,tj+1-α1,iLvxi,tj+qi+α2,iLvxi,tj+1-α2,iLvxi+1,tj-LN,KVij≤LN,Kvxi,tj-qi-Lvξi,tj-qi+Lvηi,tj-LN,KΦij+qi-α1,i1-α1,i+qi+α2,i1-α2,iOh2+qi-+qi+OΔt-LN,KVij≤0, where we have used α1,i=1-α2,i=Δt and (49). On the “boundary” of the Ω(2) we have (62)vxi,tj-Vij-Φij=gi-Vij-Φij≤0fori,j∈Ω1, and (63)vxi,tj-Vij-Φij=-Φij≤0fori,j∈∂Ωh. Then we have (64)vxi,tj-Vij-Φij≤0,i,j∈Ω2 by using the maximum principle (Lemma 2) to Ω(2). Hence(65)vxi,tj-Vij-Φij≤0,i,j∈Ω¯h.
Combining (60) with (65) we have(66)maxi,j∈Ω¯hvxi,tj-Vij≤maxi,j∈Ω¯hΦij≤Ch2+Δt, which completes the proof.


## 5. Numerical Experiments

In this section we perform the numerical experiments to illustrate the applicability and accuracy of the method obtained in the preceding section. Errors and convergence rates for the spline difference scheme are presented for two examples.Example 1.
Compute the value of an American put option withσ=0.1,r=0.08,d=0.03,T=1,E=25, and X=100.Example 2.
Compute the value of an American put option withσ=0.5,r=0.1,d=0.02,T=1,E=25, and X=100.The projection scheme [23] is used to solve the linear inequality system (37)-(41). Assume that Vj+1 are known; then we first solve the following discretization problem for {Vj+1/2}: (67)LN,KVij+1/2=0,i=1,⋯,N-1,V0j+1/2=E,VNj+1/2=0. Next we set (68)Vij=max⁡Vij+1/2,gi,i=0,1,⋯,N,which ensure that (38) and (39) are satisfied.The computed option valuesV and the constraints V-g with N=128 and K=64 are plotted in Figures 1–4 for Examples 1 and 2, respectively.Figure 1
Computed option valueV for Example 1.Figure 2
The constraintV-g for Example 1.Figure 3
Computed option valueV for Example 2.Figure 4
The constraintV-g for Example 2.Since the exact solutions of our examples are not available, the quadratic spline approximated solution ofN=2048,K=4096 is used as the exact solution denoted by V~(x,t). The error estimates for different N at t=0 are presented. The accuracy in the discrete maximum-norm, (69)eN,K=maxi,jVi,0N,K-V~xi,0,and the convergence rate (70)RN,K=log2eN,Ke2N,Kare computed. The error estimates and convergence rates for Examples 1 and 2 are listed in Tables 1 and 2, respectively.Table 1
Error estimates and convergence rates for Example1.
K N error rate 1024 128 2.3532e-2 2.009 256 5.8452e-3 1.914 512 1.5506e-3 2.342 1024 3.0592e-4 -Table 2
Error estimates and convergence rates for Example2.
K N error rate 1024 128 8.6028e-3 2.031 256 2.1045e-3 2.097 512 4.9176e-4 2.451 1024 8.9918e-5 -From Figures1 and 3 we can conclude that the numerical solutions are nonoscillatory, and from Figures 2 and 4 we also can conclude that the numerical solutions satisfy the constraint conditions (38) and (39). Tables 1 and 2 show that RN,K is close to 2 for sufficiently large N. These results support the convergence estimate of Theorem 3.Finally, our scheme is compared with the binomial method, analytic approximation method, and compact finite difference methods. The numerical example and numerical results in Zhao [15] are used. The parameters for the American put option are σ=0.3,T=1,r=0.04,d=0.02, and E=100. The results for the binomial method are obtained with the time mesh size Δt=0.01. The results for the analytical approximation method are obtained with the time mesh size Δt=0.02. The results for the compact finite difference methods 1, 2, and 3 are obtained with the spatial mesh size h=0.02 and the time mesh size Δt=0.0005. For our spline collocation scheme we use the time mesh size Δt=0.0005 and N=20000 mesh points for the spatial discretization which has almost same number of mesh points as that of the compact methods. The true option values are obtained with the trinomial method using the time mesh size Δt=0.00005. Table 3 shows that our scheme is more accurate than other methods.Table 3
Comparison of the American put option values att=0 for some different values of x with parameters: σ=0.3,T=1,r=0.04,d=0.02,E=100,X=400.
Stock pricex  Binomial method Analytic approx. method Compact method 1 Compact method 2 Compact method 3 Our stable scheme True values 75.9572  25.33949 25.4509 25.10042 25.32570 25.32739 25.32892 25.32986 error 0.00963 0.12104 0.22944 0.00416 0.00247 0.00094 83.9457  19.49101 19.6617 19.34597 19.49193 19.49383 19.49565 19.49691 error 0.00590 0.16479 0.15094 0.00498 0.00308 0.00126 92.7743  14.27957 14.4477 14.16375 14.25707 14.25914 14.26121 14.26265 error 0.01692 0.18505 0.09890 0.00558 0.00351 0.00144 102.5315  9.87092 10.0278 9.78167 9.83789 9.84000 9.84216 9.84354 error 0.02738 0.18426 0.06187 0.00565 0.00354 0.00138 113.3148  6.35580 6.53401 6.32881 6.36044 6.36241 6.36458 6.36558 error 0.00978 0.16843 0.03677 0.00514 0.00317 0.00100 125.2323  3.84473 3.97728 3.81244 3.82898 3.83064 3.83268 3.83337 error 0.01136 0.14391 0.02093 0.00439 0.00273 0.00069 138.4031  2.14801 2.25467 2.12653 2.13452 2.13578 2.13757 2.13784 error 0.01017 0.11683 0.01131 0.00332 0.00206 0.00027As discussed above, the optimal stopping boundary can be obtained as the maximum asset price such thatVij=gi for each tj. We plot the optimal stopping boundary for the two examples with N=K=2048 in Figures 5 and 6, respectively.Figure 5
The optimal stopping boundary for Example1.Figure 6
The optimal stopping boundary for Example2.


## 6. Conclusion and Discussion

In this paper the linear complementarity problem arising from American put option pricing is treated by a quadratic spline collocation method on a piecewise uniform mesh. The main advantage of the spline collocation method examined in the present paper lies in the possibility of simple and direct constructing of a continuous approximate solution and normalized flux between the mesh points. The suitable choice of collocation points guarantees that the spline difference operator on the piecewise uniform mesh satisfies the discrete maximum principle. Hence the scheme is maximum-norm stable. The error estimation is derived by applying the maximum principle to the discrete linear complementarity problem in two mesh sets. It is proved that the scheme is second-order convergent with respect to the spatial variable and first-order convergent with respect to the time variable. Numerical results demonstrate that the scheme is stable and accurate.For the time discretization, although the Crank-Nicolson scheme can be used to improve the truncation error scheme, the discretization scheme does not satisfy the maximum principle without additional constraints about the time sizes, which leads to the difficulty in the derivation of the error estimation for the linear complementarity problem under the difference schemes. In future we plan to overcome this difficulty in the error analysis for the Crank-Nicolson scheme in discretizing the linear complementarity problem.Since the probability distribution of realized asset returns may exhibit features such as heavy tails, volatility clustering, and volatility smile, two different classes of models have been studied in the finance literature: the jump-diffusion models [24, 25] and the stochastic volatility models [26, 27]. In future we extend this technique to construct spline collocation schemes and study the stability and error analysis of these schemes for the jump-diffusion models and the stochastic volatility models.



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