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Alternating segment explicitimplicit and implicitexplicit parallel difference method for the nonlinear Leland equation
Advances in Difference Equations volume 2016, Article number: 103 (2016)
Abstract
The nonlinear Leland equation is a BlackScholes option pricing model with transaction costs and the research of its numerical methods has theoretical significance and practical application value. This paper constructs a kind of difference scheme with intrinsic parallelismalternating segment explicitimplicit (ASEI) scheme and alternating segment implicitexplicit (ASIE) scheme based on the improved Saul’yev asymmetric scheme, explicitimplicit (EI) scheme, and implicitexplicit (IE) scheme. Theoretical analysis demonstrates that this kind of scheme is unconditional stable parallel difference scheme. Numerical experiments show that the computational accuracy of this kind of scheme is very close to the classical CrankNicolson (CN) scheme and the alternating segment CrankNicolson (ASCN) scheme. But the computational time of this kind of scheme can save nearly 81% for the classical CN scheme and save nearly 40% for the ASCN scheme. Numerical experiments confirm the theoretical analysis, showing the higher efficiency of this kind of scheme given by this paper for solving a nonlinear Leland equation.
Introduction
The BlackScholes (BS) option pricing model can be accepted by practice fields and theory fields, not only because it has abundant financial implications, but also it is linear and is a simple model. The BS model can be transformed into a heat conduction equation with a more mature theory in mathematics and can get the analytical solution of the European call option and put option pricing. However, there exist certain differences between the assumptions of the BS model and the real financial market, such as there being no transaction costs and the fixed volatility hypothesis. In order to meet the needs of the actual financial market, we need to broaden the idealized assumptions and improve the standard BS model. That has been the focus of academic research; see [1–3]. In the real financial market, because of transaction costs which one needs to pay in securities trading, using the continuous trading strategy is not realistic. So studying the option pricing model with transaction costs (nonlinear Leland model) has great financial practical significance.
The nonlinear Leland equation is one of the nonlinear BS option pricing models which need to consider transaction costs and received extensive attention of economists and applied mathematicians in the past 20 years [4–6]. Because one is unable to export the accurate analytical expression of the European option and American option pricing in the case of considering the transaction costs, many researches focus on the study of numerical solutions. In the numerical solution, in order to make the numerical scheme have good computing stability and precision, we often design implicit or half implicit difference scheme. In recent years, Ankudinova and Ehrhardt proposed the CrankNicolson (CN) scheme for solving a nonlinear BS equation (the Leland model, the BarlesSoner model, and the risk adjustment pricing model) [7]. Wu and Yang put forward the explicitimplicit (EI) and implicitexplicit (IE) difference schemes for solving the payment of dividend BS equation [8]. However, most of the schemes are calculated in a serial way and the efficiency is low.
In order to make full use of the computer advantages of multicore processors, a parallel algorithm and a parallel program design have become a necessary means to improve the computing efficiency [9]. The implicit scheme generally has good stability, but it is unfavorable for parallel computing. Inspired by the grouping explicit method [10], Zhang et al. put forward the thought that using the Saul’yev asymmetric scheme to construct a segment implicit scheme, and one properly used the alternating technology to establish a variety of explicitimplicit and pure implicit alternating parallel schemes (such as an alternating segment explicitimplicit (ASEI) scheme, an alternating segment CrankNicolson (ASCN) scheme), then one got some numerical results which contained stability and parallelism [11]. Yang et al. constructed a new kind of parallel difference schemethe alternating band CrankNicolson (ABdCN) scheme for solving the quanto option pricing model and proved that it is close to secondorder accuracy and unconditionally stable [12]. Yuan et al. had put forward a parallel difference scheme with secondorder accuracy and unconditional stability for a nonlinear parabolic equation [13]. Wang also gave a kind of alternating segment difference scheme with intrinsic parallelism for the KdV equation and proved that the scheme is linearly absolute stable [14]. Zhang showed the alternating segment explicitimplicit parallel difference scheme for a class of nonlinear evolution equations and got the result that the method has unconditional stability and parallelism [15].
For the research of the parallel difference method for solving the nonlinear Leland equation, Wu et al. presented a difference method with intrinsic parallelismthe ASCN parallel difference scheme [16]. Because of the high timeliness of the option, constructing a difference scheme with good stability and intrinsic parallelism has important practical application value. We apply the EI and IE schemes at segment interior points, and the improved asymmetric difference scheme at interior boundary points, and we get a kind of difference numerical difference scheme with intrinsic parallelismthe alternating segment explicitimplicit (ASEI) scheme and the alternating segment implicitexplicit (ASIE) scheme.
The plan of this paper is as follows. In Section 2, we construct the ASEI difference scheme for the nonlinear Leland equation. In Section 3, by using three lemmas, the unique solvability of the difference solution is discussed. We analyze the stability of the ASEI scheme in Section 4 and the accuracy in Section 5. In Section 6, the ASIE scheme is put forward by simulating the ASEI scheme and a theorem is given. Numerical examples are provided to show the effectiveness of the ASEI and ASIE schemes in Section 7. Some concluding remarks are included finally.
ASEI parallel difference method
Nonlinear Leland equation
Assuming that the underlying asset is the transaction costpaying stock, by the Δhedging principle, we can get the following nonlinear Leland equation [1–3], which we will consider for the European options:
here V is the price of a European call option (dollar), S is the price of the underlying asset, r is riskfree interest rate, σ̂ is the revised volatility, \(\hat{\sigma}^{2} = \sigma^{2} (1 + Le \operatorname{sign}(V_{SS}))\). In the revised volatility, \(Le = 2\frac{k}{\sigma} \sqrt{\frac{2}{\pi\delta t}}\) is the Leland number, σ is the volatility, k is a volume of transaction cost, δt is the time difference between the two transactions, t is the time.
The Leland equation is a definite solution problem of nonlinear partial differential equations. When \(k > \sigma\sqrt{\pi\delta t/8}\), equation (1) will become a terminal value problem of a positive parabolic equation which is an illposed problem [1, 3]. In order to transform the problem (1) into a wellposed problem, we can assume that \(k < \sigma\sqrt{\pi \delta t/8}\), and that the transaction cost should be smaller or the process of hedging risk cannot be too often.
In order to solve the equation of the European call option pricing with transaction costs by using numerical methods, equation (1) is to be satisfied on the following boundary conditions [5, 7]:

(1)
The value of the option is the payoff function i.e. \(V(S, T) = (S  K)^{ +}\).

(2)
\(\lim_{S \to\infty} \frac{V(S, t)}{S} = 1\). When S is sufficiently great, the option price is approximately \(S  K\).

(3)
If \(S(t_{0}) = 0\), then \(V(S, t) = 0\) for \(t > t_{0}\).
Hence, for the European call option, we need to solve the following equation on the domain \(\Sigma= \{ 0 \le S < \infty,0 \le t \le T\}\):
In order to be able to solve equation (2), we can substitute its variables as follows [1–3]:
Then equation (2) will be transformed into the initialboundary value problem of a partial differential equation with constant coefficients:
here \(D = \frac{\hat{\sigma}^{2}}{\sigma^{2}}\), \(L = \frac{2r}{\sigma^{2}}\), \(x \in R\), \(0 \le\tau\le\tilde{T} = \frac{\sigma^{2}T}{2}\).
Meanwhile, the initial and boundary conditions will be translated into
In the specific calculation, we can select a large enough \(M^{ +}\) and a small enough \(M^{ }\) making the solving area and the boundary conditions
Construction of the ASEI scheme
Let us make a mesh partition on the area \(\Sigma_{0}\) and consider the function \(U(x, \tau)\) at the discrete set of points
Here h is the space step, p is the time step, and m, n are the number of grid points in the x direction and τ direction, respectively. We use \(U_{i}^{j}\) to denote the solution of (3) at a finite difference point \((x_{i},\tau_{j})\). In order to construct the ASEI scheme, we give some difference schemes of equation (3). Let \(a = \frac{p(D + L)}{2h}\), \(b = \frac{pD}{h^{2}}\).
First, the classical explicit scheme is
The above scheme can be written as
Second, the classical implicit scheme is
The above scheme can be written as
At last, we present the two improved Saul’yev asymmetric schemes,
The above schemes can be written as
Among the schemes mentioned above, the classic explicit scheme (4) has the property of parallelism and is very suitable for parallel computing, but it is conditionally stable. The classic implicit scheme (5) is unconditionally stable, but it needs to solve an algebraic equation which cannot be implemented on a parallel computer [11]. The improved Saul’yev asymmetric schemes (6), (7) are convenient to parallel computing, but they are conditionally stable (see Figure 1).
The ASEI scheme which we constructed is combined with the advantages of the above schemes and the design is as follows:
Let \(m  1 = Nl \), here N is a positive odd number, l is a positive integer (N, \(l \ge3\)) and we divide the points on each time level into N sections. And on the odd level, we arrange the computation according to the rule of ‘the explicit segment  the implicit segment  the explicit segment’. When it turns to the even level, the rule changes into ‘the implicit segment  the explicit segment  the implicit segment’ thus making the implicit segment and the explicit segment doing alternatively at different time levels.
For realizing the parallel computing of the ASEI scheme, for \(i_{0} \ge0\), we consider the calculation of the implicit segment point \((i_{0} + i, j + 1)\), \(i = 1, 2, \ldots, l\). The left endpoint \((i_{0} + 1, j + 1)\) of the implicit segment is calculated with the improved Saul’yev scheme (6), the right endpoint \((i_{0} + l, j + 1)\) is calculated with the improved Saul’yev scheme (7), and the ‘interior points’ \((i_{0} + i, j + 1)\), \(i = 2, 3, \ldots,l  1\), are calculated with the classical implicit scheme (5), leading to the following implicit segment (see Figure 2).
here, \(i_{0} = l, 2l, \ldots, (N  2)l\).
In order to improve the calculation accuracy, the implicit segment will be translated into
when \(i_{0} = 0\) and
when \(i_{0} = (N  1)l\).
The explicit segment is
We use to denote the classical explicit scheme, to denote the classical implicit scheme, to denote the two improved Saul’yev asymmetric schemes. Let \(m = 26\), \(l = 5\), \(N = 5\) and let the schematic of the ASEI scheme be as given (see Figure 3).
A complete calculation step of the ASEI scheme is as follows. For odd level:
For even level, we just switch the segment implicit scheme and the segment explicit scheme of the odd level to calculate \(U_{i}^{j + 1}\).
The ASEI scheme can also be expressed as
where \(U^{j} = (U_{2}^{j}, U_{3}^{j}, \ldots, U_{m  1}^{j}, U_{m}^{j})^{T}\), \(b_{1}^{j} = ( (b a)U_{1}^{j}, 0, \ldots, 0, (b + a)U_{m + 1}^{j} )^{T}\), \(j = 1, 2,3, \ldots, n + 1\),
in which
and \(Q_{l'}\) (\(l' = l,l  2\)) is a \(l' \times l'\) zero matrix.
Existence and uniqueness of the ASEI scheme solution
In order to discuss the existence and the uniqueness of the ASEI scheme solution, we need to introduce the following three lemmas.
Lemma 1
If \(\rho> 0\) and \((C + C^{T})\) is a nonnegative (or positive) definite, then \((\rho I + C)^{  1}\) exists, and
Lemma 2
If \((C + C^{T})\) is a nonnegative (or positive) definite, for any \(\rho\ge0\), then
Lemma 3
\(G_{1}\) and \(G_{2}\) in the ASEI scheme (12) for solving the nonlinear Leland equation are nonnegative matrices.
Proof
We only need to prove \(G_{1} + G_{1}^{T}\) and \(G_{2} + G_{2}^{T}\) are nonnegative matrices. Because of
we know that \(G_{l'}^{(i)} + (G_{l'}^{(i)})^{T}\) is a diagonally dominant matrix and the diagonal elements of \(G_{l'}^{(i)} + (G_{l'}^{(i)})^{T}\) are nonnegative real numbers. Therefore, \(G_{l'}^{(i)} + (G_{l'}^{(i)})^{T}\) is a nonnegative matrix. In the same way, \(\bar{G}_{l + 1}^{(1)} + (\bar{G}_{l + 1}^{(1)})^{T}\) and \(\bar{G}_{l + 1}^{(\frac{N + 1}{2})} + (\bar{G}_{l + 1}^{(\frac{N + 1}{2})})^{T}\) are also nonnegative matrices. Therefore, \(G_{1}\) and \(G_{2}\) are nonnegative matrices. □
From the initial conditions and the boundary conditions of the nonlinear Leland equation, we know the difference solution of the first time layer. Assuming the value \(U_{i}^{2j}\) of the \((2j)\)th time layer is known, the value \(U_{i}^{2j + 1}\) of the \((2j + 1)\)th time layer waits for calculating. From the ASEI scheme (12), the matrix equation for calculating the value of the \((2j + 1)\)th time layer is
Apparently the right of equation (13) is known and \((I + G_{1})^{  1}\) exists by Lemma 3 and Lemma 1. Then equation (13) has a unique solution.
In the same way, applying the ASEI scheme to calculate the value of the \((2j + 2)\)th time layer, the matrix equation is
We could also prove that the matrix equation (14) has a unique solution. Then we could get the following.
Theorem 1
The solution of the ASEI scheme (12) for solving a nonlinear Leland equation exists and is unique.
Stability of the ASEI scheme
By eliminating \(U^{j + 1}\) from equation (12), we obtain
here Y is the growth matrix of the ASEI scheme. The growth matrix of the ASEI scheme is
From Lemmas 13, we can get the following inequality easily:
So
Therefore we have the following theorem.
Theorem 2
The ASEI scheme (12) for solving the nonlinear Leland equation is absolutely stable.
Accuracy of the ASEI scheme
We take the inside points without interior boundary points as ‘interior points’. From the segment construction of the ASEI scheme, we know that the ASEI scheme uses the classic EI scheme at an ‘interior point’ of odd and even levels, and it uses the two improved Saul’yev asymmetric schemes at the ‘interior boundary points’. The truncation error of the classic EI scheme is of second order in time and space [8]. The ASEI scheme just has a finite number of ‘interior boundary points’, so the overall accuracy of the ASEI scheme is close to that of the CN scheme.
The truncation error of the ASEI scheme at the ‘interior boundary points’ will be given in the following. We denote the truncation error as \(T_{1}(p, h)\) when we use (6), we denote the truncation error as \(T_{2}(p, h)\) when we use (7), and we let each point of (6) and (7) be expanded as the Taylor series at the point \((x_{i  1},\tau_{j})\), \((x_{i + 1},\tau_{j})\). Then we get
where \(\alpha+ \beta= 4\). Because of
we can get
Noticing that \(T_{1}(p, h)\) and \(T_{2}(p, h)\) contain the same form as regards the expression of the function, respectively, but we have the reversed symbol. For these items we have the following:
This part of the ‘interior boundary point’ can be offset when the ASEI scheme alternatively uses (6) and (7) at different times. Ultimately, we can get the following theorem.
Theorem 3
The truncation error of the ASEI scheme (12) for solving a nonlinear Leland equation at interior points is \(O(p^{2} + h^{2})\), and at the improved Saul’yev asymmetric schemes (6), (7) of interior boundary points it is \(O(p + h^{2})\).
Hence the error of the points which are near the interior boundary point is bigger than that of the other interior points. The result will be proved in the following numerical experiments.
ASIE parallel difference method
Imitating the method constructed in the ASEI scheme, we give the ASIE scheme for solving the nonlinear Leland equation.
On the odd level, we arrange the computation according to the rule of ‘the implicit segmentthe explicit segmentthe implicit segment’. When it turns to the even level, the rule changes into ‘the explicit segmentthe implicit segmentthe explicit segment’. Getting the ASIE difference scheme for solving the nonlinear Leland equation, we have
here \(j = 1, 3, 5, \ldots\) ; \(G_{1}\), \(G_{2}\) and \(b_{1}\) are as in the above definition.
Imitating the analytical and proved method of the ASEI scheme (12), we have the following theorem.
Theorem 4
The ASIE scheme (15) for solving a nonlinear Leland equation is uniquely solvable, absolutely stable, its truncation error is \(O(p^{2} + h^{2})\) at the interior points, and it is \(O(p + h^{2})\) at the interior boundary points.
Numerical experiments
Numerical experiments will be done in Matlab 2008a, based on the Intel Core i54200 CPU@1.60GHz. We use the ASEI scheme (12) and the ASIE scheme (15) of this paper, the ASCN scheme in [16] and the classic CN scheme to calculate European call option prices with transaction costs. For the nonlinear Leland equation (1) it is very difficult to obtain an analytical solution [3, 4]. Therefore, we will let the numerical solution of the CN scheme approximately substitute the exact solution of a European call option pricing problem with transaction costs and compare these various difference schemes.
Example
We consider a European call option on stocks with transaction cost. Assuming the initial price of the underlying stock is 70 dollars, the strike price of an option is 50 dollars, the riskfree interest rate is 0.01 per year, the deadline of the option is 6 months, the volatility is 0.2 per year, the ratio of the transaction cost is 0.02, δt is \(1/12\).
Solution
We use the following symbols:
Let
First of all, we give the numerical solutions of the ASEI and ASIE schemes.
From Figure 4 and Table 1 we can see that the numerical solutions of the ASEI and ASIE schemes are very close to those of the CN and ASCN schemes.
Second, we regard the solution \(U_{i}^{j}\) of the classical CN scheme as the control solution and the solution \(\bar{U}_{i}^{j}\) of the other schemes as perturbation solutions. Let the grid ratio be \(r1 = \frac{p}{h^{2}}\) and give the absolute error (AE) under the different r1. The definition of AE is as follows:
Observing Figures 5, 6 and Tables 2, 3, we see that the AE of the numerical solutions between he ASEI, ASIE schemes, and CN scheme has the same magnitudes as that of the ASCN scheme, showing that the accuracy of the ASEI and the ASIE schemes is close to that of the ASCN scheme. Because grid points correspond with the stock price (40$ or 100$) near the interior boundary and the AEs of these points are bigger than that of other points, this accords with the theory (see the details in Section 3 and Theorem 3). In addition, when r1 is increasing, the ASEI and ASIE schemes still have a good accuracy.
Thirdly, we will give the proof of stability and the convergence order of the ASEI and the ASIE schemes. We analyze the sum of the relative error at every time level (SRET) and the convergence order in the temporal direction (COT) and the spatial direction (COS). The definitions of SRET, COT, and COS are as follows:
The error of the \(L^{2}\) measurement norm is defined as follows:
From Figure 7 we can see that the SRET of the ASEI and ASIE schemes is larger in the beginning and decreasing along with the movement of the time step; and it is bounded. This shows that the ASEI and ASIE schemes have better stability.
Table 4 and Table 5 show that the convergence order of the ASEI and ASIE schemes in the temporal direction is approaching \(O(p^{2})\) and in the spatial direction it is \(O(h^{2})\).
Next, observing Table 1, the computing times of the ASEI and ASIE schemes (0.5195s, 0.4995s) are less than that of the CN and ASCN schemes (5.2358s, 1.2225s). In order to better compare the computing efficiency of the several difference schemes, we choose different points at the space grid and let \(m = 101,301,501,701,901,1{,}001\), \(n = 1{,}000\). Because the calculated amount of the ASIE scheme is the same as that of the ASIE scheme, we just need to compare the ASEI scheme, the ASCN scheme, and the CN scheme, and the results are in Figure 8 and Table 6.
From Figure 8 and Table 6 we see that when the number of grid points we need calculated is greater than a certain range, the ASEI and ASIE schemes of this paper show a clear superiority in computation time. With the increase of the grid number, the computing time of the difference schemes rises for the nonlinear Leland equation. But the increased amplitude of the computing time of the CN scheme is greater than that of the ASEI, ASCN schemes. The computing time of the ASEI scheme saves nearly 81% for the CN scheme by calculating and saves nearly 40% for the ASCN scheme, showing the computing efficiency of the ASEI scheme is best.
As is well known, the parallel scheme has superiority in computing time. But when the amount of calculation data is small, the impact of the data communication on the cycle can reduce the computing efficiency. For programming of the ASEI scheme in our example, we, respectively, adopt the serial for loop and the parallel parfor loop. For the serial for loop, numerical array and the loop body are performed in the same Matlab process, so there are no data communication problems. But, for a parallel parfor loop, numerical arrays are created in the Matlab client, while parallel computing of the parfor loop body is finished under the Matlab worker, so numerical arrays need to be transmitted from the Matlab client to the Matlab worker. Because of taking up time and processor resources in data communication, we need to consider the data communication problem in parallel programming [18].
Last, we give the computation time of the ASEI scheme in the case of the singlecore cpu and quadcore cpu. The result is in Figure 9.
Figure 9 and the first line of Table 6 show that when the number of grid points is less than a certain range, the serial scheme is more effective than the parallel scheme, meaning that the data communication problems have an effect on the execution efficiency of the programming in the case of small data quantity (grid points). And when we have a larger amount of data (grid points), the influence of the loop body execution is greater than that of the data communication, meaning that using the parallel computing is more effective.
In the practical application, in order to make the numerical results more precise, we tend to a dense mesh and the number of space points becomes higher. The ASEI parallel method has obvious localization characteristics in computing and communications and is very suitable for largescale parallel computing in a distributed storage system on the application.
Conclusion
For the nonlinear Leland equation, this paper constructs the ASEI and ASIE parallel difference schemes with unconditional stability and high accuracy characteristics. Theoretical analysis gets the result that the numerical solutions of the ASEI and ASIE schemes are very close to that of the CN and ASCN schemes. Under the same computing accuracy, the ASEI and ASIE schemes are greatly improved as regards the computing efficiency. Numerical experiment demonstrates that the computing time of the ASEI and ASIE schemes save nearly 81% for the CN scheme and saved nearly 40% for the ASCN scheme, showing the practicability of this kind of parallel difference schemes for solving a nonlinear Leland equation.
The ASEI and ASIE schemes given by this paper can be extended to solve other nonlinear BS models with transaction costs, such as the BarlesSoner model and the risk adjustment pricing model, and they can better solve the timeliness problem of the option pricing.
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Acknowledgements
This work was supported in part by the National Natural Science Foundation of China (No. 11371135), the Special Research Funds for the Central Universities of China (Nos. 2014ZZD10, 13QN30).
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MSC
 65M06
 65Y05
Keywords
 nonlinear Leland equation
 alternating segment explicitimplicit (ASEI) scheme
 alternating segment implicitexplicit (ASIE) scheme
 parallel computing
 numerical experiments