A new kind of parallel finite difference method for the quanto option pricing model
- Xiaozhong Yang^{1},
- Lifei Wu^{1}Email author and
- Yuying Shi^{1}
https://doi.org/10.1186/s13662-015-0643-z
© Yang et al. 2015
Received: 16 June 2015
Accepted: 23 September 2015
Published: 9 October 2015
Abstract
The quanto option pricing model is an important financial derivatives pricing model; it is a two-dimensional Black-Scholes (B-S) equation with a mixed derivative term. The research of its numerical solutions has theoretical value and practical application significance. An alternating band Crank-Nicolson (ABdC-N) difference scheme for solving the quanto options pricing model was constructed. It is constituted of the classical implicit scheme, the explicit scheme and the Crank-Nicolson scheme, it has the following advantages: parallelism, high precision, and unconditional stability. Numerical experiments and theoretical analysis all show that ABdC-N scheme can be used to solve the quanto options pricing problems effectively.
Keywords
1 Introduction
The multi-asset options pricing model (the multi-dimensional Black-Scholes equation) is a famous financial mathematics basic model; its numerical solutions had played a significant role in promoting a lot of financial derivatives pricing methods. Therefore, the numerical solutions have attracted more and more attention from applied mathematicians and economists. With the rapid development of multi-core and cluster technology, parallel algorithms have become one of the mainstream technologies improving the numerical calculation efficiency. The research of parallel numerical difference methods for solving a multi-asset options pricing problem has basic scientific significance. This is so because the option pricing has higher time requirements from the need of practical application. Therefore, over the past 20 years an efficient numerical solution of the multi-asset options pricing model has been the focus of academic research [1].
The analytical solution is very complex, difficult to quickly solve, so numerical solutions were usually used to compute option pricing models in the real financial market, for example, the Monte-Carlo method, the binary tree method, and the finite difference method, etc. [1, 2]. The considered computing speed and accuracy, and the finite difference method was usually used in the real financial market.
In recent years, the study of finite difference methods for solving the dual currency option pricing model has made a lot of progress. An implicit scheme for the multi-asset option pricing model had been made by Gilli et al. (2002), but a calculation of this scheme was relatively complex; one needed to solve algebra equations which contained a large tridiagonal block matrix [4]. Khaliq et al. (2008) had given a kind of difference method for solving the 2D B-S equation [5]; the method needs to use the penalty function approach. So the method was difficult on using parallel computing on a computer. Yang and Zhou (2011) had put forward a rapid AOS difference method of quanto option pricing model [6], but the calculation accuracy of the method is not ideal, because the error of its mixed derivative term is not ideal. In addition, most of those schemes had applied serial calculation. When the computing grid points or dimension of equation required is large, a higher order algebraic equation \(Ax=b\) should be solved. The efficiency of the calculation process is not ideal, and it is difficult to meet the requirements of the options as regards time.
About the parallel finite difference scheme, Evans and Abdullah (1983) had put forward an alternating group explicit (AGE) scheme based on the Saul’yev asymmetric format [7]. The AGE scheme not only was good to keep the stability of numerical calculation, but it also has good parallel properties. Then Zhang (1991) had established a variety of alternating segment explicit-implicit (ASE-I) schemes and alternating segment Crank-Nicolson (ASC-N) schemes [8], and they had got some research results which contained stability and the parallelism. Now, the research of this method has been extended to solve many development equations. For example, Wang (2006) had given a kind of alternating segment difference scheme with intrinsic parallelism for the KdV equation [9]. Sheng et al. (2007) had constructed two kinds of difference formats with intrinsic parallelism for a linear parabolic equation [10], and they proved that the format is unconditionally stable and we have second-order convergence. Yuan (2007) has put forward a parallel difference scheme with second-order accuracy and unconditional stability for a nonlinear parabolic system [11].
For the quanto option pricing model (2D B-S equation), we used the classical implicit scheme, the explicit scheme, and the Crank-Nicolson scheme, constructed a parallel difference scheme-alternating band Crank-Nicolson (ABdC-N) scheme, which is unconditionally stable, and which is close to second-order accuracy. Numerical experiments show that the method is effective.
2 ABdC-N difference scheme
2.1 Initial-boundary value condition of 2D B-S equation
2.2 Construction of the ABdC-N scheme
Let Δx, Δy, Δτ be the steps of x, y, and τ, respectively. Here, \(\Delta x=(\ln{S_{1 \max}}-\ln{S_{1 \min}})/m\), \(\Delta y=(\ln{S_{2 \max}}-\ln{S_{2 \min}})/n\), \(\Delta\tau=T/nt\), m, n, nt are positive integers. \(x_{i}=\ln{S_{1 \min}}+i\Delta x\), \(y_{j}=\ln{S_{2 \min}}+j\Delta y\), \(\tau_{k}=k\Delta\tau\), \(i=0, 1,\ldots,m\), \(j=0, 1,\ldots,n\), \(k=0, 1,\ldots,nt\). For convenience, let \(h=\Delta x=\Delta y\). We use \(V^{k}_{i,j}\) to denote the solution of (2) at point \((x_{i}, y_{j}, \tau_{k})\).
Here, (3) is called the universal difference scheme (θ-scheme).
As is well known, when \(\theta=0\), (3) is a classical explicit scheme, which has a natural parallelism, but its stability condition is more demanding. When \(\theta=1\), (3) is the classical implicit scheme, which has unconditional stability. When \(\theta=0.5\), (3) is a classical Crank-Nicolson scheme, which is of second-order accuracy and has unconditional stability. But the implicit scheme’s and the C-N scheme’s computing times are longer.
The design of ABdC-N is as follows.
Assume the value of the kth time layer \(V^{k}_{i,j} \) (\(i=1,2,\ldots ,m-1\)) is known, the value of the \(k+1\)th time layer \(V^{k+1}_{i,j}\) waits for calculating. Assume s is a positive integer and \(1<2s\leq m-1\), \(I_{l}\), \(l=1,2,\ldots,2s\) are 2s positive integers, which meet \(1\leq I_{1}< I_{2}<\cdots<I_{2s}\leq m-1\).
When k is an even number, at point \(x_{i,j} \) (\(i=I_{1}, I_{3},\ldots, I_{2s-1}\)), we apply the classical explicit scheme (\(\theta=0\)) to calculate \(V^{k+1}_{i,j}\); at point \(x_{i,j} \) (\(i=I_{2}, I_{4},\ldots, I_{2s}\)), we apply the classical implicit scheme (\(\theta=1\)) to calculate \(V^{k+1}_{i,j}\). At the remaining points, we apply the classical C-N scheme (\(\theta=0.5\)) to calculate \(V^{k+1}_{i,j}\). When k is an odd number, at point \(x_{i,j} \) (\(i=I_{2}, I_{4},\ldots, I_{2s}\)), we apply the implicit scheme (\(\theta=0\)) to calculate \(V^{k+1}_{i,j}\); at point \(x_{i,j} \) (\(i=I_{1}, I_{3},\ldots, I_{2s-1}\)), we apply the explicit scheme (\(\theta=1\)) to calculate \(V^{k+1}_{i,j}\). At the remaining points, we apply the classical C-N scheme (\(\theta =0.5\)) to calculate \(V^{k+1}_{i,j}\).
3 Existence and uniqueness of the ABdC-N scheme solution
Based on the above analysis, we will get the following theorem.
Theorem 1
The ABdC-N scheme (4) for solving the quanto option pricing model is uniquely solvable.
4 Stability and convergence of the ABdC-N scheme
Lemma 1
If \(\rho>0\) and \(C+C^{T}\) is a non-negative (or positive) matrix, then \((\rho E+C)^{-1}\) exists, and \(\|(\rho E-C)(\rho E+C)^{-1}\|_{2}\leq1\).
Lemma 2
\(D_{1}G\), \(D_{2}G\) in the growth matrix of the ABdC-N scheme for solving the quanto option pricing model are non-negative matrices.
Proof
Theorem 2
The ABdC-N scheme (4) for solving the quanto option pricing model is unconditionally stable.
Due to the Lax theorem [14], we can get a corollary.
Corollary 1
The ABdC-N scheme (4) for solving the quanto option pricing model is convergent.
5 Accuracy of the ABdC-N scheme
From the construction of the ABdC-N scheme, we take inside points without interior boundary points as ‘interior point’. Because the ABdC-N scheme (4) is applied to the C-N scheme at the interior point, the truncation error of the interior point is of second order.
The truncation error of interior boundary points will be analyzed in the following. The ABdC-N scheme (4) alternatively applies the classical explicit scheme and the implicit scheme at the interior boundary points.
The two classical schemes approximate the analytical solution from either side, respectively. It had been proved that the numerical solution of the classical explicit scheme \(\theta=0\) is greater than the analytical solution, and the numerical solution of the classical implicit scheme \(\theta=1\) is less than the analytical solution. Therefore, alternatively applying them can improve the calculation accuracy. For example, the truncation error of the ‘Explicit-Implicit scheme’ or the ‘Implicit-Explicit scheme’ is of second order in time and space, and unconditionally stable. In the ABdC-N scheme (4) for solving the quanto option pricing model is alternatively applied the classical explicit and the implicit scheme at the interior boundary point; then the truncation error of interior boundary point also can achieve a second order.
Theorem 3
The truncation error of the ABdC-N scheme (4) for solving the quanto option pricing model is \(O(\Delta\tau^{2}+h^{2})\).
In order to make have the ABdC-N scheme a better parallelism, usually we take s constant. Then we can use \((s+1)\) CPU for parallel computing. Every CPU applies the Thomas method to solve a tridiagonal equation. Usually, we take \(I_{2l}-I_{2l-1}\) and \(I_{2l+1}-I_{2l}\) (\(0< l\leq s\)) constant for different l.
6 Numerical experiments
Numerical experiments will be done in Matlab 2008a, based on Intel core i5-2400 CPU@3.10GHz. The comparison is between the ABdC-N scheme and the classical C-N scheme, referring to computing accuracy and computing time.
Example
We consider an American investor buying a Nikkei index call option. Assuming the current price of Nikkei is 20,000 yen, the dividend rate of the Nikkei is 0.03, the volatility of the Nikkei is 0.2, the exchange rate of Japanese yen against dollar is 0.01, the volatility of the exchange rate is 0.1, the correlation coefficient is 0.2, the risk-free rates of American and Japan are 0.08 and 0.04, respectively. The strike price of an option is 15,000 yen. We consider the deadline of the option to be one year (12 months), and the final exchange rate is the spot exchange rate.
Comparison of analytical and numerical solution
Scheme | 12 months | Relative error | CPU time |
---|---|---|---|
Analytical solution | 53.521809 | - | - |
Classical C-N scheme | 51.331487 | 4.09% | 158.157s |
ABdC-N scheme | 52.284465 | 2.49% | 18.037s |
Error analysis of classical C-N scheme
Grid | CPU time | RMSE | RRMSE |
---|---|---|---|
20 × 20 | 0.836s | 0.1450 | 2.900 |
30 × 30 | 7.259s | 0.0999 | 2.997 |
40 × 40 | 38.07s | 0.0762 | 3.048 |
50 × 50 | 158.15s | 0.0616 | 3.080 |
Error analysis of the ABdC-N scheme
Grid | CPU time | RMSE | RRMSE |
---|---|---|---|
20 × 20 | 1.410s | 0.0795 | 1.590 |
30 × 30 | 2.095s | 0.0547 | 1.641 |
40 × 40 | 5.649s | 0.0417 | 1.668 |
50 × 50 | 18.307s | 0.0336 | 1.680 |
In terms of computation time, from Tables 1, 2, and 3, the computing time of the ABdC-N scheme of the quanto option has a big advantage compared with the classical C-N scheme (except grid 20 × 20). When the number of grid points is smaller, the impact of the data communication on the cycle can greatly reduce the computation efficiency. So when the grid is \(20 \times 20\), the CPU time of the C-N scheme is smaller than the ABdC-N scheme. The computing time (CPU time) of the ABdC-N scheme is 28.86%, 14.83%, 11.57% of the C-N scheme for grids \(30 \times 30\), \(40 \times 40\), \(50 \times 50\), respectively. When the number of grid points is larger, the advantages of parallel computing of the ABdC-N scheme are obviously superior. The computation time of the ABdC-N scheme can save about 88% compared with the classical C-N scheme for grid \(50 \times 50\). Therefore, the ABdC-N scheme can be more effective to solve option pricing problems than the classical C-N scheme.
Comprehensively considering the computing efficiency and the computing accuracy, the ABdC-N scheme can be more effective to solve the quanto option pricing problems.
7 Conclusion
An alternating band Crank-Nicolson (ABdC-N) difference scheme for solving the quanto options pricing model (2D B-S equation) has been constructed. The computing accuracy, stability, and convergence of the ABdC-N scheme have been analyzed. The result of the numerical experiments is consistent with the theoretical analysis. The ABdC-N scheme has an ideal computing accuracy and computing efficiency, and it can be more effective to solve the quanto options pricing problems.
Declarations
Acknowledgements
This work is sponsored by the project National Science Foundation of China (No. 11371135, 11271126), the Fundamental Research Funds for the Central Universities (Nos. 13QN30, 2014ZZD10).
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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