A combined compact difference scheme for option pricing in the exponential jump-diffusion models
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Abstract
In the present paper, starting with the Black–Scholes equations, whose solutions are the values of European options, we describe the exponential jump-diffusion model of Levy process type. Here, a jump-diffusion model for a single-asset market is considered. Under this assumption the value of a European contingency claim satisfies a general “partial integro-differential equation” (PIDE). With a combined compact difference (CCD) scheme for the spatial discretization, a high-order method is proposed for solving exponential jump-diffusion models. The method is sixth-order accurate in space and second-order accurate in time. A known analytical solution to the model is used to evaluate the performance of the numerical scheme.
Keywords
Black–Scholes equation Combined compact difference (CCD) Jump-diffusion model Option pricingMSC
65M06 65M12 47G20 91B281 Introduction
The investigation of problems related to finance has been one of the major fields of research and studies in recent decades, and due to its widespread applications in financial institutions and industries, there has been a noticeable change and growth in this field. Since the beginning of the twentieth century, a variety of option pricing models have been introduced, among them the Black–Scholes model is the most important and popular one [1, 2, 3, 4]. Unfortunately, the Black–Scholes model is based on the assumption that the price movements have no jumps and is governed by the Brownian motion, which is inconsistent with empirical evidences. Nevertheless, this assumption can be relaxed and replaced by a more realistic one which accepts that the price movements follow a Poisson or Levy process with jumps [5, 6, 7]. Based on this new assumption, a model can be derived but a closed form of its analytical solution can be found only under the boundary conditions of European options. Thus, for other boundary conditions, in particular boundary conditions for American options, numerical tools must be applied [8, 9, 10, 11, 12].
There might be many numerical techniques that can be applied for the exponential jump-diffusion models, but the most popular one is the finite difference method. This is because its implementation is simple and its rate of convergence can be improved [13, 14]. On the other hand, this technique has some weaknesses, for example, it might not be stable and the points on its related grid must be located in specified locations. However, one can overcome these weaknesses by using the methods for which no grid is needed (i.e., methods in which radial basis functions or spectral basis functions are used). d’Halluin, Forsyth, and Labahn [15] and d’Halluin, Forsyth, and Vetzal [16] proposed an implicit method of the Crank–Nicolson type combined with a penalty method for pricing American options under the Merton model. In [15] the authors showed that the fixed point iteration at each time step converges to the solution of a linear system of discrete penalized equations. Also, Kwon and Lee [17] proposed a method constructed on three-time levels, and the operator splitting method was used to treat the American constraints. These numerical methods use iterative techniques to solve resulting linear systems involving dense coefficient matrices. Recently, many numerical methods have been applied successfully for solving financial problems (see, e.g., [18, 19, 20, 21, 22, 23]).
Nowadays, due to the creation of complex problems, in particular in financial mathematics, classical methods are not able to produce reasonable results in a satisfactory computational time. For decreasing computational time, some authors have proposed exploiting parallel programming [24]. On the other hand, applying some higher-order but simple methods, such as combined compact difference (CCD) method, is becoming more interesting. In 1998, Chu and Fan [25] proposed a CCD method for solving 1D and 2D steady convection-diffusion equations. This method is an implicit three-point scheme, and its accuracy is sixth-order of local truncated approximation [25]. For solving partial differential equations (PDEs) by the CCD scheme, the first- and second-order derivatives are computed together with the function values of unknowns at grid points. We refer the reader to [26, 27, 28] for further discussions.
For the sake of completeness, first of all, we review the Black–Scholes integro model, governed by the Poisson process, and its analytic solution for specific boundary conditions that evaluate the prices of special options. Then, by changing some of the variables that transform the Black–Scholes integro model into an integro-diffusion model, we prepare the ground to solve the model numerically. After that, we develop a compact finite difference method, which leads to a triple coefficient matrices, to solve exponential jump-diffusion models. To do this, we first explain and then define our CCD algorithm for solving PIDEs, which we implement using the jump-diffusion model. In this paper, we show that our new model is convergent and its rate of convergence is quite reasonable and high, and it is stable. Finally, after solving our model, we discuss its numerical results and its applications in reasonably estimating different kinds of options prices. The rest of the paper is organized as follows. In Sect. 2, we introduce exponential jump-diffusion models and option pricing problems for European and American options. In Sect. 3, we derive our CCD scheme. The von Neumann stability analysis is carried out for the proposed CCD method in Sect. 4. In Sect. 5, numerical experiments are conducted to show the performance of the presented method. The paper ends with a brief conclusion in Sect. 6.
2 Exponential jump-diffusion model
3 Construction of the method
4 Stability analysis
Lemma 1
Proof
Theorem 1
The proposed CCD method is unconditionally stable.
Proof
5 Numerical results
5.1 European options
A comparison between the approximate solution and the exact solution corresponding to the European call options under the Merton model with \(\lambda =0\)
S | Approximate solution | Exact solution | \(L_{\infty }\) error |
---|---|---|---|
90 | 0.36577 | 0.36646 | 6.9931e−04 |
100 | 3.63286 | 3.63507 | 2.2097e−03 |
110 | 11.50600 | 11.50588 | 1.2608e−04 |
A comparison between the approximate solution and the exact solution corresponding to the European call options under the Merton model with \(\lambda =0.1\)
K | S | CN | CCD | Exact | \(L_{\infty }\) (CN) | \(L_{\infty }\) (CCD) |
---|---|---|---|---|---|---|
100 | 90 | 0.54524 | 0.52751 | 0.52764 | 1.7602e−02 | 1.2718e−04 |
130 | 32.28254 | 32.28218 | 32.28218 | 3.6643e−04 | 4.5527e−06 | |
170 | 71.96107 | 71.96065 | 71.96065 | 4.1983e−04 | 2.3599e−06 |
Order of convergence for the European put options under the Merton model by the CCD method with \(\lambda =0.1\) and \(S=30\)
(M,N) | (128,25) | (256,200) | (512,1600) | (1024,12800) |
---|---|---|---|---|
\(E=L_{\infty }\) error | 1.1753e-04 | 3.4297e − 06 | 5.9131e − 08 | 9.2704e − 10 |
\(R=\frac{E(M, N)}{E(2M, 8N)}\) | – | 34.2681 | 58.0014 | 63.7844 |
Order = log_{2}R | – | 5.0988 | 5.8580 | 5.9941 |
Check the validity of Theorem 1 for the European call options under the Merton model: \(\lambda =0\), \(T=1\) and \(S=30\)
(h,k) | \(E=L_{\infty }\) error |
---|---|
(0.15,0.1) | 8.5571e − 05 |
(0.075,0.025) | 3.0810e − 06 |
(0.0375,0.0063) | 2.4753e − 07 |
(0.0187,0.0016) | 9.1614e − 09 |
5.2 American options
Values of American put options obtained by the CCD method under the Merton model. The reference values are 10.003822 at \(S = 90\), 3.241251 at \(S = 100\), and 1.419803 at \(S = 110\) [16]. The truncated domain is \([-1.5, 1.5]\) and \(\lambda =0.1\), \(M=128\), and \(N=25 \)
Order of convergence of the CCD method for American put options under the Merton model with \(\lambda =0.1\)
(M,N) | (32,10) | (64,80) | (128,640) | (256,5120) |
---|---|---|---|---|
\(E=L_{\infty }\) | 4.0571e−02 | 1.8523e − 03 | 5.7864e − 05 | 1.1350e − 6 |
\(R=\frac{E(M, N)}{E(2M, 8N)}\) | – | 21.9030 | 32.0113 | 50.9815 |
Order = log_{2}R | – | 4.4531 | 5.001 | 5.6719 |
6 Conclusions
We have offered a compact difference method for pricing options when the underlying asset process follows a regime-switching jump-diffusion model. This model is more efficient than other models because of its efficiency and high accuracy. To support the model, we have prepared several numerical examples to show that the suggested algorithm computes accurate values in comparison to the test solution. This method is shown to be second- and sixth-order accurate in time and space, respectively.
Notes
Acknowledgements
Authors would like to appreciate the handling editor and the referees for his supports and their useful comments, respectively.
Availability of data and materials
Not applicable.
Authors’ information
Rahman Akbari is currently a PhD student of Financial Mathematics, Reza Mokhtari is currently an Associate Professor of Numerical Analysis, and Mohammad Taghi Jahandideh is currently an Assistant Professor of Mathematical Analysis at the Department of Mathematical Sciences, Isfahan University of Technology, Isfahan 84156-83111, Iran.
Authors’ contributions
All authors contributed significantly to this manuscript, and they read and approved the final manuscript.
Funding
Not applicable.
Competing interests
Authors confirm that SpringerOpen’s guidance on competing interests has been read and there are not any financial and non-financial competing interests for them.
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