Homotopy perturbation transform method for pricing under pure diffusion models with affine coefficients
⁎Corresponding author. pindzaedson@yahoo.fr (Edson Pindza),
-
Received: ,
Accepted: ,
This article was originally published by Elsevier and was migrated to Scientific Scholar after the change of Publisher.
Peer review under responsibility of King Saud University.
Abstract
Most existing multivariate models in finance are based on diffusion models. These models typically lead to the need of solving systems of Riccati differential equations. In this paper, we introduce an efficient method for solving systems of stiff Riccati differential equations. In this technique, a combination of Laplace transform and homotopy perturbation methods is considered as an algorithm to the exact solution of the nonlinear Riccati equations. The resulting technique is applied to solving stiff diffusion model problems that include interest rates models as well as two and three-factor stochastic volatility models. We show that the present approach is relatively easy, efficient and highly accurate.
Keywords
Diffusion model
Interest rate
Stochastic volatility
Stiffness
Laplace transform
Homotopy perturbation method
1 Introduction
Stochastic processes have taken over the world of financial modelling. Starting with simple Geometric Brownian Motion well described by Bachelier (1900), to more sophisticated processes for better fitness and calibration of market fluctuations. A huge range of papers have considered Lévy processes as their driving force, they usually take the umbrella of jump diffusion process. Basically the state process
Various types of diffusion models have been introduced in the literature and used in different sectors. They differ in the model parameter while preserving some properties that are essential in asset price dynamics such as mean reversion.
In one-factor models the volatility process is a deterministic function of t, this includes the class of affine term structures that is encountered in interest rates and option pricing contexts, see Duffie (2005), Björk (2004), Black and Scholes (1973) and Merton (1974).
Heston (1993) introduces a two-factor model where the volatility is now stochastic, to accommodate the implied volatility smile encountered in the financial markets as shown by Kotzé et al. (2015). Bates (1996) extends the Heston model in adding a up jumps into the diffusion process to model a huge flux of information that can occur in the market.
Fang (2000) brings a three-factor model with jump diffusion process that looks quite stable. The model takes its roots from the Bates model to which an extra parameter is added: the long-term volatility, which is important for instruments like bonds that have long term maturity. Banks are often interested in this parameter. Christoffersen et al. (2009) consider a three-factor model with no jumps, but the state process is driven by a deterministic drift, plus two Brownian motions. However, as the number of factors increases one must expect the model to become more robust, but less realistic. In this paper we present a detailed framework of a class of asset prices whose pay-off at a future time (the maturity time) T is of the form
In addition we introduce a modified form of homotopy perturbation and Laplace transform methods to value financial models of diffusion type with affine coefficients. These models lead to the need of solving systems of Riccati differential equations. Laplace transform method (LTM) alone is incapable of handling such equations, instead some variants of the LTM prove to better handle nonlinear differential equations. Among those variants we cite the Laplace decomposition algorithm (Khuri, 2001; Khan, 2009) and the H2LTM, (Fatoorehchi and Abolghasemi, 2016) which are obtained with the help of the Adomian decomposition and Adomian polynomials respectively. Another successful variant of LTM is obtained in coupling it with variation iteration methods. Alawad et al. (2013) used it to solve space–time fractional telegraphic equation as it allowed them to overcome the difficulty arising from finding Lagrange multipliers. LTM has also known great success when combined with differential transform methods on solving non–homogenous equations, see Alquran et al. (2012). On the other hand, the homotopy perturbation method is a combination of the classical perturbation technique and the homotopy technique whose origin is in topology , more on homotopy can be found in Hilton (1953), but not restricted to small parameters as it occurs with traditional perturbation methods. For example, the HPM requires neither small parameters nor linearisation, but only few iterations to obtain highly accurate solutions. The standard homotopy perturbation method was proposed by He (1999) as a powerful tool to approach various kinds of nonlinear problems. It can also be viewed as a special case of Homotopy Analysis Method (HAM) proposed by Liao (1992, 1997). For the past decade, many improvements on HAM have been introduced, one of it being the Homotopy Analysis Transform Method (HATM) which is basically a HAM coupled with a Laplace transform. The method is very powerful, fast converging and accurate. Recently, it has been applied in many different areas of science including fluid dynamics, wave theory (Kumar et al., 2014a, 2015b, 2016b), quantum physics, see Kumar (2014), and many more, with extension to fractional cases. Kumar et al. (2014c) applied this method on Volterra integral equation to obtain good quality results. Another improvement of HAM is obtained by coupling it with the Samudu transform which gives rise to the Homotopy Analysis Samudu Transform Method (HASTM), see Kumar and Sharma (2016); Kumar et al., 2016a for more details. Likewise, the Samudu transform has also been introduced in HPM to generate the Homotopy Perturbation Samudu Transform Method (HPSTM), see Singh et al. (2013); Singh et al., 2014b. Singh et al. (2014a) used the method successfully to get analytical and numerical solutions of nonlinear fractional equations found in the area of biological population model. Also, Kamdem (2014) proposed a generalised integral transform based on the homotopy perturbation method where various integral transforms were used. In this paper we are interested in the combination of the HPM with Laplace transform giving rise to the Homotopy Perturbation Transform Method (HPTM). The method has shown success already in obtaining solutions of the Navier–Stokes equations (Kumar et al., 2015a), gas dynamics equations coming from fluid dynamics in the case of fractional differential equation as explored by Kumar et al. (2012). The method has also shown success in solving KdV equations arising in wave theory (Goswami et al., 2016) as well as Fokker-Planck equations commonly found in solid-state physics (Kumar, 2013). Another useful application of the technique is found in Kumar et al. (2014b) in which the authors derive the price of a plain vanilla call option of European type under Black–Scholes model in the financial market.
This paper is structured as follows: Section 2 reviews the formalism of diffusion models with emphasis on the case with affine coefficients. In Section 3, we introduce the basic concept of homotopy perturbation transform method (HPTM). In Section 4, we describe the solution procedure of the HPTM for interest rate models, especially the two and three-factor stochastic volatility models. Finally, the conclusions are presented in Section 5.
2 Mathematical description of affine models
Consider the financial market model
Under an equivalent martingale measure
Let us consider an auxiliary process
Under no-arbitrage conditions, the discounted pay-off
In Affine framework as described by Cont and Tankov (2004), we consider
Theorem 2.1 see Duffie et al., 2000
Under technical conditions, if the pay-off function is chosen such that
Given the pay-off
We can apply the above result on the auxiliary process to obtain
Using the Affine framework coupled with the fact that
Implying,
Finally we have,
Eq. (2.7) together with initial condition (2.8) is a nonlinear system of ODE of Riccati type. In general, Riccati equations do not have analytical solutions, hence numerical methods have to be used. In addition, these equations have been reported to be stiff. Hence the use of explicit methods will require a high mesh refinement to produce acceptable solutions. This will result in an increase in the computational cost. In this article we propose a Laplace Transform Homotopy Perturbation method to circumvent the stiffness problem.
3 Basic idea of homotopy perturbation transform method
We first define the Laplace transform (LT) and its inverse transform, and list useful properties employed in this paper.
Laplace transform of
The inverse Laplace transform is evaluated on a contour
The contour
One useful property of LT for this paper is:
To illustrate the basic ideas of this method, let us consider the following system of nonlinear partial differential equations
In order to solve the system of differential Eq. (3.4) by means of the homotopy perturbation transform method, we construct the homotopy
We apply the Laplace transform on both sides of the homotopy Eq. (3.8) to obtain
Using the differential property of the Laplace transform we have
By applying the inverse Laplace transform on both sides of (3.13), we have
Assuming that the solutions of Eq. (3.7) can be expressed as a power series of p
Then substituting Eq. (3.15) into Eq. (3.14), we get
Comparing coefficients of p with the same power leads to
Assuming that the initial approximation has the form
The utility of HPTM is shown by its applications on Affine diffusion problems.
4 Numerical experiments
4.1 Interest rate models
We first look at affine term structure models that are found in interest rate models (see Björk, 2004). Here the state process is the interest rate itself r following the dynamics:
For a zero coupon bond that pays 1 at maturity T, we see that its price at any time
We investigate numerical solutions of the system (4.1) by means of the HPTM. To this end we construct the following homotopy
Applying the Laplace transform on both sides of (4.2), we have
Using the differential property of the Laplace transform we have
By applying inverse the Laplace transform on both sides of Eq. (4.4) and after algebraic simplification we get
Suppose the solution of Eq. (4.2) to have the following form
We consider a particular case of the Vasicek model (see Björk, 2004). This model is obtained from Eq. (4.1) when the parameters are
The Taylor expansions of both
To obtain the numerical solution of the (4.1) for the Vasicek model, we assume
The polynomials
Table 1 illustrates the convergence of the HPTM. At
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1.19416E−5 | 9.66072E−5 | 1.55847E−3 | 4.36287E-3 | 1.30838E-2 |
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5.29545E−8 | 8.61305E−7 | 3.53106E−5 | 1.39979E−4 | 6.10400E−4 | |
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2.72792E−9 | 8.86011E−8 | 9.02825E−6 | 4.98717E−5 | 3.08174E−4 |
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3.1584E−12 | 2.05922E−10 | 5.30832E−8 | 4.14078E−7 | 3.70637E−6 | |
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3.84661E−13 | 4.88715E−11 | 3.13493E−8 | 3.40916E−7 | 4.32664E−6 |
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8.13539E−17 | 2.92069E−14 | 4.63624E−11 | 7.1053E−10 | 1.30249E−8 | |
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1.11102E−14 | 1.99155E−15 | 7.16094E−11 | 1.53097E−9 | 3.98281E−8 |
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1.91192E−16 | 5.38435E−16 | 2.64043E−14 | 7.95826E−13 | 2.98415E−11 |
Fig. 1 (a) shows that the exact and numerical solutions of Eq. (4.1) are in good agreement. The bond price behaviour as a function of
In this experiment, we consider the Cox-Ingerson-Ross (CIR) model (see Björk, 2004). This model is obtained from Eq. (4.1) when the parameters are

- (a) Exact and numerical solutions of 4.1 and (b) Bond price process for the Vasicek model when
Using the initial conditions
Again we observe that the Taylor expansion and the solution of the CIR model computed by the HPTM are in good agreement. Table 2 records the error for different values of j and different values of
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1.43085E−5 | 1.16921E−4 | 4.02833E−4 | 9.74221E−4 | 1.9403E−3 |
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4.95614E−7 | 8.31425E−6 | 4.40686E−5 | 1.45623E−4 | 3.71224E−4 | |
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1.07357E−10 | 5.13301E−9 | 5.16706E−8 | 2.71528E−7 | 9.93647E−7 |
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2.00642E−10 | 1.30547E−8 | 1.51097E−7 | 8.62187E−7 | 3.33847E−6 | |
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2.68126E−12 | 1.90102E−10 | 3.28522E−9 | 2.49695E−8 | 1.20746E−7 |
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1.28341E−13 | 3.43621E−11 | 9.14227E−10 | 9.47091E−9 | 5.84918E−8 | |
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1.22378E−12 | 7.96574E−13 | 5.11847E−12 | 7.03396E−11 | 5.42104E−10 |
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7.29208 E−16 | 1.16398E−14 | 6.42225E−13 | 1.09941E−11 | 9.83115E−11 |
Fig. 2 shows the behaviour of the parameters

- (a) Parameters
4.2 Two-factor stochastic volatility model
One of the most well-known stochastic volatility models is the Heston model described by Duffie et al. (2000). Under constant interest rate r, the stock price has dynamics driven by
The stochastic differential Eq. (4.18) can be written as
Let
Under the risk free equivalent martingale measure
Referring to Affine settings, we see that
Eq. (2.3) is now two-dimensional and referred to as a two-factor model. By choosing the pay-off function in the form
Resulting in
The exact solution is given by
Applying the Laplace transform on both sides of (4.24), we have
Using the differential property of the Laplace transform we have
By applying the inverse Laplace transform on both sides of (4.26) and after algebraic simplification, we have (see Fig. 4)
Suppose the solution of Eq. (4.27) to have the following form
Assuming
Here the parameters are chosen randomly as

- (a)

- (a) Asset price behaviour with respect to
If we consider
4.3 Three-factor stochastic volatility model
We consider the three-factors Heston model also considered by Duffie et al. (2000) where the state process
We know the solution is given by Eq. (4.1) ie. with
After algebraic simplifications, we end up with the following system of ordinary differential equations of Riccati type
To solve Eq. (4.33) by the HPTM, we construct the following homotopy
Applying the Laplace transform on both sides of (4.34), we have
Using the differential property of the Laplace transform we have
By applying the inverse Laplace transform on both sides of (4.36) and after algebraic simplification we have
Suppose the solution of Eq. (4.36) to have the following form
Assuming

- (a)
Defining the error function at order j to be
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9.44787E−10 | 2.88458E−7 | 8.75677E−6 | 1.02925E−4 | 7.17550E−4 |
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2.92738E−8 | 1.00574E−5 | 3.40898E−4 | 4.44356E−3 | 3.42980E−2 |
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5.24351E−10 | 3.73261E−7 | 2.00047E−5 | 3.72887E−4 | 3.91183E−3 |
5 Conclusion
In the present work, we proposed a combination of the Laplace transform method and the homotopy perturbation method to solve nonlinear systems of stiff Riccati differential equations arising in finance. We have discussed the methodology for the construction of these schemes and studied their performance on one, two and three-factor diffusion models with affine coefficients. The solution of these Riccati systems of equations by means of the homotopy perturbation transform method converges rapidly to the exact solution as the number of truncated term increases. The HPTM is an effective mathematical tool which can play a very important role in the field of finance.
Acknowledgments
E. Pindza and C.R. Bambe Moutsinga are thankful to Brad Welch for the financial support through RidgeCape Capital.
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