The Using Neural Network and Finite Difference Method for Option Pricing under Black-Scholes-Vasicek Model

Subject Areas : Financial Engineering

Mahdiye Mohmmadi ¹ , Elham Dastranj ² , Abdolmajid Abdolbaghi Ataabadi ³ , Hossein Sahebi fard ⁴

1 -
2 -
3 -
4 -

Received: 2024-05-01 Accepted : 2024-12-11 Published : 2025-01-17

Keywords: Option Pricing , Neural Networks , Finite Difference Method , Black-Scholes-Vasicek Model,

Abstract :

In this paper, the European option pricing is done using neural networks in the Black-Scholes-Vasicek market. The general purpose of this research is to compare the accuracy of neural network and Black-Scholes-Vasicek models for the pricing of call options. In the sequel, the finite difference method is applied to find approximate solutions of partial differential equation related to option pricing in the considered market. In the design of the artificial neural network required for this research, the parameters of the Black-Scholes-Vasicek model have been used as network inputs, as well as 720 data from the daily price of stock options available in the Tehran Stock Exchange market (in 1400) as the network output. The approximate solutions obtained in this article, which were carried out by two methods of neural networks and finite differences on the Tehran stock exchange based on the daily price of stock options, are shown that neural networks are more accurate method comparing with finite difference. The comparison of pricing results using neural networks with real prices in the assumed market is presented and shown via diagram, as well. In this paper, the European option pricing is done using neural networks in the Black-Scholes-Vasicek market. The general purpose of this research is to compare the accuracy of neural network and Black-Scholes-Vasicek models for the pricing of call options. In the sequel, the finite difference method is applied to find approximate solutions of partial differential equation related to option pricing in the considered market. In the design of the artificial neural network required for this research, the parameters of the Black-Scholes-Vasicek model have been used as network inputs, as well as 720 data from the daily price of stock options available in the Tehran Stock Exchange market (in 1400) as the network output. The approximate solutions obtained in this article, which were carried out by two methods of neural networks and finite differences on the Tehran stock exchange based on the daily price of stock options, are shown that neural networks are more accurate method comparing with finite difference. The comparison of pricing results using neural networks with real prices in the assumed market is presented and shown via diagram, as well.

References:

[1] Bennell, J., Sutcliffe, C., Black-Scholes versus neural networks in pricing FTSE 100 options, Intelligent Systems in Accounting, Finance & Management: International Journal, 2000; 12(4), 243-260. Doi: 10.1002/isaf.254.
[2] Björk, T., Arbitrage theory in continuous time. Oxford university press, 2009.
[3] Dastranj, E., Sahebi Fard, H., Abdolbaghi, A., Hejazi, S, R., Power option pricing under the unstable conditions (Evidence of power option pricing under fractional Heston model in the Iran gold mar-ket), Physica A: Statistical Mechanics and its Applications, 2020; 537, 122690. Doi: 10.1016/j.physa.2019.122690
[4] Dastranj, E., Sahebi Fard, H., Exact solutions and numerical simulation for Bakstein-Howison model, Computational Methods for Differential Equations, 2022; 10(2), 461-474. Doi:10.22034/cmde.2021.42640.1834.
[5] Dura, G., Mosneagu, A. M., Numerical approximation of Black-Scholes equation, Annals of the Alexandru Ioan Cuza University-Mathematics, 2010; 56(1), 39-64. Doi:10.2478/v10157-010-0004-x.
[6] François-Lavet, V., Henderson, P., Islam, R., Bellemare, M. G., Pineau, J., An introduction to deep rein-forcement learning, Foundations and Trends® in Machine Learning, 2018;11(3-4): 219-354. Doi:10.1561/2200000071.
[7] Hull, J.c, Risk management and financial institutions, John Wiley & Sons, 2012.
[8] MacBeth, J. D., Merville, L, J., An empirical examination of the Black-Scholes call option pricing model, The journal of finance, 1979; 34(5), 1173-1186. Doi:10.2307/2327242
[9] Pagnottoni, P., Neural network models for Bitcoin option pricing, Frontiers in Artificial Intelligence, 2019; 2: 5. Doi:10.3389/frai.2019.00005
[10] Ren, P., Parametric estimation of the Heston model under the indirect observability framework, Doctoral dissertation, University of Houston. 2014.
[11] Sahebi Fard, H., Easapour, H., Dastranj, E., Stochastic analysis to find exact solutions of rainbow option under 2-D Black-Scholes model, Journal of Interdisciplinary Mathematics, accepted.
[12] Sahebi Fard, H., Dastranj, E., Abdolbaghi Ataabadi, A., Analytical and numerical solutions for the pricing of a combination of two financial derivatives in a market under Hull-White model, Advances in Mathematical Finance and Applications, 2022; 7(4):1013-1023. Doi: 10.22034/AMFA.2021.1902303.1447
[13] Kaustubh, Y., Formulation of a rational option pricing model using artificial neural networks, South-eastCon, IEEE, 2021, Doi:10.20944\preprints202012.0426.v1
[14] Shahvaroughi Farahani, M., Prediction of interest rate using artificial neural network and novel meta-heuristic algorithms, Iranian Journal of Accounting, Auditing and Finance, 2021; 5(1): 1-30. Doi:10.22067/ijaaf.2021.67957.1017
[15] Roul, P., Goura, V, P., A compact finite difference scheme for fractional Black-Scholes option pricing model, Applied Numerical Mathematics, 2021; 166: 40-60. Doi:10.1016/j.apnum.2021.03.017
[16] Mitra, S.K., An option pricing model that combines neural network approach and Black-Scholes formula, Global Journal of Computer Science and Technology, 2012; 12(4): 7-15.
[17] Chalasani, P., and Jha, S., Steven Shreve: Stochastic Calculus and Finance, Lecture notes, 1997;1-343.
[18] Wang, X., He, X., Bao, Y., Zhao, Y., Parameter estimates of Heston stochastic volatility model with MLE and consistent EKF algorithm, Science China Information Sciences, 2018; 6:1-17.‏
[19] Araghi, M., Dastranj, E., Abdolbaghi Ataabadi, A., Sahebi Fard, H., Option pricing with artificial neural network in a time dependent market, Advances in Mathematical Finance and Applications, 2024; 2(2): 723.‏
[20] Zohrabi, A., Tehrani, R., Souri, A., Sadeghi Sharif, S.J., Provide an improved factor pricing model using
neural networks and the gray wolf optimization algorithm, Advances in Mathematical Finance and
Applications, 2023; 8(1):23-45. Doi: 10.22034/AMFA.2022.1942547.1641.
[21] Nikumaram, H., Rahnamay Roodposhti, F., Zanjirdar, M., The explanation of risk and expected rate of return by using of Conditional Downside Capital Assets Pricing Model, Financial knowledge of securities analysis,2008;3(1):55-77
[22] Zamanpour,A., Zanjirdar, M., Davodi Nasr,M, Identify and rank the factors affecting stock portfolio optimization with fuzzy network analysis approach, Financial Engineering And Portfolio Management,2021;12(47): 210-236

Full-Text:



Original Research

 

The Using Neural Network and Finite Difference Method for Option Pricing under Black-Scholes-Vasicek Model Mahdiyeh Mohammadia, Elham Dastranjb, Abdolmajid Abdolbaghi Ataabadic,*, Hossein Sahebi Fardd   a,b,d Department of Mathematics, Faculty of Mathematical Sciences, Shahrood University of Technology, Shahrood, Semnan, Iran cDepartment of Management, Faculty of Industrial Engineering and Management, Shahrood University of Technology, Shahrood, Semnan, Iran

 

Article Info Article history: Received   2024-05-01 Accepted   2024-12-11   Keywords: Option Pricing Neural Networks Finite Difference Method  Black-Scholes-Vasicek Model		Abstract
		In this paper, the European option pricing is done using neural networks in the Black-Scholes-Vasicek market. The general purpose of this research is to compare the accuracy of neural network and Black-Scholes-Vasicek models for the pricing of call options. In the sequel, the finite difference method is applied to find approximate solutions of partial differential equation related to option pricing in the considered market. In the design of the artificial neural network required for this research, the parameters of the Black-Scholes-Vasicek model have been used as network inputs, as well as 720 data from the daily price of stock options available in the Tehran Stock Exchange market (in 1400) as the network output. The approximate solutions obtained in this article, which were carried out by two methods of neural networks and finite differences on the Tehran stock exchange based on the daily price of stock options, are shown that neural networks are more accurate method comparing with finite difference. The comparison of pricing results using neural networks with real prices in the assumed market is presented and shown via diagram, as well.

 

1 Introduction

    Option pricing is considered one of the important issues in financial markets, which has significant growth in recent years and solve this issue has absorbed traders and investors to these markets. An option is a type of contract that gives the holder the option to buy or sale of an asset for a certain period of time [7]. Nowadays derivative instruments are used in the stock market for trading, and one of the most important derivative instruments is the option, which is considered as a privilege for its holder.

A lot of models have been presented for option pricing and are available in literature. The Black-Scholes model is known as the first model in European option pricing, which was presented by Fisher Black and Mayron Scholes. In this model, the price of a stock was simulated with a geometric Brownian motion. In the Black-Scholes model, it is assumed that the probability distribution of the future price of underlying asset is log-normal, but the information obtained from the real market dedicates that the mentioned probability distribution has a heavier tail [3]. This model had maintained its efficiency for a long time despite its defects such as the stability of stock price volatility and the absence of transaction costs. But after five years and with the collapse of the financial markets, it lost its efficiency, and it was used as a basis for models [8]. Vasicek model is one of the stochastic interest rate models that enables mean reversion. This model was introduced by Uldrich Vasicek, which was a type of single factor short term rate and described interest rate changes using a kind of risk in the market. The separation of the stochastic interest rate from other underlying asset prices is one of the important features of Vasicek model. In this model if the financial prices increase indefinitely, the interest rate will not increase indefinitely [12].

In numerical methods for solving boundary value problems and equations with partial derivatives, finite differences are one of the most widely used methods. The basis of the finite difference method is local approximations of partial derivatives, which are obtained by expanding the Taylor series from lower orders. This method is simple in terms of definition and implementation and works on simple and uniform areas like rectangles.  Since 1928, the finite difference method has been used to find approximate solutions of partial differential equations (PDE) and ordinary differential equations (ODE). Bull, Thomson and Jordan are among the researchers who worked in this field. Finite difference methods with discretization of time interval and spatial domain are considered a numerical technique for solving differential equations. At any point of the grid, we are faced with an unknown value or a function value, so it is possible that the approximation of the differential equation will eventually become a device that can be solved by a suitable algorithm [15].

Finite difference method is widely used in financial researches. In [11] numerical solution for multi-assets option pricing (rainbow option) under two dimensional Black-Scholes model are presented via finite difference method. Results available in [9] dedicates the pricing of Bitcoin options sheets are systematically overpriced by classical methods, while there is a significant improvement in price prediction using neural network models. In [4], European options with transaction cost under some Black-Scholes markets using the finite difference method, are presented. In [12] the finite difference method is applied to find numerical solutions for zero-coupon bond option pricing. 

The results in [14] show using the neural network, an accurate prediction can be obtained to evaluate the result, which makes sure that it will be favorable in different sectors and earn money. One of the most important concepts in artificial intelligence are neural networks, which are widely used in various sciences, including financial sciences and markets. Parallel learning and processing is one of the most important features of artificial neural networks. The main part of neural networks is machine learning. This paper is organized in 7 sections. In section 2 the Theoretical Fundamentals and Research Background are stated. In section 3 the considered market equipped with combination of Black-Scholes and Vasicek model is presented. Section 4 is devoted to a numerical solution of partial differential equation related to European option pricing in Black-Scholes-Vasicek market using finite difference method. In section 5, the neural networks used in the article are described. Neural network prediction and finite difference method results for desired option pricing under the Black-Scholes-Vasicek model are compared in section 6 by assuming 720 stock option prices in the Tehran stock market. The paper is concluded in section 7. 

 

 

 

2 Theoretical Fundamentals and Research Background

      Neural network science researchers, McCulloch and Pitts, conducted studies on the internal communication ability of a neuron model. The result of their work was to present a computational model based on a simple pseudo-neuronal element. At that time, other scientists such as Donald Hebb were also working on the rules of adaptation in neuronal systems. Donald Hebb proposed a learning rule for adapting connections between artificial neurons. Verbus published a backpropagation algorithm in his doctoral dissertation, and finally Rosenblatt rediscovered this technique [9]. In [1], the Black-Scholes model and neural networks were compared and the results show that the neural network is more accurate for option pricing than the Black-Scholes model. In this article, finite difference method and neural network are used in the market of Black-Scholes-Vasicek for European option pricing. The purpose was a comparison of efficiency of these two models in option pricing issue. So 120 options were priced by these two methods. In this paper, the partial differential equation related to European option pricing in the Black-Scholes-Vasicek market is approximated by the finite difference method. It has been done on a number of options in the Tehran Stock Exchange market in 2021-2022. In the sequel, a neural network was designed using Black-Scholes-Vasicek model parameters, and using it, pricing on options was presented. The results obtained from the pricing by the two mentioned methods indicates that neural networks have higher accuracy in comparison with the finite difference method. Combining artificial neural networks and two Black-Scholes and Heston models in [13], it is concluded that artificial neural networks can be considered as an efficient alternative to existing quantitative models for option pricing. 

The researchers' findings show that risk premium was a determining factor in explaining changes in investors' expected rate of return, and that there was a conditional relationship between the Downside Beta and expected return. Therefore, to explain the relationship between risk and return, one must pay attention to the market direction [21].Other findings research, through a regular and logical process based on the judgment method in a survey of 14 experts in the field of capital market investment and a quantitative and multivariate model of fuzzy network analysis, to assess the level of importance, ranking and refining the effective factors. Portfolio optimization was undertaken. Based on the analysis, the variables of profit volatility, return on capital, company value, market risk, stock profitability, financial structure, liquidity and survival index can be introduced as the most important factors affecting the optimization of the stock portfolio [22].

3 The Model

    In the 1980s and 1990s, the Black-Scholes model played an essential and important role in the success of financial engineering because option pricing is one of the most important topics in financial economics, and the Black-Scholes model has revolutionized this pricing method. For this reason, in this section, this model is investigated, then option pricing under the Vasicek model is presented.

The Black-Scholes model is a model that includes two assets; riskless asset () and risky asset () which respectively have the following dynamics [2].

 

where  is interest rate,  is volatility of risky underlying assets,  is standard Wiener process and  is drift.

Vasicek model is a financial mathematical model to describe the interest rate improvement, which is a single factor short term rate and describes the interest rate volatility using a kind of risk in the market. Instantaneous interest rate in Vasicek model is calculated as follows.

 

= 1	(1)
	(2)

 

The parameters of eq. (3) define as follow.

·       (Long-term mean of return process); All curves of  are close to the mean value.

·        (Speed of return to the mean return process); Assuming it is non-negative, the parameter  determines the speed at which the curves converge to .

·        (Instantaneous volatilities); It measures the stochastic numbers are entered into the system in real time. The relationship of  with stochastic numbers is direct, as  increases with the increase of stochastic numbers.

·       (Wiener process); It is stochastic and independent of modeling in the market risk factor. suppose there exists a constant  that implies the correlation between two processes  and ,

 

(3)

 

Using Ito’s lemma, the pde of Black-Scholes-Vasicek model obtained as follow.

 

(4)

 

In eq. (4),  is the option price equation [17].

 

3.1 Parameter estimation

     At first, the following points have been considered in our computation; the initial value of , called , is always positive. which denotes the rate of return to the mean of the underlying asset price process, σ the underlying asset price volatility process, and  the long-term mean of the underlying asset price process get positive values. Feller's condition, which is expressed as  [10].
Theorem 1: The MLE estimation for calculating parameters  and  in each time step is

 

(5)

 

(6)

 

(7)

 

where  and  is an arbitrary positive number [10].

Proof: Conditioned on  ,  is subject to the following Gaussian distribution [18]:

(8)

 

N (  ,   ),

(9)

 

Denote the likelihood function as

=  +  [

(10)

Where  Taking the logarithm of the above likelihood function (11), omitting the constant variables, the logarithm likelihood function can be written as

	(11)

Taking the derivative of the function  with respect to the parameters, ab, a, , respectively, we can obtain

(12)

 

To obtain the MLE of the parameter  , the derivatives of the parameters are set as follows:

(13)

 

Then the equations in (14) implies that

(14)

 

(15)

 

(16)

 

With an ingenious technique, setting , sorting (15) and (16), we can get

(17)

 

The above equations can be solved as

	(18)
	(19)

 

(20)

 

Then according to (17),

(21)

 

thus

(22)

 

With , , , we can get that

.

(23)

 

Then Eqs.  are the estimations of a, , , b by MLE, respectively. Since the MLE of the parameters are derived through the Gaussian distribution of  rather than the approximation of chisquare distribution, the estimation method is called MLE [18].

To calculate the parameters in Vasicek model, first  is selected using equation (8). Then substituting  into equation (6),  is obtained. To calculate ,  is needed as well, which is obtained from the relationship of the deviation of the daily returns of the underlying asset in the entire period. Parameters and  are used as input layer nodes of the neural network.

 

4 Finite Difference Method

     In this section, finite difference method is used to construct a numerical simulation for pde (5). In relation (5),  denotes maturity time which decreases from  to zero. To simplify finite difference method calculations, changing the variable  as, the backward equation has been transformed to a forward equation [5]. Note that  increases from zero to .

 

(24)

 

Replacing  with , eq. (5) is rewritten as

 

(25)

---