A Metaheuristic Approach to Optimize European Call Function with Boundary Conditions

Najeeb Alam KHAN; Oyoon Abdul RAZZAQ; Tooba HAMEED

doi:10.21078/JSSI-2018-260-09

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Journal of Systems Science and Information ›› 2018, Vol. 6 ›› Issue (3) : 260-268. DOI: 10.21078/JSSI-2018-260-09

A Metaheuristic Approach to Optimize European Call Function with Boundary Conditions

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Abstract

The purpose of this paper is to investigate the pricing European call option valuation problems under the exercise price, maturity, risk-free interest rate, and the volatility function. An advance methodology, Chebyshev simulated annealing neural network (ChSANN), is enforced for the Black-Scholes (B-S) model with boundary conditions. Our scheme is stable and easy to implement on B-S equation, for arbitrary volatility and arbitrary interest rate values. Also, the comparative results demonstrate that the attained approximate solutions are converging towards the exact solution. The graphical results show that the increasing flow of the European call option as the exponential increase takes place in assets. The presented algorithm can be further applied to other financial models with certain boundary conditions. The algorithm of the method shows that the approach can also be easily employed on time-fractional B-S equation.

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neural network / numerical approximation / Chebyshev polynomials

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Najeeb Alam KHAN , Oyoon Abdul RAZZAQ , Tooba HAMEED. A Metaheuristic Approach to Optimize European Call Function with Boundary Conditions. Journal of Systems Science and Information, 2018, 6(3): 260-268 https://doi.org/10.21078/JSSI-2018-260-09

1 Introduction

The mathematical tools are pragmatically effective in modeling different financial aspects, such as asset and option pricing theories, risk and asset management, etc. have lead the path of financial mathematics. Among various tools, differential equations play the key role in modeling different financial problems. In this regard, in 1970s, a groundbreaking research was performed by Black and Scholes^[1]. They collaboratively pioneered the famous Black-Scholes equation (B-S) for the option pricing. Using the partial differential equations, they demonstrated the possibility to construct a portfolio made up of the stock and the bond that perfectly replicates the option. These options comprise the American and European call and put options. In the literature, several investigations have been made for the evaluation of these options. For instance, Cen, et al.^[2] discussed the generalized B-S equation with European options using finite difference method, Kumar, et al.^[3] presented homotopy analysis method and homotopy perturbation method to illustrate analytically the time fractional B-S equation with European options. Moreover, Company, et al.^[4] utilized semidiscretization technique to obtain numerical solutions of nonlinear B-S equation with transaction cost, Liu, et al.^[5] provided approximate solutions of fractional B-S equation with European options by employing the variational iteration method, etc^[6-10]. This paper aims to numerically discuss the B-S model, described as

\frac{\partial E_{c}}{\partial t} + \frac{1}{2} σ^{2} S^{2} \frac{\partial^{2} E_{c} (S, t)}{\partial S^{2}} + r (t) S \frac{\partial E_{c} (S, t)}{\partial S} - r (t) E_{c} (S, t) = 0, t \in (0, T)

(1)

with

E_{c} (0, t) = 0, E_{c} (S, t) \approx S, a s S \to \infty,

(2)

and

E_{c} (S, t) = max (S - K, 0) .

(3)

In the above equation,

E_{c} (S, t)

defines the values of European call option,

r

depending on asset price

S

at time

t

is the risk free interest rate,

K

denotes the exercise price of the option,

σ

shows the volatility function of underlying asset and

T

is the maturity time. Here, we exercise a metaheuristic approach, Chebyshev-simulated annealing neural network (ChSANN), to optimize the European call function of the B-S equation (1)–(3). This technique was firstly discussed by Khan, et al.^[11], to solve fractional differential equations. This approach is based on the Chebyshev neural network (ChNN) approximation plus the simulated annealing (SA) optimization^[12-14]. The ChNN has been widely employed in solving integer as well as fractional order differential equations. Mall, et al.^[15] employed ChNN to overcome the singularity and obtain effective solutions of Lane-Emden equations, Shaik, et al.^[16] utilized ChNN to estimate the unknown nonlinearities of twin rotor multi-input-multi-output system and many others^[17-20]. Whereas, the implication of simulated annealing, based on the characteristics of the metal annealing process, is increasing in optimizing the dynamical problems globally. In this endeavor, generally, after the approximations of European call function by ChNN, the simulated annealing algorithm is processed to optimize the system globally. Moreover, some examples of B-S equations with different states of the volatility of the assets, maturity time and expiration price are interpreted through the obtained numerical and graphical results.

2 The Chebyshev Simulated Annealing Artificial Neural Network

The Chebyshev simulated annealing artificial neural network comprises two well-known methodologies, namely, Chebyshev neural network and simulated annealing. The ChNN functions as the approximate solution to the problem while SA meets the purposes of optimizing and training the unknowns of the system. Hence, the two structural parts of the erected artificial neural network are comprehensively defined below.

2.1 Numerical Transformation

The numerical transformation is carried out by expanding each input data to several terms of Chebyshev polynomials of the second kind

T (x)

. The Chebyshev polynomials are widely used for function expansion, as its expansions converge more rapidly than the other orthogonal polynomials. These polynomials may be obtained by the following recursive formula

T_{m + 1} (x) = 2 x T_{m} (x) - T_{m - 1} (x),

(4)

where

T_{0} (x) = 1

T_{1} (x) = 2 x

and

T_{m} (x)

denotes the

m

th Chebyshev polynomial of the second kind. In ChNN to get more effective results the dimension of the input is increased through Chebyshev polynomial. For the activation of the output function, here we consider the tangent hyperbolic function as

N (x, p) = \tan (ξ)

(5)

with input data

x

and

ξ

be the linear weighted sum of Chebyshev polynomials that can be expressed as

ξ = \sum_{j = 1}^{m} p_{j} T_{j - 1} (x),

(6)

where

T_{j - 1} (x)

denote the

(j - 1)

th Chebyshev polynomial, and

p_{j}

are the unknown weight vectors of the ChNN.

2.2 Training Process

In the training process, the simulated annealing algorithm is exercised to optimize the conditional equation and update the weights, until the network learns the training data. The SA scheme features the following components:

(a) A space of random points (RP) in a specified domain say

[a, T]

R P = \frac{(T - a) i}{N}, i = 0, 1, \dots, N,

(7)

which is used as the sampling points for the iteration.

(b) The trial solution of the unknown function

y (x)

\hat{y} (x, p) = G (x) + F (x, N (x, p)),

(8)

which is the combination of,

G (x)

that satisfies the initial/boundary conditions and

F (x, N (x, p))

that contains the output of ChSANN with unknown parameters

p

R (x_{i}, p)

, which is constructed by taking the residual of the respective differential equation.

(d) Boltzmann probability distribution, which is an exponential function expressed as

P_{d} = e^{\frac{- R}{k T}},

(9)

where

k = 1.38 \times 10^{23}

is Boltzmann constant and

T = 1

is taken as the initial state. This assigns probability for each value of the conditional equation.

The SA scheme starts with a random selection of a point from the set, which is generated by using Equation (7) and calculates the trial solution of the function at this point. Then, undergoes the construction of residual

R (x_{i}, p)

by substituting the trial solution in the governing differential equation. Next, Boltzmann probability is calculated by manipulating the value of

R (x_{i}, p)

in the probability distribution (9). The obtained value of probability will suggest the extent of accuracy of the approximated values of the unknowns. Repetitively, after training all the sampling points of the given space, the value of the error function with the greatest probability, i.e., 1, is undertaken to be the global minimum value. Additionally, the values of the unknown parameters of that probability will be considered as the most convergent approximations towards the exact solutions among the other calculated values of the unknown parameters in the space.

3 Algorithm Implementation on B-S Equation

The B-S model specified in Equations (1)

\sim

(3) is cumbersome, due to the presence of the differentiation and multiplication of

S

, for direct implementation of the method. Therefore, firstly, it is made dimensionless by taking the following suppositions:

S = K e^{x}, t = T - \frac{τ}{σ^{2} / 2}, E_{c} (S, t) = K v (x, τ) .

(10)

Hence, Equation (1) reduces to

\frac{\partial v (x, τ)}{\partial τ} = \frac{\partial^{2} v (x, τ)}{\partial x^{2}} + (ϖ - 1) \frac{\partial v (x, τ)}{\partial x} - ϖ v (x, τ), x \in (0, a], τ \in (0, b],

(11)

where

ϖ = \frac{2 r}{σ^{2}}

signifies the balance between the interest rate, the dimensionless time to expiry

\frac{1}{2} σ^{2} T

and the variability of stock returns. While with

K = 1

, the initial condition becomes

v (x, 0) = max (e^{x} - 1, 0) .

(12)

This B-S equation has been analytically evaluated using the homotopy perturbation method and homotopy analysis method with fractional order derivative^[3] and homotopy perturbation method with integer order derivatives of the functions in [9]. Here, for the first time we will discuss its numerical solutions by using ChSANN. Since, the governing model is in space, let the trial solution for the function be defined as

\hat{v} (x, τ, p, q) = G (x, τ) + F (x, τ, N (ξ)),

(13)

where

\begin{array}{rcl} G (x, τ) = v (a, b) + (x - a) \frac{\partial v (a, b)}{\partial x} + (τ - b) \frac{\partial v (a, b)}{\partial τ}, \end{array}

(14)

\begin{array}{rcl} F (x, τ, N (ξ)) = (x - a)^{2} (τ - b) N (ξ), \end{array}

(15)

and activation function can be constructed with the tangent hyperbolic function, the same as defined in Equation (5) with

ξ = \sum_{j = 2}^{m} p_{j} T_{j - 1} (x) + \sum_{j = 2}^{m} q_{j} T_{j - 1} (τ) .

(16)

Substituting Equation (13) in Equation (11) we attain the residual function as

R (x, τ, p, q) = \frac{\partial \hat{v} (x, τ, p, q)}{\partial τ} - \frac{\partial^{2} \hat{v} (x, τ, p, q)}{\partial x^{2}} - (ϖ - 1) \frac{\partial \hat{v} (x, τ, p, q)}{\partial x} + ϖ \hat{v} (x, τ, p, q) .

To obtain the unknown parameter vectors

p

and

q

we discretize

x

and

τ

into

N

subintervals

(0, a]

and

(0, b]

then, optimizes the following error function

min E (p, q) = \frac{1}{N^{2}} \sum_{l = 1}^{N} \sum_{d = 1}^{N} (R (x_{d}, τ_{l}, p_{j}, q_{j}))^{2} .

(17)

Using the sampling points specified in Equation (7) and following the algorithm in Subsection 2.2, we obtain the optimized values of the unknown parameters

p_{j}

and

q_{j}

for

j = 1, 2, \dots, n

. On replacing these values in Equation (13) the approximate solution of

v (x, τ)

is acquired. All the numerical manipulation of SA is carried out using Mathematica 10. Consequently, the graphical solutions of

v (x, τ)

for different values of stock market parameters, such as interest rate and maturity time are plotted.

Figures 1 and 2 represent the two-dimensional flow of

v (x, τ)

with respect to the values of

x

and

τ

. From Figure 1 it can be demonstrated that, when 4% interest rate is applied with 0.2 volatility, the value of the European call option becomes

v (x, τ) > $ 6.3

million, as the exponential increase in the assets occurs, but its value becomes less than the initial stage as the time moves towards the completion of a year. On the other hand, in Figure 2 with the

10 %

risk-free interest rate, the value of the European call option becomes

v (x, τ) > $ 153

million, as the exponential increase in the assets occurs, with the same volatility. In addition, with respect to time,

v (x, τ)

moves in decreasing direction till the completion of six months and after that keeps on increasing but remains less than the starting value during the period of one year. Furthermore, Figures 3 and 4 depict the increasing pattern of

v (x, τ)

at different time periods

τ = 1, 1.5, 2

years and different risk-free interest rates, 4% and 10%, respectively. In Figure 3, at the end of the different periods of year,

v (x, τ)

reaches the maximum of

$ 7.5

millions with 4% interest rate. Whereas, Figure 4 shows the maximum value of

v (x, τ)

$ 130

millions, with 10% interest rate.

Figure 1 Approximations of $v (x, τ)$ using ChSANN, for $r = 0.04$ , $σ = 0.2$ and $k = 2$

Full size|PPT slide

Figure 2 Approximations of $v (x, τ)$ using ChSANN, for $r = 0.1$ , $σ = 0.2$ and $k = 5$

Full size|PPT slide

Figure 3 Flow patterns of $v (x, τ)$ using ChSANN, for $r = 0.04$ , $σ = 0.2$ and $k = 2$ , at different values of time period

Full size|PPT slide

Figure 4 Flow patterns of $v (x, τ)$ using ChSANN, for $r = 0.1$ , $σ = 0.2$ and $k = 5$ , at different values of time period

Full size|PPT slide

Moreover, Figures 5 to 8 elucidate the meticulousness of the ChSANN method with first five Chebyshev polynomials. In Figures 5 and 6, the approximated solutions are plotted against the exact solutions at

τ = 1

r = 4 %

10 %

, respectively. The overlapping of the curves clarifies the convergence of the approximated solutions of

v (x, τ)

and the efficiency of the proposed method. Figures 7 and 8 plot the absolute errors, obtained by measuring the absolute difference between approximated and exact values. The plotted values in the form of

10^{- γ}

, for any positive integer

γ

, represent the proficiency of the working precision of the calculated values.

Figure 5 Exact versus approximated solutions of $v (x, τ)$ , for $r = 0.04$ , $σ = 0.2$ and $k = 2$ , at $τ = 1$

Full size|PPT slide

Figure 6 Exact versus approximated solutions of $v (x, τ)$ , for $r = 0.1$ , $σ = 0.2$ and $k = 5$ , at $τ = 1$

Full size|PPT slide

Figure 7 Absolute error for $v (x, τ)$ , for $r = 0.04$ , $σ = 0.2$ and $k = 2$ , at $τ = 1$

Full size|PPT slide

Figure 8 Absolute error for $v (x, τ)$ , for $r = 0.1$ , $σ = 0.2$ and $k = 5$ , at $τ = 1$

Full size|PPT slide

4 Conclusion

In this work, a metaheuristic method was exerted to optimize a European call option pricing model. In this regard, we undertook the Black-Scholes equation with boundary conditions and employed Chebyshev simulated annealing neural network algorithm to discuss its numerical and analytical results. Thus, from the facts and figures of entire study, following clinching points are scrutinized:

1) The structure of B-S equation being in the form of the well-known heat equation, made the valuation of European call option pricing problems easy to tackle.

2) The ChSANN magnificently optimized and approximated the solutions of B-S equation with boundary conditions.

3) The graphical depictions of the European options, attained successfully by using ChSANN, enabled to illustrate the behavior of the options under different circumstances of the stock market parameters.

4) Increasing the number of terms of the Chebyshev neural network, increases the effectiveness of the approximated solution towards the exact solution (see Table 1).

Table 1 Global optimum mean square errors $E$ for different numbers of Chebyshev polynomials $m$

$m$	$k = 5$
3	2.197 $\times 10^{- 5}$
4	3.898 $\times 10^{- 6}$
5	3.095 $\times 10^{- 7}$

5) Simulated annealing produces the values of unknown parameters of the network more efficiently than the other non-probabilistic approaches that exists in the literature.

6) The proposed approach also has the capability to solve the dynamical problems in

ℜ^{n}

, of integer as well as fractional order.

References

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Abstract
Key words
Cite this article
1 Introduction
2 The Chebyshev Simulated Annealing Artificial Neural Network
2.1 Numerical Transformation
2.2 Training Process
3 Algorithm Implementation on B-S Equation
Figure 1 Approximations of $v (x, τ)$ using ChSANN, for $r = 0.04$ , $σ = 0.2$ and $k = 2$
Figure 2 Approximations of $v (x, τ)$ using ChSANN, for $r = 0.1$ , $σ = 0.2$ and $k = 5$
Figure 3 Flow patterns of $v (x, τ)$ using ChSANN, for $r = 0.04$ , $σ = 0.2$ and $k = 2$ , at different values of time period
Figure 4 Flow patterns of $v (x, τ)$ using ChSANN, for $r = 0.1$ , $σ = 0.2$ and $k = 5$ , at different values of time period
Figure 5 Exact versus approximated solutions of $v (x, τ)$ , for $r = 0.04$ , $σ = 0.2$ and $k = 2$ , at $τ = 1$
Figure 6 Exact versus approximated solutions of $v (x, τ)$ , for $r = 0.1$ , $σ = 0.2$ and $k = 5$ , at $τ = 1$
Figure 7 Absolute error for $v (x, τ)$ , for $r = 0.04$ , $σ = 0.2$ and $k = 2$ , at $τ = 1$
Figure 8 Absolute error for $v (x, τ)$ , for $r = 0.1$ , $σ = 0.2$ and $k = 5$ , at $τ = 1$
4 Conclusion
Table 1 Global optimum mean square errors $E$ for different numbers of Chebyshev polynomials $m$
References

Received	Accepted	Published
2017-05-31	2017-12-07	2018-06-25
Issue Date
2018-06-25

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Abstract

Key words

Cite this article

1 Introduction

2 The Chebyshev Simulated Annealing Artificial Neural Network

2.1 Numerical Transformation

2.2 Training Process

3 Algorithm Implementation on B-S Equation

Figure 1 Approximations of $v (x, τ)$ using ChSANN, for $r = 0.04$ , $σ = 0.2$ and $k = 2$

Figure 2 Approximations of $v (x, τ)$ using ChSANN, for $r = 0.1$ , $σ = 0.2$ and $k = 5$

Figure 3 Flow patterns of $v (x, τ)$ using ChSANN, for $r = 0.04$ , $σ = 0.2$ and $k = 2$ , at different values of time period

Figure 4 Flow patterns of $v (x, τ)$ using ChSANN, for $r = 0.1$ , $σ = 0.2$ and $k = 5$ , at different values of time period

Figure 5 Exact versus approximated solutions of $v (x, τ)$ , for $r = 0.04$ , $σ = 0.2$ and $k = 2$ , at $τ = 1$

Figure 6 Exact versus approximated solutions of $v (x, τ)$ , for $r = 0.1$ , $σ = 0.2$ and $k = 5$ , at $τ = 1$

Figure 7 Absolute error for $v (x, τ)$ , for $r = 0.04$ , $σ = 0.2$ and $k = 2$ , at $τ = 1$

Figure 8 Absolute error for $v (x, τ)$ , for $r = 0.1$ , $σ = 0.2$ and $k = 5$ , at $τ = 1$

4 Conclusion

Table 1 Global optimum mean square errors $E$ for different numbers of Chebyshev polynomials $m$

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Abstract

Key words

Cite this article

1 Introduction

2 The Chebyshev Simulated Annealing Artificial Neural Network

2.1 Numerical Transformation

2.2 Training Process

3 Algorithm Implementation on B-S Equation

Figure 1 Approximations of v(x,τ) using ChSANN, for r=0.04, σ=0.2 and k=2

Figure 2 Approximations of v(x,τ) using ChSANN, for r=0.1, σ=0.2 and k=5

Figure 3 Flow patterns of v(x,τ) using ChSANN, for r=0.04, σ=0.2 and k=2, at different values of time period

Figure 4 Flow patterns of v(x,τ) using ChSANN, for r=0.1, σ=0.2 and k=5, at different values of time period

Figure 5 Exact versus approximated solutions of v(x,τ), for r=0.04, σ=0.2 and k=2, at τ=1

Figure 6 Exact versus approximated solutions of v(x,τ), for r=0.1, σ=0.2 and k=5, at τ=1

Figure 7 Absolute error for v(x,τ), for r=0.04, σ=0.2 and k=2, at τ=1

Figure 8 Absolute error for v(x,τ), for r=0.1, σ=0.2 and k=5, at τ=1

4 Conclusion

Table 1 Global optimum mean square errors E for different numbers of Chebyshev polynomials m

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References

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Footnotes

Figure 1 Approximations of $v (x, τ)$ using ChSANN, for $r = 0.04$ , $σ = 0.2$ and $k = 2$

Figure 2 Approximations of $v (x, τ)$ using ChSANN, for $r = 0.1$ , $σ = 0.2$ and $k = 5$

Figure 3 Flow patterns of $v (x, τ)$ using ChSANN, for $r = 0.04$ , $σ = 0.2$ and $k = 2$ , at different values of time period

Figure 4 Flow patterns of $v (x, τ)$ using ChSANN, for $r = 0.1$ , $σ = 0.2$ and $k = 5$ , at different values of time period

Figure 5 Exact versus approximated solutions of $v (x, τ)$ , for $r = 0.04$ , $σ = 0.2$ and $k = 2$ , at $τ = 1$

Figure 6 Exact versus approximated solutions of $v (x, τ)$ , for $r = 0.1$ , $σ = 0.2$ and $k = 5$ , at $τ = 1$

Figure 7 Absolute error for $v (x, τ)$ , for $r = 0.04$ , $σ = 0.2$ and $k = 2$ , at $τ = 1$

Figure 8 Absolute error for $v (x, τ)$ , for $r = 0.1$ , $σ = 0.2$ and $k = 5$ , at $τ = 1$

Table 1 Global optimum mean square errors $E$ for different numbers of Chebyshev polynomials $m$