Solving Black-Scholes Equation for European Option Pricing Using Laplace Adomian Decomposition Method

Abstract

In finance and investment, the Black-Scholes equa- tion is a very well-known model for pricing European call option. This equation is a partial differential equation which looks a little like the diffusion equation. In this paper, the Laplace Adomian Decomposition Method (LADM) is used to find the solution of the Black-Scholes equation with the boundary conditions for European option pricing.

This decom- position method is an effective procedure for getting solutions without linearization, discretization, perturbation theory or other restrictive assumptions.

The numerical example of the solution of the Black-Scholes equation with the LADM shows that the solution forms an infinite series that is convergent and easily calculated. The results presented in graphical form, show that this method is accurate and efficient for option pricing. Index Terms—pricing call option, partial differential equa- tion, Black-Scholes, Laplace-Adomian decomposistion method (LADM), stock option.

Introduction

IN recent years, investment has grown rapidly in thefinancial and economic fields. This is indicated by the increasing number of investors and funds involved in in- vestment activities, as well as the increasing diversity of financial derivative products developed.

Financial derivatives are investment instruments which are derived from a financial asset. Thus the value depends on the price of the financial asset, for example is an option contract [1].

Options are rights, not obligations, which are owned by the holder to call or put an investment instrument, in this case, is an underlying financial asset, at a certain price for a certain period or date [2]. Options can be used for hedging and speculation.

Based on the implementation, the options consist of American and European type options. The American type option can be excercised at any time up to the maturity, while the European type option can be excercised only on the maturity itself.

Therefore, most options traded are American options, but European options are generally easier to analyze than American options. Keep in mind that the option gives the holder the right to make a call or put. The holder does not have to do this right. One very well-known technique for pricing option is a binomial tree which assumes time follows a discrete ap- proach [3]. This tree illustrates the direction of movement of stocks during the option period; it has the probability of Ira Sumiati, Magister Program of Department of Mathematics, Faculty of Mathematics and Natural Science, Universitas Padjadjaran, Bandung, Indonesia. (irasumiati@gmail.com) Endang Rusyaman, Department of Mathematics, Faculty of Mathematics and Natural Science, Universitas Padjadjaran, Bandung, Indonesia. (rusya- man@unpad.ac.id) Sukono, Department of Mathematics, Faculty of Mathematics and Natural Science, Universitas Padjadjaran, Bandung, Indonesia. (sukono@unpad.ac.id) Aceng Sambas, Department of Mechanical Engineering, Universitas Muhammadiyah Tasikmalaya, Indonesia. (aceng@umtas.ac.id) going up or down. Another model or technique that is also very well known for pricing option is the partial differential Black-Scholes equation with a continuous time approach [4].

The main idea of this model is to determine the price of a European type call option with the underlying asset being a stock without dividend payment. Black-Scholes partial differential equations can be solved by the Merlin transformation approach [5] and semi dis- cretization techniques with high transaction costs on options [6]. The homotopy perturbation, homotopy analysis [7], [8] and variation iteration method [9], [10] can be used in solving the Black-Scholes equation and the boundary conditions for European option pricing problems are fast, accurate and easily calculated.

The finite difference method ensures that the scheme is stable for any volatility and interest rates and shows that the approach is accurate and effective for solving the Black-Scholes equation [11], [12]. The projected differential transformation method which is a modification of the classical differential transformation method is ap- plied to solve the Black-Scholes equation, the results are converged faster to the exact form of related solutions and involve less computation [13]. Another numerical method used to find the solution for the Black-Scholes equation is the Adomian decomposition method. Solutions for equations are calculated in the form of convergent power series with easily computed components and obtain efficient recursive relationships, where nonlinear forms are decomposed into Adomian polynomials [14], [15], [16].

Adomian decomposition method is one of the tool that can be used to solve differential equations, including nonlinear partial differential equations. This method was first intro- duced by George Adomian to solve a system of stochastic equations [17]. This decomposition method can be an ef- fective procedure for obtaining analytical solutions without linearization or weak nonlinear assumptions, perturbation, or restrictive assumptions on stochastic cases [18].

This method can be used to solve integral, differential and integral- differential equations. Differential equations that can be solved by this method can be an integer or fractional, or- dinary or partial, with an initial value or boundary problems, with coefficients or constants variable, linear or nonlinear, homogeneous or nonhomogeneous [19], [20], [21], [22]. This decomposition method is also a powerful and useful technique for solving heat [23], waves [24], Fokker-Plank [25] and Riccati equation [26]. The numerical scheme of Laplace transformation based on the modified Adomian decomposition method can be used to solve nonlinear dif- ferential equations. The main advantage of this technique is that solutions are expressed as non-series to converge quickly to the actual solution [27]. The Laplace Adomian decomposition method can be used to solve Bratu problems [28], Burgers [29] and Kundu-Eckhaus equation [30]. Thus in this study, the Laplace Adomian Decomposition Method (LADM) is used to solve the Black-Scholes equa- tion, then applies the solution obtained in numerical solution for pricing option.

Black-Scholes Equation

This section discusses the Black-Scholes equation for pricing option. The option value is denoted by V (S, t) is a function that depends on the current price of the under- lying asset (S) and time (t), where C(S, t) and P (S, t) respectively are call and put options. Option values also depend on volatility of underlying asset prices (σ), strike price (E), maturity date (T ) and free-risk interest rate (r). Black-Scholes partial differential equation for pricing option can be written [7], [31]

∂V ∂t + 1 2 σ2S2 ∂2V ∂S2 + rS ∂V ∂S − rV = 0. (1)

The Black-Scholes equation for pricing call options based on Eq. (1) can be rewritten as follows

∂C ∂t + 1 2 σ2S2 ∂2C ∂S2 + rS ∂C ∂S − rC = 0 (2)

with

C(0, t) = 0, C(S, t) ∼ S as S →∞

and

C(S, T ) = max{S − E, 0}.

Furthermore, Eq. (2) looks a little like diffusion equation and has a backward form, with final data given at t = T , thus to be converted into a forward form, set

S = Eex, t = T − 2τ σ2 , C(S, t) = Ev(x, τ). (3)

Use Eq. (3) and the partial derivatives of C is

∂C ∂t = ∂Ev ∂τ ∂τ ∂t = −Eσ 2 2 ∂v ∂τ ∂C ∂S = ∂Ev ∂x ∂x ∂S = E S ∂v ∂x ∂2C ∂S2 = ∂ ∂S ( E S ∂v ∂x ) = − E S2 ∂v ∂x + E S2 ∂2v ∂x2 .

By substitution Eq. (3) dan partial derivatives C to Black- ScholesEq. (2), thus obtained

∂v ∂τ − ∂ 2v ∂x2 − ( 2r σ2 − 1 ) ∂v ∂x + 2r σ2 v = 0.

Suppose k = 2rσ2 , then the equation above can be written

∂v ∂τ = ∂2v ∂x2 + (k − 1)∂v ∂x − kv (4)

with the initial condition v(x, 0) = max{ex − 1, 0}. This system of equations contains two dimensionless pa- rameters that is k represents the balance between the free- risk interest rate and the variance of stock returns, and the dimensionless time to expiry σ 2 2 T . Even though there are four dimensional parameters, E, T , σ and r, in the original statement of the problem for pricing option.

Laplace Adomian Decomposition Method

In this section, Laplace Adomian Decomposition Method (LADM) discussed to solve partial diferential equation. Given a partial diferential equation as follows

Mtu(x, t) +Nu(x, t) +Ru(x, t) = g(x, t) (5)

where Mt = ∂∂t , N is a nonlinear operator, R is a linear oper- ator and g is a non-homogeneous term that is u-independent, with initial condition u(x, 0) = f(x). Solving for Mtu(x, t), Eq. (5) can be written

Mtu(x, t) = g(x, t)−Nu(x, t)−Ru(x, t). (6)

The Laplace transform is the transformation of the integral function of a real variable t to the function of a complex variable s. Laplace transform can be used to find solutions to differential equations by turning them into algebraic equa- tions [32], [33]. Before using the Adomian decomposition method combined with Laplace transformation, first explain some basic definitions as follows. Definition 1 Suppose that f is a real or complex function of variables t > 0 and s is a real or complex parameter.

Laplace transform is defined

F (s) = L[f(t)] = ∞∫ e−stf(t)dt = lim b→∞ b∫ e−stf(t)dt

where the limit value exists and finite. If L[f(t)] = F (s),

then the Laplace transform inverse is denoted as

L−1[F (s)] = f(t), t ≥ 0.

Based on Definition 1, for f(t) = tn where t ≥ 0, Laplace

transform f(t) is

L[tn] = n! sn+1 , s > 0

and Laplace transform for f (n)(t) is

L[f (n)(t)] = snF (s)− n∑ k=1 sn−kf (n−1)(0).

LADM apply Laplace transform [27], [30] to Eq. (6), obtaining

L[Mtu(x, t)] = L[g(x, t)−Nu(x, t)−Ru(x, t)]

or equivalent with

su(x, s)− u(x, 0) = L[g(x, t)−Nu(x, t)−Ru(x, t)]. (7)

By ubstitutingintial condition, Eq. (7) can be written

u(x, s) = f(x) s + 1 s L[g(x)]− 1 s L[Nu(x, t)+Ru(x, t)] (8)

furthermore, apply inverse Laplace transform to Eq. (8)

u(x, t) = f(x)+L−1 [ 1 s L[g(x)]− 1 s L[Nu(x, t) +Ru(x, t)] ] . (9)

Adomian decomposition method assumes that u(x, t) can be decomposed into an infinite series [18], [22]

u(x, t) = ∞∑ n=0 un(x, t) (10)

and nonlinear term Nu(x, t) is decomposed become

Nu(x, t) = ∞∑ n=0 An (11)

where An = An(u0, u1, ..., un) are the Adomianpolynomi- als defined by

An = 1 n! dn dλn [ N ( n∑ k=0 λkuk )] λ=0 ;n = 0, 1, 2, ...

with λ is a parameter. Substituting Eq. (10) and Eq. (11) to Eq. (9)

∞∑ n=0 un = f(x)+L −1 [ 1 s L[g(x)]− 1 s L [ ∞∑ n=0 An +R ∞∑ n=0 un ]] . (12)

therefore based on Eq. (12), a recursive relation of solution is obtained

u0(x, t) = f(x) + L −1 [ 1 s L[g(x)] ] , un+1(x, t) = −L−1 [ 1 s L [An +Run] ] , n = 0, 1, 2, · · · .

Hence, an approximate solution of Eq. (5) is

u(x, t) ≈ k∑ n=0 un(x, t)

where

lim k→∞ k∑ n=0 un(x, t) = u(x, t).

The Adomian decomposition method that is combined with the Laplace transform needs less work in comparison with the standard Adomian decomposition method. The decomposition procedure of Adomian will be easily and efficient technique, without linearization or discretiaztion of the problem. The approximate solution is found in the form of a convergent series with easily computed components and convergence quickly to the exact solution [29], [30].

Numerical Example

Example 1. Consider a Black–ScholesEq. (4)

∂v ∂τ = ∂2v ∂x2 + (k − 1)∂v ∂x − kv

with the initial condition v(x, 0) = max{ex − 1, 0}. To solve Eq. (4) using algoritmLADM, we can write

∂v ∂τ = Ψv(x, τ) (13) with Ψv = ∂ 2v ∂x2 + (k− 1) ∂v ∂x

−kv, where Ψ loads linear and nonlinear operators.

Applying Laplace transform to Eq. (13) and substituting initial condition, so

v(x, s) = max{ex − 1, 0} s + 1 s L[Ψv(x, τ)]. (14)

Furthermore, applying inverse Laplace transform to Eq.

(14), thus obtained

v(x, τ) = max{ex − 1, 0}+ L−1 [ 1 s L[Ψv(x, τ)] ] . (15)

Thus the decomposition solusiton in the form as follow

∞∑ n=0 vn(x, τ) = max{ex − 1, 0}+ L−1 [ 1 s L [ ∞∑ n=0 Ψvn ]] .

Then the recursive relation of Black-Scholes equation solution is given below

v0 = max{ex − 1, 0},

vn+1 = L −1 [ 1 s L [Ψvn] ] ;n = 0, 1, 2, · · ·

thus obtained

v1 = L −1 [ 1 s L [Ψv0] ] = L−1 [ 1 s L [kmax{ex, 0} − kmax{ex − 1, 0}] ] = L−1 [ kmax{ex, 0} − kmax{ex − 1, 0} s2 ] = kτ max{ex, 0} − kτ max{ex − 1, 0}

because ∂v1∂x = kτ max{e x, 0} − kτ max{ex, 0} = 0, so

v2 = L −1 [ 1 s L [Ψv1] ] = L−1 [ −k2 max{ex, 0}+ k2 max{ex − 1, 0} s3 ] = −1 2 (kτ)2 max{ex, 0}+ 1 2 (kτ)2 max{ex − 1, 0}

v3 = L −1 [ 1 s L [Ψv2] ] = L−1 [ k3 max{ex, 0} − k3 max{ex − 1, 0} s4 ] = 1 6 (kτ)3 max{ex, 0} − 1 6 (kτ)2 max{ex − 1, 0}

...

Hence the solution of this problem in an infinite series form given by

v(x, τ) = lim k→∞ k∑ n=0 vn(x, τ) = max{ex − 1, 0}e−kτ + max{ex, 0}(1− e−kτ ) with ex = SE , τ = σ2 2 (T − t) and k = 2r σ2 ,

where S is asset price, t is time or date, E is excercise price, T is maturity date, σ is volatility of asset price and r is interest rate. Based on the solution above, the call option price formula is obtained by the Black-Scholes equation using LADM and substitution of equation 3 is

C(S, t) = Emax { S E , 0 } e−r(T−t) + Emax { S E , 0 } (1− e−r(T−t)).

The exact solution or we called the classical Black-Scholes model for pricing call option is given in [4], [31]

C(S, t) = SN(d1)−Ee−r(T−t)N(d2)

with d1 = ln SE + ( r + σ 2 2 ) (T − t) √ T − t and d2 = d1 − σ √ T − t

where N(d) is the cumulative normal density function.

Following surfaces on Figure 1 shows the solutions v(x, τ) from Black-Scholes Eq. (4) with the initial condition. Figure 2 shows the call option price C of a stock variable S from the solution of Black-Scholes equation using LADM compared with exact solution, where an excercise price E = 5 and a risk-free interest rate r = 0.05 during a three-month option contract, even Figure 3 for six months and Figure 4 for one year.

Based on the Mean Absolute Error, each error for the different option contract periods (3 months, 6 months and 1 year), respectively is 2%, 4% and 7%. In Figures 3 and 4, it can be seen that for the option periods of 6 months and 1 year, the exact solution or classical Black-Scholes model for option prices on stock prices 4.9 and 4.8 has a jumping point, while the Black-Scholes equation solution using LADM has smoother graphics.

Example 2. Consider the following generalized Black- Scholes equation as follows

∂v ∂t + 0.08(2 + sinx)2x2 ∂2v ∂x2 + 0.06x ∂v ∂x − 0.06v = 0 (16)

with the initial condition v(x, 0) = max{x− 25e−0.06, 0}.

Using LADM, the recursive relation of generalized Black- Scholes equation solution is given below

v0 = max{x− 25e−0.06, 0},

vn+1 = L −1 L [ 0.08(2 + sinx)2x2 ∂ 2vn ∂x2 + 0.06x ∂vn ∂x − 0.06vn ] s 

thus obtained

v1 = L −1 [ L [ −0.06x+ 0.06 max{x− 25e−0.06, 0} ] s ] = L−1 [ −0.06x+ 0.06 max{x− 25e−0.06, 0} s2 ] = −0.06xt+ 0.06tmax{x− 25e−0.06, 0}

because ∂v1∂x = 0, so

v2 = L −1 [ L [ −(0.06)2xt+ (0.06)2tmax{x− 25e−0.06, 0} ] s ] = −1 2 (0.06t)2x+ 1 2 (0.06t)2 max{x− 25e−0.06, 0}

v3 = L −1 [ L [ − 12 (0.06) 3xt2 + 12 (0.06) 3t2 max{x− 25e−0.06, 0} ] s ] = −1 6 (0.06t)3x+ 1 6 (0.06t)3 max{x− 25e−0.06, 0}

...

Hence the solution of this problem in an infinite series form given by

v(x, t) = lim k→∞ k∑ n=0 vn(x, t) = max{x− 25e−0.06, 0}e0.06t + x(1− e0.06t).

Following surfaces on Figure 5 shows the solutions v(x, t) from generalized Black-ScholesEq. (16) with the initial condition.

Conclusion

The Laplace Adomian decomposition method shows an effective and easy to use the technique in solving partial differential equations, especially in this paper for solving the Black-Scholes equation. This method also produces nu- merical solutions that converge quickly. The solution Black- Scholes equation using LDAM is compared with the exact solution show that the solution is accurate, have a small error and smoother than the exact solution.

Acknowledgment

Acknowledgments are conveyed to the Director General of Higher Education of the Republic of Indonesia, and Chan- cellor, Director of the Directorate of Research, Community Engagement and Innovation, and the Dean of the Faculty of Mathematics and Natural Sciences, Universitas Padjadjaran, who have provided the Master Thesis Research Grant. This grant is intended to support the implementation of research and publication of master students with contract number: 2892/UN6.D/LT/2019.

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Cham, Switzerland: Springer International Publishing, 2019. Ira Sumiati was born in Bandung, West Java, on March 3rd, 1996. She is currently a Magister student of Mathematics Program at the Universitas Padjadajran, Indonesia since 2018. She received her Bachelor in Mathematics from the same university, in 2018. Her current research focuses on fractional, mathematical analysis and financial. Endang Rusyaman was born in Tasikmalaya, West Java, on April 8th 1961. He obtained bachelor degree in Mathematics from Padjadjaran University, Bandung. Subsequently, he continued his master study on Mathematical Analysis at Bandung Institute of Technology.

In 2010, he completed his Doctoral study on Mathematical Analysis at Padjadjaran University Bandung. Currently, he is working as a lecturer and researcher at Department of Mathematics, Faculty of Mathematics and Natural Science, Padjadjaran University. His research interest is mathematical analysis, particularly fractional differential equation. He already published several publications in international scientific journals, such as in Journal of Physics entitled The Convergence Of The Order Sequence And The Solution Function Sequence On Fractional Partial Differential Equation and in Advances in Social Science, Education and Humanities Research, entitled Fractional Differential Equation as a Models of Newton Fluids for Stress and Strain Problems. Sukono (Member), was born in Ngawi, East Java, Indonesia on April 19, 1956. Master’s in Actuarial Sciences at Instutut Teknologi Bandung, Indonesia in 2000, and PhD in Financial Mathematics at the Universitas Gajah Mada, Yogyakarta Indonesia in 2011.

The current work is the Chairman of the Masters Program in Mathematics, Faculty of Mathematics and Natural Sciences, Universitas Padjadjaran, Bandung Indonesia. The research is in the field of financial mathematics and actuarial science. One previous publication is titled of Credit Scoring for Cooperative of Financial Services Using Logistic Regression Estimated by Genetic Algorithm on International Journal of Applied Mathematical Sciences, Vol. 8, 2014, no. 1, 45 57. Dr. Sukono is a member of Indonesian Mathematical Society (IndoMS), member of Indonesian Operations Research Association (IORA), and in IAENG is a new member has been received on February 2016. Aceng Sambas is currently a Lecturer at the Muhammadiyah University of Tasikmalaya, Indonesia since 2015. He received his M.Sc in Mathematics from the Universiti Sultan Zainal Abidin (UniSZA), Malaysia in 2015. His current research focuses on dynamical systems, chaotic signals, electrical engineering, computational science, signal processing, robotics, embedded systems and artificial intelligence.