key: cord-031396-cb97rcbk
authors: Saratha, S. R.; Sai Sundara Krishnan, G.; Bagyalakshmi, M.; Lim, Chee Peng
title: Solving Black–Scholes equations using fractional generalized homotopy analysis method
date: 2020-09-04
journal: Comp
DOI: 10.1007/s40314-020-01306-4
sha: 
doc_id: 31396
cord_uid: cb97rcbk

This paper aims to solve the Black–Scholes (B–S) model for the European options pricing problem using a hybrid method called fractional generalized homotopy analysis method (FGHAM). The convergence region of the B–S model solutions are clearly identified using h-curve and the closed form series solutions are produced using FGHAM. To verify the convergence of the proposed series solutions, sequence of errors are obtained by estimating the deviation between the exact solution and the series solution, which is increased in number of terms in the series. The convergence of sequence of errors is verified using the convergence criteria and the results are graphically illustrated. Moreover, the FGHAM approach has overcome the difficulties of applying multiple integration and differentiation procedures while obtaining the solution using well-established methods such as homotopy analysis method and homotopy perturbation method. The computational efficiency of the proposed method is analyzed using a comparative study. The advantage of the proposed method is shown with a numerical example using the comparative study between FGHAM and Monte Carlo simulation. Using the numerical example, analytical expression for the implied volatility is derived and the non-local behavior is studied for the various values of the fractional parameter. The results of FGHAM are statistically validated with the exact solution and the other existing computational methods.

European option valuation under transaction costs. Section 5 presents an analysis of the non-linear generalized fractional B-S equation governing European option valuation under transaction costs. Section 6 presents an analysis of fractional B-S option pricing equation and also compares with Monte Carlo simulation. Section 7 validates the statistical analysis for the above numerical examples. Section 8 draws the concluding remarks and presents the suggestions for future research.

This section discusses some basic definitions of fractional calculus used in this study.

Definition 2.1 The Riemann-Liovillie fractional integrals (Samko et al. 1993 ) of the left and right sides are defined for any function φ(x) ∈ L 1 (a, b) as:

Definition 2.2 The Riemann integral on the half axis (Samko et al. 1993) subjects to a variable limit which can be expressed as:

The left-and right-handed Riemann-Liouville fractional derivatives (Samko et al. 1993 ) of order α, 0 < α < 1, in the interval [a, b] are defined as:

Definition 2.4 The Caputo fractional derivative of order α is defined as:

Definition 2.5 The Mittag-Leffler function, which is a generalization of the exponential function, is defined as:

where α ∈ C, R(α) > 0.

The continuous function f : R → R, t → f (t) has a fractional derivative of order kα. For any positive integer k and for any α, 0 < α < 1, the Taylor series of fractional order is given by:

where (1 + αk) = (αk)! .

, y(0) = 0 is given by:

This section introduces the fractional G-transform and discusses some of the properties proposed by Saratha et al. (2020) .

Definition 2.7 Let g(t) be any time-domain function defined for t > 0. Then, the fractional G-transform of order α of g(t) is denoted by G α [g(t) ] and is defined as:

where E α is the Mittag-Leffler function.

The fractional G-transform satisfies the following properties:

If the fractional-order Laplace Transform of a function g(t) is L α {g(t)} = F α (s), then the fractional G-transform of order α of g(t) is:

Theorem 2.2 (Scaling property) If G α [g(t)] = H α (u), then:

where a is a non-zero constant.

Theorem 2.3 If G α [g(t)] = H α (u), then:

Theorem 2.4 If G α [g(t)] = H α (u), then:

The systematic procedure for the FGHAM (Saratha et al. 2020 ) is given in the next section.

Consider a fractional time non-linear partial differential equation with the following initial condition:

where D α is the fractional differential operator D α = ∂ α ∂t α , R is the linear differential operator , N is the non-linear differential operator, and g(x, t) is the source term. The following systematic procedure steps are used to solve the non-linear fractional differential equations:

Step 1: Using fractional G-transform, Eq. (3.1) is transformed to:

(3.2)

Step 2: Applying the derivative property of fractional G-transform, Eq. (3.2) is expressed as:

Step 3: Decomposing the non-linear terms in Eq. (3.3), the following homotopy is constructed:

where s ∈ [0, 1] is an embedding parameter and φ(x, t; s) is a real function of x, t, and s, while h is a non-zero auxiliary parameter, H (x, t) = 0 is an auxiliary function, v 0 (x, t) is an initial guess of v(x, t), and φ(x, t; s) is an unknown function. Equation (3.4) is called the zeroth-order deformation equation. In (3.4), if s = 0 and s = 1, then φ(x, t, 0) = v 0 (x, t) and φ(x, t, 1) = v(x, t), respectively. If s ∈ [0, 1], then the solution is transferred from v 0 (x, t) to v(x, t).

Step 4: Deriving the n th -order deformation equation in (3.5) as in (Liao 1992) :

(3.5)

Step 5: Using the Inverse G-transform on both the sides of Eq. (3.5), the following equation is obtained:

(3.7) and χ n = 0 n ≤ 1 1 n > 1.

Step 6: The following solution is obtained:

(3.8)

It is essential to evaluate the convergence of the series solution obtained in Eq. (3.8) by FGHAM as in Bagyalakshmi et al. (2016) . The approximate solution of (3.1) is obtained as v app(k) (x, t) = k n=0 v n (x, t) from (3.8) by truncating the terms for n = k + 1, k + 2, . . . ∞. Then, the exact solution of Eq. (3.1) is represented as:

where ev k (x, t) is the error function. Generally, the absolute error is defined as ev k (

To establish convergence of equation (3.8), it is necessary to show that the sequence Ev k (x, t) is a convergent sequence. Since the sequence is bounded below, it is sufficient to prove that the sequence Ev k (x, t) is monotonically decreasing. As such, the convergence criteria are

Ev k (x,t) < 1 for k < p. Using the following algorithm, convergence of the iterative solution v app(k) (x, t) to the exact solution v(x, t) is shown below:

The following section solves the fractional B-S equation and generalized fractional B-S equation using FGHAM. The obtained solutions are compared with those of the exact solution along with statistical validation. The results indicate an excellent agreement with some existing methods.

Considering the fractional B-S equation:

with the initial condition: v(x, 0) = max(e x − 1, 0). Equation (4.1) contains parameter k = 2r σ 2 , where k represents a balance between the interest rate and variability of stock returns, with the dimensionless time to expiry σ 2 2T . The other four dimensionless parameters are the exercise price E, expiry T , volatility of the underlying asset σ 2 , and risk-free interest rate r as in the original problem. Applying the fractional G-transform on both the sides of Eq. (4.1):

Solving the above equation for n = 1, 2, 3, . . .:

Similarly, v 4 , v 5 , . . . are estimated and the series solution is obtained, that is:

If h = −1, Eq. (4.2) can be expressed as: Table 1 shows the absolute errors subject to some particular points α = 1 and x = 0.5. This proves the convergence of the series solution of (4.1). Figure 1 depicts a comparison of the absolute errors for the different sequences of partial sums. 

The convergence region is obtained using the h-curve. Figure 2 shows the convergence region of Eq. (4.1) between − 2 and 1. Figure Figure 5 depicts the financial pricing derivatives subjects to the different settings of the fractional parameter α = 0.25, 0.5, 0.75 and 1. Table 2 provides the pricing option derivatives using the fractional parameter α = 1, which depicts a good agreement with the results of FGHAM, exact solution, RPS, and CFADM, respectively. Table 3 and 4 provide the pricing option derivatives using the fractional parameter α = 0.75, α = 0.5, and depict a good agreement with the results of FGHAM, MFDTM, RPS, and CFADM, respectively. Case 2. Consider the Vanilla call option (Company et al. 2008 ) with parameter σ = 0.2, r = 0.01,α = 1, τ = 1 year, then k = 5. The solution of equation (4.6) is obtained as:

(4.6) Equation (4.6) is the exact solution for the given equation. Table 5 shows the absolute errors with respect to some particular points α = 1 and x = 1. This proves the convergence of the series solution of (4.1). Figure 6 shows a comparison of the approximate absolute errors subject to the different sequences of partial sums. 

The convergence region is obtained using the h-curve. Figure 7 shows that the convergence region of Eq. (4.1) is between − 2 and 2. Figure 8 indicates that the FGHAM results almost coincide with those of HAM, HPM, MFDTM, RPS, CFADM, and the exact solution for the B-S equation. Figure 9 illustrates the solution for the B-S equation subject to the various settings of the fractional parameter α = 0.25, 0.5, 0.75, 1,.

Figures 10 depicts the financial pricing derivatives for the different settings of the fractional parameter α = 0.25, 0.5, 0.75 and 1, respectively. Table 6 provides the pricing option derivatives using fractional parameter α = 1, depicts a good agreement among the results of FGHAM, the exact solution, RPS, and CFADM respectively.

Tables 7 and 8 provide the pricing option derivatives using fractional parameter α = 0.75, α = 0.5, and depict a good agreement among the results of FGHAM, MFDTM, RPS, and CFADM, respectively.

The implied volatility, an important financial parameter, which plays a vital role in pricing option problem. Generally, due to the mathematical structure of the integer order B-S formula, Table 2 Comparison of the results for the B-S equation using the fractional parameter Table 3 Comparison of the results for the B-S equation using the fractional parameter Table 4 Comparison of the results for the B-S equation using the fractional parameter the analytical expression to estimate the implied volatility cannot be obtained in the closed form. However, it is worth to mention that the presence of fractional parameter α in the fractional B-S mathematical model has an advantage of memory-less property. Thus, the non-local behavior of the implied volatility in terms of option can be analytically estimated in the closed form for the various values of the fractional parameter α.

Comparison of the results for the B-S equation using fractional parameter 

Comparison of the results for the B-S equation using fractional parameter The implied volatility function σ (x, t) with respect to the fractional parameter α is denoted by σ α (x, t) and can be derived from (4.1) as shown below: (Dadachanji 2015) .

(4.7)

Using Eq. (4.7), the implied volatility σ α is estimated for the various values of fractional parameter α and the results are shown in Tables 9 and 10. Even though the implied volatility is assumed to be a constant, the observation from the Tables 9 and 10 clearly reveals the pre-local change in the implied volatility. This result illustrates the importance of the fractional-order B-S model over the integer order in analyzing the non-local behavior of the implied volatility.

Even though the B-S model is considered with the constant volatility due to the presence of the fractional parameter α, the non-local behavior of implied volatility [α ∈ (0, 1)] is observed and the results are presented in Tables 9 and 10.

Considering the following generalized B-S equation:

Applying the fractional G-transform on both the sides of Eq. (5.1):

Solving the above equation for n = 1, 2, 3, . . .

Similarly, v 4 , v 5 , . . . are estimated and the series solution is obtained as÷: Table 11 shows the absolute errors with respect to some particular points α = 1 and x = 5. This proves the convergence of series solution of Eq. (5.1). Figure 11 depicts a comparison of the approximate absolute errors subject to the different sequence of the partial sums. 

The convergence region is obtained using the h-curve. Figure 12 shows that the convergence region of equation (4.1) is between -2 and 2. Figure 13 indicates that the FGHAM results coincide with those of HAM, HPM, MFDTM, RPS, CFADM, and the exact solution v(x, t) of the B-S equation. Figure 14 illustrates the solution v(x, t) for the B-S equation subject to the various settings of the fractional parameter α = 0.25, 0.5, 0.75, 1, respectively. Figure 15 depicts the information on the financial pricing derivatives using different settings of the fractional parameter α = 0.25, 0.5, 0.75 and 1, respectively. Table 12 provides the pricing option derivatives using fractional parameter α = 1, which depicts a good agreement among the results of FGHAM, the exact solution, RPS, and CFADM, respectively.

Tables 13 and 14 provide the pricing option derivatives subject to the fractional parameter α = 0.75, α = 0.5, which depict a good agreement among the results of FGHAM, MFDTM, RPS, and CFADM, respectively.

Considering the following fractional Black-Scholes option pricing equation:

subject to the initial condition:

v(x, 0) = max (Ax − B, 0) .

Applying the fractional G-transform on both the sides of the equation (6.1):

Comparison of the results for the B-S equation using fractional parameter 

Comparison of results for the B-S equation using fractional parameter 

Comparison of the results for the B-S equation using the fractional parameter 

Solving the above equation for n = 1, 2, 3, . . 

Similarly, v 4 , v 5 , . . . are estimated and the series solution is obtained as:

If h = −1, Eq. (6.2) can be expressed as:

] + · · · (6.3)

The convergence region is obtained using the h-curve. Figure 16 shows that the convergence region of equation is between − 4 and 1. Figure 17 indicates that the solution v(x, t) for the B-S equation subject to the various settings of the fractional parameter α = 0.25, 0.5, 0.75 and 1, respectively. Figure 18 depicts the financial pricing derivatives for the different settings of the fractional parameter α = 0.25, 0.5, 0.75 and 1, respectively.

To verify the accuracy of the proposed method, the numerical results of the call option obtained using FGHAM for α = 1 which is compared with the results estimated using Monte Carlo simulation (MATLAB) and the results are shown in Table 15 . From Table 15 , it is evident that the numerical results obtained by FGHAM show an excellent agreement with the results estimated using Monte Carlo simulation.

The statistical significance pertaining to the difference among the mean results obtained by FGHAM, exact solution, RPS, and CFADM for the B-S equation using fractional parameter 

The FGHAM approach has been successfully applied to solve the fractional non-linear B-S equation governing European option pricing. Using various plots of h-curves, convergence region of the solution is identified and closed form series solutions are obtained using Mittag-Leffler function, which clearly reveals the financial process. The suitable conver- ciency of the proposed method is verified. The analytical expression for the implied volatility is derived and the non-local behavior is studied for the various values of the fractional parameter. The statistical analysis has been carried out using the solutions obtained from the various methods to ascertain the effectiveness of the FGHAM approach and to tackle the B-S pricing model in financial studies. From the experimental analysis, it is evident that the proposed method accurately predicts the solution of the B-S model. The application of FGHAM can be extended to solve the B-S model with the time variable coefficients to analyze the financial process in future. Moreover, FGHAM can be applied to solve different problems, such as the Navier-Stokes equation, epidemic models, and Pandemic model like COVID-19. The application of FGHAM can also be extended to analyze complicated non-linear differential equations and fractional differential equations that arise in different fields of science and engineering.

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