Int. J. Math. And Appl., 7(3)(2019), 23–27

ISSN: 2347-1557

Available Online: http://ijmaa.in/Applications•ISSN:234

7-15

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International Journal ofMathematics And its Applications

Analytical Solution of Fractional BSM Differential

Equation for ML-Payoff Function Using GDTM

S J Ghevariya∗

Department of Mathematics, Sardar Patel University, Vallabh Vidyanagar-388120, Gujarat, India.

Abstract: This paper contributes to the analytical solution of fractional Black-Scholes-Merton (BSM) differential equation to obtainEuropean option pricing formula for Modified Log-payoff (ML-Payoff) function, max{S ln

(SK

), 0} using Generalized Dif-

ferential Transform Method (GDTM). It turns out that the BSM formula for ML-Payoff function using GDTM is quiteclose to the closed form solution of BSM model for ML-Payoff function.

MSC: 91B25

Keywords: BSM differential equation, Fractional derivative, Generalized Differential Transform Method, ML-Payoff functions.

c© JS Publication.

1. Introduction

In the last more than five decades, finance is one of the fastest developing branch in the corporate world. There are various

option pricing formulas have been used in the financial markets. Many researchers solved Black-Scholes-Merton (BSM)

differential equation to derive BSM option pricing formulas for the various payoff functions [3, 8]. This indicates the

demand of exact or closed form solution of BSM differential equation. There are many methods for pricing options; namely

Finite difference, Fast Fourier transform, Mellin transform, Projected Differential Transform Method (PDTM), etc. The

closed form solution of BSM differential equation for the ML-payoff function max{S ln(SK

), 0)} has been derived [2]. Note

that ML-Payoff function is a realistic modification of Paul Wilmott’s log payoff function [13, P.149]. The famous BSM model

is for plain vanilla payoff function. The analytical solution of fractional BSM differential equation for plain vanilla payoff

function has been derived using Generalized Differential Transform Method (GDTM) [14]. The BSM model for ML-Payoff

function is quite close to the BSM model for plain vanilla payoff function [4]. In section-2, definitions of fractional derivative

and GDTM of function of two variables and some of its properties have been discussed. In section-3, analytical solution of

fractional BSM differential equation for ML-Payoff function has been derived through GDTM. In section-4, the comparisons

of values of BSM models using GDTM and closed form solutions of BSM models have been discussed.

2. Generalized Differential Transform Method

In this section, we mention definitions of fractional derivative of function of one and two variables as well as GDTM. Also

some of the properties of GDTM are mentioned which will be used in the next section.

∗ E-mail: sjghevaria@gmail.com

23

Analytical Solution of Fractional BSM Differential Equation for ML-Payoff Function Using GDTM

Definition 2.1 ([9]). A function, g : R+ → R is in Cν , ν ∈ R, if there exists a number q > ν such that g(x) = xqg1(x),

where g1(x) ∈ C[0,∞). Further it is in Cnν , if fn ∈ Cν , where n ∈ N.

Definition 2.2 ([10]). Let g ∈ Cn−1. Then Caputo fractional derivative of g of order α, where n− 1 < α ≤ n is given by

Dαg(x) =1

Γ(n− α)

∫ x

0

(x− t)n−α−1fn(t)dt,

Further the Caputo time fractional derivative (see [1]) of u(x, t) of order α > 0 is defined as

∂αv(x, t)

∂tα=

1

Γ(n−α)

∫ t0

(t− τ)n−α−1 ∂nu(x,τ)∂τn

dτ, n− 1 < α < n

∂nu(x,t)∂tn

, n = α ∈ N.

Now we give definitions and properties of GDTM for functions of two variables. Consider a function of two variables v(x, τ)

which can be written as product of two single valued functions, v(x, τ) = f(x)g(τ) then we have

v(x, τ) =

∞∑k=0

Fα(k)(x− x0)kα∞∑h=0

Gβ(h)(τ − τ0)hβ

=

∞∑k=0

∞∑h=0

Vα,β(k, h)(x− x0)kα(τ − τ0)hβ , (1)

where 0 < α, β ≤ 1, Vα,β(k, h) = Fα(k)Gβ(h) is called spectrum of v(x, τ). The generalized differential transform (GDT) of

v(x, τ) is given by

Vα,β(k, h) =1

Γ(αk + 1)Γ(βh+ 1)[(Dα

x0)k(Dβτ0)hv(x, τ)](x0,τ0),

where (Dαx0)k = Dα

x0 ·Dαx0 ·D

αx0 · · ·D

αx0 (k-times).

The following are some properties for GDTM of function of two variables. Let Uα,β(k, h), Vα,β(k, h) and Wα,β(k, h) be GDT

of functions u(x, τ), v(x, τ) and w(x, τ), respectively, then for any c1, c2 ∈ R and polynomial p(x), we have

(i). If u(x, τ) = c1v(x, τ) + c2w(x, τ), then Uα,β(k, h) = c1Vα,β(k, h) + c2Wα,β(k, h)

(ii). If u(x, τ) = Dβτ0v(x, τ), where 0 < β ≤ 1, then Uα,β(k, h) = Γ(β(h+1)+1)

Γ(βh+1)Vα,β(k, h+ 1)

3. BSM Models for ML-Payoff Functions Through GDTM

In this section, we derive analytical solution of fractional BSM partial differential equation for ML-Payoff through GDTM.

Note that the call option will be exercised, when it is in the money (S > K), while the put option will be exercised, if it is

out of the money (S < K).

Theorem 3.1. The analytical solution of fractional BSM differential equation for ML-payoff function, max{S ln( SK

), 0}

using GDTM is given by

C(S, t) = S

[ln( SK

)+

(r + σ2

2)

Γ(α+ 1)(T − t)

], if S ≥ K.

Proof. The BSM partial differential equation with the boundary conditions for a European call option C(S, t) is given by

∂C

∂t+

1

2σ2S2 ∂

2C

∂S2+ rS

∂C

∂S− rC = 0 (2)

24

S J Ghevariya

with C(0, t) = 0, C(S, t) → S when S → ∞ and C(S, T ) = max{Sln( SK

), 0}, where S is the value of asset price, r is the

risk free interest rate, σ is the volatility of asset price, T is the maturity date and K is the striking price. Taking S = Kex,

t = T − τ12σ2 , C(S, t) = Kv(x, τ), the Equation (2) becomes

∂v

∂τ=∂2v

∂x2+ (p− 1)

∂v

∂x− pv. (3)

where p = r12σ2 . Also C(S, T ) = max{S ln( S

K), 0} imply

v(x, 0) = max{xex, 0}. (4)

Now the time fractional BSM differential equation of equation (3) will be

∂βv

∂τβ=∂2v

∂x2+ (p− 1)

∂v

∂x− pv. (5)

where 0 < β ≤ 1. Now from equations (4) and (5) along properties of GDTM with α = 1, gives

Γ(β(h+ 1) + 1)

Γ(βh+ 1)V1,β(k, h+ 1) = (k + 1)(k + 2)V1,β(k + 2, h) + (p− 1)(k + 1)V1,β(k + 1, h)− pV1,β(k, h) (6)

and

V1,β(x, 0) = max{xex, 0}, (7)

where h = 0, 1, 2... By equating series from equation (7) with equation (1), we get

V1,β(0, 0) = 0 and V1,β(k, 0) =1

(k − 1)!, k ∈ N. (8)

From equations (6) and (8), we get

V1,β(k, 1) =p+ 1

k! Γ(β + 1)and V1,β(k, h) = 0, (9)

where k ∈ N ∪ {0} and h ∈ N. Now from equations (9) and (1), we get

v(x, τ) = ex(x+

p+ 1

Γ(β + 1)τ

)

But S = Kex, t = T − τ12σ2 , C(S, t) = Kv(x, τ), the equation (3) reduces to

C(S, t) = S

[ln( SK

)+

(r + σ2

2)

Γ(β + 1)(T − t)

], if S ≥ K.

This completes the proof.

By similar argument, we can derive the analytical solution of fractional BSM differential equation for the ML-Payoff function,

max{S ln(KS

), 0} through GDTM which is given in the next theorem. Further, the closed form solutions for ML-Payoff

functions are given.

Theorem 3.2. The analytical solution of fractional BSM differential equation for ML-payoff function, max{S ln(KS

), 0}

using GDTM is given by

P (S, t) = S

[ln(KS

)−

(r + σ2

2)

Γ(β + 1)(T − t)

], if S < K.

25

Analytical Solution of Fractional BSM Differential Equation for ML-Payoff Function Using GDTM

Theorem 3.3 ([2, Theorem 2.1]). The closed form solutions of BSM differential equation for ML-payoff functions are given

by

C1(S, t) = S

[η(d1)σ

√T − t+

(ln( SK

)+(r +

1

2σ2)

(T − t))N(d1)

]and

P1(S, t) = S

[η(d1)σ

√T − t−

(ln( SK

)+(r +

1

2σ2)(T − t))N(−d1)

],

where

d1 =ln( S

K) + (r + 1

2σ2)(T − t)

σ√T − t

, η(d1) =1√2π

e−d212 and N(x) =

∫ x

−∞η(x)dx.

4. Comparisons

In this section, we compare the values of BSM models for ML-payoff functions from GDTM with exact values from closed

form solutions for different values of β, namely β = 0.5 and β = 1. Now we fix the striking price K = 100, the maturity

time T = 0.5 and the risk free interest rate r = 0.08; using these, we draw the graphs of call and put option values in the

money and out of the money, respectively, against asset prices and volatility.

0.2

0.3

0.4

0.5

s

110

120

130

S

10

20

30

40

50

C

0.2

0.3

0.4

0.5

s

70

80

90

S

-10

0

10

20

P

� C with GDTM (β = 0.5) � P with GDTM (β = 0.5)

� Exact values � Exact values

Graph 1 Graph 2

0.2

0.3

0.4

0.5

s

110

120

130

S

10

20

30

40

50

C

0.2

0.3

0.4

0.5

s

70

80

90

S

0

10

20

P

� C with GDTM (β = 1.0) � P with GDTM (β = 1.0)

� Exact values � Exact values

Graph 3 Graph 4

5. Conclusion

From the comparisons of all the graphs for different values of β, we can conclude that the values from GDTM are very close

to the values from closed form solutions. So the GDTM is quite effective and simple method to get accurate solution of

fractional BSM differential equation for ML-Payoff functions.

26

S J Ghevariya

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