Applied Mathematics E-Notes, 13(2013), 243-248 c
Available free at mirror sites of http://www.math.nthu.edu.tw/ amen/
ISSN 1607-2510
Analytical Solution Of The Black-Scholes Equation
By Using Variational Iteration Method
Hamid Rouhparvary, Mehdi Salamatbakhshz
Received 11 September 2013
Abstract
In this paper, we shall use the variational iteration method to solve the BlackScholes equation and will obtain a closed form of the solution. In this method,
the problems are initially approximated with possible unknowns, then a correction
functional is constructed by a general Lagrange multiplier, which can be identi…ed
optimally via the variational theory.
1
Introduction
Options are widely used on markets and exchanges. The pricing of options is a central
problem in …nancial investment. The famous Black-Scholes model [9, 12] is a convenient
way to calculate the price of an option. The equation assumes the existence of perfect
capital markets and the security prices are log normally distributed or, equivalently,
the log-returns are normally distributed. To these, one adds the assumptions that
trading in all securities is continuous and that the distribution of the rates of return is
stationary. The linear parabolic partial di¤erential equation of Black-Scholes equation
for valuing an option with value u is
2
ut +
2
x2 uxx + (r
)xux
ru = 0;
(1)
where r is the risk-free rate,
is the volatility, and
is the dividend yield. The
numerical solution of this equation has been of paramount interest due to the governing
partial di¤erential equation, which is very di¢ cult to generate stable and accurate
solutions. In this paper, we research the equation (1) with a view to obtaining the
analytical solution to the terminal condition
u(x; T ) = g(x);
(2)
by using the variational iteration method (VIM) [1, 2, 4, 7, 8, 10, 11]. We suppose
throughout that g has derivatives of all orders.
Mathematics Subject Classi…cations: 35K10.
of Mathematics, Saveh Branch, Islamic Azad University, Saveh 39187-366, Iran.
z Department of Mathematics, Saveh Branch, Islamic Azad University, Saveh 39187-366, Iran.
y Department
243
244
2
Analytical Solution of the Black-Scholes Equation
Variational Iteration Method
In this method, the solution of a di¤erential equation with a linearization assumption
is used as an initial approximation, then a more precise approximation at some special
point can be obtained. To illustrate this method, consider the di¤erential equation in
the formal form
Lu(x; t) + N u(x; t) = g(x; t);
(3)
where L is a linear operator, N is a nonlinear operator, and g(x; t) is an inhomogeneous
term.
According to the VIM, we can construct a correctional functional as
Z t
un+1 (x; t) = un (x; t) +
(Lun (x; s) + N u
~n (x; s) g(x; s)) ds;
(4)
0
where is a general Lagrangian multiplier [5, 6], and can be identi…ed optimally via
the variational theory, the subscript n denotes the n-th order approximation, u
~n is
considered as a restricted variation [5, 6], i.e., u
~n = 0. Eq. (4) is called a correction
functional. The successive approximations un+1 , n 0, of the solution u will be readily
obtained by suitable choice of trial function u0 . Consequently, the solution is given as
u(x; t) = lim un (x; t):
n!1
3
(5)
Applications
In this section, we illustrate the proposed method to Black-Scholes equation.
3.1
Case I
According to the VIM, we construct a correction functional for Eq. (1) in the form
Z T
2
x2 (~
un )xx + (r
)x(~
un )x r(~
un ) ds; (6)
un+1 (x; t) = un (x; t) +
(un )s +
2
t
where is a general Lagrange multiplier, and u
~ denotes the restricted variation, i.e.,
u
~ = 0.
The correction functional (6)
Z T
2
x2 (~
un )xx + (r
)x(~
un )x r(~
un ) ds
un+1 (x; t) = un (x; t) +
(un )s +
2
t
Z T
= un (x; t) +
(un )s ds
t
=
un (x; t)
un (x; s)js=t
Z
T
0
t
yields the following stationary condition
0
1
(s) = 0;
(s)js=t = 0:
un (x; s)ds = 0;
H. Rouhparvar and M. Salamatbakhsh
245
Therefore, the Lagrange multiplier is
(s) = 1:
Substituting this value of the Lagrange multiplier into the functional (6) gives the
iteration formula
Z T
2
x2 (un )xx + (r
un+1 (x; t) = un (x; t) +
(un )s +
)x(un )x r(un ) ds: (7)
2
t
We start with an initial approximation: u0 (x; t) = g(x), and using the iteration formula
(7), we obtain the following successive approximations
u1 (x; t) = g(x) + (T
u2 (x; t)
t)
= g(x) + (T
1
+ (T
8
+(4r2
+(4r
t)
8r + 4
4
2
2
2
+4
4
g (x) ;
2 00
g (x)
4 2 )xg 0 (x)
(4r2
+ 4r
2 00
1
)xg 0 (x) + x2
2
rg(x) + x(r
t)2 4r2 g(x)
2
1
)xg 0 (x) + x2
2
rg(x) + x(r
8
2
+2
)x3 g (3) (x) + x4
..
.
)x2 g 00 (x)
i
4 (4)
g (x) ;
4
i.e.,
" 2k ( m
n
X
X X ( 1)(m v)
un (x; t) =
v!(m v)!
m=0 v=0
k=0
where n
u(x; t)
2
v
2
k
+ r (v
1)
v
)
m (m)
x g
(x)
#
t)k
(T
k!
;
0. Then, the exact solution is given as series
=
=
lim un (x; t)
" 2k ( m
1
X
X X ( 1)(m v)
v!(m v)!
m=0 v=0
n!1
2
2
k=0
LEMMA 1. Assume
(m
2k
X
X ( 1)(m v)
'k (x) =
v!(m v)!
m=0 v=0
v
k
+ r (v
1)
v
)
m (m)
x g
#
(x)
t)k
(T
k!
(8)
2
2
v
k
+ r (v
1)
v
)
xm g (m) (x):
The de…ned function 'k (x) satis…es the recursion
2
2
x2 '00k (x) + (r
)x'0k (x)
:
r'k (x) = 'k+1 (x) for all k 2 N0 :
(9)
246
Analytical Solution of the Black-Scholes Equation
PROOF. See [Lemma 3.1, 3].
THEOREM 1. The function u(x; t) de…ned by
1
X
u(x; t) =
'k (x)
t)k
(T
k!
k=0
;
(10)
satis…es equation (1).
PROOF. Substituting u(x; t) into the equation (1) gives
2
ut +
=
1
X
(r
k=0
1
X
=
k=1
1
X
=
=
x2 uxx + (r
'k (x)
k=0
1
X
+
2
k=1
1
X
3.2
0:
t)(k
k!
)x'0k (x)
'k (x)
'k (x)
'k (x)
k=1
=
k(T
)xux
ru
1)
+
1
X
k=0
t)k
(T
k!
2
2
1
X
x2 '00k (x)
r'k (x)
k!
t)k
(T
k!
k=0
1
t)k
(T
(T t)(k 1) X 2 2 00
+
( x 'k (x) + (r
(k 1)!
2
(T t)(k 1)
+
(k 1)!
(T t)(k 1)
+
(k 1)!
k=0
1
X
k=0
1
X
'k+1 (x)
'k (x)
k=1
)x'0k (x)
r'k (x))
t)k
(T
k!
t)k
(T
k!
(T t)(k 1)
(k 1)!
Case II
According to the VIM, we construct a correction functional for Eq. (1) in the following
form
Z T
2
un+1 (x; t) = un (x; t) +
(un )s run +
x2 (~
un )xx + (r
)x(~
un )x ds: (11)
2
t
The correction functional for (11) is
Z T
un+1 (x; t) = un (x; t) +
(un )s
2
run +
2
t
=
un (x; t) +
Z
un (x; t)
)x(~
un )x ds
T
((un )s
t
=
x2 (~
un )xx + (r
un (x; s)js=t
run ) ds
Z
t
T
0
(s)
r (s)
un (x; s)ds = 0;
H. Rouhparvar and M. Salamatbakhsh
247
which yields the following stationary condition
0
(s) + r (s)
1
(s)js=t
=
0;
=
0:
Hence, the Lagrange multiplier is
(s) = er(s
t)
:
Substituting this value of the Lagrange multiplier into the functional (11) gives the
iteration formula
Z T
2
x2 (un )xx + (r
)x(un )x r(un ) ds:
un+1 (x; t) = un (x; t) +
er(s t) (un )s +
2
t
(12)
We start with an initial approximation: u0 (x; t) = g(x), and using the iteration formula
(12), we obtain the following successive approximations
1
u1 (x; t) = g(x) + (er(T
r
u2 (x; t)
t)
1)
2
r
= g(x)
1
er(T
t)
1
+ r(T
2
)xg 02
rg(x) + x(r
+
1
)xg 0 (x) + x2
2
rg(x) + x(r
2
2
t)er(T
+(4r
8r + 4
2
4
2
2
+4
+ 4r
2
4
3 (3)
)x g
g (x) ;
t)
g 00 (x)
1
1 er(T t) + r(T t)er(T
4r2
[4r2 g(x) (4r2 4 2 )xg 0 (x)
+(4r2
2 00
8
2
t)
+2
(x) + x
4
)x2 g 00 (x)
4 4 (4)
g
(x)];
..
.
where un (x; t) is an approximation of the solution. In fact, the cases I and II show
‡exibility of the VIM method where it can be used in solving the di¤erential equations.
4
Conclusions
In this work, the VIM is applied to obtain the solution of Black-Scholes equation in
two cases. A theoretical analysis and closed form of the solution was presented for the
Black-Scholes equation. The results clearly indicate the reliability and accuracy of the
proposed technique.
Acknowledgement. The authors would like to thank anonymous referees for their
helpful comments.
248
Analytical Solution of the Black-Scholes Equation
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