Download Full Issue in PDF - Academy Publisher

1584 JOURNAL OF COMPUTERS, VOL. 8, NO. 6, JUNE 2013
~
T
T
{ r∈Θ
}
r
P w r ≥minw r ≥ p
The optimal problem reduces to a convex second-order
cone program[18].
T
2 T
max w μ − δ Σ
1/ w 1 = 1,
l ≤ w ≤ u
w
{ }
2
According to the Central Limit Theorem, it is
^
concluded that the sample mean μ is approximately
normally distributed. That is ,it follows:
^
⎛ Σ ⎞
μ ~ N ⎜ μ,
⎟
⎝ n ⎠
Similarly, the ellipsoidal uncertainty set for the
mean μ can be expressed as
Θ
μ
−1
⎪⎧
^
⎛ Σ
^
⎛ ⎞ ⎞ ⎛ ⎞ ⎪⎫
2
= ⎨μ
: ⎜ μ − μ ⎟⎜
⎟ ⎜ μ − μ ⎟ ≤ κ ⎬
⎪⎩ ⎝ ⎠⎝
n ⎠ ⎝ ⎠ ⎪⎭
where κ = q / 1−
q for some q ∈[0,1)
The problem reduces to
⎪⎧
1/ 2
^
⎛ Σ
^ 1/ 2
⎪⎫
T ⎞
T
max⎨w
μ−κ
⎜ ⎟ w −δ
Σ w w 1 = 1, l ≤ w ≤ u⎬
w
⎪⎩
⎝ n ⎠
2
2
⎪⎭
See[19] the problem is finally reduced to
⎪⎧
^
^ 1 / 2
⎪⎫
T
1 / 2
T
max⎨w
μ − κ Ω w − δ Σ w w 1 = 1, l ≤ w ≤ u⎬
w
2
⎪⎩
2
⎪⎭
where
Σ 1 Σ T Σ
Ω = − 11
n T Σ n n
1 1
n
In this paper, firstly, we consider the uncertain set for
return mean. We define r′ is the estimation of real
value r , the uncertainly set I as
{ : i
′ i i i
′ i }
I = r r −s ≤r ≤ r + s for mean μ .
According to Anna[20], the robust counterpart:
rx ≥ r
∑
min
i i p
can be transferred to the following form:
⎧∑rx ′
i i
−∑sm i i
≥rp
⎪
⎨ mi
≥ xi
⎪
⎩ si
≥ 0
The robust counterpart of objective function is
n
m
⎛ ⎧
C P ⎫⎞
max ⎜∑⎨xr i i
+ ∑ { βik Rik + γik Rik}
⎬⎟
x, βγ ,
⎝ i= 1 ⎩ k=
1
⎭⎠
let x αβγ
is the share of stock and options.
The goal is to determine the solution of above problem
under the constraints.
Ⅲ.MONTE CARLO SIMULATION AND EMPIRICAL EXAMPLE
A. Monte Carlo Method and the Simulation Process
Comparing with other numerical methods, Monte
Carlo simulation has two major advantages: first, more
flexible, easy to implement and improvement; secondly,
the simulation of estimation error and convergence speed
in solving the problem has strong independence of
dimension. European option because of its execution time
is fixed, not to be executed in advance, therefore it only
need to calculate the earnings of the option of each
sample path at expiration date, which is available by
Matlab programming. [21-23] discuss the application of
the simulation methods in various area. Monte Carlo
method can overcome the obstacle and we use it further
to improve the accuracy of simulated price with the
enhancement of reduction variate technique for more
complex options whose payoff function is dependent on
the underlying asset path and sum of asset is more than
one.
Now, we illustrate the key steps in Monte Carlo. It is
saw that to draw samples of the terminal stock price
S( T ) it suffices to have a mechanism for drawing
samples from the standard normal distribution. For now
we simply assume the ability to produce a sequence
Z1,
Z2 of independent standard normal random
variables. Given a mechanism for generating the Z
i
, we
rT
can estimate E⎡
−
e ( ST
− K )
+ ⎤ using the following
⎣
⎦
algorithm:
For i = 1, 2n
generate
Z
i
⎛⎡
1 ⎤ ⎞
Si
T = S0
exp⎜⎢
r− σ +
2 ⎥
T σ TZi⎟
⎝⎣
⎦ ⎠
set ( )
2
C = e S −K
−rT
set ( ) +
^
i
set n = ( + + + )
T
C C C C n
1 2 n
/
For any n ≥ 1, the estimator C n is unbiased, in the
sense that its expectation is the target quantity:
^
⎛ ⎞
− rT
+
E⎜Cn
⎟= C = E⎡
⎣e ( ST
−K)
⎤
⎝ ⎠
⎦
The estimator is strongly consistent meaning that as
n →∞.
In this paper, we suppose z = z()
t is a random
process, the change in a very small time interval Δt
is
expressed as Δ z . If Δz
satisfies that Δ z = ε Δ t where
ε ∼ N ( 0,1)
. For different time interval Δ t , Δz
are
independent, then call z = z()
t follows Wiener process.
Suppose the stock price follows ds = μsdt + σ sdz ,
where dz is the Standard Brown motion. In the practical
^
© 2013 ACADEMY PUBLISHER