OPTION PRICING FOR WEIGHTED AVERAGE
OF ASSET PRICES
Hiroshi Inoue
1
1∗
, Masatoshi Miyake2†, Satoru Takahashi
1‡
School of M anagement, T okyo U niversity of Science,
Kuki-shi Saitama 346-8512, Japan
2
Department of Industrial Administration, T okyo U niversity of Science,
N oda-shi Chiba 278-8510, Japan
Abstract
Average options are path-dependent and have payoffs which depend on the average price over a fixed period leading up to the maturity date. This option is of interest and important for thinly-traded
assets since price manipulation is prohibited, and both the investor
and issuer may enjoy a certain degree of protection from the caprice
of the market. There are several nice results for the average options
with different approaches. In this paper, we consider to propose a
more general weight instead of usual simple average, for which it may
be possible to control the weights in the light of unexpected situation incurred. An analytical form of option pricing along with Monte
Carlo simulations is obtained, and a numerical example is presented.
In particular, our methodologies are based on Kemna & Vorst, Vorst
and Levy, so that some comparisons for their results are made.
Keywords. Weighted average option, Weighted sums, Conditional
expectation, Geometric average, Arithmetic average
1
INTRODUCTION
Concerning the study of option pricing F.Black, M.Sholes and R.Merton
made a major breakthrough in the past and the Black-Sholes model is still
∗
inoue@ms.kuki.tus.ac.jp, +81-480-21-7636
toshimiya1@yahoo.co.jp
‡
satoru@ms.kuki.tus.ac.jp
†
1
used in various situations, in particular, in European option pricing problems.
Average options are fully path-dependent and have payoffs which depend
on the average price of the underlying asset over a fixed period leading up to
the maturity date. Since no general analytical solution for the price of the average option is known, a variety of techniques have been developed to analyze
average options. There is historically enormous literature devoted to work of
this option. Among them Bergman(1981) studies average rate options but
only considers options with a zero strike price. Kemna and Vorst(1990,1992)
propose a Monte Carlo methodology which employs the corresponding geometric option as a control variable. Carverhill and Clewlow (1990) use the
Fourier Transformation to evaluate numerically the necessary convolutions
of density functions. Ruttiens (1990) and Vorst (1990) employing the solution to the corresponding geometric average problem improves the speed
of calculation. On the other hand, Levy(1992) proposes a more accurate
method which relies on the assumption that the distribution of sum of log
normal variables is itself well approximated at least to a first order by the log
normal. Also, the author assumes that the valuation of average option becomes possible for typical range of volatility experienced. The idea of such
options are of particular interest and importance for thinly-traded assets
since price manipulation is inhibited. This option is in general considered
path dependent options for which the payoff at maturity date depends on
the history of the prices the underlying asset takes.
In this paper, we are interested in pricing for which options payoff depends on the weighted sums of prices but not on simply the average price
of underlying assets. As Kemna and Vorst mentioned in [1] that if a standard European call option is based on an asset which remains low in price
during a large part of the final time period and rises significantly at maturity, the firm would not have been able to generate sufficient revenues to
pay the high premium to the option. In that situation a pricing method
based on standard asset values may not be sufficient so that more general
weighted average option rather than the simple average option will be seriously considered. But, as Henderson(2004) pointed out, we don’t know to
what stochastic process the average of exchange over a fixed period follows
even though the path of exchange follows Brownian motion. Therefore, it
will be needed to find probability distribution for the underlying asset and
the prices of it. Thus, we extend the results for average options to pric2
ing problems based on the weighted sums of prices. Average option has
usually three different variables as explained later so that the problem is
how to solve three dimensionally stochastic differential equations. Wilmot
at el(2002) reduce three to two dimensional stochastic differential equations
and use transformation of variables, applying Black-Sholes formula. Vecef
at el(2001) and Henderson(2004) assume that asset price follows submartingale. Then, submartingale is considered as sum of martingale and increasing
process and hence it is possible to apply Black-Sholes formula even though
the stochastic process about average of exchange is not known.
This paper is organized as follows. Section2 provides a brief description
of simple average option. In Section3 a more general weighted average option is proposed and the analytical form is obtained. Some corrections for
arithmetic and geometric valuations are made with numerical examples in
Section4. Section5 provides another valuation formula based on moments
for arithmetic weighted option. In Section6 these methodologies developed
in the previous sections are compared with numerical examples. Concluding
remarks are given in Section7.
2
SIMPLE AVERAGE OPTION
We assume that a perfect security market which is open continuously,
offers a constant riskless interest rate r to borrowers and lenders, in which
no transaction costs and /or taxes are incurred.
Let the underlying asset price S(t) at time t follow the geometric Brownian motion process
dS(t) = µS(t)dt + σS(t)dW (t),
where dW (t) stands for a Wiener process which has a normal distribution
with the mean 0 the variance dt, µ is the drift parameter and σ is the
volatility parameter.
The standard partial differential equation for the option price C can be
derived by hedging arguments([10],[11]):
1
Ct + σ 2 S 2 Css + r(SCs − C) = 0
2
where Ct , Cs are the first order partial derivatives with respect to t and S
and Css a second order partial derivative with respect to S.
3
For t ∈ [T0 , T ] consider the variable A(t) as
1
A(t) =
t − T0
∫
t
S(τ )dτ
T0
where T is the maturity date. Then A(t) is defined as the part of the
final average up to time t and the payoff on the option can be expressed
as Max (A(t) − K, 0), where K is the exercise price of the average option.
Kemna and Vorst derive the expression for the value of the average option
C(S(t), A(t), t) = e−r(T −t) E # [Max (A(t) − K, 0)]
with appropriate boundary conditions, where E # indicates the conditional
expectation with respect to S(t), A(t) and t.
Also, log value of S(τ ) after τ = T − T0 period passed is expressed as
log S(τ ) = log S(T0 ) + (r − 1/2σ 2 )τ + σW (τ )
where σW (τ ) ∼ N (0, σ 2 τ ).
There are two types of defining geometric average formulae to find the
average prices of the asset, which are continuous and discrete cases;
(
SC = exp
(
SD =
N
∏
)
∫ T
1
log S(τ )dτ
T − T0 T0
) N1
S(Ti )
i=1
Under the assumption that the asset prices follow lognormal distribution the
distribution of average price does not have desirable nature to be analyzed
but geometric average possesses, and hence some adjustment will be needed
to obtain more appropriate and reasonable option pricing. If asset prices are
taken as just simple average its variation the asset possesses are almost lost.
We are interested in the situation for which a European call option based
on an asset remains low in price during a large part of the final time period
and rises significantly at maturity. In such a circumstance the firm would
cope with this problem for some way.
4
3
A PRICING FORMULA FOR WEIGHTED AVERAGE OPTION
Thus, we want to propose more general way, in which weighted average
rather than the simple average to control its variation is considered. In this
study, we find a way to obtain the weighted sums of prices which depend on
each asset price S(T i ) at time Ti . Let the weight and the exercise price at
Ti be Wi , K, respectively. Denote by Ti = T0 + ih for i = 1, 2, · · ·, N where
h = (T − T0 ) /N . Define the weighted average
A(Tm ) =
m
∑
Wi S(Ti ) with
i=1
m
∑
Wi = 1 ,
Wi > 0, i = 1, 2, · · ·, N
i=1
for 1 < m < N . Then, the weighted average option is characterized by
the payoff function at time Tm given by Max(A(Tm ) − K, 0) for a call and
Max(K − A(Tm ), 0) for a put option.
Over a fixed contract date to the maturity date Tm we consider the
following weighted average defined below
SAWA (Tm ) =
m
∑
i=1
(1 + ah)i
∑N
k=1
S (Ti ) ,
(1 + ah)k
−∞ < a < ∞
(1)
In this expression a variable a to control the weights is incorporated and
hence it may cope with more different situations incurred by underlying
asset.
But the expression above is not suitable since each asset price S(T i ) is
assumed to be log normally distributed and hence SAWA (Tm ) is a weighted
sum of lognormally distributed. Then, SAWA (Tm ) is no longer lognormally
distributed. Therefore, for a possible way to develop an approximation another type of weighted average option needs to be considered. The option
based on geometric weighted average is
m
∏
(1 + ah)i
> 0, i = 1, 2, · · ·, N
S (Ti )wi with wi = ∑N
k
(1
+
ah)
i=1
k=1
(2)
The final payoff at maturity of a call option on the geometric average is,
denoting by CGWA the values of the option,
SGWA (Tm ) =
CGWA = e−r(T −T0 ) E # [Max (SGWA (Tm ) − K, 0)] .
5
(3)
Finally, the log value of geometric weighted average of (2) for continuous
case is defined as
∫ T
eaτ
log S (τ ) dτ,
∫T
as ds
e
T0
T0
and hence log SGWA (T ) is obtained
log SGWA (T ) =
eaT
a
− eaT0
∫
T
eaτ log S(τ )dτ .
(4)
T0
∫t
1
Remark 1 In the simple average, A(t) = T −T
S(τ )dτ for T0 ≤ t ≤ T ,
T0
0
the partial differential equation for the option price is derived by using arbitrage arguments, in which the factor relevant to A(t) has been incorporated
[1], while the case of log SGWA (T ) may not be possible. However, since the
function of option price contains the variables, S(t), A(t) we are not able to
find an explicit formula for the value of option.
Remark 2 Kemna and Vorst show that when the price is based on the
geometric average instead of the usual arithmetic average an analytic pricing
formula can be derived, to gives a lower bound for the value of a call option
based on the arithmetic average.
In finding the option prices for SAWA (T ) and SGWA (T ), since it is known
that a geometric average is always lower than an arithmetic average [14],
the prices obtained from the geometric average is usually underestimated so
that some adjustments are needed to have suitable values. Since SGWA (T )
is a product of lognormally distributed variables it is also log normally distributed with the mean
(
)
1 2
(aT − 1) eaT − (aT0 − 1) eaT0
r− σ ,
E [log SGWA (T )] = log S (T0 ) +
a (eaT − eaT0 )
2
(5)
and the variance is
V ar [log SGWA (T )] =
(2aT − 3) e2aT + (2aT0 − 1) e2aT0 − (4aT0 − 4) ea(T +T0 ) 2
σ .
2a (eaT − eaT0 )2
(6)
6
Thus, log SGWA (T ) follows normal distribution
(
(
)
(aT − 1) eaT − (aT0 − 1) eaT0
1 2
log SGWA (T ) ∼ N log S (T0 ) +
r− σ ,
a (eaT − eaT0 )
2
)
2aT
2aT0
a(T +T0 )
(2aT − 3) e
+ (2aT0 − 1) e
− (4aT0 − 4) e
2
σ .
2a (eaT − eaT0 )2
Note: When a is considered as a variable holding T − T0 constant the
expected values and variance above are obtained as a → 0.
1
E[log SGWA (T )] = log S(T0 ) +
2
(
)
1 2
r − σ (T − T0 )
2
1
V ar[log SGWA (T )] = σ 2 (T − T0 ).
3
Then, following Jarrow and Rudd[9] and Vorst[2], we obtain the values for
geometric weighted average for call and put options,
(
[
dC
e S (T0 ) N (d) − KN d −
σ
CGWA = e
×
eaT − eaT0
)]
√
(2aT − 3) e2aT + (2aT0 − 1) e2aT0 − (4aT0 − 4) ea(T +T0 )
2a
−r(T −T0 )
(7)
[
(
dC
−e S (T0 ) N (−d) + KN −d +
σ
PGWA = e
×
aT
e − eaT0
)]
√
(2aT − 3) e2aT + (2aT0 − 1) e2aT0 − (4aT0 − 4) ea(T +T0 )
2a
−r(T −T0 )
where
(
)
(aT − 1) eaT − (aT0 − 1) eaT0
1 2
d =
r− σ
a (eaT − eaT0 )
2
2aT
2aT0
(2aT − 3) e
+ (2aT0 − 1) e
− (4aT0 − 4) ea(T +T0 ) 2
∗∗
d =
σ
2a (eaT − eaT0 )2
∗
1
dC = d∗ + d∗∗
2
,
d=
log (S (T0 ) /K) + d∗ + d∗∗
√
d∗∗
Thus, we have obtained an analytic expression for CGWA which satisfies our
7
reguirements, and hence we can find a Monte Carlo estimate of the weighted
average option to make some adjustments.
Remark 3 On the other hand, the corresponding formulae for the standard
average of geometric average option without control variable a for a call and
a put become
[
(
CGA = e−r(T −T0 ) edC S(T0 )N (d) − KN
[
−r(T −T0 )
PGA = e
where
4
4.1
√
d−σ
(
−e S(T0 )N (−d) + KN
dC
1
(T − T0 )
3
√
−d + σ
)]
)]
1
(T − T0 )
3
(
)
1 2
1
r − σ (T − T0 ) + σ 2 (T − T0 )
2
6
(
)
log(S(T0 )/K) + 12 r − 12 σ 2 (T − T0 ) + 31 σ 2 (T − T0 )
√
d=
σ 13 (T − T0 )
1
dC =
2
NUMERICAL COMPUTATIONS
Adjustments For The Differences
There exist some differences in option premiums between arithmetic
weighted average and geometric weighted average in Monte Carlo simulation so that we need to find the difference, adding it to the results CGWA
which was based on (1) and (2). To proceed numerical computation the period [0, T ] is divided into n subintervals. Then, it is distributed as a normal
distribution with the mean (r − σ 2 /2)T /n, variance T /σ 2 /n. Therefore, the
random sequence S(T1 ), ..., S(Tn ) can be generated as follows
(
1
log S(Ti ) = log S(Ti−1 ) + r − σ 2
2
)
T
+σ
n
√
T
zi ,
n
where zi is assumed to be drawn from the standard normal distribution.
Then, the premium values for the arithmetic weighted sums and geometric weighted sums are given with (1) and (2)
C̃AWA = e−r(T −T0 ) E # [Max (SAWA (Tm ) − K, 0)]
C̃GWA = e−r(T −T0 ) E # [Max (SGWA (Tm ) − K, 0)].
8
(8)
Thus, by Monte Carlo simulation the values of realizations of C̃AWA , C̃GWA
are calculated. Noting that a geometric average SGWA (T ) is always less than
an arithmetic average SAWA (T ) [Beckenbach and Bellman,1971] and hence
CGWA is a lower bound for the call option value CAWA . As a result, the
following relation holds,
{
}
CAWA ≤ CGWA + e−r(T −T0 ) E # [SAWA (T )] − E # [SGWA (T )]
Therefore, the upper bound can be considered as a correction of the theoretical price for the difference between the expectations of the arithmetic
average and the geometric average.
On the other hand, the proposed approximation implies to correct the
exercise price K for the difference these two expectations [Vorst,1992].
The arithmetic average SAWA (T ) for continuous case is expressed as
SAWA (T ) =
eaT
a
− eaT0
∫
T
eaτ S(τ )dτ .
(9)
T0
With some manipulation the expected value of SAWA (T ) is derived
aS (T0 )
E [SAWA (T )] = aT
e − eaT0
#
(
e(r+a)T − e(r+a)T0
r+a
)
.
(10)
0
.
Note: As a → 0 E # [SAWA (T )] becomes S(T0 ) er(T−e
−T0 )
rT
rT
As a result, since the expected value of geometric average E # [SGWA (T )] is
the form of (5), by replacing K by K − E # [SAWA (T )] + E # [SGWA (T )] we
have
(
)
CAWA = e−r(T −T0 ) E # [Max SGWA (T ) − (K − E # [SAWA (T )] + E # [SGWA (T )]), 0 ].
In other words, the formulae for option price whose exercise price is adjusted
are obtained by
[
(
dC
∗
e S (T0 ) N (d) − K N d −
σ
CAWA = e
×
aT
e − eaT0
)]
√
2aT
2aT
a(T
+T
)
0
0
(2aT − 3) e
+ (2aT0 − 1) e
− (4aT0 − 4) e
,
2a
−r(T −T0 )
9
and
[
(
dC
∗
−e S (T0 ) N (−d) + K N −d +
σ
×
PAWA = e
aT
e − eaT0
)]
√
(2aT − 3) e2aT + (2aT0 − 1) e2aT0 − (4aT0 − 4) ea(T +T0 )
, (11)
2a
−r(T −T0 )
where dC and d are same as in (7) in the previous section, and
( (r+a)T
)
e
− e(r+a)T0
aS (T0 )
K = K − aT
e − eaT0
r+a
(
)
(
aT
(aT − 1) e − (aT0 − 1) eaT0
1 2
+ S (T0 ) exp
r− σ
a (eaT − eaT0 )
2
)
(2aT − 3) e2aT + (2aT0 − 1) e2aT0 − (4aT0 − 4) ea(T +T0 ) 2
+
σ .
4a (eaT − eaT0 )2
∗
(12)
Remark 4 In relation to Remark 3, the exercise price for standard average of geometric average option for a call and put is expressed as
erT − erT0
K = K−S(T0 )
+S(T0 ) exp
r(T − T0 )
∗
4.2
( (
)
)
1
1 2
1 2
r − σ (T − T0 ) + σ (T − T0 ) .
2
2
6
Numerical Example
First, option premiums for geometric weighted average based on section3
and section4 are compared with usual plain option when a is allowed to vary.
Let S=100, volatility=20%, non-risk rate= 0.5%, T =1year. Simulation was
performed 15000 times and observations 60 times.
It is observed from figure1 through figure3 that premium values derived
by (7) show some changes, according as the variable a varies for different
relationships of S and K.
For S > K the premium values for geometric average for a call and a
put are 7.3298, 2.4376, respectively, while those for plain options are 10.788
and 5.314, showing much difference for the both results. Also, note that
the values for geometric weighted average, at a = 0, become equivalent to
that for simple geometric average and become equivalent to that of plain
option as a → +∞. For S = K the similarity of the results are observed
except that the premium value for call is slightly higher than that for put
at any value of a as it is expected, and the opposite figure is also obtained
for S < K.
10
Figure 1: Premium values for S > K(95)
Figure 2: Premium values for S = K(100)
Figure 3: Premium values for S < K(105)
11
Next, the analytical expression obtained in previous section, (7) is used
along with the Monte Carlo simulation to find appropriate premium values
for call option. We obtain the option premiums for more weights placed
on around maturity date and for more weights around contract date. Each
value for geometric weighted average (C̃GWA ) and arithmetic average options
(C̃AWA ) is computed for T =6, 12 and σ=10%, 20%, 30% and adding the
difference to the value (CGWA ) calculated by analytical formula (7), obtaining
the appropriate values of option (CA ). Table1 through Table3 show the
results of this process.
It is observed that the results of CA for both cares are much smaller in
standard deviations than those of Monte Carlo simulation.
Notice that the expression of exercise price K ∗ used in (11) is different
from that in (7), replacing by (12). The results are slightly higher than the
analytical method.
5
A PRICING FORMULA BASED ON MOMENTS
As we worked above, options involving the arithmetic average may not
have closed-form solutions if the conventional assumption of a geometric
diffusion is specified for the underlying price process.
Levy[6] proposes a more accurate method which relies on the assumption
that the distribution of sum of log normal variables is itself well approximated at least to a first order by the lognormal. He mentions that the
valuation of average option becomes possible for typical range of volatility
experienced. Turnbull and Wakeman[15] also recognize the suitability of the
log normal as a first-order approximation.
Thus, since first and second moments for arithmetic weighted average are
possible to obtain we first define a new variable which follows lognormal distribution and corresponds up to first and second moments of the arithmetic
weighted average. Then, option price obtained for such variable defined is
nothing but that for plain option and hence Black-Scholes formula may be
applicable.
The value of S (τ ) after τ = T − T0 period passed is expressed as
1 2
S (τ ) = S (T0 ) e(r− 2 σ )τ +σW (τ )
where σW (τ ) ∼ N (0, σ 2 τ ). We also found the first moment of SAWA (T ) as
12
(10),
aS (T0 )
E [SAWA (T )] = aT
e − eaT0
(
e(r+a)T − e(r+a)T0
r+a
)
.
(13)
Then, the second moment becomes
)2 ]
∫ T
a
eaτ S (τ ) dτ
eaT − eaT0 T0
(
)2
aS (T0 )
= 2 aT
(14)
e − eaT0
(
)
2
2
2
2
e(2(r+a)+σ )T − e(2(r+a)+σ )T0
e(r+a)T e(r+a+σ )T0 − e(2(r+a)+σ )T0
−
.
(2 (r + a) + σ 2 ) (r + a + σ 2 )
(r + a) (r + a + σ 2 )
[
]
E (SAW A (T ))2 = E
[(
On the other hand, a new variable, X (T ) we now define is considered to
have the same price as that of underlying asset after T period. Assume
σW (τ ) ∼ N (0, σ 2 τ ). Denoting rx , σx by drift rate and volatility for the
variable X (τ ) , respectively, define
1 2
X (τ ) = S (T0 ) e(rx − 2 σx )τ +σx W (τ )
Then, the first and second moments for X (T ) can be expressed as
[
]
2
E (X (T ))2 = (S (T0 ))2 e(2rx +σx )(T −T0 ) .
(15)
These results along with (13) , (14) give the expression
E [X (T )] = S (T0 ) erx (T −T0 )
rx (T −T0 )
S (T0 ) e
,
aS (T0 )
= aT
e − eaT0
(
e(r+a)T − e(r+a)T0
r+a
)
.
From this we obtain
1
log
rx =
T − T0
(
eaT
a
− eaT0
(
e(r+a)T − e(r+a)T0
r+a
))
,
and for σx we have
2
(S (T0 )) e
(2rx +σx2 )(T −T0 )
(
)2
aS (T0 )
= 2 aT
e − eaT0
(
2
2
e(2(r+a)+σ )T − e(2(r+a)+σ )T0
2
2
e(r+a)T e(r+a+σ )T0 − e(2(r+a)+σ )T0
−
(2 (r + a) + σ 2 ) (r + a + σ 2 )
(r + a) (r + a + σ 2 )
13
)
.
Hence, we have
√
σx =
1
log A − 2rx
T − T0
where
(
)2
a
A = 2 aT
e − eaT0
(
2
2
e(2(r+a)+σ )T − e(2(r+a)+σ )T0
2
2
e(r+a)T e(r+a+σ )T0 − e(2(r+a)+σ )T0
−
(2 (r + a) + σ 2 ) (r + a + σ 2 )
(r + a) (r + a + σ 2 )
)
Therefore, the approximation formulae for pricing of atithmetic weithted
average are described below,
(
)
√
CAWA = S (T0 ) e(rx −r)(T −T0 ) N d + σx T − T0 − Ke−r(T −T0 ) N (d)
(
)
√
PAWA = −S (T0 ) e(rx −r)(T −T0 ) N −d − σx T − T0 + Ke−r(T −T0 ) N (−d)
(16)
where
log (S (T0 ) /K) + (rx − σx2 /2) (T − T0 )
√
σx T − T 0
(
))
( (r+a)T
a
1
e
− e(r+a)T0
rx =
log aT
T − T0
e − eaT0
r+a
√
1
σx =
log A − 2rx
T − T0
(
)2
a
A = 2 aT
e − eaT0
)
(
2
2
2
2
e(2(r+a)+σ )T − e(2(r+a)+σ )T0
e(r+a)T e(r+a+σ )T0 − e(2(r+a)+σ )T0
−
(2 (r + a) + σ 2 ) (r + a + σ 2 )
(r + a) (r + a + σ 2 )
d=
14
The differences of premiums between geometric weighted average and
arithmetic weighted average are showed for each case of different strike
prices, remaining periods and volatilities. The values of the difference depicted on Figures 4,5 and 6 are computed by subtracting the premium for
geometric weighted average from that for arithmetic weighted average at
each value of a.
Let S=100, σ=20%, non-risk rate=0.5%, and T =1 year. The variable a is
allowed to vary −30 to 30. For exercise prices, K=95,100,105 the difference
of premiums for the two different weighted averages show similar patterns
while the value peak at a=0. (Fig.4) Also, none of the shape of patterns
is symmetrical about a=0, having more heavy tail on positive large values
of a. Holding the parameters, S, σ, r being the same as above the exercise
price is fixed with K=100. The patterns are quite different for remaining
periods T =1 month, T =6 months, T =12 months. The shoter the remaining
period becomes the lower the difference is observed. (Fig.5)
Different volatilities, σ =10,20,30% show different values of differences.
It is noted, in particular, that the value of the difference is more than 0.08
at around a=0.
15
Figure 4: Differences for exercise prices
Figure 5: Differences for remaining periods
Figure 6: Differences for volatilities
16
6
COMPARISON FOR THREE METHODOLOGIES
In order to compare there methodologies developed for weighted average
call option in the previous sections we set the following numerical example
The weights vary -20 to 20, implying more weights placed on contract date
and maturity date. Let S=100 with K=95,100,105. The premium values for
T =6,12 and for σ=10%,20%,30% are presented in Table1 through Table3.
Simulation was performed 15000 times and observations 60 times.
The standard deviations, as given in parentheses, for the approximation,
are very small due to the variance reduction technique.
It can be observed that there are much differences of option premiums between for more weights placed on around maturity date and for more weights
placed on around contract date.
More weights placed on around maturity date lead to higher average values
of call option prices, in particular, higher volatilities and longer remaining
periods make the values conspicuous.
Though three methodologies for the values of call option premium don’t
show noticeable difference, the difference being occuring only in the third
decimal, in many cases, the premium values for Levy shows the highest in
all methodologies for the case where more weights are placed on contract
date. However, for the situation where more weights are placed on maturity
date Kemna & Vorst record higher values of premium than those of other
ways.
17
Table1: Premiums for call option with volatility, 10%
T=6
Analysis
a
-20
-10
0
10
20
Monte Carlo
Approximation
CGWA
C̃GWA
C̃AWA
CA
CB
CC
95
5.0002
4.9995
5.0109 (0.01142)
5.0116 (0.00008)
5.0127
5.0127
100
0.6355
105
0.0005
0.5608
0.5667 (0.00686)
0.6414 (0.00008)
0.6417
0.6419
0.0001
0.0001 (0.00004)
0.0005 (0.00001)
0.0005
0.0005
95
5.0196
5.0123
5.0348 (0.01710)
5.0421 (0.00015)
5.0429
5.0430
100
0.8964
0.8389
0.8504 (0.01022)
0.9080 (0.00015)
0.9083
0.9088
105
0.0116
0.0075
0.0083 (0.00092)
0.0124 (0.00007)
0.0119
0.0120
K
volatility=10%
95
5.2605
5.2609
5.2958 (0.02656)
5.2954 (0.00025)
5.2978
5.2985
100
1.6671
1.6714
1.6935 (0.01763)
1.6891 (0.00026)
1.6880
1.6895
105
0.2466
0.2428
0.2520 (0.00676)
0.2559 (0.00022)
0.2515
0.2522
95
5.7598
5.7279
5.7464 (0.04150)
5.7783 (0.00016)
5.7788
5.7795
100
2.4665
2.4967
2.5097 (0.03028)
2.4795 (0.00017)
2.4783
2.4793
105
0.7497
0.7578
0.7649 (0.01669)
0.7568 (0.00015)
0.7546
0.7553
95
5.9366
5.9696
5.9800 (0.04512)
5.9469 (0.00009)
5.9464
5.9467
100
2.7042
2.6948
2.7018 (0.03287)
2.7112 (0.00009)
2.7105
2.7109
105
0.9277
0.9069
0.9107 (0.01926)
0.9314 (0.00008)
0.9305
0.9308
CA :the results based on Kemna & Vorst
CB :the results based on Vorst
CC :the results based on Levy
T=12
Analysis
CGWA
Monte Carlo
C̃GWA
Approximation
C̃AWA
CA
CB
CC
a
K
95
4.9877
4.9876
4.9979 (0.00991)
4.9981 (0.00007)
5.0002
5.0002
-20
100
0.6339
0.5875
0.5928 (0.00591)
0.6392 (0.00007)
0.6401
0.6403
105
0.0005
0.0000
0.0000 (0.00001)
0.0005 (0.00000)
0.0005
0.0005
95
5.0078
5.0065
5.0292 (0.01616)
5.0305 (0.00015)
5.0324
5.0324
100
0.9002
0.8991
0.9110 (0.00972)
0.9120 (0.00015)
0.9127
0.9132
105
0.0121
0.0056
0.0063 (0.00075)
0.0129 (0.00007)
0.0125
0.0125
95
5.6861
5.6146
5.6761 (0.03522)
5.7477 (0.00050)
5.7539
5.7566
100
2.3773
2.4335
2.4793 (0.02572)
2.4231 (0.00053)
2.4192
2.4235
105
0.6880
0.6925
0.7204 (0.01400)
0.7160 (0.00052)
0.7047
0.7079
95
6.9035
6.9072
6.9263 (0.06022)
6.9226 (0.00018)
6.9213
6.9223
100
3.8822
3.9117
3.9261 (0.04823)
3.8966 (0.00019)
3.8947
3.8959
105
1.9120
1.8546
1.8642 (0.03360)
1.9216 (0.00017)
1.9195
1.9205
95
7.0604
7.0345
7.0445 (0.06246)
7.0704 (0.00010)
7.0692
7.0697
100
4.0602
4.0769
4.0845 (0.04971)
4.0678 (0.00010)
4.0664
4.0670
105
2.0695
2.0393
2.0442 (0.03578)
2.0745 (0.00009)
2.0733
2.0738
-10
0
10
20
volatility=10%
18
Table2: Premiums for call option with volatility, 20%
T=6
Analysis
CGWA
a
-20
-10
0
10
20
Monte Carlo
C̃GWA
K
Approximation
C̃AWA
CA
CB
CC
volatility=20%
95
5.0317
4.9911
5.0333 (0.02452)
5.0738 (0.00032)
5.0788
5.0791
100
1.0897
1.0897
1.1129 (0.01167)
1.2689 (0.00026)
1.2703
1.2714
105
0.0843
0.0496
0.0545 (0.00282)
0.0891 (0.00022)
0.0872
0.0876
95
5.2127
5.1387
5.2126 (0.03515)
5.2866 (0.00066)
5.2949
5.2964
100
1.7445
1.6757
1.7236 (0.01819)
1.7923 (0.00055)
1.7908
1.7937
105
0.3062
0.2703
0.2913 (0.00851)
0.3272 (0.00057)
0.3186
0.3202
95
6.2345
6.1986
6.3110 (0.04657)
6.3470 (0.00104)
6.3548
6.3622
100
3.2269
3.1948
3.2806 (0.03571)
3.3127 (0.00105)
3.3077
3.3168
105
1.4007
1.3785
1.4406 (0.02403)
1.4628 (0.00104)
1.4445
1.4523
95
7.6159
7.5850
7.6485 (0.08416)
7.6794 (0.00079)
7.6772
7.6828
100
4.8030
4.7674
4.8188 (0.05318)
4.8545 (0.00060)
4.8485
4.8547
105
2.8111
2.8320
2.8719 (0.04824)
2.8510 (0.00067)
2.8418
2.8475
95
8.0535
7.9386
7.9744 (0.08906)
8.0893 (0.00043)
8.0852
8.0881
100
5.2776
5.2982
5.3267 (0.05887)
5.3061 (0.00033)
5.3015
5.3046
105
3.2549
3.2171
3.2382 (0.05426)
3.2759 (0.00035)
3.2716
3.2745
T=12
Analysis
CGWA
Monte Carlo
C̃GWA
Approximation
C̃AWA
CA
CB
CC
a
K
95
5.0191
4.9517
4.9908 (0.01955)
5.0582 (0.00027)
5.0662
5.0665
-20
100
1.2427
1.1046
1.1264 (0.01079)
1.2645 (0.00025)
1.2672
1.2683
105
0.0841
0.0381
0.0410 (0.00151)
0.0870 (0.00017)
0.0870
0.0874
95
5.2024
5.1140
5.1891 (0.02987)
5.2775 (0.00058)
5.2889
5.2904
100
1.7498
1.6976
1.7451 (0.01732)
1.7974 (0.00053)
1.7987
1.8018
105
0.3114
0.3013
0.3199 (0.00716)
0.3300 (0.00054)
0.3246
0.3263
95
7.3298
7.1401
7.3469 (0.06259)
7.5366 (0.00215)
7.5467
7.5696
100
4.5378
4.5549
4.7339 (0.05199)
4.7168 (0.00222)
4.6972
4.7230
-10
0
10
20
volatility=20%
105
2.5910
2.5618
2.7026 (0.04018)
2.7319 (0.00217)
2.6966
2.7206
95
10.1135
10.1648
10.2333 (0.11209)
10.1820 (0.00079)
10.1716
10.1799
100
7.4981
7.3616
7.4178 (0.08530)
7.5543 (0.00066)
7.5452
7.5539
105
5.4165
5.2848
5.3326 (0.08439)
5.4644 (0.00076)
5.4533
5.4618
95
10.4562
10.4238
10.4601 (0.11747)
10.4926 (0.00042)
10.4851
10.4890
100
7.8538
7.8199
7.8497 (0.09020)
7.8835 (0.00035)
7.8773
7.8815
105
5.7633
5.7128
5.7382 (0.08958)
5.7887 (0.00040)
5.7819
5.7859
19
Table3: Premiums for call option with volatility, 30%
T=6
Analysis
CGWA
a
-20
-10
0
10
20
Monte Carlo
C̃GWA
Approximation
C̃AWA
K
CA
CB
CC
volatility=30%
95
5.2404
5.1311
5.2137 (0.03188)
5.3231 (0.00066)
5.3361
5.3380
100
1.8433
1.6475
1.7006 (0.02089)
1.8963 (0.00069)
1.8980
1.9014
105
0.3688
0.2458
0.2682 (0.00801)
0.3911 (0.00059)
0.3850
0.3870
95
5.6799
5.6136
5.7619 (0.04426)
5.8282 (0.00139)
5.8425
5.8492
100
2.5682
2.4380
2.5434 (0.03168)
2.6736 (0.00142)
2.6706
2.6798
105
0.8820
0.7933
0.8589 (0.01888)
0.9476 (0.00138)
0.9285
0.9356
95
7.4831
7.4566
7.6898 (0.06684)
7.7163 (0.00243)
7.7216
7.7478
100
4.7412
4.7524
4.9502 (0.05515)
4.9390 (0.00248)
4.9182
4.9471
105
2.7985
2.7105
2.8681 (0.04254)
2.9561 (0.00242)
2.9184
2.9455
95
9.7057
9.6864
9.8193 (0.10915)
9.8387 (0.00162)
9.8289
9.8480
100
7.1090
7.0136
7.1272 (0.09569)
7.2226 (0.00158)
7.2081
7.2282
105
5.0627
5.0624
5.1606 (0.08282)
5.1609 (0.00158)
5.1393
5.1588
95
10.4007
10.5240
10.5997 (0.11910)
10.4764 (0.00092)
10.4645
10.4743
100
7.8316
7.8259
7.8891 (0.10511)
7.8948 (0.00086)
7.8838
7.8939
105
5.7666
5.7509
5.8059 (0.09252)
5.8217 (0.00089)
5.8079
5.8179
T=12
Analysis
CGWA
Monte Carlo
C̃GWA
Approximation
C̃AWA
CA
CB
CC
a
K
95
5.2273
5.0147
5.0910 (0.02792)
5.3037 (0.00057)
5.3229
5.3247
-20
100
1.8387
1.7059
1.7526 (0.01796)
1.8855 (0.00062)
1.8933
1.8967
105
0.3679
0.3580
0.3767 (0.00618)
0.3866 (0.00056)
0.3841
0.3861
95
5.6704
5.4332
5.5831 (0.04166)
5.8203 (0.00137)
5.8415
5.8486
100
2.5739
2.4708
2.5743 (0.02999)
2.6773 (0.00138)
2.6819
2.6915
105
0.8907
0.7609
0.8227 (0.01666)
0.9525 (0.00132)
0.9400
0.9475
95
9.1539
9.1167
9.5614 (0.09173)
9.5986 (0.00505)
9.5867
9.6637
100
6.6044
6.6530
7.0539 (0.08247)
7.0054 (0.00532)
6.9493
7.0308
-10
0
10
20
volatility=30%
105
4.6207
4.6679
5.0196 (0.07041)
4.9724 (0.00534)
4.8840
4.9634
95
13.4815
13.6146
13.7646 (0.17207)
13.6315 (0.00198)
13.5990
13.6271
100
11.0697
10.8667
10.9977 (0.15406)
11.2007 (0.00191)
11.1707
11.1994
105
9.0130
9.1666
9.2867 (0.14602)
9.1330 (0.00189)
9.0985
9.1270
95
14.0172
13.8045
13.8821 (0.17542)
14.0947 (0.00100)
14.0755
14.0891
100
11.6160
11.5369
11.6082 (0.16278)
11.6873 (0.00101)
11.6664
11.6803
105
9.5552
9.3776
9.4395 (0.15065)
9.6171 (0.00100 )
9.5982
9.6119
20
7
CONCLUSION
In this study, a method for the valuation of options based on weighted average instead of simple average was proposed, extending the classical results
to more general weighted average option. In particular, a control variable
to adjust the weights is incorporated in the weights so that the proposed
formulae for option premiums may cope with more different situations and
phenomena incurred by underlying assets.
We calculated and compared the prices of an average option based on
weighted average in two different ways. (i) Monte Carlo simulation with
some adjustments for analytical formula of geometric average option. (ii)
use of a new variable whose first and second moments correspond up to those
of arithmetic average, which makes B-S formulae applicable to use.
As a future work the values of options over a multi-period will be considered for which each value for the period may be observed as usual time series
and hence we can analyze the option problems with time series models.
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22