Ewa Dziawgo

English

Research Papers in Economics

© Copyright by The Nicolaus Copernicus University Scientific Publishing House
 DYNAMIC  ECONOMETRIC  MODELS 
Vol. 6 – Nicolaus Copernicus University – Toruń –  2004 
 
 
 
 
 
Ewa Dziawgo 
Nicolaus Copernicus University in Toruń 
 
 
 
Approximation of Basket Call Option Price  
 
 
1. Introduction 
 
  Exotic options are characterized by the income structure different from the 
structure of standard options. The exotic options are traded on the over-the 
counter derivatives market. Basket options are exotic, correlative options. Their 
pricing encompasses the correlations among the base assets behaviour. 
The aim of the paper is to present the approximation methods of the basket call 
option price. This article presents pricing of the basket call option on USD, 
EUR and GBP. In the article a tree-month basket call options was priced. Based 
on the selected basket call options the analysis of the basket options price and 
the average price of the standard options was carried out.  
 
 
2. Basket options and their pricing 
 
  The basket options are issued on more than one base instrument. Their price 
is directly related to the correlation coefficient of the base instruments. The base 
instrument of the basket options is the basket composed of a few instruments: 
shares, currencies or indexes. In 1993, in the field of basket options pricing, 
Gentle suggested the approximation of arithmetic average by the properly 
weighted geometric mean (Weron, Weron (1998)). 
 Let  St
i (for i = 1,2,...,k) be the price processes of  k underlying financial in-
struments (which are shares). The process St
i is described by the geometric 
rownian motion:  B
    
  ( )
i
t i
i
t
i
t dB rdt S dS σ + = , (1) 
where: © Copyright by The Nicolaus Copernicus University Scientific Publishing House
Ewa Dziawgo  196
BBt – one dimensional Brownian motion with respect to the martingale measure 
Q, 
r – risk-free rate of interest, 
R i ∈ σ , is a volatility of share price
1. 
T
   
he equation (1) shows, that the share price amounts to: 
  [ ] t r B S S i
i
t i
i
t ) 5 , 0 ( exp
2
0 σ σ − + = . (2) 
 
The payoff function of the basket call option end value, whose base instrument 
is composed of  k shares, amounts to
2: 
  ,   (3) 
+
=
⎟ ⎟
⎠
⎞
⎜ ⎜
⎝
⎛
− = ∑
k
i
i
T i T K S w f
1
where: 
K – option exercise price,  
ST 
i – price of asset „i” at time T,  
T – time to the option expiry date,  
k – number of assets on the underlying option,  
wi – weight of asset  „i”, where:   and  .  0 ≥ i w 1
1
= ∑
=
k
i
i w
The modified weights for the fixed t∈(0;T) amount to: 
  ,
) , (
) , ( ~
1 1 ∑ ∑
= =
= =
k
j
S j
S i
k
j
j
t j
i
t i
i
T t F w
T t F w
S w
S w
w
j
i
         for i = 1,2,...,k 
where: 
) , ( T t F i S – forward price in time t of the fixed i – on the asset with the expiry 
date T, 
i
t
t T r
S S e T t F i
) ( ) , (
− = . 
The payoff function of the basket option for the modified weights amounts to: 
  ,   (4) 
+
= =
⎟ ⎟
⎠
⎞
⎜ ⎜
⎝
⎛
− ⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
= ∑ ∑ K S w T t F w f
k
i
i
T i
k
j
S j T j
~ ~ ~ ) , (
1 1
where: 
                                                      
1 The volatility of the shares is calculated as a standard deviation of the return rate 
on shares. 
2   
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
− =
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
− = ∑ ∑
=
+
=
k
i
i
T i
k
i
i
T i T K S w K S w f
1 1
0 ; max© Copyright by The Nicolaus Copernicus University Scientific Publishing House
 Approximation of Basket Call Option Price  197
 
) , (
~
T t F
S
S
i S
i
T i
T =               for i = 1,2, ..., k,  
 
∑ ∑
=
− −
=
= =
k
j
j
t j
t T r
k
j
S j S w
K e
T t F w
K
K
j
1
) (
1
) , (
~
. 
 
The arbitrage price of the basket call option Ct
B in time t amounts to: 
 
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
⎟ ⎟
⎠
⎞
⎜ ⎜
⎝
⎛
− ⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
=
= ⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
⎟ ⎟
⎠
⎞
⎜ ⎜
⎝
⎛
− ⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
=
+
= =
+
= =
− −
∑ ∑
∑ ∑
t
k
i
i
T i
Q
k
j
j
t j
t
k
i
i
T i
Q
k
j
S j
t T r B
t
F K S w E S w
F K S w E T t F w e C j
~ ~ ~
~ ~ ~ ) , (
1 1
1 1
) (
     (5) 
The weighted arithmetic mean   is replaced with the properly weighted 
geometric mean. Then 
∑
=
k
i
i
T iS w
1
~ ~
     (6)  ()
+
= =
⎟ ⎟
⎠
⎞
⎜ ⎜
⎝
⎛
− ⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
= ∏ ∑ K S E S w C
k
i
w i
T
Q
k
j
j
t j
B i ˆ ~
1
~
1
0
(
is the approximation of the basket call option price
3, 
where: 
    () ⎟ ⎟
⎠
⎞
⎜ ⎜
⎝
⎛
− + = ∑ ∏
= =
k
i
i
T i
k
i
w i
T
Q S w S E K K
i
1 1
~ ~ ~ ~ ~ ˆ
Because: 
 
()
()
) ~ 5 , 0 exp(
~ 5 , 0 ~ exp ) 5 , 0 exp(
) 5 , 0 ( exp( ~
1
2
1
2
1 1
~
2
1
~
0
2
0
1
~
∑
∑ ∑ ∏
∏ ∏
=
= = =
= =
− =
= ⎟ ⎟
⎠
⎞
⎜ ⎜
⎝
⎛
− = − =
= ⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛ − +
=
k
i
i i T
k
i
i i
k
i
i
i
T i
k
i
w
i
i
T i
k
i
w
rT i
i
i
T i
i k
i
w i
T
w T
w T w B T B
e S
T r B S
S
i
i
i
σ η
σ σ σ σ
σ σ
   (7) 
where: 
  ,  ,   ∑
=
=
k
i
i
i
T i T w B
1
~ σ η T v T w w Var
k
j i
j i j i ij T
2
1 ,
~ ~ ) ( = =∑
=
σ σ ρ η
                                                       
3  At time t = 0. © Copyright by The Nicolaus Copernicus University Scientific Publishing House
Ewa Dziawgo  198
∑
=
=
k
j i
j i j i ij w w v
1 ,
2 ~ ~ σ σ ρ ,  j i, ρ – correlation coefficient between base instruments. 
Then the expected values of the weighted geometric mean and weighted arith-
metic mean amount to: 
     (8) 
() () c w v T Var E
w v T w T E
k
i
i i T T
Q
k
i
i i
k
i
i i T
Q
= ⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
⎟ ⎟
⎠
⎞
⎜ ⎜
⎝
⎛
− = − ⋅
⋅ ⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
⎟ ⎟
⎠
⎞
⎜ ⎜
⎝
⎛
− = ⎟ ⎟
⎠
⎞
⎜ ⎜
⎝
⎛
−
∑
∑ ∑
=
= =
1
2 2
1
2 2
1
2
~ 5 , 0 exp ( 5 , 0 exp
~ 5 , 0 exp ) ~ 5 , 0 exp(
σ η η
σ σ η
and 
     (9) 
()
1 ~ ~
~ ~ ~
1 1
0 0
1
0
1
= = =
= = ⎟ ⎟
⎠
⎞
⎜ ⎜
⎝
⎛
∑ ∑
∑ ∑
= =
+ − −
=
− −
=
k
i
i
k
i
rT rT i i
i
k
i
i
T
rT Q i
i
k
i
i
T i
Q
w e S S w
S e E S w S w E
         (10)  () 1
~ ~ ~
1 1
~
− =
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
−∑ ∏
= =
c S w S E
k
i
i
T i
k
i
w i
T
Q i
 
From the formulae (7), (8) and (10) it can be obtained: 
 
   (11)  () () ()
+
+
=
− + − − = ⎟ ⎟
⎠
⎞
⎜ ⎜
⎝
⎛
− ∏ ) 1
~
( ) ( 5 , 0 exp ˆ ~
1
~
c K Var c E K S E T T
Q
k
i
w i
T
Q i η η
 
Substituting formula (11) in formula (6) the approximation of the basket call 
ption price is obtained
4:  o
   
     (12)  () ( ) ( 1
~
) ( 2 1
1
d N c K d cN S w C
k
j
j
t j
B
t − + − ⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
= ∑
=
( )
                                                     
 
where: 
 
4 It can be concluded from the following (using the following notation: T η ξ = ,a =c 
and  ):   1
~
− + = c K b
Let ξ  be a variable in space  ) , , ( Q F Ω  which has a normal distribution with zero mean 
and variance σ >0. Then for the positive real numbers a and b, the following is true:  
() ), ( ) ( ) 5 , 0 exp(
2 σ σ ξ − − = − −
+
h bN h aN b a E
Q  
where:  σ σ 5 , 0 ln
1 + ⎟
⎠
⎞
⎜
⎝
⎛ =
−
b
a
h ;  (Musiela, Rutkowski (1998)). © Copyright by The Nicolaus Copernicus University Scientific Publishing House
 Approximation of Basket Call Option Price  199
 
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡ −
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
− = ∑ ∑
= = 2
) ( ~ ~ ~ exp
1
2
1 ,
t T
w w w c
k
j
j j
k
j i
j i j i ij σ σ σ ρ , 
 
()
t T v
t T v c K c
d
−
− + − + −
=
) (
2
1
1
~
ln ln
2
1 , 
 
()
t T v
t T v c K c
d
−
− − − + −
=
) (
2
1
1
~
ln ln
2
2 , 
N(d) – cumulative probability function of the standardised normal distribution. 
 
 
3. Examples  
 
  The empirical illustration is concerned with the pricing of the three-month 
European basket options with USD, EUR and GBP currencies as base instru-
ments. The analysis is carried out for the call options and relates to the period 
between: 04.01.2003 and 04.04.2003. In the calculations the identical weights 
for the base instruments were accepted, namely in the case of basket options on 
two currencies wi = 0.5 (for i =1,2). Moreover, for the basket option on the three 
base instruments wi = 
3
1
(for i =1,2,3).  
  Fig. 1 shows the change in the basket call option price based on USD and 
EUR. In the mentioned period the empirical correlation coefficient between the 
currencies amounts to 0.54. Fig. 1 also illustrates the variation in the average 
price of standard options on USD and EUR.  
0
0,05
0,1
0,15
0,2
0,25
0,3
0,35
0,4
0,45
0
6
.
0
1
1
6
.
0
1
2
9
.
0
1
1
0
.
0
2
2
0
.
0
2
0
4
.
0
3
1
4
.
0
3
2
6
.
0
3
Date
P
r
i
c
e
 
[
P
L
N
]
st. op. av. pr.
bas. op. pr.
 
Fig. 1. Variation in the basket option price (on USD and EUR currencies) and standard 
option average price (on EUR and USD currencies)  
Source: Author’s calculations. 
 © Copyright by The Nicolaus Copernicus University Scientific Publishing House
Ewa Dziawgo  200
  Fig. 2 illustrates the change in the basket option price based on USD and 
GBP. The empirical correlation coefficient between the currencies amounts to 
0.088. Fig. 2, furthermore, presents the change in the average price of standard 
options on USD and GBP. 
0
0,05
0,1
0,15
0,2
0,25
0,3
0,35
0,4
0
6
.
0
1
1
6
.
0
1
2
9
.
0
1
1
0
.
0
2
2
0
.
0
2
0
4
.
0
3
1
4
.
0
3
2
6
.
0
3
Date
P
r
i
c
e
 
[
P
L
N
]
st. op. av. pr.
bas. op. pr.
 
Fig. 2. Variation in the basket option price (on USD and GBP currencies) and standard 
option average price (on GBP and USD currencies)  
Source: Author’s calculations. 
 
  The variation in the basket option price based on EUR and GBP is shown in 
Fig. 3. That graph also depicts the change in the average price of standard op-
tions on EUR and GBP. The empirical correlation coefficient between the cur-
rencies amounts to -0.03. 
0
0,1
0,2
0,3
0,4
0,5
0
6
.
0
1
1
5
.
0
1
2
7
.
0
1
0
5
.
0
2
1
4
.
0
2
2
5
.
0
2
0
6
.
0
3
1
7
.
0
3
2
6
.
0
3
0
4
.
0
4
Date
P
r
i
c
e
 
[
P
L
N
]
st. op. av. pr.
bas. op. pr.
 
 Fig. 3. Variation in the basket option price (on GBP and EUR currencies) and standard 
option average price (on EUR and GBP currencies)  
Source: Author’s calculations. 
  
 
  Fig. 4 presents the change in the basket option price based on three curren-
cies: USD, EUR and GBP. Fig. 4 also exhibits the change in the average price 
of standard options on USD, EUR and GBP. © Copyright by The Nicolaus Copernicus University Scientific Publishing House
 Approximation of Basket Call Option Price  201
0
0,05
0,1
0,15
0,2
0,25
0,3
0,35
0,4
0,45
0
6
.
0
1
1
6
.
0
1
2
9
.
0
1
1
0
.
0
2
2
0
.
0
2
0
4
.
0
3
1
4
.
0
3
2
6
.
0
3
Date
P
r
i
c
e
 
[
P
L
N
]
st. op. av. pr.
bas.  op. pr.
 
Fig. 4. Variation in the basket option price (on GBP, USD and EUR currencies) and 
standard option average price (on GBP, USD and EUR currencies) 
Source: Author’s calculations. 
 
 
4. Conclusions 
 
  The analysis of the variation in option prices depicted in graphs clearly indi-
cates that the prices of basket options are lower than the prices of standard op-
tions. This is true both for the positive and negative correlation coefficient. Tak-
ing into consideration the cost incurred by the investor in order to hedge against 
the risk of change in the base instruments prices, it should be stated that basket 
options are more attractive than standard options. 
 
 
References 
 
Hull, C. J. (1997), Options, Futures and Other Derivatives, Prentice-Hall International, 
Inc. 
Musiela, M., Rutkowski, M. (1998), Martingale Methods in Financial Modelling, 
Springer-Verlag, Berlin. 
Neftci, S. N. (1996), An Introduction to the Mathematics of Financial Derivatives, Aca-
demic Press, San Diego. 
Weron, A., Weron, R. (1998), Inżynieria finansowa (Financial engineering), Wydaw-
nictwo Naukowo-Techniczne, Warszawa. 

Approximation of Basket Call Option Price

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