Applied Mathematics, 2011, 2, 461-464
doi:10.4236/am.2011.24058 Published Online April 2011 (http://www.SciRP.org/journal/am)
Pricing European Call Currency Option Based on
Fuzzy Estimators
Xing Yu, Hongguo Sun, Guohua Chen
The Departme nt of Mathematics & Appli ed Mat hematics, Humanities & Science and Technology
Institute of Hunan Loudi, Loudi, China
Received December 30, 2010; revised March 2, 2011; accepted March 5, 2011
Abstract
In this paper we present an application of fuzzy estimators method to price European call currency option.
We make use of fuzzy estimators for the volatility of exchange rate which based on statistical data to obtain
the fuzzy pattern of G-K model. A numerical example is presented to get the

-level closed intervals of the
European call currency option fuzzy price.
Keywords: Currency Option, Fuzzy Estimators, Fuzzy Volatility, G-K Model
1. Introduction
In financial markets there are great fluctuations, thus the
element of vagueness an d uncertainty is frequent. For in-
stant, on the foreign exchange market, th e spot exchang e
rates fluctuate from time to time according to the finan-
cial market effects and may occur imprecisely; therefore,
it is natural to consider the fuzzy foreign interest rate and
the domestic interest rate, fuzzy volatility and fuzzy ex-
change rate. Practically, many financial investors are
concerned with the currency options price range, then at
the last of the paper, we get the

-level close intervals
of the European call currency option fuzzy price.
Since the closed form solution of the European cur-
rency options piecing model was derived by Garman and
Kohlhagen (1983) based on Black and Scholes (1973),
many methodologies for the currency options pricing
have been proposed by using modification of Garman-
Kohlhagen (G-K) model, such as Amin and Jarrow
(1991), Heston (1993) , Bates (1996), Ekvall et al. (1997),
Lim et al. (1998) , Rosenberg (1998 ), Sarwar and Krenh-
biel (2000) [1], Bollen and Rasiel (2003) [2]. In this pa-
per we present an app lication o f fuzzy estimators method
to price European call currency option. We make use of
fuzzy estimators for the volatility o f exchang e rate which
based on statistical data to obtain the fuzzy pattern of
G-K model. We use the fuzzy estimators based on con-
findence intervals introdu ced by Papadopoulos and Sfiris
in [3] in order to estimate the volatility of exchange rate
having sample data. This paper is organized as follows.
In Section 2, the notions of fuzzy number and the arith-
metics of fuzzy numbers are introduced. In Section 3, the
fuzzy patterns of G-K model. In Section 4, the optimal
model is provided in order to obtain the belief degrees of
given option price. Finally, the empirical study is per-
formed.
2. Fuzzy Sets Preliminaries
2.1. Fuzzy Set Theory
Now we remind some facts about fuzzy sets and numbers
[4]:
Let X be a universal set and
A
 be a fuzzy subset of
X, we denote b y A

its membership function


:0,1
AX
 , and by

:A Ax


 
 the

-level
set of, where is the closure of the set. Let a
 be a fuzzy
number. Then, under our assumptions, the

-level set
a

 is a closed interval, which can be denoted
by ,
LU
aaa







.
The four arithmetic operations on closed intervals are
defined as follows:






,, ,abdea db e

 






,, ,ab deaebd

 





 

,,min,,,,min,,,
and provided that 0,
a bdeadae bdbeadae bdbe
de







X. YU ET AL. 462






if,,,0, t hen,,,abdeab deadbe
 
11
,,,and provided that 0,ab deabde
de

 


2.2. Triangular Fuzzy Number
The membership function of a triangular fuzzy number
a
 is defined by :





,if
,if
0, otherwise
L
cL LC
aRRCCR
x
aaa axa
x
axaa axa 
 

 



Which is denoted by


;;
L
CR
aaaa
.The

-level
set of a
 is then:


1,1
LC RC
aaaaa

    







1, 1
LU
L
CRC
aaaaaa


  

3. Fuzzy Currency Options Pricing Model
3.1. Estimate Fuzzy Volatility from Fuzzy
Estimators of 2

—Fuzzy Estimators
Based on Confidence Intervals
Following the proposition proved by Papadopoulos and
Sfiris in [3], we consider the fuzzy volatility as:

22
2
1
2
22
2
1
22 1
1if
11 22
11
21
22 1
1if
11 22
11
21
ns s
x
s
x
n
xns s
sx
x
n

 

 





 



 


 






 



 


 




its

-cut is the following:

22
2,
22
11
11
ss
uu
nn









(1)
where
11
122 2 u




 
 


(2)
3.2. The G-K Model and Its Fuzzy Pattern
The G-K model for a European call currency option with
expiry date T and strike price K. t
s
denotes the spot
exchange rate at time


0,tT, and t c denotes the
price of a currency option at time t. The G-K model is as:


12
12
,
rr
tt
cseNd KeNdTt




 (3)


 
2
121
21
ln2 ,
t
dsKrr
dd






 (4)
where 1
r denotes the foreign interest rate, 2 r denotes
the domestic interest rate,

denotes the volatility, and
N stands for the cumulative distribution function of a
standard normal random variable

0, 1N.
Under the considerations of fuzzy exchange rate t
s
,
fuzzy interest rate 1 r
, 2
r
, fuzzy volatility

, the fuzzy
price is denoted as t c
. The fuzzy patterns of G-K model
are described as the interval:











12
12
() UL
LU
Lrr
L
tt
cseNdKeNd








 (5)











12
12
() LU
UL
Urr
U
tt
cseNdKeNd










 (6)
where



 





2
121
ln 2
LLLU LU
t
dsKrr
  











(7)



 





2
121
ln 2
UUUL UL
t
dsKrr
  










 
(8)

X. YU ET AL. 463
Table 1. The price intervals for a given belief degree.

c



c



c


0.8 [0.5475, 0.5502] 0.87 [0.5479, 0.5497] 0.94 [0.5484, 0.5492]
0.81 [0.5475, 0.5501] 0.88 [0.5480, 0.5496] 0.95 [0.5484, 0.5491]
0.82 [0.5476, 0.5500] 0.89 [0.5480, 0.5495] 0.96 [0.5486, 0.5491]
0.83 [0.5476, 0.5499] 0.9 [0.5482, 0.5495] 0.97 [0.5486, 0.5490]
0.84 [0.5477, 0.5499] 0.91 [0.5482, 0.5494] 0.98 [0.5487, 0.5490]
0.85 [0.5478, 0.5498] 0.92 [0.5483, 0.5494] 0.99 [0.5487, 0.5488]
0.86 [0.5479, 0.5498] 0.93 [0.5483, 0.5493] 1.00 [0.5488, 0.5488]
 

21
LL
U
dd



 

 (9)
 

21
UU
L
dd



 

 (10)
3.3. To Calculation the Fuzzy Currency Option
Steps
Calculating the fuzzy currency option as the following
steps:
Step 1. We compute the Estimate fuzzy volatility from
fuzzy estimators of 2

based the history data following
(1)-(2).
Step 2. On the hypothesis of the fuzzy interest and
fuzzy exchange rate are triangular fuzzy numbers, and
we obtain the

-cuts of fuzzy currency option which
are closed intervals following (1) -(2), (5)-(10).
4. Empirical Example
In this section, our model is tested with the daily market
price data of EUR/CNY. These market data come from
[5] and cover the period 9-27-2010 to 11-15-2010. We
calculate 20.013287,s the spot exchange rate is
around 9.2418, and assume the strike price is K = 9.12
with one year to expiry. Let 0.01

. The triangular
fuzzy numbers are

11.09%,1.1%,1.11% ;r


22.4%,2.5%,2.6% ;r



9.2398,9.2418,9.2438
t
s
.
Because the closed form of volatility interval based on
samples are very hard to get, so we adopt numerical al-
gorithm to solve optimization model. For example, we
average the interval [0.8, 1] into 20 equal parts, that is
0.8,0.81,0.82, ,1

. Based on every

, we calcu-
late the intervals of option price as Table 1, from which,
we find that with

increasing ,the left interval in-
creasing while the right interval decreasing, and both
intersects when 1

 where is the situation matches
the actual price by Equations (3,4). For 0.95


, it
means that the call currency option price will lie in the
closed interval [0.5484, 0.5491] with belief degree 0.95.
From another point of view, if a financial analyst is
comfortable with this belief degree 0.95, then he (she)
can pick any value from the closed interval [0.5484,
0.5491] as the optio n p ri ce fo r his(her)later use.
5. Conclusions
Owing to the fluctuation of financial market from time to
time, the data sometimes cannot be expected in a precise
sense. Therefore, the fuzzy sets theory provides a useful
tool for conquering this kind of impreciseness. The fuzzy
pattern of G-K model is proposed in this paper. Under
the considerations of fuzzy estimators for the volatility of
exchange rate which based on statistical data, the Euro-
pean call currency option price turns into a fuzzy number.
This makes the financial analyst who can pick any Eu-
ropean call currency option with an acceptable belief
degree for he (her) later use and get the interval for the
price with the give n bel i e f degree.
6. Acknowledgments
Supported by Youth Fund of Hunan Institute of Humani-
ties Science and technology (2 008QN013).
7. References
[1] G. Sarwar and T. Krehbiel, “Empirical Performance of
Alternative Piecing Models of Currency Options,” The
Journal of Futures Markets, Vol. 20, No. 2, March 2000,
pp. 265-291.
doi:10.1002/(SICI)1096-9934(200003)20:3<265::AID-F
UT4>3.0.CO;2-4
[2] P. B. N. Bollen and E. Rasiel, “The Performance of Al-
ternative Valuation Models in the OTC Currency Options
Market,” Journal of International Money and Finance,
Vol. 22, No. 1, February 2003, pp. 33-64.

X. YU ET AL. 464
doi:10.1016/S0261-5606(02)00073-6
[3] A. Thavaneswaran, J. Singh and S. S. Appadoo, “Option
Pricing for Some Stochastic Volatility Models,” The
Journal of Risk Finance, Vol. 7, No. 4, 2006, pp. 425-445.
doi:10.1108/15265940610688982
[4] G. J. Klir and B. Yuan, “Fuzzy Sets and Fuzzy Logic:
Theory and Applications,” Prentice Hall, Englewood
Cliffs, 1995.
[5] http://www.123cha.com/hl/?q=100&from=EUR&to=CN
Y&s=EURCNY#symbol=EURCNY=X;range=3m