Pricing Currency Option Based on the Extension Principle and Defuzzification via Weighting Parameter Identification
We present a fuzzy version of the Garman-Kohlhagen (FG-K) formula for pricing European currency option based on the extension principle. In order to keep consistent with the real market, we assume that the interest rate, the spot exchange rate, and the volatility are fuzzy numbers in the FG-K formula. The conditions of a basic proposition about the fuzzy-valued functions of fuzzy subsets are modified. Based on the modified conditions and the extension principle, we prove that the fuzzy price obtained from the FG-K formula for European currency option is a fuzzy number. To simplify the trade, the weighted possibilistic mean (WPM) value with a weighting function is adopted to defuzzify the fuzzy price to a crisp price. The numerical example shows our method makes the α-level set of fuzzy price smaller, which decreases the fuzziness. The example also indicates that the WPM value has different approximation effects to real market price by taking different values of weighting parameter in the weighting function. Inspired by this example, we provide a method, which can identify the optimal parameter.
With the fast growing of the trading volume in the foreign exchange market, the trading of currency option has increased. It is well know that currency option manages the risk of the foreign exchange market. Hence, an appropriate formula for pricing currency option is becoming extremely significant. Garman and Kohlhagen (1983)  derived the closed-form formula (G-K formula) for pricing European currency option by the method of Black and Scholes (1973) . There are six variables in the G-K formula, that is, the spot exchange rate, the volatility of spot exchange rate, the domestic and foreign risk-free interest rate, the strike price, and the time to maturity date. However, some variables in the G-K formula are assumed as constants, which is inconsistent with the empirical phenomena, such as the volatility smile. According to these empirical phenomena, many researchers have been devoted to modify the G-K model, such as Chesney and Scott (1989) , Amin and Jarrow (1991) , Heston (1993) , Bates (1996) , Sarwar and Krehbiel (2000) , and Carr and Wu (2007) .
In general, the data, for instance, the spot exchange rate, the domestic or foreign risk-free interest rate, cannot be recorded or collected precisely. Usually, these financial tools may have different values in different commercial banks and financial institutions. Meanwhile, there are differences between the buy (bid) and the sale (ask) price for these financial tools, which are similar to bid-ask spreads in dealer markets for the stock. The existence of bid-ask spreads implies that we cannot precisely get the true market price. The bid-ask spreads are considered to be natural bands to represent the uncertainty around the market price, which is similar to fuzzy number. Furthermore, expert opinions or statistical estimators of market parameters may be expressed as fuzzy numbers. Therefore, the fuzzy set theory proposed by Zadeh (1965)  is useful to model such uncertainty. In a parallel development, the fuzzy set theory is applied to option pricing. Yoshida (2003) [10, 11] discussed the valuation of European call and put options in fuzzy environment. Muzzioli and Torricelli (2004)  discussed the pricing of European options in a multiperiod binomial model. Wu (2007)  obtained the fuzzy pattern of Black-Scholes formula using the extension principle. Xu et al. (2009)  and Zhang et al. (2012)  investigated a jump-diffusion option pricing model under fuzzy environment. Liu (2009)  applied fuzzy approach in  to the G-K formula of currency option. Xu et al. (2010)  and Guerra et al. (2011)  calculated the Greek letters of currency option and stock option under uncertainty environment, respectively, where the Greek letters are useful tools for managing option risk for an option writer. But the division operation of fuzzy numbers in Liu , Xu et al.  is invalid occasionally. In this paper, we adopt the extension principle to obtain the fuzzy version of G-K formula for European currency option.
Owing to the vague of the currency market data as we just described above, some input variables, especially the interest rate, the spot exchange rate, and the volatility in the G-K formula, cannot always be expected in a precise sense. Therefore, it is reasonable to assume these input variables in the G-K formula are fuzzy numbers. Then, the fuzzy version of the G-K formula for pricing European currency option is obtained via the extension principle. It is found that one of the conditions of the basic proposition in  is not verified. Actually, it is difficult to be verified. So we modify the conditions such that they can be satisfied easily. Based on the modified conditions and the extension principle, we prove that the fuzzy price obtained from the FG-K formula for European currency option is a fuzzy number. Meanwhile, we obtain the explicit expressions of the endpoints of every -level set (the closed interval) of the fuzzy price for European currency option. In so doing, the investor can pick any value from some closed interval (-level set) of the fuzzy price with an acceptable belief degree as European currency option trading price. To simplify the trade convenient, the weighted possibilistic mean (WPM) value with weighting function  is adopted to defuzzify the fuzzy price. Usually, the process of defuzzification is to find a crisp number that synthesizes the fuzzy price. Compared with the methods in [16, 18], the numerical example shows our method makes the -level set of fuzzy price smaller, which decreases the fuzziness. The reason of the differences is also discussed. The example also indicates that the WPM value has different approximation effects to real market price by taking different values of weighting parameter in the weighting function. Inspired by this example, we provide a method, which can identify the the optimal parameter.
The rest of the paper is organized as follows. In Section 2, some terminologies, notations of fuzzy number, and the main results of [16, 18] are introduced. The fundamental theories about the fuzzy-valued functions of fuzzy subsets are proposed in Section 3. In Section 4, the fuzzy version of the G-K formula for European currency option is obtained by the extension principle. About the fuzzy price for European currency option, defuzzification via weighting parameter identification and numerical analysis are discussed in Section 5. Finally, the conclusions are stated in Section 6.
In this section, we will introduce some definitions and propositions that will be used in the sequel. In order to weaken the conditions of the basic proposition in , we modify the definition of fuzzy number by another definitions of the -level set and the support set of fuzzy subset. Let be the universe of discourse. Throughout this paper, the universe set is assumed to be the set of all real number () endowed with a usual topology.
Definition 1 (see ). A fuzzy subset of is a set of ordered pairs , where is called membership function of .
The concept of fuzzy subset was first introduced by . The value can be interpreted as the membership degree of a point to the set .
Definition 2 (see ). A fuzzy subset of is called normal if there exists at least one element such that .
Definition 3 (see ). Let be a fuzzy subset of . The support of , denoted by , is the crisp set .
Definition 4 (see ). Let be a fuzzy subset of . The -level set of , denoted by , is the crisp set , where .
Definition 5 (see ). A fuzzy subset of is called convex if for all its -level sets are (crisp) convex sets.
Alternatively, a fuzzy subset of is a convex fuzzy subset if and only if for all ; that is, is a quasi-concave function.
Definition 6 (see ). A real-valued function is called upper semicontinuous at a point if for every sequence that converges to . If is upper semicontinuous at every point in , we say that is upper semicontinuous.
A real-valued function is upper semicontinuous if and only if is closed for all .
Definition 7 (fuzzy number). Let be a fuzzy subset of . Then is called a fuzzy number if the following conditions are satisfied:(i) is a normal and convex fuzzy subset,(ii)its membership function is upper semicontinuous,(iii)the is bounded.
Note that if is a fuzzy number, then the -level set , for all , is a compact (closed and bounded in ) and convex set; that is, is a bounded and closed interval for all . Then, the -level set of is denoted as for all , and especially. For convenience, we consider the fuzzy input variables as triangular fuzzy numbers; that is, the graph of the membership function looks like triangles.
Definition 8. A fuzzy number is a triangular fuzzy number if its membership function is characterized as follows: Here, is the support set and the membership function has a peak at . The triangular fuzzy number usually is denoted as .
Definition 9 (see ). Let be a fuzzy number with , . A function is said to be a weighting function if is nonnegative, monotone increasing and satisfies the normalization condition: We define the -weighted possibilistic mean (WPM) value of the fuzzy number as
Next, the decomposition theory is introduced which establishes an important connection between fuzzy sets and crisp sets. That is, we can get the expression of membership function of a fuzzy set if all its -level sets are available.
Let denote all the fuzzy subsets of . Assume , . Let and , where is Cartesian product. A fruitful and powerful tool of fuzzy set theory for calculating the fuzzy-valued functions on fuzzy sets is the extension principle.
Proposition 11 (extension principle [23–25]). Let be a real-valued function and let be fuzzy subsets of . Then one can induce a fuzzy-valued function according to the real-valued function . In fact, is a fuzzy subset of . Then the membership function of is characterized as follows: where .
Remark 12. It is necessary to point out that there is a term in (5), which does not exist in [23–25]. Since , the term does not affect the proposition. However, it plays an important role in our results.
Proposition 13 (see ). Let be a nonempty compact set in . If is upper semicontinuous on , then attains maximum over and if is lower semicontinuous on , then attains minimum over .
Proposition 14. (i) [26, Theorem 4.25] Let be a function from one metric space to another . If is continuous on a compact subset of , then the image is a compact subset of ; in particular, is a closed and bounded in .
(ii) [26, Theorem 4.37] Let be a function from one metric space to another . Let be a connected subset of . If is continuous on , then the is a connected subset of .
3. Expressing of -Level Set of the Fuzzy-Valued Functions of Fuzzy Numbers
In this section, we will establish the fundamental proposition for this paper, which based on the decomposition theorem, the extension principle, and some properties about continuous function. The following proposition is a result about the fuzzy-valued functions of fuzzy subsets.
Proposition 15. Let be a continuous, surjective, real-valued function and let be fuzzy subsets of . Let be a fuzzy-valued function induced by via the extension principle defined in (5). Suppose that(i) is upper semicontinuous on , ,(ii) is a bounded set of ,then
Proof. (a) For , (8) holds from Definition 4 and the property of surjection of .
For , let . Then there exists such that . Thus . It follows that which means that On the other hand, is upper semicontinuous on , then is a closed subset of , . Therefore, is a closed subset of . Since is a closed subset of , we have is upper semicontinuous on . Let , that is, Since is a closed subset of and is a compact set of and is upper semicontinuous on , we conclude from Proposition 13 that there exists such that and . Therefore , that is, , . This means Therefore, which completes the proof of (a).
(b) Let , we conclude from Definition 3 that there exists a sequence which satisfies and , that is, Since is closed and is bounded and closed, then is compact. Moreover, we deduce that is upper semicontinuous on according to the proof of (a). Consequently, we conclude from Proposition 13 that there exist such that where and . Since is compact, for each , there exists a subsequence of such that as . Notice that We have , where . Then For the converse, let . Then there exist sequences , , which satisfy and as , where and . Recall the continuity of , we have This completes the proof.
Remark 16. One of the conditions of Proposition in  is that is compact . This one is not verified there. Actually, it is difficult to be verified. Meanwhile, (8) only holds for . We change the above conditions to (i) is upper semicontinuous on , ; (ii) is bounded of and; (iii) is a continuous, surjective, real-valued function. Conditions (i)–(iii) can be easily satisfied. At the same time, (8) holds for and (9) is also obtained.
The following is the fundamental proposition for discussing the Garman-Kohlhagen formula for European currency option via the extension principle.
Proposition 17. Let be a continuous, surjective, real-valued function and let be fuzzy numbers. Let be a fuzzy-valued function induced by via the extension principle defined in (5). Then is a fuzzy number and its -level set is
Proof. Let for all . Then it follows from Definition 7 that is an -dimensional bounded interval; that is, is a compact and connected subset of . Next, we conclude from Propositions 15 and 14 that(i) is a compact and connected subset of for all ; (ii) is closed and convex set; (iii).Furthermore,(i′) the membership function of is upper semicontinuous; (ii′) is a convex fuzzy subset; (iii′) is bounded.
Obviously, is a normal fuzzy subset, as is nonempty. Therefore, from Definition 7 we conclude that is a fuzzy number and its -level set is (21). This completes the proof.
4. Fuzzy Currency Options Pricing Formula via the Extension Principle
4.1. The G-K Formula for European Currency Option
The closed-form solution for European currency option pricing formula has been derived by Garman and Kohlhagen . The G-K formula for European call currency option with the expiry date and the strike price is described as follows. Let , denote the spot exchange rate, and the price of European call currency option at time , respectively, then where , , , , , , denote the foreign interest rate, the domestic interest rate, the volatility of spot exchange rate and the standard Gaussian cumulative distribution function, respectively. Let then for all . Furthermore, the price of European put option at time with same expiry date and strike price can be obtained by the following put-call parity relationship (Musiela and Rutkowski ): Let for all , then
4.2. The Fuzzy Version of the G-K Formula for European Currency Option
As the discussion in Section 1, it is suitable to represent the four input variables in the G-K formula by fuzzy numbers. Thus, we give the following assumptions.
Assumption 1. (i)The spot exchange rate at time is fuzzy number ; (ii)the domestic interest rate is fuzzy number ; (iii)the foreign interest rate is fuzzy number ; (iv)the volatility of spot exchange rate is fuzzy number .
Based on Assumption 1, (24) and the extension principle (Proposition 11), the original price for European call currency option at time turns into fuzzy subset with the membership function where . Since the function satisfies conditions of Proposition 17, we conclude that the fuzzy price at time is a fuzzy number. Furthermore, its -level set is for all . On the other hand, from Definition 7, the -level set of is a closed interval Notice that the -level set of , , , and are , , , and , respectively. Therefore, from (29) and (30), the left endpoint and the right-end point can be displayed as Moreover, according to the sensitivity analysis (Greek letters) of the G-K formula , we have This mean is increasing with respect to , , and and is decreasing with respect to . Therefore, from (31), we conclude that On the other hand, from Proposition 10, the membership function of can also be written as Finally, (28), (33), and (34) constitute the fuzzy version of the G-K formula for European call currency option.
Similarly, the conclusions about European put currency option are as follows, through the above discussions.
The original price of European put currency option at time turns into fuzzy subset with the membership function under Assumption 1 where is as above. Furthermore, the fuzzy price at time is a fuzzy number and the -level set of is a closed interval Moreover, the left endpoint and the right endpoint can be displayed as On the other hand, the membership function of can also be written as
5. Defuzzification via Weighting Parameter Identification and Numerical Analysis
In this section, we defuzzify the fuzzy price for European currency option (). Then an numerical example is presented. Up to now, it is proved that the fuzzy price for European currency option is a fuzzy number under Assumption 1. For practical purposes, it may be convenient to find a crisp number that synthesizes the fuzzy number (). This type of problem is known as the “defuzzification procedure" in the literature. Defuzzification is the process to select an appropriate crisp value based on a fuzzy set such that the selected crisp value may represent the fuzzy set in some sense.
Of all the methods of defuzzification, two most popular methods are the center of area (COA) and the mean of maxima (MOM) which require explicit membership function. Now, the membership function of () cannot be expressed explicitly; it can be deduced by the decomposition theorem (see (34) or (38)) or the extension principle (see (28) or (35)) only. Since its -level sets of () are available (see (33) or (37)), the weighted possibilistic mean value of Fullér and Majlender  about fuzzy number is fited to be the crisp number that we needed (see Definition 9) in the following.
Next, the proposed fuzzy version of the G-K formula (28) is tested with the daily market price data of EUR/USD currency option. To compare with the results in [16, 18] conveniently, the market data from  are used in Example 18. In the following example, we assumed that the weighting function is , where is the belief degree and is the belief degree of weighting parameter.
Example 18. A European EUR/USD currency call option is studied on march 16, 2006. For the EUR/USD currency option, the spot exchange rate is , the 3-month volatility is around , the domestic 3-month interest rate is around , the foreign 3-month interest rate is around , and the strike price is with 3-month to expiry. The market price of this currency option is USD. The fuzzy foreign and domestic interest rates, the fuzzy spot exchange rate, and the fuzzy volatility are assumed to triangular fuzzy numbers: , , , and , respectively.
The fuzzy price of European call currency option can be obtained by (28). Table 1 gives the -level set of the fuzzy price for different belief degrees and the -weighted possibilistic mean value for different belief degrees of weighting parameters , respectively. In fact, if , , , and , then European call currency option price is by the G-K formula (24). This situation matches the observation with belief degree .
Remark 19. There are some differences between the -level sets of fuzzy call option price in this paper and those in [16, 18], which are shown in Table 1. The reason is that the division operation of fuzzy numbers in [16, 18] is invalid occasionally. Table 1 also shows that is the best approximation to the real price when .
Next, assuming the currency option prices at dates are available, how to acquire a more accurate at date is an important question. Based on minimizing the mean square error between the market prices and WPMs, a method of weighting parameter identification is presented as follows.
Frame:(i)collect the market price data: ; (ii)calculate the performance index ; (iii)solve and optimal solution is obtained and; (iv)calculate , and let .
In this paper, the Fuzzy version of the Garman-Kohlhagen (FG-K) formula for pricing European currency option via the extension principle is proposed. In the FG-K formula, the interest rate, the spot exchange rate, and the volatility in conventional G-K formula are replaced by fuzzy numbers. So the FG-K formula is able to keep consistent with the real market. We prove that the fuzzy price for European currency option is a fuzzy number. And then the fuzzy price () is defuzzified by the weighted possibilistic mean value. The example shows that the fuzzy price in this paper is more accurate than the ones in [16, 18] and the fuzzy prices in [16, 18] are just approximations to our fuzzy price for European currency option. In addition, the example indicates that the WPM value has different approximation effects to real market price by taking different values of weighting parameter in the weighting function. So an identification method is presented. And the optimal parameter is obtained if we have already known some market prices for European currency option. The method in this paper can be extended to other option pricing. How to find out different kinds of weighting function in defuzzification requires a further study.
This work was supported by 863 Program (Grant no. 2007AA02Z208), 973 Program (Grant no. 2007CB714304), and the National Natural Science Foundation of China (Grants nos. 10871033 and 11171050).
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