- Email: [email protected]
Pricing European options based on the fuzzy pattern of Black–Scholes formula
Available online at www.sciencedirect.com
Computers & Operations Research 31 (2004) 1069 – 1081
www.elsevier.com/locate/dsw
Pricing European options based on the fuzzy pattern of Black–Scholes formula Hsien-Chung Wu Department of Information Management, National Chi Nan University, Puli, Nantou 545, Taiwan
Abstract The application of fuzzy sets theory to the Black–Scholes formula is proposed in this paper. Owing to the 1uctuation of 2nancial market from time to time, some input parameters in the Black–Scholes formula cannot always be expected in the precise sense. Therefore, it is natural to consider the fuzzy interest rate, fuzzy volatility and fuzzy stock price. The fuzzy pattern of Black–Scholes formula and put–call parity relationship are then proposed in this paper. Under these assumptions, the European option price will turn into a fuzzy number. This makes the 2nancial analyst who can pick any European option price with an acceptable belief degree for the later use. In order to obtain the belief degree, an optimization problem has to be solved. ? 2003 Elsevier Ltd. All rights reserved. Keywords: Black–Scholes formula; European call option; European put option; Fuzzy numbers; Fuzzy random variables; Optimization; Put–call parity
1. Introduction Since the closed-form solution of the European call option was derived by Black and Scholes [1], many methodologies for the option pricing have been proposed by using the modi2cation of Black– Scholes formula. The input parameters of the Black–Scholes formula are usually regarded as the precise real-valued data; that is, the input data are considered as the real numbers. However, in the real world, some parameters in the Black–Scholes formula cannot always be expected in a precise sense. For instance, the risk-free interest rate r may occur imprecisely. Therefore, the fuzzy sets theory proposed by Zadeh [2] may be a useful tool for modeling this kind of imprecise problem. The book of collected papers edited by Ribeiro et al. [3] gave the applications of using fuzzy sets theory to the discipline called 2nancial engineering. E-mail address: [email protected] (H.-C. Wu). 0305-0548/$ - see front matter ? 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0305-0548(03)00065-0
1070
H.-C. Wu / Computers & Operations Research 31 (2004) 1069 – 1081
Let X be a universal set and A be a subset of X . We can de2ne a characteristic function A : X → {0; 1} with respect to A by 1 if a ∈ A; A (a) = 0 if a ∈ A: Zadeh [2] introduced the concept of fuzzy subset A˜ of X by extending the characteristic function. A fuzzy subset A˜ of X is de2ned by its membership function A˜ : X → [0; 1] which is viewed as an extension of characteristic function. The value A˜(a) can be interpreted as the membership degree of a point a in the set “A.” Under some suitable conditions for the membership function, the fuzzy set is then termed as a fuzzy number (in Section 2). Let a be a real number. The fuzzy number a˜ corresponding to a can be interpreted as “around a.” The graph of the membership function a˜(x) is bell shaped and a˜(a) = 1. It means that the membership degree a˜(x) is close to 1 when the value x is close to a. The possibility theory derived from fuzzy sets theory is always compared with the probability theory. The detailed discussion can be referred to Zadeh [4] and Dubois and Prade [5]. Zadeh [4] illustrates an example to tell the diHerence between possibility distribution and probability distribution. Let us describe it as follows. Consider the statement “John ate X eggs for breakfast,” where X = {1; 2; : : :}. A possibility distribution and probability distribution can be associated with the variable X . The possibility distribution X (x) can be interpreted as the possibility with which John can eat x eggs for breakfast. The probability distribution PX (x) is interpreted as the probability with which John can eat x eggs for breakfast, or alternatively, the probability distribution PX (x) is determined by observing the number of eggs ate by John for 100 days. The obvious diHerence is that P (x) = 1 and x X x X (x) is not necessarily equal to 1. For instance, the values of X (x) and PX (x) can be shown in the following table: x
1
2
3
4
5
6
7
8
X (x) PX (x)
1.0 0.1
1.0 0.8
1.0 0.1
0.9 0.0
0.8 0.0
0.6 0.0
0.4 0.0
0.2 0.0
We observe that a high degree of possibility does not imply a high degree of probability. The most possible number of eggs ate by John is 1, 2, or 3 while the most probable number of eggs ate by John is 2. Therefore, the possibility theory and probability theory cannot be substituted for each other. However, the interesting research topic for combining the probability theory and fuzzy sets theory can complement each other. For instance, the notion of fuzzy random variable that unites the randomness and fuzziness has been introduced by Kwakernaak [6] and Puri and Ralescu [7]. In the real world, the data sometimes cannot be recorded or collected precisely. For instance, the water level of a river cannot be measured in an exact way because of the 1uctuation, and the price of a product 1uctuate from time to time according to the market eHects. Therefore, the fuzzy sets theory naturally provides an appropriate tool in modeling the imprecise models. For instance, the more appropriate way to describe the water level is to say that the water level is around 30 m. ˜ Now let us consider another The phrase “around 30 m” should be regarded as a fuzzy number 30.
H.-C. Wu / Computers & Operations Research 31 (2004) 1069 – 1081
1071
instance, considering to ask a group of economists for predicting the rate of economical growth in the next year. Their statements may be like “approximately 5%,” “should be 4 – 6%” or “may be below 7%,” etc. All of those statements can be characterized as fuzzy sets. Suppose that we introduce a selection procedure which will determine the probability that each economist will be selected. Then for each possible selection, their statements are fuzzy sets. Thus, the association of fuzzy sets and probabilities forms the notion of fuzzy random variable. Roughly speaking, the observations of random variables are real numbers and the observations of fuzzy random variables are fuzzy numbers. The concept of fuzzy random variable was introduced by Kwakernaak [6] and Puri and Ralescu [7]. The occurrence of fuzzy random variable makes the combination of randomness and fuzziness more persuasive, since the probability theory can be used to model uncertainty, and the fuzzy sets theory can be used to model imprecision. Now it is the right time to explain the reasons for considering the imprecise (fuzzy) input data of the Black–Scholes formula. In the usual approach, there are two ways to describe the risk-free interest rate. One assumes the risk-free interest rate as a constant, and another one assumes the stochastic interest rate. As we just described above, the theories for modeling the randomness and fuzziness cannot be substituted for each other. Therefore, the methodology for considering the fuzzy interest rate will be totally diHerent from that of considering the stochastic interest rate. As a matter of fact, in the general case, it is still possible to take into account the imprecisely stochastic interest rate for the options pricing. Under this situation, the imprecisely stochastic interest rate can be regarded as a fuzzy random variable. Therefore, it means that the fuzzy interest rate is the corresponding consideration of the constant interest rate under fuzzy environment, and the imprecisely stochastic interest rate (i.e. the fuzzy random variable) is the corresponding consideration of stochastic interest rate under fuzzy environment. Only the fuzzy interest rate is under investigated in this paper as the beginning step for studying the options pricing by using the fuzzy sets theory. The study for the imprecisely stochastic interest rate will be the future research. Next we provide the motivation why we need to take into account the fuzzy interest rate in the Black–Scholes formula. In the constant interest rate approach, when the 2nancial analyst tries to price an option, the interest rate is assumed as a constant. However, the interest rate may have the diHerent values in the diHerent commercial banks and 2nancial institutions. Therefore, the choice of a reasonable interest rate may cause a dilemma (Note that the situation for the diHerent interest rates in the diHerent commercial banks and 2nancial institutions may not be regarded as a kind of randomness). But one thing can be sure is that the diHerent interest rates may be around a 2xed value within a short period of time. For instance, the interest rates may be around 5% (The phrase “around 5%” might have problems to be modeled by using the probability theory. Of course, it might be possible to enforce to model this phrase “around 5%” by using the probability theory. However, under this situation, the structure of the interest rate could be very complicated. Then it might cause the diOculties for ˜ when the further analysis). In this case, the interest rate may be regarded as a fuzzy number 5% 2nancial analyst tries to price an European call option using the Black–Scholes formula. On the other hand, there are two approaches used to obtain the volatilities in order to apply the Black–Scholes formula, the so-called historical volatility approach and implied volatility approach. Therefore, the choice of a reasonable volatility approach may also cause a dilemma. The other reason is that the 2nancial market 1uctuates from time to time. It is a little unreasonable to pick a 2xed volatility to price an option at this time and use this option price for the later use, since the later volatility has already changed. In this case, it is natural to regard the volatility as an imprecise
1072
H.-C. Wu / Computers & Operations Research 31 (2004) 1069 – 1081
(fuzzy) data. Of course, the same reasons as described in the case of interest rate, the methodology for considering the fuzzy volatility will be completely diHerent from that of considering the stochastic volatility. As a matter of fact, in the general case, the stochastic volatility can also be extended by considering the imprecisely stochastic volatility. In this case, the imprecisely stochastic volatility can be modeled as a fuzzy random variable. This will be the future research. The usual approach for options pricing is to pick a 2xed stock price to price an option by applying the Black–Scholes formula. However, this option price will be used for the further decision making by a 2nancial analyst within a short period of time. The problem is that the stock price will be changed within this short period of time because the stock price 1uctuates very irregular (assumed to be a Brownian motion). Therefore, it is a little unreasonable to pick a 2xed stock price to price an option at this time and then use this option price for later use, since the later stock price has already changed. Therefore, it is also natural to assume the stock price as a fuzzy number. If this assumption does not convince the 2nancial analysts, then they still can take the stock price as a real number and just consider the fuzzy interest rate and fuzzy volatility. Under this situation, the methodology proposed in this paper is still applicable since the real numbers are the special case of the fuzzy numbers. Although we have described how the fuzzy interest rate, fuzzy volatility and fuzzy stock price occur in the real world, some of those three input data can still be taken as the real numbers (it is also called the crisp numbers in the fuzzy literature) if the 2nancial analyst can make sure that those data occur in a crisp sense. In this case, the methodology proposed in this paper is still applicable since the real numbers (crisp numbers) are the degenerated case (special case) of the fuzzy numbers. Now, under the considerations of fuzzy interest rate, fuzzy volatility and fuzzy stock price, the option price will turn into a fuzzy number. The price of a European call option can now be interpreted as The fuzzy “around $3.5,” for instance. The phrase “around $3.5” is regarded as a fuzzy number 3:5. number 3:5 is in fact a function de2ned on R into [0; 1], and denoted by 3:5 (r). Given any value r0 , the function value 3:5 (r ) will be interpreted as the belief degree of closeness to value 3:5. The 0 graph of this function 3:5 (r) will be bell shaped. It means that the closer the value r0 to 3:5 is, the higher the belief degree is. Therefore, the 2nancial analyst can pick any value which is around 3:5 with an acceptable belief degree as the option price for his (her) later use. In order to obtain the belief degree of any given option price, an optimization problem will be solved. An eOcient computational procedure is proposed in this paper to solve this optimization problem. This paper is organized as follows. In Section 2, the notions of fuzzy number and the arithmetics of fuzzy numbers are introduced. In Section 3, the notion of fuzzy random variable is introduced. In Section 4, the fuzzy patterns of the Black–Scholes formula and put–call parity relationship are proposed. In Section 5, the computational procedure is provided in order to obtain the belief degrees of any given option prices. Finally, the conclusions of this paper is depicted in the 2nal Section 6. 2. Fuzzy numbers Let X be a universal set and A˜ be a fuzzy subset of X . We denote by A˜ = {x: A˜(x) ¿ } ˜ where A˜0 is the closure of the set {x: A˜(x) = 0}. A˜ is called a northe -level set of A, mal fuzzy set if there exists x such that A˜(x) = 1. A˜ is called a convex fuzzy set if A˜( x + (1 − )y) ¿ min{A˜(x); A˜(y)} for ∈ [0; 1]. (That is, A˜ is a quasi-concave function.)
H.-C. Wu / Computers & Operations Research 31 (2004) 1069 – 1081
1073
In this paper, the universal set X is assumed to be a real number system; that is, X = R. Let f be a real-valued function de2ned on R. f is said to be upper semicontinuous, if {x: f(x) ¿ } is a closed set for each . Or equivalently, f is upper semicontinuous at y if and only if ∀ ¿ 0; ∃ ¿ 0 such that |x − y| ¡ implies f(x) ¡ f(y) + . a˜ is called a fuzzy number if the following conditions are satis2ed: (i) a˜ is a normal and convex fuzzy set. (ii) Its membership function a˜ is upper semicontinuous. (iii) The -level set a˜ is bounded for each ∈ [0; 1]. From Zadeh [2], A˜ is a convex fuzzy set if and only if its -level set A˜ ={x: A˜(x) ¿ } is a convex set for all . Therefore, if a˜ is a fuzzy number, then the -level set a˜ is a compact (closed and bounded in R) and convex set; that is, a˜ is a closed interval. The -level set of a˜ is then denoted by a˜ = [a˜L ; a˜U ]. A fuzzy number a˜ is said to be nonnegative if a˜(x) = 0 for x ¡ 0. It is easy to see that if a˜ is a nonnegative fuzzy numbers then a˜L and a˜U are all nonnegative real numbers for all ∈ [0; 1]. The following proposition is useful for further discussion. Proposition 2.1 (Zadeh [8, Resolution identity]). Let A˜ be a fuzzy set with membership function A˜ and A˜ = {x: A˜(x) ¿ }. Then A˜(x) = sup 1A˜ (x); ∈[0;1]
where 1A is an indicator function of set A, i.e., 1A (x) = 1 if x ∈ A and 1A (x) = 0 if x ∈ A. Note that the -level set A˜ of A˜ is a crisp (usual) set. a˜ is called a crisp number with value m if its membership function is 1 if x = m; a˜(x) = 0 otherwise: It is denoted by a˜ ≡ 1˜{m} . It is easy to see that (1˜{m} )L = (1˜{m} )U = m for all ∈ [0; 1]. We see that the real numbers are the degenerated case (special case) of the fuzzy numbers when the real numbers are regarded as the crisp numbers. Now it is the right time to introduce the arithmetics of any two fuzzy numbers. Let “ ” be a ˜ The membership function of binary operation ⊕, , ⊗ or between two fuzzy numbers a˜ and b. ˜ a˜ b is de2ned by a˜b˜(z) =
sup
{(x;y): x◦y=z }
min{a˜(x); b˜(y)};
where the binary operations = ⊕, , ⊗ or correspond to the binary operations ◦ = +; −, × or / according to the “Extension Principle” in Zadeh [2]. Let “ int ” be a binary operation ⊕int , int or ⊗int between two closed intervals [a; b] and [c; d]. Then [a; b] int [c; d] is de2ned by {z ∈ R: z = x ◦ y; ∀x ∈ [a; b]; ∀y ∈ [c; d]; [a; b] int [c; d] ≡ where “ ◦ ” is an usual binary operation +; − or ×}
1074
H.-C. Wu / Computers & Operations Research 31 (2004) 1069 – 1081
and [a; b] int [c; d] ≡ {z ∈ R: z = x=y; ∀x ∈ [a; b]; ∀y ∈ [c; d]} if the interval [c; d] does not contain zero. Then the following well-known results are not hard to prove. ˜ a˜ b˜ and a˜ ⊗ b˜ are also fuzzy Proposition 2.2. Let a˜ and b˜ be two fuzzy numbers. Then a˜ ⊕ b; numbers and their -level sets are ˜ = a˜ ⊕int b˜ = [a˜L + b˜L ; a˜U + b˜U ]; (a˜ ⊕ b) ˜ = a˜ int b˜ = [a˜L − b˜U ; a˜U − b˜L ]; (a˜ b) ˜ = a˜ ⊗int b˜ = [min{a˜L b˜L ; a˜L b˜U ; a˜U b˜L ; a˜U b˜U }; max{a˜L b˜L ; a˜L b˜U ; a˜U b˜L ; a˜U b˜U }] (a˜ ⊗ b) for all ∈ [0; 1]. If the -level set b˜ of b˜ does not contain zero for all ∈ [0; 1], then a˜ b˜ is also a fuzzy number and its -level set is ˜ = a˜ int b˜ = [min{a˜L = b˜L ; a˜L = b˜U ; a˜U = b˜L ; a˜U = b˜U }; max{a˜L = b˜L ; a˜L = b˜U ; a˜U = b˜L ; a˜U = b˜U }] (a˜ b) for all ∈ [0; 1]. Let F denote the set of all fuzzy subsets of R. Let f(x) be a nonfuzzy real-valued function from R into R and A˜ be a fuzzy subset of R. By the extension principle in Zadeh [8], the fuzzy-valued ˜ A) ˜ is a fuzzy subset function f˜ : F → F can be induced by the nonfuzzy function f(x); that is, f( ˜ ˜ of R. The membership function of f(A) is de2ned by f( ˜ A) ˜ (r) =
sup
{x: r=f(x)}
A˜(x):
The following proposition is very useful for discussing the fuzzy pattern of the Black–Scholes formula. Proposition 2.3. Let f(x) be a real-valued function and A˜ be a fuzzy subset of R. The function f(x) can induce a fuzzy-valued function f˜ : F→F via the extension principle. Suppose that the membership function A˜ of A˜ is upper semicontinuous and {x: r = f(x)} is a compact set (it will be ˜ A) ˜ A)) ˜ is (f( ˜ = {f(x): x ∈ A˜ }. a closed and bounded set in R) for all r, then the -level set of f( Proof. If r ∈ {f(x): x ∈ A˜ }, then there exists an x such that r = f(x) and x ∈ A˜ ; that is, A˜(x) ¿ . ˜ ˜ ˜ ˜ ˜ Thus, f( ˜ A) ˜ (r) = sup{x: r=f(x)} A˜(x) ¿ implies r ∈ (f(A)) . It says that {f(x): x ∈ A } ⊆ (f(A)) . ˜ A)) ˜ , then sup{x: r=f(x)} A˜(x) ¿ ; that is, there exists an x such that On the other hand, if r ∈ (f( A˜(x) ¿ and r = f(x), since {x: r = f(x)} is a compact set and A˜(x) is upper semicontinuous (Using the fact that an upper semicontinuous function assumes maximum over a compact set in Bazarra and Shetty [9]. Therefore, r ∈ {f(x): x ∈ A˜ }. This completes the proof.
H.-C. Wu / Computers & Operations Research 31 (2004) 1069 – 1081
1075
3. Fuzzy random variables The stock price S is a stochastic process, i.e., the stock price St at time t is a random variable. Therefore, if the fuzzy stock price is considered at time t in the Black–Scholes formula, then it is natural to introduce the notion of fuzzy random variable. Roughly speaking, the usual random variable assumes real numbers and the fuzzy random variable assumes fuzzy numbers; that is, the observations of random variables are real numbers and the observations of fuzzy random variables are fuzzy numbers. Let ( ; A) be a measurable space and (R; B) be a Borel measurable space. Let f : → P(R) (set of all subsets of R) be a set-valued function. According to Aumann [10], f is called measurable if and only if each of the set {(x; y): y ∈ f(x)} is A × B-measurable. Let F(R) denote the set of all fuzzy numbers. If f˜ : → F(R) is a fuzzy-valued function, then f˜ is a set-valued function for all ∈ [0; 1]. f˜ is called (fuzzy-valued) measurable if and only if f˜ is (set-valued) measurable for all ∈ [0; 1]. Let ( ; A; P) be a probability space (a complete -2nite measure space). Then X : → R is a random variable if X is (A; B)-measurable function. Thus, it is natural to propose the following de2nition. De nition 3.1. Let X˜ : →F(R) be a fuzzy-valued function. X˜ is called a fuzzy random variable if X˜ is measurable. Then the following proposition holds true obviously (Puri and Ralescu [7] and KrRatschmer [11]). Proposition 3.1. Let X˜ : →F(R) be a fuzzy-valued function. X˜ is a fuzzy random variable if and only if X˜ L and X˜ U are random variables for all ∈ [0; 1]. 4. Fuzzy pattern of Black–Scholes formula The well-known Black–Scholes formula for European call option written on a stock S with expiry date T and strike price K (Black and Scholes [1]) is described as follows. Let the function c be given by the formula c(s; t; K; r; ) = sN (d1 ) − Ke−rt N (d2 ); where d1 = and
ln(s=K) + (r + 2 =2)t √ t
√ d2 = d1 − t
and N stands for the cumulative distribution function of a standard normal random variable N (0; 1). Let Ct denote the price of a European call option at time t ∈ [0; T ]. Then Ct = c(St ; T − t; K; r; )
1076
H.-C. Wu / Computers & Operations Research 31 (2004) 1069 – 1081
for all t ∈ [0; T ]. Furthermore, the price Pt of a European put option at time t with the same expiry date T and strike price K can be obtained by the following put–call parity relationship (Musiela and Rutkowski [12]) Ct − Pt = St − Ke−r(T −t) for all t ∈ [0; T ]. Under the considerations of fuzzy interest rate r, ˜ fuzzy volatility ˜ and fuzzy stock price S˜ (here the fuzzy stock price is assumed to be a nonnegative fuzzy number), the fuzzy pattern of the Black– Scholes formula is described as follows. Let the fuzzy-valued function c˜ be given by the formula ˜ c( ˜ s;˜ t; K; r; ˜ ) ˜ = (s˜ ⊗ N˜ (d˜ 1 )) (1˜{K } ⊗ e−r˜⊗1{t} ⊗ N˜ (d˜ 2 ));
where d˜ 1 = [ln(s˜ 1˜{K } ) ⊕ (r˜ ⊕ ( ˜ ⊗ ˜ 1˜{2} )) ⊗ 1˜{t } ] ( ˜ ⊗ and d˜ 2 = d˜ 1 ( ˜ ⊗
1˜{t } )
1˜{t } ):
Since the strike price K and time t are real numbers, they are displayed as the crisp numbers 1˜{K } and 1˜{t } with values K and t, respectively. Then the fuzzy price of a European call option at time t is given by the formula C˜ t = c( ˜ S˜ t ; T − t; K; r; ˜ ); ˜ where S˜ t and C˜ t are fuzzy random variables, for all t ∈ [0; T ]. Furthermore, the fuzzy price P˜ t of a European put option at time t with the same expiry date T and strike price K can be obtained by the following fuzzy pattern of put–call parity relationship: ˜ P˜ t = C˜ t S˜ t ⊕ (1˜{K } ⊗ e−r˜⊗1{T −t} )
(1)
for all t ∈ [0; T ]. According to the “Resolution identity” in Proposition 2.1, the membership function of C˜ t is given by C˜ t (c) = sup 1(C˜ t ) (c); 06 61
(2)
where (C˜ t ) is the -level set of the fuzzy price C˜ t of a European call option at time t. Now it is the right time to display the left- and right-end points of the closed interval (C˜ t ) = [(C˜ t )L ; (C˜ t )U ]: From Proposition 2.3, since the cumulative distribution function N (x) is increasing, the -level set ˜ is given by of N˜ (d) ˜ = {N (x): x ∈ d˜ } = {N (x): d˜ L 6 x 6 d˜ U } = [N (d˜ L ); N (d˜ U )]: (N˜ (d))
(3)
e− x
is a decreasing function and ln x is an increasing function, the -level sets of Similarly, since − r˜⊗1˜{t } ˜ e and ln(S t 1˜{K } ) are then given by U L ˜ (e−r˜⊗1{t} ) = {e−x : x ∈ (r˜ ⊗ 1˜{t } ) } = {e−x : r˜L t 6 x 6 r˜U t} = [e−r˜ t ; e−r˜ t ]
(4)
H.-C. Wu / Computers & Operations Research 31 (2004) 1069 – 1081
1077
and (ln(S˜ t 1˜{K } )) = [ln((S˜ t )L =K)); ln((S˜ t )U =K))]
(5)
by using Propositions 2.2 and 2.3. Then the left-end point (C˜ t )L and right-end point (C˜ t )U of the closed interval (C˜ t ) will be displayed as follows by using Proposition 2.2 and Eqs. (3), (4), and (5) (note that, since the fuzzy stock prices are assumed to be nonnegative fuzzy numbers, (S˜ t )L and (S˜ t )U are clearly nonnegative for all ∈ [0; 1]): L (C˜ t )L = (S˜ t )L N ((d˜ 1 )L ) − Ke−r˜ (T −t) N ((d˜ 2 )U )
and (C˜ t )U = (S˜ t )U N ((d˜ 1 )U ) − Ke−r˜ (T −t) N ((d˜ 2 )L ); U
where ln((S˜ t )L =K)) + (r˜L + ( ˜ L )2 =2)(T − t) √ ; ˜ U T − t ln((S˜ t )U =K)) + (r˜U + ( ˜ U )2 =2)(T − t) √ ; (d˜ 1 )U = ˜ L T − t √ (d˜ 2 )L = (d˜ 1 )L − ˜ U T − t (d˜ 1 )L =
and
√ (d˜ 2 )U = (d˜ 1 )U − ˜ L T − t:
According to the fuzzy pattern of put–call parity relationship in (1), the fuzzy price P˜ t of a European put option at time t can be derived similarly. The left-end point (P˜ t )L and right-end point (P˜ t )U of the closed interval (P˜ t ) are given by (P˜ t )L = (C˜ t )L − (S˜ t )U + Ke−r˜
U
(T −t)
(6)
(P˜ t )U = (C˜ t )U − (S˜ t )L + Ke−r˜ (T −t)
(7)
and L
for all ∈ [0; 1] using Proposition 2.2 again. 5. Computational method and example Given a European call option price c of the fuzzy price C˜ t at time t, we plan to know its membership value , i.e., its belief degree. If the 2nancial analysts are comfortable with this membership value , then it will be reasonable to take the value c as the European option price at time t. In this case, the 2nancial analysts can accept the value c as the European option price at time t with belief degree . The membership function of fuzzy price C˜ t of a European call option at time t is given in Eq. (2). Therefore, given any European call option price c, its belief degree can be obtained by solving the
1078
H.-C. Wu / Computers & Operations Research 31 (2004) 1069 – 1081
following optimization problem: (MP1)
max
subject to
(C˜ t )L 6 c 6 (C˜ t )U 0 6 6 1:
Since g( ) = (C˜ t )L is an increasing function of and h( ) = (C˜ t )U is a decreasing function of , the following procedure is proposed in order to make the optimization problem (MP1) easier to be solved. The optimization problem (MP1) can be rewritten as (MP2)
max
subject to
g( ) 6 c h( ) ¿ c 0 6 6 1:
Since g( ) = (C˜ t )L 6 (C˜ t )U = h( ), one of the constraints g( ) 6 c or h( ) ¿ c can be discarded in the following ways: (i) If g(1) 6 c 6 h(1) then C˜ t (c) = 1. (ii) If c ¡ g(1) then the constraint h( ) ¿ c is redundant, since h( ) ¿ h(1) ¿ g(1) ¿ c for all ∈ [0; 1] using the fact that h( ) is decreasing and g( ) 6 h( ) for all ∈ [0; 1]. Thus, the following relaxed optimization problem will be solved: (MP3)
max
subject to
g( ) 6 c 0 6 6 1:
(iii) If c ¿ h(1) then the constraint g( ) 6 c is redundant, since g( ) 6 g(1) 6 h(1) 6 c for all ∈ [0; 1] using the fact that g( ) is increasing and g( ) 6 h( ) for all ∈ [0; 1]. Thus, the following relaxed optimization problem will be solved: (MP4)
max
subject to
h( ) ¿ c 0 6 6 1:
Since g( ) is continuous and increasing, problem (MP3) can be solved using the following algorithm (bisection search). Step Step Step and go Step
1: Let be the tolerance and 0 be the initial value. Set ← 0 , low ← 0 and up ← 1. 2: Find g( ) = (C˜ t )L . If g( ) 6 c then go to Step 3 otherwise go to Step 4. 3: If c−g( ) ¡ then EXIT and the maximum is otherwise set low ← , ← (low+up)=2 to Step 2. 4: Set up ← , ← (low + up)=2 and go to Step 2.
H.-C. Wu / Computers & Operations Research 31 (2004) 1069 – 1081
1079
For problem (MP4), it is enough to consider the equivalent constraint −h( ) 6 − c; since h( ) is decreasing and continuous, i.e., −h( ) is increasing and continuous. Thus, the above algorithm is still applicable for solving problem (MP4). The membership function of a triangular fuzzy number a˜ is de2ned by (r − a1 )=(a2 − a1 ) a˜(r) = (a3 − r)=(a3 − a2 ) 0
if a1 6 r 6 a2 ; if a2 ¡ r 6 a3 ; otherwise;
which is denoted by a˜ = (a1 ; a2 ; a3 ) (The graph of the membership function a˜(r) looks like a triangle). The triangular fuzzy number a˜ can be expressed as “around a2 ” or “being approximately equal to a2 ”. a2 is called the core value of a, ˜ and a1 and a3 are called the left- and right-spread values of a, ˜ respectively. The -level set (a closed interval) of a˜ is then a˜ = [(1 − )a1 + a2 ; (1 − )a3 + a2 ]; that is, a˜L = (1 − )a1 + a2
and
a˜U = (1 − )a3 + a2 :
Example 5.1. Consider a European call option on a stock with strike price $30 and with 3 months to expiry. Suppose that the current stock price is around $33, the stock price volatility is around 10% and the risk-free interest rate is around 5% per annum with continuous compounding. Assume that t = 0 and T = 0:25. The fuzzy interest rate r, ˜ fuzzy volatility ˜ and fuzzy stock price S˜ 0 are assumed to be triangular fuzzy numbers r=(0:048; ˜ 0:05; 0:052), S˜ 0 =(32; 33; 34) and =(0:08; 0:1; 0:12), respectively. ˜ The current fuzzy price C 0 of a European call option can be obtained. Then the following table gives the belief degrees C˜ 0 (c) for diHerent European call option prices c by solving the optimization problems (MP3) and (MP4) using the computational procedure proposed above. c 2.58 2.78 2.98 3.18 3.38 3.39 3.59 3.79 3.99 4.19 C˜ 0 (c) 0.7793 0.8338 0.8888 0.9441 0.9996 0.9976 0.9421 0.8870 0.8323 0.7782 For example, if the European call option price $3.59 is taken, then its belief degree is 0:9421. Therefore, if a 2nancial analyst is comfortable with this belief degree 0:9421, then he (she) can take this option price $3.59 for his (her) later use. If the European call option price is taken as c = 3:3813, then its belief degree will be 1:00. As a matter of fact, if we take r = 0:05, S0 = 33 and = 0:1, then the European call option price will be $3.3813. This situation matches the observation that c = 3:3813 has belief degree 1:00.
1080
H.-C. Wu / Computers & Operations Research 31 (2004) 1069 – 1081
The following table gives the -level closed intervals (C˜ 0 ) of the fuzzy price C˜ 0 of a European call option.
(C˜ 0 )
0.99 0.98 0.97 0.96 0.95 0.94 0.93 0.92 0.91 0.90
[3:3453; 3:4174] [3:3092; 3:4534] [3:2732; 3:4895] [3:2372; 3:5256] [3:2011; 3:5617] [3:1650; 3:5978] [3:1289; 3:6340] [3:0928; 3:6702] [3:0567; 3:7064] [3:0205; 3:7427]
For = 0:95, it means that the call option price will lie in the closed interval [3:2011; 3:5617] with belief degree 0.95. From another point of view, if a 2nancial analyst is comfortable with this belief degree 0.95, then he (she) can pick any value from the closed interval [3:2011; 3:5617] as the option price for his (her) later use. The fuzzy price of a European put option can be obtained from Eqs. (6) and (7), and the belief degrees and the -closed intervals can also be evaluated similarly. 6. Conclusions Owing to the 1uctuation of 2nancial market from time to time, some input data in the Black– Scholes formula cannot always 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 Black–Scholes formula and put–call parity relationship are then proposed in this paper. Under the considerations of fuzzy interest rate, fuzzy volatility and fuzzy stock price, the European option price turns into a fuzzy number. This makes the 2nancial analyst who can pick any European option price with an acceptable belief degree for his (her) later use. A computational procedure is also proposed in order to obtain the belief degree. Since the price of an American call option without dividends is the same as the price of a European call option, the methodology proposed in this paper may also be applicable for the American call option without dividends. References [1] Black F, Scholes M. The pricing of options and corporate liabilities. Journal of Political Economy 1973;81:637–59. [2] Zadeh LA. Fuzzy sets. Information and Control 1965;8:338–53. [3] Ribeiro RA, Zimmermann H-J, Yager RR, Kacprzyk J. editors. Soft computing in 2nancial engineering. Heidelberg: Physica-Verlag, 1999.
H.-C. Wu / Computers & Operations Research 31 (2004) 1069 – 1081 [4] [5] [6] [7] [8] [9] [10] [11] [12]
1081
Zadeh LA. Fuzzy sets as a basis for a theory of possibility. Fuzzy Sets and Systems 1978;1:3–28. Dubois D, Prade H. Possibility theory. New York: Plenum Press, 1988. Kwakernaak H. Fuzzy random variables I: De2nitions and theorems. Information Sciences 1978;15:1–29. Puri ML, Ralescu DA. Fuzzy random variables. Journal of Mathematical Analysis and Applications 1986;114: 409–22. Zadeh LA. The concept of linguistic variable and its application to approximate reasoning I, II and III. Information Sciences 1975;8:199–249. 8:301–57; 9:43–80. Bazarra MS, Shetty CM. Nonlinear programming. New York: Wiley, 1993. Aumann RJ. Integrals of set-valued functions. Journal of Mathematical Analysis and Applications 1965;12:1–12. KrRatschmer V. A uni2ed approach to fuzzy random variables. Fuzzy Sets and Systems 2001;123:1–9. Musiela M, Rutkowski M. Martingale methods in 2nancial modelling. Berlin, Heidelberg: Springer, 1997.
Computers & Operations Research 31 (2004) 1069 – 1081
www.elsevier.com/locate/dsw
Pricing European options based on the fuzzy pattern of Black–Scholes formula Hsien-Chung Wu Department of Information Management, National Chi Nan University, Puli, Nantou 545, Taiwan
Abstract The application of fuzzy sets theory to the Black–Scholes formula is proposed in this paper. Owing to the 1uctuation of 2nancial market from time to time, some input parameters in the Black–Scholes formula cannot always be expected in the precise sense. Therefore, it is natural to consider the fuzzy interest rate, fuzzy volatility and fuzzy stock price. The fuzzy pattern of Black–Scholes formula and put–call parity relationship are then proposed in this paper. Under these assumptions, the European option price will turn into a fuzzy number. This makes the 2nancial analyst who can pick any European option price with an acceptable belief degree for the later use. In order to obtain the belief degree, an optimization problem has to be solved. ? 2003 Elsevier Ltd. All rights reserved. Keywords: Black–Scholes formula; European call option; European put option; Fuzzy numbers; Fuzzy random variables; Optimization; Put–call parity
1. Introduction Since the closed-form solution of the European call option was derived by Black and Scholes [1], many methodologies for the option pricing have been proposed by using the modi2cation of Black– Scholes formula. The input parameters of the Black–Scholes formula are usually regarded as the precise real-valued data; that is, the input data are considered as the real numbers. However, in the real world, some parameters in the Black–Scholes formula cannot always be expected in a precise sense. For instance, the risk-free interest rate r may occur imprecisely. Therefore, the fuzzy sets theory proposed by Zadeh [2] may be a useful tool for modeling this kind of imprecise problem. The book of collected papers edited by Ribeiro et al. [3] gave the applications of using fuzzy sets theory to the discipline called 2nancial engineering. E-mail address: [email protected] (H.-C. Wu). 0305-0548/$ - see front matter ? 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0305-0548(03)00065-0
1070
H.-C. Wu / Computers & Operations Research 31 (2004) 1069 – 1081
Let X be a universal set and A be a subset of X . We can de2ne a characteristic function A : X → {0; 1} with respect to A by 1 if a ∈ A; A (a) = 0 if a ∈ A: Zadeh [2] introduced the concept of fuzzy subset A˜ of X by extending the characteristic function. A fuzzy subset A˜ of X is de2ned by its membership function A˜ : X → [0; 1] which is viewed as an extension of characteristic function. The value A˜(a) can be interpreted as the membership degree of a point a in the set “A.” Under some suitable conditions for the membership function, the fuzzy set is then termed as a fuzzy number (in Section 2). Let a be a real number. The fuzzy number a˜ corresponding to a can be interpreted as “around a.” The graph of the membership function a˜(x) is bell shaped and a˜(a) = 1. It means that the membership degree a˜(x) is close to 1 when the value x is close to a. The possibility theory derived from fuzzy sets theory is always compared with the probability theory. The detailed discussion can be referred to Zadeh [4] and Dubois and Prade [5]. Zadeh [4] illustrates an example to tell the diHerence between possibility distribution and probability distribution. Let us describe it as follows. Consider the statement “John ate X eggs for breakfast,” where X = {1; 2; : : :}. A possibility distribution and probability distribution can be associated with the variable X . The possibility distribution X (x) can be interpreted as the possibility with which John can eat x eggs for breakfast. The probability distribution PX (x) is interpreted as the probability with which John can eat x eggs for breakfast, or alternatively, the probability distribution PX (x) is determined by observing the number of eggs ate by John for 100 days. The obvious diHerence is that P (x) = 1 and x X x X (x) is not necessarily equal to 1. For instance, the values of X (x) and PX (x) can be shown in the following table: x
1
2
3
4
5
6
7
8
X (x) PX (x)
1.0 0.1
1.0 0.8
1.0 0.1
0.9 0.0
0.8 0.0
0.6 0.0
0.4 0.0
0.2 0.0
We observe that a high degree of possibility does not imply a high degree of probability. The most possible number of eggs ate by John is 1, 2, or 3 while the most probable number of eggs ate by John is 2. Therefore, the possibility theory and probability theory cannot be substituted for each other. However, the interesting research topic for combining the probability theory and fuzzy sets theory can complement each other. For instance, the notion of fuzzy random variable that unites the randomness and fuzziness has been introduced by Kwakernaak [6] and Puri and Ralescu [7]. In the real world, the data sometimes cannot be recorded or collected precisely. For instance, the water level of a river cannot be measured in an exact way because of the 1uctuation, and the price of a product 1uctuate from time to time according to the market eHects. Therefore, the fuzzy sets theory naturally provides an appropriate tool in modeling the imprecise models. For instance, the more appropriate way to describe the water level is to say that the water level is around 30 m. ˜ Now let us consider another The phrase “around 30 m” should be regarded as a fuzzy number 30.
H.-C. Wu / Computers & Operations Research 31 (2004) 1069 – 1081
1071
instance, considering to ask a group of economists for predicting the rate of economical growth in the next year. Their statements may be like “approximately 5%,” “should be 4 – 6%” or “may be below 7%,” etc. All of those statements can be characterized as fuzzy sets. Suppose that we introduce a selection procedure which will determine the probability that each economist will be selected. Then for each possible selection, their statements are fuzzy sets. Thus, the association of fuzzy sets and probabilities forms the notion of fuzzy random variable. Roughly speaking, the observations of random variables are real numbers and the observations of fuzzy random variables are fuzzy numbers. The concept of fuzzy random variable was introduced by Kwakernaak [6] and Puri and Ralescu [7]. The occurrence of fuzzy random variable makes the combination of randomness and fuzziness more persuasive, since the probability theory can be used to model uncertainty, and the fuzzy sets theory can be used to model imprecision. Now it is the right time to explain the reasons for considering the imprecise (fuzzy) input data of the Black–Scholes formula. In the usual approach, there are two ways to describe the risk-free interest rate. One assumes the risk-free interest rate as a constant, and another one assumes the stochastic interest rate. As we just described above, the theories for modeling the randomness and fuzziness cannot be substituted for each other. Therefore, the methodology for considering the fuzzy interest rate will be totally diHerent from that of considering the stochastic interest rate. As a matter of fact, in the general case, it is still possible to take into account the imprecisely stochastic interest rate for the options pricing. Under this situation, the imprecisely stochastic interest rate can be regarded as a fuzzy random variable. Therefore, it means that the fuzzy interest rate is the corresponding consideration of the constant interest rate under fuzzy environment, and the imprecisely stochastic interest rate (i.e. the fuzzy random variable) is the corresponding consideration of stochastic interest rate under fuzzy environment. Only the fuzzy interest rate is under investigated in this paper as the beginning step for studying the options pricing by using the fuzzy sets theory. The study for the imprecisely stochastic interest rate will be the future research. Next we provide the motivation why we need to take into account the fuzzy interest rate in the Black–Scholes formula. In the constant interest rate approach, when the 2nancial analyst tries to price an option, the interest rate is assumed as a constant. However, the interest rate may have the diHerent values in the diHerent commercial banks and 2nancial institutions. Therefore, the choice of a reasonable interest rate may cause a dilemma (Note that the situation for the diHerent interest rates in the diHerent commercial banks and 2nancial institutions may not be regarded as a kind of randomness). But one thing can be sure is that the diHerent interest rates may be around a 2xed value within a short period of time. For instance, the interest rates may be around 5% (The phrase “around 5%” might have problems to be modeled by using the probability theory. Of course, it might be possible to enforce to model this phrase “around 5%” by using the probability theory. However, under this situation, the structure of the interest rate could be very complicated. Then it might cause the diOculties for ˜ when the further analysis). In this case, the interest rate may be regarded as a fuzzy number 5% 2nancial analyst tries to price an European call option using the Black–Scholes formula. On the other hand, there are two approaches used to obtain the volatilities in order to apply the Black–Scholes formula, the so-called historical volatility approach and implied volatility approach. Therefore, the choice of a reasonable volatility approach may also cause a dilemma. The other reason is that the 2nancial market 1uctuates from time to time. It is a little unreasonable to pick a 2xed volatility to price an option at this time and use this option price for the later use, since the later volatility has already changed. In this case, it is natural to regard the volatility as an imprecise
1072
H.-C. Wu / Computers & Operations Research 31 (2004) 1069 – 1081
(fuzzy) data. Of course, the same reasons as described in the case of interest rate, the methodology for considering the fuzzy volatility will be completely diHerent from that of considering the stochastic volatility. As a matter of fact, in the general case, the stochastic volatility can also be extended by considering the imprecisely stochastic volatility. In this case, the imprecisely stochastic volatility can be modeled as a fuzzy random variable. This will be the future research. The usual approach for options pricing is to pick a 2xed stock price to price an option by applying the Black–Scholes formula. However, this option price will be used for the further decision making by a 2nancial analyst within a short period of time. The problem is that the stock price will be changed within this short period of time because the stock price 1uctuates very irregular (assumed to be a Brownian motion). Therefore, it is a little unreasonable to pick a 2xed stock price to price an option at this time and then use this option price for later use, since the later stock price has already changed. Therefore, it is also natural to assume the stock price as a fuzzy number. If this assumption does not convince the 2nancial analysts, then they still can take the stock price as a real number and just consider the fuzzy interest rate and fuzzy volatility. Under this situation, the methodology proposed in this paper is still applicable since the real numbers are the special case of the fuzzy numbers. Although we have described how the fuzzy interest rate, fuzzy volatility and fuzzy stock price occur in the real world, some of those three input data can still be taken as the real numbers (it is also called the crisp numbers in the fuzzy literature) if the 2nancial analyst can make sure that those data occur in a crisp sense. In this case, the methodology proposed in this paper is still applicable since the real numbers (crisp numbers) are the degenerated case (special case) of the fuzzy numbers. Now, under the considerations of fuzzy interest rate, fuzzy volatility and fuzzy stock price, the option price will turn into a fuzzy number. The price of a European call option can now be interpreted as The fuzzy “around $3.5,” for instance. The phrase “around $3.5” is regarded as a fuzzy number 3:5. number 3:5 is in fact a function de2ned on R into [0; 1], and denoted by 3:5 (r). Given any value r0 , the function value 3:5 (r ) will be interpreted as the belief degree of closeness to value 3:5. The 0 graph of this function 3:5 (r) will be bell shaped. It means that the closer the value r0 to 3:5 is, the higher the belief degree is. Therefore, the 2nancial analyst can pick any value which is around 3:5 with an acceptable belief degree as the option price for his (her) later use. In order to obtain the belief degree of any given option price, an optimization problem will be solved. An eOcient computational procedure is proposed in this paper to solve this optimization problem. This paper is organized as follows. In Section 2, the notions of fuzzy number and the arithmetics of fuzzy numbers are introduced. In Section 3, the notion of fuzzy random variable is introduced. In Section 4, the fuzzy patterns of the Black–Scholes formula and put–call parity relationship are proposed. In Section 5, the computational procedure is provided in order to obtain the belief degrees of any given option prices. Finally, the conclusions of this paper is depicted in the 2nal Section 6. 2. Fuzzy numbers Let X be a universal set and A˜ be a fuzzy subset of X . We denote by A˜ = {x: A˜(x) ¿ } ˜ where A˜0 is the closure of the set {x: A˜(x) = 0}. A˜ is called a northe -level set of A, mal fuzzy set if there exists x such that A˜(x) = 1. A˜ is called a convex fuzzy set if A˜( x + (1 − )y) ¿ min{A˜(x); A˜(y)} for ∈ [0; 1]. (That is, A˜ is a quasi-concave function.)
H.-C. Wu / Computers & Operations Research 31 (2004) 1069 – 1081
1073
In this paper, the universal set X is assumed to be a real number system; that is, X = R. Let f be a real-valued function de2ned on R. f is said to be upper semicontinuous, if {x: f(x) ¿ } is a closed set for each . Or equivalently, f is upper semicontinuous at y if and only if ∀ ¿ 0; ∃ ¿ 0 such that |x − y| ¡ implies f(x) ¡ f(y) + . a˜ is called a fuzzy number if the following conditions are satis2ed: (i) a˜ is a normal and convex fuzzy set. (ii) Its membership function a˜ is upper semicontinuous. (iii) The -level set a˜ is bounded for each ∈ [0; 1]. From Zadeh [2], A˜ is a convex fuzzy set if and only if its -level set A˜ ={x: A˜(x) ¿ } is a convex set for all . Therefore, if a˜ is a fuzzy number, then the -level set a˜ is a compact (closed and bounded in R) and convex set; that is, a˜ is a closed interval. The -level set of a˜ is then denoted by a˜ = [a˜L ; a˜U ]. A fuzzy number a˜ is said to be nonnegative if a˜(x) = 0 for x ¡ 0. It is easy to see that if a˜ is a nonnegative fuzzy numbers then a˜L and a˜U are all nonnegative real numbers for all ∈ [0; 1]. The following proposition is useful for further discussion. Proposition 2.1 (Zadeh [8, Resolution identity]). Let A˜ be a fuzzy set with membership function A˜ and A˜ = {x: A˜(x) ¿ }. Then A˜(x) = sup 1A˜ (x); ∈[0;1]
where 1A is an indicator function of set A, i.e., 1A (x) = 1 if x ∈ A and 1A (x) = 0 if x ∈ A. Note that the -level set A˜ of A˜ is a crisp (usual) set. a˜ is called a crisp number with value m if its membership function is 1 if x = m; a˜(x) = 0 otherwise: It is denoted by a˜ ≡ 1˜{m} . It is easy to see that (1˜{m} )L = (1˜{m} )U = m for all ∈ [0; 1]. We see that the real numbers are the degenerated case (special case) of the fuzzy numbers when the real numbers are regarded as the crisp numbers. Now it is the right time to introduce the arithmetics of any two fuzzy numbers. Let “ ” be a ˜ The membership function of binary operation ⊕, , ⊗ or between two fuzzy numbers a˜ and b. ˜ a˜ b is de2ned by a˜b˜(z) =
sup
{(x;y): x◦y=z }
min{a˜(x); b˜(y)};
where the binary operations = ⊕, , ⊗ or correspond to the binary operations ◦ = +; −, × or / according to the “Extension Principle” in Zadeh [2]. Let “ int ” be a binary operation ⊕int , int or ⊗int between two closed intervals [a; b] and [c; d]. Then [a; b] int [c; d] is de2ned by {z ∈ R: z = x ◦ y; ∀x ∈ [a; b]; ∀y ∈ [c; d]; [a; b] int [c; d] ≡ where “ ◦ ” is an usual binary operation +; − or ×}
1074
H.-C. Wu / Computers & Operations Research 31 (2004) 1069 – 1081
and [a; b] int [c; d] ≡ {z ∈ R: z = x=y; ∀x ∈ [a; b]; ∀y ∈ [c; d]} if the interval [c; d] does not contain zero. Then the following well-known results are not hard to prove. ˜ a˜ b˜ and a˜ ⊗ b˜ are also fuzzy Proposition 2.2. Let a˜ and b˜ be two fuzzy numbers. Then a˜ ⊕ b; numbers and their -level sets are ˜ = a˜ ⊕int b˜ = [a˜L + b˜L ; a˜U + b˜U ]; (a˜ ⊕ b) ˜ = a˜ int b˜ = [a˜L − b˜U ; a˜U − b˜L ]; (a˜ b) ˜ = a˜ ⊗int b˜ = [min{a˜L b˜L ; a˜L b˜U ; a˜U b˜L ; a˜U b˜U }; max{a˜L b˜L ; a˜L b˜U ; a˜U b˜L ; a˜U b˜U }] (a˜ ⊗ b) for all ∈ [0; 1]. If the -level set b˜ of b˜ does not contain zero for all ∈ [0; 1], then a˜ b˜ is also a fuzzy number and its -level set is ˜ = a˜ int b˜ = [min{a˜L = b˜L ; a˜L = b˜U ; a˜U = b˜L ; a˜U = b˜U }; max{a˜L = b˜L ; a˜L = b˜U ; a˜U = b˜L ; a˜U = b˜U }] (a˜ b) for all ∈ [0; 1]. Let F denote the set of all fuzzy subsets of R. Let f(x) be a nonfuzzy real-valued function from R into R and A˜ be a fuzzy subset of R. By the extension principle in Zadeh [8], the fuzzy-valued ˜ A) ˜ is a fuzzy subset function f˜ : F → F can be induced by the nonfuzzy function f(x); that is, f( ˜ ˜ of R. The membership function of f(A) is de2ned by f( ˜ A) ˜ (r) =
sup
{x: r=f(x)}
A˜(x):
The following proposition is very useful for discussing the fuzzy pattern of the Black–Scholes formula. Proposition 2.3. Let f(x) be a real-valued function and A˜ be a fuzzy subset of R. The function f(x) can induce a fuzzy-valued function f˜ : F→F via the extension principle. Suppose that the membership function A˜ of A˜ is upper semicontinuous and {x: r = f(x)} is a compact set (it will be ˜ A) ˜ A)) ˜ is (f( ˜ = {f(x): x ∈ A˜ }. a closed and bounded set in R) for all r, then the -level set of f( Proof. If r ∈ {f(x): x ∈ A˜ }, then there exists an x such that r = f(x) and x ∈ A˜ ; that is, A˜(x) ¿ . ˜ ˜ ˜ ˜ ˜ Thus, f( ˜ A) ˜ (r) = sup{x: r=f(x)} A˜(x) ¿ implies r ∈ (f(A)) . It says that {f(x): x ∈ A } ⊆ (f(A)) . ˜ A)) ˜ , then sup{x: r=f(x)} A˜(x) ¿ ; that is, there exists an x such that On the other hand, if r ∈ (f( A˜(x) ¿ and r = f(x), since {x: r = f(x)} is a compact set and A˜(x) is upper semicontinuous (Using the fact that an upper semicontinuous function assumes maximum over a compact set in Bazarra and Shetty [9]. Therefore, r ∈ {f(x): x ∈ A˜ }. This completes the proof.
H.-C. Wu / Computers & Operations Research 31 (2004) 1069 – 1081
1075
3. Fuzzy random variables The stock price S is a stochastic process, i.e., the stock price St at time t is a random variable. Therefore, if the fuzzy stock price is considered at time t in the Black–Scholes formula, then it is natural to introduce the notion of fuzzy random variable. Roughly speaking, the usual random variable assumes real numbers and the fuzzy random variable assumes fuzzy numbers; that is, the observations of random variables are real numbers and the observations of fuzzy random variables are fuzzy numbers. Let ( ; A) be a measurable space and (R; B) be a Borel measurable space. Let f : → P(R) (set of all subsets of R) be a set-valued function. According to Aumann [10], f is called measurable if and only if each of the set {(x; y): y ∈ f(x)} is A × B-measurable. Let F(R) denote the set of all fuzzy numbers. If f˜ : → F(R) is a fuzzy-valued function, then f˜ is a set-valued function for all ∈ [0; 1]. f˜ is called (fuzzy-valued) measurable if and only if f˜ is (set-valued) measurable for all ∈ [0; 1]. Let ( ; A; P) be a probability space (a complete -2nite measure space). Then X : → R is a random variable if X is (A; B)-measurable function. Thus, it is natural to propose the following de2nition. De nition 3.1. Let X˜ : →F(R) be a fuzzy-valued function. X˜ is called a fuzzy random variable if X˜ is measurable. Then the following proposition holds true obviously (Puri and Ralescu [7] and KrRatschmer [11]). Proposition 3.1. Let X˜ : →F(R) be a fuzzy-valued function. X˜ is a fuzzy random variable if and only if X˜ L and X˜ U are random variables for all ∈ [0; 1]. 4. Fuzzy pattern of Black–Scholes formula The well-known Black–Scholes formula for European call option written on a stock S with expiry date T and strike price K (Black and Scholes [1]) is described as follows. Let the function c be given by the formula c(s; t; K; r; ) = sN (d1 ) − Ke−rt N (d2 ); where d1 = and
ln(s=K) + (r + 2 =2)t √ t
√ d2 = d1 − t
and N stands for the cumulative distribution function of a standard normal random variable N (0; 1). Let Ct denote the price of a European call option at time t ∈ [0; T ]. Then Ct = c(St ; T − t; K; r; )
1076
H.-C. Wu / Computers & Operations Research 31 (2004) 1069 – 1081
for all t ∈ [0; T ]. Furthermore, the price Pt of a European put option at time t with the same expiry date T and strike price K can be obtained by the following put–call parity relationship (Musiela and Rutkowski [12]) Ct − Pt = St − Ke−r(T −t) for all t ∈ [0; T ]. Under the considerations of fuzzy interest rate r, ˜ fuzzy volatility ˜ and fuzzy stock price S˜ (here the fuzzy stock price is assumed to be a nonnegative fuzzy number), the fuzzy pattern of the Black– Scholes formula is described as follows. Let the fuzzy-valued function c˜ be given by the formula ˜ c( ˜ s;˜ t; K; r; ˜ ) ˜ = (s˜ ⊗ N˜ (d˜ 1 )) (1˜{K } ⊗ e−r˜⊗1{t} ⊗ N˜ (d˜ 2 ));
where d˜ 1 = [ln(s˜ 1˜{K } ) ⊕ (r˜ ⊕ ( ˜ ⊗ ˜ 1˜{2} )) ⊗ 1˜{t } ] ( ˜ ⊗ and d˜ 2 = d˜ 1 ( ˜ ⊗
1˜{t } )
1˜{t } ):
Since the strike price K and time t are real numbers, they are displayed as the crisp numbers 1˜{K } and 1˜{t } with values K and t, respectively. Then the fuzzy price of a European call option at time t is given by the formula C˜ t = c( ˜ S˜ t ; T − t; K; r; ˜ ); ˜ where S˜ t and C˜ t are fuzzy random variables, for all t ∈ [0; T ]. Furthermore, the fuzzy price P˜ t of a European put option at time t with the same expiry date T and strike price K can be obtained by the following fuzzy pattern of put–call parity relationship: ˜ P˜ t = C˜ t S˜ t ⊕ (1˜{K } ⊗ e−r˜⊗1{T −t} )
(1)
for all t ∈ [0; T ]. According to the “Resolution identity” in Proposition 2.1, the membership function of C˜ t is given by C˜ t (c) = sup 1(C˜ t ) (c); 06 61
(2)
where (C˜ t ) is the -level set of the fuzzy price C˜ t of a European call option at time t. Now it is the right time to display the left- and right-end points of the closed interval (C˜ t ) = [(C˜ t )L ; (C˜ t )U ]: From Proposition 2.3, since the cumulative distribution function N (x) is increasing, the -level set ˜ is given by of N˜ (d) ˜ = {N (x): x ∈ d˜ } = {N (x): d˜ L 6 x 6 d˜ U } = [N (d˜ L ); N (d˜ U )]: (N˜ (d))
(3)
e− x
is a decreasing function and ln x is an increasing function, the -level sets of Similarly, since − r˜⊗1˜{t } ˜ e and ln(S t 1˜{K } ) are then given by U L ˜ (e−r˜⊗1{t} ) = {e−x : x ∈ (r˜ ⊗ 1˜{t } ) } = {e−x : r˜L t 6 x 6 r˜U t} = [e−r˜ t ; e−r˜ t ]
(4)
H.-C. Wu / Computers & Operations Research 31 (2004) 1069 – 1081
1077
and (ln(S˜ t 1˜{K } )) = [ln((S˜ t )L =K)); ln((S˜ t )U =K))]
(5)
by using Propositions 2.2 and 2.3. Then the left-end point (C˜ t )L and right-end point (C˜ t )U of the closed interval (C˜ t ) will be displayed as follows by using Proposition 2.2 and Eqs. (3), (4), and (5) (note that, since the fuzzy stock prices are assumed to be nonnegative fuzzy numbers, (S˜ t )L and (S˜ t )U are clearly nonnegative for all ∈ [0; 1]): L (C˜ t )L = (S˜ t )L N ((d˜ 1 )L ) − Ke−r˜ (T −t) N ((d˜ 2 )U )
and (C˜ t )U = (S˜ t )U N ((d˜ 1 )U ) − Ke−r˜ (T −t) N ((d˜ 2 )L ); U
where ln((S˜ t )L =K)) + (r˜L + ( ˜ L )2 =2)(T − t) √ ; ˜ U T − t ln((S˜ t )U =K)) + (r˜U + ( ˜ U )2 =2)(T − t) √ ; (d˜ 1 )U = ˜ L T − t √ (d˜ 2 )L = (d˜ 1 )L − ˜ U T − t (d˜ 1 )L =
and
√ (d˜ 2 )U = (d˜ 1 )U − ˜ L T − t:
According to the fuzzy pattern of put–call parity relationship in (1), the fuzzy price P˜ t of a European put option at time t can be derived similarly. The left-end point (P˜ t )L and right-end point (P˜ t )U of the closed interval (P˜ t ) are given by (P˜ t )L = (C˜ t )L − (S˜ t )U + Ke−r˜
U
(T −t)
(6)
(P˜ t )U = (C˜ t )U − (S˜ t )L + Ke−r˜ (T −t)
(7)
and L
for all ∈ [0; 1] using Proposition 2.2 again. 5. Computational method and example Given a European call option price c of the fuzzy price C˜ t at time t, we plan to know its membership value , i.e., its belief degree. If the 2nancial analysts are comfortable with this membership value , then it will be reasonable to take the value c as the European option price at time t. In this case, the 2nancial analysts can accept the value c as the European option price at time t with belief degree . The membership function of fuzzy price C˜ t of a European call option at time t is given in Eq. (2). Therefore, given any European call option price c, its belief degree can be obtained by solving the
1078
H.-C. Wu / Computers & Operations Research 31 (2004) 1069 – 1081
following optimization problem: (MP1)
max
subject to
(C˜ t )L 6 c 6 (C˜ t )U 0 6 6 1:
Since g( ) = (C˜ t )L is an increasing function of and h( ) = (C˜ t )U is a decreasing function of , the following procedure is proposed in order to make the optimization problem (MP1) easier to be solved. The optimization problem (MP1) can be rewritten as (MP2)
max
subject to
g( ) 6 c h( ) ¿ c 0 6 6 1:
Since g( ) = (C˜ t )L 6 (C˜ t )U = h( ), one of the constraints g( ) 6 c or h( ) ¿ c can be discarded in the following ways: (i) If g(1) 6 c 6 h(1) then C˜ t (c) = 1. (ii) If c ¡ g(1) then the constraint h( ) ¿ c is redundant, since h( ) ¿ h(1) ¿ g(1) ¿ c for all ∈ [0; 1] using the fact that h( ) is decreasing and g( ) 6 h( ) for all ∈ [0; 1]. Thus, the following relaxed optimization problem will be solved: (MP3)
max
subject to
g( ) 6 c 0 6 6 1:
(iii) If c ¿ h(1) then the constraint g( ) 6 c is redundant, since g( ) 6 g(1) 6 h(1) 6 c for all ∈ [0; 1] using the fact that g( ) is increasing and g( ) 6 h( ) for all ∈ [0; 1]. Thus, the following relaxed optimization problem will be solved: (MP4)
max
subject to
h( ) ¿ c 0 6 6 1:
Since g( ) is continuous and increasing, problem (MP3) can be solved using the following algorithm (bisection search). Step Step Step and go Step
1: Let be the tolerance and 0 be the initial value. Set ← 0 , low ← 0 and up ← 1. 2: Find g( ) = (C˜ t )L . If g( ) 6 c then go to Step 3 otherwise go to Step 4. 3: If c−g( ) ¡ then EXIT and the maximum is otherwise set low ← , ← (low+up)=2 to Step 2. 4: Set up ← , ← (low + up)=2 and go to Step 2.
H.-C. Wu / Computers & Operations Research 31 (2004) 1069 – 1081
1079
For problem (MP4), it is enough to consider the equivalent constraint −h( ) 6 − c; since h( ) is decreasing and continuous, i.e., −h( ) is increasing and continuous. Thus, the above algorithm is still applicable for solving problem (MP4). The membership function of a triangular fuzzy number a˜ is de2ned by (r − a1 )=(a2 − a1 ) a˜(r) = (a3 − r)=(a3 − a2 ) 0
if a1 6 r 6 a2 ; if a2 ¡ r 6 a3 ; otherwise;
which is denoted by a˜ = (a1 ; a2 ; a3 ) (The graph of the membership function a˜(r) looks like a triangle). The triangular fuzzy number a˜ can be expressed as “around a2 ” or “being approximately equal to a2 ”. a2 is called the core value of a, ˜ and a1 and a3 are called the left- and right-spread values of a, ˜ respectively. The -level set (a closed interval) of a˜ is then a˜ = [(1 − )a1 + a2 ; (1 − )a3 + a2 ]; that is, a˜L = (1 − )a1 + a2
and
a˜U = (1 − )a3 + a2 :
Example 5.1. Consider a European call option on a stock with strike price $30 and with 3 months to expiry. Suppose that the current stock price is around $33, the stock price volatility is around 10% and the risk-free interest rate is around 5% per annum with continuous compounding. Assume that t = 0 and T = 0:25. The fuzzy interest rate r, ˜ fuzzy volatility ˜ and fuzzy stock price S˜ 0 are assumed to be triangular fuzzy numbers r=(0:048; ˜ 0:05; 0:052), S˜ 0 =(32; 33; 34) and =(0:08; 0:1; 0:12), respectively. ˜ The current fuzzy price C 0 of a European call option can be obtained. Then the following table gives the belief degrees C˜ 0 (c) for diHerent European call option prices c by solving the optimization problems (MP3) and (MP4) using the computational procedure proposed above. c 2.58 2.78 2.98 3.18 3.38 3.39 3.59 3.79 3.99 4.19 C˜ 0 (c) 0.7793 0.8338 0.8888 0.9441 0.9996 0.9976 0.9421 0.8870 0.8323 0.7782 For example, if the European call option price $3.59 is taken, then its belief degree is 0:9421. Therefore, if a 2nancial analyst is comfortable with this belief degree 0:9421, then he (she) can take this option price $3.59 for his (her) later use. If the European call option price is taken as c = 3:3813, then its belief degree will be 1:00. As a matter of fact, if we take r = 0:05, S0 = 33 and = 0:1, then the European call option price will be $3.3813. This situation matches the observation that c = 3:3813 has belief degree 1:00.
1080
H.-C. Wu / Computers & Operations Research 31 (2004) 1069 – 1081
The following table gives the -level closed intervals (C˜ 0 ) of the fuzzy price C˜ 0 of a European call option.
(C˜ 0 )
0.99 0.98 0.97 0.96 0.95 0.94 0.93 0.92 0.91 0.90
[3:3453; 3:4174] [3:3092; 3:4534] [3:2732; 3:4895] [3:2372; 3:5256] [3:2011; 3:5617] [3:1650; 3:5978] [3:1289; 3:6340] [3:0928; 3:6702] [3:0567; 3:7064] [3:0205; 3:7427]
For = 0:95, it means that the call option price will lie in the closed interval [3:2011; 3:5617] with belief degree 0.95. From another point of view, if a 2nancial analyst is comfortable with this belief degree 0.95, then he (she) can pick any value from the closed interval [3:2011; 3:5617] as the option price for his (her) later use. The fuzzy price of a European put option can be obtained from Eqs. (6) and (7), and the belief degrees and the -closed intervals can also be evaluated similarly. 6. Conclusions Owing to the 1uctuation of 2nancial market from time to time, some input data in the Black– Scholes formula cannot always 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 Black–Scholes formula and put–call parity relationship are then proposed in this paper. Under the considerations of fuzzy interest rate, fuzzy volatility and fuzzy stock price, the European option price turns into a fuzzy number. This makes the 2nancial analyst who can pick any European option price with an acceptable belief degree for his (her) later use. A computational procedure is also proposed in order to obtain the belief degree. Since the price of an American call option without dividends is the same as the price of a European call option, the methodology proposed in this paper may also be applicable for the American call option without dividends. References [1] Black F, Scholes M. The pricing of options and corporate liabilities. Journal of Political Economy 1973;81:637–59. [2] Zadeh LA. Fuzzy sets. Information and Control 1965;8:338–53. [3] Ribeiro RA, Zimmermann H-J, Yager RR, Kacprzyk J. editors. Soft computing in 2nancial engineering. Heidelberg: Physica-Verlag, 1999.
H.-C. Wu / Computers & Operations Research 31 (2004) 1069 – 1081 [4] [5] [6] [7] [8] [9] [10] [11] [12]
1081
Zadeh LA. Fuzzy sets as a basis for a theory of possibility. Fuzzy Sets and Systems 1978;1:3–28. Dubois D, Prade H. Possibility theory. New York: Plenum Press, 1988. Kwakernaak H. Fuzzy random variables I: De2nitions and theorems. Information Sciences 1978;15:1–29. Puri ML, Ralescu DA. Fuzzy random variables. Journal of Mathematical Analysis and Applications 1986;114: 409–22. Zadeh LA. The concept of linguistic variable and its application to approximate reasoning I, II and III. Information Sciences 1975;8:199–249. 8:301–57; 9:43–80. Bazarra MS, Shetty CM. Nonlinear programming. New York: Wiley, 1993. Aumann RJ. Integrals of set-valued functions. Journal of Mathematical Analysis and Applications 1965;12:1–12. KrRatschmer V. A uni2ed approach to fuzzy random variables. Fuzzy Sets and Systems 2001;123:1–9. Musiela M, Rutkowski M. Martingale methods in 2nancial modelling. Berlin, Heidelberg: Springer, 1997.