Black-Scholes Fuzzy Numbers as Indexes of Performance

We use the set of propositions of some previous papers to define a fuzzy version of the Black-Scholes value where the risk free instantaneous interest intensity, the volatility and the initial stock price are fuzzy numbers whose parameters are built with statistical financial data. With our Black-Scholes fuzzy numbers we define indexes of performance varing in time. As an example, with data of the Italian Stock Exchange on MIB30, we see that in 2004 and 2006 our indexes are negative, that is, they are indexes of the refuse to invest and this refuse increased. So, on November 11, 2006 we could forecast that the market will become with more risk: the risk of loss will increase. Now, on January 25, 2010, we know that this forecast has happened. Obviously, the parameters of our Black-Scholes fuzzy numbers can be valued also with incomplete, possibilistic data. With respect to the probabilistic one, our fuzzy method is more simple and immediate to have a forecast on the financial market.


Introduction
In this paper, we use the fuzzy theory of Zadeh [1], the set of propositions used in some previous papers of us [2][3][4] and statistical data, to have a fuzzy version of the Black-Scholes (B-S) value for a European call option, where the istantaneous intensity of the risk-free interest, the standard deviation of the instantaneous intensity of return from the underlying security (volatility), and the initial stock price are fuzzy numbers built with statistical financial data.Obviously, a similar method can be used if one has incomplete possibilistic data.
With a theorem we deduce that our B-S fuzzy sets are fuzzy numbers.
Then we give a statistical example: with the data of the Italian MIB 30, that is, with its time series, we build a B-S fuzzy number which for every belief degree gives a B-S value.
With the support of our B-S fuzzy numbers we define indexes of the performance of the market varing in time.In particular, in 2004 and 2006 we see that these indexes are negative, that is, they are indexes of the refuse to invest.So, comparing our indexes of 2004 and 2006, on November 11, 2006 we could forecast for future loss in the gamble on the increase of the MIB30.The forecast happened.This paper is organized as follows.In Section 2 we set the preliminaries and give the theorems which link the Zadeh's extension principle to the construction of fuzzy numbers with cut-functions in accordance with the theorems in [2,3,5] and Zadeh's identity; inSection 3 we define three fuzzy numbers which we deduce by probabilistic densities to use for our applications; in Section 4 we give our model of the Black-Scholes fuzzy numbers and its theorem; and in Section 5 we put the applications and cues for further researchs.

Preliminaries and Theorems
Let(Ω, A), be a probabilizable space, where Ω is the set of real numbers, A = B(Ω) is the Borel's σ-field.
We may identify a subset A of Ω (event) with its characteristic function, where A is the support of χ A : ( A fuzzy subset A (fuzzy event) is identified with its membership function, f , (see Zadeh [1]): a membership degree of ω as element of A; (ii) a belief degree that ω happens.
The y-cut of f, stated at the level y ∈ [0, 1], is defined by: We say that f is a fuzzy number if f is convex, that is, all its y-cuts are closed and bounded real intervals; f is normal, that is, there exists ω 0 ∈ Ω such that f (ω 0 ) = 1; f is upper semi-continuous (usc); f is compactly supported on the real line.We denote by N the set of fuzzy numbers.
where f −l is increasing and is called the left cut-function of f , f −r is decreasing and is called the right cut-function of f .f −l , f −r are left-continuous because f ∈ N (see [2, page 5]).
In this case, we have the following Zadeh's identity: where χ is the characteristic function of the real interval [ f −l (y), f −r (y)].
Zadeh's Extension Principle.Let x := (x 1 , . . ., x n ) ∈ R n , h(x) a continuous deterministic function with values in Ω: let h be a deterministic function and f 1 , f 2 , . . ., f n , n be fuzzy numbers.The Zadeh's extension principle (Z.e.pr.) gives a belief degree to every value ω = h(x), that is, the Z.e.pr.defines a belief degree or membership function, h, by h(ω) := sup So, from the deterministic function h, we have that the Z.e.pr.induces a fuzzy number, that is, and h is obtained by the deterministic function h and the fuzzy numbers f 1 , . . ., f n , that is, This Proposition is an extension obtained by induction by a Proposition of Biacino-Lettieri (see [5,Section 2.4)] and it is used in our previous papers [2][3][4].
From Proposition 1 we have the following proposition.
Proposition 2. Let h(x 1 , . . ., x n ) be a continuous deterministic function increasing with respect to x i and decreasing with respect to x j and let f 1 , . . ., f n be in N , then we have that the membership function h is in N if we have that and is increasing; and is decreasing; Then the membership function h has the following shape and it is an element of N , where χ is the characteristic function of the real interval [ h −l (y), h −r (y)].
Proof.If f 1 , •, f n ∈ N then we have that for all y ∈ [0, 1] their cuts are the following compact (closed and bounded) intervals, i ∈ {1, . . .n}: On the other hand, by Proposition 1, from the hypothesis of that is h is in N and by Zadeh's identity we have that h has the following shape:

Examples of Fuzzy Numbers Useful for Our Applications
For our option price fuzzy numbers we can use, for example, one of the following fuzzy numbers.
(1) Triangular fuzzy numbers where a < b < c.The graph of f looks like a triangle.In our cases, the support [a, c] may be calculated by a statistical time series and b may be equal to its arithmetical mean or the value of the time series with the bigger frequence, but also with possibilistic data.
In this case the y-cuts are given by (2) Gaussian fuzzy set where the mean μ and the standard deviation σ are calculated using a statistical time series.For all y ∈]0, 1] the cut-functions are given by The belief degree of ω = μ is equal to g(μ) = 1 which is the max of g.In fact the derivative of g is given by which is equal to zero if ω = μ.The function g has a symmetric and campanulate graph.The flex points are μ − σ, μ + σ.In fact the second derivative of g is given by The function g is not in N because its support is not compact.
(3) Gaussian fuzzy number (with compact support) where μ and σ are calculated with the statistical time series or with possibilistic data.We choose 3.9 because the statistical tables are up to this number; the support of g is [μ − 3.9σ, μ + 3.9σ] which influences the results.We can use, for example, 4.1 to have a larger support [μ − 4.1σ, μ + 4.1σ].

Black-Scholes Fuzzy Numbers
Let k be the strike price, δ the annual istantaneous intensity of the risk-free interest, a the initial market price of the underlying security, t the expiration date, σ 2 the variance of the annual istantaneous intensity of the random return from the underlying security, then the classical Black-Scholes value for a European call option at time zero is given by where N(x) is the cumulative distribution function for a standard normal random variable (r.v.): also and Note that if the volatility σ is very big then we may have d 1 > 3.9 hence d 2 < −3.9.In this case the values of the cumulative distribution function of the standard normal r.v.become N(d 1 ) = 1, N(d 2 ) = 0, respectively, and so we have that the price of the call is equal to the price of the underlying security at time zero (this is the extreme superior value of the call) (see [6,7]).
The price of the European call (26) depends in particular on the initial market price of the underlying security, a, and on the strike price, k.If we consider (26) varing in time, then its fuzzy versions give us indexes of performance as we see in Proposition 3 and Section 5 where we make a statistical application on MIB30.

instead of the corresponding crisp ones, then the fuzzy version of the Black-Scholes formula, h = c, is in N if and only if (i) the left cut-function of c given by
is increasing; (ii) the right cut-function of c given by is decreasing; (iii) the left cut-function of c is less than the right: Moreover, with the left and right cut-functions we may write c: and c is an element of N .
Then we obtain the fuzzy price of the call, c by the Zadeh's extension principle: On the other hand, if we know the fuzzy numbers a, δ, σ ∈ N then we can write their left and right cut-functions Taking into account that d 1 has σ at the denominator, we have that for every fixed belief degree, y ∈ [0, 1], the cut functions of the fuzzy numbers d 1 and d 2 are given by In this way, the left cut-functions are increasing and the right cut-functions are decreasing.
Since c is increasing with respect to a, d 1 , and δ and decreasing with respect to d 2 , by Proposition 2, we have that the left and right cut-functions of c are given by Moreover we have c −l (y) ≤ c −r (y) and by Zadeh's identity For y = 0, we have that the support of the fuzzy Black-Schose number c is In particular we have that the real number represent the pessimistic and optimistic performance on the future if we gamble on the increase of the price.The real numbers c −l (0), c −r (0) are two synthesis of the fuzzy parameters which we use, so they may represent indexes of the performance of the market of the underlying security.Obviously if the underlying security is a market index, as the MIB30, then c −l (0), c −r (0) may represent indexes of the performance of the market.Obviously these indexes vary in time.We will see that their variations help us to evaluate and to forecast the future of the market.

Fuzzy Numbers with Parameters Valued with Statistical Observations
Here we build a fuzzy set using statistical observations of financial prices.We have a time series of the daily prices, From this time series we can build a fuzzy number, σ, to represent the fuzzy volatility.We can do this in many ways.One of these is to calculate the arithmetical mean and the variance of the volatility time series: and to take the Gaussian fuzzy number with mean μ and variance σ 2 .Another way is using a triangular fuzzy number to represent the fuzzy volatility.In this case we need to know the support.We can do this reading the time series.In the case of the "MIB 30" volatility we have respectively.So we have the interval [0.1575, 0.436079], but if we think that the standard deviation has in the interval [0.15, 0.50] and the value with more belief degree is 0.25, we may take as triangular fuzzy annual volatility of the "MIB 30", the following fuzzy number: The y-cuts are given by For building the fuzzy istantaneous intensity of the riskfree interest rate for the same duration of the option, we read (1) the risk-free interest rates of the Italian Banks in the real interval i 1 ∈ [0.015, 0.018] then it follows that δ 1 ∈ [0.0149, 0.017]; (2) the interest rates of the "Buoni Ordinari del Tesoro" i 2 ∈ [0.02, 0.025] ⇒ δ 2 ∈ [0.0198, 0.02469].
So, we give a great belief degree to these data if we take the basic points of the triangle in this way: b = δ l (1) = 0.02, a = 0.014 = δ l (0), c = 0.025 = δ r (0) so the support of the triangular fuzzy number δ is the interval [0.014, 0.025]: The y-cut functions are δ −l (y) = 0.014 + 0.006y; δ −r (y) = 0.025 − 0.005y.On October 30, 2006 the "MIB 30" had the following prices: the last price 39500, the max 40000 and min 38000.We can use these prices to build the triangular fuzzy number, a in this way: With belief degree y = 1 we have that our MIB30 is equal to a −l (1) = a −r (1) = 39500.The values 38000 and 40000 have membership degrees equal to zero.
But if we want to make evaluations on the future we may say that in the future with belief degree y = 0.5 the min and max price of MIB30 will be equal to a −l (0.50) = 38000 + 1500 • 0.50 = 38750; a −r (0.50) = 40000 − 500 • 0.50 = 39750. (58)
cut functions are a −l y = 38000 + 1500y, a −r y = 40000 − 500y.(57) number h( f 1 , . . ., f n ) is equal to the h of the y-cuts of f 1 , . . ., f n : Proposition 1.If f 1 , . . ., f n are in N and h is a continuous deterministic function, then (for all y ∈ [0, 1]) the y-cut of the fuzzy