#### Abstract

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.

#### 1. Introduction

In this paper, we use the fuzzy theory of Zadeh [1], the set of propositions used in some previous papers of us [2–4] 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 3we 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.

#### 2. Preliminaries and Theorems

Letbe a probabilizable space, where is the set of real numbers, is the Borel's -field.

We may identify a subset of (event) with its characteristic function,

where is the support of :

A fuzzy subset (fuzzy event) is identified with its membership function, , (see Zadeh [1]):

that is, the support of is ; represents (i) a membership degree of as element of ; (ii) a belief degree that happens.

* The y-cut of f*, stated at the level , is defined by:

We say that is a * fuzzy number* if is convex, that is, all its *y*-cuts are closed and bounded real intervals; is normal, that is, there exists such that ; is upper semi-continuous (usc); is compactly supported on the real line. We denote by the set of fuzzy numbers.

If then

where is increasing and is called the * left cut-function* of , is decreasing and is called the * right cut-function of *. , are left-continuous because (see [2, page ]). In this case, we have the following Zadeh's identity:

where is the characteristic function of the real interval .

*Zadeh's Extension Principle*

Let , a continuous deterministic function with values in :
let be a deterministic function and be fuzzy numbers. The Zadeh's extension principle (Z.e.pr.) gives a belief degree to every value , that is, the Z.e.pr. defines a belief degree or membership function, , by

So, from the deterministic function , we have that the Z.e.pr. induces a fuzzy number, that is,

and is obtained by the deterministic function and the fuzzy numbers , that is,

Proposition 1. *If are in and is a continuous deterministic function, then (for all ) the -cut of the fuzzy number is equal to the of the -cuts of :
*

This Proposition is an extension obtained by induction by a Proposition of Biacino-Lettieri (see [5, Section)] and it is used in our previous papers [2–4].

From Proposition 1 we have the following proposition.

Proposition 2. *Let be a continuous deterministic function increasing with respect to and decreasing with respect to and let be in , then we have that the membership function is in if we have that*(i)*the left cut-function is given by
and is increasing; *(ii)*the right cut-function is given by
and is decreasing; *(iii)*for all , and the equal holds if . **Then the membership function has the following shape**
and it is an element of , where is the characteristic function of the real interval .*

*Proof. *If then we have that for all their cuts are the following compact (closed and bounded) intervals, :
On the other hand, by Proposition 1, from the hypothesis of continuity of it follows that is a compact interval. So, if the left cut-function is increasing and the right is decreasing, and , then is equal to the following
that is is in and by Zadeh's identity we have that has the following shape:

#### 3. 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.

Triangular fuzzy numbers

where . The graph of looks like a triangle. In our cases, the support may be calculated by a statistical time series and 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

with

Gaussian fuzzy set

where the mean and the standard deviation are calculated using a statistical time series.

For all the cut-functions are given by

The belief degree of is equal to which is the max of . In fact the derivative of is given by

which is equal to zero if . The function has a symmetric and campanulate graph.

The flex points are . In fact the second derivative of is given by

if and only if that is, if The function is not in because its support is not compact.

(3) Gaussian fuzzy number (with compact support)

where and are calculated with the statistical time series or with possibilistic data. We choose because the statistical tables are up to this number; the support of is which influences the results. We can use, for example, to have a larger support .

#### 4. Black-Scholes Fuzzy Numbers

Let be the strike price, the annual istantaneous intensity of the risk-free interest, the initial market price of the underlying security, the expiration date, 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 is the cumulative distribution function for a standard normal random variable (r.v.):

also

and

Note that if the volatility is very big then we may have hence . In this case the values of the cumulative distribution function of the standard normal r.v. become , 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, , and on the strike price, . 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.

Proposition 3 (on black-scholes fuzzy numbers). *If we take these fuzzy numbers
**
instead of the corresponding crisp ones, then the fuzzy version of the Black-Scholes formula, is in if and only if *(i)*the left cut-function of given by
is increasing;*(ii)*the right cut-function of given by
is decreasing;*(iii)*the left cut-function of is less than the right:**
Moreover, with the left and right cut-functions we may write :
**
and is an element of .*

*Proof. *We suppose to know these three fuzzy numbers
that is, for all we suppose to know the belief degree of the fuzzy event and analogously we know the belief degree and of the fuzzy events and respectively.

Then we obtain the fuzzy price of the call, by the Zadeh’s extension principle:

according to the Black-Scholes formula
On the other hand, if we know the fuzzy numbers then we can write their left and right cut-functions
Taking into account that has at the denominator, we have that for every fixed belief degree, , the cut functions of the fuzzy numbers and are given by

In this way, the left cut-functions are increasing and the right cut-functions are decreasing.

Since is increasing with respect to , , and and decreasing with respect to , by Proposition 2, we have that the left and right cut-functions of are given by

Moreover we have and by Zadeh's identity
For , we have that the support of the fuzzy Black-Schose number is
In particular we have that the real number is negative if and is positive if . The two prices
represent the pessimistic and optimistic performance on the future if we gamble on the increase of the price.

The real numbers 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 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.

#### 5. 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,

of the “MIB 30” from the first of April 2001 to the first of October 2006 (for 67 months) and we want to build a fuzzy option price for an option, of the first of October 2006, with strike price 39500, maturity on 11 days.

From the security time series of the daily price we build the time series of the instantaneous intensity of returns from the security (using continuous compounding),

Every month we have a daily arithmetical mean of . So, from the observation of months we have means:

and daily arithmetical variances of :

So we have a time series of 67 standard deviations, that is, * the volatility time series*:

From this time series we can build a fuzzy number, 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 and variance .

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

(1)the min variance of the volatility in August 2006: 0.000068;(2)the max daily variance of the volatility in September 2002: 0.000521.We have to multiply by 365 for obtaining the min variance per annum:

and the max variance per annum:

The min and max standard deviations per annum were

respectively.

So we have the interval , but if we think that the standard deviation has in the interval and the value with more belief degree is , we may take as triangular fuzzy annual volatility of the “MIB 30”, the following fuzzy number:

The *y*-cuts are given by

; .

So with belief degree we have that the annual volatility is in the real interval because

For building * the fuzzy istantaneous intensity* of the risk-free interest rate for the same duration of the option, we read

So, we give a great belief degree to these data if we take the basic points of the triangle in this way: so the support of the triangular fuzzy number is the interval :

The *y*-cut functions are ; .

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, in this way:

The *y*-cut functions are

With belief degree we have that our MIB30 is equal to . 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 the min and max price of MIB30 will be equal to

*Statistical Fuzzy Black-Scholes*

Using the triangular fuzzy numbers , , , built with statistical data, we may write the cut functions of the triangular fuzzy Black-Scholes, , on “MIB 30 with strike price 39000, maturity on 11 days”.*The left cut-function* of is given by

where
and ; *The right cut-function* of is

where
and .

Now we have to calculate the basic point of the triangle: the min value of the call with belief degree zero, , the mean value with belief degree 1, , and the max value of the call with belief degree zero .

The value of the cut function with the max belief degree, , is given by

So we have to calculate and :

with the value of distribution function:
Now we need :
with
Substituing these value in we have
So the value of the call with max belief degree is

Now we calculate the minimum value of the call, that is, , that is

we need
passing to the numbers we have
and we have
now we need also so we have to calculate :
the value of the distribution function is given by
Now we calculate :
Now we can calculate our minimum value of the call
To have the max value of the call, , we have already , now we need , where , so
and

Substituing, we have our max value of the call:

and our * 2006-Black-Scholes* triangular fuzzy number is given by
where , , .

Therefore, the 2006-Black-Scholes cut functions are given by for all

The graph of looks like a triangle with support: These evaluations of MIB30 are with the prices of November 11, 2006 when the prices were increased up to 41034 Euro.

Let us see these evaluations also on January 1, 2004 when the MIB30 prices were lower than 2006. So, we take the strike price of the B-S call equal to 27500 and the following cut functions values of initial prices, ; ; and . Therefore, in the same way of the previous calculus, we have the following cut-functions values of the B-S fuzzy number , and . Indeed, we have

Now, let us compare the supports of our Black-Scholes fuzzy numbers in these two different dates. On January 1, 2004 the support of was On November 11, 2006, the support of was So in 2004–2006 the max B-S value increases. But the min decreases:
these negative values can represent indexes of the investor refuse the possibility to buy the stocks of MIB30. The minimum price of the B-S changes from , to : the investor refuse increased in 2004–2006. So, on November 11, 2006 we could forecast that the market will become with more risks: the risk of loss will increase. The forecast happened, on January 25, 2010 we know that the market loss increases.

The MIB30 is an index of the Italian Stock Exchange so the max of these intervals are indexes of optimistic performance, but the minimum of these B-S intervals are indexes of pessimistic performance on the future. We see that the pessimistic performance becomes more pessimistic.

Further research can be made with Gaussian fuzzy numbers and dynamic program using Matlab.