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.

In this paper, we use the fuzzy theory of Zadeh [

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

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

Let

We may identify a subset

where

A fuzzy subset

that is,

We say that

If

where

where

Let

So, from the deterministic function

If

This Proposition is an extension obtained by induction by a Proposition of Biacino-Lettieri (see [

From Proposition

Let

the left cut-function

the right cut-function

for all

Then the membership function

If

For our option price fuzzy numbers we can use, for example, one of the following fuzzy numbers.

where

In this case the

with

where the mean

For all

The belief degree of

which is equal to zero if

The flex points are

(3) Gaussian fuzzy number (with compact support)

where

Let

also

and

Note that if the volatility

The price of the European call (

If we take these fuzzy numbers

the left cut-function of

the right cut-function of

the left cut-function of

We suppose to know these three fuzzy numbers

Then we obtain the fuzzy price of the call,

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

Since

The real numbers

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

and

So we have a time series of 67 standard deviations, that is,

From this time series we can build a fuzzy number,

and to take the Gaussian fuzzy number with mean

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

the min variance of the volatility in August 2006: 0.000068;

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

The

So with belief degree

For building

the risk-free interest rates of the Italian Banks in the real interval

the interest rates of the “Buoni Ordinari del Tesoro”

So, we give a great belief degree to these data if we take the basic points of the triangle in this way:

The

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,

The

With belief degree

But if we want to make evaluations on the future we may say that in the future with belief degree

Using the triangular fuzzy numbers

Now we have to calculate the basic point of the triangle: the min value of the call with belief degree zero,

The value of the cut function with the max belief degree,

So we have to calculate

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

Substituing, we have our max value of the call:

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

The graph of

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,

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.