Recently, the derivatives have reached an evident importance in many of the world financial and commodity markets. Some authors estimate that the value of the amount traded on derivatives is greater than the value of the underlying assets. In particular, the financial derivatives which are more diffused are options and futures.

This introduction aims to answer the request by operators, by financial (and not financial) institutions and by stock and commodity markets of pricing in a correct way such financial tools.

In the following, the main methodologies used for pricing some of the so-called options of second generation will be illustrated. Note that such methodologies are already implemented in a FORTRAN code for parallel computing.

The pricing of any financial asset (shares, bonds, derivatives, ...) depends on the characteristics of the market in which this same financial asset is accounted for. In particular, in the following it will be assumed that the considered financial market is describable by the classical hypothesis set by Black and Scholes. These hypothesis can be summarized as follows:

1. absence of transaction costs and frictions;

2. infinite divisibility of the underlying financial activities;

3. possibility to sell short and/or to issue risky financial activity;

4. neutrality to the risk of the economic operators;

5. "behaviour" of the prices of the underlying financial asset modelled by a log-Normal random walk.

In particular, the assumptions 1., 2. and 3. are not completely realistic but, however, are acceptable approximations for significant economic-financial operators. Moreover, the assumption 5. basically affirms the impredictability of the returns of the considered financial asset.

Note that the classical hypothesis of absence of dividend payment on the underlying financial asset is relaxed.

Among the options that can be priced in the above framework by the implemented software package, there are the European call and put belonging to the following families:

1. the "vanilla" options,

2. the PACKAGES options giving, at expiration, a payoff equal to that of a portfolio constituted by a proper combination of "vanilla" options, the underlying activity and financial riskless assets,

3. and 4. the FORWARD START options and the TANDEM ones giving, at expiration, the right to buy or to sell an other option,

5. the BINARY options that, at expiration, give (or less) a prefixed amount,

6. the LOOKBACK options giving, at expiration, a payoff equal to the maximum price reached by the underlying activity in the preceeding exercise period decreased by the exercise price,

1. the ASIAN options giving, at expiration, the right to buy or to sell the underlying activity to a price that is function of the (arithmetic or geometric) average of the prices reached by the same underlying activity in the preceeding exercise period,
2. the SPREAD options giving, at expiration, a payoff which depends on the difference between the prices of two assets and
3. and 10. the DUAL STRIKE options and the PORTFOLIO ones.

Note that, with the exception of the ASIAN, the SPREAD, the DUAL STRIKE and the PORTFOLIO options, each of the options previously listed can be priced by a closed formula. In general, for the ASIAN, the SPREAD, ... ones such a closed formula is not available. In this case, the pricing of the option can be obtained by the recourse to proper numerical methods, that will be illustrated in the following section.

1. Numerical Methods for the Option Pricing

The numerical methods for the option pricing presented in literature can be grouped in the three following families:

1. Montecarlo methods,

2. tree based methods and

3. methods based on the approximation of the probability density of the average of the prices.

Options pricing based on these methods is articulated like follows:

Step 1: a large number of trajectories of the prices of the underlying activity is simulated;

Step 2: for each of the simulated trajectories of the prices, the average is calculated;

Step 3: for each of the calculated average, the value at expiration of the option is determined;

Step 4: each of the previously determined values is discounted by the riskless rate at the current instant;

Step 5: the current value of the option is determined by the average of all the discounted values calculated in the Step 4.

The numerical methods for pricing the options belonging to this second family is an opportune generalization of those, based on the binomial tree, developed by Cox and Rubinstein. The application of these methods, also giving good results, is strongly limited by its computational (exponential) complexity. Because of this, their use is possible only for options whose expiration is relatively close. Note that, concerning these methods, some aspects have been originally developed.

The methods for the option pricing belonging to this last family differ from those presented up to now. In fact, instead than numerically simulating trajectories of the prices, they use the approximations (express by a closed formula) of the probability density of the average of the prices. Such approximations are determined by an opportune series expansion. Note that, also concerning these methods, some aspects have been originally developed.

In this section some experimental results are proposed.

* The time to maturity of this option is 87 stock exchange days.

** The numerical method used is the Montecarlo one and the number of the simulations used is 10,000. Notice that the computation time grows linearly with the simulation number.

*** The numerical method used is the Montecarlo one and the number of the simulations used is 50,000.