3. Options


3.1 Calculation of Option Prices and Implied Volatilities

A calculation of option prices is possible by using one of the following functions:


997 optstart ()
starting program to calculate option prices or implied volatilities
1000 bitree (vers, task)
calculates option prices using the Binomial tree
{opvv,sel,ingred} = 1003 bs1 (task)
calculates option prices using the Black-Scholes formula
1006 mcmillan (eopv, sel, task, ingred)
calculates option prices using the McMillan formula
1009 american ()
starting program to calculate option prices for american options
1012 european ()
starting program to calculate option prices or implied volatilities for european options
1015 asset (vers)
auxiliary quantlet to calculate option prices for american options

The interactive option pricing quantlet 1018 optstart is simply invoked by typing


  optstart()

in the XploRe command line. A selection box appears which starts the interactive option pricing procedure.


1027

Simply select the method you want to use. If you wish to calculate the option price analytically, choose Black/Scholes & MC Millan; if you want XploRe to calculate it numerically, choose Binomial Tree. Let's choose Black/Scholes & MC Millan. In any case you will be asked whether you want to compute the price of an European (an option which can be executed only at a given date) or an American option (that can be executed anytime).


1035

In this example we have chosen American. This is the kind of option that is usually traded e.g. in the USA or in Germany. The next decision is about the underlying asset (stock or exchange rate).


1039

In our example we are regarding a stock as the underlying asset. In this setting you are questioned if you like to have dividends included in the stock and of what kind you want them to be. (If you choose Exchange Rate here, the next two menu items will be skipped.) Then you are asked whether you like to compute the price of an option or the implied volatility. Now we are ready to enter the parameters needed for the computation of the option prices. These are Price of the Underlying Asset, Exercise Price, Domestic Interest Rate per Year and Volatility per Year in percent as well as the Time to Expiration in years.


1043

In case you have chosen a dividend payment, one more window will appear where you are asked to put the amount of the dividend.


1047

Finally you can choose the kind of option you like to calculate. Let's say we wanted to know the price of a call option:


1051

The price of our American call option on the given stock in the scenario (chosen through the corresponding parameters) with fixed dividend is now displayed in the XploRe output window. In case you have chosen a stock as underlying asset even the price of the European call option is displayed (in case you have not chosen a dividend, the price of a European call option equals that of an American call option):


  [1,]  

  [2,] -------------------------------------

  [3,]  The Price of Your European Call-Option 

  [4,]  on Given Stock with fixed Dividend is

  [5,]  27.3669

  [6,] -------------------------------------

  [7,]  



  [1,]  

  [2,] -------------------------------------

  [3,]  The Price of Your American Call-Option 

  [4,]  on Given Stock with fixed Dividend is

  [5,]  27.4641

  [6,] -------------------------------------

  [7,]


3.2 Option Price Determining Factors


1064 influence ()
displays the influence of price determining parameters on options

The quantlet 1067 influence measures and visualizes the influence of different factors on the prices of options. It is simply started by typing


  influence()

in the command line of XploRe. The option prices are calculated with the Black-Scholes formula. After starting the quantlet the following window appears:

1076

You may enter the different parameters needed to simulate the diffusion process. Next select the influence variables -- you may select up to two variables. The following example demonstrates the use of just one variable:


1080

In this example we would like to calculate the influence of the exercise price on the option price. You must set the lower and upper bound for your chosen variable.


1084

After pushing the OK button you will be asked for what kind of option the influence is to be calculated.

If you choose for example a Put option you will obtain the following graph which shows the influence of the factor (exercise price) on the price of the option:


1088

Using 1090 influence you can also select two variables as the following example demonstrates:


1095

After selecting the two variables you wish to compute, XploRe asks you to set the lower and upper bound for both variables.

If you choose e.g. Put you will obtain a three-dimensional graphic with the two selected influence factors (exercise price and time to expiration) and the price of the option. You may turn the graphic around by using the cursor buttons.

\includegraphics[scale=0.6]{finnew2.ps}


3.3 Greeks


1109 greeks ()
calculates and displays the different indices which are used for trading with options

The interactive function 1112 greeks calculates and displays the different indices used for analyzing and trading with options. You start it by


  data=greeks()

The first step is to enter the asset's basic data:


1117

Next, you have to select the variables you want to analyze (at most two), e.g.


1121

Select the ranges for the values of the chosen variables:


1125

1129

Now you can choose the index you are interested in:


1133

After telling the program the kind of option you want, the quantlet 1135 greeks will produce a graphical output window for your result:

\includegraphics[scale=0.6]{finnew3.ps}



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