! Computing the value of an option using the Black

 Scholes formula (see "The Pricing of Options and

 Corporate Liabilities", Journal of Political

 Economy, May-June, 1973);

SETS:

! We have 27 weeks of prices P( t), LOGP( t) is log

 of prices;

  WEEK/1..27/: P, LOGP;

ENDSETS

 

DATA:

! Weekly prices of National Semiconductor;

  P = 26.375, 27.125, 28.875, 29.625, 32.250,

      35.000, 36.000, 38.625, 38.250, 40.250,

      36.250, 41.500, 38.250, 41.125, 42.250,

      41.500, 39.250, 37.500, 37.750, 42.000,

      44.000, 49.750, 42.750, 42.000, 38.625,

      41.000, 40.750;

 

! The current share price;

  S = 40.75;

 

! Time until expiration of the option, expressed

 in years;

  T = .3644;

 

! The exercise price at expiration;

  K = 40;

 

! The yearly interest rate;

  I = .163;

ENDDATA

 

SETS:

! We will have one less week of differences;

  WEEK1( WEEK)| &1 #LT# @SIZE( WEEK): LDIF;

ENDSETS

 

! Take log of each week's price;

  @FOR( WEEK: LOGP = @LOG( P));

 

! and the differences in the logs;

  @FOR( WEEK1( J): LDIF( J) =

   LOGP( J + 1) - LOGP( J));

 

! Compute the mean of the differences;

  MEAN = @SUM( WEEK1: LDIF)/ @SIZE( WEEK1);

 

! and the variance;

  WVAR = @SUM( WEEK1: ( LDIF - MEAN)^2)/

   ( @SIZE( WEEK1) - 1);

 

! Get the yearly variance and standard deviation;

  YVAR = 52 * WVAR;

  YSD = YVAR^.5;

 

! Here is the Black-Scholes option pricing formula;

  Z = (( I + YVAR/2) *

   T + @LOG( S/ K))/( YSD * T^.5);

 

! where VALUE is the expected value of the option;

  VALUE = S *@PSN( Z) - K *@EXP( - I * T) *

   @PSN( Z - YSD *T^.5);

 

! LDIF may take on negative values;

  @FOR( WEEK1: @FREE( LDIF));

 

! The price quoted in the Wall Street Journal for

 this option when there were 133 days left was

 $6.625;

Model: OPTION