MODEL:
! Computing the value of an option using the Black
Scholes formula (see "The Pricing of Options and
Corporate Liabilities", Journal of Political
Economy, MayJune, 1973);
! Keywords: Black/Scholes, option pricing,
derivative/financial;
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);
! LDIF may take on negative values;
@FREE( LDIF(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 BlackScholes option pricing formula;
@FREE(Z);
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);
! The price quoted in the Wall Street Journal for
this option when there were 133 days left was
$6.625;
