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					                   COURSE 7 COMMON CORE CASE STUDY

        VALUATION OF EQUITY OPTIONS AND IMPLIED VOLATILITY

                Material Given To Candidates Near The End Of The Session



REPORT ON OPTION OPTIONS

There is limited flexibility in the range of option contracts currently trading on the
market. The objective has been to estimate call option prices for a wider range of strike
prices and expiry dates than is currently available in the market.

The option price estimation is based on the Black-Scholes option pricing formula for
non-dividend paying stocks. The formula is

C0  S 0 N ( d1 )  Xe rT N ( d 2 )
                         ln( S 0 / X )  ( r   2 / 2) T
where             d1 
                                        T
and               d 2  d1   T         ,
and where
      C0          = Current option value
         S0       = Current stock price
         X        = Exercise price
         r        = Risk-free interest rate (annualized continuously compounded)
         T        = Time to maturity or expiry of option in years
                 = Standard deviation of the annualized continuously compounded rate of
                    return on the stock
         ln       = Natural logarithm function
         e        = 2.71828, the base of the natural logarithm function
         N(d)     = P[Z  d ] , where Z has a standard normal distribution

The parameters needed to do a calculation are all readily available, except for .

S0   18.625 is today’s closing price
r  4 ln(1.0127582)=.05071 for 13-week Treasury Bills
r  2 ln(1.012619)=.05171 for 26-week Treasury Bills
Estimation of  from historical data

                                                                            P
The spreadsheet stock.xls gives daily closing prices for the past year , t is the closing
price on day t. The daily return for day t (from the close of day t-1 to the close of day t)
    R P/ P 1
is t       t  t
                 . Since  is the standard deviation of the annualized continuously
compounded rate of return on the stock, we first estimate the standard deviation of the
ln( Rt ) values. The spreadsheet “stock.xls” contains 252 consecutive trading day daily
closing prices for the stock. We can estimate  using all or some of the past year’s data.
The following table summarizes the estimated values of  using several different time
periods, all ending today (the number of days is calendar time). Within the spreadsheet,
                                                                   R P/ P 1
from the original column of daily closing stock prices, columns t         t   t
                                                                                 and ln( Rt )
values are generated. The EXCEL function STDEV is then applied to calculate the
standard deviation of various sequential groups of data points from the ln( Rt ) column,
ending with today’ ln( Rt ) value. This results in an estimate of the standard deviation of
the daily continuously compounded return of the stock. With 252 trading days per
calendar year, the estimate of  is the daily standard deviation multiplied by 252 .
The resulting estimates of  are given in the following table.


Estimated Volatility (Standard Deviation)

30-day                  0.753168
60-day                  0.625336
90-day                  0.603594
120-day                 0.653869
180-day                 0.649999
1-year                  0.728998
TABLE 1


When calculating call option prices using the formula, the value of  used will be the
value from the table above for the associated expiry date. The continuously conpounded
risk-free rate for options expiring in 13 weeks or less will be r=.05071 , for options
expiring in 26 weeks or more the risk-free rate will be r=.05171, and for options expiring
between 13 and 26 weeks from now, the risk-free rate will be determined using linear
interpolation between the 13 and 16 week rates. The following table summarizes the call
option prices for various strike price and expiry date combinations, the prices having
been calculated using the Black-Scholes formula. Calculations were done withn EXCEL
using the NORMDIST function for normal distribution cdf values.
Table of call option prices based on  estimated from historical data

           Time to Expiry            1 mo       2 mo      3 mo      4 mo      6 mo        1 yr
Exercise Price                       .753168 .625336 .603594 .653869 .649999 .728998
        12                                  6.7     6.786     7.054     7.132      7.474       8.758
        15                               3.981      4.191     4.738     4.892      5.426       7.163
   18.625                                1.648      1.963     2.714     2.925      3.564       6.657
        22                               0.578      0.837     1.531     1.741         2.4      4.555
        25                              0.1972       0.36     0.894     1.077       1.67       3.784
TABLE 2



Estimation of  from implied volatility

In the Black-Scholes formula, if the call option price C0 is known along with all other
model parameters except , then it is possible to solve for  using some approximation
method. The bisection algorithm can be applied, as can the Newton-Rapshon method.
The values of
 the other parameters are found in the same way is in the application of the formula using
estimates of  based on historical data. The following are the implied values of  from
the known market prices of the option contracts currently being traded.

Strike Price          Expiry    Days     Option Price               Implied Volatility
         15           21-Aug-98        9      3.875                      1.193
       17.5           21-Aug-98        9         1.5                     0.664
         20           21-Aug-98        9      0.375                      0.695
         20           18-Sep-98       37           1                     0.644
         20           16-Oct-98       65    1.5625                       0.654
         20           15-Jan-99      156       2.75                       0.65
       22.5           16-Oct-98       37      0.875                       0.89
TABLE 3

The implied volatility seems to be more closely related to the option strike price than the
time to maturity (an illustration of the volatility “smile”). This seems to suggest that the
value of  used in the option pricing formula ought to reflect this market pricing
phenomenon. Now we use the following implied volatility values (calculating an average
volatility for the strike price of 20) with interpolation between strike prices to recalculate
the entries in Table 2.

Strike Price          Implied Volatility
        12                 1.828
        15                 1.193
    18.625                 0.662
        22                 0.844
        25                 1.120
TABLE 4
Option Prices Using Implied Volatility
           Time to Expiry         1 mo       2 mo      3 mo      4 mo      6 mo       1 yr
Exercise Price                 
        12                 1.828       7.537      8.53     9.344    10.036    11.169       13.463
        15                 1.193       4.587     5.427     6.103     6.679      7.644       9.736
   18.625                  0.662       1.454     2.073     2.552     2.959      3.645       5.192
        22                 0.844       0.741     1.453      2.04      2.55      3.419       5.383
        25                  1.12       0.697     1.568     2.317     2.976      4.104       6.623
TABLE 5


Comparing Tables 2 and 5, it appears that using estimates of  based on historical data
may be less appropriate for use in the option pricing formula when the strike price is
significantly different than the current stock price. On the other hand, implied volatility
values become suspect if we try to extrapolate beyond the range of strike prices currently
being traded in the market. Correct valuations are likely to lie somewhere between the
valuations presented in these tables.

				
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