calculate volatility

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 ) where and and where C0 S0 d1  ln( S 0 / X )  ( r   2 / 2) T  T , d 2  d1   T X r T  ln e = = = = = = N(d) Current option value Current stock price Exercise price Risk-free interest rate (annualized continuously compounded) Time to maturity or expiry of option in years Standard deviation of the annualized continuously compounded rate of return on the stock = Natural logarithm function = 2.71828, the base of the natural logarithm function = P[Z  d ] , where Z has a standard normal distribution The parameters needed to do a calculation are all readily available, except for .  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 S0 Estimation of  from historical data The spreadsheet stock.xls gives daily closing prices for the past year , Pt 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 t t is 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 t t from the original column of daily closing stock prices, columns 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 60-day 90-day 120-day 180-day 1-year 0.753168 0.625336 0.603594 0.653869 0.649999 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 Exercise Price 12 15 18.625 22 25  1 mo 2 mo 3 mo 4 mo 6 mo 1 yr .753168 .625336 .603594 .653869 .649999 .728998 6.7 6.786 7.054 7.132 7.474 8.758 3.981 4.191 4.738 4.892 5.426 7.163 1.648 1.963 2.714 2.925 3.564 6.657 0.578 0.837 1.531 1.741 2.4 4.555 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 15 17.5 20 20 20 20 22.5 Expiry Days Option Price 21-Aug-98 9 3.875 21-Aug-98 9 1.5 21-Aug-98 9 0.375 18-Sep-98 37 1 16-Oct-98 65 1.5625 15-Jan-99 156 2.75 16-Oct-98 37 0.875 Implied Volatility 1.193 0.664 0.695 0.644 0.654 0.65 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 12 15 18.625 22 25 Implied Volatility 1.828 1.193 0.662 0.844 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.