Derivatives in Plain Words by Frederic Lau, with a ... - HKU Libraries

di = [ln(IOO/IOO) + (0.06 + O.P/2) x 0.25] / (0.! x 0.25 I/2 )= 0.3260 or 0.33; N(0.33) = 0.6293di = 0.3260 - O.I x 0.25 l/2= 0.2760 or 0.28; N(0.28) = 0.6103Co = 100 x 0.6293 - (100 x e-° 06x025 ) x 0.6103= 62.93 - 60.12 = $2.81If the volatility is 15 percent instead of 10 percent, with other factorsunchanged, the value of the option will be higher at $3.96.The Black-Scholes formula is not perfect. It makes some assumptions. Forexample, it assumes that the underlying stock does not pay dividends. Andit values American options not so accurately because of the early exercisecause. However, it is considered an easy-to-get approximation of the priceof an option.Applying the Black-Scholes formula is much easier than understanding theconcepts behind it From the above equations, we can first solve for di andd2, and then Co. All you have to do is to plug all the necessary numbersinto the equations. With the technology nowadays, you can get the optionvalue in a second using the Black-Scholes formula.For data inputs, we can obtain the stock price, strike price, risk-free interestrate, and time to maturity of the option very easily. But how do we getthe standard deviation of a stock?IMPLIED VOLATILITYThe standard deviation (or the volatility) of a stock cannot be readily observedlike the other inputs. Market practitioners usually use the historical data,scenario analysis or prices of other options to measure a stock's standarddeviation. Because there is no uniform way to derive a stock's volatility, thetrue price of the option and the option price calculated using the Black-Scholes formula can be different.Delta and Volatility