When examining the value of the 70 million Microsoft Stock Options, we started by examining the value by using the Black-Scholes Valuation method. We were given that the time period would be T=6 years, and that both the strike price (K) and the Stock Price (S0) were equal to 66.625. When examining the case provided, it gave the data for both volatility and the risk-free rate in 2000. The volatility for this case in 2000 was 0.33 and the risk-free rate was 6.20%.
Next, we calculated d1 and d2 in order to evaluate the price of the call option. D1 and d2 were both calculated at 0.63427 and -0.17406, respectively. This was determined by using the formulas to find d1 and d2 using the Black-Scholes method of valuation.
Lastly, using the Black-Scholes method, the price of the call following all of these conditions was determined to by $29.31. So, the value of 70 million options is simply 70 million multiplied by the price of the call, $29.31. This was found to be $2,052,045,496.29. This data can be seen in Table 1 in the appendix.
Next, I examined the value of the 70 million options when T followed the distribution of: 10% at 4 years, 20% at 5 years, 40% at 6 years, 20% at 7 years, and 10% at 8 years. Following this distribution set, and calculating the summation of all 70 million options at each different time, T, gave a smaller valuation when compared to the value of all 70 million options at one time, T=6 ($2,040,710,000.00 vs all at T=6, $2,052,045,496.29). The following distribution of the call options at different times, their values at those times, and the summation of all the values can be found in Table 2 of the appendix. The Black-Scholes valuation of the options was then recalculated using two different historical volatilities calculated using the daily return data for Microsoft in 1999 and in 2000-2005. To determine this volatility, the continuously