Question

CEO of Company X has 30 million shares of stocks and 30 million shares of options (of Company X). The stock

return volatility () is 0.50, dividend yield (d) is 0.0, riskfree rate (r) is 0.05, and maturity (T) is 5 years. Following
table shows stock price of Company X during 2016.
Date Jan. Feb. Mar. Apr. May. Jun. Jul. Aug. Sep. Oct. Nov. Dec.
Price $15.2 $10.0 $22.7 $25.5 $20.0 $23.0 $18.8 $18.6 $15.5 $18.3 $20.0 $20.4
Suppose that options were granted at-the-money on February 2016.
a. Find option’s delta (change in one share of option due to $1 increase in stock price) as of December 2016?
b. Find increase in CEO wealth for $1 increase in Company X’s stock price as of December 2016?

Answer 1

Without the exercise price of the options and the Black-Scholes model inputs, we cannot calculate the precise option's delta or the exact increase in CEO wealth for a $1 increase in Company X's stock price. To compute the increase in wealth, one would typically use the options' delta and multiply it by the number of shares–but key data is missing for this calculation.

Step-by-step explanation:

The delta of an option as of December 2016, given the specified conditions, would need to be calculated using a formula from the Black-Scholes model, which is not provided here. Therefore, we cannot provide a direct answer to the delta question without additional information such as the exercise price of the options.

As for the increase in CEO wealth for a $1 increase in company X's stock price, the total wealth increase would be the sum of the increase in value of the shares and the change in value of the options due to the $1 increase in stock price, which is impacted by the delta of the options. To calculate the increase in CEO's wealth due to a $1 increase in stock price multiply the change in stock price by the sum of the shares and options delta. However, without knowing the options' strike price and the Black-Scholes inputs, we cannot calculate the precise wealth increase.