Final answer:
The hedge ratio for a call option on ILG stock with a strike price of $160 is 0.4, indicating that 0.4 shares of ILG should be purchased for each call option sold in order to hedge the position.
Step-by-step explanation:
To determine the number of shares of ILG stock to purchase to hedge each call option, we need to calculate the hedge ratio, which is also known as the option's delta.
The delta measures how much an option's price is expected to change per a one-dollar change in the price of the underlying asset.
For a call option with a strike price of $160, the value of the option in the up-state (when the stock price is $180) is $180 - $160 = $20, and in the down-state (when the stock price is $90), the value of the option is $0 because the stock price is below the strike price, and the option would not be exercised.
Thus, the change in option price is $20 in the up-state and $0 in the down-state, while the change in stock price is $180 - $130 = $50 up and $90 - $130 = -$40 down. Now we calculate the hedge ratio:
Hedge Ratio = Change in Option Price / Change in Stock Price
Hedge Ratio Up-State = $20 / $50 = 0.4
Hedge Ratio Down-State = $0 / -$40 = 0 (ignoring the negative because the option wouldn't be exercised)
Therefore, the hedge ratio is 0.4, which means you need to purchase 0.4 shares of ILG to hedge each call option sold.