Question

The share price of Acme is $100. Six months from now, it will be either $150 with probability of.70 or $90 with probability .30. A call option exists on the stock that can be exercised only at the end of 6 months at an exercise price of $120.

If you wished to establish a perfectly hedged position, what would you do on the basis of the facts just presented?

Answer 1

To perfectly hedge a position with Acme's stock and the call option, one would use delta hedging by writing an option and buying 0.5 shares of Acme stock for every option written.

Step-by-step explanation:

If you wished to establish a perfectly hedged position given the future scenarios of Acme's stock and the call option, you would employ the concept of delta hedging. Delta hedging is a strategy that ensures your position remains neutral to small movements in the stock price by adjusting the number of shares held against the options written.

Here's how you would apply this:

1. Determine the payoffs: If the stock price goes to $150, the call option with an exercise price of $120 will be worth $30 (150-120). If the stock price goes to $90, the option will expire worthless, so its value is $0.
2. Calculate the hedge ratio (also known as the 'delta'): The change in option price divided by the change in stock price. Here, the option price moves from $30 to $0 when the stock price moves from $150 to $90, resulting in a delta of (30-0) / (150-90) = 30/60 = 0.5.
3. Implement the hedge: For every option you write, buy 0.5 shares of Acme stock.

This strategy would result in a perfectly hedged position that is indifferent to whether the stock price rises to $150 or falls to $90 after six months.