Q:
A portfolio hedge is purchased in the form of a spread.
1) The call is an option on ABC Corp. with a $3 premium and a $55 exercise.
2) The put is an option on ABC Corp. with a $1 premium and a $45 exercise.
What is the value of the hedge with a 20% probability of ABC E(v) at $40, a 60% E(v) at $50 (current price), and a 20% E(v) at $60?
A:
First, calculate the net gains/losses under each probability.

At 50, the value is: cost of the spread ($1 x 100 share put) + ($3 x 100 share call). So 100 + 300 = $400 cost (or -400) of the spread.
1) The $50 per share level has no gain or loss with a cost of -400, so at $50 per share the profit/loss is -$400.

2) At $60 per share, the put is worth zero since the price of the stock is more than the put's exercise price (45). The call is worth 60-55 or $500 for 100 shares with a cost of -400, or a net of $100, so at $60 per share the profit/loss is $100.

3) At $40 per share, the call won't be exercised because the ABC stock price is under the exercise price of the call ($55). The put is worth 45-40, or $500 with a cost of -400, so at $40 per share the profit/loss is $100.

Finally, sum the probabilities:

.6 * -400 = -240
.2 * 100 = 20
.2 * 100 = 20

The value of the hedge turned out to be -$200. What would it be if there were a 40% chance of stability (no gain or loss) and a 40% chance the put would be exercised at an actual of 30? Knowing how these sensitivities work, and the direction ups and downs take the end result, is key to quick responses on many of the CFA questions!

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