Question

A collar is established by buying a share of stock for $45, buying a 6?month put option with exercise price $38, and writing a 6?month call option with exercise price $52. On the basis of the volatility of the stock, you calculate that for a strike price of $38 and expiration of 6 months, N(d1) = .7314, whereas for the exercise price of $52, N(d1) = 0.6192.

a. What will be the gain or loss on the collar if the stock price increases by $1? (Round your answer to 2 decimal places. Omit the "$" sign in your response.)

b-1. What happens to the delta of the portfolio if the stock price becomes very large? (Omit the "$" sign in your response.) Delta of the portfolio approaches:

b-2. What happens to the delta of the portfolio if the stock price becomes very small? (Omit the "$" sign in your response.) Delta of the portfolio approaches:

Answer 1

a) In the given case, we have bought stock at 45, and our position in options is -

Long Put (X = 38); Short Call (X = 52).

If the price of the stock goes up by 1, i.e. the market price of stock becomes 46, then the put will lapse as we will not wish to sell our stock at 38 when the market price is 46. Similarly, the short call will lapse as the buyer of the call will not buy the stock at 52 from us when the market price is 46. So what we get due to options is the call premium and what we pay is the put premium. However, since the stock is up by 1 and we are long on stock, we can sell it and gain 1.

b-1) Delta refers to the sensitivity of the option price to changes in the price of the underlying. When the underlying does not pay any dividend, Delta of call = N(d₁) and delta of put = N(-d₁). If the stock price becomes very large, say 100, the put will lapse i.e. value of put = 0 and the short call will be exercised by the buyer (Value of call = 100-52 = 48 for the buyer). We will result in huge loss. Here, Delta of call i.e. N(d₁) = 0.7314 i.e. every $1 change in price of stock generates 0.7314 change in call value. Continuing with our example - the change in value of stock is (100-45) ie 55 then delta will change by 55*0.7314 = 40.227. Delta of put = 1- 0.6192 = 0.3808 ie. $1 change in price of stock brings 0.3808 change in put value. The overall delta of the portfolio does not move to the extremes because of having opposite positions in call and put.

b-2) When the stock price becomes very small, the short call lapses and the long put is exercised. Positions with positive delta will go up when underlying goes up and positions with negative delta will go down when underlying goes down. The overall portfolio delta is somewhat balanced and does not move to extremes because of having opposite positions in call and put.