Question

A collar is established by buying a share of stock for \(50, buying a six-month put option with exercise price \)45, and writing a six-month call option with exercise price \(55. Based on the volatility of the stock, you calculate that for an exercise price of \)45 and maturity of six months, N (d₁) = .60, whereas for the exercise price of \(55, N (d₁) = .35.

a. What will be the gain or loss on the collar if the stock price increases by \)1?

b. What happens to the delta of the portfolio if the stock price becomes very large? Very small?

Short answer

a. loss on the short put is $.40

b. delta of the collar approaches zero; delta of the collar approaches zero

Step-by-step solution

Calculation of gain or loss on the collar ‘a’

	Delta
Short	1.00
Short call	-N(d1) = -.35
Long put (0.60-1)	N(d1) – 1= -.40
Total (gain)	0.25

On an increase of stock price by $1, the value of the collar increases by $.25. Hence, the value of the stock would be $1 more while the loss on the short put is $.40, and the call written is a liability that increases by $.35.

Calculation of the delta of the portfolio if the stock price is very large ‘b’

When S becomes very large, the delta of the collar approaches zero. Both N(d1) terms approach 1.0 so that the delta for the short call position approaches -1.0 and for the long put position approaches zero.

Calculation of the delta of the portfolio if the stock price is zero ‘b’

When S becomes zero, then the delta of the collar also approaches zero. Both N(d1) terms approach 0 so that the delta for the long put position approaches -1.0 and for delta for long put position approaches -1.0.