The current price of a non-dividend-paying stock is $40. Over the next year it is expected to rise to $46 or fall to $34. Assume the risk-free rate is zero. An investor buys a collar on the stock (i.e., buy 1 share of the stock, buy 1 European put option on the stock, and write 1 European call option on the stock). Both options have a one-year maturity and a strike price of $40. Which of the following is the hedge ratio of the collar position?

Question

BlackTom AI · Accepted Answer

We start by restating the scenario and the hedge components to understand how the hedge ratio should be determined.
The investor constructs a collar on 1 share of the stock by: (i) buying 1 share of the stock, (ii) buying 1 European put option with strike 40, and (iii) writing 1 European call option with strike 40. All instruments have a 1-year maturity and the current stock price is 40 with a zero risk-free rate.
To assess the hedge ratio, we analyze the deltas of each component, since the hedge ratio is essentially the net delta of the position.
Option-by-option delta intuition:
Option 1 (0.50): If the overall hedge ratio were 0.50, the net delta of the position would be +0.50 per share. That would imply the collar leaves some directional risk unhedged, which contradicts the typical collar structure designed to be delta-neutral when the stock is at-the-money and the put and call have the same strike.
Option 2 (-0.50): A net delta of -0.50 would imply the collar significantly increases exposure to price rises, which again is inconsistent with the intended hedging effect of combining a long put and a short call around a long stock position.
Option 3 (0.00): This is the key idea. The delta of the long stock is +1. The long put with strike equal to the current price at initiation has a negative delta (approximately -0.5 when at-the-money), and the short call with the same strike has a negative delta (approximately -0.5 for a short at-the-money call). Summing deltas: +1 (stock) + (-0.5) (put) + (-0.5) (short call) ≈ 0. Thus the net delta is zero, yielding a delta-neutral collar.
Option 4 (2.00): A net delta of +2 would correspond to twice the exposure of the stock, which is inconsistent with holding only 1 share of the stock and the paired options designed to hedge that exposure.
Option 5 (1.00): A net delta of +1 would imply you still have full long-stock exposure after accounting for the options, which would defeat the hedging intent of a collar that stabilizes delta around the current price.
In summary, the delta of the collar position hinges on the combination of +1 from the stock and approximately -0.5 from the long put and -0.5 from the short call, yielding a net delta very close to zero. The only option that aligns with this delta-neutral outcome is the one that expresses a net hedge ratio of zero.

2025FallDYN-T-FIN530-86763-86762 Quiz 6 - "Black-Scholes-Merton Model and Dynamic Hedging"

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