Assume a stock with current stock price of 90 and an American put option written on this stock with exercise price 75. The time to maturity is one year. The factor for the stock's downward movement is 2/3 per year; for the upward movement u=1.5 holds. The risk-free rate is 10.00% per year.Construct a risk-free hedge portfolio which is composed of m stocks and one long put written against the stock. Which statement regarding the hedge portfolio is most likely true?

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BlackTom AI · Accepted Answer

Question restatement: We are given a stock with current price S0 = 90, and an American put option with strike K = 75 expiring in one year. The stock moves in a binomial fashion with downward factor d = 2/3 and upward factor u = 1.5. The risk-free rate is 10% per year. We are asked to construct a risk-free hedge portfolio composed of m shares of the stock and one long put option written against the stock, and to identify which statement about the hedge portfolio is most likely true.

First, compute the possible stock prices in one year:
- If the stock moves up: Su = S0 * u = 90 * 1.5 = 135
- If the stock moves down: Sd = S0 * d = 90 * (2/3) = 60

Next, determine the payoff of the put option in each state:
- Payoff of the put when up (Pu): max(K − Su, 0) = max(75 − 135, 0) = 0
- Payoff of the put when down (Pd): max(K − Sd, 0) = max(75 − 60, 0) = 15

Now write the value of the hedge portfolio in each state. The portfolio consists of m shares and one long put:
- In the up state: V_u = m * Su + Pu = m * 135 + 0 = 135m
- In the down state: V_d = m * Sd + Pd = m * 60 + 15

To achieve a risk-free (state-independent) payoff, set V_u = V_d:
135m = 60m + 15
Solve for m: 135m − 60m = 15 ⇒ 75m = 15 ⇒ m = 15/75 = 0.20
Thus, the hedge portfolio should hold 0.20 shares of the stock for every long put written against the stock.

With m = 0.20, the common payoff in both states is: V = 135 * 0.20 = 27 in the up state, and V = 60 * 0.20 + 15 = 12 + 15 = 27 in the down state. Therefore, the hedge portfolio has a risk-free payoff of 27 in one year, irrespective of whether the stock goes up or down.

Interpretation of the result: The key idea is to choose the number of shares m so that the combined payoff of the stock position and the put position replicates a risk-free payoff. Here, the required m is 0.20, which aligns with the intuitive goal of balancing the downward payoff from the stock with the put’s payoff in the down state, while the up-state payoff comes entirely from the stock portion since the put expires worthless when the stock goes up.

Why this differs from other possible allocations: If you choose any m other than 0.20, the up-state and down-state payoffs will not match, leaving residual risk. For example, if m > 0.20, the up-state payoff would exceed the down-state payoff, creating a mismatch; if m < 0.20, the down-state payoff would exceed the up-state payoff. The single long put is crucial here because it provides the extra 15 payoff in the down state, enabling the two-state payoffs to align when paired with the right amount of stock.

Notes on the given answer options: The statement that describes a hedge of 0.20 shares of stock and one long put aligns with the replication calculation above. If any other options propose different m or omit the put, they would fail the requirement for a state-independent payoff, given the provided up/down prices and the put’s payoffs.

Important caveat: This analysis presumes a one-period binomial model and that the put is held to maturity (as opposed to being exercised early in the case of an American put). In practice, early exercise features could affect the hedge if dividends or other factors are present, but under standard binomial-hedging assumptions without dividends, the above replication holds for the one-period horizon described.

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