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?单项选择题
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Question

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?单项选择题

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BlackTom AI · Accepted Answer

We start by laying out the given binomial-like one-period hedge problem in plain terms. The current stock price is S0 = 90, the American put has strike K = 75, time to maturity T = 1 year. The up and down factors are u = 1.5 and d = 2/3, so the future stock prices are Su = 90 × 1.5 = 135 and Sd = 90 × (2/3) = 60. The risk-free rate is r = 10% per year, so the gross risk-free factor over the period is 1 + r = 1.10.

We are asked to construct a risk-free hedge portfolio consisting of m shares of the stock and one long put. A risk-free hedge means that the value of the portfolio in the up state and the down state after one year must be the same (so that it can grow risk-free to its future value at rate r).

Step 1: Impose the risk-free condition on the portfolio values in each state.
- In the up state, the portfolio value is: V_u = m × Su + P_u. Since Pu = max(K − Su, 0) and Su = 135 > K = 75, Pu = 0. So V_u = m × 135.
- In the down state, the portfolio value is: V_d = m × Sd + P_d. Here Sd = 60 and Pd = max(K − Sd, 0) = max(75 − 60, 0) = 15. So V_d = m × 60 + 15.

Step 2: Set V_u equal to V_d to ensure risk-freeness:
m × 135 = m × 60 + 15
This simplifies to 75m = 15, so m = 15 / 75 = 0.20.

Thus, the hedge uses 0.20 shares of stock for each long put. This makes both future portfolio values equal to V_u = V_d = 0.20 × 135 = 27.

Step 3: Interpret the result in terms of the statement options.
- The hedge portfolio consists of 0.20 long stock(s) and 1 long put. This is precisely the pattern we derived: a small long position in the stock combined with one long put yields a risk-free payoff over the one-period horizon.
- The numeric payoff in either state would be 27 at T = 1 year (before discounting), corresponding to a present value of 27 / 1.10 ≈ 24.55, which would then equal the cost of establishing this hedge today: 0.20 × 90 + P0, where P0 is the price of the put today. While the exact put’s current price P0 is not specified in the question, the structural hedge (0.20 shares plus one put) is the correct risk-free combination given the up and down movements and the payoff profile of the put.

Why other (nonexistent) options would be wrong in a typical question like this: if someone proposed a different m, say m = 0, the down state payoff would be 15 while the up state payoff would be 0, which are not equal, so the portfolio would not be risk-free. If someone proposed m = 0.5, we’d get V_u = 67.5 and V_d = 30 + 15 = 45, again unequal, so not risk-free. The key is solving m from the equality of the up- and down-state payoffs, which yields m = 0.20, exactly matching the stated option.

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