The current price of a non-dividend-paying stock is $100. Over the next year it is expected to rise to $120 or fall to $80. Assume the risk-free rate is 10% per year with annual compounding. An investor buys a AMERICAN put option on the stock with a strike price of $110 and maturity of one year. What is the value of the option?

Question

BlackTom AI · Accepted Answer

First, restate the problem in my own words to ensure understanding: We have a non-dividend-paying stock currently priced at $100. Over the next year, the stock can go up to $120 or down to $80. The risk-free rate is 10% per year with annual compounding. An American put option with strike $110 and 1 year to maturity is available. The task is to determine the value of this option.

Option 1 (0.00): This would imply the put is worthless today. Consider that the strike is $110 and the current stock price is $100, so exercising now would yield an immediate payoff of $110 - $100 = $10. Since you can exercise immediately if you hold an American put, a value of 0.00 is clearly inconsistent with this immediate exercise payoff. Therefore, this choice is incorrect.

Option 2 (6.82): This value corresponds to the risk-neutral expected payoff discounted back, without considering the possibility of early exercise. To explore this, compute the risk-neutral probability p* using the binomial framework: u = 1.20, d = 0.80, R = 1 + 0.10 = 1.10, so p* = (R - d) / (u - d) = (1.10 - 0.80) / (1.20 - 0.80) = 0.30 / 0.40 = 0.75. The option payoffs at maturity are: if up to 120, put payoff is max(110 - 120, 0) = 0; if down to 80, put payoff is max(110 - 80, 0) = 30. The risk-neutral expected payoff is 0.75*0 + 0.25*30 = 7.5. Discounting by R gives 7.5 / 1.10 ≈ 6.82. However, this ignores the important feature of American options: the possibility of early exercise. Since the build-up above does not account for the value of exercising now, this value is incomplete for an American put. Hence, while 6.82 is a relevant figure for a European-style value, it cannot be the final value for an American put.

Option 3 (10.00): The immediate exercise value of an American put is max(K - S0, 0) = max(110 - 100, 0) = 10. Since you can exercise immediately (the stock pays no dividend, so early exercise value is not offset by any dividends), the rational choice for an American put holder at t=0 is to exercise now if the continuation value is less than 10. The continuation value computed via risk-neutral pricing was about 6.82, which is less than 10. Therefore, exercising now yields a higher payoff than waiting, making 10 the correct value of the American put in this one-period model. This aligns with the principle that an American put on a non-dividend-paying stock is often optimally exercised early when the immediate exercise payoff exceeds the continuation value. Consequently, this option is correctly valued at 10.

Option 4 (40.00): A value of 40 would imply an unrealistically high premium given the potential future payoffs (the put can only be worth up to the strike minus the current price in worst-case scenarios, i.e., up to 10 if exercised now, or up to 30 at maturity in the down state, but discounted expectations do not support a 40). This is inconsistent with the payoffs and the discounting implications in the one-period binomial framework. Therefore, this choice is incorrect.

Putting these observations together, the option’s value should reflect the higher immediate exercise value of 10 rather than the purely discounted expected payoff of 6.82, because early exercise is optimal for this American put in this setup. Hence, the best-supported choice is the 10.00 value.

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