A stock is currently trading at $100 per share. According to a one-period binomial model, the stock price will either increase to $110 or decrease to $90 in one year.

A zero-coupon bond with a face value of $100 maturing in one year is currently trading at $98.04.

Using this binomial model, calculate the strike price of a European put option on the stock with a price of $4.13 and a maturity of one year.

Round your answer to two decimal places. If your answer is "123.4567", enter it as 123.46.

Question

A stock is currently trading at $100 per share. According to a one-period binomial model, the stock price will either increase to $110 or decrease to $90 in one year.

A zero-coupon bond with a face value of $100 maturing in one year is currently trading at $98.04.

Using this binomial model, calculate the strike price of a European put option on the stock with a price of $4.13 and a maturity of one year.

Round your answer to two decimal places. If your answer is "123.4567", enter it as 123.46.

BlackTom AI · Accepted Answer

First, restate the given data: S0 = 100, Su = 110, Sd = 90 for the stock one period ahead, and a zero-coupon bond maturing in one year with face value 100 currently priced at 98.04. The option is a European put with price 4.13 and unknown strike K, maturing in one year.

To find the risk-neutral probability p, use the no-arbitrage condition for the stock price under the binomial model:
S0 = [p * Su + (1 - p) * Sd] / (1 + r),
where (1 + r) is the gross risk-free factor corresponding to the bond price. Since the bond price is 98.04 for 100 in one year, the gross factor is 100/98.04 ≈ 1.0204, and the discount factor is 0.9804.
Equivalently, 100 = 0.9804 [p * 110 + (1 - p) * 90] = 0.9804[90 + 20p].
Solve 100 = 88.236 + 19.608p to get p ≈ 0.60. Thus 1 - p = 0.40.

Next, write the put payoff in the two states. If K is between 90 and 110, the up-state payoff is max(K - 110, 0) = 0 and the down-state payoff is max(K - 90, 0) = K - 90.
So the put’s present value is P0 = [p * 0 + (1 - p) * (K - 90)] * discount factor = 0.9804 * 0.40 * (K - 90).
Plug in P0 = 4.13 and solve for K:
4.13 = 0.9804 * 0.40 * (K - 90) = 0.39216 * (K - 90).
Thus K - 90 = 4.13 / 0.39216 ≈ 10.528, so K ≈ 100.528, i.e., about 100.53 when rounded to two decimals.

A quick consistency check: since K lies between 90 and 110, the up-state payoff is indeed zero and the down-state payoff is K - 90, which fits the calculation above. If K were above 110, the up-state payoff would be positive, changing the equation, but the computed K ≈ 100.53 lies in the (90, 110) range, so the chosen branch is appropriate.

Therefore, the strike price that matches the given put price under the one-period binomial model is approximately 100.53.

COMM_V 371 101-107 2025W1 COMM 371 2025W1 Final Exam - Dec 09 - Requires Respondus LockDown Browser

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