Stock A is currently trading at $100 per share. A binomial model indicates that in one year, the stock price will be either $120 or $80. At the moment, the effective one year interest rate is 5.21% (APR compounded annually).

Using a one-period binomial model, calculate the price of a European put option on one share of Stock A, with a strike price of $95 and one-year maturity.

Enter your final answer rounded to two decimal places. For example, enter 1.23 if your answer is $1.234, and enter -1.23 if your answer is -$1.234.

Question

Stock A is currently trading at $100 per share. A binomial model indicates that in one year, the stock price will be either $120 or $80. At the moment, the effective one year interest rate is 5.21% (APR compounded annually).

Using a one-period binomial model, calculate the price of a European put option on one share of Stock A, with a strike price of $95 and one-year maturity.

Enter your final answer rounded to two decimal places. For example, enter 1.23 if your answer is $1.234, and enter -1.23 if your answer is -$1.234.

BlackTom AI · Accepted Answer

To price the European put in a one-period binomial model, I start by identifying the up and down factors from the stock price movements. Here, the current stock price S0 = 100. If the stock moves up to Su = 120 and down to Sd = 80, the up factor is u = Su/S0 = 120/100 = 1.20 and the down factor is d = Sd/S0 = 80/100 = 0.80.
Next, I determine the risk-free rate for one year in effective terms. The problem states the effective one-year interest rate is r = 5.21%, so the gross rate is 1 + r = 1.0521.
I compute the risk-neutral probability p using the formula p = [(1 + r) − d] / (u − d). Substituting the numbers gives p = (1.0521 − 0.80) / (1.20 − 0.80) = 0.2521 / 0.40 = 0.63025.
Then I calculate the option payoffs at the two possible end nodes for a European put with strike K = 95:
- If the stock goes up to 120, the put payoff is max(K − Su, 0) = max(95 − 120, 0) = 0.
- If the stock goes down to 80, the put payoff is max(K − Sd, 0) = max(95 − 80, 0) = 15.
I form the expected payoff under the risk-neutral measure: E^Q[payoff] = p × 0 + (1 − p) × 15 = (1 − 0.63025) × 15 = 0.36975 × 15 = 5.54625.
Finally, I discount this expected payoff back to the present using the gross risk-free rate: P0 = E^Q[payoff] / (1 + r) = 5.54625 / 1.0521.
Carrying out the division yields P0 ≈ 5.273, which rounds to 5.27 when expressed to two decimal places.
In summary, the one-period binomial price of the European put with K = 95 is approximately 5.27. The reasoning hinges on constructing a risk-neutral world, computing the payoffs at each node, weighting by the risk-neutral probability, and discounting back at the risk-free rate.

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