You observe the following prices European options on a non-dividend-paying stock:

Current stock price: $20

Strike price (both options): $22

Time to maturity: 1 year

Option prices (each option is written on 1 share):

European call price: $1.23

European put price: $1.98

You know that both options are correctly priced.

Using these prices, compute the implied one-year effective risk-free interest rate.

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

You observe the following prices European options on a non-dividend-paying stock:

Current stock price: $20

Strike price (both options): $22

Time to maturity: 1 year

Option prices (each option is written on 1 share):

European call price: $1.23

European put price: $1.98

You know that both options are correctly priced.

Using these prices, compute the implied one-year effective risk-free interest rate.

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

We start from the put-call parity for European options on a non-dividend-paying stock: C - P = S0 - K / (1 + r), where r is the annual effective risk-free rate and T = 1 year.

Given data: S0 = 20, K = 22, C = 1.23, P = 1.98, T = 1 year.

Compute the left-hand side: C - P = 1.23 - 1.98 = -0.75.

Set this equal to the right-hand side: -0.75 = 20 - 22 / (1 + r).

Rearrange to solve for the discount term: 22 / (1 + r) = 20 + 0.75 = 20.75.

Invert to solve for (1 + r): 1 + r = 22 / 20.75 ≈ 1.0608.

Subtract 1 to obtain r: r ≈ 0.0608, which corresponds to about 6.08% per year.

To express with two decimal places, we interpret the result as approximately 0.06 (i.e., 6.00%). Since the instruction uses a decimal format without a percent sign, the rounded value to two decimals is 0.06.

Sanity check: the result is a positive risk-free rate consistent with the call price being less than the stock price and the put price being relatively high, aligning with parity under a positive r.

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