Act as a finance professor at a prestigious university who teaches derivatives. To follow is a draft practice question for students learning stock option valuation. After solving the question, make su

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Claude-3-Opus ·

Act as a finance professor at a prestigious university who teaches derivatives. To follow is a draft practice question for students learning stock option valuation. After solving the question, make suggestions to improve the question.

Here is the question: While the risk-free rate is 3.0% per annum, a European call option with a strike price of $50.00 has one year to maturity. If the underlying non-dividend paying stock price is $50.00 (i.e., the option is exactly at the money) with a volatility of 20.0% per annum, what is the value of the option?

Claude-3-Opus

Poe

Great! Let's solve this problem step by step using the Black-Scholes model for European call option pricing.

Given:

Risk-free rate (r) = 3.0% per annum
Strike price (K) = $50.00
Time to maturity (T) = 1 year
Underlying stock price (S) = $50.00
Volatility (σ) = 20.0% per annum
The stock pays no dividends

Step 1: Calculate d1 and d2
d1 = (ln(S/K) + (r + σ²/2)T) / (σ√T)
d1 = (ln(50/50) + (0.03 + 0.2²/2)1) / (0.2√1)
d1 = (0 + 0.05) / 0.2
d1 = 0.25

d2 = d1 - σ√T
d2 = 0.25 - 0.2√1
d2 = 0.05

Step 2: Calculate N(d1) and N(d2)
Using a standard normal distribution table or calculator, find the cumulative probabilities:
N(d1) = N(0.25) = 0.5987
N(d2) = N(0.05) = 0.5199

Step 3: Calculate the call option price
C = SN(d1) - Ke^(-rT)N(d2)
C = 50 × 0.5987 - 50e^(-0.03 × 1) × 0.5199
C = 29.935 - 25.512
C = $4.42 (rounded to the nearest cent)

Therefore, the value of the European call option is $4.42.

Suggestions to improve the question:

Consider providing the Black-Scholes formula or a hint to guide students towards using the appropriate pricing model.
You could ask students to interpret the meaning of the calculated option price in the context of the problem.
To increase the difficulty, you could ask students to calculate the option's implied volatility or the option's delta.
You may want to include a question about the assumptions behind the Black-Scholes model and their implications for the calculated option price.
Consider asking students to discuss how the option price would change if one of the given parameters (e.g., risk-free rate, volatility, or time to maturity) were to change.

These suggestions can help make the question more comprehensive and challenging, encouraging students to think critically about option pricing and the factors that influence it.

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