A great question for my derivatives students! Let's dive into the solution.

To value the European call option, we can use the Black-Scholes model, which is a widely used and well-established framework for pricing options. The Black-Scholes formula for a European call option is:

C = SN(d1) - Ke^(-rt)N(d2)

where:
C = value of the call option
S = current stock price = $50.00
K = strike price = $50.00
r = risk-free rate = 3.0% per annum = 0.03
t = time to maturity = 1 year
σ = volatility of the underlying stock = 20.0% per annum = 0.20
N(d1) and N(d2) are cumulative distribution functions of the standard normal distribution
d1 = (ln(S/K) + (r + σ^2/2)t) / (σ√t)
d2 = d1 - σ√t

First, let's calculate d1 and d2:

d1 = (ln(50/50) + (0.03 + 0.20^2/2) × 1) / (0.20 × √1) = 0.15
d2 = 0.15 - 0.20 × √1 = -0.05

Next, we need to find the values of N(d1) and N(d2) using a standard normal distribution table or a calculator. For d1 = 0.15, N(d1) ≈ 0.5596, and for d2 = -0.05, N(d2) ≈ 0.4801.

Now, we can plug in the values to the Black-Scholes formula:

C = 50 × 0.5596 - 50 × e^(-0.03 × 1) × 0.4801 ≈ 4.08

So, the value of the European call option is approximately $4.08.

Suggestions to improve the question:

1. Add more context: While the question provides the necessary inputs, it would be helpful to add some context about the underlying stock and the market conditions. For example, you could mention that the stock is a technology company or that the market is experiencing high volatility.
2. Vary the inputs: To make the question more challenging, you could vary the inputs, such as the strike price, time to maturity, or volatility. This would require students to think more critically about how these changes affect the option value.
3. Ask for additional calculations: Consider asking students to calculate other Greeks, such as delta, gamma, or theta, in addition to the option value. This would help them understand the sensitivity of the option to changes in the underlying parameters.
4. Provide a twist: Introduce a twist, such as a dividend payment or a non-standard option feature, to make the question more realistic and challenging. For example, you could ask students to value an American call option or a barrier option.
5. Consider alternative models: While the Black-Scholes model is a fundamental tool, you could ask students to value the option using alternative models, such as the binomial model or a Monte Carlo simulation. This would help them understand the strengths and limitations of different approaches.

By incorporating these suggestions, you can create a more comprehensive and engaging question that challenges students to think critically about derivatives and their applications.

Solution:

To value this European call option, we can utilize the Black-Scholes model.

Given parameters:

• Risk-free rate (r) = 3.0% per annum
• Strike price (K) = $50.00
• Time to maturity (T) = 1 year
• Current stock price (S) = $50.00
• Volatility (σ) = 20.0% per annum

Black-Scholes formula for a European call option:

C = S * N(d1) - K * e^(-rT) * N(d2)

Where:

• C = Call option price
• S = Current stock price
• K = Strike price
• r = Risk-free rate
• T = Time to maturity
• e = Exponential term (approximately 2.71828)
• N(x) = Cumulative standard normal distribution function
• d1 = (ln(S/K) + (r + σ^2/2) * T) / (σ * sqrt(T))
• d2 = d1 - σ * sqrt(T)

Calculating d1 and d2:

d1 = (ln(50/50) + (0.03 + 0.2^2/2) * 1) / (0.2 * sqrt(1)) = 0.135
d2 = 0.135 - 0.2 * sqrt(1) = -0.065

Using a standard normal distribution table or calculator, we find:

• N(d1) = 0.5528
• N(d2) = 0.4750

Calculating the call option price:

C = 50 * 0.5528 - 50 * e^(-0.03 * 1) * 0.4750
C = $4.77

Therefore, the value of the European call option is $4.77.

Suggestions for Improvement:

• Add complexity: The current question is a basic application of the Black-Scholes model. To challenge students further, consider incorporating early exercise features (American options), dividends, or more complex payoff structures.
• Comparative statics: Ask students to analyze how the option price would change with variations in one or more input parameters (e.g., volatility, time to maturity). This encourages a deeper understanding of the model's sensitivities.
• Real-world application: Introduce a scenario based on a real company and its stock option chain. This helps students connect theoretical concepts to practical situations.
• Open-ended discussion: Pose questions that encourage critical thinking and debate. For example, ask students to discuss the limitations of the Black-Scholes model or alternative valuation methods.

By incorporating these suggestions, you can create a more engaging and comprehensive learning experience for your students.