Question 1 (Level I – Derivatives, Option Pricing)**
Using the Black-Scholes-Merton model, what is the value of a **3-year European call option** on the S&P 500 with a strike price of 4,000?
**Given:**
- Current S&P 500 (S₀) = 4,000
- Strike price (K) = 4,000
- Risk-free rate (r) = 3% (continuous)
- Dividend yield (q) = 1.5%
- Volatility (σ) = 22%
- Time to maturity (T) = 3
**Black-Scholes Inputs:**
- \( d_1 = \frac{\ln(S_0 / K) + (r - q + 0.5σ^2)T}{σ \sqrt{T}} \)
- \( d_2 = d_1 - σ \sqrt{T} \)
**Calculate the call option value (C):**
\( C = S_0 e^{-qT} N(d_1) - K e^{-rT} N(d_2) \)
**What is the closest value of the call option?**
A) $450
B) $550
C) $650
D) $750
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We need to value a three-year European call option on the S and P five hundred using the Black-Scholes-Merton model. The current stock price and strike price are both four thousand dollars. We have a risk-free rate of three percent, dividend yield of one point five percent, and volatility of twenty-two percent. We'll use the Black-Scholes formula to calculate the option value.
Let's calculate d one step by step. First, we compute the natural logarithm of S zero over K, which equals ln of one, giving us zero. Next, we calculate the drift term: r minus q plus zero point five sigma squared, times T. This equals zero point zero one five plus zero point zero two four two, times three, which gives us zero point one one seven six. The denominator is sigma times square root of T, which is zero point two two times square root of three, approximately zero point three eight one one. Therefore, d one equals zero point one one seven six divided by zero point three eight one one, which is approximately zero point three zero eight six.
Now we calculate d two using the formula d two equals d one minus sigma times square root of T. This gives us zero point three zero eight six minus zero point three eight one one, which equals negative zero point zero seven two five. Next, we find the standard normal probabilities. N of d one equals N of zero point three zero eight six, which is approximately zero point six two one two. N of d two equals N of negative zero point zero seven two five, which is approximately zero point four seven one one. Finally, we calculate the discount factors: e to the negative q T equals zero point nine five six, and e to the negative r T equals zero point nine one four.
Now we calculate the final call option value using the Black-Scholes formula. We substitute all our calculated values: C equals four thousand times zero point nine five six times zero point six two one two, minus four thousand times zero point nine one four times zero point four seven one one. This simplifies to three thousand eight hundred twenty four times zero point six two one two, minus three thousand six hundred fifty six times zero point four seven one one. Computing these products gives us two thousand three hundred seventy five point four minus one thousand seven hundred twenty two point three, which equals six hundred fifty three point one dollars. Comparing this to the given options, six hundred fifty three point one is closest to option C, which is six hundred fifty dollars.
To summarize what we've learned: The Black-Scholes-Merton model provides a systematic approach to value European options. We calculated d one and d two using the given parameters, found the standard normal probabilities, and applied the Black-Scholes formula to get a call option value of six hundred fifty three dollars and ten cents. This matches closest with option C, six hundred fifty dollars. The Black-Scholes model remains a cornerstone of modern derivatives pricing in financial markets.