When the non-dividend paying stock price is $20, the strike price is $20, the risk-free rate is 6%, the volatility is 20% and the time to maturity is 3 months which of the following is the price of a European call option on the stock

Question

BlackTom AI · Accepted Answer

We start by identifying the standard Black-Scholes formula components for a European call: C = S0 * N(d1) - K * e^{-rT} * N(d2).

First, compute the key parameters:
- S0 = 20, K = 20, r = 0.06, T = 0.25 years (3 months), sigma = 0.20.
- d1 = [ln(S0/K) + (r + 0.5*sigma^2)*T] / (sigma * sqrt(T))
- d2 = d1 - sigma * sqrt(T)
- Since ln(20/20) = 0, and (r + 0.5*sigma^2)*T = (0.06 + 0.5*0.04)*0.25 = (0.06 + 0.02)*0.25 = 0.08*0.25 = 0.02, we have:
  d1 = 0.02 / (0.2 * 0.5) = 0.02 / 0.1 = 0.2.
- Then d2 = 0.2 - 0.2*0.5 = 0.2 - 0.1 = 0.1.

Now evaluate the components:
- The term S0 * N(d1) becomes 20 * N(0.2).
- The term K * e^{-rT} * N(d2) becomes 20 * e^{-(0.06)(0.25)} * N(0.1). Numerically e^{-0.015} ≈ 0.985, so this is approximately 19.70 * N(0.1).

With these, the expression for the call price is:
C ≈ 20 * N(0.2) - 19.70 * N(0.1).

Now analyze each option:
- Option a: 20N(0.1) - 19.7N(0.2).
  This swaps d1 and d2 in the N() functions. Since d1 = 0.2 and d2 = 0.1, this option mixes the probabilities incorrectly, yielding an incorrect weighting of the N() terms.
- Option b: 20N(0.2) - 19.7N(0.1).
  This matches the correct structure derived from the Black-Scholes formula with the calculated d1 and d2, using N(d1) with S0 and N(d2) with K e^{-rT}.
- Option c: 19.7N(0.2) - 20N(0.1).
  Here both coefficients (19.7 and 20) are swapped relative to the correct formula, and the order of N(d1) and N(d2) factors is reversed, which is inconsistent with the standard pricing expression.
- Option d: 19.7N(0.1) - 20N(0.2).
  This places the larger coefficient with N(0.1) and the smaller with N(0.2), effectively misplacing both the coefficients and the arguments of N(), again not aligning with the Black-Scholes result.

In summary, the only expression that aligns with the Black-Scholes pricing formula using the computed d1 = 0.2 and d2 = 0.1 is the one that uses 20N(0.2) minus 19.7N(0.1). The remaining options misplace either the d terms, the coefficients, or both, leading to incorrect pricing.

FIN 413 (LEC B01 B02 B03 Winter 2025) Quiz 5

When the non-dividend paying stock price is $20, the strike price is $20, the risk-free rate is 6%, the volatility is 20% and the time to maturity is 3 months which of the following is the price of a European call option on the stock

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