Question11 Apple stock currently sells for $40. The strike price on a call option with 6 months until expiration is $45. The continuously compounded risk-free rate is 4.0% and the Apple’s stock standard deviation is 0.40. Calculate the price of the call option on Apple stock using the Black-Scholes option pricing model. (Note: Use the provided Standard Normal Table to obtain the correct value from the options below. If you use your calculator you will not find any of the values below.) A. $2.65 B. $3.07 C. $5.53 D. $15.37 ResetMaximum marks: 2 Flag question undefined

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BlackTom AI · Accepted Answer

We begin by restating the given data to ensure we’re applying the Black-Scholes formula correctly: S = 40, K = 45, T = 0.5 years, risk-free rate r = 0.04 (continuously compounded), and annual volatility sigma = 0.40. The option is a plain-vanilla call.

Key intermediate quantities:
- d1 = [ln(S/K) + (r + 0.5*sigma^2)*T] / [sigma*sqrt(T)]
- d2 = d1 - sigma*sqrt(T)
- The call price C = S*N(d1) - K*e^{-rT}*N(d2), where N() is the standard normal CDF.

Step-by-step evaluation of each option:
Option A: $2.65
- If we compute the Black-Scholes terms with the given inputs, we first estimate ln(S/K) = ln(40/45) ≈ ln(0.8889) ≈ -0.1173.
- The term (r + 0.5*sigma^2)*T = (0.04 + 0.5*(0.40)^2) * 0.5 = (0.04 + 0.5*0.16) * 0.5 = (0.04 + 0.08) * 0.5 = 0.12 * 0.5 = 0.06.
- So the numerator for d1 is ≈ -0.1173 + 0.06 = -0.0573.
- The denominator is sigma*sqrt(T) = 0.40 * sqrt(0.5) ≈ 0.40 * 0.7071 ≈ 0.2828.
- Thus d1 ≈ -0.0573 / 0.2828 ≈ -0.2027. Then d2 = d1 - 0.2828 ≈ -0.4855.
- Using standard normal values, N(d1) ≈ N(-0.203) ≈ 0.420, and N(d2) ≈ N(-0.486) ≈ 0.313.
- With e^{-rT} ≈ e^{-0.04*0.5} ≈ e^{-0.02} ≈ 0.9802, the price computes as C ≈ 40*0.420 - 45*0.9802*0.313 ≈ 16.8 - 13.8 ≈ 3.0. This rough result is in the neighborhood of option values near 3, not as low as 2.65, but given the instruction to use the provided standard table values, the precise table lookup can shift the result slightly toward the 2.65 figure.
- The form and components are consistent: the call price is the stock-present value portion minus the discounted strike portion, both weighted by normal probabilities.

Option B: $3.07
- This value lies close to the rough Black-Scholes estimate we computed (~3.0). The difference arises from the exact normal CDF values used via the standard table (and possibly rounding in ln, T, or e^{-rT}). If the table you’re required to use provides slightly different N(d1) and N(d2) values, the computed price can move to about 3.07.
- In practice, small shifts in N(d1) and N(d2) (on the order of a few percentage points) can produce a few tenths of difference in C when S and K are relatively close and volatility is moderate.

Option C: $5.53
- A price around 5.53 would imply a substantially larger N(d1) payoff or a much smaller discounting effect, which would require either a larger S/K advantage, lower strike, higher volatility, or a longer time to expiration than given. Since we’re at S = 40, K = 45, T = 0.5, with the specified r and sigma, the call payoff component is not large enough to justify a price in the 5+ range. The d1 and d2 values above do not support such a high premium under Black-Scholes for these inputs.

Option D: $15.37
- A price this high would indicate a deep-in-the-money situation or extremely favorable parameters, neither of which aligns with S = 40, K = 45, T = 0.5, r = 0.04, sigma = 0.40. The magnitude of the call price is implausible here because even with a high N(d1) and N(d2), the strike-to-stock gap and short time to expiration cap the premium well below the mid-teens. This option overstates the value under the given assumptions.

Synthesis across options:
- The Black-Scholes framework with the stated inputs tends to produce a call price in the low to mid single digits for this out-of-the-money call with six months to go and moderate volatility. The approximate calculation hovering around 3.0 aligns with the general intuition that out-of-the-money calls have value primarily from time value and volatility, but the distance to the strike and the short horizon limit the premium.
- Among the listed options, the values that fall within a reasonable range given the computation and typical table-derived values are those near 3.0, while the 5.53 figure is too high for these parameters and 15.37 is clearly excessive. The 2.65 option is somewhat lower than the rough estimate, which can occur depending on the exact N(d1) and N(d2) values taken from the specific standard normal table the problem instructs you to use.

If you reference the standard normal table as instructed and carry through the same steps with the exact table values, you should observe that the resulting price aligns with one of the choices that is closest to your computed C, recognizing that small differences arise from table rounding and the precise N(d1), N(d2) values.

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