You have the following information (make sure to round d1 and d2 to two decimals):

S = 85

K = 100

r = 5%

t = 9 months

implied volatility (standard deviation) = 50%

The value of the call is closest to:

Question

You have the following information (make sure to round d1 and d2 to two decimals):

S = 85

K = 100

r = 5%

t = 9 months

implied volatility (standard deviation) = 50%

The value of the call is closest to:

BlackTom AI · Accepted Answer

We start by identifying the given inputs and the target: S = 85, K = 100, r = 5% (annual), t = 9 months = 0.75 years, and implied volatility sigma = 50% = 0.50. The instruction asks to round d1 and d2 to two decimals before proceeding.

Step 1: Compute d1 and d2 using the Black-Scholes formulas with two-decimal rounding for the intermediate steps.
- First, compute ln(S/K) = ln(85/100) = ln(0.85) ≈ -0.1625.
- Next, compute (r + 0.5*sigma^2) = 0.05 + 0.5*(0.50)^2 = 0.05 + 0.125 = 0.175.
- Multiply by t: 0.175 * 0.75 = 0.13125.
- Numerator for d1: ln(S/K) + (r + 0.5*sigma^2)*t ≈ -0.1625 + 0.13125 = -0.03125.
- Denominator: sigma * sqrt(t) = 0.50 * sqrt(0.75) ≈ 0.50 * 0.8660 ≈ 0.4330.
- Therefore, d1 ≈ -0.03125 / 0.4330 ≈ -0.0722. Rounding to two decimals gives d1 ≈ -0.07.
- Then d2 = d1 - sigma*sqrt(t) ≈ -0.0722 - 0.4330 ≈ -0.5052. Rounding to two decimals gives d2 ≈ -0.51.

Step 2: Look up the normal CDF values for d1 and d2, or approximate them.
- N(d1) with d1 ≈ -0.07 is about N(-0.07) ≈ 0.472 (since N(0) = 0.5 and small negative z shifts slightly below 0.5).
- N(d2) with d2 ≈ -0.51 is about N(-0.51) ≈ 0.306.

Step 3: Compute the call price under Black-Scholes:
- The term S * N(d1) ≈ 85 * 0.472 ≈ 40.1.
- The discount factor e^{-r t} = e^{-0.05 * 0.75} = e^{-0.0375} ≈ 0.9632.
- The term K * e^{-r t} * N(d2) ≈ 100 * 0.9632 * 0.306 ≈ 100 * 0.294 ≈ 29.4.
- Put together: C ≈ 40.1 − 29.4 ≈ 10.7.

Step 4: Compare with the provided answer choices.
- 8.50: This would imply a substantially lower value than our estimate. It misses the roughly 10.7 range by about 2.2.
- 10.50: This is very close to the calculated value of about 10.7, lying within the typical rounding/approximation differences from using approximate N(d) values and the two-decimal rounding for d1/d2.
- 11.50: This would be higher than the computed price by roughly 1.0, which is plausible if N(d1) were a touch larger or N(d2) a bit smaller, but our current approximations keep the price below this mark.
- 9.50: This is notably lower than the estimate of around 10.7 and would require a sizable change in the inputs or the probability terms to justify.

Step 5: Synthesize the reasoning about the options as a whole. The central driver of the call value here is the intrinsic-like gap between the stock price and strike (S vs K) combined with time value and volatility. High volatility (sigma = 0.5) and a non-negligible time to expiration (0.75 years) lift the call price above zero, but the stock is below the strike, so the call is not in-the-money and gains are driven by time value and volatility rather than intrinsic value. Given the calculated price around 10.7, the option that best matches that ballpark among the choices is the one near 10.50, while the others diverge more noticeably from the estimate.

Notes on potential sources of small differences: rounding d1 and d2 to two decimals, slight variations in the N/d1 and N/d2 approximations, and using a slightly different numerical approximation for e^{−rt} can shift the final price by a few tenths. These kinds of small discrepancies are common in hand calculations versus a precise numerical Black-Scholes implementation.

fin_412_120258_251367 Mid-term Test 2

You have the following information (make sure to round d1 and d2 to two decimals): S = 85 K = 100 r = 5% t = 9 months implied volatility (standard deviation) = 50% The value of the call is closest to:

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