How do we apply the Black-Scholes-Merton formula to compute $d_1$ and $d_2$ for a specific set of parameters?

Conditions

• The parameters are $S_0=60$, $X=65$, $r=11\%$, $\sigma=45\%$, and maturity is 4 months.
• Time $T$ must be expressed in years, so $T = 4/12$.
• Values of $d_1$ and $d_2$ must be rounded to the nearest hundredth before table lookup.

Reasoning, step by step

1. Convert the maturity of 4 months into years: $T = 4/12$.
2. Substitute the parameters into the $d_1$ formula: $d_1 = \frac{\ln(60/65) + (11\% + 45\%^2/2)\times 4/12}{45\%\times \sqrt{4/12}}$.
3. Calculate the numerator and denominator to find $d_1 \approx -0.0371$.
4. Substitute $d_1$ and the parameters into the $d_2$ formula: $d_2 = -0.0371 - 45\%\times \sqrt{4/12}$.
5. Compute the result to find $d_2 = -0.1871$.
6. Round $d_1$ and $d_2$ to two decimal places (e.g., -0.04 and -0.19) before looking up their $N(\cdot)$ values in the standard normal distribution table.

Example

The video demonstrates this exact calculation: substituting the given values yields $d_1 \approx -0.0371$ and $d_2 = -0.1871$. The instructor explicitly reminds viewers that when looking up these values in the table, they must first be rounded to the nearest hundredth.

Common misconceptions

• Forgetting to convert the time to maturity from months to years before substituting into the formula.
• Using the unrounded values of $d_1$ and $d_2$ directly in the standard normal distribution table, which can lead to lookup errors.
• Confusing the percentage values (e.g., 11%) with their decimal equivalents (0.11) during calculation.

Watch the explanation

Explore next

• How are the intermediate variables $d_1$ and $d_2$ calculated in the Black-Scholes-Merton formula?
• Do you need to compute $N(d)$ yourself when solving option pricing problems?