How are the intermediate variables $d_1$ and $d_2$ calculated in the Black-Scholes-Merton formula?

Conditions

• $S_0 > 0$ and $X > 0$ to evaluate the natural logarithm $\ln(S_0/X)$.
• $\sigma > 0$ to serve as a valid denominator.
• $T > 0$ for the square root $\sqrt{T}$ to be meaningful.

Reasoning, step by step

1. Identify the current stock price $S_0$, strike price $X$, risk-free rate $r$, volatility $\sigma$, and time to maturity $T$.
2. Compute the natural logarithm of the price ratio: $\ln(S_0/X)$.
3. Calculate the drift term: $(r + \sigma^2/2)T$.
4. Add the logarithmic term and the drift term to form the numerator of $d_1$.
5. Calculate the denominator of $d_1$: $\sigma\sqrt{T}$.
6. Divide the numerator by the denominator to find $d_1$.
7. Subtract $\sigma\sqrt{T}$ from $d_1$ to find $d_2$.
8. Round $d_1$ and $d_2$ to the nearest hundredth before looking up their values in the standard normal distribution table.

Example

In the video's example, the instructor writes $d_1 = \frac{\ln(S_0/X) + (r + \sigma^2/2)T}{\sigma\sqrt{T}}$ and then writes $d_2 = d_1 - \sigma\sqrt{T}$, explicitly stating that $d_2$ need not be recomputed from scratch but is obtained by subtracting the same denominator quantity from $d_1$.

Common misconceptions

• Attempting to calculate $d_2$ using a separate, complex formula instead of deriving it from $d_1$.
• Forgetting to round $d_1$ and $d_2$ to two decimal places before consulting the standard normal distribution table.
• Confusing the variance $\sigma^2$ with the standard deviation $\sigma$ in the numerator of $d_1$.

Watch the explanation

Explore next

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