D1 and D2 Calculation

Black-Scholes d1 and d2:

\[ d_1 = \frac{\ln(S/K) + (r + \sigma^2/2)t}{\sigma \sqrt{t}} \] \[ d_2 = d_1 - \sigma \sqrt{t} \]

Stock Price (S):

USD

Strike Price (K):

USD

Risk-free Rate (r):

decimal

Volatility (σ):

decimal

Time to Maturity (t):

years

d1:

d2:

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1. What are d1 and d2?

d1 and d2 are intermediate variables in the Black-Scholes option pricing model. They are used to calculate the probabilities that determine the price of European call and put options.

2. How Does the Calculator Work?

The calculator uses the Black-Scholes formulas:

\[ d_1 = \frac{\ln(S/K) + (r + \sigma^2/2)t}{\sigma \sqrt{t}} \] \[ d_2 = d_1 - \sigma \sqrt{t} \]

Where:

\( S \) — Current stock price (USD)
\( K \) — Strike price (USD)
\( r \) — Risk-free interest rate (decimal)
\( \sigma \) — Volatility (decimal)
\( t \) — Time to maturity (years)

Explanation: d1 represents the expected return of the stock exceeding the strike price, while d2 adjusts for the volatility over the time period.

3. Importance of d1 and d2

Details: These values are crucial in option pricing as they appear in the cumulative normal distribution functions that determine the option's price. They represent risk-adjusted probabilities in the Black-Scholes model.

4. Using the Calculator

Tips: Enter all values in consistent units (USD for prices, decimal for rates and volatility, years for time). All values must be positive.

5. Frequently Asked Questions (FAQ)

Q1: Why are d1 and d2 important?
A: They represent the probabilities used in the Black-Scholes formula to calculate option prices and Greeks.

Q2: What do negative d1/d2 values mean?
A: Negative values indicate the stock price is below the strike price adjusted for interest and volatility.

Q3: How does volatility affect d1 and d2?
A: Higher volatility increases the absolute value of d1 but decreases d2 (since d2 = d1 - σ√t).

Q4: What time units should be used?
A: Time should be in years (e.g., 0.5 for 6 months, 0.25 for 3 months).

Q5: Can this be used for American options?
A: No, these formulas are specifically for European options. American options require different calculations.