1. What Are Black-Scholes d1 and d2?

d1 and d2 are intermediate variables in the Black-Scholes option pricing model that incorporate the stock price, strike price, time to expiration, risk-free rate, and volatility. They are crucial for calculating option prices and Greeks.

2. How Does the Calculator Work?

The calculator uses the Black-Scholes formulas:

Where:

• \( S \) — Current stock price (USD)
• \( K \) — Option strike price (USD)
• \( r \) — Risk-free interest rate (decimal)
• \( \sigma \) — Volatility of the stock (decimal)
• \( t \) — Time to expiration (years)

Explanation: d1 represents the expected return of the stock exceeding the strike price, while d2 adjusts for the volatility over the option's life.

3. Importance of d1 and d2

Details: These values are used to calculate option prices (via N(d1) and N(d2)), delta (N(d1)), and other Greeks. They quantify the moneyness of an option adjusted for volatility and time.

4. Using the Calculator

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

5. Frequently Asked Questions (FAQ)

Q1: What does a higher d1 value indicate?
A: Higher d1 suggests the option is more likely to be in-the-money at expiration, all else equal.

Q2: Why does d2 subtract volatility?
A: d2 accounts for the uncertainty in the stock's path to expiration, adjusting the probability of exercise.

Q3: What are typical d1/d2 values?
A: Values range from -∞ to +∞. Near 0 indicates at-the-money, positive for in-the-money, negative for out-of-the-money.

Q4: How does time affect d1 and d2?
A: More time increases both terms (unless r is negative), but d2 increases less due to the volatility adjustment.

Q5: Can d1/d2 be used for American options?
A: They're primarily for European options, but can approximate American options when early exercise isn't optimal.