Black And Scholes Model Calculator

Black-Scholes Put Option Formula:

\[ Put = K e^{-rT} N(-d2) - S N(-d1) \]

Strike Price (K):

USD

Risk-Free Rate (r):

decimal

Time to Maturity (T):

years

N(-d2):

probability

Spot Price (S):

USD

N(-d1):

probability

Put Option Price:

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1. What is the Black-Scholes Model?

The Black-Scholes model is a mathematical model for pricing options contracts. It calculates the theoretical price of European-style options using current stock prices, expected dividends, the option's strike price, expected interest rates, time to expiration, and expected volatility.

2. How Does the Calculator Work?

The calculator uses the Black-Scholes put option formula:

\[ Put = K e^{-rT} N(-d2) - S N(-d1) \]

Where:

\( K \) — Strike price (USD)
\( r \) — Risk-free interest rate (decimal)
\( T \) — Time to maturity (years)
\( N(-d2) \) — Cumulative standard normal distribution of -d2
\( S \) — Spot price of the underlying asset (USD)
\( N(-d1) \) — Cumulative standard normal distribution of -d1

Explanation: The formula calculates the present value of the strike price minus the present value of the expected stock price if it's below the strike price at expiration.

3. Importance of Option Pricing

Details: Accurate option pricing is crucial for traders, investors, and financial institutions to determine fair value, hedge positions, and assess risk.

4. Using the Calculator

Tips: Enter all required parameters in the correct units. Strike and spot prices must be positive, probabilities between 0-1, and time must be positive.

5. Frequently Asked Questions (FAQ)

Q1: What assumptions does the Black-Scholes model make?
A: It assumes log-normal distribution of stock prices, no dividends, no transaction costs, constant volatility, and risk-free interest rate.

Q2: What's the difference between European and American options?
A: European options can only be exercised at expiration, while American options can be exercised anytime before expiration.

Q3: How accurate is the Black-Scholes model?
A: It's widely used but has limitations, especially for deep in/out-of-the-money options and long-dated options.

Q4: What is implied volatility?
A: The volatility parameter that makes the model price equal to the market price of an option.

Q5: Can this model price other types of options?
A: The basic model is for European options. Modified versions exist for American options, dividends, etc.