Black Scholes Pricing Model Calculator

Black-Scholes Formula:

\[ C = S N(d_1) - K e^{-rT} N(d_2) \] \[ d_1 = \frac{\ln(S/K) + (r + \sigma^2/2)T}{\sigma\sqrt{T}} \] \[ d_2 = d_1 - \sigma\sqrt{T} \]

Current Stock Price (S):

USD

Strike Price (K):

USD

Risk-Free Rate (r):

decimal

Time to Maturity (T):

years

Volatility (σ):

decimal

Option Price (C):

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1. What is the Black-Scholes Model?

The Black-Scholes model is a mathematical model for pricing an options contract. Developed in 1973 by Fischer Black and Myron Scholes, it provides a theoretical estimate of the price of European-style options.

2. How Does the Calculator Work?

The calculator uses the Black-Scholes formula:

\[ C = S N(d_1) - K e^{-rT} N(d_2) \] \[ d_1 = \frac{\ln(S/K) + (r + \sigma^2/2)T}{\sigma\sqrt{T}} \] \[ d_2 = d_1 - \sigma\sqrt{T} \]

Where:

\( C \) — Call option price
\( S \) — Current stock price
\( K \) — Strike price
\( r \) — Risk-free interest rate
\( T \) — Time to maturity
\( \sigma \) — Volatility of the stock
\( N() \) — Standard normal cumulative distribution function

Explanation: The formula calculates the theoretical value of options using stock price, strike price, time to expiration, risk-free rate, and volatility.

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 fields in the specified units. Stock price and strike price in USD, risk-free rate and volatility as decimals (e.g., 0.05 for 5%), and time in years.

5. Frequently Asked Questions (FAQ)

Q1: What types of options does this price?
A: The Black-Scholes model prices European options which can only be exercised at expiration. For American options, other models like Binomial are needed.

Q2: What are the model's assumptions?
A: Key assumptions include: no dividends, efficient markets, no transaction costs, constant volatility, and log-normal distribution of stock prices.

Q3: How accurate is the model?
A: While widely used, the model has limitations. It tends to misprice deep in/out of the money options and doesn't account for dividends or early exercise.

Q4: What is implied volatility?
A: The volatility value that makes the model price equal to the market price. Traders often work backwards from market prices to calculate implied volatility.

Q5: Can this price put options?
A: The calculator shows call pricing. Put prices can be derived using put-call parity: \( P = C - S + Ke^{-rT} \).