Theoretical Options Price Calculator

Black-Scholes Model:

\[ C = S_0 e^{-qT} N(d_1) - K e^{-rT} N(d_2) \] \[ P = K e^{-rT} N(-d_2) - S_0 e^{-qT} N(-d_1) \] \[ d_1 = \frac{\ln(S_0/K) + (r - q + \sigma^2/2)T}{\sigma\sqrt{T}} \] \[ d_2 = d_1 - \sigma\sqrt{T} \]

S (stock price, $):

K (strike price, $):

T (time to expiration, years):

years

r (risk-free rate, decimal):

decimal

σ (volatility, decimal):

decimal

q (dividend yield, decimal):

decimal

Option Type:

Theoretical 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 factors including stock price, strike price, time to expiration, risk-free rate, volatility, and dividend yield.

2. How Does the Calculator Work?

The calculator uses the Black-Scholes formula:

Where:

$ C $ — Call option price
$ P $ — Put option price
$ S_0 $ — Current stock price
$ K $ — Strike price
$ T $ — Time to expiration (years)
$ r $ — Risk-free interest rate
$ \sigma $ — Volatility
$ q $ — Dividend yield
$ N() $ — Cumulative normal distribution function

Explanation: The model assumes lognormal distribution of stock prices and no arbitrage opportunities in efficient markets.

3. Importance of Theoretical Price Calculation

Details: Calculating theoretical option prices helps traders identify mispriced options, assess fair value, and make informed trading decisions in options markets.

4. Using the Calculator

Tips: Enter all required parameters in appropriate units. Ensure volatility and rates are entered as decimals (e.g., 0.20 for 20%). All values must be positive.

5. Frequently Asked Questions (FAQ)

Q1: What are the main assumptions of the Black-Scholes model?
A: The model assumes constant volatility, no dividends (unless specified), European exercise style, efficient markets, and lognormal price distribution.

Q2: How accurate is the Black-Scholes model?
A: While widely used, the model has limitations and may not perfectly predict market prices due to its simplifying assumptions, particularly regarding constant volatility.

Q3: Can this model price American options?
A: The standard Black-Scholes model is for European options only. American options require more complex models that account for early exercise features.

Q4: What is implied volatility?
A: Implied volatility is the volatility value that, when input into the Black-Scholes model, gives a theoretical price equal to the market price of the option.

Q5: How does dividend yield affect option prices?
A: Higher dividend yields generally decrease call option prices and increase put option prices, as dividends reduce the expected future stock price.