Black-Scholes Option Pricing

 Black-Scholes Option Pricing

Calculate the Black-Scholes option price.

Parameters:
    S (float): Current stock price
    K (float): Option strike price
    T (float): Time to expiration in years
    r (float): Risk-free interest rate
    sigma (float): Stock volatility
    option_type (str): Type of option, 'call' or 'put'

Returns:
    float: Black-Scholes option price

    ## Black-Scholes Option Pricing Model

The Black-Scholes model is a mathematical model used for pricing European-style options. It was developed by Fischer Black, Myron Scholes, and Robert Merton and has become a standard tool in the financial industry for valuing options.

Calculation

The Black-Scholes formula calculates the theoretical price of an option using the following parameters:

• S: Current stock price
• K: Strike price of the option
• T: Time to expiration in years
• r: Risk-free interest rate
• σ: Volatility of the underlying stock
• Option Type: Whether it's a call or put option

The formula for calculating the price of a call option is:

C = S * N(d1) - K * e^(-r * T) * N(d2)

And for a put option: P = K * e^(-r * T) * N(-d2) - S * N(-d1)

Where:

• N(d1) and N(d2) are cumulative distribution functions of the standard normal distribution.
• d1 and d2 are calculated as follows:

Interpretation

• Option Price: The calculated price of the option using the Black-Scholes model.
• Implied Volatility: Volatility implied by the option price. Higher implied volatility suggests higher uncertainty or expected price swings in the underlying stock.
• Delta: Sensitivity of the option price to changes in the underlying stock price. Delta values range from -1 to 1 for put options and from 0 to 1 for call options. Higher delta values indicate higher sensitivity to changes in the stock price.
• Gamma: Rate of change of delta with respect to the stock price. Gamma measures how delta changes as the underlying stock price changes.
• Theta: Rate of change of the option price with respect to time. Theta measures the erosion of the option's value as time passes.
• Vega: Sensitivity of the option price to changes in implied volatility. Vega measures the impact of volatility changes on the option price.

Recommendation

• Valuation: Use the calculated option price as a reference for buying or selling options. Compare the calculated price with the market price to identify potential trading opportunities.
• Risk Management: Consider the implied volatility, delta, gamma, theta, and vega to assess the risk associated with the option position and adjust your strategy accordingly.
• Market Conditions: Keep an eye on market conditions, news, and events that may impact the underlying stock price and volatility to make informed trading decisions.