Black-Scholes Option Pricing Model

About:

Basic Black-Scholes Algorithm

• provides theoretical estimate of the price of European call/put option

Formula:

• $ S $ = current price of stock
• $ K $ = strike price of the option
• $ T $ = time to maturity in years
• $ r $ = risk-free interest rate
• $ \sigma $ = volatility of stock (the standard deviation of returns)

European Call Option:

$$C = S \cdot N(d_1) - K e^{-rT} \cdot N(d_2)$$

European Put Option:

$$P = K e^{-rT} \cdot N(-d_2) - S \cdot N(-d_1)$$

Where: $$d_1 = \frac{\ln(S/K) + \left(r + \frac{\sigma^2}{2}\right) T}{\sigma \sqrt{T}}$$ $$d_2 = d_1 - \sigma \sqrt{T}$$ $$N(d) = \text{cumulative distribution function of the standard normal distribution}$$

Newton Raphson (Implied Volatility):

• $ f(\sigma) = \text{BlackScholes}(\sigma) - \text{MarketPrice} = 0 $
• $$\sigma_{\text{new}} = \sigma - \frac{f(\sigma)}{f'(\sigma)} = \sigma - \frac{\text{BlackScholes}(\sigma) - \text{MarketPrice}}{\text{Vega}(\sigma)}$$
• $ \text{Vega}(\sigma) = \text{rate of change of BlackScholes option price wrt changes in volatility } \sigma $

Option Greeks:

• Delta ($ \Delta $) = measures price sensitivity to underlying:
  • Call option: $ \Delta $ ranges from 0 to 1
  • Put option: $ \Delta $ ranges from -1 to 0
• Gamma ($ \Gamma $) = measures rate of change of delta
• Vega ($ \nu $) = measures sensitivity to volatility
• Theta ($ \Theta $) = measures sensitivity to time decay
• Rho ($ \rho $) = measures sensitivity to interest rates

Formula for Option Greeks:

• For Calls: $$\Delta = N(d_1)$$
• For Puts: $$\Delta = N(d_1) - 1$$
• where $ N() $ is the standard normal cumulative distribution function
• $$\Gamma = \frac{N'(d_1)}{S \sigma \sqrt{T}}$$
• where $ N'(d_1) $ is the standard normal probability density function
• $$\nu = S \sqrt{T} N'(d_1)$$

Tasks:

• Implement basic Black Scholes Algotithm
• Read options list from a csv file and ouput the call and put option price using the black scholes algorithm
• Implement implied volatility where we reverse the black sholes algo given a market option price find out what the implied volatility value is
• Visualise Option Prices using matplotlib such as option price vs stock price, option price vs strike price, option price vs volatility, option price vs time to maturity
• Implement simple Steamlit app that shows these results
• Add real time market data integration using Yahoo Finance API to retrieve real-world stock data
• Add a SQL database backend to store and export user stock option queries