Black-Scholes Documentation

Overview

The black_scholes function implements the Black-Scholes-Merton model, which is widely used to price European call and put options on non-dividend paying stocks. This model provides not only the option price but also key risk sensitivities ("Greeks"), including Delta, Gamma, Vega, Theta, and Rho.
The formula for the option pricing and the calculation of the Greeks depend on several factors including the current stock price (S), the strike price (K), time to expiration (T), the risk-free interest rate (r), and volatility (sigma).
The function returns a dictionary of the calculated option price and Greeks, rounded to two decimal places.

Key Assumptions of the Black-Scholes Model

• Efficient Markets: the model assumes that markets are efficient, meaning that there are no arbitrage opportunities. Prices move in a continuous and random manner, reflecting all available information.


• Lognormal Distribution of Stock Prices: The Black-Scholes model assumes that the underlying asset price follows a lognormal distribution. This implies that stock prices can’t fall below zero and have a skewed distribution, with more potential for upward movement than downward movement.


• Constant Risk-Free Rate: The risk-free interest rate (r) is assumed to be constant throughout the life of the option. This means that there are no significant changes in monetary policy or other factors affecting risk-free returns over the option's duration


• Constant Volatility: The volatility (sigma) of the underlying asset is assumed to be constant during the life of the option. Volatility represents the asset's price fluctuation, and in the real world, this assumption is often considered one of the major limitations of the Black-Scholes model


• European Option Style: The model assumes the option is European-style, meaning it can only be exercised at the expiration date. This differs from American-style options, which can be exercised at any point before or at expiration.


• No Dividends: The model assumes the underlying asset does not pay dividends during the life of the option. This is a key limitation, as many real-world assets, such as stocks, do pay dividends.

Function Parameters

• S: Current price of the underlying asset.
• K: Strike price of the option.
• T: Time to expiration in years.
• r: Risk-free interest rate
• σ: Volatility of the underlying asset (annualized standard deviation of the asset’s returns).
• Option type: The type of option, either 'call' or 'put'

Explanation of Key Calculations

1. d1 and d2: These values are intermediate variables used in the calculation of both the option price and the Greeks. They represent the standard deviations (adjusted by time) from the current price to the strike price under the lognormal assumption
   • \( d_1 = \frac{\ln\left(\frac{S}{K}\right) + \left(r + \frac{\sigma^2}{2}\right) T}{\sigma \sqrt{T}} \)
   • \( d_2 = \frac{\ln\left(\frac{S}{K}\right) + \left(r - \frac{\sigma^2}{2}\right)T}{\sigma \sqrt{T}} = d_1 - \sigma \sqrt{T}\)
2. Option price:
   • \(c = S N\left(d_1\right) - Ke^{-rT} N\left(d_2\right)\)
   • \(p = Ke^{-rT} N\left(-d_2\right) - S N\left(-d_1\right)\)
3. Greeks:
   • Delta measures the sensitivity of the option price to changes in the underlying asset price
     • \(delta_{call} = N\left(d_1\right)\)
     • \(delta_{put} = N\left(d_1\right)\) - 1
   • Gamma measures the sensitivity of Delta to changes in the underlying asset price
     • \(gamma = \frac{N\left(d_1\right)}{S\sigma\sqrt{T}}\)
   • Vega measures sensitivity to volatility
     • \(vega = \frac{S N\left(d_1\right)\sqrt{T}}{100}\)
   • Theta measures the sensitivity to time decay
     • \(theta_{ call} = - \frac{S N^{'}\left(d_1\right)\sigma}{2\sqrt{T}} - rKe^{-rT}N\left(d_2\right)\)
     • \(theta_{ put} = - \frac{S N^{'}\left(d_1\right)\sigma}{2\sqrt{T}} + rKe^{-rT}N\left(-d_2\right)\)
   • Rho measures sensitivity to the risk-free interest rate
     • \(rho_{call} = KTe^{-rT}N\left(d_2\right)\)
     • \(rho_{putt} = -KTe^{-rT}N\left(-d_2\right)\)