Black-Scholes Options Pricing & Greeks
The Black-Scholes-Merton model is the foundational framework for pricing European-style options. While sophisticated quantitative funds use more complex variations (like local volatility models or binomial trees for American options), Black-Scholes remains the benchmark for implied volatility and the calculation of "the Greeks."
e.g., 0.5 for 6 months
Call Price
Put Price
The Greeks
| Greek | Call | Put |
|---|---|---|
| Delta (Δ) | ||
| Gamma (Γ) | ||
| Theta (Θ) | ||
| Vega (ν) | ||
| Rho (ρ) |
Decoding the Model
The Black-Scholes model assumes that the underlying asset follows a log-normal distribution, modeled as Geometric Brownian Motion (the same stochastic process used in our Monte Carlo Simulator).
The Greeks Explained
Derivatives traders do not generally trade the price of the option; they trade the Greeks.
- Delta (Δ): The rate of change of the option price with respect to the price of the underlying asset. A call option has a Delta between 0 and 1.
- Gamma (Γ): The rate of change of Delta. High Gamma means the Delta is highly sensitive to the underlying price (usually highest at-the-money).
- Theta (Θ): The time decay. This calculates the amount the option price decreases per day as expiration approaches.
- Vega (ν): Sensitivity to volatility. This is arguably the most important metric for options traders, as implied volatility is the only unknown input in the Black-Scholes equation.
The Limits of the Model
Like all quantitative models, Black-Scholes makes assumptions that fail in reality:
- Constant Volatility: Black-Scholes assumes volatility is constant over the life of the option. In reality, options markets exhibit a "volatility smile," where out-of-the-money options price in higher implied volatility due to crash risk.
- European Exercise: This calculator models European options, which can only be exercised at expiration. American options (which can be exercised early) require a binomial or trinomial tree to price accurately.
- Log-Normal Returns: The model assumes markets are continuous and returns are log-normally distributed. Actual markets have fat tails and discrete jumps (gaps), which is why stress-testing with VaR and managing sizing with the Kelly Criterion is mandatory.