Introduction
Options are complex financial instruments used for hedging, speculating, and portfolio optimization. Understanding the pricing of options through models like the Black-Scholes and Binomial Tree models is fundamental for traders and investors. This blog also explores the Greeks, which are measures of the sensitivity of the price of derivatives to a change in underlying parameters.
Black-Scholes Model for European Option Pricing
The Black-Scholes model provides a theoretical estimate for the price of European-style options. The formula for a call option is:
C = S0 * N(d1) – X * e^(-rT) * N(d2)
Where:
- S0 = current stock price,
- X = strike price of the option,
- r = risk-free interest rate,
- T = time to expiration,
- N() = cumulative normal distribution function,
- d1 = (ln(S0/X) + (r + σ²/2) * T) / (σ * sqrt(T)),
- d2 = d1 – σ * sqrt(T),
- σ = volatility of the stock.
The put option formula can be derived similarly using put-call parity.
Binomial Tree Model for American Options
The Binomial Tree model is useful for pricing American options, which can be exercised at any time before expiration. This model uses a discrete-time lattice to model different paths that the price of the underlying asset might take:
Option value = max(early exercise value, hold option value)
The calculations involve a recursive process through the tree, from expiration to the present, determining the value of the option at each node by considering the exercise versus hold decisions.
The Greeks: Delta, Gamma, Theta, Vega, and Rho
The “Greeks” measure different sensitivities in an option’s price:
- Delta (Δ): Measures the rate of change of the option price with respect to changes in the underlying asset’s price. For call options, Delta = N(d1).
- Gamma (Γ): Measures the rate of change in Delta with respect to changes in the underlying asset’s price. Gamma = N'(d1) / (S0 * σ * sqrt(T)).
- Theta (Θ): Measures the sensitivity of the option price to the passage of time. Theta can be different for calls and puts.
- Vega (ν): Measures sensitivity to volatility. Vega = S0 * N'(d1) * sqrt(T).
- Rho (ρ): Measures the sensitivity of the option price to the interest rate. For call options, Rho = X * T * e^(-rT) * N(d2).
Conclusion
Understanding the pricing of options and the factors that influence their price is crucial for anyone involved in trading or investing in options. The Black-Scholes and Binomial Tree models provide frameworks for estimating option prices, while the Greeks offer insights into the risk and sensitivity of options to various factors.
