Black-Scholes Model
Quick Definition
The foundational mathematical model for pricing European options, using stock price, strike, time, volatility, and risk-free rate as inputs.
What Is Black-Scholes Model?
The Black-Scholes model (also called Black-Scholes-Merton), published by Fischer Black, Myron Scholes, and Robert Merton in 1973, revolutionized options pricing by providing a closed-form formula for European option values. The model assumes constant volatility, continuous trading, no dividends, log-normal price distribution, and a constant risk-free rate. Despite these simplifying assumptions, it remains the foundation of modern options theory. The formula takes five inputs: current stock price, strike price, time to expiration, risk-free interest rate, and volatility. The model also gives rise to the Greeks (delta, gamma, theta, vega, rho), which measure option price sensitivity. Scholes and Merton received the 1997 Nobel Prize in Economics for this work.
Black-Scholes Model Example
- 1Using Black-Scholes with S=$100, K=$105, T=30 days, σ=25%, r=5%, the model prices a European call at approximately $1.45
- 2A trader notices the market price of an option exceeds the Black-Scholes theoretical value, suggesting the market implies higher volatility than the input used
Related Terms
Binomial Model
An options pricing model that uses a discrete-time tree of possible price paths to value options, especially useful for American-style options.
Implied Volatility (IV)
The market's forecast of the likely magnitude of future price movements, derived from current option prices using pricing models.
Options Greeks
A set of risk measures (delta, gamma, theta, vega, rho) that quantify how an option's price responds to changes in various market factors.
Delta (Options)
A Greek that measures how much an option's price changes for a $1 move in the underlying asset, also approximating the probability of expiring in the money.
Vega (Options)
The Greek that measures an option's sensitivity to changes in implied volatility of the underlying asset.
Call Option
A contract giving the holder the right, but not the obligation, to buy an underlying asset at a specified price within a specified time period.
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