The Black-Scholes Model
Introduction
The seminal work of Fischer Black and Myron Scholes in 1973 produced an elegant closed form solution for pricing European style call options on stock.
Assumptions under which the formula was derived include:
- the option can only be exercised on the expiry date (European style);
- the underlying instrument pays a constant dividend yield;
- there are no taxes, margins or transaction costs;
- the risk free interest rate is constant;
- the price volatility of the underlying instrument is constant; and
- the price movements of the underlying instrument follow a lognormal distribution.
The model assumes that movements in the underlying follow a geometric Brownian motion with constant drift and volatility. The constant volatility of the Black-Scholes model corresponds to the assumption that the underlying asset follows a lognormal stochastic process.
Technical Details
The Black-Scholes formula for a European option is
where
and
where
Analysis Supported
Fair value, implied volatility, implied underlying price, and the implied strike price are calculated for European options as well as several risk statistics including delta, gamma, theta, vega, rho of the rate etc. for:
- European call or put option
- European option on securities that pay a continuous dividend yield
- European option on securities with discrete cash flows
- European option with both a repo rate and discrete dividends
- European foreign exchange options
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