- Sometimes called the Black-Scholes-Merton model. Pioneering 1973 model for
option valuation based upon the assumption that stock prices follow a random
process known as Geometric Brownian motion. Under this model, the call option
price is given by
- In the formula,
- C = call option price, S = stock price, K = strike price, r = an interest
- q = dividend yield, T = time to option expiration,
instantaneous volatility of returns, log(...) is the natural
logarithm, (...) is the
cumulative normal function.
- A security that gives the owner the right, but not the obligation, to
purchase the underlying security at a fixed price (the strike price K)
either (i) at any time until an expiration date (American-style), or (ii) on
the expiration date (European-style). On expiration, the call option is
worth max(S - K, 0).
- The Greeks are the derivatives of the option price with respect to
various parameters, such as the stock price and volatility. For example,
the Greek symbol (pronounced
delta) refers to the change in the option
price divided by a small change in the underlying stock price, holding all
the other parameters fixed.
- A reference to whether or not the option would have any intrinsic
value if exercised today. For a call option, the option is in-the-money if
S > K, is at-the-money if S = K, and is out-of-the money if S < K, where S
is today's stock price and K is the option strike. The relations are
reversed for a put.
- A security that gives the owner the right, but not the obligation, to sell
the underlying security at a fixed price (the strike price K) either (i) at
any time until an expiration date (American-style), or (ii) on the expiration
date (European-style). On expiration, the put option is worth max(K - S, 0).
- A graph, plotting the implied volatility of a series of options with the
same expiration vs. the strike price. Sometimes, instead of the strike
price, the x-axis is a dimensionless moneyness, such as S/K or
log(F/K), where F = the price of the stock for forward delivery on the
- A measure of the variability of price returns. (The price returns of a
stock are the relative changes
The instantaneous volatility
is the parameter that appears in the
Black-Scholes' formula: it's the square root of the expected variance of
, per unit time, as the time
interval shrinks to zero. The implied
volatility is that value
of the Black-Scholes' parameter
that equates the Black-Scholes model price to some given price. The
given price is either a market price or some other model price.
Stochastic volatility is a generalization where the instantaneous
volatility becomes a random variable
which is then described by a stochastic process model.
CBOE glossary (opens in a new window)