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Home of the OC Calculator ### Black-Scholes' model

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 rate
q = dividend yield, T = time to option expiration, =  instantaneous volatility of returns, log(...) is the natural logarithm, (...) is the cumulative normal function.

### Call option

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).

### Greeks

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.

### In-the-money/out-of-the-money/at-the-money

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.

### Put option

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).

### Smile or skew

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 option expiration.

### Volatility

A measure of the variability of price returns. (The price returns of a stock are the relative changes S/S). The instantaneous volatility is the parameter that appears in the Black-Scholes' formula: it's the square root of the expected variance of ( S/S) , 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.

### Other Glossaries

CBOE glossary
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