Definition

Black Scholes is the name of the mathematical Options Pricing model used to calculate the theoretical price of European Style Options. It was devised by Fischer Black and Myron Scholes in 1973.

Black Scholes is based on a number of key assumptions:

The option cannot be exercised until Expiration (i.e. European Style).
The Underlying stock (or asset) does not pay any dividends.
All market participants have the same level of knowledge about the market for the Underlying (i.e. no inside information which enables them to gain an edge over others).
The interest rate for cash (or so-called 'risk free rate') remains constant during the lifetime of the option.
The Volatility of the Underlying asset or commodity also remains constant.
No other costs (e.g. commissions) are included.

The components of the model are:

S0 = Underlying price
X = Strike Price
Ïƒ = Volatility
r = risk-free interest rate (per annum)
q = dividend yield (per annum)
t = time to Expiration

The calculation is expressed as follows:

d1 and d2 are as below:

The Black Scholes Options Pricing model enables a number of key variables about Options to be derived which are collectively known as The Greeks: Delta, Gamma, Vega, Theta, Rho. These allow the parameters about an option to be monitored independently of the price and can provide Options traders with more precise information about potential risks than the price alone.

For example, Delta is the rate at which the price of an option changes based on the price of the Underlying asset or commodity and can therefore be used as an approximation of probability - the higher the Delta, the more likely that the option will expire In The Money. Theta measures the rate of time decay and a higher Theta value implies that an option will lose Extrinsic Value more quickly than an option with a low Theta. The separate glossary entry on The Greeks (and each individual entry for them) discusses the topic in more detail.

While Black Scholes forms the theoretical basis underpinning Options Pricing, it is not without shortcomings and criticisms. The most well known issue is the so-called Volatility Smile. This is the tendency for further Out Of The Money Options to have a higher Implied Volatility, where Black Scholes is based on the Implied Volatility curve being level across all prices. Some Options traders and investors such as Nassim Taleb and Warren Buffet have argued that the model is flawed and (in the case of Taleb) that others may have discovered it before Fischer Black and Myron Scholes published their paper.

A number of calculators are available on-line to help investors to work out the price of options using the Black Scholes model. See the directory entry about calculators and tools for more information.
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Contributed by: Ralph Windsor

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Introduction to the Black Scholes Formula

Khan Academy video describing the Black Scholes model.

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Black Scholes Original 1973 Paper (PDF)
https://www.cs.princeton.edu/courses/archive/fall09/cos323/papers/black_scholes73.pdf

The original Black Scholes 1973 paper on pricing options contracts (PDF download).

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Blackâ€“Scholes model
https://en.wikipedia.org/wiki/Black%E2%80%93Scholes_model

Wikipedia article with a detailed overview of the Blackâ€“Scholes model.

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Option Pricing Models and the "Greeks"

Article by options tools software developer, Peter Hoadley.

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Black-Scholes - a Critical Examination and Suggestions for Alternative Methods
http://www.cboeoptionshub.com/2012/11/22/black-scholes-a-critical-examination-and-suggestions-for-alternative-methods-by-michael-c-thomsett-2012-11-22-4-00/

Discussion of the shortcomings of Black Scholes and some suggestions for improving it by options trader and educator, Michael Thomsett.

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Video

Introduction to the Black Scholes Formula

Khan Academy video describing the Black Scholes model.

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Informed Trades guide to Black Scholes option pricing model.

Black-Scholes Model of Option Pricing Explained

Anton Theunissen from New York Institute of Finance describes the history, mechanics, and application of the Black Scholes options pricing model

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