Editing Black–Scholes model (section)

== Fundamental hypotheses ==
The Black–Scholes model assumes that the market consists of at least one risky asset, usually called the stock, and one riskless asset, usually called the [[money market]], cash, or [[Bond (finance)|bond]].

The following assumptions are made about the assets (which relate to the names of the assets):
* Risk-free rate: The rate of return on the riskless asset is constant and thus called the [[risk-free interest rate]].
* Random walk: The instantaneous log return of the stock price is an infinitesimal [[random walk]] with drift; more precisely, the stock price follows a [[geometric Brownian motion]], and it is assumed that the drift and volatility of the motion are constant. If drift and volatility are time-varying, a suitably modified Black–Scholes formula can be deduced, as long as the volatility is not random.
* The stock does not pay a [[dividend]].<ref name="div_yield" group="Notes">Although the original model assumed no dividends, trivial extensions to the model can accommodate a continuous dividend yield factor.</ref>

The assumptions about the market are:
* No [[arbitrage]] opportunity (i.e., there is no way to make a riskless profit).
* Ability to borrow and lend any amount, even fractional, of cash at the riskless rate.
* Ability to buy and sell any amount, even fractional, of the stock (this includes [[short selling]]).
* The above transactions do not incur any fees or costs (i.e., [[frictionless market]]).

With these assumptions, suppose there is a derivative security also trading in this market. It is specified that this security will have a certain payoff at a specified date in the future, depending on the values taken by the stock up to that date. Even though the path the stock price will take in the future is unknown, the derivative's price can be determined at the current time. For the special case of a European call or put option, Black and Scholes showed that "it is possible to create a [[Hedge (finance)|hedged position]], consisting of a long position in the stock and a short position in the option, whose value will not depend on the price of the stock".<ref>{{cite journal |author=Black, Fischer |author2=Scholes, Myron |title=The Pricing of Options and Corporate Liabilities |journal=Journal of Political Economy |volume=81 |issue=3 |pages=637–654 |doi=10.1086/260062 |year=1973 |s2cid=154552078}}</ref> Their dynamic hedging strategy led to a partial differential equation which governs the price of the option. Its solution is given by the Black–Scholes formula.

Several of these assumptions of the original model have been removed in subsequent extensions of the model. Modern versions account for dynamic interest rates (Merton, 1976),{{Citation needed |date=November 2010}} [[transaction cost]]s and taxes (Ingersoll, 1976),{{Citation needed |date=November 2010}} and dividend payout.<ref name="merton 1973">{{cite journal |last=Merton |first=Robert |title=Theory of Rational Option Pricing |journal=Bell Journal of Economics and Management Science |volume=4 |issue=1 |pages=141–183 |doi=10.2307/3003143 |jstor=3003143 |year=1973 |hdl=10338.dmlcz/135817 |hdl-access=free}}</ref>