# Black–Scholes model

The Black–Scholes Template:IPAc-en or Black–Scholes–Merton model is a mathematical model of a financial market containing certain derivative investment instruments. From the model, one can deduce the Black–Scholes formula, which gives a theoretical estimate of the price of European-style options. The formula led to a boom in options trading and legitimised scientifically the activities of the Chicago Board Options Exchange and other options markets around the world. lt is widely used, although often with adjustments and corrections, by options market participants.:751 Many empirical tests have shown that the Black–Scholes price is "fairly close" to the observed prices, although there are well-known discrepancies such as the "option smile".:770–771

The Black–Scholes model was first published by Fischer Black and Myron Scholes in their 1973 paper, "The Pricing of Options and Corporate Liabilities", published in the Journal of Political Economy. They derived a partial differential equation, now called the Black–Scholes equation, which estimates the price of the option over time. The key idea behind the model is to hedge the option by buying and selling the underlying asset in just the right way and, as a consequence, to eliminate risk. This type of hedging is called delta hedging and is the basis of more complicated hedging strategies such as those engaged in by investment banks and hedge funds.

Robert C. Merton was the first to publish a paper expanding the mathematical understanding of the options pricing model, and coined the term "Black–Scholes options pricing model". Merton and Scholes received the 1997 Nobel Prize in Economics for their work. Though ineligible for the prize because of his death in 1995, Black was mentioned as a contributor by the Swedish Academy.

The model's assumptions have been relaxed and generalized in many directions, leading to a plethora of models that are currently used in derivative pricing and risk management. It is the insights of the model, as exemplified in the Black-Scholes formula, that are frequently used by market participants, as distinguished from the actual prices. These insights include no-arbitrage bounds and risk-neutral pricing. The Black-Scholes equation, a partial differential equation that governs the price of the option, is also important as it enables pricing when an explicit formula is not possible.

The Black–Scholes formula has only one parameter that cannot be observed in the market: the average future volatility of the underlying asset. Since the formula is increasing in this parameter, it can be inverted to produce a "volatility surface" that is then used to calibrate other models, e.g. for OTC derivatives.

## The Black-Scholes world

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.

Now we make assumptions on the assets (which explain their names):

• (riskless rate) The rate of return on the riskless asset is constant and thus called the risk-free interest rate.
• (random walk) The instantaneous log returns of the stock price is an infinitesimal random walk with drift; more precisely, it is a geometric Brownian motion, and we will assume its drift and volatility is constant (if they are time-varying, we can deduce a suitably modified Black–Scholes formula quite simply, as long as the volatility is not random).
• The stock does not pay a dividend.[Notes 1]

Assumptions on the market:

• There is no arbitrage opportunity (i.e., there is no way to make a riskless profit).
• It is possible to borrow and lend any amount, even fractional, of cash at the riskless rate.
• It is possible 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 holding, suppose there is a derivative security also trading in this market. We specify that this security will have a certain payoff at a specified date in the future, depending on the value(s) taken by the stock up to that date. It is a surprising fact that the derivative's price is completely determined at the current time, even though we do not know what path the stock price will take in the future. For the special case of a European call or put option, Black and Scholes showed that "it is possible to create a 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". Their dynamic hedging strategy led to a partial differential equation which governed 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){{ safesubst:#invoke:Unsubst||date=__DATE__ |$B= {{#invoke:Category handler|main}}{{#invoke:Category handler|main}}[citation needed] }}, transaction costs and taxes (Ingersoll, 1976){{ safesubst:#invoke:Unsubst||date=__DATE__ |$B= {{#invoke:Category handler|main}}{{#invoke:Category handler|main}}[citation needed] }}, and dividend payout.

## Notation

Let

$S$ , be the price of the stock, which will sometimes be a random variable and other times a constant (context should make this clear).
$V(S,t)$ , the price of a derivative as a function of time and stock price.
$C(S,t)$ the price of a European call option and $P(S,t)$ the price of a European put option.
$K$ , the strike price of the option.
$r$ , the annualized risk-free interest rate, continuously compounded (the force of interest).
$\mu$ , the drift rate of $S$ , annualized.
$\sigma$ , the standard deviation of the stock's returns; this is the square root of the quadratic variation of the stock's log price process.
$t$ , a time in years; we generally use: now=0, expiry=T.
$\Pi$ , the value of a portfolio.
$N(x)={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{x}e^{-{\frac {z^{2}}{2}}}\,dz$ .
$N'(x)={\frac {1}{\sqrt {2\pi }}}e^{-{\frac {x^{2}}{2}}}$ ## The Black–Scholes equation

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As above, the Black–Scholes equation is a partial differential equation, which describes the price of the option over time. The equation is:

${\frac {\partial V}{\partial t}}+{\frac {1}{2}}\sigma ^{2}S^{2}{\frac {\partial ^{2}V}{\partial S^{2}}}+rS{\frac {\partial V}{\partial S}}-rV=0$ ## Criticism

Espen Gaarder Haug and Nassim Nicholas Taleb argue that the Black–Scholes model merely recasts existing widely used models in terms of practically impossible "dynamic hedging" rather than "risk", to make them more compatible with mainstream neoclassical economic theory. They also assert that Boness in 1964 had already published a formula that is "actually identical" to the Black–Scholes call option pricing equation. Edward Thorp also claims to have guessed the Black–Scholes formula in 1967 but kept it to himself to make money for his investors. Emanuel Derman and Nassim Taleb have also criticized dynamic hedging and state that a number of researchers had put forth similar models prior to Black and Scholes. In response, Paul Wilmott has defended the model.

British mathematician Ian Stewart published a criticism in which he suggested that "the equation itself wasn't the real problem" and he stated a possible role as "one ingredient in a rich stew of financial irresponsibility, political ineptitude, perverse incentives and lax regulation" due to its abuse in the financial industry.