Mostrar código-fonte de Black-Scholes

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A fórmula '''Black-Scholes''' é um dos modelos mais usados para estabelecer o pricing de opções.

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The term '''Black–Scholes''' refers to three closely related concepts:
* The '''Black–Scholes model''' is a mathematical model of the market for an equity, in which the equity's price is a [[stochastic process]].
* The '''Black–Scholes PDE''' is an equation which (in the model) must be satisfied by the price of a derivative on the equity.
* The '''Black–Scholes formula''' is the result obtained by applying the Black-Scholes PDE to [[option style|European]] put and call options.

[[Robert C. Merton]] was the first to publish a paper expanding our mathematical understanding of the [[Option (finance)|options]] pricing model and coined the term "Black-Scholes" options pricing model, by enhancing work that was published by [[Fischer Black]] and [[Myron Scholes]]. The paper was first published in [[1973]]. The foundation for their research relied on work developed by scholars such as [[Louis Bachelier]], [[A. James Boness]], [[Sheen T. Kassouf]], [[Edward O. Thorp]], and [[Paul Samuelson]]. The fundamental insight of Black-Scholes is that the option is implicitly priced if the stock is traded.

Merton and Scholes received the 1997 [[Nobel Prize in Economics]] for this and related work; though ineligible for the prize because of his death in 1995, Black was mentioned as a contributor by the Swedish academy.

==O modelo==
Os pressupostos chave do modelo de Black–Scholes são:

*O preço do [[activo subjacente]] ao longo do tempo, ''S''<sub>''t''</sub>, segue um [[movimento Browniano geométrico]] com deriva <tex> \mu </tex> e [[Volatilidade (finanças)|volatilidade]] <tex> \sigma </tex> constantes :
:<tex> dS_t = \mu S_t\,dt + \sigma S_t\,dW_t \, </tex>
*É possível [[venda a descoberto|vender a descoberto]] o activo subjacente.
*Não há oportunidades de [[arbitragem|arbitragem]].
*A negociação é realizada continuamente no tempo.
*Não há impostos nem [[comissão de negociação|comissões de negociação]].
*Todos os activos são infinitamente divisíveis (por exemplo, é possível comprar <tex>\ \frac{1}{73}\ </tex> unidades de um título.
*É possível emprestar e pedir emprestado dinheiro a uma [[taxa de juro sem risco|taxa de juro]] constante.
*O título não paga dividendos (ver abaixo uma extensão destes conceitos que lida com os pagamentos de dividendos).

As hipóteses anteriores conduzem à fórmula

<center><tex> C(S,T) = S\Phi(d_1) - Ke^{-rT}\Phi(d_2) \, </tex></center>


para o preço de uma [[European call option|opção de compra do tipo europeu]]
*com preço de exercício ''K''
*sobre um título (acção) com preço actual ''S'' (''i.e.'', o direito a comprar uma acção pelo preço ''K'' numa data futura fixa)
*a taxa de juro (constante) é denotada por ''r''
*a [[Volatility (finance)|volatilidade]] é constante e representada por <tex> \sigma </tex>.
Na fórmula anterior:
<center><tex>d_1 = \frac{\ln(S/K) + (r + \sigma^2/2)T}{\sigma\sqrt{T}}\ \ \ \ \mbox{e}\ \ \ \ <tex> d_2 = \frac{\ln(S/K) + (r - \sigma^2/2)T}{\sigma\sqrt{T}} = d_1 - \sigma\sqrt{T}. </tex></center>

onde <tex> \Phi </tex> é a [[cumulative distribution function|função distribuição]] da [[standard normal|normal standard]].

O preço de uma [[put option|opção de venda]] é calculado a partir do resultado anterior utilizando a paridade [[put-call parity|put-call]]:

:<tex> P(S,T) = Ke^{-rT}\Phi(-d_2) - S\Phi(-d_1). \, </tex>

O cáculo das [[The Greeks|Gregas]] no modelo de Black–Scholes model está feito na tabela seguinte:

{| border="1" cellspacing="0" cellpadding="10"
! !! Calls !! Puts
|-
! delta || <tex> \Phi(d_1) \, </tex> || <tex> - \Phi( - d_1) = \Phi(d_1)-1\, </tex>
|-
! gamma ||colspan="2"| <tex> \frac{\varphi(d_1)}{S\sigma\sqrt{T}} \, </tex>
|-
! vega ||colspan="2"| <tex> S \varphi(d_1) \sqrt{T} \, </tex>
|-
! theta || <tex> - \frac{S \varphi(d_1) \sigma}{2 \sqrt{T}} - rKe^{-rT}\Phi(d_2) \, </tex> || <tex> - \frac{S \varphi(d_1) \sigma}{2 \sqrt{T}} + rKe^{-rT}\Phi(-d_2) \, </tex>
|-
! rho || <tex> KTe^{-rT}\Phi(d_2)\, </tex> || <tex> -KTe^{-rT}\Phi(-d_2)\, </tex>
|}

Aqui, <tex> \varphi </tex> é a [[probability density function|função densidade de probabilidade]] standard da distribuição normal.
Note-se que as fórmulas para a gamma e a vega são as mesmas, quer se trate de uma call ou de uma put.

Isto é consequência directa da [[Rácio_put/call|paridade put-call]].

Na prática, algumas gregas estão cotadas numa escala própria, no sentido de interpretar mais facilmente a informação nelas contida. Por exemplo, rho aparece dividido por 10.000 (alteração da taxa em 1pb), vega por 100 (1 vol point change), e theta por 365 ou 252 (decrescimento diário, considerando dias de calendário ou dias de negociação num ano, respectivamente).

==Generalizações do modelo==
Este modelo pode ser extendido a outros de forma a ter taxas e volatilidades variáveis (mas perfeitamente determinadas). O modelo pode ainda ser usado na avaliação de [[opção europeia|opções do tipo Europeu]] sobre activos que paguem dividendos. Neste caso, existem soluções exactas se o valor do dividendo for dado em percentagem do valor do activo subjacente. [[opção americana|Opções do tipo Americano]] e opções sobre acções que paguem um valor de dividendos conhecido (no curto prazo, é um valor mais realista do que saber a percentagem que corresponde ao valor do dividendo) são mais difíceis de avaliar, mas existe uma série de técnicas disponíveis para tal: reticulados, redes... 

===Activos que pagam dividendos de forma contínua no tempo===
Para as opções sobre índices (tal como o [[FTSE 100 Index|FTSE]] onde cada uma das cem empresas que o constituem podem pagar dividendos semestralmente e em datas independentes umas das outra, é razoável considerar, para simplificar, que os dividendos são pagos de forma ''contínua no tempo'' e que o dividendo é proporcional ao valor do índice.

Modelemos o dividendo pago no período <tex>[t,t+dt]</tex> por <tex> qS_t\ ,dt </tex>
em que 
* <tex> q \ </tex> é a taxa de dividendo;
* <tex> S_t \ </tex> é o valor do índice no instante ''t'';
* <tex> dt \ </tex> é o período de tempo considerado.
 
De acordo com esta formulação, num contexto livre de arbitragem como no modelo de Black-Sholes pode mostrar-se que:

:<tex> C(S_0,T) = e^{-rT}(F\Phi(d_1) - K\Phi(d_2)) \, </tex>

onde

:<tex> F = S_0 e^{(r - q)T} \, </tex>

é o preço ''forward'' utilizado nos cálculos de ''d''<sub>1</sub> and ''d''<sub>2</sub>:

:<tex> d_1 = \frac{\ln(F/K) + (\sigma^2/2)T}{\sigma\sqrt{T}} </tex>

:<tex> d_2 = d_1 - \sigma\sqrt{T}. </tex>

A mesma fórmula é usada para avaliar opções de taxas de juro de divisas, notando que, neste caso, ''q'' denota a taxa de juro sem risco entre divisas e ''S'' é a taxa de juro ''spot''.
Este é o modelo de '''Garman–Kohlhagen''' (1983).

===Instruments paying discrete proportional dividends===
It is also possible to extend the Black–Scholes framework to options on instruments paying discrete proportional dividends. This is useful when the option is struck on a single stock. 

A typical model is to assume that a proportion <tex> \delta </tex> of the stock price is paid out at pre-determined times ''t''<sub>1</sub>, ''t''<sub>2</sub>, .... The price of the stock is then modelled as

:<tex> S_t = S_0(1 - \delta)^{n(t)}e^{ut + \sigma W_t} </tex>

where ''n''(''t'') is the number of dividends that have been paid by time ''t''. 

The price of a call option on such a stock is again

:<tex> C(S_0,T) = e^{-rT}(F\Phi(d_1) - K\Phi(d_2)) \, </tex>

where now

:<tex> F = S_0(1 - \delta)^{n(T)}e^{rT} \, </tex>

is the forward price for the dividend paying stock.

==Black–Scholes in practice==
===The volatility smile===
{{main|Volatility smile}}

All the parameters in the model other than the volatility — the time to maturity, the [[strike price|strike]], the risk-free rate, and the current underlying price — are unequivocally observable. Furthermore, under normal circumstances the option's theoretical value is a monotonic increasing function of the volatility. This means there is a one-to-one relationship between the option price and the volatility.
By computing the implied volatility for traded options with different strikes and maturities, we can test the Black-Scholes model.
If the Black–Scholes model held, then the implied volatility for a particular stock would be the same for all strikes and maturities.
In practice, the [[volatility surface]] (the three-dimensional graph of implied volatility against strike and maturity) is not flat.
The typical shape of the implied volatility curve for a given maturity depends on the underlying instrument. Equities tend to have skewed curves: implied volatility is higher for low strikes, and slightly lower for high strikes. Currencies tend to have more symmetrical curves, with implied volatility lowest [[at-the-money]], and higher volatilities in both wings. Commodities often have the reverse behaviour to equities, with higher implied volatility for higher strikes.

Despite the existence of the volatility smile (and the violation of all the other assumptions of the Black-Scholes model), the Black-Scholes
PDE and Black-Scholes formula are still used extensively in practice. A typical approach is to regard the volatility surface as a fact about the market, and use an implied volatility from it in a Black-Scholes valuation model. This has been described as using "the wrong number in the wrong formula to get the right price" [Rebonato 1999]. This approach also gives usable values for the hedge ratios (the Greeks).

Even when more advanced models are used, traders prefer to think in terms of volatility as it allows them to evaluate and compare options of different maturities, strikes, and so on.

===Valuing bond options===
Black–Scholes cannot be applied directly to [[bond (finance)|bond securities]] because of the [[pull to par|pull-to-par]] problem. As the bond reaches its maturity date, all of the prices involved with the bond become known, thereby decreasing its volatility, and the simple Black–Scholes model does not reflect this process. A large number of extensions to Black–Scholes, beginning with the [[Black model]], have been used to deal with this phenomenon.

===Interest rate curve and short stock rate===
One difficulty that often arises in practice is how to derive the proper interest rate to use as an input. The deposit rate for a risk-free bond maturing on the option's expiration date is, in general, not observable in the market. Instead, an [[yield curve|interest rate curve]] is used. Composed of market-quoted interest rates of various maturities, the curve provides an estimate of the risk-free rate of appropriate maturity for the option being priced.

Another issue arises when [[short (finance)|short stock]] is to be used as part of the hedging portfolio. This is because your broker typically pays you some rate that is less than the risk-free rate on the proceeds of the short stock sale. In addition, when a stock is hard to borrow, the rate you receive on the short sale proceeds can go down and even be negative. That is, you might have to pay your broker interest on the proceeds from your short sale as an inducement to lend you the shares you have sold short. In these cases, the correct interest rate to use in the model should be adjusted to account for this effect.

For example, your broker pays you the Fed funds overnight rate less 0.85% (85 basis points) on your short stock proceeds. You have no existing position in IBM, but you are considering purchasing IBM Jan08 100 Calls. Because you would ordinarily sell IBM short to hedge this purchase, you will need to borrow IBM shares from your broker. From your interest rate curve, you determine the proper risk-free rate for a theoretical bond expiring on January 19th, 2008 is 5.05%. Therefore, the correct interest rate to use in the Black-Scholes model is 4.2%. Now, assume that you are considering the same trade, but in the symbol HLYS, which is hard to borrow. Your broker will only pay you 2% less than the overnight rate on proceeds from a short sale in HLYS stock. Now, the correct rate to use in the Black-Scholes model is 3.05%.

==Formula derivation==
===Elementary derivation===
Let ''S''<sub>0</sub> be the current price of the underlying stock and ''S'' the price when the option matures at time ''T''. Then ''S''<sub>0</sub> is known, but ''S'' is a [[random variable]]. Assume that

:<tex> X \equiv \ln(S/S_0) \, </tex>

is a [[normal random variable]] with [[mean]] <tex> uT </tex> and [[variance]] <tex> \sigma^2 T </tex>. It follows that the mean of ''S'' is

:<tex> \mathbb{E}\left[ S \right] = S_0 e^{qT} \, </tex>

for some constant ''q'' (independent of ''T'').
Now a simple no-arbitrage argument shows that the theoretical future value of a derivative paying one share of the stock at time ''T'', and so with payoff ''S'', is

:<tex> S_0 e^{rT} \, </tex>

where ''r'' is the risk-free interest rate. This suggests making the identification ''q'' = ''r'' for the purpose of pricing derivatives. Define the theoretical value of a derivative as the [[NPV|present value]] of the [[expected value|expected]] payoff in this sense.  For a call option with exercise price ''K'' this discounted expectation (using [[risk-neutral measure|risk-neutral probabilities]]) is

:<tex> C(S_0,T) = e^{-rT} \mathbb{E}\left[ \max(S - K,0) \right]. \, </tex>

The derivation of the formula for ''C'' is facilitated by the following [[lemma (mathematics)|lemma]]: Let ''Z'' be a [[standard normal]] random variable and let ''b'' be an [[extended real number]]. Define

:<tex> Z^+(b) = \begin{cases} Z & \mbox{se }Z{\g}b \\ -\infty & \mbox{se }Z\le{b} \end{cases}. </tex>

If ''a'' is a positive real number, then

:<tex> \mathbb{E}\left[e^{aZ^+(b)}\right] = e^{a^2/2}\Phi(-b + a) </tex>

where <tex>\Phi</tex> is the standard normal [[cumulative distribution function]]. In the special case ''b'' = &minus;&infin;, we have

:<tex> \mathbb{E}\left[e^{aZ}\right] = e^{a^2/2}. </tex>

Now let

:<tex> Z = \frac{X - uT}{\sigma\sqrt{T}} </tex>

and use the [[corollary]] to the lemma to verify the statement above about the mean of ''S''. Define

:<tex> S^+ = \begin{cases} S & \mbox{se }S\g{K} \\ 0 & \mbox{se }S\le{K} \end{cases} </tex>

:<tex> X^+ = \ln(S^+/S_0) \, </tex>

and observe that

:<tex> \frac{X^+ - uT}{\sigma\sqrt{T}} = Z^+(b) </tex>

for some ''b''. Define

:<tex> K^+ = \begin{cases} K & \mbox{se }S\g{K} \\ 0 & \mbox{se }S\le{K} \end{cases} </tex>

and observe that

:<tex> \max(S - K,0) = S^+ - K^+. \, </tex>

The rest of the calculation is straightforward.

Although the elementary derivation leads to the correct result, it is incomplete as it cannot explain, why the formula refers to the riskfree interest rate while a higher rate of return is expected from risky investments. This limitation can be overcome using the risk-neutral probability measure, but the concept of risk-neutrality and the related theory is far from elementary.

===PDE based derivation===
In this section we derive the [[partial differential equation]] (PDE) at the heart of the Black–Scholes model via a no-arbitrage or [[delta hedging|delta-hedging]] argument; for more on the underlying logic, see the discussion at [[Rational pricing#Delta hedging|rational pricing]]. 

The presentation given here is [[Proof theory#Formal and informal proof|informal]] and we do not worry about the validity of moving between ''dt'' meaning a small increment in time and ''dt'' as a [[derivative]].

====The Black–Scholes PDE====
As per the model assumptions above, we assume that the [[underlying]] (typically the stock) follows a [[geometric Brownian motion]]. That is,

:<tex> dS_t = \mu S_t\,dt + \sigma S_t\,dW_t \, </tex>

where ''W''<sub>''t''</sub> is Brownian. 

Now let ''V'' be some sort of option on ''S''—mathematically ''V'' is a function of ''S'' and ''t''. ''V''(''S'', ''t'') is the value of the option at time ''t'' if the price of the underlying stock at time ''t'' is ''S''. The value of the option at the time that the option matures is known. To determine its value at an earlier time we need to know how the value evolves as we go backward in time. By [[Itō's lemma]] for two variables we have

:<tex> dV = \left(\mu S \frac{\partial V}{\partial S} + \frac{\partial V}{\partial t} + \frac{1}{2}\sigma^2 S^2\frac{\partial^2 V}{\partial S^2}\right)dt + \sigma S \frac{\partial V}{\partial S}\,dW. </tex>

Now consider a trading strategy under which one holds one option and continuously trades in the stock in order to hold &minus;&part;''V''/&part;''S'' shares. 
At time ''t'', the value of these holdings will be

:<tex> \Pi = V - S\frac{\partial V}{\partial S}. </tex>

The composition of this portfolio, called the [[delta hedging|delta-hedge]] portfolio, will vary from time-step to time-step. Let ''R'' denote the accumulated profit or loss from following this strategy. Then over the time period [''t'', ''t'' + ''dt''], the instantaneous profit or loss is

:<tex> dR = dV - \frac{\partial V}{\partial S}\,dS. </tex>

By substituting in the equations above we get

:<tex> dR = \left(\frac{\partial V}{\partial t} + \frac{1}{2}\sigma^2 S^2\frac{\partial^2 V}{\partial S^2}\right)dt. </tex>

This equation contains no ''dW'' term. That is, it is entirely riskless ([[delta neutral]]). Thus, given that there is no arbitrage, the rate of return on this portfolio must be equal to the rate of return on any other riskless instrument. Now assuming the risk-free rate of return is ''r'' we must have over the time period [''t'', ''t'' + ''dt'']

:<tex> r\Pi\,dt = dR = \left(\frac{\partial V}{\partial t} + \frac{1}{2}\sigma^2 S^2\frac{\partial^2 V}{\partial S^2}\right)dt. </tex>

If we now substitute in for <tex> \Pi </tex> and divide through by ''dt'' we obtain the '''Black–Scholes PDE''':

:<tex> \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. </tex>

This is the law of evolution of the value of the option. With the assumptions of the Black–Scholes model, this equation holds whenever ''V'' has two derivatives with respect to ''S'' and one with respect to ''t''.

====Other derivations of the PDE====
Above we used the method of [[arbitrage]]-free pricing ("[[Rational pricing#Arbitrage free pricing|delta-hedging]]") to derive a PDE governing option prices given the Black–Scholes model. It is also possible to use a [[Rational pricing#Risk neutral valuation|risk-neutrality]] argument. This latter method gives the price as the [[expected value|expectation]] of the option payoff under a particular [[probability measure]], called the [[risk-neutral measure]], which differs from the real world measure.

====Solution of the Black–Scholes PDE====

We now show how to get from the general Black–Scholes PDE to a specific valuation for an option. Consider as an example the Black–Scholes price of a [[call option]] on a stock currently trading at price ''S''. The option has an exercise price, or strike price, of ''K'', i.e. the right to buy a share at price ''K'', at ''T'' years in the future. The constant interest rate is ''r'' and the constant stock volatility is <tex> \sigma </tex>. Now, for a call option the PDE above has [[boundary condition]]s

:<tex> V(0,t) = 0 \, </tex> for all ''t''

:<tex> V(S,t) \sim S \, </tex> as <tex> S \rightarrow \infty \, </tex>

:<tex> V(S,T) = \max(S - K,0). \, </tex>

The last condition gives the value of the option at the time that the option matures. The solution of the PDE gives the value of the option at any earlier time. In order to solve the PDE we transform the equation into a [[heat equation|diffusion equation]] which may be solved using standard methods. To this end we introduce the change-of-variable transformation

:<tex> x = \ln(S/K) + (r - \sigma^2/2)(T - t) \, </tex>

:<tex> \tau = T - t \, </tex>

:<tex> u = Ve^{r(T - t)}. \, </tex>

Then the Black–Scholes PDE becomes a diffusion equation

:<tex> \frac{\partial u}{\partial \tau} = \frac{\sigma^2}{2} \frac{\partial^2 u}{\partial x^2}. </tex>

The terminal condition <tex> V(S,T) = \max(S - K,0) </tex> now becomes an initial condition

:<tex> u(x,0) = u_0(x) \equiv K\max(e^x - 1,0). \, </tex>

Using the standard method for solving a diffusion equation we have

:<tex> u(x,\tau) = \frac{1}{\sigma\sqrt{2\pi\tau}}\int_{-\infty}^{\infty}u_0(y)e^{-(x - y)^2/(2\sigma^2\tau)}\,dy. </tex>

After some algebra we obtain

:<tex> u(x,\tau) = Ke^{x + \sigma^2\tau/2}\Phi(d_1) - K\Phi(d_2) </tex>

where

:<tex> d_1 = \frac{x + \sigma^2\tau}{\sigma\sqrt{\tau}} </tex>

:<tex> d_2 = \frac{x}{\sigma\sqrt{\tau}} </tex>

and <tex>\Phi</tex> is the [[standard normal]] [[cumulative distribution function]].

Substituting for ''u'', ''x'', and <tex> \tau </tex>, we obtain the value of a call option in terms of the Black–Scholes parameters:

:<tex> V(S,t) = S\Phi(d_1) - Ke^{-r(T - t)}\Phi(d_2) \, </tex>

where

:<tex> d_1 = \frac{\ln(S/K) + (r + \sigma^2/2)(T - t)}{\sigma\sqrt{T - t}} </tex>

:<tex> d_2 = d_1 - \sigma\sqrt{T - t}. </tex>

The formula for the price of a [[put option]] follows from this via [[put-call parity]].

==Remarks on notation==
The reader is warned of the inconsistent notation that appears in this article.  Thus the letter ''S'' is used as:

:(1) a [[constant]] denoting the current price of the stock
:(2) a real [[variable]] denoting the price at an arbitrary time
:(3) a [[random variable]] denoting the price at maturity
:(4) a [[stochastic process]] denoting the price at an arbitrary time

It is also used in the meaning of (4) with a subscript denoting time, but here the subscript is merely a mnemonic.

In the partial derivatives, the letters in the numerators and denominators are, of course, real variables, and the partial derivatives themselves are, initially, real functions of real variables. But after the substitution of a stochastic process for one of the arguments they become stochastic processes.

The Black–Scholes PDE is, initially, a statement about the stochastic process ''S'', but when ''S'' is reinterpreted as a real variable, it becomes an ordinary PDE. It is only then that we can ask about its solution.

The parameter ''u'' that appears in the discrete-dividend model and the elementary derivation is not the same as the parameter <tex> \mu </tex> that appears elsewhere in the article. For the relationship between them see [[Geometric Brownian motion]].

==See also==
*[[Black model]], a variant of the Black–Scholes option pricing model.
*[[Binomial options model]], which is a discrete [[numerical method]] for calculating option prices.
*[[Monte Carlo option model]], using [[simulation]] in the valuation of options with complicated features.
*[[Financial mathematics]], which contains a list of related articles.
*[[Heat equation]], to which the Black–Scholes PDE can be transformed.
*[[Mathematical Finance Programming in TI-Basic]], which contains some [[TI-Basic]] programs to perform stochastic calculus with [[Texas Instruments]] [[Calculators]]
*[[QuantLib]], which includes an open-source implementation of Black-Scholes in C++
*[[real options analysis]]

==References==
===Primary references===
*{{cite journal|title=The Pricing of Options and Corporate Liabilities|last=Black|first=Fischer|coauthors=Myron Scholes|journal=Journal of Political Economy|date=1973|volume=81|issue=3|pages=637-654}} [http://links.jstor.org/sici?sici=0022-3808%28197305%2F06%2981%3A3%3C637%3ATPOOAC%3E2.0.CO%3B2-P] (Black and Scholes' original paper.)
*{{cite journal|title=Theory of Rational Option Pricing|last=Merton|first=Robert C.|journal=Bell Journal of Economics and Management Science|date=1973|volume=4|issue=1|pages=141-183}} [http://links.jstor.org/sici?sici=0005-8556%28197321%294%3A1%3C141%3ATOROP%3E2.0.CO%3B2-0&origin=repec]

===Historical and sociological aspects===
* {{cite book|title=Capital Ideas: The Improbable Origins of Modern Wall Street|last=Bernstein|first=Peter|
id=ISBN 0-02-903012-9|publisher=The Free Press}}
*{{cite journal|title=An Equation and its Worlds: Bricolage, Exemplars, Disunity and Performativity in Financial Economics|last=MacKenzie|first=Donald|journal=Social Studies of Science|date=2003|volume=33|issue=6|pages=831-868}} [http://sss.sagepub.com/cgi/content/abstract/33/6/831]
*{{cite journal|title=Constructing a Market, Performing Theory: The Historical Sociology of a Financial Derivatives Exchange|last=MacKenzie|first=Donald|coauthors=Yuval Millo|journal=American Journal of Sociology|date=2003|volume=109|issue=1|pages=107-145}} [http://www.journals.uchicago.edu/AJS/journal/issues/v109n1/060259/brief/060259.abstract.html]
* {{cite book|title=An Engine, not a Camera: How Financial Models Shape Markets|last=MacKenzie|first=Donald|
id=ISBN 0-262-13460-8|publisher=MIT Press}}

==External links==
===Discussion of the model===
*[http://www.riskglossary.com/link/option_pricing_theory.htm Option pricing theory], Overview and links to more detailed articles on specific models, riskglossary.com
*[http://www.global-derivatives.com/options/black-scholes.php The Black–Scholes Model], global-derivatives.com
*[http://www.finance24.com/register/help/mmx_school/displayarticlewide.asp?ArticleID=271869 Options pricing using the Black-Scholes Model], [[Investment Analysts Society of Southern Africa]]
*[http://www.ftsmodules.com/public/texts/optiontutor/eappch6.htm The Black–Scholes Option Pricing Model], optiontutor
*[http://www.mayin.org/ajayshah/PDFDOCS/Shah1997_bms.pdf Black, Merton, and Scholes: Their work and its consequences], by Ajay Shah
*[http://bradley.bradley.edu/~arr/bsm/model.html A Study of Option Pricing Models], Prof. Kevin Rubash
*[http://www.risklatte.com/features/quantsKnow050905.php Black-Scholes in English], risklatte.com

===Variations on the model===
*[http://www.riskglossary.com/articles/black_scholes_1973.htm Options on non-dividend-paying stocks (Black Scholes)], riskglossary.com
*[http://www.riskglossary.com/articles/merton_1973.htm Options on stock indexes and continuous dividend-paying stocks], riskglossary.com
*[http://www.riskglossary.com/articles/garman_kohlhagen_1983.htm Foreign exchange options], riskglossary.com
*[http://www.riskglossary.com/articles/black_1976.htm Options on forwards (the Black model)], riskglossary.com
*[http://www.esomanager.com  Employee Stock Option Valuation], esomanager.com

===Derivation and solution===
*[http://www.quantnotes.com/fundamentals/options/riskneutrality.htm The risk neutrality derivation of the Black-Scholes Equation], quantnotes.com
*[http://www.quantnotes.com/fundamentals/options/black-scholes.htm Arbitrage-free pricing derivation of the Black-Scholes Equation], quantnotes.com, or [http://www.sjsu.edu/faculty/watkins/blacksch.htm an alternative treatment], Prof. Thayer Watkins
*[http://www.quantnotes.com/fundamentals/options/solvingbs.htm Solving the Black-Scholes Equation], quantnotes.com
*[http://www.physics.uci.edu/%7Esilverma/bseqn/bs/bs.html Solution of the Black–Scholes Equation Using the Green's Function], Prof. Dennis Silverman
*[http://homepages.nyu.edu/~sl1544/KnownClosedForms.pdf Solution via risk neutral pricing or via the PDE approach using Fourier transforms] (includes discussion of other option types), Simon Leger
*[http://www.quantnotes.com/fundamentals/options/thegreeks-delta.htm Step-by-step derivation of delta from the Black-Scholes equation] (site also contains step-by-step derivations of some of [[The Greeks|the other greeks]]), quantnotes.com
*[http://planetmath.org/encyclopedia/AnalyticSolutionOfBlackScholesPDE.html Step-by-step solution of the Black-Scholes PDE], planetmath.org.
*[http://pegasus.rutgers.edu/~gangdong/docs/BSM.pdf Analytical and Numerical Solution of Black-Scholes PDE], Gang Dong.

===Tests of the model===
*[http://www.findarticles.com/p/articles/mi_m3937/is_1996_March-April/ai_18367627 Anomalies in option pricing: the Black–Scholes model revisited], New England Economic Review, March-April, 1996

===Computer implementations===

*Sourcecode 
**[http://www.espenhaug.com/black_scholes.html Black–Scholes in Multiple Languages], espenhaug.com
**[http://www.vbnumericalmethods.com/financial.asp VBA sourcecode for Black Scholes and related models], vbnumericalmethods.com
**[http://www.global-derivatives.com/code/vba/BSEuro-Greeks.txt VBA sourcecode for Black Scholes and Greeks], global-derivatives.com

*Excel
**[http://www.optiontradingtips.com/pricing/free-spreadsheet.html Option Pricing Spreadsheet with documented VBA], OptionTradingTips.com
**[http://www.quantnotes.com/softwaredata/excelvba/BSEuropeanCalculator.xls Excel spreadsheet with VBA sourcecode], quantnotes.com
**[http://www.researchkitchen.co.uk/blog/archives/77 Excel implementation and tutorial], researchkitchen.co.uk

*Real Time
**[http://cdmurray80.googlepages.com/optiongreeks Surface Plots of Black-Scholes Greeks], Chris Murray
**[http://www.cba.ua.edu/~rpascala/revertGBM/BSOPMRForm.php Real-time calculator of Call and Put Option prices when the underlying follows a Mean-Reverting Geometric Brownian Motion], by Razvan Pascalau, Univ. of Alabama
**[http://www.sitmo.com/live/OptionVanilla.html Online real-time Black &amp; Scholes calculator, including all greeks], sitmo.com

===Historical===
*[http://www.nobel.se/economics/laureates/1997/press.html The Sveriges Riksbank (Bank of Sweden) Prize in Economic Sciences in Memory of Alfred Nobel for 1997]
*[http://www.pbs.org/wgbh/nova/stockmarket/ Trillion Dollar Bet]—Companion Web site to a Nova episode originally broadcast on February 8, 2000. ''"The film tells the fascinating story of the invention of the Black-Scholes Formula, a mathematical Holy Grail that forever altered the world of finance and earned its creators the 1997 Nobel Prize in Economics."''
*[http://www.bbc.co.uk/science/horizon/1999/midas.shtml BBC Horizon] A TV-programme on the so-called [[Midas formula]] and the bankruptcy of [[Long-Term Capital Management]] ([[LTCM]])




[[Categoria:Teorias]][[Categoria:Derivados]]