BlackScholes

A fórmula BlackScholes é um dos modelos mais usados para estabelecer o pricing de opções.
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 BlackScholes PDE to European put and call options.
Robert C. Merton was the first to publish a paper expanding our mathematical understanding of the options pricing model and coined the term "BlackScholes" 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 BlackScholes 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.
Índice
O modelo
Os pressupostos chave do modelo de Black–Scholes são:
 O preço do activo subjacente ao longo do tempo, S_{t}, segue um movimento Browniano geométrico com deriva e volatilidade constantes :
 É possível vender a descoberto o activo subjacente.
 Não há oportunidades de arbitragem.
 A negociação é realizada continuamente no tempo.
 Não há impostos nem comissões de negociação.
 Todos os activos são infinitamente divisíveis (por exemplo, é possível comprar unidades de um título.
 É possível emprestar e pedir emprestado dinheiro a uma 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
para o preço de uma 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 volatilidade é constante e representada por .
Na fórmula anterior:
onde é a função distribuição da normal standard.
O preço de uma opção de venda é calculado a partir do resultado anterior utilizando a paridade putcall:
O cáculo das Gregas no modelo de Black–Scholes model está feito na tabela seguinte:
Calls  Puts  

delta  
gamma  
vega  
theta  
rho 
Aqui, é a função densidade de probabilidade standard da distribuição normal. Notese 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 paridade putcall.
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 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. 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 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 por em que
 é a taxa de dividendo;
 é o valor do índice no instante t;
 é o período de tempo considerado.
De acordo com esta formulação, num contexto livre de arbitragem como no modelo de BlackSholes pode mostrarse que:
onde
é o preço forward utilizado nos cálculos de d_{1} and d_{2}:
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 of the stock price is paid out at predetermined times t_{1}, t_{2}, .... The price of the stock is then modelled as
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
where now
is the forward price for the dividend paying stock.
Black–Scholes in practice
The volatility smile
All the parameters in the model other than the volatility — the time to maturity, the strike, the riskfree 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 onetoone 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 BlackScholes 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 threedimensional 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 atthemoney, 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 BlackScholes model), the BlackScholes PDE and BlackScholes 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 BlackScholes 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 securities because of the pulltopar 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 riskfree bond maturing on the option's expiration date is, in general, not observable in the market. Instead, an interest rate curve is used. Composed of marketquoted interest rates of various maturities, the curve provides an estimate of the riskfree rate of appropriate maturity for the option being priced.
Another issue arises when 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 riskfree 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 riskfree rate for a theoretical bond expiring on January 19th, 2008 is 5.05%. Therefore, the correct interest rate to use in the BlackScholes 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 BlackScholes model is 3.05%.
Formula derivation
Elementary derivation
Let S_{0} be the current price of the underlying stock and S the price when the option matures at time T. Then S_{0} is known, but S is a random variable. Assume that
is a normal random variable with mean and variance . It follows that the mean of S is
for some constant q (independent of T). Now a simple noarbitrage 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
where r is the riskfree interest rate. This suggests making the identification q = r for the purpose of pricing derivatives. Define the theoretical value of a derivative as the present value of the expected payoff in this sense. For a call option with exercise price K this discounted expectation (using riskneutral probabilities) is
The derivation of the formula for C is facilitated by the following lemma: Let Z be a standard normal random variable and let b be an extended real number. Define
If a is a positive real number, then
where is the standard normal cumulative distribution function. In the special case b = −∞, we have
Now let
and use the corollary to the lemma to verify the statement above about the mean of S. Define
and observe that
for some b. Define
and observe that
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 riskneutral probability measure, but the concept of riskneutrality 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 noarbitrage or deltahedging argument; for more on the underlying logic, see the discussion at rational pricing.
The presentation given here is 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,
where W_{t} 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
Now consider a trading strategy under which one holds one option and continuously trades in the stock in order to hold −∂V/∂S shares. At time t, the value of these holdings will be
The composition of this portfolio, called the deltahedge portfolio, will vary from timestep to timestep. 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
By substituting in the equations above we get
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 riskfree rate of return is r we must have over the time period [t, t + dt]
If we now substitute in for and divide through by dt we obtain the Black–Scholes PDE:
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 arbitragefree pricing ("deltahedging") to derive a PDE governing option prices given the Black–Scholes model. It is also possible to use a riskneutrality argument. This latter method gives the price as the expectation of the option payoff under a particular probability measure, called the riskneutral 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 . Now, for a call option the PDE above has boundary conditions
 for all t
 as
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 diffusion equation which may be solved using standard methods. To this end we introduce the changeofvariable transformation
Then the Black–Scholes PDE becomes a diffusion equation
The terminal condition now becomes an initial condition
Using the standard method for solving a diffusion equation we have
After some algebra we obtain
where
and is the standard normal cumulative distribution function.
Substituting for u, x, and , we obtain the value of a call option in terms of the Black–Scholes parameters:
where
The formula for the price of a put option follows from this via putcall 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 discretedividend model and the elementary derivation is not the same as the parameter 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 TIBasic, which contains some TIBasic programs to perform stochastic calculus with Texas Instruments Calculators
 QuantLib, which includes an opensource implementation of BlackScholes in C++
 real options analysis
References
Primary references
 Black, Fischer; Myron Scholes (1973). "The Pricing of Options and Corporate Liabilities". Journal of Political Economy 81 (3): 637654. [1] (Black and Scholes' original paper.)
 Merton, Robert C. (1973). "Theory of Rational Option Pricing". Bell Journal of Economics and Management Science 4 (1): 141183. [2]
Historical and sociological aspects
 Bernstein, Peter. Capital Ideas: The Improbable Origins of Modern Wall Street. The Free Press. ISBN 0029030129.
 MacKenzie, Donald (2003). "An Equation and its Worlds: Bricolage, Exemplars, Disunity and Performativity in Financial Economics". Social Studies of Science 33 (6): 831868. [3]
 MacKenzie, Donald; Yuval Millo (2003). "Constructing a Market, Performing Theory: The Historical Sociology of a Financial Derivatives Exchange". American Journal of Sociology 109 (1): 107145. [4]
 MacKenzie, Donald. An Engine, not a Camera: How Financial Models Shape Markets. MIT Press. ISBN 0262134608.
External links
Discussion of the model
 Option pricing theory, Overview and links to more detailed articles on specific models, riskglossary.com
 The Black–Scholes Model, globalderivatives.com
 Options pricing using the BlackScholes Model, Investment Analysts Society of Southern Africa
 The Black–Scholes Option Pricing Model, optiontutor
 Black, Merton, and Scholes: Their work and its consequences, by Ajay Shah
 A Study of Option Pricing Models, Prof. Kevin Rubash
 BlackScholes in English, risklatte.com
Variations on the model
 Options on nondividendpaying stocks (Black Scholes), riskglossary.com
 Options on stock indexes and continuous dividendpaying stocks, riskglossary.com
 Foreign exchange options, riskglossary.com
 Options on forwards (the Black model), riskglossary.com
 Employee Stock Option Valuation, esomanager.com
Derivation and solution
 The risk neutrality derivation of the BlackScholes Equation, quantnotes.com
 Arbitragefree pricing derivation of the BlackScholes Equation, quantnotes.com, or an alternative treatment, Prof. Thayer Watkins
 Solving the BlackScholes Equation, quantnotes.com
 Solution of the Black–Scholes Equation Using the Green's Function, Prof. Dennis Silverman
 Solution via risk neutral pricing or via the PDE approach using Fourier transforms (includes discussion of other option types), Simon Leger
 Stepbystep derivation of delta from the BlackScholes equation (site also contains stepbystep derivations of some of the other greeks), quantnotes.com
 Stepbystep solution of the BlackScholes PDE, planetmath.org.
 Analytical and Numerical Solution of BlackScholes PDE, Gang Dong.
Tests of the model
 Anomalies in option pricing: the Black–Scholes model revisited, New England Economic Review, MarchApril, 1996
Computer implementations
 Sourcecode
 Black–Scholes in Multiple Languages, espenhaug.com
 VBA sourcecode for Black Scholes and related models, vbnumericalmethods.com
 VBA sourcecode for Black Scholes and Greeks, globalderivatives.com
 Excel
 Option Pricing Spreadsheet with documented VBA, OptionTradingTips.com
 Excel spreadsheet with VBA sourcecode, quantnotes.com
 Excel implementation and tutorial, researchkitchen.co.uk
 Real Time
 Surface Plots of BlackScholes Greeks, Chris Murray
 Realtime calculator of Call and Put Option prices when the underlying follows a MeanReverting Geometric Brownian Motion, by Razvan Pascalau, Univ. of Alabama
 Online realtime Black & Scholes calculator, including all greeks, sitmo.com
Historical
 The Sveriges Riksbank (Bank of Sweden) Prize in Economic Sciences in Memory of Alfred Nobel for 1997
 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 BlackScholes Formula, a mathematical Holy Grail that forever altered the world of finance and earned its creators the 1997 Nobel Prize in Economics."
 BBC Horizon A TVprogramme on the socalled Midas formula and the bankruptcy of LongTerm Capital Management (LTCM)