A fórmula Black-Scholes 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 Black-Scholes 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 "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.
- 1 O modelo
- 2 Generalizações do modelo
- 3 Black–Scholes in practice
- 4 Formula derivation
- 5 Remarks on notation
- 6 See also
- 7 References
- 8 External links
Os pressupostos chave do modelo de Black–Scholes são:
- O preço do activo subjacente ao longo do tempo, St, 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:
O cáculo das Gregas no modelo de Black–Scholes model está feito na tabela seguinte:
Aqui, é a 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 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 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 Black-Sholes pode mostrar-se que:
é o preço forward utilizado nos cálculos de d1 and d2:
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 pre-determined times t1, t2, .... 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
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 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 securities because of the 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 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 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%.
Let S0 be the current price of the underlying stock and S the price when the option matures at time T. Then S0 is known, but S is a random variable. Assume that
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
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 present value of the expected payoff in this sense. For a call option with exercise price K this discounted expectation (using risk-neutral probabilities) is
If a is a positive real number, then
where is the standard normal cumulative distribution function. In the special case b = −∞, we have
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 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 argument; for more on the underlying logic, see the discussion at rational pricing.
The Black–Scholes PDE
where Wt 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 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
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 risk-free 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 arbitrage-free pricing ("delta-hedging") to derive a PDE governing option prices given the Black–Scholes model. It is also possible to use a risk-neutrality argument. This latter method gives the price as the 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 . Now, for a call option the PDE above has boundary conditions
- for all t
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 change-of-variable 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
Substituting for u, x, and , we obtain the value of a call option in terms of the Black–Scholes parameters:
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 that appears elsewhere in the article. For the relationship between them see Geometric Brownian motion.
- 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
- Black, Fischer; Myron Scholes (1973). "The Pricing of Options and Corporate Liabilities". Journal of Political Economy 81 (3): 637-654.  (Black and Scholes' original paper.)
- Merton, Robert C. (1973). "Theory of Rational Option Pricing". Bell Journal of Economics and Management Science 4 (1): 141-183. 
Historical and sociological aspects
- Bernstein, Peter. Capital Ideas: The Improbable Origins of Modern Wall Street. The Free Press. ISBN 0-02-903012-9.
- MacKenzie, Donald (2003). "An Equation and its Worlds: Bricolage, Exemplars, Disunity and Performativity in Financial Economics". Social Studies of Science 33 (6): 831-868. 
- MacKenzie, Donald; Yuval Millo (2003). "Constructing a Market, Performing Theory: The Historical Sociology of a Financial Derivatives Exchange". American Journal of Sociology 109 (1): 107-145. 
- MacKenzie, Donald. An Engine, not a Camera: How Financial Models Shape Markets. MIT Press. ISBN 0-262-13460-8.
Discussion of the model
- Option pricing theory, Overview and links to more detailed articles on specific models, riskglossary.com
- The Black–Scholes Model, global-derivatives.com
- Options pricing using the Black-Scholes 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
- Black-Scholes in English, risklatte.com
Variations on the model
- Options on non-dividend-paying stocks (Black Scholes), riskglossary.com
- Options on stock indexes and continuous dividend-paying 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 Black-Scholes Equation, quantnotes.com
- Arbitrage-free pricing derivation of the Black-Scholes Equation, quantnotes.com, or an alternative treatment, Prof. Thayer Watkins
- Solving the Black-Scholes 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
- Step-by-step derivation of delta from the Black-Scholes equation (site also contains step-by-step derivations of some of the other greeks), quantnotes.com
- Step-by-step solution of the Black-Scholes PDE, planetmath.org.
- Analytical and Numerical Solution of Black-Scholes PDE, Gang Dong.
Tests of the model
- Anomalies in option pricing: the Black–Scholes model revisited, New England Economic Review, March-April, 1996
- Black–Scholes in Multiple Languages, espenhaug.com
- VBA sourcecode for Black Scholes and related models, vbnumericalmethods.com
- VBA sourcecode for Black Scholes and Greeks, global-derivatives.com
- Real Time
- Surface Plots of Black-Scholes Greeks, Chris Murray
- 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
- Online real-time Black & Scholes calculator, including all greeks, sitmo.com
- 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 Black-Scholes 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 TV-programme on the so-called Midas formula and the bankruptcy of Long-Term Capital Management (LTCM)