Black–Scholes

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 a partial differential 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 solving the Black-Scholes PDE for 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.

The model

The key assumptions of the Black–Scholes model are:

• The price of the underlying instrument S_t follows a geometric Brownian motion with constant drift Failed to parse (Missing texvc executable; please see math/README to configure.): \mu

and volatility Failed to parse (Missing texvc executable; please see math/README to configure.):  \sigma 

Failed to parse (Missing texvc executable; please see math/README to configure.): dS_t = \mu S_t\,dt + \sigma S_t\,dW_t \,

• It is possible to short sell the underlying stock.
• There are no arbitrage opportunities.
• Trading in the stock is continuous.
• There are no transaction costs or taxes.
• All securities are perfectly divisible (e.g. it is possible to buy 1/100th of a share).
• It is possible to borrow and lend cash at a constant risk-free interest rate.
• The stock does not pay a dividend (see below for extensions to handle dividend payments).

The above lead to the following formula for the price C of a European call option with exercise price K on a stock currently trading at price S, i.e., the right to buy a share of the stock at price K after T years. The constant interest rate is r, and the constant stock volatility is Failed to parse (Missing texvc executable; please see math/README to configure.): \sigma .

Failed to parse (Missing texvc executable; please see math/README to configure.): C(S,T) = S\Phi(d_1) - Ke^{-rT}\Phi(d_2) \,

where

Failed to parse (Missing texvc executable; please see math/README to configure.): d_1 = \frac{\ln(S/K) + (r + \sigma^2/2)T}{\sigma\sqrt{T}}

Failed to parse (Missing texvc executable; please see math/README to configure.): d_2 = \frac{\ln(S/K) + (r - \sigma^2/2)T}{\sigma\sqrt{T}} = d_1 - \sigma\sqrt{T}.

Here Failed to parse (Missing texvc executable; please see math/README to configure.): \Phi

is the standard normal cumulative distribution function.

The price of a put option may be computed from this by put-call parity and simplifies to

Failed to parse (Missing texvc executable; please see math/README to configure.): P(S,T) = Ke^{-rT}\Phi(-d_2) - S\Phi(-d_1). \,

The Greeks under the Black–Scholes model are calculated below:

Here, Failed to parse (Missing texvc executable; please see math/README to configure.): \varphi

is the standard normal probability density function. Note that the gamma and vega formulas are the same for calls and puts.

This can be seen directly from put-call parity.

In practice, some sensitivities are usually quoted in scaled-down terms, to match the scale of likely changes in the parameters. For example, rho is often reported divided by 10,000 (1bp rate change), vega by 100 (1 vol point change), and theta by 365 or 252 (1 day decay based on either calendar days or trading days per year).

Extensions of the model

The above model can easily be extended to have non-constant (but deterministic) rates and volatilities. The model may also be used to value European options on instruments paying dividends. In this case, closed-form solutions are available if the dividend is a known proportion of the stock price. American options and options on stocks paying a known cash dividend (in the short term, more realistic than a proportional dividend) are more difficult to value, and a choice of solution techniques is available (for example lattices and grids).

Instruments paying continuous yield dividends

For options on indexes (such as the FTSE where each of 100 constituent companies may pay a dividend twice a year and so there is a payment nearly every business day), it is reasonable to make the simplifying assumption that dividends are paid continuously, and that the dividend amount is proportional to the level of the index.

The dividend payment paid over the time period [t, t + dt] is then modelled as

Failed to parse (Missing texvc executable; please see math/README to configure.): qS_t\,dt

for some constant q (the dividend yield).

Under this formulation the arbitrage-free price implied by the Black–Scholes model can be shown to be

Failed to parse (Missing texvc executable; please see math/README to configure.): C(S_0,T) = e^{-rT}(F\Phi(d_1) - K\Phi(d_2)) \,

where now

Failed to parse (Missing texvc executable; please see math/README to configure.): F = S_0 e^{(r - q)T} \,

is the modified forward price that occurs in the terms d₁ and d₂:

Failed to parse (Missing texvc executable; please see math/README to configure.): d_1 = \frac{\ln(F/K) + (\sigma^2/2)T}{\sigma\sqrt{T}}

Failed to parse (Missing texvc executable; please see math/README to configure.): d_2 = d_1 - \sigma\sqrt{T}.

Exactly the same formula is used to price options on foreign exchange rates, except that now q plays the role of the foreign risk-free interest rate and S is the spot exchange rate. This is the Garman–Kohlhagen model (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 Failed to parse (Missing texvc executable; please see math/README to configure.): \delta

of the stock price is paid out at pre-determined times t1, t2, .... The price of the stock is then modelled as

Failed to parse (Missing texvc executable; please see math/README to configure.): S_t = S_0(1 - \delta)^{n(t)}e^{ut + \sigma W_t}

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

Failed to parse (Missing texvc executable; please see math/README to configure.): C(S_0,T) = e^{-rT}(F\Phi(d_1) - K\Phi(d_2)) \,

where now

Failed to parse (Missing texvc executable; please see math/README to configure.): F = S_0(1 - \delta)^{n(T)}e^{rT} \,

is the forward price for the dividend paying stock.

Black–Scholes in practice

The Black–Scholes–Merton model is empirically wrong—prices do not follow a stationary log-normal process—but Black–Scholes pricing is widely used in practice, as it is robust: it is wrong, but a useful approximation, like Newtonian mechanics. It is used both as a quoting convention and a basis for more refined models.

Most fundamentally, rather than assuming a volatility a priori and computing prices from it, one generally goes backward: given prices in the market, one backs out an implied volatility of options at given tenors and given strikes, obtaining a volatility surface. Thus Black–Scholes gives a coordinate transformation from the price domain to the volatility domain: rather than quoting option prices in terms of dollars per unit (which are hard to compare across strikes and tenors), one generally quotes option prices in terms of implied volatility.

Similarly, since implied volatility is not constant, but is close to constant (especially for small moves), the Black–Scholes Greeks are close to accurate, and thus if one hedges ones book using Black–Scholes Greeks, one will be close to correctly hedged.

The Black–Scholes approximation breaks down most severely for far OTM options, as the pricing of these correspond to the likelihood of tail events, which are not captured by Brownian motion.

The volatility smile

Main article: 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: compared to at-the-money, implied volatility is substantially 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

In practice, interest rates are not constant - they vary by tenor, giving an interest rate curve which may be interpolated to pick an appropriate rate to use in the Black-Scholes formula. Another consideration is that interest rates vary over time. This volatility may make a significant contribution to the price, especially of long-dated options.

Short stock rate

It is not free to take a short stock position. Similarly, it may be possible to lend out a long stock position for a small fee. In either case, this can be treated as a continuous dividend for the purposes of a Black-Scholes valuation.

Formula derivation

Elementary derivation

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

Failed to parse (Missing texvc executable; please see math/README to configure.): X \equiv \ln(S/S_0) \,

is a normal random variable with mean Failed to parse (Missing texvc executable; please see math/README to configure.): uT

and variance Failed to parse (Missing texvc executable; please see math/README to configure.):  \sigma^2 T 

. It follows that the mean of S is

Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbb{E}\left[ S \right] = S_0 e^{qT} \,

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

Failed to parse (Missing texvc executable; please see math/README to configure.): S_0 e^{rT} \,

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

Failed to parse (Missing texvc executable; please see math/README to configure.): C(S_0,T) = e^{-rT} \mathbb{E}\left[ \max(S - K,0) \right]. \,

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

Failed to parse (Missing texvc executable; please see math/README to configure.): Z^+(b) = \begin{cases} Z & \mbox{if }Z>b \\ -\infty & \mbox{otherwise} \end{cases}.

If a is a positive real number, then

Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbb{E}\left[e^{aZ^+(b)}\right] = e^{a^2/2}\Phi(-b + a)

where Failed to parse (Missing texvc executable; please see math/README to configure.): \Phi

is the standard normal cumulative distribution function. In the special case b = −∞, we have

Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbb{E}\left[e^{aZ}\right] = e^{a^2/2}.

Now let

Failed to parse (Missing texvc executable; please see math/README to configure.): Z = \frac{X - uT}{\sigma\sqrt{T}}

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

Failed to parse (Missing texvc executable; please see math/README to configure.): S^+ = \begin{cases} S & \mbox{if }S>K \\ 0 & \mbox{otherwise} \end{cases}

Failed to parse (Missing texvc executable; please see math/README to configure.): X^+ = \ln(S^+/S_0) \,

and observe that

Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{X^+ - uT}{\sigma\sqrt{T}} = Z^+(b)

for some b. Define

Failed to parse (Missing texvc executable; please see math/README to configure.): K^+ = \begin{cases} K & \mbox{if }S>K \\ 0 & \mbox{otherwise} \end{cases}

and observe that

Failed to parse (Missing texvc executable; please see math/README to configure.): \max(S - K,0) = S^+ - K^+. \,

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 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,

Failed to parse (Missing texvc executable; please see math/README to configure.): dS_t = \mu S_t\,dt + \sigma S_t\,dW_t \,

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

Failed to parse (Missing texvc executable; please see math/README to configure.): 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.

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

Failed to parse (Missing texvc executable; please see math/README to configure.): \Pi = V - S\frac{\partial V}{\partial S}.

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

Failed to parse (Missing texvc executable; please see math/README to configure.): dR = dV - \frac{\partial V}{\partial S}\,dS.

By substituting in the equations above we get

Failed to parse (Missing texvc executable; please see math/README to configure.): dR = \left(\frac{\partial V}{\partial t} + \frac{1}{2}\sigma^2 S^2\frac{\partial^2 V}{\partial S^2}\right)dt.

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]

Failed to parse (Missing texvc executable; please see math/README to configure.): 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.

If we now substitute in for Failed to parse (Missing texvc executable; please see math/README to configure.): \Pi

and divide through by dt we obtain the Black–Scholes PDE:

Failed to parse (Missing texvc executable; please see math/README to configure.): \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.

This is the law of evolution of the value of the option. With the assumptions of the Black–Scholes model, this partial differential equation holds whenever V is twice differentiable with respect to S and once with respect to t.

Other derivations of the PDE

Above we used the method of arbitrage-free pricing ("delta-hedging") to derive some 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 Failed to parse (Missing texvc executable; please see math/README to configure.): \sigma . Now, for a call option the PDE above has boundary conditions

Failed to parse (Missing texvc executable; please see math/README to configure.): V(0,t) = 0 \,

for all t

Failed to parse (Missing texvc executable; please see math/README to configure.): V(S,t) \sim S \,

as Failed to parse (Missing texvc executable; please see math/README to configure.):  S \rightarrow \infty \, 

Failed to parse (Missing texvc executable; please see math/README to configure.): V(S,T) = \max(S - K,0). \,

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

Failed to parse (Missing texvc executable; please see math/README to configure.): x = \ln(S/K) + (r - \sigma^2/2)(T - t) \,

Failed to parse (Missing texvc executable; please see math/README to configure.): \tau = T - t \,

Failed to parse (Missing texvc executable; please see math/README to configure.): u = Ve^{r(T - t)}. \,

Then the Black–Scholes PDE becomes a diffusion equation

Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{\partial u}{\partial \tau} = \frac{\sigma^2}{2} \frac{\partial^2 u}{\partial x^2}.

The terminal condition Failed to parse (Missing texvc executable; please see math/README to configure.): V(S,T) = \max(S - K,0)

now becomes an initial condition

Failed to parse (Missing texvc executable; please see math/README to configure.): u(x,0) = u_0(x) \equiv K\max(e^x - 1,0). \,

Using the standard method for solving a diffusion equation we have

Failed to parse (Missing texvc executable; please see math/README to configure.): 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.

After some algebra we obtain

Failed to parse (Missing texvc executable; please see math/README to configure.): u(x,\tau) = Ke^{x + \sigma^2\tau/2}\Phi(d_1) - K\Phi(d_2)

where

Failed to parse (Missing texvc executable; please see math/README to configure.): d_1 = \frac{x + \sigma^2\tau}{\sigma\sqrt{\tau}}

Failed to parse (Missing texvc executable; please see math/README to configure.): d_2 = \frac{x}{\sigma\sqrt{\tau}}

and Failed to parse (Missing texvc executable; please see math/README to configure.): \Phi

is the standard normal cumulative distribution function.

Substituting for u, x, and Failed to parse (Missing texvc executable; please see math/README to configure.): \tau , we obtain the value of a call option in terms of the Black–Scholes parameters:

Failed to parse (Missing texvc executable; please see math/README to configure.): V(S,t) = S\Phi(d_1) - Ke^{-r(T - t)}\Phi(d_2) \,

where

Failed to parse (Missing texvc executable; please see math/README to configure.): d_1 = \frac{\ln(S/K) + (r + \sigma^2/2)(T - t)}{\sigma\sqrt{T - t}}

Failed to parse (Missing texvc executable; please see math/README to configure.): d_2 = d_1 - \sigma\sqrt{T - t}.

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 Failed to parse (Missing texvc executable; please see math/README to configure.): \mu

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.
• real options analysis
• Black Shoals, a financial art piece

References

Primary references

• Black, Fischer; Myron Scholes (1973). "The Pricing of Options and Corporate Liabilities". Journal of Political Economy 81 (3): 637-654.  [1] (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.  [2]

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.  [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): 107-145.  [4]
• MacKenzie, Donald. An Engine, not a Camera: How Financial Models Shape Markets. MIT Press. ISBN 0-262-13460-8. 

External links

Discussion of the model

• The Black–Scholes Model, global-derivatives.com
• Options pricing using the Black-Scholes Model, Investment Analysts Society of Southern Africa
• 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
• The Black–Scholes Option Pricing Model, optiontutor
• Inside Wall Street's Black Hole by Michael Lewis, March 2008 Issue of portfolio.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.

Tests of the model

• Anomalies in option pricing: the Black–Scholes model revisited, New England Economic Review, March-April, 1996
• Why We Have Never Used the Black-Scholes-Merton Option Pricing Formula, Nassim Taleb and Espen Haug
• The illusions of dynamic replication, Emanuel Derman and Nassim Taleb

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, global-derivatives.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 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
  • Javascript Black & Scholes calls calculator, with greeks and profitability of some operations, epx.com.br

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 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)
• BBC Horizon - The Midas Formula (1999). Documentary available at Stage6.

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