Қаржы нарықтарының математикалық моделі

Mathematical model of financial markets
The Black–Scholes ,/b//l//æ//k// /'/ʃ//oʊ//l//z/ or Black–Scholes–Merton model is a mathematical model for the dynamics of a financial market containing derivative investment instruments. From the parabolic partial differential equation in the model, known as the Black–Scholes equation, one can deduce the Black–Scholes formula, which gives a theoretical estimate of the price of European style options and shows that the option has a unique price given the risk of the security and its expected return (instead replacing the security's expected return with the risk neutral rate). The equation and model are named after economists Fischer Black and Myron Scholes. Robert C. Merton, who first wrote an academic paper on the subject, is sometimes also credited. The main principle behind the model is to hedge the option by buying and selling the underlying asset in a specific way to eliminate risk. This type of hedging is called "continuously revised delta hedging" and is the basis of more complicated hedging strategies such as those used by investment banks and hedge funds. The model is widely used, although often with some adjustments, by options market participants. The model's assumptions have been relaxed and generalized in many directions, leading to a plethora of models that are currently used in derivative pricing and risk management. The insights of the model, as exemplified by the Black–Scholes formula, are frequently used by market participants, as distinguished from the actual prices. These insights include no arbitrage bounds and risk neutral pricing (thanks to continuous revision). Further, the Black–Scholes equation, a partial differential equation that governs the price of the option, enables pricing using numerical methods when an explicit formula is not possible. The Black–Scholes formula has only one parameter that cannot be directly observed in the market: the average future volatility of the underlying asset, though it can be found from the price of other options. Since the option value (whether put or call) is increasing in this parameter, it can be inverted to produce a "volatility surface" that is then used to calibrate other models, e. g. for OTC derivatives.

Economists Fischer Black and Myron Scholes demonstrated in 1968 that a dynamic revision of a portfolio removes the expected return of the security, thus inventing the risk neutral argument. They based their thinking on work previously done by market researchers and practitioners including Louis Bachelier, Sheen Kassouf and Edward O. Thorp. Black and Scholes then attempted to apply the formula to the markets, but incurred financial losses, due to a lack of risk management in their trades. In 1970, they decided to return to the academic environment. After three years of efforts, the formula—named in honor of them for making it public—was finally published in 1973 in an article titled "The Pricing of Options and Corporate Liabilities", in the Journal of Political Economy. Robert C. Merton was the first to publish a paper expanding the mathematical understanding of the options pricing model, and coined the term "Black–Scholes options pricing model". The formula led to a boom in options trading and provided mathematical legitimacy to the activities of the Chicago Board Options Exchange and other options markets around the world. Merton and Scholes received the 1997 Nobel Memorial Prize in Economic Sciences for their work, the committee citing their discovery of the risk neutral dynamic revision as a breakthrough that separates the option from the risk of the underlying security. Although ineligible for the prize because of his death in 1995, Black was mentioned as a contributor by the Swedish Academy.

It is possible to have intuitive interpretations of the Black–Scholes formula, with the main subtlety being the interpretation of and why there are two different terms. For related discussion and graphical representation see Datar–Mathews method for real option valuation. The equivalent martingale probability measure is also called the risk neutral probability measure. Note that both of these are probabilities in a measure theoretic sense, and neither of these is the true probability of expiring in the money under the real probability measure. To calculate the probability under the real ("physical") probability measure, additional information is required—the drift term in the physical measure, or equivalently, the market price of risk.

A standard derivation for solving the Black–Scholes PDE is given in the article Black–Scholes equation. The Feynman–Kac formula says that the solution to this type of PDE, when discounted appropriately, is actually a martingale. Thus the option price is the expected value of the discounted payoff of the option. Computing the option price via this expectation is the risk neutrality approach and can be done without knowledge of PDEs. Note the expectation of the option payoff is not done under the real world probability measure, but an artificial risk neutral measure, which differs from the real world measure. For the underlying logic see section "risk neutral valuation" under Rational pricing as well as section "Derivatives pricing: the Q world" under Mathematical finance; for details, once again, see Hull.

The above model can be extended for variable (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).

The problem of finding the price of an American option is related to the optimal stopping problem of finding the time to execute the option. Since the American option can be exercised at any time before the expiration date, the Black–Scholes equation becomes a variational inequality of the form:

together with where denotes the payoff at stock price and the terminal condition:
In general this inequality does not have a closed form solution, though an American call with no dividends is equal to a European call and the Roll–Geske–Whaley method provides a solution for an American call with one dividend; see also Black's approximation. Barone Adesi and Whaley is a further approximation formula. Here, the stochastic differential equation (which is valid for the value of any derivative) is split into two components: the European option value and the early exercise premium. With some assumptions, a quadratic equation that approximates the solution for the latter is then obtained. This solution involves finding the critical value, , such that one is indifferent between early exercise and holding to maturity. Bjerksund and Stensland provide an approximation based on an exercise strategy corresponding to a trigger price. Here, if the underlying asset price is greater than or equal to the trigger price it is optimal to exercise, and the value must equal , otherwise the option "boils down to: (i) a European up and out call option and (ii) a rebate that is received at the knock out date if the option is knocked out prior to the maturity date". The formula is readily modified for the valuation of a put option, using put–call parity. This approximation is computationally inexpensive and the method is fast, with evidence indicating that the approximation may be more accurate in pricing long dated options than Barone Adesi and Whaley.

Despite the lack of a general analytical solution for American put options, it is possible to derive such a formula for the case of a perpetual option meaning that the option never expires (i. e., ). In this case, the time decay of the option is equal to zero, which leads to the Black–Scholes PDE becoming an ODE:Let denote the lower exercise boundary, below which is optimal for exercising the option. The boundary conditions are:The solutions to the ODE are a linear combination of any two linearly independent solutions:For , substitution of this solution into the ODE for yields:Rearranging the terms gives:Using the quadratic formula, the solutions for are:In order to have a finite solution for the perpetual put, since the boundary conditions imply upper and lower finite bounds on the value of the put, it is necessary to set , leading to the solution From the first boundary condition, it is known that:Therefore, the value of the perpetual put becomes:The second boundary condition yields the location of the lower exercise boundary:To conclude, for , the perpetual American put option is worth:

By solving the Black–Scholes differential equation with the Heaviside function as a boundary condition, one ends up with the pricing of options that pay one unit above some predefined strike price and nothing below. In fact, the Black–Scholes formula for the price of a vanilla call option (or put option) can be interpreted by decomposing a call option into an asset or nothing call option minus a cash or nothing call option, and similarly for a put—the binary options are easier to analyze, and correspond to the two terms in the Black–Scholes formula.

This pays out one unit of cash if the spot is above the strike at maturity. Its value is given by:

This pays out one unit of cash if the spot is below the strike at maturity. Its value is given by:

This pays out one unit of asset if the spot is above the strike at maturity. Its value is given by:

This pays out one unit of asset if the spot is below the strike at maturity. Its value is given by:

Since a binary call is a mathematical derivative of a vanilla call with respect to strike, the price of a binary call has the same shape as the delta of a vanilla call, and the delta of a binary call has the same shape as the gamma of a vanilla call.

Black–Scholes cannot be applied directly to bond securities because of pull to par. 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. See .

In practice, interest rates are not constant—they vary by tenor (coupon frequency), 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. This is simply like the interest rate and bond price relationship which is inversely related.

Taking a short stock position, as inherent in the derivation, is not typically free of cost; equivalently, it is 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, provided that there is no glaring asymmetry between the short stock borrowing cost and the long stock lending income.

Espen Gaarder Haug and Nassim Nicholas Taleb argue that the Black–Scholes model merely recasts existing widely used models in terms of practically impossible "dynamic hedging" rather than "risk", to make them more compatible with mainstream neoclassical economic theory. They also assert that Boness in 1964 had already published a formula that is "actually identical" to the Black–Scholes call option pricing equation. Edward Thorp also claims to have guessed the Black–Scholes formula in 1967 but kept it to himself to make money for his investors. Emanuel Derman and Taleb have also criticized dynamic hedging and state that a number of researchers had put forth similar models prior to Black and Scholes. In response, Paul Wilmott has defended the model. In his 2008 letter to the shareholders of Berkshire Hathaway, Warren Buffett wrote: "I believe the Black–Scholes formula, even though it is the standard for establishing the dollar liability for options, produces strange results when the long term variety are being valued The Black–Scholes formula has approached the status of holy writ in finance If the formula is applied to extended time periods, however, it can produce absurd results. In fairness, Black and Scholes almost certainly understood this point well. But their devoted followers may be ignoring whatever caveats the two men attached when they first unveiled the formula." British mathematician Ian Stewart, author of the 2012 book entitled In Pursuit of the Unknown: 17 Equations That Changed the World, said that Black–Scholes had "underpinned massive economic growth" and the "international financial system was trading derivatives valued at one quadrillion dollars per year" by 2007. He said that the Black–Scholes equation was the "mathematical justification for the trading"—and therefore—"one ingredient in a rich stew of financial irresponsibility, political ineptitude, perverse incentives and lax regulation" that contributed to the financial crisis of 2007–08. He clarified that "the equation itself wasn't the real problem", but its abuse in the financial industry. When dealing with options whose underlying can go negative, practitioners may use a different model such as the Bachelier model or simply add a constant offset to the prices.

(Black and Scholes' original paper.)

Derman, Emanuel. "My Life as a Quant" John Wiley & Sons, Inc. 2004. Mandelbrot & Hudson, "The (Mis)Behavior of Markets" Basic Books, 2006. Szpiro, George G., Pricing the Future: Finance, Physics, and the 300 Year Journey to the Black–Scholes Equation; A Story of Genius and Discovery (New York: Basic, 2011) 298 pp. Taleb, Nassim. "Dynamic Hedging" John Wiley & Sons, Inc. 1997. Thorp, Ed. "A Man for all Markets" Random House, 2017.

Solution of the Black–Scholes Equation Using the Green's Function, Prof. Dennis Silverman
The Black–Scholes Equation Expository article by mathematician Terence Tao.