The Black–Scholes 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.
History Louis Bachelier's thesis in 1900 was the earliest publication to apply Brownian motion to derivative pricing, though his work had little impact for many years and included important limitations for its application to modern markets. In the 1960's Case Sprenkle, James Boness, Paul Samuelson, and Samuelson's Ph.D. student at the time Robert C. Merton all made important improvements to the theory of options pricing.
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.Taleb, 1997. pp. 91, 110–111.Mandelbrot & Hudson, 2006. pp. 9–10. They based their thinking on work previously done by market researchers and practitioners including the work mentioned above, as well as work by 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.Mandelbrot & Hudson, 2006. p. 74 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.Mandelbrot & Hudson, 2006. pp. 72–75.Derman, 2004. pp. 143–147.Thorp, 2017. pp. 183–189. 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 options trading, led by Cboe Global Markets.
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.
Fundamental hypotheses The Black–Scholes model assumes that the market consists of at least one risky asset, usually called the stock, and one riskless asset, usually called the money market, cash, or bond.
The following assumptions are made about the assets (which relate to the names of the assets): Risk-free rate: The rate of return on the riskless asset is constant and thus called the risk-free interest rate. Random walk: The instantaneous log return of the stock price is an infinitesimal random walk with drift; more precisely, the stock price follows a geometric Brownian motion, and it is assumed that the drift and volatility of the motion are constant. If drift and volatility are time-varying, a suitably modified Black–Scholes formula can be deduced, as long as the volatility is not random. The stock does not pay a dividend.Although the original model assumed no dividends, trivial extensions to the model can accommodate a continuous dividend yield factor.
The assumptions about the market are: No arbitrage opportunity (i.e., there is no way to make a riskless profit in excess of the risk-free rate). Ability to borrow and lend any amount, even fractional, of cash at the riskless rate. Ability to buy and sell any amount, even fractional, of the stock (this includes short selling). The above transactions do not incur any fees or costs (i.e., frictionless market).
With these assumptions, suppose there is a derivative security also trading in this market. It is specified that this security will have a certain payoff on a specified future date, depending on the values of the stock up to that date. Even though the path the stock price will take in the future is unknown, the derivative's price can be determined at the current time. For the special case of a European call or put option, Black and Scholes showed that "it is possible to create a hedged position, consisting of a long position in the stock and a short position in the option, whose value will not depend on the price of the stock". Their dynamic hedging strategy led to a partial differential equation which governs the price of the option. Its solution is given by the Black–Scholes formula.
Several of these assumptions of the original model have been removed in subsequent extensions of the model. Modern versions account for dynamic interest rates (Merton, 1976), transaction costs and taxes (Ingersoll, 1976), and dividend payout.
Notation At time t, in particular: is the price of a European call option and is the price of a European put option. is the time of option expiration. is the time until maturity: . is the strike price of the option, also known as the exercise price.
denotes the standard normal cumulative distribution function:
denotes the standard normal probability density function:
Black–Scholes equation
The Black–Scholes equation is a parabolic partial differential equation that describes the price of the option, where is the price of the underlying asset and is time:
A key financial insight behind the equation is that one can perfectly hedge the option by buying and selling the underlying asset and the bank account asset (cash) in such a way as to "eliminate risk". This implies that there is a unique price for the option given by the Black–Scholes formula (see the next section).
Black–Scholes formula
The Black–Scholes formula calculates the price of European put and call options. This price is consistent with the Black–Scholes equation. This follows since the formula can be obtained by solving the equation for the corresponding terminal and boundary conditions:
The value of a call option for a non-dividend-paying underlying stock in terms of the Black–Scholes parameters is:
The price of a corresponding put option based on put–call parity with discount factor is:
Alternative formulation Introducing auxiliary variables allows for the formula to be simplified and reformulated in a form that can be more convenient (this is a special case of the Black '76 formula):
where:
is the discount factor
is the forward price of the underlying asset, and
Given put–call parity, which is expressed in these terms as:
the price of a put option is:
Interpretation 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.
The formula can be interpreted by first decomposing a call option into the difference of two binary options: an asset-or-nothing call minus a cash-or-nothing call (long an asset-or-nothing call, short a cash-or-nothing call). A call option exchanges cash for an asset at expiry, while an asset-or-nothing call just yields the asset (with no cash in exchange) and a cash-or-nothing call just yields cash (with no asset in exchange). The Black–Scholes formula is a difference of two terms, and these two terms are equal to the values of the binary call options. These binary options are less frequently traded than vanilla call options, but are easier to analyze.
Thus the formula:
breaks up as:
where is the present value of an asset-or-nothing call and is the present value of a cash-or-nothing call. The D factor is for discounting, because the expiration date is in future, and removing it changes present value to future value (value at expiry). Thus is the future value of an asset-or-nothing call and is the future value of a cash-or-nothing call. In risk-neutral terms, these are the expected value of the asset and the expected value of the cash in the risk-neutral measure.
A naive, and slightly incorrect, interpretation of these terms is that is the probability of the option expiring in the money , multiplied by the value of the underlying at expiry F, while is the probability of the option expiring in the money multiplied by the value of the cash at expiry K. This interpretation is incorrect because either both binaries expire in the money or both expire out of the money (either cash is exchanged for the asset or it is not), but the probabilities and are not equal. In fact, can be interpreted as measures of moneyness (in standard deviations) and as probabilities of expiring ITM (percent moneyness), in the respective numéraire, as discussed below. Simply put, the interpretation of the cash option, , is correct, as the value of the cash is independent of movements of the underlying asset, and thus can be interpreted as a simple product of "probability times value", while the is more complicated, as the probability of expiring in the money and the value of the asset at expiry are not independent. More precisely, the value of the asset at expiry is variable in terms of cash, but is constant in terms of the asset itself (a fixed quantity of the asset), and thus these quantities are independent if one changes numéraire to the asset rather than cash.
If one uses spot S instead of forward F, in instead of the term there is which can be interpreted as a drift factor (in the risk-neutral measure for appropriate numéraire). The use of d− for moneyness rather than the standardized moneyness in other words, the reason for the factor is due to the difference between the median and mean of the log-normal distribution; it is the same factor as in Itō's lemma applied to geometric Brownian motion. In addition, another way to see that the naive interpretation is incorrect is that replacing by in the formula yields a negative value for out-of-the-money call options.