
In
mathematical finance
Mathematical finance, also known as quantitative finance and financial mathematics, is a field of applied mathematics, concerned with mathematical modeling in the financial field.
In general, there exist two separate branches of finance that req ...
, the Black–Scholes equation, also called the Black–Scholes–Merton equation, is a
partial differential equation
In mathematics, a partial differential equation (PDE) is an equation which involves a multivariable function and one or more of its partial derivatives.
The function is often thought of as an "unknown" that solves the equation, similar to ho ...
(PDE) governing the price evolution of derivatives under the
Black–Scholes model
The Black–Scholes or Black–Scholes–Merton model is a mathematical model for the dynamics of a financial market containing Derivative (finance), derivative investment instruments. From the parabolic partial differential equation in the model, ...
. Broadly speaking, the term may refer to a similar PDE that can be derived for a variety of
options, or more generally,
derivatives.
Consider a stock paying no dividends. Now construct any derivative that has a fixed maturation time
in the future, and at maturation, it has payoff
that depends on the values taken by the stock at that moment (such as European call or put options). Then the price of the derivative satisfies
:
where
is the price of the option as a function of stock price ''S'' and time ''t'', ''r'' is the risk-free interest rate, and
is the
volatility of the stock.
The key financial insight behind the equation is that, under the model assumption of a
frictionless market, one can perfectly
hedge the option by buying and selling the
underlying
In finance, a derivative is a contract between a buyer and a seller. The derivative can take various forms, depending on the transaction, but every derivative has the following four elements:
# an item (the "underlier") that can or must be bou ...
asset in just the right way and consequently “eliminate risk". This hedge, in turn, implies that there is only one right price for the option, as returned by the
Black–Scholes formula.
Financial interpretation
The equation has a concrete interpretation that is often used by practitioners and is the basis for the common derivation given in the next subsection. The equation can be rewritten in the form:
:
The left-hand side consists of a "time decay" term, the change in derivative value with respect to time, called
theta
Theta (, ) uppercase Θ or ; lowercase θ or ; ''thē̂ta'' ; Modern: ''thī́ta'' ) is the eighth letter of the Greek alphabet, derived from the Phoenician letter Teth 𐤈. In the system of Greek numerals, it has a value of 9.
Gree ...
, and a term involving the second spatial derivative
gamma
Gamma (; uppercase , lowercase ; ) is the third letter of the Greek alphabet. In the system of Greek numerals it has a value of 3. In Ancient Greek, the letter gamma represented a voiced velar stop . In Modern Greek, this letter normally repr ...
, the convexity of the derivative value with respect to the underlying value. The right-hand side is the riskless return from a long position in the derivative and a short position consisting of
shares of the underlying asset.
Black and Scholes' insight was that the portfolio represented by the right-hand side is riskless: thus the equation says that the riskless return over any infinitesimal time interval can be expressed as the sum of theta and a term incorporating gamma. For an option, theta is typically negative, reflecting the loss in value due to having less time for exercising the option (for a European call on an underlying without dividends, it is always negative). Gamma is typically positive and so the gamma term reflects the gains in holding the option. The equation states that over any infinitesimal time interval the loss from theta and the gain from the gamma term must offset each other so that the result is a return at the riskless rate.
From the viewpoint of the option issuer, e.g. an investment bank, the gamma term is the cost of hedging the option. (Since gamma is the greatest when the spot price of the underlying is near the strike price of the option, the seller's hedging costs are the greatest in that circumstance.)
Derivation
Per the model assumptions above, the price of the
underlying asset (typically a stock) follows a
geometric Brownian motion
A geometric Brownian motion (GBM) (also known as exponential Brownian motion) is a continuous-time stochastic process in which the logarithm of the randomly varying quantity follows a Brownian motion (also called a Wiener process) with drift. It ...
.
That is
:
where ''W'' is a stochastic variable (
Brownian motion
Brownian motion is the random motion of particles suspended in a medium (a liquid or a gas). The traditional mathematical formulation of Brownian motion is that of the Wiener process, which is often called Brownian motion, even in mathematical ...
). Note that ''W'', and consequently its infinitesimal increment ''dW'', represents the only source of uncertainty in the price history of the stock. Intuitively, ''W''(''t'') is a
process
A process is a series or set of activities that interact to produce a result; it may occur once-only or be recurrent or periodic.
Things called a process include:
Business and management
* Business process, activities that produce a specific s ...
that "wiggles up and down" in such a random way that its expected change over any time interval is 0. (In addition, its
variance
In probability theory and statistics, variance is the expected value of the squared deviation from the mean of a random variable. The standard deviation (SD) is obtained as the square root of the variance. Variance is a measure of dispersion ...
over time ''T'' is equal to ''T''; see ); a good discrete analogue for ''W'' is a
simple random walk. Thus the above equation states that the infinitesimal rate of return on the stock has an expected value of ''μ'' ''dt'' and a variance of
.
The payoff of an option (or any derivative contingent to stock )
at maturity is known. To find its value at an earlier time we need to know how
evolves as a function of
and
. By
Itô's lemma
In mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. ...
for two variables we have
:
Now consider a portfolio
consisting of a short option and
long shares at time
. The value of these holdings is
:
As
changes with time, the position in
is continually updated. We implicitly assume that the portfolio contains a cash account to accommodate buying and selling shares
, making the portfolio
self-financing. Therefore, we only need to consider the total profit or loss from changes in the values of the holdings:
:
Substituting
and
into the expression for
:
:
Over a time period