In
mathematical finance
Mathematical finance, also known as quantitative finance and financial mathematics, is a field of applied mathematics, concerned with mathematical modeling of financial markets.
In general, there exist two separate branches of finance that require ...
, the Black–Scholes equation is a
partial differential equation
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function.
The function is often thought of as an "unknown" to be solved for, similarly to h ...
(PDE) governing the price evolution of a
European call In finance, the style or family of an option is the class into which the option falls, usually defined by the dates on which the option may be exercised. The vast majority of options are either European or American (style) options. These options� ...
or
European put 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 investment instruments. From the parabolic partial differential equation in the model, known as the Black� ...
. Broadly speaking, the term may refer to a similar PDE that can be derived for a variety of
options
Option or Options may refer to:
Computing
*Option key, a key on Apple computer keyboards
*Option type, a polymorphic data type in programming languages
*Command-line option, an optional parameter to a command
*OPTIONS, an HTTP request method
...
, or more generally,
derivatives.
For a European call or put on an underlying stock paying no dividends, the equation is:
:
where ''V'' 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
Frictionless can refer to:
* Frictionless market
* Frictionless continuant
* Frictionless sharing
* Frictionless plane
The frictionless plane is a concept from the writings of Galileo Galilei. In his 1638 '' The Two New Sciences'', Galileo prese ...
, one can perfectly
hedge
A hedge or hedgerow is a line of closely spaced shrubs and sometimes trees, planted and trained to form a barrier or to mark the boundary of an area, such as between neighbouring properties. Hedges that are used to separate a road from adjoi ...
the option by buying and selling the
underlying
In finance, a derivative is a contract that ''derives'' its value from the performance of an underlying entity. This underlying entity can be an asset, index, or interest rate, and is often simply called the "underlying". Derivatives can be use ...
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 of the Black–Scholes PDE
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'', and a term involving the second spatial derivative ''gamma'', 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.
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 of the Black–Scholes PDE
The following derivation is given in
Hull's ''Options, Futures, and Other Derivatives''.
That, in turn, is based on the classic argument in the original Black–Scholes paper.
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 i ...
. That is
:
where ''W'' is a stochastic variable (
Brownian motion
Brownian motion, or pedesis (from grc, πήδησις "leaping"), is the random motion of particles suspended in a medium (a liquid or a gas).
This pattern of motion typically consists of random fluctuations in a particle's position insi ...
). 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 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 expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbe ...
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, Itô's lemma or Itô's formula (also called the Itô-Doeblin formula, especially in French literature) is an identity used in Itô calculus to find the differential of a time-dependent function of a stochastic process. It serves ...
for two variables we have
:
Now consider a certain portfolio, called the
delta-hedge portfolio, consisting of being short one option and long
shares at time
. The value of these holdings is
:
Over the time period