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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 T in the future, and at maturation, it has payoff K(S_T) 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 :\begin \frac + \frac\sigma^2 S^2 \frac + rS\frac - rV = 0 \\ V(T, s) = K(s) \quad \forall s \end where V(t, S) is the price of the option as a function of stock price ''S'' and time ''t'', ''r'' is the risk-free interest rate, and \sigma 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: :\frac + \frac\sigma^2 S^2 \frac = rV - rS\frac 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 :dS = \mu S\,dt + \sigma S\,dW\, 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 \sigma^2 dt . The payoff of an option (or any derivative contingent to stock ) V(S,T) at maturity is known. To find its value at an earlier time we need to know how V evolves as a function of S and t. 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 :dV = \left(\mu S \frac + \frac + \frac\sigma^2 S^2 \frac\right)dt + \sigma S \frac\,dW Now consider a portfolio \Pi consisting of a short option and / long shares at time t. The value of these holdings is :\Pi = -V + \fracS As \frac changes with time, the position in S is continually updated. We implicitly assume that the portfolio contains a cash account to accommodate buying and selling shares S, making the portfolio self-financing. Therefore, we only need to consider the total profit or loss from changes in the values of the holdings: :d\Pi = -dV + \fracdS Substituting dS and dV into the expression for d\Pi: :d \Pi = \left(-\frac - \frac\sigma^2 S^2 \frac\right)d t Over a time period ,t+\Delta t/math>, for \Delta t small enough, we see that :\Delta \Pi = \left(-\frac - \frac\sigma^2 S^2 \frac\right)\Delta t Note that the d W terms have vanished. Thus uncertainty has been eliminated and the portfolio is effectively riskless, i.e. a delta-hedge. The rate of return on this portfolio must be equal to the rate of return on any other riskless instrument; otherwise, there would be opportunities for arbitrage. Now assuming the risk-free rate of return is r we must have over the time period ,t+\Delta t/math>: :\Delta \Pi = r\Pi\,\Delta t If we now substitute our formulas for \Delta\Pi and \Pi we obtain: :\left(-\frac - \frac\sigma^2 S^2 \frac\right)\Delta t = r\left(-V + S\frac\right)\Delta t Simplifying, we arrive at the Black–Scholes partial differential equation: :\frac + rS\frac + \frac\sigma^2 S^2 \frac = rV With the assumptions of 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, ...
, this second order partial differential equation holds for any type of option as long as its price function V is twice differentiable with respect to S and once with respect to t.


Alternative derivation

Here is an alternative derivation that can be utilized in situations where it is initially unclear what the hedging portfolio should be. (For a reference, see 6.4 of Shreve vol II). In the Black–Scholes model, assuming we have picked the risk-neutral probability measure, the underlying stock price ''S''(''t'') is assumed to evolve as a geometric Brownian motion: : \frac = r\ dt + \sigma dW(t) Since this stochastic differential equation (SDE) shows the stock price evolution is Markovian, any derivative on this underlying is a function of time ''t'' and the stock price at the current time, ''S''(''t''). Then an application of Itô's lemma gives an SDE for the discounted derivative process e^V(t, S(t)), which should be a martingale. In order for that to hold, the drift term must be zero, which implies the Black—Scholes PDE. This derivation is basically an application of the Feynman–Kac formula and can be attempted whenever the underlying asset(s) evolve according to given SDE(s).


Solving methods

Once the Black–Scholes PDE, with boundary and terminal conditions, is derived for a derivative, the PDE can be solved numerically using standard methods of numerical analysis, such as a type of
finite difference method In numerical analysis, finite-difference methods (FDM) are a class of numerical techniques for solving differential equations by approximating Derivative, derivatives with Finite difference approximation, finite differences. Both the spatial doma ...
. In certain cases, it is possible to solve for an exact formula, such as in the case of a European call, which was done by Black and Scholes. The solution is conceptually simple. Since in the Black–Scholes model, the underlying stock price S_t follows a geometric Brownian motion, the distribution of S_T, conditional on its price S_t at time t, is a log-normal distribution. Then the price of the derivative is just discounted expected payoff E S_t /math>, which may be computed analytically when the payoff function K is analytically tractable, or numerically if not. To do this for a call option, recall the PDE above has
boundary condition In the study of differential equations, a boundary-value problem is a differential equation subjected to constraints called boundary conditions. A solution to a boundary value problem is a solution to the differential equation which also satis ...
s :\begin C(0, t) &= 0\textt \\ C(S, t) & \sim S - K e^\textS \rightarrow \infty \\ C(S, T) &= \max\ \end The last condition gives the value of the option at the time that the option matures. Other conditions are possible as ''S'' goes to 0 or infinity. For example, common conditions utilized in other situations are to choose delta to vanish as ''S'' goes to 0 and gamma to vanish as ''S'' goes to infinity; these will give the same formula as the conditions above (in general, differing boundary conditions will give different solutions, so some financial insight should be utilized to pick suitable conditions for the situation at hand). The solution of the PDE gives the value of the option at any earlier time, \mathbb\left max\\right/math>. To solve the PDE we recognize that it is a Cauchy–Euler equation which can be transformed into a
diffusion equation The diffusion equation is a parabolic partial differential equation. In physics, it describes the macroscopic behavior of many micro-particles in Brownian motion, resulting from the random movements and collisions of the particles (see Fick's l ...
by introducing the change-of-variable transformation :\begin \tau &= T - t \\ u &= Ce^ \\ x &= \ln\left(\frac\right) + \left(r - \frac\sigma^2\right)\tau \end Then the Black–Scholes PDE becomes a
diffusion equation The diffusion equation is a parabolic partial differential equation. In physics, it describes the macroscopic behavior of many micro-particles in Brownian motion, resulting from the random movements and collisions of the particles (see Fick's l ...
:\frac = \frac\sigma^\frac The terminal condition C(S, T) = \max\ now becomes an initial condition :u(x, 0) = u_0(x) := K(e^ - 1) = K\left(e^-1\right)H(x) , where ''H''(''x'') is the
Heaviside step function The Heaviside step function, or the unit step function, usually denoted by or (but sometimes , or ), is a step function named after Oliver Heaviside, the value of which is zero for negative arguments and one for positive arguments. Differen ...
. The Heaviside function corresponds to enforcement of the boundary data in the ''S'', ''t'' coordinate system that requires when ''t'' = ''T'', :C(S,\,T)=0\quad \forall\;S < K , assuming both ''S'', ''K'' > 0. With this assumption, it is equivalent to the max function over all ''x'' in the real numbers, with the exception of ''x'' = 0. The equality above between the max function and the Heaviside function is in the sense of distributions because it does not hold for ''x'' = 0. Though subtle, this is important because the Heaviside function need not be finite at ''x'' = 0, or even defined for that matter. For more on the value of the Heaviside function at ''x'' = 0, see the section "Zero Argument" in the article
Heaviside step function The Heaviside step function, or the unit step function, usually denoted by or (but sometimes , or ), is a step function named after Oliver Heaviside, the value of which is zero for negative arguments and one for positive arguments. Differen ...
. Using the standard
convolution In mathematics (in particular, functional analysis), convolution is a operation (mathematics), mathematical operation on two function (mathematics), functions f and g that produces a third function f*g, as the integral of the product of the two ...
method for solving a
diffusion equation The diffusion equation is a parabolic partial differential equation. In physics, it describes the macroscopic behavior of many micro-particles in Brownian motion, resulting from the random movements and collisions of the particles (see Fick's l ...
given an initial value function, ''u''(''x'', 0), we have :u(x, \tau) = \frac\int_^dy , which, after some manipulation, yields :u(x, \tau) = Ke^N(d_+) - KN(d_-) , where N(\cdot) is the standard normal
cumulative distribution function In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable X, or just distribution function of X, evaluated at x, is the probability that X will take a value less than or equal to x. Ever ...
and :\begin d_+ &= \frac \left left(x + \frac \sigma^\tau\right) + \frac \sigma^2 \tau\right\\ d_- &= \frac \left left(x + \frac \sigma^\tau\right) - \frac \sigma^2 \tau\right. \end These are the same solutions (up to time translation) that were obtained by Fischer Black in 1976.See equation (16) in Reverting u, x, \tau to the original set of variables yields the above stated solution to the Black–Scholes equation. :The asymptotic condition can now be realized. :u(x,\,\tau) \overset Ke^x, which gives simply ''S'' when reverting to the original coordinates. :\lim_ N(x) = 1 .


See also

*
Bachelier model The Bachelier model is a model of an asset price under Brownian motion presented by Louis Bachelier on his PhD thesis ''The Theory of Speculation'' (''Théorie de la spéculation'', published 1900). It is also called "Normal Model" equivalently ( ...
- uses arithmetic Brownian motion instead of geometric


References

{{DEFAULTSORT:Black-Scholes equation Mathematical finance Financial models Partial differential equations