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A geometric Brownian motion (GBM) (also known as exponential Brownian motion) is a continuous-time
stochastic process In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables. Stochastic processes are widely used as mathematical models of systems and phenomena that ap ...
in which the
logarithm In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a number  to the base  is the exponent to which must be raised, to produce . For example, since , the ''logarithm base'' 10 ...
of the randomly varying quantity follows a
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 ...
(also called a
Wiener process In mathematics, the Wiener process is a real-valued continuous-time stochastic process named in honor of American mathematician Norbert Wiener for his investigations on the mathematical properties of the one-dimensional Brownian motion. It is ...
) with
drift Drift or Drifts may refer to: Geography * Drift or ford (crossing) of a river * Drift, Kentucky, unincorporated community in the United States * In Cornwall, England: ** Drift, Cornwall, village ** Drift Reservoir, associated with the village ...
. It is an important example of stochastic processes satisfying a
stochastic differential equation A stochastic differential equation (SDE) is a differential equation in which one or more of the terms is a stochastic process, resulting in a solution which is also a stochastic process. SDEs are used to model various phenomena such as stock p ...
(SDE); in particular, it is used 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 ...
to model stock prices in 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� ...
.


Technical definition: the SDE

A stochastic process ''S''''t'' is said to follow a GBM if it satisfies the following
stochastic differential equation A stochastic differential equation (SDE) is a differential equation in which one or more of the terms is a stochastic process, resulting in a solution which is also a stochastic process. SDEs are used to model various phenomena such as stock p ...
(SDE): : dS_t = \mu S_t\,dt + \sigma S_t\,dW_t where W_t is a Wiener process or Brownian motion, and \mu ('the percentage drift') and \sigma ('the percentage volatility') are constants. The former is used to model deterministic trends, while the latter term is often used to model a set of unpredictable events occurring during this motion.


Solving the SDE

For an arbitrary initial value ''S''0 the above SDE has the analytic solution (under Itô's interpretation): : S_t = S_0\exp\left( \left(\mu - \frac \right)t + \sigma W_t\right). The derivation requires the use of Itô calculus. Applying Itô's formula leads to : d(\ln S_t) = (\ln S_t)' d S_t + \frac (\ln S_t)'' \,dS_t \,dS_t = \frac -\frac \,\frac \, dS_t \, dS_t where dS_t \, dS_t is the quadratic variation of the SDE. : d S_t \, d S_t \, = \, \sigma^2 \, S_t^2 \, d W_t^2 + 2 \sigma S_t^2 \mu \, d W_t \, d t + \mu^2 S_t^2 \, d t^2 When d t \to 0 , d t converges to 0 faster than d W_t, since d W_t^2 = O(d t) . So the above infinitesimal can be simplified by : d S_t \, d S_t \, = \, \sigma^2 \, S_t^2 \, dt Plugging the value of dS_t in the above equation and simplifying we obtain : \ln \frac = \left(\mu -\frac\,\right) t + \sigma W_t\,. Taking the exponential and multiplying both sides by S_0 gives the solution claimed above.


Properties

The above solution S_t (for any value of t) is a log-normally distributed
random variable A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the po ...
with
expected value In probability theory, the expected value (also called expectation, expectancy, mathematical expectation, mean, average, or first moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean of a ...
and
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 ...
given by :\operatorname(S_t)= S_0e^, :\operatorname(S_t)= S_0^2e^ \left( e^-1\right). They can be derived using the fact that Z_t = \exp\left(\sigma W_t - \frac\sigma^2 t\right) is a martingale, and that : \operatorname\left \exp\left(2\sigma W_t - \sigma^2 t\right) \mid \mathcal_s\right= e^ \exp\left(2\sigma W_s - \sigma^2 s\right),\quad \forall 0 \leq s < t. The
probability density function In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) ca ...
of S_t is: : f_(s; \mu, \sigma, t) = \frac\, \frac\, \exp \left( -\frac \right). To derive the probability density function for GBM, we must use the Fokker-Planck equation to evaluate the time evolution of the PDF: : + mu(t,S)p(t,S)= sigma^(t,S)p(t,S) \quad p(0,S) = \delta(S) where \delta(S) is the
Dirac delta function In mathematics, the Dirac delta distribution ( distribution), also known as the unit impulse, is a generalized function or distribution over the real numbers, whose value is zero everywhere except at zero, and whose integral over the enti ...
. To simplify the computation, we may introduce a logarithmic transform x = \log (S/S_), leading to the form of GBM: :dx = \left(\mu - \sigma^\right)dt + \sigma dW Then the equivalent Fokker-Planck equation for the evolution of the PDF becomes: : + \left(\mu - \sigma^\right) = \sigma^, \quad p(0,x) = \delta(x) Define V=\mu-\sigma^/2 and D=\sigma^/2. By introducing the new variables \xi = x-Vt and \tau = Dt, the derivatives in the Fokker-Planck equation may be transformed as: :\begin\partial_p &= D\partial_p - V\partial_p \\ \partial_p &= \partial_p \\ \partial_^p &= \partial_^p \end Leading to the new form of the Fokker-Planck equation: : = , \quad p(0,\xi) = \delta(\xi) However, this is the canonical form of the
heat equation In mathematics and physics, the heat equation is a certain partial differential equation. Solutions of the heat equation are sometimes known as caloric functions. The theory of the heat equation was first developed by Joseph Fourier in 1822 for ...
. which has the solution given by the
heat kernel In the mathematical study of heat conduction and diffusion, a heat kernel is the fundamental solution to the heat equation on a specified domain with appropriate boundary conditions. It is also one of the main tools in the study of the spectr ...
: :p(\tau,\xi) = \exp\left(- \right) Plugging in the original variables leads to the PDF for GBM: :p(t,S) = \exp\left\ When deriving further properties of GBM, use can be made of the SDE of which GBM is the solution, or the explicit solution given above can be used. For example, consider the stochastic process log(''S''''t''). This is an interesting process, because in the Black–Scholes model it is related to the log return of the stock price. Using
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 ...
with ''f''(''S'') = log(''S'') gives : \begin d\log(S) & = f'(S)\,dS + \frac f'' (S)S^2\sigma^2 \, dt \\ pt& = \frac \left( \sigma S\,dW_t + \mu S\,dt\right) - \frac\sigma^2\,dt \\ pt&= \sigma\,dW_t +(\mu-\sigma^2/2)\,dt. \end It follows that \operatorname \log(S_t)=\log(S_0)+(\mu-\sigma^2/2)t. This result can also be derived by applying the logarithm to the explicit solution of GBM: : \begin \log(S_t) &=\log\left(S_0\exp\left(\left(\mu - \frac \right)t + \sigma W_t\right)\right)\\ pt& =\log(S_0) +\left(\mu - \frac \right)t + \sigma W_t. \end Taking the expectation yields the same result as above: \operatorname \log(S_t)=\log(S_0)+(\mu-\sigma^2/2)t .


Simulating sample paths

# Python code for the plot import numpy as np import matplotlib.pyplot as plt mu = 1 n = 50 dt = 0.1 x0 = 100 np.random.seed(1) sigma = np.arange(0.8, 2, 0.2) x = np.exp( (mu - sigma ** 2 / 2) * dt + sigma * np.random.normal(0, np.sqrt(dt), size=(len(sigma), n)).T ) x = np.vstack( p.ones(len(sigma)), x x = x0 * x.cumprod(axis=0) plt.plot(x) plt.legend(np.round(sigma, 2)) plt.xlabel("$t$") plt.ylabel("$x$") plt.title( "Realizations of Geometric Brownian Motion with different variances\n $\mu=1$" ) plt.show()


Multivariate version

GBM can be extended to the case where there are multiple correlated price paths. Each price path follows the underlying process :dS_t^i = \mu_i S_t^i\,dt + \sigma_i S_t^i\,dW_t^i, where the Wiener processes are correlated such that \operatorname(dW_^i \,dW_^j) = \rho_ \, dt where \rho_ = 1. For the multivariate case, this implies that :\operatorname(S_t^i, S_t^j) = S_0^i S_0^j e^\left(e^-1\right).


Use in finance

Geometric Brownian motion is used to model stock prices in the Black–Scholes model and is the most widely used model of stock price behavior. Some of the arguments for using GBM to model stock prices are: *The expected returns of GBM are independent of the value of the process (stock price), which agrees with what we would expect in reality. *A GBM process only assumes positive values, just like real stock prices. *A GBM process shows the same kind of 'roughness' in its paths as we see in real stock prices. *Calculations with GBM processes are relatively easy. However, GBM is not a completely realistic model, in particular it falls short of reality in the following points: *In real stock prices, volatility changes over time (possibly
stochastically Stochastic (, ) refers to the property of being well described by a random probability distribution. Although stochasticity and randomness are distinct in that the former refers to a modeling approach and the latter refers to phenomena themselve ...
), but in GBM, volatility is assumed constant. *In real life, stock prices often show jumps caused by unpredictable events or news, but in GBM, the path is continuous (no discontinuity). Apart from modeling stock prices, Geometric Brownian motion has also found applications in the monitoring of trading strategies.


Extensions

In an attempt to make GBM more realistic as a model for stock prices, one can drop the assumption that the volatility (\sigma) is constant. If we assume that the volatility is a
deterministic Determinism is a philosophical view, where all events are determined completely by previously existing causes. Deterministic theories throughout the history of philosophy have developed from diverse and sometimes overlapping motives and cons ...
function of the stock price and time, this is called a local volatility model. If instead we assume that the volatility has a randomness of its own—often described by a different equation driven by a different Brownian Motion—the model is called a stochastic volatility model.


See also

*
Brownian surface A Brownian surface is a fractal surface generated via a fractal elevation function. As with Brownian motion, Brownian surfaces are named after 19th-century biologist Robert Brown. Example For instance, in the three-dimensional case, where two ...


References


External links


Geometric Brownian motion models for stock movement except in rare events.Excel Simulation of a Geometric Brownian Motion to simulate Stock Prices
*
Non-Newtonian calculus website

Trading Strategy Monitoring: Modeling the PnL as a Geometric Brownian Motion
{{DEFAULTSORT:Geometric Brownian Motion Wiener process Non-Newtonian calculus Articles with example Python (programming language) code