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.
[1] It is an important example of stochastic processes satisfying a stochastic differential equation (SDE); in particular, it is used in mathematical finance to model stock prices in the Black–Scholes model.
A stochastic process St is said to follow a GBM if it satisfies the following stochastic differential equation (SDE): where
is a Wiener process or Brownian motion, and
The former parameter is used to model deterministic trends, while the latter parameter models unpredictable events occurring during the motion.
For an arbitrary initial value S0 the above SDE has the analytic solution (under Itô's interpretation): The derivation requires the use of Itô calculus.
Applying Itô's formula leads to where
in the above equation and simplifying we obtain Taking the exponential and multiplying both sides by
are real constants and for an initial condition
, is called an Arithmetic Brownian Motion (ABM).
This was the model postulated by Louis Bachelier in 1900 for stock prices, in the first published attempt to model Brownian motion, known today as Bachelier model.
As was shown above, the ABM SDE can be obtained through the logarithm of a GBM via Itô's formula.
Similarly, a GBM can be obtained by exponentiation of an ABM through Itô's formula.
(for any value of t) is a log-normally distributed random variable with expected value and variance given by[2] They can be derived using the fact that
is a martingale, and that The probability density function of
is: To derive the probability density function for GBM, we must use the Fokker-Planck equation to evaluate the time evolution of the PDF: where
To simplify the computation, we may introduce a logarithmic transform
, leading to the form of GBM: Then the equivalent Fokker-Planck equation for the evolution of the PDF becomes: Define
which has the solution given by the heat kernel: Plugging in the original variables leads to the PDF for GBM: 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(St).
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 with f(S) = log(S) gives It follows that
This result can also be derived by applying the logarithm to the explicit solution of GBM: Taking the expectation yields the same result as above:
GBM can be extended to the case where there are multiple correlated price paths.
For the multivariate case, this implies that A multivariate formulation that maintains the driving Brownian motions
[4] Some of the arguments for using GBM to model stock prices are: However, GBM is not a completely realistic model, in particular it falls short of reality in the following points: Apart from modeling stock prices, Geometric Brownian motion has also found applications in the monitoring of trading strategies.
[5] In an attempt to make GBM more realistic as a model for stock prices, also in relation to the volatility smile problem, one can drop the assumption that the volatility (
If we assume that the volatility is a deterministic function of the stock price and time, this is called a local volatility model.
A straightforward extension of the Black Scholes GBM is a local volatility SDE whose distribution is a mixture of distributions of GBM, the lognormal mixture dynamics, resulting in a convex combination of Black Scholes prices for options.
[3][6][7][8] 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 for example the Heston model.
