For an arbitrary initial valueS0 the above SDE has the analytic solution : The derivation requires the use of Itô calculus. Applying Itô's formula leads to where is the quadratic variation of the SDE. When, converges to 0 faster than, since. So the above infinitesimal can be simplified by Plugging the value of in the above equation and simplifying we obtain Taking the exponential and multiplying both sides by gives the solution claimed above.
Properties
The above solution is a log-normally distributedrandom variable with expected value and variance given by 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 is the Dirac delta function. 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 and. By introducing the new variables and, the derivatives in the Fokker-Planck equation may be transformed as: Leading to the new form of the Fokker-Planck equation: However, this is the canonical form of the heat equation. 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. This is an interesting process, because in the Black–Scholes model it is related to the log return of thestock price. Using Itô's lemma with f = log 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: .
import numpy as np import matplotlib.pyplot as plt mu = 1 n = 50 dt = 0.1 x0 = 100 np.random.seed sigma = np.arange x = np.exp * dt + sigma * np.random.normal.T x = np.vstack x = x0 * x.cumprod plt.plot plt.legend plt.xlabel plt.ylabel plt.title
Multivariate version
GBM can be extended to the case where there are multiple correlated price paths. Each price path follows the underlying process where the Wiener processes are correlated such that where. For the multivariate case, this implies that
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, 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, 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.
Extensions
In an attempt to make GBM more realistic as a model for stock prices, one can drop the assumption that the volatility is constant. If we assume that the volatility is a deterministic 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.
