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.

Contents

• 
  1 Technical definition: the SDE
  


• 
  2 Solving the SDE
  


• 
  3 Properties
  


• 
  4 Simulating sample paths
  


• 
  5 Multivariate version
  


• 
  6 Use in finance
  


• 
  7 Extensions
  


• 
  8 See also
  


• 
  9 References
  


• 
  10 External links
  

Technical definition: the SDE

A stochastic process S_t is said to follow a GBM if it satisfies the following stochastic differential equation (SDE):

dSt=μStdt+σStdWt{displaystyle dS_{t}=mu S_{t},dt+sigma S_{t},dW_{t}}

where Wt{displaystyle W_{t}} is a Wiener process or Brownian motion, and μ{displaystyle mu } ('the percentage drift') and σ{displaystyle 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 S₀ the above SDE has the analytic solution (under Itô's interpretation):

St=S0exp⁡((μ−σ22)t+σWt).{displaystyle S_{t}=S_{0}exp left(left(mu -{frac {sigma ^{2}}{2}}right)t+sigma W_{t}right).}

To arrive at this formula, we will divide the SDE by St{displaystyle S_{t}} in order to have our choice random variable be on only one side. From there we write the previous equation in Itō integral form:

∫0tdStSt=μt+σWt,assuming W0=0.{displaystyle int _{0}^{t}{frac {dS_{t}}{S_{t}}}=mu ,t+sigma ,W_{t},,qquad {text{assuming }}W_{0}=0,.}

Of course, dStSt{displaystyle {frac {dS_{t}}{S_{t}}}} looks related to the derivative of ln⁡St{displaystyle ln S_{t}}. However, St{displaystyle S_{t}} is an Itō process which requires the use of Itō calculus. Applying Itō's formula leads to

d(ln⁡St)=dStSt−121St2dStdSt{displaystyle d(ln S_{t})={frac {dS_{t}}{S_{t}}}-{frac {1}{2}},{frac {1}{S_{t}^{2}}},dS_{t}dS_{t}}

Where dStdSt{displaystyle dS_{t}dS_{t}} is the quadratic variation of the SDE. This can also be written as d[S]t{displaystyle d[S]_{t}} or ⟨S.⟩t.{displaystyle leftlangle S_{.}rightrangle _{t},.}. In this case we have:

dStdSt=σ2St2dt.{displaystyle dS_{t}dS_{t},=,sigma ^{2},S_{t}^{2},dt.}

Plugging the value of dSt{displaystyle dS_{t}} in the above equation and simplifying we obtain

ln⁡StS0=(μ−σ22)t+σWt.{displaystyle ln {frac {S_{t}}{S_{0}}}=left(mu -{frac {sigma ^{2}}{2}},right)t+sigma W_{t},.}

Taking the exponential and multiplying both sides by S0{displaystyle S_{0}} gives the solution claimed above.

Properties

The above solution St{displaystyle S_{t}} (for any value of t) is a log-normally distributed random variable with expected value and variance given by^[2]

E(St)=S0eμt,{displaystyle mathbb {E} (S_{t})=S_{0}e^{mu t},}

Var⁡(St)=S02e2μt(eσ2t−1).{displaystyle operatorname {Var} (S_{t})=S_{0}^{2}e^{2mu t}left(e^{sigma ^{2}t}-1right).}

They can be derived using the fact that Zt=exp⁡(σWt−12σ2t){displaystyle Z_{t}=exp left(sigma W_{t}-{frac {1}{2}}sigma ^{2}tright)} is a martingale, and that

E[exp⁡(2σWt−σ2t)|Fs]=eσ2(t−s)exp⁡(2σWs−σ2s),∀0≤s<t.{displaystyle mathbb {E} left[exp left(2sigma W_{t}-sigma ^{2}tright)|{mathcal {F}}_{s}right]=e^{sigma ^{2}(t-s)}exp left(2sigma W_{s}-sigma ^{2}sright),quad forall 0leq s<t.}

The probability density function of St{displaystyle S_{t}} is:

fSt(s;μ,σ,t)=12π1sσtexp⁡(−(ln⁡s−ln⁡S0−(μ−12σ2)t)22σ2t).{displaystyle f_{S_{t}}(s;mu ,sigma ,t)={frac {1}{sqrt {2pi }}},{frac {1}{ssigma {sqrt {t}}}},exp left(-{frac {left(ln s-ln S_{0}-left(mu -{frac {1}{2}}sigma ^{2}right)tright)^{2}}{2sigma ^{2}t}}right).}

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 with f(S) = log(S) gives

dlog⁡(S)=f′(S)dS+12f′′(S)S2σ2dt=1S(σSdWt+μSdt)−12σ2dt=σdWt+(μ−σ2/2)dt.{displaystyle {begin{alignedat}{2}dlog(S)&=f^{prime }(S),dS+{frac {1}{2}}f^{prime prime }(S)S^{2}sigma ^{2},dt\&={frac {1}{S}}left(sigma S,dW_{t}+mu S,dtright)-{frac {1}{2}}sigma ^{2},dt\&=sigma ,dW_{t}+(mu -sigma ^{2}/2),dt.end{alignedat}}}

It follows that Elog⁡(St)=log⁡(S0)+(μ−σ2/2)t{displaystyle mathbb {E} 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:

log⁡(St)=log⁡(S0exp⁡((μ−σ22)t+σWt))=log⁡(S0)+(μ−σ22)t+σWt.{displaystyle {begin{alignedat}{2}log(S_{t})&=log left(S_{0}exp left(left(mu -{frac {sigma ^{2}}{2}}right)t+sigma W_{t}right)right)\&=log(S_{0})+left(mu -{frac {sigma ^{2}}{2}}right)t+sigma W_{t}.end{alignedat}}}

Taking the expectation yields the same result as above: Elog⁡(St)=log⁡(S0)+(μ−σ2/2)t{displaystyle mathbb {E} log(S_{t})=log(S_{0})+(mu -sigma ^{2}/2)t}.

Simulating sample paths

#python code for the plot



import numpy as np

import pandas as pd

import matplotlib.pyplot as plt



mu=1

n=50

dt=0.1

x0=100

x=pd.DataFrame()

np.random.seed(1)



for sigma in np.arange(0.8,2,0.2):

    step=np.exp((mu-sigma**2/2)*dt)*np.exp(sigma*np.random.normal(0,np.sqrt(dt),(1,n)))

    temp=pd.DataFrame(x0*step.cumprod())

    x=pd.concat([x,temp],axis=1)



x.columns=np.arange(0.8,2,0.2)

plt.plot(x)

plt.legend(x.columns)

plt.xlabel('t')

plt.ylabel('X')

plt.title('Realizations of Geometric Brownian Motion with different variancesn 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

dSti=μiStidt+σiStidWti{displaystyle 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 E(dWtidWtj)=ρi,jdt{displaystyle mathbb {E} (dW_{t}^{i}dW_{t}^{j})=rho _{i,j}dt} where ρi,i=1{displaystyle rho _{i,i}=1}.

For the multivariate case, this implies that

Cov(Sti,Stj)=S0iS0je(μi+μj)t(eρi,jσiσjt−1){displaystyle mathrm {Cov} (S_{t}^{i},S_{t}^{j})=S_{0}^{i}S_{0}^{j}e^{(mu _{i}+mu _{j})t}left(e^{rho _{i,j}sigma _{i}sigma _{j}t}-1right)}.

Use in finance

Main article: Black–Scholes model

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.^[3]

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.^[3]
  


• 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), 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).

Extensions

In an attempt to make GBM more realistic as a model for stock prices, one can drop the assumption that the volatility (σ{displaystyle sigma }) 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.

See also

• Brownian surface

References

1. 
   ^ Ross, Sheldon M. (2014). "Variations on Brownian Motion". Introduction to Probability Models (11th ed.). Amsterdam: Elsevier. pp. 612–14. ISBN 978-0-12-407948-9..mw-parser-output cite.citation{font-style:inherit}.mw-parser-output q{quotes:"""""""'""'"}.mw-parser-output code.cs1-code{color:inherit;background:inherit;border:inherit;padding:inherit}.mw-parser-output .cs1-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-lock-limited a,.mw-parser-output .cs1-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Lock-gray-alt-2.svg/9px-Lock-gray-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Lock-red-alt-2.svg/9px-Lock-red-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration{color:#555}.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration span{border-bottom:1px dotted;cursor:help}.mw-parser-output .cs1-hidden-error{display:none;font-size:100%}.mw-parser-output .cs1-visible-error{font-size:100%}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration,.mw-parser-output .cs1-format{font-size:95%}.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-right{padding-right:0.2em}
   


2. 
   ^ Oksendal, Bernt K. (2002), Stochastic Differential Equations: An Introduction with Applications, Springer, p. 326, ISBN 3-540-63720-6
   


3. 
   ^ ^ab Hull, John (2009). "12.3". Options, Futures, and other Derivatives (7 ed.).
   

External links

• Geometric Brownian motion models for stock movement except in rare events.


• R and C# Simulation of a Geometric Brownian Motion


• Excel Simulation of a Geometric Brownian Motion to simulate Stock Prices


• 
  "Interactive Web Application: Stochastic Processes used in Quantitative Finance".
  

v t e Stochastic processes
Discrete time	Bernoulli process Branching process Chinese restaurant process Galton–Watson process Independent and identically distributed random variables Markov chain Moran process Random walk Loop-erased Self-avoiding Biased Maximal entropy
Continuous time	Bessel process Birth–death process Brownian motion Bridge Excursion Fractional Geometric Meander Cauchy process Contact process Continuous-time random walk Cox process Diffusion process Empirical process Feller process Fleming–Viot process Gamma process Hunt process Interacting particle systems Itô diffusion Itô process Jump diffusion Jump process Lévy process Local time Markov additive process McKean–Vlasov process Ornstein–Uhlenbeck process Poisson process Compound Non-homogeneous Point process Schramm–Loewner evolution Semimartingale Sigma-martingale Stable process Superprocess Telegraph process Variance gamma process Wiener process Wiener sausage
Both	Branching process Galves–Löcherbach model Gaussian process Hidden Markov model (HMM) Markov process Martingale Differences Local Sub- Super- Random dynamical system Regenerative process Renewal process Stochastic chains with memory of variable length White noise
Fields and other	Dirichlet process Gaussian random field Gibbs measure Hopfield model Ising model Potts model Boolean network Markov random field Percolation Pitman–Yor process Point process Cox Poisson Random field Random graph
Time series models	Autoregressive conditional heteroskedasticity (ARCH) model Autoregressive integrated moving average (ARIMA) model Autoregressive (AR) model Autoregressive–moving-average (ARMA) model Generalized autoregressive conditional heteroskedasticity (GARCH) model Moving-average (MA) model
Financial models	Black–Derman–Toy Black–Karasinski Black–Scholes Chen Constant elasticity of variance (CEV) Cox–Ingersoll–Ross (CIR) Garman–Kohlhagen Heath–Jarrow–Morton (HJM) Heston Ho–Lee Hull–White LIBOR market Rendleman–Bartter SABR volatility Vašíček Wilkie
Actuarial models	Bühlmann Cramér–Lundberg Risk process Sparre–Anderson
Queueing models	Bulk Fluid Generalized queueing network M/G/1 M/M/1 M/M/c
Properties	Càdlàg paths Continuous Continuous paths Ergodic Exchangeable Feller-continuous Gauss–Markov Markov Mixing Piecewise deterministic Predictable Progressively measurable Self-similar Stationary Time-reversible
Limit theorems	Central limit theorem Donsker's theorem Doob's martingale convergence theorems Ergodic theorem Fisher–Tippett–Gnedenko theorem Large deviation principle Law of large numbers (weak/strong) Law of the iterated logarithm Maximal ergodic theorem Sanov's theorem
Inequalities	Burkholder–Davis–Gundy Doob's martingale Kunita–Watanabe
Tools	Cameron–Martin formula Convergence of random variables Doléans-Dade exponential Doob decomposition theorem Doob–Meyer decomposition theorem Doob's optional stopping theorem Dynkin's formula Feynman–Kac formula Filtration Girsanov theorem Infinitesimal generator Itô integral Itô's lemma Kolmogorov continuity theorem Kolmogorov extension theorem Lévy–Prokhorov metric Malliavin calculus Martingale representation theorem Optional stopping theorem Prokhorov's theorem Quadratic variation Reflection principle Skorokhod integral Skorokhod's representation theorem Skorokhod space Snell envelope Stochastic differential equation Tanaka Stopping time Stratonovich integral Uniform integrability Usual hypotheses Wiener space Classical Abstract
Disciplines	Actuarial mathematics Econometrics Ergodic theory Extreme value theory (EVT) Large deviations theory Mathematical finance Mathematical statistics Probability theory Queueing theory Renewal theory Ruin theory Statistics System on Chip design Stochastic analysis Time series analysis Machine learning
List of topics Category

This page is only for reference, If you need detailed information, please check here