Difference between revisions of "Geometric Brownian motion"

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en>Zfeinst
m (does not add any understanding by saying it is 'like' brownian motion with drift)
 
 
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:<math> dS_t = \mu S_t\,dt + \sigma S_t\,dW_t </math>
:<math> dS_t = \mu S_t\,dt + \sigma S_t\,dW_t </math>


where <math> W_t </math> is a [[Wiener process|Wiener process or Brownian motion]] and <math> \mu </math> ('the percentage drift') and <math> \sigma </math> ('the percentage volatility') are constants. The latter term is often used to model a set of unpredictable events occurring during this motion, while the former is used to model deterministic trends.
where <math> W_t </math> is a [[Wiener process|Wiener process or Brownian motion]] and <math> \mu </math> ('the percentage drift') and <math> \sigma </math> ('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==
==Solving the SDE==
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: <math> \int_0^t \frac{d S_t}{S_t} = \mu \, t + \sigma\, W_t \,, \qquad\text{assuming }W_0=0\,.</math>
: <math> \int_0^t \frac{d S_t}{S_t} = \mu \, t + \sigma\, W_t \,, \qquad\text{assuming }W_0=0\,.</math>


Of course, <math>\frac{d S_t}{S_t}</math> looks like having a lot to do with the derivative of <math>\ln S_t</math>; however, <math>S_t</math> being an Itō process, we need to use [[Itō calculus]]: by [[Itō's formula]], we have  
Of course, <math>\frac{d S_t}{S_t}</math> looks related to the derivative of <math>\ln S_t</math>; however, <math>S_t</math> being an Itō process, we need to use [[Itō calculus]]: by [[Itō's formula]], we have  


: <math>d(\ln S_t) = \frac{d S_t}{S_t} -\frac{1}{2} \, \sigma^2 \, dt\,.</math>
: <math>d(\ln S_t) = \frac{d S_t}{S_t} -\frac{1}{2} \, \sigma^2 \, dt\,.</math>
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Plugging back to the equation we got from the SDE, we obtain  
Plugging back to the equation we got from the SDE, we obtain  


: <math>\ln \frac{S_t}{S_0} = \left(\mu -\frac{1}{2}\,\sigma^2\right) t + \sigma W_t\,.</math>
: <math>\ln \frac{S_t}{S_0} = \left(\mu -\frac{\sigma^2}{2}\,\right) t + \sigma W_t\,.</math>


Exponentiating gives the solution claimed above.
Exponentiating gives the solution claimed above.


==Properties of GBM==
==Properties==


The above solution <math> S_t </math> (for any value of t) is a [[log-normal distribution|log-normally distributed]] [[random variable]] with [[expected value]] and [[variance]] given by<ref>{{Citation
The above solution <math> S_t </math> (for any value of t) is a [[log-normal distribution|log-normally distributed]] [[random variable]] with [[expected value]] and [[variance]] given by<ref>{{Citation
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  | isbn = 3-540-63720-6
  | isbn = 3-540-63720-6
}}</ref>  
}}</ref>  
:<math>\mathbb{E}(S_t)= S_0e^{\mu t},</math>  
:<math>\mathbb{E}(S_t)= S_0e^{\mu t},</math>  
:<math>\operatorname{Var}(S_t)= S_0^2e^{2\mu t} \left( e^{\sigma^2 t}-1\right),</math>
:<math>\operatorname{Var}(S_t)= S_0^2e^{2\mu t} \left( e^{\sigma^2 t}-1\right),</math>
that is the [[probability density function]] of a ''S<sub>t</sub>'' is:
that is the [[probability density function]] of a ''S<sub>t</sub>'' is:
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Taking the expectation yields the same result as above:  <math>\mathbb{E} \log(S_t)=\log(S_0)+(\mu-\sigma^2/2)t</math>.
Taking the expectation yields the same result as above:  <math>\mathbb{E} \log(S_t)=\log(S_0)+(\mu-\sigma^2/2)t</math>.


== Multivariate Geometric Brownian motion ==
== Multivariate version ==


GBM can be extended to the cast where there are multiple correlated price paths.
GBM can be extended to the case where there are multiple correlated price paths.


Each price path follows the underlying process
Each price path follows the underlying process
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For the multivariate case, this implies that
For the multivariate case, this implies that
:<math>\mathrm{Cov}(S_{t}^i, S_{t}^j) = e^{(\mu_i + \mu_j) t }\left(e^{\rho_{i,j} \sigma_i \sigma_j t}-1\right)</math>.
:<math>\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}-1\right)</math>.


==Use of GBM in finance==
==Use in finance==
{{main|Black–Scholes model}}
{{main|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.<ref name="Hull">{{cite book|title=Options, Futures, and other Derivatives|edition=7|first=John|last=Hull|year=2009|chapter=12.3}}</ref>   
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.<ref name="Hull">{{cite book|title=Options, Futures, and other Derivatives|edition=7|first=John|last=Hull|year=2009|chapter=12.3}}</ref>   


Some of the arguments for using GBM to model stock prices are:
Some of the arguments for using GBM to model stock prices are:
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However, GBM is not a completely realistic model, in particular it falls short of reality in the following points:
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 [[stochastic volatility|stochastically]]), but in GBM, volatility is assumed constant.
*In real stock prices, volatility changes over time (possibly [[stochastic volatility|stochastically]]), but in GBM, volatility is assumed constant.
*In real stock prices, returns are usually not normally distributed (real stock returns have higher [[kurtosis]] ('fatter tails'), which means there is a higher chance of large price changes).<ref>{{cite book|title=Paul Wilmott on Quantitative Finance|edition=2|first=Paul|last=Wilmott|year=2006|chapter=16.4}}</ref>
*In real stock prices, returns are usually not normally distributed (real stock returns have higher [[kurtosis]] ('fatter tails'), which means there is a higher chance of large price changes. In addition, returns have negative [[skewness]]).<ref>{{cite book|title=Paul Wilmott on Quantitative Finance|edition=2|first=Paul|last=Wilmott|year=2006|chapter=16.4}}</ref>


==Extensions of GBM==
==Extensions==
In an attempt to make GBM more realistic as a model for stock prices, one can drop the assumption that the volatility (<math>\sigma</math>) 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.
In an attempt to make GBM more realistic as a model for stock prices, one can drop the assumption that the volatility (<math>\sigma</math>) 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.


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== External links ==
== External links ==
* [http://www.math.nyu.edu/financial_mathematics/content/02_financial/02.html Geometric Brownian motion models for stock movement except in rare events.]
* [http://www.math.nyu.edu/financial_mathematics/content/02_financial/02.html Geometric Brownian motion models for stock movement except in rare events.]
*[http://forevermar.com/Brownian.php R and C# Simulation of a Geometric Brownian Motion ]
*[http://excelandfinance.com/simulation-of-stock-prices/brownian-motion/ Excel Simulation of a Geometric Brownian Motion to simulate Stock Prices ]
{{Stochastic processes}}


{{DEFAULTSORT:Geometric Brownian Motion}}
{{DEFAULTSORT:Geometric Brownian Motion}}
[[Category:Stochastic processes]]
[[Category:Stochastic processes]]
[[de:Geometrische brownsche Bewegung]]
[[it:Moto browniano geometrico]]
[[he:תנועה בראונית גאומטרית]]
[[ja:幾何ブラウン運動]]
[[pt:Movimento browniano geométrico]]
[[ru:Геометрическое броуновское движение]]
[[sv:Geometrisk Brownsk rörelse]]
[[uk:Геометричний броунівський рух]]
[[zh:几何布朗运动]]

Latest revision as of 18:20, 5 March 2014

Two sample paths of Geometric Brownian motion, with different parameters. The blue line has larger drift, the green line has larger variance.

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.

Technical definition: the SDE

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 percentage drift') and ('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 Itō's interpretation):

To arrive at this formula, let us divide the SDE by , and write it in Itō integral form:

Of course, looks related to the derivative of ; however, being an Itō process, we need to use Itō calculus: by Itō's formula, we have

Plugging back to the equation we got from the SDE, we obtain

Exponentiating gives the solution claimed above.

Properties

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

that is the probability density function of a St is:

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 Itō'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: .

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

{{#invoke:main|main}} 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 stock prices, returns are usually not normally distributed (real stock returns have higher kurtosis ('fatter tails'), which means there is a higher chance of large price changes. In addition, returns have negative skewness).[4]

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.

See also

References

  1. {{#invoke:citation/CS1|citation |CitationClass=book }}
  2. {{#invoke:citation/CS1|citation |CitationClass=citation }}
  3. 3.0 3.1 {{#invoke:citation/CS1|citation |CitationClass=book }}
  4. {{#invoke:citation/CS1|citation |CitationClass=book }}

External links

Template:Stochastic processes