Geometric Brownian motion

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

:$dS\_t\; =\; \backslash mu\; S\_t\backslash ,dt\; +\; \backslash sigma\; S\_t\backslash ,dW\_t$

where $W\_t$ is a Wiener process or Brownian motion, and $\backslash mu$ ('the percentage drift') and $\backslash sigma$ ('the percentage volatility') are constants.

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):

: $S\_t\; =\; S\_0\backslash exp\backslash left(\; \backslash left(\backslash mu\; -\; \backslash frac\{\backslash sigma^2\}\{2\}\; \backslash right)t\; +\; \backslash sigma\; W\_t\backslash right).$

The derivation requires the use of Itô calculus. Applying Itô's formula leads to

: $d(\backslash ln\; S\_t)\; =\; (\backslash ln\; S\_t)\text{'}\; d\; S\_t\; +\; \backslash frac\{1\}\{2\}\; (\backslash ln\; S\_t)\text{'}\text{'}\; \backslash ,dS\_t\; \backslash ,dS\_t$

= \frac{d S_t}{S_t} -\frac{1}{2} \,\frac{1}{S_t^2} \, dS_t \, dS_t

where $dS\_t\; \backslash ,\; dS\_t$ is the quadratic variation of the SDE.

:$d\; S\_t\; \backslash ,\; d\; S\_t\; \backslash ,\; =\; \backslash ,\; \backslash sigma^2\; \backslash ,\; S\_t^2\; \backslash ,\; d\; W\_t^2\; +\; 2\; \backslash sigma\; S\_t^2\; \backslash mu\; \backslash ,\; d\; W\_t\; \backslash ,\; d\; t\; +\; \backslash mu^2\; S\_t^2\; \backslash ,\; d\; t^2$

When $d\; t\; \backslash to\; 0$, $d\; t$ converges to 0 faster than $d\; W\_t$,

since $d\; W\_t^2\; =\; O(d\; t)$. So the above infinitesimal can be simplified by

:$d\; S\_t\; \backslash ,\; d\; S\_t\; \backslash ,\; =\; \backslash ,\; \backslash sigma^2\; \backslash ,\; S\_t^2\; \backslash ,\; dt$

Plugging the value of $dS\_t$ in the above equation and simplifying we obtain

: $\backslash ln\; \backslash frac\{S\_t\}\{S\_0\}\; =\; \backslash left(\backslash mu\; -\backslash frac\{\backslash sigma^2\}\{2\}\backslash ,\backslash right)\; t\; +\; \backslash sigma\; W\_t\backslash ,.$

Taking the exponential and multiplying both sides by $S\_0$ gives the solution claimed above.

The process for $X\_t\; =\; \backslash ln\; \backslash frac\{S\_t\}\{S\_0\}$, satisfying the SDE

: $d\; X\_t\; =\; \backslash left(\backslash mu\; -\backslash frac\{\backslash sigma^2\}\{2\}\backslash ,\backslash right)\; dt\; +\; \backslash sigma\; dW\_t\backslash ,\; ,$

or more generally the process solving the SDE

: $d\; X\_t\; =\; m\backslash ,\; dt\; +\; v\backslash ,\; dW\_t\backslash ,\; ,$

where $m$ and $v\; >0$ are real constants and for an initial condition $X\_0$, 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.

The above solution $S\_t$ (for any value of t) is a log-normally distributed random variable with expected value and variance given by{{Citation

| title = Stochastic Differential Equations: An Introduction with Applications

|publisher=Springer

|author = Øksendal, Bernt K.

| year = 2002

| pages = 326

| isbn = 3-540-63720-6

}}

:$\backslash operatorname\{E\}(S\_t)=\; S\_0e^\{\backslash mu\; t\},$

:$\backslash operatorname\{Var\}(S\_t)=\; S\_0^2e^\{2\backslash mu\; t\}\; \backslash left(\; e^\{\backslash sigma^2\; t\}-1\backslash right).$

They can be derived using the fact that $Z\_t\; =\; \backslash exp\backslash left(\backslash sigma\; W\_t\; -\; \backslash frac\{1\}\{2\}\backslash sigma^2\; t\backslash right)$ is a martingale, and that

:

The probability density function of $S\_t$ is:

: $f\_\{S\_t\}(s;\; \backslash mu,\; \backslash sigma,\; t)\; =\; \backslash frac\{1\}\{\backslash sqrt\{2\; \backslash pi\}\}\backslash ,\; \backslash frac\{1\}\{s\; \backslash sigma\; \backslash sqrt\{t\}\}\backslash ,\; \backslash exp\; \backslash left(\; -\backslash frac\{\; \backslash left(\; \backslash ln\; s\; -\; \backslash ln\; S\_0\; -\; \backslash left(\; \backslash mu\; -\; \backslash frac\{1\}\{2\}\; \backslash sigma^2\; \backslash right)\; t\; \backslash right)^2\}\{2\backslash sigma^2\; t\}\; \backslash right).$

{{Collapse top|title=Derivation of GBM probability density function}}

To derive the probability density function for GBM, we must use the Fokker-Planck equation to evaluate the time evolution of the PDF:

:$\{\backslash partial\; p\backslash over\{\backslash partial\; t\}\}\; +\; \{\backslash partial\backslash over\{\backslash partial\; S\}\}[\backslash mu(t,S)p(t,S)]\; =\; \{1\backslash over\{2\}\}\{\backslash partial^\{2\}\backslash over\{\backslash partial\; S^\{2\}\}\}[\backslash sigma^\{2\}(t,S)p(t,S)],\; \backslash quad\; p(0,S)\; =\; \backslash delta(S)$

where $\backslash delta(S)$ is the Dirac delta function. To simplify the computation, we may introduce a logarithmic transform $x\; =\; \backslash log\; (S/S\_\{0\})$, leading to the form of GBM:

:$dx\; =\; \backslash left(\backslash mu\; -\; \{1\backslash over\{2\}\}\backslash sigma^\{2\}\backslash right)dt\; +\; \backslash sigma\; dW$

Then the equivalent Fokker-Planck equation for the evolution of the PDF becomes:

:$\{\backslash partial\; p\backslash over\{\backslash partial\; t\}\}\; +\; \backslash left(\backslash mu\; -\; \{1\backslash over\{2\}\}\backslash sigma^\{2\}\backslash right)\{\backslash partial\; p\backslash over\{\backslash partial\; x\}\}\; =\; \{1\backslash over\{2\}\}\backslash sigma^\{2\}\{\backslash partial^\{2\}p\backslash over\{\backslash partial\; x^\{2\}\}\},\; \backslash quad\; p(0,x)\; =\; \backslash delta(x)$

Define $V=\backslash mu-\backslash sigma^\{2\}/2$ and $D=\backslash sigma^\{2\}/2$. By introducing the new variables $\backslash xi\; =\; x-Vt$ and $\backslash tau\; =\; Dt$, the derivatives in the Fokker-Planck equation may be transformed as:

:

Leading to the new form of the Fokker-Planck equation:

:$\{\backslash partial\; p\backslash over\{\backslash partial\backslash tau\}\}\; =\; \{\backslash partial^\{2\}p\backslash over\{\backslash partial\; \backslash xi^\{2\}\}\},\; \backslash quad\; p(0,\backslash xi)\; =\; \backslash delta(\backslash xi)$

However, this is the canonical form of the heat equation. which has the solution given by the heat kernel:

:$p(\backslash tau,\backslash xi)\; =\; \{1\backslash over\{\backslash sqrt\{4\backslash pi\; \backslash tau\}\}\}\backslash exp\backslash left(-\{\backslash xi^\{2\}\backslash over\{4\backslash tau\}\}\; \backslash right)$

Plugging in the original variables leads to the PDF for GBM:

:$p(t,S)\; =\; \{1\backslash over\{S\backslash sqrt\{2\backslash pi\; \backslash sigma^\{2\}t\}\}\}\backslash exp\backslash left\backslash \{-\{\backslash left[\backslash log(S/S\_\{0\})-\backslash left(\; \backslash mu\; -\; \{1\backslash over\{2\}\}\backslash sigma^\{2\}\backslash right)t\; \backslash right]^\{2\}\backslash over\{2\backslash sigma^\{2\}t\}\}\; \backslash right\backslash \}$

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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

:$$

\begin{alignat}{2}

d\log(S) & = f'(S)\,dS + \frac{1}{2} f'' (S)S^2\sigma^2 \, dt \\[6pt]

& = \frac{1}{S} \left( \sigma S\,dW_t + \mu S\,dt\right) - \frac{1}{2}\sigma^2\,dt \\[6pt]

&= \sigma\,dW_t +(\mu-\sigma^2/2)\,dt.

\end{alignat}

It follows that $\backslash operatorname\{E\}\; \backslash log(S\_t)=\backslash log(S\_0)+(\backslash mu-\backslash sigma^2/2)t$.

This result can also be derived by applying the logarithm to the explicit solution of GBM:

:$$

\begin{alignat}{2}

\log(S_t) &=\log\left(S_0\exp\left(\left(\mu - \frac{\sigma^2}{2} \right)t + \sigma W_t\right)\right)\\[6pt]

& =\log(S_0) +\left(\mu - \frac{\sigma^2}{2} \right)t + \sigma W_t.

\end{alignat}

Taking the expectation yields the same result as above: $\backslash operatorname\{E\}\; \backslash log(S\_t)=\backslash log(S\_0)+(\backslash mu-\backslash sigma^2/2)t$.

1. 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([np.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()

GBM can be extended to the case where there are multiple correlated price paths.Musiela, M., and Rutkowski, M. (2004), Martingale Methods in Financial Modelling, 2nd Edition, Springer Verlag, Berlin.

Each price path follows the underlying process

:$dS\_t^i\; =\; \backslash mu\_i\; S\_t^i\backslash ,dt\; +\; \backslash sigma\_i\; S\_t^i\backslash ,dW\_t^i,$

where the Wiener processes are correlated such that $\backslash operatorname\{E\}(dW\_\{t\}^i\; \backslash ,dW\_\{t\}^j)\; =\; \backslash rho\_\{i,j\}\; \backslash ,\; dt$ where $\backslash rho\_\{i,i\}\; =\; 1$.

For the multivariate case, this implies that

:$\backslash operatorname\{Cov\}(S\_t^i,\; S\_t^j)\; =\; S\_0^i\; S\_0^j\; e^\{(\backslash mu\_i\; +\; \backslash mu\_j)\; t\; \}\backslash left(e^\{\backslash rho\_\{i,j\}\; \backslash sigma\_i\; \backslash sigma\_j\; t\}-1\backslash right).$

A multivariate formulation that maintains the driving Brownian motions $W\_t^i$ independent is

:$dS\_t^i\; =\; \backslash mu\_i\; S\_t^i\backslash ,dt\; +\; \backslash sum\_\{j=1\}^d\; \backslash sigma\_\{i,j\}\; S\_t^i\backslash ,dW\_t^j,$

where the correlation between $S\_t^i$ and $S\_t^j$ is now expressed through the $\backslash sigma\_\{i,j\}\; =\; \backslash rho\_\{i,j\}\backslash ,\; \backslash sigma\_i\backslash ,\; \backslash sigma\_j$ terms.

{{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.{{cite book|title=Options, Futures, and other Derivatives|edition=7|first=John|last=Hull|year=2009|chapter=12.3}}

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.
• 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).

Apart from modeling stock prices, Geometric Brownian motion has also found applications in the monitoring of trading strategies.{{cite journal |last1=Rej |first1=A. |last2=Seager |first2=P. |last3=Bouchaud |first3=J.-P. |title=You are in a drawdown. When should you start worrying? |journal=Wilmott |date=January 2018 |volume=2018 |issue=93 |pages=56–59 |doi=10.1002/wilm.10646 |arxiv=1707.01457 |s2cid=157827746 |url=https://onlinelibrary.wiley.com/doi/abs/10.1002/wilm.10646}}

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 ($\backslash sigma$) 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. 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.Fengler, M. R. (2005), Semiparametric modeling of implied volatility, Springer Verlag, Berlin. DOI https://doi.org/10.1007/3-540-30591-2{{cite journal | title = Lognormal-mixture dynamics and calibration to market volatility smiles | first1 = Damiano | last1 = Brigo | author-link1 = Damiano Brigo | first2 = Fabio | last2 = Mercurio | author-link2 = Fabio Mercurio | pages = 427–446 | journal = International Journal of Theoretical and Applied Finance | volume = 5 | year = 2002 | issue = 4 | doi = 10.1142/S0219024902001511 }}Brigo, D, Mercurio, F, Sartorelli, G. (2003). Alternative asset-price dynamics and volatility smile, QUANT FINANC, 2003, Vol: 3, Pages: 173 - 183, {{ISSN|1469-7688}} 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.{{cite journal | title = A closed-form solution for options with stochastic volatility with applications to bond and currency options | first = Steven L. | last = Heston | author-link1 = Steven L. Heston | pages = 327–343 | journal = Review of Financial Studies | volume = 6 | issue = 2 | year = 1993 | jstor = 2962057 | doi = 10.1093/rfs/6.2.327| s2cid = 16091300 }}

• Brownian surface

{{Reflist}}

• [https://web.archive.org/web/20120130222949/http://math.nyu.edu/financial_mathematics/content/02_financial/02.html Geometric Brownian motion models for stock movement except in rare events.]
• [https://web.archive.org/web/20170402233858/http://excelandfinance.com/simulation-of-stock-prices/brownian-motion/ Excel Simulation of a Geometric Brownian Motion to simulate Stock Prices ]
• {{cite web | url=http://turingfinance.com/interactive-stochastic-processes/ | title=Interactive Web Application: Stochastic Processes used in Quantitative Finance | access-date=2015-07-03 | archive-date=2015-09-20 | archive-url=https://web.archive.org/web/20150920231636/http://turingfinance.com/interactive-stochastic-processes/ | url-status=dead }}
• [https://sites.google.com/site/nonnewtoniancalculus/ Non-Newtonian calculus website]
• [https://portfoliooptimizer.io/blog/trading-strategy-monitoring-modeling-the-pnl-as-a-geometric-brownian-motion/ Trading Strategy Monitoring: Modeling the PnL as a Geometric Brownian Motion]

{{Stochastic processes}}

{{DEFAULTSORT:Geometric Brownian Motion}}

Category:Wiener process

Category: Non-Newtonian calculus

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