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.
where $\delta (S)$ is the Dirac delta function. To simplify the computation, we may introduce a logarithmic transform $x=\log(S/S_{0})$, leading to the form of GBM:
Define $V=\mu -\sigma ^{2}/2$ and $D=\sigma ^{2}/2$. By introducing the new variables $\xi =x-Vt$ and $\tau =Dt$, the derivatives in the Fokker-Planck equation may be transformed as:
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
Taking the expectation yields the same result as above: $\operatorname {E} \log(S_{t})=\log(S_{0})+(\mu -\sigma ^{2}/2)t$.
Simulating sample paths
edit
# Python code for the plotimportnumpyasnpimportmatplotlib.pyplotaspltmu=1n=50dt=0.1x0=100np.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()
Multivariate version
edit
GBM can be extended to the case where there are multiple correlated price paths.^{[3]}
where the correlation between $S_{t}^{i}$ and $S_{t}^{j}$ is now expressed through the $\sigma _{i,j}=\rho _{i,j}\,\sigma _{i}\,\sigma _{j}$ terms.
Use in finance
edit
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.^{[4]}
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.^{[4]}
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.^{[5]}
Extensions
edit
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 ($\sigma$) 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.^{[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.^{[9]}
^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.
^Øksendal, Bernt K. (2002), Stochastic Differential Equations: An Introduction with Applications, Springer, p. 326, ISBN 3-540-63720-6
^ ^{a}^{b}Musiela, M., and Rutkowski, M. (2004), Martingale Methods in Financial Modelling, 2nd Edition, Springer Verlag, Berlin.
^ ^{a}^{b}Hull, John (2009). "12.3". Options, Futures, and other Derivatives (7 ed.).
^Rej, A.; Seager, P.; Bouchaud, J.-P. (January 2018). "You are in a drawdown. When should you start worrying?". Wilmott. 2018 (93): 56–59. arXiv:1707.01457. doi:10.1002/wilm.10646. S2CID 157827746.
^Fengler, M. R. (2005), Semiparametric modeling of implied volatility, Springer Verlag, Berlin. DOI https://doi.org/10.1007/3-540-30591-2
^Brigo, Damiano; Mercurio, Fabio (2002). "Lognormal-mixture dynamics and calibration to market volatility smiles". International Journal of Theoretical and Applied Finance. 5 (4): 427–446. 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
^Heston, Steven L. (1993). "A closed-form solution for options with stochastic volatility with applications to bond and currency options". Review of Financial Studies. 6 (2): 327–343. doi:10.1093/rfs/6.2.327. JSTOR 2962057. S2CID 16091300.
External links
edit
Geometric Brownian motion models for stock movement except in rare events.
Excel Simulation of a Geometric Brownian Motion to simulate Stock Prices
"Interactive Web Application: Stochastic Processes used in Quantitative Finance". Archived from the original on 2015-09-20. Retrieved 2015-07-03.
Non-Newtonian calculus website
Trading Strategy Monitoring: Modeling the PnL as a Geometric Brownian Motion