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| [[File:GeoBM.png|thumb|500px|right|Two sample paths of Geometric Brownian motion, with different parameters. The blue line has larger drift, the green line has larger variance.]]
| | This really is especially important. If you make half-hearted efforts to lose fat, you may be never actually going to succeed. So develop a strong determination, which this really is what we want and you'll do anything it takes to achieve the objective.<br><br>The Grapefruit Diet. The grapefruit diet is one of the number one diets to lose weight fast. People participating in this diet plan lose a typical of 10 pounds inside lower than 2 weeks. The grapefruit diet is low inside calories plus low inside fat. During this diet, grapefruit is consumed at every meal with a small serving of greens plus lean protein. The grapefruit diet is amazing for fast weight reduction, yet should just be selected short-term to get rid of a limited amount of weight, such as 10 to 15 pounds.<br><br>The ideal time to do any cardio exercise is initial thing each morning before we eat anything. What happens is the fact that because you haven't consumed anything during the evening, because you're asleep, your body has to utilize calories that are stored in the body plus burning calories is what you want if you need to lose weight.<br><br>More lately, certain companies are making it even easier for a breastfeeding mom to start losing weight. Slim Fast and Weight Watchers have a special diet plan only for nursing moms. The South Beach Diet may moreover be employed when you simply skip over Phase One of the plan. Many diets have now altered their plans to suit a nursing mother's requirements.<br><br>11 Say no to white carbohydrates. This includes white rice, white flour treatments like bread, pastries plus starchy greens like [http://safedietplansforwomen.com/how-to-lose-weight-fast how to lose weight fast] potatoes. Instead loads up the body on fiber rich foods plus lean protein.<br><br>Cardio is an aerobic exercise that you do for hours at slow - moderate pace with an intensity level where you are able to still talk. It does burn fat nevertheless only throughout the exercise bout. When you stop, the fat-burning effect stops.<br><br>The little secrets reported above are nothing yet small changes in the diet which prove to be surprisingly efficient in terms of the weight reduction efforts. Along with implementing these changes we equally have to recognize much more about a complete fat loss system which may promote effective fat reduction. |
| 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 [[stochastic drift|drift]].<ref>{{cite book|title=Introduction to Probability Models|first=Sheldon M.|last=Ross|year=2007|chapter=10.3.2}}</ref> 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]].
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| ==Technical definition: the SDE==
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| A stochastic process ''S''<sub>''t''</sub> is said to follow a GBM if it satisfies the following [[stochastic differential equation]] (SDE):
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| :<math> dS_t = \mu S_t\,dt + \sigma S_t\,dW_t </math>
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| 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.
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| 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. | |
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| ==Solving the SDE==
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| For an arbitrary initial value ''S''<sub>0</sub> the above SDE has the analytic solution (under [[Itō calculus|Itō's interpretation]]):
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| : <math> S_t = S_0\exp\left( \left(\mu - \frac{\sigma^2}{2} \right)t + \sigma W_t\right).</math>
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| To arrive at this formula, let us divide the SDE by <math> S_t </math>, and write it in [[Itō integral]] form:
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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>
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| 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
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| : <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
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| : <math>\ln \frac{S_t}{S_0} = \left(\mu -\frac{\sigma^2}{2}\,\right) t + \sigma W_t\,.</math>
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| Exponentiating gives the solution claimed above.
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| ==Properties==
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| 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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| | title = Stochastic Differential Equations: An Introduction with Applications
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| |publisher=Springer
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| |author=Oksendal, Bernt K.
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| | year = 2002
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| | pages = 326
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| | isbn = 3-540-63720-6
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| }}</ref>
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| :<math>\mathbb{E}(S_t)= S_0e^{\mu t},</math>
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| :<math>\operatorname{Var}(S_t)= S_0^2e^{2\mu t} \left( e^{\sigma^2 t}-1\right),</math>
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| that is the [[probability density function]] of a ''S<sub>t</sub>'' is:
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| : <math>f_{S_t}(s; \mu, \sigma, t) = \frac{1}{\sqrt{2 \pi}}\, \frac{1}{s \sigma \sqrt{t}}\, \exp \left( -\frac{ \left( \ln s - \ln S_0 - \left( \mu - \frac{1}{2} \sigma^2 \right) t \right)^2}{2\sigma^2 t} \right).</math>
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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''<sub>''t''</sub>). 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
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| :<math> | |
| \begin{alignat}{2}
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| d\log(S) & = f^\prime(S)\,dS + \frac{1}{2}f^{\prime\prime} (S)S^2\sigma^2 \, dt \\
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| & = \frac{1}{S} \left( \sigma S\,dW_t + \mu S\,dt\right) - \frac{1}{2}\sigma^2\,dt \\
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| &= \sigma\,dW_t +(\mu-\sigma^2/2)\,dt.
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| \end{alignat}
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| </math>
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| It follows that <math>\mathbb{E} \log(S_t)=\log(S_0)+(\mu-\sigma^2/2)t</math>.
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| This result can also be derived by applying the logarithm to the explicit solution of GBM:
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| :<math>
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| \begin{alignat}{2}
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| \log(S_t) &=\log\left(S_0\exp\left(\left(\mu - \frac{\sigma^2}{2} \right)t + \sigma W_t\right)\right)\\&
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| =\log(S_0)+\left(\mu - \frac{\sigma^2}{2} \right)t + \sigma W_t.
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| \end{alignat}
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| </math>
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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>.
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| == Multivariate version ==
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| GBM can be extended to the case where there are multiple correlated price paths.
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| Each price path follows the underlying process
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| :<math>dS_{t}^i = \mu_i S_{t}^i\,dt + \sigma_i S_{t}^i\,dW_{t}^i</math>,
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| where the Wiener processes are correlated such that <math> \mathbb{E}(dW_{t}^i dW_{t}^j) = \rho_{i,j} dt</math> where <math>\rho_{i,i} = 1</math>. | |
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| For the multivariate case, this implies that
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| :<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>.
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| ==Use in finance==
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| {{main|Black–Scholes model}}
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| 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>
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| Some of the arguments for using GBM to model stock prices are:
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| *The expected returns of GBM are independent of the value of the process (stock price), which agrees with what we would expect in reality.<ref name="Hull" />
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| *A GBM process only assumes positive values, just like real stock prices.
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| *A GBM process shows the same kind of 'roughness' in its paths as we see in real stock prices.
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| *Calculations with GBM processes are relatively easy.
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| However, GBM is not a completely realistic model, in particular it falls short of reality in the following points:
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| *In real stock prices, volatility changes over time (possibly [[stochastic volatility|stochastically]]), but in GBM, volatility is assumed constant.
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| *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>
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| ==Extensions==
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| 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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| ==See also==
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| *[[Brownian surface]]
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| == References ==
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| {{Reflist}}
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| == External links ==
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| * [http://www.math.nyu.edu/financial_mathematics/content/02_financial/02.html Geometric Brownian motion models for stock movement except in rare events.]
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| *[http://forevermar.com/Brownian.php R and C# Simulation of a Geometric Brownian Motion ]
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| *[http://excelandfinance.com/simulation-of-stock-prices/brownian-motion/ Excel Simulation of a Geometric Brownian Motion to simulate Stock Prices ]
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| {{Stochastic processes}}
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| {{DEFAULTSORT:Geometric Brownian Motion}}
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| [[Category:Stochastic processes]]
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This really is especially important. If you make half-hearted efforts to lose fat, you may be never actually going to succeed. So develop a strong determination, which this really is what we want and you'll do anything it takes to achieve the objective.
The Grapefruit Diet. The grapefruit diet is one of the number one diets to lose weight fast. People participating in this diet plan lose a typical of 10 pounds inside lower than 2 weeks. The grapefruit diet is low inside calories plus low inside fat. During this diet, grapefruit is consumed at every meal with a small serving of greens plus lean protein. The grapefruit diet is amazing for fast weight reduction, yet should just be selected short-term to get rid of a limited amount of weight, such as 10 to 15 pounds.
The ideal time to do any cardio exercise is initial thing each morning before we eat anything. What happens is the fact that because you haven't consumed anything during the evening, because you're asleep, your body has to utilize calories that are stored in the body plus burning calories is what you want if you need to lose weight.
More lately, certain companies are making it even easier for a breastfeeding mom to start losing weight. Slim Fast and Weight Watchers have a special diet plan only for nursing moms. The South Beach Diet may moreover be employed when you simply skip over Phase One of the plan. Many diets have now altered their plans to suit a nursing mother's requirements.
11 Say no to white carbohydrates. This includes white rice, white flour treatments like bread, pastries plus starchy greens like how to lose weight fast potatoes. Instead loads up the body on fiber rich foods plus lean protein.
Cardio is an aerobic exercise that you do for hours at slow - moderate pace with an intensity level where you are able to still talk. It does burn fat nevertheless only throughout the exercise bout. When you stop, the fat-burning effect stops.
The little secrets reported above are nothing yet small changes in the diet which prove to be surprisingly efficient in terms of the weight reduction efforts. Along with implementing these changes we equally have to recognize much more about a complete fat loss system which may promote effective fat reduction.