# Correlated geometric brownian motion matlab

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Quantitative Finance Stack Exchange is a question and answer site for finance professionals and academics. It only takes a minute to sign up. Is it true see herefootnote 2, p. Here is the general approach you can follow to generate two correlated random variables. So we have:. Sign up to join this community.

The best answers are voted up and rise to the top. Home Questions Tags Users Unanswered. Two correlated brownian motions Ask Question. Asked 4 years, 7 months ago. Active 1 year, 7 months ago. Viewed 13k times. If this is true, we could easily simulate them in Python by doing: import numpy as np import matplotlib.

Basj Basj 1 1 gold badge 6 6 silver badges 19 19 bronze badges. Active Oldest Votes. AdB 4 4 silver badges 16 16 bronze badges. Ric Ric Do you think my simulation code is ok? It seems ok. By summing the increments cumulative sumit seems that it would leed to the same formula as yours. You need the square-root because constant multiplicators enter variance with their square.

Edited in my question. Then if you integrate you have the same as I have. Neeraj Neeraj 2, 9 9 silver badges 28 28 bronze badges. You have to cumsum them to get brownian motion. Can you include code to plot the two correlated brownian motions? More generally I think your method is more or less the same as the one I used in Python in my original post, but it's interesting to have it in R for learning purpose, thanks!

I have added R code for simulating two geometric Brownian motion.

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I hope it would help you.By using our site, you acknowledge that you have read and understand our Cookie PolicyPrivacy Policyand our Terms of Service. Stack Overflow for Teams is a private, secure spot for you and your coworkers to find and share information. Unfortunately, it has not been vectorized. The easiest way to do what you want is to use a for loop:. This is rather slow and inefficient. You will need to modify the function a lot to vectorize it.

One thing that would improve performance is if you at least removed the plotting code from inside the function and ran that separately after the loop.

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For comparatively short small sets of simulations looping over your code or executing the above should do. For heavy duty simulations you may benefit from Horchler's promised speed advantage. Learn more. Asked 7 years, 1 month ago. Active 5 years, 2 months ago. Viewed 18k times. However, I would like to generate 1, simulations and to be to display them in a graph.

The Overflow Bugs vs. Featured on Meta. Responding to the Lavender Letter and commitments moving forward. Visit chat. Related Hot Network Questions. Question feed. Stack Overflow works best with JavaScript enabled.Documentation Help Center. Asset returns are simulated as the proportional increments of constant drift, constant volatility stochastic processes, thereby approximating continuous-time geometric Brownian motion.

This example shows the distinction between the Exact and Expected methods of simulation.

Consider a portfolio of five assets with the following expected returns, standard deviations, and correlation matrix based on daily asset returns where ExpReturn and Sigmas are divided by to convert percentages to returns.

Assume that there are trading days in a calendar year, and simulate two sample paths realizations of daily returns over a two-year period. To illustrate the distinction between methods, simulate two paths by each method, starting with the same random number state. Compare the mean and covariance of RetExact with the inputs ExpReturn and ExpCovarianceyou will observe that they are almost identical.

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At this point, RetExact and RetExpected are both byby-2 arrays. Now assume an equally weighted portfolio formed from the five assets and create arrays of portfolio returns in which each column represents the portfolio return of the corresponding sample path of the simulated returns of the five assets.

3 1 Correlated Brownian Motion

Finally, convert the simulated portfolio returns to prices and plot the data. In particular, note that since the Exact method matches expected return and covariance, the terminal portfolio prices are virtually identical for each sample path.

This is not true for the Expected simulation method. Although this example examines portfolios, the same methods apply to individual assets as well. Thus, Exact simulation is most appropriate when unique paths are required to reach the same terminal prices. Recall that portsim simulates correlated asset returns over an interval of length dtgiven by the equation. The time increment dt is determined by the optional input RetIntervalseither as an explicit input argument or as a unit time increment by default.

This point is often misunderstood. To illustrate the interplay among ExpReturnExpCovarianceand RetIntervalsconsider a portfolio of five assets with the following expected returns, standard deviations, and correlation matrix based on daily asset returns. Convert the correlations and standard deviations to a covariance matrix of daily returns. Assume trading days per calendar year, and simulate a single sample path of daily returns over a four-year period. Resetting the random number generator to its initial state, you can reproduce the results.

Assume an equally weighted portfolio and compute portfolio returns associated with each simulated return series. Comparison of the data reveals that PortRet1 and PortRet2 are identical.

This example shows how to simulate a univariate geometric Brownian motion process. In addition to verifying Hull's example, it also graphically illustrates the lognormal property of terminal stock prices by a rather large Monte Carlo simulation. Simulate the daily price process for this stock over the course of one full calendar year trading days.

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RetIntervals is expressed in years, consistent with the fact that ExpReturn and ExpCovariance are annualized. Also, ExpCovariance is entered as a variance rather than the more familiar standard deviation volatility.

Set the random number generator state, and simulate 10, trials realizations of stock returns over a full calendar year of trading days. The squeeze function reformats the output array of simulated returns from a -by- 1 -by- array to more convenient -by- array. Recall that portsim is fundamentally a multivariate simulation engine. Display the sample density function of the terminal stock price after one calendar year.

From the sample density function, the lognormal distribution of terminal stock prices is apparent. ExpCovariance must be symmetric and positive semidefinite no negative eigenvalues. If ExpCovariance is not a symmetric positive semidefinite matrix, use nearcorr to create a positive semidefinite matrix for a correlation matrix. If NumObs is entered as the empty matrix []the length of RetIntervals is used.

If RetIntervals is not specified, all intervals are assumed to have length 1. Optional Type of Monte Carlo simulation, specified as a character vector with one of the following values:. The expected values of the sample mean and covariance are equal to the input mean ExpReturn and covariance ExpCovariance specifications.

For either Methodthe sample mean and covariance returned are appropriately scaled by RetIntervals.Documentation Help Center. Creates and displays geometric Brownian motion GBM models, which derive from the cev constant elasticity of variance class.

Specifically, this model allows the simulation of vector-valued GBM processes of the form.

## $$y=\rho x + e_i$$

X t is an NVars -by- 1 state vector of process variables. D is an NVars -by- NVars diagonal matrix, where each element along the main diagonal is the corresponding element of the state vector X t. V is an NVars -by- NBrowns instantaneous volatility rate matrix. Specifying an array indicates a static non-time-varying parametric specification. This array fully captures all implementation details, which are clearly associated with a parametric form.

Specifying a function provides indirect support for virtually any static, dynamic, linear, or nonlinear model. This parameter is supported via an interface, because all implementation details are hidden and fully encapsulated by the function. You can specify combinations of array and function input parameters as needed. Moreover, a parameter is identified as a deterministic function of time if the function accepts a scalar time t as its only input argument.

Otherwise, a parameter is assumed to be a function of time t and state X t and is invoked with both input arguments. Name is a property name and Value is its corresponding value.

Name must appear inside single quotes ''. The GBM object has the following Properties :. StartState — Initial state at StartTime. Correlation — Access function for the Correlation input, callable as a function of time. Drift — Composite drift-rate function, callable as a function of time and state.

Diffusion — Composite diffusion-rate function, callable as a function of time and state. Simulation — A simulation function or method. Return — Access function for the input argument Returncallable as a function of time and state.The generated paths are suitable to be used in the Monte-Carlo approach to pricing options on a basket of assets.

Note that the primary purpose of the code presented here is to show how to efficiently generate the correlated asset paths. The code contains no error checking and as such it is not suitable for inclusion into a larger application without appropriate modifications. The following is code for generating a 3-dimensional matrix where each row represents a time step, each column represent a seperate simulation for a specific asset and the 3 rd dimension represents different assets in the basket. Equation 1: Stock Price Evolution Equation.

The following MATLAB code gives an example of how to use the function AssetPathsCorrelatedincluding creating and customizing a plot showing a subset of the generated price paths. Back To Top Option Pricing. MATLAB Function: AssetPathsCorrelated The following is code for generating a 3-dimensional matrix where each row represents a time step, each column represent a seperate simulation for a specific asset and the 3 rd dimension represents different assets in the basket.Sign in to comment.

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## How to implement a Correlated Brownian Motion correctly

Search MathWorks. MathWorks Answers Support. Open Mobile Search. Trial software. You are now following this question You will see updates in your activity feed. You may receive emails, depending on your notification preferences. How to implement a Correlated Brownian Motion correctly.

Nina on 12 Dec Vote 0. Edited: Walter Roberson on 29 Apr I have trouble implementing a Correlated Brownian Motion. The following code is running, but does not return the expected values. Does anybody know what I did wrong? Thankful for any advice or hint! Walter Roberson on 20 Apr Speaking of Arsenal and Manchester United, they are featured in the marquee match of the weekend on Saturday afternoon at 12:30 pm ET. This is great news for draw bettors, especially since the 3-way moneylines are all very close.

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Read on for our free betting tips, predictions and analysis. Free TipsOur tip will be online shortly. Preview AnalysisThere will be no hiding places with any number of these only knowing one way of running, from the front.

Coneygree, American and Genie In Abottle all hold excellent chances on the formbook, but with others in here who go from the front, this could end up a carve up falling into the lap of something from the back. Total Recall looked like a horse miles ahead of his mark when winning on his debut for Willie Mullins, he will get a perfect tow into the race but is plenty short enough given how competitive a handicap this is.

A token choice goes to POTTERS LEGEND who has shaped on a number of occasions as if he is better than his current mark. He can be smuggled round off the strong pace, with first time blinkers an interesting change from cheekpieces. Struck into himself on his seasonal return so that is easily excused, he is the class act in the race. This contest can be won with topweight, he looks a big price with conditions to suit.

Made a pleasing return to action at Kempton over a trip too short. Rated as high as 167 over hurdles in his pomp, there is still some room from this 157 level here. The ground might not be quite as deep as he would like here and he is surrounded by other front runners. Should have the class from this sort of mark, but the race may not be run to suit.

This clearly will have been the long term plan for his this season but a career high mark is going to take plenty of defying. He acts on any ground, giving the impression that the step up in trip is going to suit him. He has cheekpieces back on for the first time in this country.

They didn't work well in France so that is a slight concern. That horse won the Troytown from 148 so that makes this mark of 150 look very tempting. Faces a lot of competition for the lead however which isn't a plus for his chances.

Six pounds higher than that in a deeper contest, he doesn't look one of the more likely ones to challenge.