Simulating multiple correlated stochastic processes

[Max-118]

Hi,

I am trying to simulate the price dynamics of 4 different assets given the correlation among them (assuming all asset returns follow geometric Brownian motion).

I have implemented the monte-carlo simulation using Quantlib classes, such as StochasticProcessArray, MultiPathGenerator, etc.

However, I am not so sure how the StochasticProcessArray implementation ensures the correlation among the 4 assets is guaranteed, given the input 4x4 correlation matrix is positive defintie. Could someone help explain this from a theoretical perspective? or point me to the related reference?

Thanks!

Best regards,
Max


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[Max-118]

Hi,

I studied the code further. And I realized that in StochasticProcessArray class, the correlation matrix L is decomposed into a lower triangular using SalvagingAlgorithm::Spectral. Then the new triangular matrix is applied to the array of independent normal random variables.

I am not familiar with the "Spectral" analysis. However, I want to ask:
Given correlation matrix is symmetric and positive semidefinite, can I use SalvagingAlgorithm::none method (which is essentially a Cholesky decomposition) instead? And in this case, would the final results be equivelant for SalvagingAlgorithm::Spectral and SalvagingAlgorithm::none methods?

On Thu, Jun 26, 2008 at 12:08 AM, Max <[hidden email]> wrote:

Hi,

I am trying to simulate the price dynamics of 4 different assets given the correlation among them (assuming all asset returns follow geometric Brownian motion).

I have implemented the monte-carlo simulation using Quantlib classes, such as StochasticProcessArray, MultiPathGenerator, etc.

However, I am not so sure how the StochasticProcessArray implementation ensures the correlation among the 4 assets is guaranteed, given the input 4x4 correlation matrix is positive defintie. Could someone help explain this from a theoretical perspective? or point me to the related reference?

Thanks!

Best regards,
Max

... [show rest of quote]

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[xionghan]

Hi Max,
 
The Spectral method is just an eigenvalue decomposition. It is also called Principle Component Analysis.
 
Best,
Han

---

Date: Thu, 26 Jun 2008 01:28:30 +0800
From: [hidden email]
To: [hidden email]; [hidden email]
Subject: Re: [Quantlib-users] Simulating multiple correlated stochastic processes

Hi,

I studied the code further. And I realized that in StochasticProcessArray class, the correlation matrix L is decomposed into a lower triangular using SalvagingAlgorithm::Spectral. Then the new triangular matrix is applied to the array of independent normal random variables.

I am not familiar with the "Spectral" analysis. However, I want to ask:
Given correlation matrix is symmetric and positive semidefinite, can I use SalvagingAlgorithm::none method (which is essentially a Cholesky decomposition) instead? And in this case, would the final results be equivelant for SalvagingAlgorithm::Spectral and SalvagingAlgorithm::none methods?

On Thu, Jun 26, 2008 at 12:08 AM, Max <max.zou@...> wrote:

Hi,

I am trying to simulate the price dynamics of 4 different assets given the correlation among them (assuming all asset returns follow geometric Brownian motion).

I have implemented the monte-carlo simulation using Quantlib classes, such as StochasticProcessArray, MultiPathGenerator, etc.

However, I am not so sure how the StochasticProcessArray implementation ensures the correlation among the 4 assets is guaranteed, given the input 4x4 correlation matrix is positive defintie. Could someone help explain this from a theoretical perspective? or point me to the related reference?

Thanks!

Best regards,
Max

... [show rest of quote]

... [show rest of quote]

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[Luigi Ballabio]

On Thu, 2008-06-26 at 01:28 +0800, Max wrote:
> Given correlation matrix is symmetric and positive semidefinite, can I
> use SalvagingAlgorithm::none method (which is essentially a Cholesky
> decomposition) instead?

Yes, you can.

> And in this case, would the final results be equivelant for
> SalvagingAlgorithm::Spectral and SalvagingAlgorithm::none methods?

It should be. It would be nice if you made the experiment and reported
back here...

Luigi


--

Ninety percent of everything is crap.
--- Theodore Sturgeon



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[Max-118]

Hi Luigi,

I have just done some basic test on a basket call option on the average return of 4 assets (using Sobol Quasi-random number generator). And the correlation matrix given is positive semi-definite. The following compares the fair values produced by Cholesky and Spectral decompositions for different number of iterations. (see attached chart).

no. of trails    Cholesky       Spectral         relative difference
500               0.14439479    0.14822383    2.6518%
1000             0.14844382    0.15099720    1.7201%
3000             0.15228533    0.15291014    0.4103%
5000             0.15295908    0.15328304    0.2118%
7000             0.15288972    0.15356331    0.4406%
10000           0.15333001    0.15370840    0.2468%
15000           0.15339209    0.15373016    0.2204%
30000           0.15374863    0.15387913    0.0849%
60000           0.15390326    0.15396157    0.0379%
90000           0.15393888    0.15402227    0.0542%
120000         0.15395743    0.15402453    0.0436%
150000         0.15396883    0.15405399    0.0553%
200000         0.15399083    0.15405433    0.0412%
300000         0.15400569    0.15404896    0.0281%
600000         0.15403359    0.15405266    0.0124%


As you can see, the two methods converge at higher number iterations; and Spectral decomposition seems to have higher rate of convergence rate. Is this behavior expected? and any existing literature explains the difference between them?

Best regards,
Max

On Tue, Jul 1, 2008 at 6:46 PM, Luigi Ballabio <[hidden email]> wrote:

On Thu, 2008-06-26 at 01:28 +0800, Max wrote:
> Given correlation matrix is symmetric and positive semidefinite, can I
> use SalvagingAlgorithm::none method (which is essentially a Cholesky
> decomposition) instead?

Yes, you can.

> And in this case, would the final results be equivelant for
> SalvagingAlgorithm::Spectral and SalvagingAlgorithm::none methods?

It should be. It would be nice if you made the experiment and reported
back here...

Luigi


--

Ninety percent of everything is crap.
--- Theodore Sturgeon

... [show rest of quote]

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