# Tag Info

## Hot answers tagged random-variables

12

There are a number of different tests that are generally used to compare samples to different distributions, such as Jarque-Bera, Anderson-Darling, and Kolmogorovâ€“Smirnov (see this related question). In your case, with just the standard deviation and mean, there isn't a whole lot to say. You need to assume a distribution (e.g. normal). You would be able ...

8

Two ways: Model the returns using an Ornstein-Uhlenbeck process You can control the variance of the residual noise in the process to your desired level of correlation. Conceptually you inject gaussian noise into the synthetic OU process to satisfy your requirement. For example, let's say you have time-series A which is what you are modelling. Time-series ...

8

There are certainly (short-rate) models which assume bounded interest rates. I suppose I should clarify - the design of the model prohibits negative interest rates. Further, some models asymptotically reach some target, or mean rate which is considered mean reversion, the most famous perhaps the Vasicek. Short rate models where rates cannot go negative: Cox-...

6

Be careful, remember that the mean and the standard deviation don't tell you the whole story: http://en.wikipedia.org/wiki/Anscombe%27s_quartet

6

A discrete-time model only works in no-arbitrage land with discrete asset values. Furthermore, the number of allowable asset values per timestep is limited by the number of available securities. The tree is the classic example of this. Binomial trees "work", but if you make a one-step trinomial tree, you will find that you can no longer form a risk-free ...

5

Yes, this technique is called moment matching variance reduction and it may indeed lead to a form of variance reduction. The first and second order moments correspond to the mean and the variance of the distribution. You can extend to higher order moments, which is of course more difficult to implement and creates some extra overhead. The mean can be adjust ...

4

Wilmott Forums - "How can I simulate correlated random numbers?" Generating correlated normal variates Random Correlated Series Generator (using R) All found with a Google search for "how to generate random correlated series".

3

You could create a rescaled stochastic indicator from your randomly generated, correlated series. 1) use whatever software/methodology you want to create your random series with 0.85 correlation to the original data. 2) find the maximum and minimum values of this new series and rescale the series to range between 0 and 1 using this formula; (series_value - ...

3

Exact Discretization of the Solution to the Geometric Brownian Motion Stochastic Differential Equation Let $P_{t}$ represent the time series of market prices of the underlying, $\mu$ be its mean continuous log-return, $\sigma$ be its instantaneous volatility and $W_{t}$ be a Wiener process. Here is the stochastic differential equation for the geometric ...

3

Gaussian random variable is another name for Normal random variable. It is called Gaussian because Carl Friedrich Gauss discovered many properties of the Normal distribution. A linear combination of Gaussian random variables is another random variable, not necessarily Gaussian itself, that you get by adding and subtracting Gaussian random variables. Lets ...

3

Mersenne Twister is currently the most used PRNG in the quant world. It was even incorporated in C++11 so it can be considered standard nowadays. Any PRNG with reasonable statistical quality shall perform well (equivalently) for pricing, so that differences relate more to convenience (speed, parallelizability etc..). If the statistical quality is poor then ...

3

I assume, this is not for real-time display, so you can use the price from future. If this is not the case, this answer is irrelevant. I don't know about a standard technique, but this is my suggestion: $p_{noise} = p_{current} + \nu * (p_{future} - p_{current})$ where $p_{future}$ is future price for some horizon, and $\nu$ is a zero-mean Gaussian noise.

2

I have seen a technique which uses frequency domain and does pretty much what (I think) you are trying to do. The author does not give the complete details, so you might have to contact him for that, or take a look at the (free) software he has developed. Link here: http://blog.quantumfading.com/2009/08/24/historical-data-randomization-using-the-frequency-...

2

This looks to me like a range accrual. Let $t_1, \ldots, t_n$, where $0 < t_1 < \cdots < t_n$ be business days that are being considered. We compute \begin{align*} E\left(\sum_{i=1}^n \pmb{1}_{b_1 < S_{t_i} < b_2} \right) &=\sum_{i=1}^n E\left(\pmb{1}_{b_1 < S_{t_i} < b_2} \right)\\ &=\sum_{i=1}^n \left[E\big(\pmb{1}_{S_{t_i} >...

1

First I provide a brief description of Halton sequences. A Halton sequence is a deterministic sequence of numbers that provides well-spaced 'draws' from an interval and provides negative correlation between simulated probability for individuals. Generation is based on a prime number Sequence is constructed based on finer and finer prime-based divisions of ...

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Maybe the accepted answer to this earlier thread and the more detailed description on my blog might be of use to you. Within the FFT you could just manipulate the higher frequency components to create your synthetic microstructure noise.

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Further to my comments to Samik R's answer above, here is the link to my blog post where my attempts to recreate the method linked to in said answer are given. Edit to include more information per Tal Fishman's comment Octave .oct function code #include <octave/oct.h> #include <octave/dColVector.h> #include <octave/CNDArray.h> #include "...

1

If you want to make the returns "random", then you will have to generate the whole price paths that meet your correlation criteria and then discard results that don't fit your price criteria. This isn't random. If I know how prices evolve, then I can generate a killer trading rule. I suggest using historical data. These data are easy to obtain with the ...

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