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Let $u_t$ be the random walk $$ u_t = u_{t-i} + \varepsilon_t $$ where $\mathrm{E}[\varepsilon_t]=0$ and $\mathrm{var}[\varepsilon_t]=\sigma^2$ , i.e. $\varepsilon_t$ is stationary. Now let $$X_t = \alpha u_t +\nu_t$$ and $$Y_t = \beta u_t + \eta_t$$ where $\nu_t$ and $\eta_t$ are stationary processes similar to $\varepsilon_t$ Then both $X_t$ and ...


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Well that is a very good question. Once I realised of the issue with EMD I investigated all wavelets transforms I could access to in Matlab, all seemed to suffer either some end point distorsion or what is called Time phase distorsion for all one-step-ahead forecast where a lag of one sample appear in the forecast at some segments of the test data. I did ...


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Looking for the same issue, I found an article by de Jong(1997). In section 2 you can find a method for estimation of covariances and correlations between irregularly spaced data. Also look at the article by Jonas Andersson where some interpolation methods and method form de Jong are presented and compared together. Hope it helps.


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In my case (and I work mostly with natural gas) what I do in the calibration is to use the real value of $\Delta t$ from the historical data, and measure the time in days. In this way, $\Delta t=1$ in most cases, and $\Delta t=1$ in the weekends, so that you take into account the invisible changes in the markets during the weekends. It also help after the ...


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Given that other corporate events are reasonably modelled through regression models (compare The Detection of Earnings Manipulation I would try for using an regression approach. I believe a more recent and related paper has been published but I don't seem to find it at this time. Edit: and now I did - Earnings Manipulation and Expected Returns That said, ...


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Time is expressed in fractions of year in the GBM formula. Therefore, $T=1$ year and $\Delta t = 1/m$. Considered that you have $253$ observations, I would use $m = 253$, so the second option as Drew suggested. In general, using 253 or 365 days in a year depends on how you consider reality: do you think that when markets are closed (i.e. weekends) the price ...


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The second one will be the best estimate. Also, a smaller timestep usually corresponds to a smaller bias. But I agree, the answer is not obvious. You should be careful about increasing $T$ though, because for negative drifts there is a threshold value ($2\mu + \sigma^2 < 0$) beyond which the variance of the price process stops increasing. It's an ...


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For both time-series, just plot the log returns. You will see that one is not a Random-Walk .. the S&P500 since you will get values that far beyond the normal distribution. Just watch this video by Benoit Mandelbrot (starting at 11min:54sec). Looking at both graphs, your eyes can fool you making you believe that both are generated by Random Walks...


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I think the main difference even in this little example is the gain-loss asymmetry which is a known stylized fact: When you look at the big bump both time series posses your artificial one is perfectly symmetric whereas the real one takes longer for going up and then crashes in a relatively shorter time frame. This is a known phenomenon in real financial ...


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You have just luckily created 1 path of the random walk by chance that fitted the S&P. You can create another random walk and it will look much different. The efficient markets hypothesis predicts that stock prices behave as random walks, so it is likely that S&P looks similar to that. However, one cannot predict the next step to make a profit, ...



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