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Maybe this could also be a comment but I think an it is not possible to answer this question with a 'yes and here is how you do it'. It has been tried, e.g. by me for a university research project. In this research we focused primarily on aggregation of returns and the main problem was the tractability of the resulting distributions and expressions, also ...


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It could be much more simple: if you use the method of moments (MM) then you estimate the mean and the variance and for example the kurtosis of your sample. Then you fit the parameters to these statistics. Alternatively you use maximum-likelihood (MLE). For MM: from wikipedia you get the mean and the variance. In your notation you can fit $b = \bar{r}$ so ...


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The consensus nowadays is that stable distributions are not a well fit, although they do possess heavy tails. In particular Cauchy has too fat tails. The reasons for this are disparate, however the first that comes to mind is that empirically longer horizons show a decrease in tail thickness, approaching normality for 1-year returns (although this has been ...


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Historical returns are not to be used 'untreated' for the calculation of option prices. The expectation that you will be using in Monte Carlo will take the form $$ C(K,T) = E^Q\{D(T)\ \max[0, S_T-K, 0]\} $$ where $T$ is the maturity, $K$ is the strike price, $S$ is the stock price and $D$ is the discount factor. But the expectation is taken under the 'risk ...


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I think modelling hedge fund returns is a very interesting yet demanding task. Your model will have to strike a balance between the tangibility of the model on the one hand and the possibility of parameter estimation on the other. Plus I think you will encounter hedge funds that resist all modelling attempts because there strategies are just too elusive. ...


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What about this sketch of an answer: Let's put $T=1$ in your formula to simplify the notation. Then $Y_b(t)$ is a Brownian bridge where $Y_b(0)=0$ and $Y_b(1)=b$. This can be written as $Y_b(t) = b\ t + Y_0(t)$, that is to say the standard Brownian bridge (from zero to zero) with an added drift $b\ t$. The standard Brownian bridge can be written in terms ...


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If you mean by fat tails just fatter tails than the gaussian distribtuion, i.e. a distribution with finite variance, for instance the Student's t-distribution has fatter tails than the normal distribution. If you mean distributions with infinite variance, you have to have a look at Lévy distribution. In a first attempt you could just substitute the standard ...


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I think there are 2 approaches being a bit mixed up here. You can analyze the option market by looking at implied volatilities and apply Black-Scholes (BS), thus assuming that log-returns follow a Gaussian distribution. Implied volatilies are the parameters that bring together BS and market prices. Then you will observe a pattern of implied volatilies for ...


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I have not tried it myself but if i may be allowed to forward you to a link of a particular filter sold as an indicator called the Jurik MA. If you check the link, there is a quote where they mention ` What we mean by a random walk is a time series produced by a cumulative sum of 5000 zero-mean, Cauchy distributed random numbers.` Also this is supposed to ...


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In my experience with forecasting, you could try a model of the form $$ X_ t = cycle_t + seasonality_t + residuum_t. $$ Sometimes it is hard to find the cycle but the seasonality could be doable if it has some natural structure (something happening in a certain month each year e.g.). Rob Hyndman explains all these things (and provides an R package) in his ...



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