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The code below pulls AAPL time series from Yahoo Finance, computes mean/std and simulates 100 paths that are 20 days long. Input: import pandas as pd import numpy as np from numpy.random import normal # bring data ticker = 'AAPL' url = 'http://real-chart.finance.yahoo.com/table.csv?s=%s' % ticker data = pd.read_csv(url, index_col='Date', parse_dates=True) ...

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If you want to use GBM in the physical measure then just calculate the standard deviation of log returns to get $\sigma$. For $\mu$ there are many choices. If you want to calibrate GBM to get option prices then you need the implied volatility and there is a package. The OP said that he/she wants to simulate paths. If you don't want to price derivatives ...

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Take a look at pinkfish. Disclaimer, I am the author. http://fja05680.github.io/pinkfish/

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Slight correction: the package in R is called rmgarch, not mgarch. It works well with rugarch, which provides a variety of univariate GARCH models. Both packages allow for parallelized computation on local cluster and return a nice and full set of fitted parameters, model specs, etc. I provided some additional links in this post.

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What you need is more mutual information rather than Shannon entropy. It is dedicated to capture the influence of one variable on another (you can think about it as a non linear version of Pearson correlations). They are closely related since the mutual information $I$ between two variables $X$ and $Y$ reads: $$I(X;Y) = H(X,Y) - H(X|Y) - H(Y|X)$$ where $H$ ...

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I calculate duration in Python using numpy, it's nice and simple: def durations(cfs, rates, price, ytm, no_coupons): import numpy as np mac_dur = np.sum([cfs[i]*i/np.power(1+rates[i],i) for i in range(len(cfs))])/price mod_dur = mac_dur/(1+ytm/no_coupons) return mac_dur, mod_dur

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Native support is very limited. TradeStation's WebAPI pretty much works with any language because it is wrapped in HTTP calls using RESTful. If a platform has an API that supports std C/C++ interfaces, you can write a wrapper to extend the API to python. Search for "Calling C from Python". It is more work to code, but otherwise your choices are very ...

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