http://ta-lib.org/
http://quantlib.org/ (C++)
http://www.jquantlib.org (Java)
JQuantLib is a free, open-source and comprehensive framework for quantitative finance, offering several mathematical and statistical tools needed for financial instrument valuation, calculation of VaR, portfolio valuation, etc. JQuantLib is written in Java and is based on QuantLib, which is written in C++. QuantLib is a de-facto reference implementation for C++ world. Following the steps of its predecessor, JQuantLib aims to become a reference implementation for Java world. JQuantLib does its best efforts to mimic as close as possible the API exposed by QuantLib, offering a smooth transition path for developers and organizations aiming to adopt Java for high performance, low latency applications.
Python and R for quantitative finance
http://www.slideshare.net/lsbardel/python-and-r-for-quantitative-finance-2409526
PyCon 2010: Python in quantitative finance
http://python.mirocommunity.org/video/1531/pycon-2010-python-in-quantitat
Mathtools.net > Java > Finance and Economics
http://www.mathtools.net/Java/Finance_and_Economics/index.html
http://sourceforge.net/projects/qff/
The Quantitative Finance Framework (QFF) supports the development of software libraries in mathematical finance. The main field of applications are the pricing of derivatives and the management of financial risks.
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