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Markowitz’s celebrated mean–variance portfolio optimization theory assumes that the means and covariances of the underlying asset returns are known. In practice, they are unknown and have to be estimated from historical data. Plugging the estimates into the efficient frontier that assumes known parameters has led to portfolios that may perform poorly and have counter-intuitive asset allocation weights; this has been referred to as the “Markowitz optimization enigma.” After reviewing different approaches in the literature to address these difficulties, we explain the root cause of the enigma and propose a new approach to resolve it. Not only is the new approach shown to provide substantial improvements over previous methods, but it also allows flexible modeling to incorporate dynamic features and fundamental analysis of the training sample of historical data, as illustrated in simulation and empirical studies.


We are the bridge between the academia and industry. We screen hundreds of highly refereed journal papers and pick the best trading ideas. We turn these ideas on paper into solid tools that you can actually use in trading. Our collection of quantitative models and algorithms are good building blocks for your strategies. By putting together these modules, you can construct quantitative trading strategies like assembling LEGO pieces. Moreover, these math models enable you to translate vague trading ideas in English into precise mathematics in Greek.


This course introduces students to quantitative trading. A “quant” portfolio manager or a trader usually starts with an intuition or a vague trading idea. Using mathematics, s/he turns the intuition into a mathematical trading model for analysis, back testing and refinement. When the quantitative investment model proves to be likely profitable after passing rigorous statistical tests, the portfolio manager implements the model on a computer system for automatic execution. In short, quantitative trading is the process where ideas are turned into mathematical models and then coded into computer programs for systematic trading. It is a science where mathematics and computer science meet. In this course, students study investment strategies from the popular academic literature and learn the fundamental mathematics and IT aspects of this emerging field.

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We are looking for your contribution to the numerical and financial analytic library. Freelance, internship, part-time and full-time are all welcome!

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Fastest Java Matrix Multiplication

Introduction Matrix multiplication occupies a central role in scientific computing with an extremely wide range of applications. Many numerical procedures in linear algebra (e.g. solving linear systems, matrix inversion, factorizations, determinants) can essentially be reduced to matrix multiplication [5, 3]. Hence, there is great interest in investigating fast matrix multiplication algorithms, to accelerate matrix multiplication (and other numerical procedures in turn). SuanShu was already the fastest in matrix multiplication and hence linear algebra per our benchmark. SuanShu v3.2.0 benchmark Starting version 3.3.0, SuanShu has implemented an advanced algorithm for even faster matrix multiplication. It makes some operations 100x times faster those of our competitors! The new benchmark can be found here: SuanShu v3.3.0 benchmark In this article, we briefly describe our implementation of a matrix multiplication algorithm that dramatically accelerates dense matrix-matrix multiplication compared to the classical IJK algorithm. Parallel IJK We first describe the method against which our new algorithm is compared against, IJK. Here is the algorithm performing multiplication for is ,  is , and  is : for (i = 1; i < = m; i ++){ for (j = 1; j <= p; j ++){ for (k = 1; k <= n; k ++){ C[i,k] += A[i,j] * B[j,k]; } } } 1234567 for (i = 1; i < = m; i ++){    for (j = 1; j <= p; j ++){        for (k = 1; k <= n; k ++){            C[i,k] += A[i,j] * B[j,k];        }    }} In Suanshu, this is implemented in parallel; the outermost loop is passed to a  ParallelExecutor . As there are often more rows than threads available, the time complexity of this parallelized IJK is still roughly the same as IJK: ,... read more
Stella XingStella XingChorys Limited
AlgoQuant greatly streamlines our modeling and backtesting process. Our family office builds our research IT infrastructure leveraging the AlgoQuant API.


Stella Xing, Owner, Chorys Limited
Allen YanAllen YanRongtong Fund Management
I am impressed with NM’s innovative mathematical approach to quantitative/algorithmic trading and their trading research technologies. Glad to work with them!


Allen Yan, Deputy CEO, Rongtong Fund Management
lfhflfhf Hedge Fund, Switzerland
As a head quant trader in a prop trading house, I found in the libraries proposed by Numerical Method Inc. an impressive set of algorithms and tools that I can use as a foundation to my in-house framework. These libraries enabled me to quickly implement research ideas by using standard tools, but also having access to more sophisticated up-to-date algorithms, as companions to our in-house algorithms.


The fact that these libraries are reliable, robust and efficiently implemented turned out to be an important gain of time for us, so we could really focus on the elaboration of our intellectual property, and the implementation of our own tool and strategies.Finally, the support and developing teams are really prompt at responding to questions, which is a real plus.


lfhf, Head Quant, Hedge Fund, Switzerland
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