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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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Using Returns in Pairs Trading

This blog article is taken from our book [1]. In most entry-level materials on pairs trading such as in [2],  a mean reverting basket is usually constructed by this relationship: , where is the price of asset at time t, the price of asset at time t, and the price of the mean reverting asset to trade. One way to find is to use cointegration. There are numerous problems in this approach as detailed in [1]. To mention a few: the identified portfolios are dense; executions involve considerable transaction costs; the resultant portfolios behave like insignificant and non-tradable noise; cointegration is too stringent and often unnecessary a requirement to satisfy. This article highlights one important problem: it is much better to work in the space of (log) returns than in the space of prices. Therefore, we would like to build a mean reverting portfolio using a similar relationship to (eq. 1) but in returns rather than in prices. The Benefits of Using Log Returns When we compare the prices of two assets, [… TODO …]   A Model for a Mean Reverting Synthetic Asset Let’s assume prices are log-normally distributed, which is a popular assumption in quantitative finance, esp. in options pricing. Then, prices are always positive, satisfying the condition of “limited liability” of stocks. The upside is unlimited and may go to infinity. [5] We have: is the return for asset between times 0 and t; likewise for asset . Instead of applying a relationship, e.g., cointegration (possible but not a very good way), to the pair on prices, we can do it on returns. This is possible because, just like prices, the returns at... 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