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Boeing Professor of Applied Math & Director of the Computational Finance & Risk Management (CFRM) Program at UW; Senior Research Advisor for Quant Funds.

Empirical mode decomposition and Hilbert spectral analysis

“Everything is Energy” — Photo by Darius Bashar on Unsplash

Market observations and empirical studies have shown that asset prices are often driven by multiscale factors, ranging from long-term economic cycles to rapid fluctuations in the short term. This suggests that financial time series are potentially embedded with different timescales.

On the other hand, nonstationary and behaviors and nonlinear dynamics are often observed in financial time series. These characteristics can hardly be captured by linear models and call for an adaptive and nonlinear approach for analysis. For decades, methods based on short-time Fourier transform have been developed and applied to nonstationary time series, but there are still challenges in capturing…


Algo Trading

Real examples, performance summary, and python package

A powerful pair, ready to pounce. Photo by Geran de Klerk on Unsplash.

Pairs trading is among the most popular trading strategies in many markets, ranging from equities and ETFs to currencies and futures markets. It involves taking simultaneous positions in two correlated assets. The idea is that while typically it is difficult to accurately capture the price evolution of a single asset, a pairs position may exhibit mean reversion that can be better modeled. In short, pairs trading is a market-neutral strategy that seeks to profit from the price convergence between the two assets.

Practical examples of mean reverting price spread include pairs of stocks/ETFs, futures and its spot, physical commodity and…


Algo Trading

From portfolio construction to optimal execution

A Deep Dive into Pairs Trading. Photo by NOAA on Unsplash

In this new python package called Machine Learning Financial Laboratory (mlfinlab) developed by Hudson & Thames, there is a module that automatically solves for the optimal trading strategies (entry & exit price thresholds) when the underlying assets/portfolios have mean-reverting price dynamics.

It covers a few mean-reverting models, including the Ornstein-Uhlenbeck (OU) model. The trading model and computations are based on the results from this journal article.

The module includes three main steps:

Model Fitting:


Algo Trading

Automatically identify assets and determine long/short positions

The long and the short. Photo by Heather M. Edwards on Unsplash

Motivated by the industry practice of pairs trading and long/short equity strategies, we study an approach that combines statistical learning and optimization to construct portfolios with mean-reverting price dynamics.

Our main objectives are:

  • Design a portfolio with mean-reverting price dynamics, with parameters estimated by maximum likelihood;
  • Select portfolios with desirable characteristics, such as high mean reversion;
  • Build a parsimonious portfolio, i.e. find a small subset from a larger collection of assets for long/short positions.

In this article, we present the full problem formulation and discuss a specialized algorithm that exploits the problem structure. …


Mountain range mirrored in a lake. Photo by Julie Lamour on Unsplash

ETFs are relatively new financial products and have gained popularity in recent years. The ETF industry now consists of more than 2,000 funds with well over $4 trillion in assets. All ETFs are traded on major exchanges like stocks, and most are designed to track an index or asset. For many investors, ETFs provide various desirable features such as liquidity, diversification, low expense ratios, and tax efficiency.

Leveraged ETFs

Within the ETF universe, some funds are designed to replicate a constant multiple (called leverage ratio) of the daily returns of a reference index. These relatively new financial products are called leveraged ETFs…


Quantitative Finance

Deception Pass, WA, USA. Photo by D. Hoefler on Unsplash

Every portfolio can be partitioned into multiple asset groups defined by asset classes, sectors, styles, and other features. A cardinality-constrained portfolio caps the number of stocks to be traded within each of these groups. These limitations arise from real-world scenarios faced by fund managers who seek to satisfy certain investment mandates or achieve their asset allocation objectives.

As an example, suppose you want to construct a portfolio by investing in stocks across m sectors you favor. And in each sector you select up to k stocks and each sector should not constitute more than q% of your portfolio. Moreover, you…


Useful information & links

Space needle and Mt. Rainier in Seattle, Washington.

The Master of Science in Computational Finance and Risk Management (MS-CFRM), housed within the Applied Math Department at University of Washington — Seattle, addresses the demand in the financial services profession for advanced quantitative and computational finance skills, and next generation risk management competencies.

For students and current professionals with strong mathematical and quantitative skills, the CFRM program provides exciting new opportunities for advancement in a new or current career pathway in the numerous fields in finance.

The CFRM program has a strong emphasis on professional & career development. We provide year-round support for our MS students with their job…


Book Description

Book Title: Employee Stock Options Exercise Timing, Hedging, and Valuation

Publisher: World Scientific

Available on Amazon

Book Description: Employee stock options (ESOs) are an integral component of compensation in the US. In fact, almost all S&P 500 companies grant options to their top executives, and the total value accounts for almost half of the total pay for their CEOs.

In view of the extensive use and significant cost of ESOs to firms, the Financial Accounting Standards Board (FASB) has mandated expensing ESOs since 2004. This gives rise to the need to create a reasonable valuation method for these options for…


Market Risk

The road through peak and trough. Photo by Sam Chang on Unsplash

A drawdown measures the distance (in %) of the portfolio value from its peak, reflecting its downside risk. If a portfolio had previously reached a peak of $100 and subsequently dropped to $90, then the portfolio experienced a 10% drawdown.

Drawdown risk is relevant to all stocks and portfolios. Investor should care about not only the magnitudes of drawdowns but also their durations.

We examine the drawdown risk sector by sector through a series of Sector ETFs. As we see below, drawdown risk may vary greatly across sectors.

As a comparison, we look at the drawdown charts for the Nasdaq…


Algo Trading

ADR price dynamics motivate a trading strategy

Looking in the same direction? Photo by SK Yeong on Unsplash

American Depositary Receipts (ADRs)

Foreign company stocks listed in US exchanges are commonly registered as American Depositary Receipts (ADRs). These are certificates, denominated in U.S. dollars, that represent shares of non-U.S. company securities. There are about 2000 ADRs traded on U.S. exchanges, namely NYSE and NASDAQ, or through the over-the-counter (OTC) market, representing shares of companies from at least 70 different countries.

ADRs are among the most direct and popular financial instruments for investing in foreign companies, and have been actively used to diversify portfolios. In 2015, foreign equity holdings — through ADRs and local shares — accounted for 19% of U.S. …

Tim Leung, Ph.D.

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