ETFs

Last updated in 2021. For latest research, please visit the research page.

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http://www.springer.com/us/book/9783319290928

From: https://amath.washington.edu/news/2017/08/25/applied-mathematical-tools-modern-financial-problems

ETFs are relatively new financial products and have gained popularity in recent years. The ETF industry now consists of more than 1,600 funds with over $2 trillion in assets. All ETFs are traded on major exchanges like stocks, and most are designed to track an index or asset. 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 new financial products are called leveraged ETFs (LETFs), and the most common leverage ratios include {+3, +2, -2, -3}. As such, LETFs offer investors easy access to trade in different asset classes with positively/negatively leveraged exposures without borrowing.

The proliferation of ETFs has generated a host of new research problems. For instance, how well do these funds track their advertised benchmarks? Is there a mathematical connection between an (L)ETF and its reference index? What is the best trading strategy for tracking a given index? In my recent papers with students, we investigate the tracking performance of a wide array of equity and commodity LETFs and find that commodity LETFs tend to underperform compared to the theoretical benchmark. We also construct a dynamic leveraged portfolio using short-term futures, and demonstrate that this portfolio tracks the benchmark better than the market-traded LETF over a long out-of-sample period.

The use of ETFs and LETFs has also led to increased trading of options written on these funds. For (L)ETFs that track the same reference index, their associated options have very similar sources of randomness. This gives rise to the question of consistent no-arbitrage (no-free-lunch) pricing of options on ETFs and LETFs. I've studied the mathematical connections among the implied volatilities derived from options prices on leveraged and unleveraged ETFs. Under a general stochastic volatility framework, we develop analytical approximations and numerical methods to price LETF options and derive mathematical formulae for price comparison across different options markets.

Figure: The implied volatilities of options written on the S&P500 ETF (SPY) and its leveraged counterparts, SSO (+2x) and SDS (-2x). At first look, they look unrelated (left panel). After a mathematical scaling, we can line them up for direct comparison (right panel).

In retrospect, my research on LETFs was first motivated by an alert issued by the U.S. Securities and Exchange Commission regarding the riskiness of LETFs, along with news articles on ETFs and my empirical observations. Looking forward, as the market of ETFs continues to grow in terms of market capitalization and product diversity, there are plenty of new problems for future research. The availability of high-frequency trading data also permits the analysis of the intraday return patterns and tracking performance of these funds compared to their reference assets.

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