ETFs are relatively new financial products and have gained popularity in recent years. The ETF industry now consists of thousands of funds with well over $5 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.

Trading decisions often depend on the trader’s subjective belief of the distribution of the asset price on a given future date. For example, if a trader anticipates a big price movement for a company stock after its earnings announcement, then perhaps a long straddle position makes sense. There are many instances like this. And as time progresses, the trader will learn more about the price distribution by observing price fluctuations.

Every portfolio can be partitioned into multiple asset groups defined by asset classes, sectors, styles, or 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.

In the US market, institutional investors own or manage a major share of public equities. For many institutional investors, orders to buy or sell stocks can be very large. Such orders are thought to be potentially costly to implement, as they create substantial immediate demand/supply and possible adverse price effects.

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.

Gold is often viewed as a safe haven asset or a hedge against market turmoils, currency depreciation, and other economic or political events. For instance, during the credit crisis, the Dow and S&P 500 declined by about 20% while gold prices rose from $850 to $1,100 per troy ounce. And then this year, S&P500 has experienced a sharp drop before returning to pre-COVID level lately. Meanwhile, gold ETF (GLD) has gained more than 25% since Feb 2020.

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.

The fundamental principle in derivatives pricing is the absence of arbitrage opportunities. Standard no-arbitrage pricing theory asserts that spot and futures prices must converge at expiration. Nevertheless, for years traders have observed significantly higher expiring futures prices for corn, wheat, and soybeans on the Chicago Board of Trade (CBOT) compared to the spot price of the physical grains.