
Why Index Tracking
A common challenge faced by many institutional and retail investors is to effectively control risk exposure to various market factors. There is a great variety of indices designed to provide different types of exposures across sectors and asset classes, including equities, fixed income, commodities, currencies, credit risk, and more.
Some of these indices can be difficult or impossible to trade directly, but investors can trade the associated financial derivatives if they are available in the market. For example, the CBOE Volatility Index (VIX), often referred to as the fear index, is not directly tradable, but investors can gain exposure to the index and potentially hedge against market turmoil by trading futures and options written on VIX.
Volatility Index (VIX)
To illustrate the benefits of having exposure to VIX, consider the 2011 U.S. credit rating downgrade by Standard and Poor’s. News of a negative outlook by S&P of the U.S. credit rating broke on April 18th, 2011. As displayed in Figure 1(a), a portfolio holding only the SPDR S&P 500 Etf (SPY) would go on to lose about 10% with a volatile trajectory for a few months past the official downgrade on August 5th, 2011.

In contrast, a hypothetical portfolio with a mix of SPY (90%) and VIX (10%) would be stable through the downgrade and end up with a positive return. Figure 1(b) shows the same pair of portfolios over the year 2014.
Both earned roughly the same 15% return though SPY alone was visibly more volatile than the portfolio with SPY and VIX. The large drawdowns (for example on October 15th, 2014) were met by rises in VIX, creating a stabilizing effect on the portfolio’s value.
This example motivates the investigation of trading strategies that directly track VIX and other indices, or achieve any pre-specified exposure with respect to an index or market factor.
Challenges of Index Tracking
Many ETFs or ETNs are advertised to provide exposure to an index by maintaining a portfolio of securities related to the index, such as futures, options, and swaps. However, some of ETFs or ETNs often miss their stated targets, and some tend to significantly underperform over time relative to the targets.

One example is the Barclay’s iPath S&P 500 VIX Short-Term Futures ETN (VXX), which is also the most popular VIX exchange-traded product.3 The failure of VXX to track VIX is well documented. In fact, most of these ETFs or ETNs follow a static strategy or time-deterministic allocation that does not adapt to the rapidly changing market.
The problem of tracking is relevant in all asset classes, and the use of derivatives is quite common. For example, many investors seek exposure to gold to hedge against market turmoil. However, direct investment in gold bullion is difficult due to storage cost. In order to gain exposure, an investor may select among a number of gold ETFs and derivatives.
A New Methodology
In our recent paper, we discuss a general methodology for index tracking and risk exposure control using derivatives. Under very general framework for the market and associated risk factors, we derive a formula that links the exposures (with respect to different risk factors) embedded in a derivatives portfolio.
The portfolio’s log-return can be decomposed as follows:

The first two terms on the right-hand side indicate that the portfolio’s log-return (LHS) is proportional to the log-returns of the index S and its driving factors Yᵢ, with the proportionality coefficients (β,ηᵢ) being equal to the desired exposures. However, the portfolio’s log return is subject to the slippage process Z.
The slippage process can be used to **** quantify the divergence of portfolio return from the target returns of the index and its factors. It is a function of not only the realized variance of the underlying factors, but also the realized covariance among the index and factors.
Slippage reveals the potential value erosion arising from the interactions among risk factors.
Index tracking can be perceived as an inverse problem to dynamic hedging of derivatives. In the traditional hedging problem, the goal is to trade the underlying assets so as to replicate the price evolution of the derivative in question, and thus, the tradability of the underlying is of crucial importance.
In our proposed paradigm, the index and stochastic factors may not be directly traded, but there exist traded derivatives written on them. We use derivatives to track or, more generally, control risk exposure with respect to the index return in a path-wise manner. Consequently, we can study the path properties resulting from various portfolios of derivatives, and quantify the portfolio’s divergence, if any, from a pre-specified benchmark. Our methodology also allows the investor to achieve leveraged or non-leveraged exposures with respect to the associated factors in the model.
Back to VIX
Motivated still further by VIX, we explore the applications and implications of our methodology under some models for VIX. In particular, our methodology and examples shed light on the connection between the mean-reverting behavior of VIX and the implications to the pricing of VIX derivatives and tracking this index and its associated factors. We consider the tracking and risk exposure control problems under the Cox-Ingersoll-Ross (CIR) model

and the two-factor Concatenated Square Root (CSQR) model

Among our findings, we derive the trading strategies using options or futures to track the VIX or achieve any exposure to the index and/or its factors. Our portfolio utilizes an explicit pathwise-adaptive strategy, as opposed to a time-deterministic one used by VXX and other ETNs.
Although we have chosen VIX as our main example, our analysis applies to other mean-reverting price processes or market factors.
References
T. Leung and B. Ward, Dynamic Index Tracking and Exposure Control Using Derivatives [pdf;link], Applied Mathematical Finance, Vol. 25, Issue 2, pp.180–212, 2018
T. Leung and B. Ward, Tracking VIX with VIX Futures: Portfolio Construction and Performance [pdf;link], in Handbook of Applied Investment Research, J. Guerard and W. Ziemba eds., World Scientific Publishing Co, 2020
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