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. 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.

We develop a methodology for index tracking and risk exposure control using financial derivatives. Under a continuous-time diffusion framework for price evolution, we present a pathwise approach to construct dynamic portfolios of derivatives in order to gain exposure to an index and/or market factors that may be not directly tradable. Among our results, we establish a general tracking condition that relates the portfolio drift to the desired exposure coefficients under any given model. We also derive a slippage process that reveals how the portfolio return deviates from the targeted return. In our multi-factor setting, the portfolio's realized slippage depends not only on the realized variance of the index, but also the realized covariance among the index and factors. We implement our trading strategies under a number of models, and compare the tracking strategies and performances when using different derivatives, such as futures and options.

Professor Tim Leung is an Associate Professor in the Department of Applied Mathematics and the Director of the Computational Finance & Risk Management (CFRM) program. He's the Chair for the Institute for Operations Research and the Management Sciences (INFORMS) Finance Section, as well as Vice Chair for the SIAM Activity Group on Financial Mathematics & Engineering (SIAG-FME). Previously, he was​​ an Assistant Professor in the Department of Applied Mathematics & Statistics at Johns Hopkins University and in the Department of Industrial Engineering & Operations Research at Columbia University​ in New York City.​ He obtained his BS from Cornell University and PhD from Princeton University. His research areas are Quantitative Finance and Optimal Stochastic Control. In 2016, he published two books, respectively, on the topics of Mean Reversion Trading and leveraged ETFs.