William Long, Ph.D. Candidate in Financial Engineering

Bio

Headshot of William LongWilliam Long holds a B.S. in pure mathematics from North Carolina State University and an M.S. in mathematical finance from UNC Charlotte. He is a Ph.D. candidate in financial engineering at Stevens Institute of Technology. With nearly a decade of experience in education and finance, he is seeking a position in the finance industry.

Skillset

He has taught classes in Python and R and is proficient in both. He also has years of experience working with LLMs, machine learning, applied statistics and data analysis. He is an excellent communicator and a natural leader, having started and run multiple clubs during his time at Stevens. He also has some experience in broadcasting and sales.

Dissertation Summary

Novel Non-Parametric Causality Test With Applications in Option Forecasts (Working title)

Summary

In late 2017, the online broker Robinhood began offering zero-commission trades on options. Since then, retail investor trading volume in options has grown rapidly, de Silva et al. (2023), with some authors claiming that this retailtrading now makes up a majority of the total options market volume, Bryzgalova et al. (2022). This shift is naturally of great interest to the financial industry, and several authors have investigated the impact of investor attention on the options market, see Bryzgalova et al. (2022), Naranjo et al. (2023), Zhan et al. (2022), etc. In particular, Choy and Wei (2022)found that delta hedged returns for attention grabbing stocks were lower the following day. Investor attention also impacts volatility, trading volume, and trading spreads, all of which are relevant for option returns, Zhan et al. (2022), Bryzgalova et al. (2022), and Naranjo et al. (2023). Thus, it is necessary to control for these factors when examining the relationship between options and investor attention.

We can view this as a specific application of the causal inference problem, i.e. determining if factor X is important when forecasting Y . In econometrics, this is often evaluated using the concept of Granger causality, Granger (1969). The test originally proposed by Clive Granger involves fitting a vector autoregression (VAR) on a data set and determiningwhich coefficients are significant. There are two key assumptions of this test: all time series are strictly stationary and that the relationships between them are linear.

However, both of these assumptions are violated in our financial application, thus we cannot use a traditional Granger causality test. First, we are studying a data set which is explicitly changing in time. Thus, we can not assume that the relevant probability distributions are time invariant, i.e. that all time series are strictly stationary. Second, even simpleoptions contracts have complicated, nonlinear payoff functions and partial derivatives, Hull and Basu (2016). Thus, we can also not assume that the relationship between investor attention and option returns is exclusively linear or even follows aparametric model. While there are several non-parametric tests for Granger causality, they are only valid for strictly stationary time series. We need a non-parametric test of Granger causality for possibly non-stationary data.

To develop a more general test, we draw inspiration from the prior literature on conditional independence testing in cross-sectional (i.e., iid) data. In particular, we focus on the non-parametric test by Cai et al. (2022), which is built on the following proposition. Let X, Y, Z be continuous random variables such that U = FX|Z(X|Z), V = FY|Z(Y|Z), W = FZ(Z) are also continuous for all values of X, Y, Z. Proposition 1 from Cai et al. (2022) states that "X ⊥ Y | Z if and only if U, V, W are mutually independent." Thus, their conditional independence test simplifies into a mutual independence test of U, V, and W. While this test was originally intended for iid data, we can adapt it to time series data. While the distribution of a non-strictly stationary time series Xt varies in time, it is a well-known result from Diebold et al. (1997) that: Ft(Xt|Ft−1) ∼ U(0, 1), where U(0, 1) is the uniform distribution from 0 to 1 and Ft−1 is the relevant information set at time t − 1. If we estimate the time-varying conditional distribution for a given time series correctly, the resulting probability integral should not only be stationary but also lack any correlations in time. This estimation can be done using an appropriate time-varying kernel density estimation method, such as the one found in Garcin et al. (2023). Thus, we are able to use both of these concepts to design a novel, non-parametric test of Granger causality, which we then use to test if investor attention conditionally Granger-causes option returns.

This leads us to how to measure investor attention. While some prior authors have used excess returns (Choy and Wei, 2022) or survey data (Borup et al., 2023), we use a more direct method to measure attention: web analytics. Since retail investors often research and discuss the stocks they are interested in online, we can use this online activity as a proxy for their interest in a stock. This method was pioneered by Da et al. (2011), who found that increased Google searches of a firm’s ticker could be used to forecast stock returns. So, we use their Abnormal Search Volume Index (ASVI) to measure investor attention in this application. In particular, we use our novel causality test to determine that these abnormal Google searches forecast shifts in option returns, even when controlling for volatility, trading volume, and trading spreads.

References

  • Borup, D., J. W. Hansen, B. D. Liengaard, and E. C. Montes Schu¨tte (2023). Quantifying investor narratives and their role during covid-19. Journal of Applied Econometrics 38 (4), 512–532.

  • Bryzgalova, S., A. Pavlova, and T. Sikorskaya (2022). Retail trading in options and the rise of the big three wholesalers.Journal of Finance forthcoming .

  • Cai, Z., R. Li, and Y. Zhang (2022). A distribution free conditional indepen- dence test with applications to causal discovery. Journal of Machine Learning Research 23 (85), 1–41.

  • Choy, S. K. and J. Wei (2022). Investor attention and option returns. Manage- ment Science.

  • Da, Z., J. Engelberg, and P. Gao (2011). In search of attention. The journal of finance 66 (5), 1461–1499.

  • de Silva, T., K. Smith, and E. C. So (2023). Losing is optional: Retail option trading and expected announcementvolatility. Available at SSRN 4050165 .

  • Diebold, F. X., T. A. Gunther, and A. Tay (1997). Evaluating density forecasts.

  • Garcin, M., J. Klein, and S. Laaribi (2023). Estimation of time-varying ker- nel densities and chronology of the impact of covid-19 on financial markets. Journal of Applied Statistics, 1–21.

  • Granger, C. W. (1969). Investigating causal relations by econometric models and cross-spectral methods. Econometrica: journal of the Econometric Society, 424–438.

  • Hull, J. C. and S. Basu (2016). Options, futures, and other derivatives. Pearson Education India.

  • Naranjo, A., M. Nimalendran, and Y. Wu (2023). Betting on elusive returns: Retail trading in complex options. Availableat SSRN 4404393 .

  • Zhan, X., B. Han, J. Cao, and Q. Tong (2022). Option return predictability.

  • The Review of Financial Studies 35 (3), 1394–1442.

Academic Advisors

Ionut Florescu and Chihoon Lee

Get in Touch
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