ESSAYS ON INFORMATION IN OPTIONS MARKETS

A DISSERTATION SUBMITTED TO THE GRADUATE SCHOOL OF BUSINESS AND THE COMMITTEE ON GRADUATE STUDIES OF STANFORD UNIVERSITY IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY

Travis L. Johnson May 2012

© 2012 by Travis Lake Johnson. All Rights Reserved. Re-distributed by Stanford University under license with the author.

This work is licensed under a Creative Commons Attribution- Noncommercial 3.0 United States License. http://creativecommons.org/licenses/by-nc/3.0/us/

This dissertation is online at: http://purl.stanford.edu/hf166pb0337

ii I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy.

Anat Admati, Primary Adviser

I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy.

Ian Martin

I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy.

Stefan Nagel

I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy.

Paul Pfleiderer

Approved for the Stanford University Committee on Graduate Studies. Patricia J. Gumport, Vice Provost Graduate Education

This signature page was generated electronically upon submission of this dissertation in electronic format. An original signed hard copy of the signature page is on file in University Archives.

iii Abstract

In the first chapter, my coauthor and I examine the information content of option and equity volumes when trade direction is unobserved. In a multimarket asymmet- ric information model, equity short-sale costs result in a negative relation between relative option volume and future firm value. In our empirical tests, firms in the lowest decile of the option to stock volume ratio (O/S) outperform the highest decile by 0.34% per week (19.3% annualized). Our model and empirics both indicate that O/S is a stronger signal when short-sale costs are high or option leverage is low. O/S also predicts future firm-specific earnings news, consistent with O/S reflecting private information. In the second chapter, I show that in many asset pricing models, the equity mar- ket’s expected return is a time-invariant linear function of its conditional variance, which can be estimated from options markets. However, I show that when the relation between conditional means and variances is state-dependent, an observer requires the combined information in multiple variance horizons to distinguish among the states and thereby reveal the equity risk premium. Empirically, I show that while the VIX by itself has little predictive power for future S&P 500 returns, the VIX term structure predicts next-quarter S&P 500 returns with a 5.2% adjusted R2.

iv Acknowledgements

The first chapter of this dissertation is co-authored with Eric C. So. Thanks to my advisor Anat Admati, and my reading committee Ian Martin, Stefan Nagel, and Paul Pfleiderer, for their comments and guidance. Thanks to an anonymous referee, Mary Barth, Darrell Duffie, Jeff Harris, Sebas- tian Infante, Charles Lee, Arthur Korteweg, Kristoffer Laursen, Ian Martin, Stefan Nagel, Paul Pfleiderer, Monika Piazzesi, Ken Singleton, and seminar participants at Boston College, Dartmouth College, Rice University, SAC Capital Advisors, Stan- ford University, University of California-Berkeley, University of Houston, University of Maryland, University of Notre Dame, University of Pennsylvania, University of Rochester, University of Texas-Austin, University of Wisconsin-Madison, and the Western Finance Association meetings, for their helpful comments and suggestions. Thanks to Data Explorers for graciously providing us institutional lending data.

v Contents

Abstract iv

Acknowledgements v

1 Introduction 1

2 The Option to Stock Volume Ratio and Future Returns 3 2.1 Introduction ...... 3 2.2 Relation to prior research ...... 6 2.3 The model ...... 10 A Equilibrium ...... 12 B Results and empirical predictions ...... 14 2.4 Empirical tests ...... 17 2.5 Additional analyses ...... 34 2.6 Conclusion ...... 37 2.7 Tables and Figures ...... 40

3 Equity Risk Premia and the VIX Term Structure 54 3.1 Introduction ...... 54 3.2 Relation to prior research ...... 56 3.3 Model ...... 61 A A three-factor model of variance and risk premia ...... 65 3.4 Empirical results ...... 71 3.5 Conclusion ...... 77 3.6 Figures and Tables ...... 78

vi A Appendices for Chapter 2 91 A.1 Simultaneous equations ...... 91 A.2 Measure of leverage ...... 93 A.3 Proofs ...... 93

B Appendices for Chapter 3 102 B.1 Conditional means and variances in prior models ...... 102 A Bansal and Yaron (2004) ...... 102 B Campbell and Cochrane (1999) ...... 103 B.2 Details of Proofs ...... 103 B.3 Continuous version of the model ...... 106 A Risk and Return in the Model ...... 108 B Matching Moments ...... 110

Bibliography 113

vii List of Tables

2.1 Descriptive statistics by year...... 42 2.2 Factor regression results by deciles of O/S, ∆O/S, and ΩO/S. . . . . 43 2.3 Fama-MacBeth multivariate regressions...... 47 2.4 Strategy alphas sorted by short-sale costs...... 49 2.5 Option volume alphas sorted by leverage...... 51 2.6 Future return skewness by deciles of call-put volume ratio...... 52 2.7 Earnings surprises and earnings announcement returns...... 53

3.1 Data and Calibrated Model Moments ...... 82 3.2 Descriptive Statistics of Term Structure ...... 83 3.3 Return Prediction Regressions ...... 84 3.4 Principal Components (Full Sample) ...... 87 3.5 Principal Components (No 2008-2009 Financial Crisis) ...... 89

B.1 Summary of Model Notation ...... 108 B.2 Data and Calibrated Model Moments ...... 111

viii List of Figures

2.1 Persistence of O/S-return relation...... 40 2.2 Cumulative hedge returns by year...... 41

3.1 Model Equity Risk Premia and the VIX Term Structure ...... 78 2 3.2 Functions integrated to compute VIX12 on January 2nd, 1996. . . . . 79 3.3 Time Series of the Term Structure ...... 80 3.4 Return predictability factors in the VIX term structure ...... 81

B.1 Risk premia and variances in Campbell and Cochrane (1999). . . . . 104

ix Chapter 1

Introduction

Classical finance theory assumes derivatives like options are redundant securities that can be perfectly replicated using the underlying asset. Transaction costs, incomplete markets, and asymmetric information are all examples of market imperfections that make accurate replication impossible, implying derivative markets are not redun- dant and may provide an alternative channel for fundamental information to reach investors. My dissertation research explores the non-redundant information about underly- ing assets provided by option markets. Pursuant to this agenda, I have two distinct chapters, each relying on option market data as a source of information about the economics of financial markets. The first chapter, “The Option to Stock Volume Ratio and Future Returns,” is co-authored with Eric So, an accounting student here Stanford. Eric and I shared responsibility for all parts of the paper: the theory, the empirics, the writing, and the editing. The paper has been accepted at the Journal of Financial Economics but hasn’t yet been published. In the paper, we examine the information content of option and equity volumes when trade direction is unobserved. In a multimarket asymmetric information model, equity short-sale costs result in a negative relation between relative option volume and future firm value. In our empirical tests, firms in the lowest decile of the option to stock volume ratio (O/S) outperform the highest decile by 0.34% per week (19.3% annualized). Our model and empirics both indicate that O/S is a stronger signal when short-sale costs are high or option leverage is low.

1 CHAPTER 1. INTRODUCTION 2

O/S also predicts future firm-specific earnings news, consistent with O/S reflecting private information. The second chapter, “Equity Risk Premia and the VIX Term Structure” is my job market paper. In it, I show that in many asset pricing models, the equity market’s expected return is a time-invariant linear function of its conditional variance, which can be estimated from options markets. However, I show that when the relation between conditional means and variances is state-dependent, an observer requires the combined information in multiple variance horizons to distinguish among the states and thereby reveal the equity risk premium. Empirically, I show that while the VIX by itself has little predictive power for future S&P 500 returns, the VIX term structure predicts next-quarter S&P 500 returns with a 5.2% adjusted R2. Chapter 2

The Option to Stock Volume Ratio and Future Returns

2.1 Introduction

In recent decades, the availability of derivative securities has rapidly expanded. This expansion is not limited to equity options and now includes a vast array of securities ranging from currency options to credit default swaps. Derivatives contribute to price discovery because they allow traders to better align their strategies with the sign and magnitude of their information. The leverage in derivative securities, combined with this alignment, creates additional incentives to generate private information. In this way, trades in derivative markets may provide more refined and precise signals of the underlying asset’s value than trades of the asset itself. Understanding how and why derivatives affect price discovery is therefore vital to understanding how information comes to be in asset prices. This study focuses on the information content of trading volumes. Observed transactions play an important role in price discovery because order flow imbalances can reflect the sign and magnitude of private information. While market makers can observe these imbalances, most outside observers cannot, which makes the problem of inferring private information more complex. Techniques to empirically estimate order flow imbalances are computationally intensive, typically requiring the pairing of intraday trades and quotes. This problem is exacerbated when agents have access

3 CHAPTER 2. OPTION TO STOCK VOLUME RATIO AND RETURNS 4

to multiple trading venues because the mapping between transactions and private information becomes more difficult to identify. In this paper, we address the inference problem of the outside observer by examining the information content of option and equity volumes when agents are privately informed but trade direction is unobserved. We provide theoretical and empirical evidence that informed traders’ private infor- mation is reflected in O/S, the ratio of total option market volume (aggregated across calls and puts) to total equity market volume. The O/S measure was first coined and studied by Roll, Schwartz, and Subrahmanyam (2010), whose findings suggest that cross-sectional and time-series variation in O/S could be driven by informed trade. As a natural extension of these findings, we examine the relation between O/S and future returns. Empirically, we find that contrasting publicly available totals of firm- specific option and equity volume portends directional prices changes, in particular that low O/S firms outperform the market while high O/S firms underperform. At the end of each week, we sort firms by O/S and compute the average return of a portfolio consisting of a short position in stocks with high O/S and a long position in stocks with low O/S. This portfolio provides an average risk-adjusted hedge return of 0.34% in the week following the formation date (19.3% annualized). If option volume is concentrated among risky firms with higher return volatility, one might anticipate the opposite result, namely that firms with higher O/S earn higher future returns. While our finding is inconsistent with this risk-based expla- nation, we take several steps to mitigate concerns that exposure to other forms of risk (liquidity risk, for example) explains the O/S-return relation. First, we show that the relation holds after controlling for exposure to the three Fama-French and momentum factors. Second, we show that the predictive power of O/S for future returns is relatively short-lived. Strategy returns rapidly decline from 0.34% in the first week following portfolio formation and become statistically insignificant beyond the sixth week. Third, to mitigate concerns that our results are driven by static firm characteristics correlated with O/S and expected returns, we show that two measures of within-firm changes in O/S also predict future returns. We argue that the negative relation between O/S and future returns is driven by short-sale costs in equity markets, which make option markets an attractive venue for traders with negative news. Motivated by this story, we model the capital allocation CHAPTER 2. OPTION TO STOCK VOLUME RATIO AND RETURNS 5

decision of privately informed traders who can trade in option and equity markets. Equity short-sale costs lead informed agents to trade options more frequently for neg- ative signals than positive ones, thus predicting a negative relation between relative option volume and future equity value. An important innovation of our paper is that this relation does not require classifying trades as being buyer- versus seller-initiated. Instead, our theoretical predictions and empirical tests rely on publicly available vol- ume totals. Having established the negative cross-sectional relation between O/S and future returns, we next test our model’s prediction that this relation is stronger when short- sale costs are high. As short-sale costs increase, informed traders are more likely to switch from equities to options for negative signals, which strengthens the O/S- return relation. We test this prediction using three different measures of short-sale costs. The first measure is derived from institutional ownership, as in Nagel (2005), and is available throughout our 1996–2010 sample window. We also use two direct measures of short-sale costs, transacted loan fees and available loan supply, from a proprietary database of institutional lending that is available on a monthly basis from 2002 through 2009. Across all three measures, we find that portfolio alphas associ- ated with O/S are generally increasing in the cost of shorting, though the statistical significance of this pattern is mixed. An additional empirical prediction arising from our model is that the O/S-return relation is weaker when option leverage is high. As option leverage increases, bid-ask spreads in options markets increase, which weakens the O/S-return relation because the bid-ask spread acts like a switching cost for traders considering the use of options to avoid short-sale costs. When option market bid-ask spreads are larger, fewer traders switch from equities to options for negative signals, and the O/S-return relation is therefore weaker. Empirically, we find that portfolio alphas associated with O/S are monotonically decreasing in option leverage. It may be initially puzzling why we do not find a relation between the ratio of call to put volume and future returns. Our model demonstrates that O/S provides a clearer signal of private information than the ratio of call to put volume because call volume could be good news (if informed traders are buying) or bad news (if informed traders are selling), and put volume is similarly ambiguous. Thus, in the absence CHAPTER 2. OPTION TO STOCK VOLUME RATIO AND RETURNS 6

of information about the sign of each trade (i.e., buy vs. sell), O/S is an indication of the sign of private information while the ratio of call to put volume is not. Our model does, however, predict a positive relation between call-put volume differences and future return skewness because informed traders buy calls (puts) for extremely good (bad) news and sell calls (puts) for moderately bad (good) news. Consistent with this prediction, we show empirically that the ratio of call volume to put volume predicts return skewness in the subsequent week. We also find that O/S predicts the sign and magnitude of earnings surprises, standardized unexplained earnings, and abnormal returns at quarterly earnings an- nouncements in the following week. These tests show that the same O/S measure we use to predict weekly returns also contains information about future earnings an- nouncements that occur in the subsequent week. This is consistent with O/S reflect- ing private information that is incorporated into equity prices following a subsequent public disclosure of the news. The rest of the paper is organized as follows. We begin in Section 2.2 by dis- cussing our results in the context of existing literature. We model the multimarket price discovery process and formalize the equilibrium strategy of informed traders in Section 2.3. In Section 2.4, we describe the data, methodology, empirical results, and robustness checks. Finally, we present results pertaining to quarterly earnings announcements in Section 2.5 and conclude in Section 2.6.

2.2 Relation to prior research

The two immediate antecedents of our work are Easley, O’Hara, and Srinivas (1998), hereafter referred to as EOS, and Roll, Schwartz, and Subrahmanyam (2010), here- after RSS. EOS contains a multimarket equilibrium model wherein privately informed traders are allowed to trade in both option and equity markets.1 The EOS model highlights conditions under which informed traders transact in both option and equity markets, and predicts that directional option volume signals private information not

1The authors point out that asymmetric information violates the assumptions underlying com- plete markets and, therefore, the option trading process is not redundant. Consistent with this idea, Bakshi, Cao, and Chen (2000) find that Standard & Poors (S&P) 500 call options frequently move in the opposite direction of equity prices. CHAPTER 2. OPTION TO STOCK VOLUME RATIO AND RETURNS 7

yet reflected in equity prices. Specifically, their model predicts that positive trades (i.e., buying calls and selling puts) are positive signals of equity value and that neg- ative trades (i.e., selling calls and buying puts) are negative signals of equity value. An interesting but otherwise unexplored empirical finding in EOS is that negative option market activity carries greater predictive power for future price changes. EOS comment on this finding in the following excerpt:

An interesting feature of our results is the asymmetry between the negative- and positive-position effects ... suggesting that options markets may be relatively more attractive venues for traders acting on ‘bad’ news. An often-conjectured role for options markets is to provide a means of avoid- ing short-sales constraints in equity markets ... Our results support this conjecture, suggesting a greater complexity to the mechanism through which negative information is impounded into stock prices [p. 458].

We provide a formal means of understanding their finding by introducing short- sale costs into a microstructure framework with asymmetric information. Like EOS, informed agents trade with a risk-neutral market maker, and can buy or sell shares of stock, buy or sell calls, or buy or sell puts. Unlike EOS, we impose short-sale costs that play a central role in determining which assets informed traders choose to trade. It is comparatively cheaper to capitalize on bearish private signals in option markets because traders can buy puts or sell calls, and in both cases they can create new option contracts without first borrowing them from a third party. In our model’s equilibrium, the costs associated with short-selling make informed traders more likely to use options for bad signals than for good ones and, as a result, high O/S indicates negative private information and low O/S indicates positive private information. Like EOS, we solve a static model and therefore need the additional assumption that some friction prevents equity prices from immediately reflecting the information in option volumes in order for the model’s prediction about the conditional mean equity value to translate into return predictability. Our main empirical prediction, that O/S is a negative cross-sectional signal of future returns, differs from EOS in that it can be tested empirically without signing the direction of trades. We predict and confirm that contrasting publicly available totals of firm-specific option and equity volume portends directional price changes. CHAPTER 2. OPTION TO STOCK VOLUME RATIO AND RETURNS 8

Empirically, our study of the relation between O/S and future returns is a natural extension of the work in RSS, which introduces the option to stock volume ratio, and coins it O/S. The authors find substantial intertemporal and cross-sectional variation in O/S, and explain a significant part of this variation in a regression framework. In particular, O/S is increasing in firm size and implied volatility but decreasing in option bid-ask spreads and institutional holdings. Our results shed additional light on the variation in O/S by examining the theoretical determinants of relative option volume when a subset of market participants is privately informed, and the empirical relation between O/S and future returns. RSS also show that O/S in the days immediately prior to announcement predicts the magnitude of returns at earnings announcements, consistent with O/S reflecting traders’ private information. Conditional on there being positive (negative) earnings news, they find that O/S predicts higher (lower) announcement returns (see Section 2.5 for more details). Our analysis builds upon this finding by demonstrating an unconditional predictive relation between the prior week’s O/S and earnings surprises. Another recent paper examining option volume is Roll, Schwartz, and Subrah- manyam (2009), which shows a positive cross-sectional relation between Tobin’s q and unscaled option volume. The authors interpret this as evidence that liquid op- tion markets increase firm value because they help complete markets and generate informed trade. Our model and empirical tests support this intuition by demonstrat- ing that option markets are an attractive venue for informed traders. The results of this paper also relate to the literature on price discovery and in- formation flow in multiple markets.2 Pan and Poteshman (2006) use proprietary Chicago Board Options Exchange (CBOE) option market data and provide strong evidence of informed trading in option markets. The authors find that sorting stocks by the amount of newly initiated positions in puts relative to calls foreshadows fu- ture returns but they conclude the predictability is not due to market inefficiencies and instead reflects the fact that their volume measure is not publicly observable. A key innovation of our paper is demonstrating that publicly available, non-directional

2Whether option markets lead equity markets or vice versa remains an open question. Anthony (1988) examines the interrelation of stock and option volumes and finds that call-option activity predicts volume in the underlying equity with a one-day lag. Similar findings are reported in Man- aster and Rendleman (1982). In contrast, Stephan and Whaley (1990) find no evidence that options lead equities. CHAPTER 2. OPTION TO STOCK VOLUME RATIO AND RETURNS 9

volume totals predict future returns. Similarly, Cremers and Weinbaum (2010) and Zhang, Zhao, and Xing (2010) find that publicly available asymmetries in implied volatility across calls and puts predict future returns. Prior research establishes that equity volume, the denominator of our primary re- turn predictor O/S, is useful by itself in predicting future returns, though the direction depends on the way volume is measured (see, e.g., Gervais, Kaniel, and Mingelgren, 2001; Lee and Swaminathan, 2000; Brennan, Chordia, and Subrahmanyam, 1998). We decouple O/S into separate measures of equity and option volume and show that past option volume is negatively related to future returns incremental to past equity volume. Other extant work uses equity volume as a conditioning variable for examin- ing the relation between past and future returns. Specifically, Lee and Swaminathan (2000) show that high (low) volume winners (losers) experience faster momentum re- versals, and Llorente et al. (2002) show that the relation between equity volume and return autocorrelation changes sign depending on the amount of informed trading for a given equity. During the 2008 financial crisis, the U.S. Securities and Exchange Commission (SEC) banned short sales for 797 ‘financial’ stocks, providing an interesting case study of the impact of short-sale costs on options markets. Both Battalio and Schultz (2011) and Grundy, Lim, and Verwijmeren (2011) find that option market spreads increased and option market volume decreased for firms subject to the ban relative to those exempt from it. A key component of our model is that option markets serve as an alternative venue for negative news when shorting is costly, and at first glance, the 2008 episode contradicts this premise. However, as emphasized in Battalio and Schultz (2011), the short-sale ban also imposed costs on option market makers who short equity, making it more difficult for them to hedge when selling puts or buying calls. In our model, increasing costs for market makers who write puts or buy calls will increase spreads and decrease volume in option markets, while banning shorts will increase both spreads and volume in options markets. Our model therefore suggests we interpret the decrease in option market volume during the 2008 ban as a result of the added costs of shorting for market makers outweighing the relocation of trades stemming from negative information to option markets. CHAPTER 2. OPTION TO STOCK VOLUME RATIO AND RETURNS 10

In modelling the relation between short-sale costs and informed trading, our paper is also related to Diamond and Verrecchia (1987), which models the impact of short- sale constraints on the speed of adjustment of security prices to private information when informed traders only have access to equity markets. In their model, short- sale constraints cause some informed parties with negative information not to trade. Thus, the absence of trade in their model is a negative signal of future firm value. In our model, trading options is an alternative to abstaining from trade when the cost of shorting is high. As a result, a high option volume ratio, rather than the absence of trade, reflects negative private information.

2.3 The model

We present a model of informed trading in both equity and options markets in the presence of short-sale costs. Informed traders build a portfolio by trading sequentially with a competitive, risk-neutral market maker. A key feature of our model is that traders must pay a lending fee to a third party when shorting stock. Because it is costly to trade on bad news in the stock market, in equilibrium the mean equity value conditional on an option trade is lower than the mean equity value conditional on a stock trade. There are three tradable assets in the model: an equity, a call option, and a put option. The stock liquidates for V˜ at time t = 2 in the future. The value of V˜ is unknown prior to t = 2, but it is common knowledge that

V˜ = µ + ˜+η, ˜ (2.1) where µ is the exogenous mean equity value, and ˜ andη ˜ are independent, normally 2 2 distributed shocks with zero mean and variances σ and ση. The call and put are both struck at µ, and both expire at time t = 2. We focus on the case of European options with a single strike price because our aim is to model the choice between trading options and trading equities, as opposed to the choice amongst options of different strikes or times to expiration. We use µ as a strike price so that calls and puts have the same leverage. Trade occurs at time t = 1, at which point a fraction α of the traders (henceforth CHAPTER 2. OPTION TO STOCK VOLUME RATIO AND RETURNS 11

“informed traders”) know the realization of ˜ but the remaining traders, and the market maker, do not. The distribution of V˜ conditional on the information that ˜ =  is: ˜ 2 V |(˜ = ) ∼ N(µ + , ση). (2.2) Informed traders are risk-neutral, and therefore value the stock, call, and put using:

E(V˜ |˜ = ) = µ +  (2.3)

    ˜   E(C|˜ = ) = Φ  + φ ση (2.4) ση ση     ˜ −  E(P |˜ = ) = −Φ  + φ ση, (2.5) ση ση respectively, where Φ is the standard normal’s cumulative distribution function, φ is its probability distribution function, and C˜ and P˜ are the values of the call and put at t = 2. We require that each trade be in exactly one type of asset, resulting in six possible trades: buy or sell stock, buy or sell calls, and buy or sell puts. At equilibrium prices, the informed traders have a strict preference among the assets for all signals other than six cutoff points.3 A fraction 1 − α of the traders are uninformed and trade for reasons outside the model, possibly a desire for liquidity, the need to hedge other investments or human capital, or a false belief that they have information. Regardless of their motivation, uninformed traders choose among the same possible transactions as the informed traders, with fractions q1, q2, q3, q4, q5, and q6 choosing to buy stock, P6 sell stock, buy calls, sell calls, buy puts, and sell puts, respectively, where i=1 qi = 1. A competitive and risk-neutral market maker posts bid and ask prices for all three 4 assets that result in zero expected profit for each trade. For notation, we write as,

3Allowing trades in bundles of multiple assets (for example, one call and two shares) complicates the analysis without changing our results or providing additional insight. Bundles serve as “inter- mediate” portfolios used by the informed trader upon receiving a signal near their indifference point between the two bundled assets. As long as bundle trades that include a short position in equities require traders to pay short-sale costs, our model still predicts that equity (option) volume reflects positive (negative) private information. 4In the model, additional market makers have no impact as long as they are risk-neutral and competitive. The return predictability evidence in this paper suggests there is some segmentation between option and equity markets, perhaps because they have different market makers. CHAPTER 2. OPTION TO STOCK VOLUME RATIO AND RETURNS 12

bs, ac, bc, ap, and bp for the ask and bid prices of the stock, call, and put, respectively. As in EOS and Glosten, Jagannathan, and Runkle (1993), trades occur sequentially and at fixed order sizes: γ shares of stock and θ options contracts. Unlike EOS, we assume throughout that θ > 2γ so that options trades allow more exposure to the underlying than stock trades, an intuition expressed in Black (1975) as well as EOS and RSS (see Appendix B for details). A critical new ingredient in our model is a short-sale cost paid by the trader to a third party who lends them the shares. The fee is a fraction ρ > 0 of the total amount shorted γbs. The lender is able to charge such a fee because they have some market power, or because there is some counterparty risk. No such fee exists when writing options because there is no need to find a contract to borrow—the market maker can create a new contract. The parameter ρ can also represent a reduced form of any cost to shorting stock; for example, recall risk or the indirect costs described in Nagel (2005). Regardless of ρ’s interpretation, the market maker pays γbs for γ 5 shares, but the trader only nets γbs(1 − ρ) from the transaction.

A Equilibrium

An equilibrium in our model consists of an optimal trading strategy for informed traders as a function of their signal, and bid-ask prices and quantities that yield zero expected profit for the market maker. In equilibrium, informed traders use the following cutoff strategy f() that maps the range of possible signals to the space of possible trades:

5It is important for our argument that option market makers do not pay the short-sale cost ρ in the course of hedging their position, and therefore embed the short-sale cost in option prices. In reality, option market makers have access to cheaper shorting than ordinary investors, and therefore the option-embedded short-sale cost is smaller than the actual short-sale cost in equity markets. CHAPTER 2. OPTION TO STOCK VOLUME RATIO AND RETURNS 13

 buy puts for  ≤ k ,  1  sell stock for  ∈ (k1, k2],  sell calls for  ∈ (k , k ],  2 3 f() = make no trade for  ∈ (k3, k4], (2.6)  sell puts for  ∈ (k4, k5],  buy stock for  ∈ (k , k ],  5 6  buy calls for  > k6.

For extremely good or bad signals, informed traders buy options despite large bid-ask spreads in these markets because they provide greater leverage. The bid-ask spreads make options unattractive for weaker signals, and so informed traders trade equities instead. For even weaker good or bad signals, however, informed traders value the stock near its unconditional mean and therefore cannot profitably trade stock. However, they value the options well below their unconditional mean because extreme outcomes occur with lower probabilities, and therefore sell options. If bid prices are below informed traders’ valuation of both a put and a call for a given signal, informed traders choose not to trade. The cutoff points ki arise endogenously in equilibrium and are chosen so that informed traders strictly prefer writing puts for all  < k1, selling stock for all k1 <  < k2, etc. Some regions can be empty in equilibrium, meaning ki = ki+1 for some i. The addition of short-sale costs shrinks the region of signals for which informed traders short stock (k1, k2].

The bid and ask prices for each asset (as, bs, ac, bc, ap, and bp) and the informed traders’ cutoff points ki are the 12 equilibrium parameters. Together they satisfy 12 equations, presented fully in Appendix A.1, which assure that the market maker’s expected profit is zero for each trade and that informed traders are indifferent between the two relevant trades at each cutoff point. CHAPTER 2. OPTION TO STOCK VOLUME RATIO AND RETURNS 14

B Results and empirical predictions

Due to the nonlinearity of the simultaneous equations, no closed-form solution for the equilibrium parameters is available. We derive our results and empirical predictions from the simultaneous equations. Our focus is on the information content of trading volumes when there are short-sale costs, so we assume throughout that ρ > 0. Proofs are in Appendix A.3.

Result 1. When each asset is equally likely to be bought or sold by an uninformed trader, the stock is worth less conditional on an option trade than it is conditional on a stock trade.

Empirical Prediction 1. Option volume, scaled by volume in the underlying equity, is negatively related to future stock returns.

The main result is that an option trade is bad news for the value of the stock and a stock trade is good news, which differs from EOS in that the conditioning variable is the location of trade rather than the direction of trade. Option volume reflects bad news because informed traders use stocks more frequently to trade on good news than bad due to the short-sale cost. Therefore, the expected equity value conditional on an option trade is lower than the unconditional mean, which is in turn lower than the expectation conditional on a stock trade. Result 1 requires that uninformed traders buy and sell each asset with equal probability, but holds regardless of how uninformed traders distribute their demand across the different assets; for example, uninformed traders may trade equities more frequently than options. To translate Result 1 into an empirical prediction, we consider the implications of our static model in a multiperiod setting. If equity markets fully incorporate the CHAPTER 2. OPTION TO STOCK VOLUME RATIO AND RETURNS 15

information revealed through options trading into their valuations, stock prices will immediately reflect the new conditional expectation of V˜ after each option trade. Otherwise, stock prices do not fully reflect the information content of options trades for the time between when the informed option market trading occurs and when the information becomes public through another channel. In this case, there will be a negative relation between option volume and subsequent returns until the public release of the information. Empirical Prediction 1 is, therefore, a joint hypothesis that (a) short-sale costs make O/S a negative cross-sectional predictor of future prices, and (b) some of the information in O/S reaches equities through other channels, such as earnings announcements, that occur after the observation of O/S.6 Our model makes no prediction about the overall volume in options and stocks together, only that option trades are bad news relative to stock trades. Our goal is to focus on informed traders’ choice between equities and options, conditional on having a signal about the future value of a firm. Therefore, our predictive measure is the ratio of option volume to equity volume, rather than unscaled option volume.

Result 2. The disparity in conditional mean equity values between option and stock trades is weakly increasing in the short-sale cost ρ.

Empirical Prediction 2. The predictive power of relative option volume for future stock returns is increasing in the cost of shorting equity.

Although Empirical Prediction 1 does not rely on cross-sectional differences in short-sale costs, if such differences exist, our model predicts that option volume is a

6A common intuition is that call volume reflects good news and put volume reflects bad news. Eq. (2.6) demonstrates that this intuition does not hold in our setting because informed traders buy calls and sell puts for good news, and buy puts and sell calls for bad news. Therefore, unless trade direction is observable, it is unclear whether call (put) volume reflects good or bad news. CHAPTER 2. OPTION TO STOCK VOLUME RATIO AND RETURNS 16

worse signal for high short-sale cost equities than low short-sale cost equities, but is still a valuable signal as long as short-sale costs exist.

Result 3. The disparity in conditional mean equity values between option and stock

∂C S θ trades is weakly decreasing in the option’s leverage λ ≡ ∂S C = 2γ .

Empirical Prediction 3. The predictive power of relative option volume for future stock returns is decreasing in the average Black-Scholes λ of options traded.

Result 3 may be surprising at first because leverage is usually an attractive feature of options. Indeed, in our model leverage allows an informed trader’s investment to be more sensitive to their private information, and therefore the overall use of op- tions by informed traders increases with leverage. However, this very attractiveness creates large bid-ask spreads in options markets, making it more expensive for in- formed traders to switch from trading equities to options to avoid the short-sale cost. Therefore, they make this switch for a smaller range of signals, which weakens the O/S-return relation. Empirically, Result 3 suggests that volume in options markets with higher leverage provides a weaker signal than volume in options markets with lower leverage. For a

∂C S measure of leverage, we use λ = ∂S C , the elasticity of C with respect to S, reflecting ∂C S θ the “bang-for-the-buck” notion of leverage. We show in Appendix A.2 that ∂S C = 2γ in our model. Empirically, we use the Black-Scholes λ because our model’s λ does not account for different strike prices. Result 3 indicates there exists a spread between conditional means of V˜ regardless of the leverage λ, but that the spread is larger for smaller values of λ. Therefore, our empirical prediction is that O/S predicts returns for all levels of λ, but most strongly for low λ. CHAPTER 2. OPTION TO STOCK VOLUME RATIO AND RETURNS 17

Result 4. Equity value has higher skewness conditional on a call trade than condi- tional on a put trade.7

Empirical Prediction 4. The ratio of call volume to put volume varies positively with the future skewness of stock returns.

Result 4 follows from the equilibrium trading strategy described in Section 2.3.1. Following the notation used to describe the informed trader’s strategy in Eq. (2.6), skewness conditional on a put trade is low because, if informed, it reflects either mod- erately good news (i.e.,  ∈ (k4, k5]) or extremely bad news (i.e.,  ≤ k1). Similarly, skewness conditional on a call trade is high because, if informed, it reflects either moderately bad (i.e.,  ∈ (k2, k3]) or extremely good news (i.e.,  > k6).

2.4 Empirical tests

The option data for this study come from the Ivy OptionMetrics database, which provides end-of-day summary statistics on all exchanged-listed options on U.S. equi- ties. The summary statistics include option volume, quoted closing prices, and option Greeks. The OptionMetrics database, and hence the sample for this study, spans from 1996 through 2010. The final sample for this study is dictated by the intersection of OptionMetrics, Compustat Industrial Quarterly, and Center for Research in Security Prices (CRSP) Daily data. We restrict the sample to firm-weeks with at least 25 call and 25 put contracts traded to reduce measurement problems associated with illiquid option markets. We require each observation to have a minimum of six months of

7 Our proof of this result requires that the uninformed trader demand for each asset γi does not approach zero. If it did, markets would begin to fail and the skewness result can reverse. CHAPTER 2. OPTION TO STOCK VOLUME RATIO AND RETURNS 18

past weekly option and equity volumes because some of our analyses involve measur- ing firms’ volumes relative to their historical averages. We also eliminate closed-end funds, real estate investment trusts, American depository receipts, and firms with a stock price below $1. The intersection of these databases and data restrictions results in 611,173 firm-weeks corresponding to approximately 730 calendar weeks and 1,660 unique firms per year. For each firm i in week w, we sum the total option and equity volumes, denoted by OPVOLi,w and EQVOLi,w, respectively. Specifically, OPVOLi,w equals the total volume in option contracts across all strikes for options expiring in the 30 trading

8 days beginning five days after the trade date. We report EQVOLi,w in round lots of 100 to make it more comparable to the quantity of option contracts that each pertain to 100 shares. We define the option to stock volume ratio, or O/Si,w, as:

OPVOLi,w O/Si,w = . (2.7) EQVOLi,w

Panel A of Table 2.1 contains descriptive statistics of O/Si,w (hereafter O/S for notational simplicity) for each year in our sample. The sample size increases substan- tially over the 1996–2010 window. The number of firm-weeks increases from 29,426 in 1997 to 45,243 in 2010.9 The remainder of Panel A presents descriptive statistics of O/S for each year of the sample. The sample mean of O/S is 5.77%, which indicates

8We exclude options expiring within five trading days to avoid measuring mechanical trading volume associated with option traders rolling forward to the next expiration date. The results are qualitatively unchanged if we include options with longer expirations. As an additional robustness check, we separate option volume into moneyness categories and find that at-the-money, in-the- money, and out-of-the-money option volumes all predict future returns once scaled by equity volume. The consistency of the O/S-return relation across moneyness categories mitigates concerns that our model omits critical determinants of the O/S-return relation by focusing on a single strike price. 9In our sample, 1996 has many fewer firm-week observations due to the requirement that six months of prior data be available for each firm. CHAPTER 2. OPTION TO STOCK VOLUME RATIO AND RETURNS 19

that there are roughly 17 times more equity round lots traded than option contracts with times to expiration between five and 35 trading days. O/S is positively skewed throughout the sample period due to a high concentration of relative option volume among a small subset of firms. Panel B of Table 2.1 presents volume characteristics by deciles of O/S. Although low O/S firms tend to be smaller, our initial data requirement of 25 calls and 25 puts traded in a week tilts our sample toward larger and more liquid firms, which mitigates, but fails to eliminate, concerns that the O/S-return relation is attributable to transaction costs. The average market capitalization of firms exceeds $2 billion in each O/S decile. VLC and VLP indicate the number of call and put contracts traded in a given week, respectively. Across all deciles of O/S, the number of call contracts traded exceeds the number of put contracts, which is consistent with calls being more liquid than puts. High O/S firms also tend to have higher levels of both option and equity volume, though equity volume changes much less across the O/S deciles. In our model, high O/S reflects negative private information, and hence our univariate trading strategy based on O/S consists of taking a short position in higher equity volume stocks (i.e., high O/S stocks) and a long position in lower equity volume stocks (i.e., low O/S stocks). This raises concerns that the predictive power of O/S could reflect compensation for taking positions in low liquidity firms. We attempt to mitigate these concerns in several ways, which are discussed in greater detail below. Panel B also presents firm characteristics by deciles of O/S. SIZE (LBM) equals the log of market capitalization (book-to-market) corresponding to firms’ most recent quarterly earnings announcement. MOMEN equals firms’ cumulative return measured over the prior six months. High O/S firms tend to be larger, have lower CHAPTER 2. OPTION TO STOCK VOLUME RATIO AND RETURNS 20

book-to-market (BM) ratios, and higher return momentum. Panel A of Table 2.2 presents time-series factor regressions for each O/S decile using capital asset pricing model (CAPM), three-factor, and four-factor risk adjust- ments. To compute weekly O/S decile returns, we sort firms by O/S at the end of each week, skip one trading day, and compute the equal-weighted return for a port- folio of all firms in each decile over the following five trading days.10 For example, when there are no trading holidays, we compute O/S from Monday through Friday of a given calendar week, skip the Friday-to-Monday return, and compute a weekly return from the close of markets on Monday to the close of markets on the following Monday. To calculate four-factor portfolio alphas, we regress the weekly excess return cor- responding to each O/S decile on the contemporaneous three Fama-French and mo- mentum factors.11 Specifically, we estimate three variants of the following regression for each O/S decile:

p f mkt f rw − rw = α + β1(rw − rw) + β2HMLw + β3SMBw + β4UMDw + w, (2.8)

p where rw is the week w return on an equal-weighted portfolio of stocks in a given

f mkt O/Si,w−1 decile. We denote the risk-free rate as rw and the market return as rw .

HMLw and SMBw correspond to the weekly returns associated with high-minus-low market-to-book and small-minus-big strategies. Similarly, UMDw equals the weekly return associated with a high-minus-low momentum strategy. The CAPM model

10This restriction is important because of non-synchronous closing times across option and equity markets. Removing this restriction does not materially affect our results. 11We compute weekly factors to match our Monday close to subsequent Monday close time-frame by first compounding the returns for each of the size/BM and size/momentum portfolios and then computing the long-short return that defines the factors, as described on Ken French’s Web site. CHAPTER 2. OPTION TO STOCK VOLUME RATIO AND RETURNS 21

mkt f omits all factors except for rw − rw and the three-factor model omits UMDw. Our main result is that the intercepts from these regressions decrease with O/S, indicating that low O/S firms outperform high O/S firms. In the four-factor regres- sion, we find that the portfolio of firms in the lowest O/S decile have a 0.19% alpha in the following week while the portfolio of firms in the highest O/S decile have a -0.15% alpha. The “1–10” row at the bottom of the table contains a statistical test for the difference of the low and high O/S decile portfolios, and shows that the 0.34% differ- ence in the four-factor alphas are highly significant (t-statistic = 5.00). The result is similar in statistical and economic magnitude for the CAPM and three-factor regres- sions, resulting in low−high alphas of 0.34% (t-statistic = 4.20) and 0.30% (t-statistic = 4.29), respectively. The final “(1+2)–(9+10)” row contains a statistical test for the difference between low and high O/S quintile portfolios, formed by combining the two lowest and two highest decile portfolios. The quintile strategy alphas attenuate relative to the decile strategy but remain economically and statistically significant for all three factor models. As predicted by our model, in addition to high O/S indicating bad news, low O/S indicates good news: a portfolio of firms with low O/S has significantly positive alphas in the week after portfolio formation. In the context of our model, low relative option volume indicates good news because informed traders use equity more (and options less) frequently for positive signals than negative ones due to the equity short-sale costs. Table 2.2 also presents the factor loadings (β coefficients from the estimates of Eq. (2.8)). We find that the low-high O/S strategy has a significantly negative loading on the market and UMD factors and a positive loading on the SMB factor. The CHAPTER 2. OPTION TO STOCK VOLUME RATIO AND RETURNS 22

negative market beta indicates that high O/S firms have more market exposure than low O/S firms, the opposite of what one would expect if the O/S-return relation reflects exposure to market risk. The remaining factor loadings confirm the univariate patterns shown in Table 2.1 in a multivariate setting: low O/S firms tend to be smaller firms with low book-to-market ratios and low momentum. One potential concern with the results in Panel A of Table 2.2 is that some firms could have consistently higher O/S and lower average returns for reasons unrelated to our information story and not captured by the four-factor risk adjustment. To address this concern, Panels B and C of Table 2.2 rexamine our return predictability tests after sorting by within-firm changes in O/S. In Panel B, we sort firms by ∆O/S, the change in O/S relative to a rolling average of past O/S for each firm. Specifically, we define ∆O/S as:

O/Si,w − O/Si ∆O/Si,w = , (2.9) O/Si where O/Si is the average O/Si,w for the firm over the prior six months. We sort the cross-section of firms by ∆O/S in each calendar week. Firms in the lowest decile of ∆O/S earn a four-factor alpha of 0.16% per week (t-statistic = 1.91). Similarly, firms in the highest decile of ∆O/S earn -0.10% per week (t-statistic = -1.38). The ∆O/S decile strategy produces an alpha of 0.27% per week (t-statistic = 4.38) and the ∆O/S quintile strategy produces an alpha of 0.17% per week (t-statistic = 3.49). The ∆O/S factor loadings are closer to zero than those for O/S, consistent with our change-based specification mitigating the influence of persistent firm characteristics that arise when sorting firms by the level of O/S. The two loadings that remain significant are SMB, which switches signs from positive in Panel A to negative in Panel B, and UMD, which remains strongly negative. CHAPTER 2. OPTION TO STOCK VOLUME RATIO AND RETURNS 23

As an alternative means of calculating abnormal levels of O/S, Panel C of Table 2.2 estimates factor-adjusted portfolio returns after sorting firms based on within-firm variation in O/S. We sort each firm-week into an ΩO/S decile by ranking it relative to the firm’s O/S time-series over the past six months (26 weeks). For example, firms in the highest ΩO/S decile have O/S above the 90th percentile of their own O/S distribution measured over the past six months. Given the large swings in O/S documented in RSS, we use six months of data to ensure that the firm’s reference distribution has a sufficiently large number of observations to capture meaningful firm- specific variation in O/S. Our choice of six months is in line with Lee and Swaminathan (2000), Llorente et al. (2002), Barber and Odean (2008), and Sanders and Zdanowicz (1992) that all use periods of six to 12 months to establish baseline levels of firm- specific volume. We find that the use of a longer reference window, such as one year, produces qualitatively similar results but reduces the number of observations available for our analyses. We also find similar results when using a shorter reference window, such as the ten weeks used in Gervais, Kaniel, and Mingelgrin (2001), however doing so reduces the robustness of ΩO/S in predicting future returns. The benefit of ΩO/S relative to ∆O/S is that it relies on firms’ own rolling distri- bution to assess abnormal levels of O/S and thus can be calculated for a single firm without reference to the cross-sectional distribution. The cost of not referencing the cross-sectional distribution is that ΩO/S is more sensitive to market-wide changes in option or equity volumes that are unrelated to firm-specific private information. Be- cause ΩO/S relies on pure time-series sorts, unlike the cross-sectional sorts in Panels A and B, there is no guarantee that we have an equal number of firms in each decile of ΩO/S in a given week. Regardless of the number of firms in each ΩO/S bin, we CHAPTER 2. OPTION TO STOCK VOLUME RATIO AND RETURNS 24

compute the weekly return of an equal-weighted portfolio of all constituent firms. We find that a portfolio consisting of a long position in the lowest decile of ΩO/S firms and a short position in highest decile of ΩO/S firms earns a four-factor alpha of 0.21% (t-statistic = 2.45), while a quintile strategy produces a four-factor alpha of 0.18% (t-statistic = 2.82). The consistency of return predictability across O/S, ∆O/S, and ΩO/S mitigates concerns that the O/S-return relation reflects compensation for a static form of risk. Our main analyses focus on the relation between O/S and weekly returns. We chose a weekly horizon, rather than daily or monthly, to balance competing concerns. Although our model does not formally define the length of a given period, the en- dogenous determination of bid-ask spreads is intuitively linked to short horizons, for example the intraday volumes in EOS and the daily O/S in RSS. On the other hand, shorter horizons are subject to the concern that the pattern of predictable returns is attributable to portfolio rebalancing costs. While we present results here pertain- ing to weekly observations of O/S and returns, untabulated results demonstrate that our inferences are unchanged when conducting the analysis using daily or monthly sampling frequencies.12 Table 2.3 presents summary statistics from weekly Fama-MacBeth regressions where the dependent variable is the firm’s return during the week after observing O/S, denoted by RET(1). Columns 1 through 3 of Panel A contain the results of regressing RET(1) on deciles of O/S. For example, in column 1 the O/S coefficient

12The main exception pertains to the concentration of the O/S-return relation among firms with high short-sale costs. Across all three horizons, we find that O/S strategy alphas are increasing in short-sale costs. However, while this result is statistically significant for the monthly and weekly horizons, it is not for the daily horizon. The absence of this effect at daily horizons is consistent with short-sale costs reflecting market frictions that prevent the information content of O/S from being reflected in equity prices in an immediate fashion. Untabulated results available from the authors upon request. CHAPTER 2. OPTION TO STOCK VOLUME RATIO AND RETURNS 25

is -0.026, indicating that firms in the highest O/S decile outperform firms in the lowest decile by an average of 0.23% (=−0.026 × 9) per week. The O/S coefficient has a corresponding t-statistic of -3.99, where standard errors are computed across weekly coefficient estimates as in Fama and MacBeth (1973). Columns 1 through 7 demonstrate that the relation between O/S and RET(1) is robust to controlling for MOMEN, log market capitalization (SIZE), and log book-to-market (LBM). Columns 2 through 7 also control for the Amihud (2002) illiquidity ratio, AMIHUD, defined as the ratio of absolute returns to total dollar volume where higher values indicate lower liquidity, and vice versa. AMIHUD is measured on a daily basis and then averaged over the six months prior to portfolio formation. Columns 3 through 6 include returns in the portfolio formation week, RET(0), to control for the possibility of weekly return reversals. Consistent with the results in Jegadeesh (1990) and Lehmann (1990), the RET(0) coefficient is significantly negative, indicating a negative relation between returns in weeks w − 1 and w. Across columns 1 through 3, the O/S coefficient is significantly negative, with the coefficients and t-statistics remaining stable across specifications. Column 4 of Panel A contains regression results where O/S is decoupled into nu- merator and denominator, option volume (OPVOL) and equity volume (EQVOL). Because EQVOL and OPVOL are highly correlated with SIZE, our regression analysis instead uses changes in EQVOL and OPVOL, denoted by ∆EQVOL and ∆OPVOL. Following Eq. (2.9), the “∆” version of each variable is the level of the variable less the average of the variable over the prior six months, all scaled by that average. Column 4 demonstrates that both the numerator and denominator contribute to predictabil- ity: the coefficient corresponding to deciles of ∆OPVOL is -0.011 (t-statistic = -2.38) CHAPTER 2. OPTION TO STOCK VOLUME RATIO AND RETURNS 26

and the coefficient corresponding to deciles of ∆EQVOL is 0.030 (t-statistic = 4.99). This is consistent with our model’s prediction that high option volume reflects nega- tive private information and high equity volume reflects positive private information, once controlling for both volume measures. The positive ∆EQVOL coefficient is also consistent with Gervais, Kaniel, and Mingelgrin (2001), which argues that abnor- mal volume garners additional visibility and therefore predicts higher future returns. Column 5 demonstrates that O/S remains negatively related to future returns after controlling for ∆OPVOL, but the ∆OPVOL coefficient is significantly positive, indi- cating that innovations in a firm’s OPVOL are positive predictors of future returns after controlling for a firm’s O/S. Finally, comparing the O/S coefficients in columns 3 and 6 or columns 2 and 7 shows that the O/S-return relation is relatively unaffected by controlling for equity volume. Taken together, the results in Panel A of Table 2.3 demonstrate a robust negative association between O/S and future equity returns, distinct from weekly return reversals, the pricing of liquidity, and the relation between equity market volume and future returns. Panel B of Table 2.3 repeats the Fama-MacBeth regressions in Panel A but with ∆O/S replacing O/S. The main result from Panel B is that ∆O/S is negatively as- sociated with future returns across all regression specifications, each controlling for a different combination of momentum, size, book-to-market, liquidity, short-term re- versal, ∆OPVOL, and ∆EQVOL. Panel C of Table 2.3 repeats the Fama-MacBeth regressions using ΩO/S, ΩOPVOL, and ΩEQVOL and yields results that are qual- itatively identical to the findings in Panel B. ΩO/S, ΩOPVOL, and ΩEQVOL rely on within-firm variation to measure how the underlying variable ranks relative to the CHAPTER 2. OPTION TO STOCK VOLUME RATIO AND RETURNS 27

firm’s historical distribution. For example, the decile of ΩOPVOL reflects a firm- week’s rank relative to the firm’s OPVOL distribution over the prior six months. In both Panels B and C, comparing columns 2 and 7 or columns 3 and 6, we find that controlling for equity volume does not significantly affect the magnitude of the O/S- return relation regardless of whether we control for RET(0). However, comparing columns 2 and 3 or columns 6 and 7, we find that controlling for RET(0) reduces the magnitude of the ∆O/S and ΩO/S coefficients regardless of whether ∆EQVOL is a regressor. Across all specifications, ∆O/S and ΩO/S are significant negative predictors of future returns, suggesting that within-firm variation in O/S reflects the direction of informed trade. Having established a robust relation between O/S, ∆O/S, and ΩO/S and future returns, we next examine the duration of return predictability associated with each measure. Fig. 2.1 shows the alphas from strategies with progressively longer delays between the observation of the O/S signal and the weekly return in question. For example, the O/S strategy with a four-week lag sorts firms by O/S measured four weeks prior to the realized return window. This results in a weekly return series that we use to compute four-factor alphas as in Eq. (2.8). Fig. 2.1 repeats this exercise with lags of 1–12 weeks, across all three measures: O/S, ∆O/S, and ΩO/S. The top graph shows weekly alphas and their 95% confidence interval. The bottom graph shows cumulative alphas. The top graph in Fig. 2.1 shows that the return predictability associated with O/S is relatively short-lived, decreasing sharply but remaining statistically signifi- cant at the 5% level in the four weeks following portfolio formation. The significant O/S-return relation disappears after the sixth week following portfolio formation, CHAPTER 2. OPTION TO STOCK VOLUME RATIO AND RETURNS 28

which is inconsistent with the O/S-return relation reflecting a static dimension of risk correlated with O/S. The persistence of return predictability suggests that it takes multiple weeks for the information content of O/S to become fully reflected in equity prices. Like O/S, both ∆O/S and ΩO/S show patterns of return predictability that are short-lived. Repeating this analysis at the 1% significance level, we again find that return predictability associated with O/S and ∆O/S persists for four and three weeks, respectively (results untabulated). We also find that ΩO/S does not predict future returns at the 1% level at any horizon, consistent with the finding in Table 2.2 that ΩO/S has the lowest predictive power for future returns among the three O/S measures. Unlike O/S and ∆O/S, the ΩO/S measure does not reference the cross-sectional distribution when assigning firms to portfolios. Thus, the weaker predictive power associated with ΩO/S is consistent with the measure being more sensitive to market-wide variation in option or equity volumes that are unrelated to private information. The bottom graph in Fig. 2.1 shows that cumulative 12-week al- phas range from 0.9% to 1.5%, where O/S outperforms ∆O/S and ∆O/S outperforms ΩO/S on a cumulative basis across all durations. Table 2.4 presents tests of Empirical Prediction 2, that the predictive power of O/S for future returns is increasing in short-sale costs. Our first measure of firm- specific short-sale costs, following Nagel (2005), is the level of residual institutional ownership RIi,q. We define RIi,q as the percentage of shares held by institutions for

firm i in quarter q, adjusted for size in cross-sectional regressions. Specifically, RIi,q equals the residual i,q from the following regression:

INSTi,q 2 logit(INSTi,q) = log( ) = αq + β1,qSIZEi,q + β2,q(SIZEi,q) + i,q, (2.10) 1 − INSTi,q CHAPTER 2. OPTION TO STOCK VOLUME RATIO AND RETURNS 29

where INSTi,q equals the fraction of shares outstanding held by institutions as re- flected in the Thomson Financial Institutional Holdings (13F) database. We calculate quarterly holdings as the sum of stock holdings of all reporting institutions for each

firm and quarter. Values of INSTi,q are winsorized at 0.0001 and 0.9999. Low levels of RIi,q (hereafter referred to as RI) correspond to high short-sale costs because stock loans tend to be scarce and, hence, short-selling is more expensive when institutional ownership is low. We match RI to a given firm-week by requiring a three-month lag between the Thomson Financial report date and the first trading day of a given week. The other two measures of short-sale costs are more direct, and rely on a pro- prietary data set of institutional lending provided to us by Data Explorers. Data Explorers aggregates information on institutional lending from several market par- ticipants including hedge funds, investment banks, and prime brokers.13 Similar to the data sets used in D’Avolio (2002) and Geczy, Musto, and Reed (2002), this data set contains monthly institutional lending data on transacted loan fees and available loan supply. The sample period is June of 2002 through December of 2009, covering approximately half of our main sample period. From this data set, we derive our sec- ond measure of firm-specific short-sale costs: LF, the value-weighted average loan fee for institutional loans occurring in the calendar month prior to the portfolio forma- tion date. Higher values of LF reflect higher short-sale costs because investors must pay the lending fee to obtain the shares necessary for shorting. The final measure of short-sale costs, LS, is the total quantity of shares available for lending, as a fraction of firms’ total shares outstanding, at the conclusion of the calendar month prior to portfolio formation. Lower values of LS correspond to higher short-sale costs because investors must first locate lendable shares before implementing a short position.

13See www.dataexplorers.com for more details regarding the data. CHAPTER 2. OPTION TO STOCK VOLUME RATIO AND RETURNS 30

Panel A presents alphas for portfolios double-sorted into short-sale cost quintiles and relative option volume quintiles, for each of the three short-sale cost measures. Within each short-sale cost quintile, we compute the four-factor alpha of a long-short strategy using extreme quintiles of each relative option volume measure (O/S, ∆O/S, and ΩO/S). For example, the entry in the RI(1) row corresponding to the O/S signal indicates that among firms in the lowest residual institutional ownership quintile, a strategy long firms in the lowest O/S quintile and short firms in the highest O/S quintile produces a weekly alpha of 0.269% (t-statistic = 2.24). The key tests of Empirical Prediction 2 are contained in the “High−low short-sale costs” rows, which examine differences in strategy alphas across extreme short-sale cost quintiles. The results in Panel A are mixed. Eight of the nine differences in strategy alphas across extreme short-sale cost portfolios are positive, indicating that O/S is a stronger signal for future returns when short-sale costs are high. However, only three of the nine are statistically significant at the 10% level, and only one of the nine is significant at the 5% level. Panel B presents analogous results for strategies relying on extreme deciles of relative option volume (rather than quintiles), sorted by terciles (rather than quintiles) of short-sale costs. We analyze 10x3 sorts because the results in Table 2.2 indicate that decile strategies produce larger alphas than quintile strategies, and because the results in Panel A show that the near-extreme quintiles of short-sale costs (2 and 4) often contain strategy alphas of different signs than the extreme quintiles (1 and 5). For example, the ΩO/S alpha is 0.098 in the lowest LF quintile but -0.105 in the second-lowest LF quintile. The results in Panel B confirm that 10x3 sorts produce a clearer difference in strategy alphas across short-sale costs. All nine of the “High−low CHAPTER 2. OPTION TO STOCK VOLUME RATIO AND RETURNS 31

short-sale costs” alphas are positive, and five of the nine are significant at the 5% level. Across both panels of Table 2.4, the differences in strategy alphas across the ex- treme short-sale cost portfolios are economically and statistically stronger for O/S than the change-based measures, ∆O/S and ΩO/S. One potential explanation is that ∆O/S and ΩO/S generate weaker strategy alphas compared to O/S (as illustrated in Table 2.3) and we therefore do not have the statistical power to distinguish alphas across short-sale cost extremes. Another potential explanation is that our change- based measures are themselves correlated with changes in short-sale costs. In unt- abulated results, we show that ∆O/S and ΩO/S are much more strongly positively correlated with the prior week’s return than O/S. As argued in Geczy, Musto, and Reed (2002) and D’Avolio (2002), short-sale costs are a decreasing function of recent returns, implying that ∆O/S and ΩO/S could be correlated with recent changes in short-sale costs that our monthly and quarterly short-sale cost measures fail to detect. To summarize, Table 2.4 provides mixed support for Empirical Prediction 2. Across three different measures of short-sale costs and the three relative option vol- ume measures, in nearly all cases the portfolio alphas associated with option volume strategies are higher for firms with high short-sale costs. These differences are more often statistically significant for the O/S strategy, and for 10x3 sorts. Collectively, the results in Table 2.4 provide some evidence that informed traders use option markets more frequently when short-sale costs are high. Consistent with Empirical Prediction 3, Table 2.5 demonstrates that the predictive power of relative option volume for future stock returns is strongest when option leverage is low. For each firm-week, leverage is defined as the open-interest-weighted CHAPTER 2. OPTION TO STOCK VOLUME RATIO AND RETURNS 32

∂C S 14 average of ∂S C , as provided by OptionMetrics, which we refer to as LM. Panel A of Table 2.5 contains factor regression results for long-short O/S quintile portfolios across quintiles of LM, where firms are independently sorted by LM and O/S. The O/S alphas are monotonically decreasing across quintiles of LM, where the difference in portfolio alphas across the extreme LM quintiles is significant at the 1% level (t- statistic = 4.42). The second and third columns of Panel A indicate that the ∆O/S and ΩO/S strategy alphas are also strongest among firms with low leverage, with both alpha differences also significant at the 1% level. Finally, similar to Table 2.4, Panel B repeats the analysis using deciles of relative option volume and terciles of option leverage. As in Panel A, portfolio alphas are concentrated among firms with low option leverage for all three option volume signals (t-statistics = 4.97, 3.81, and 1.49). The results in Table 2.5 are consistent with informed traders moving a larger portion of their bets from shorting stock to trading options when leverage is low. To address the possibility that the O/S-return relation is specific to only a sub- sample of our data, Fig. 2.2 presents annual cumulative returns to three long-short strategies assuming weekly portfolio rebalancing for each year in the sample. The first strategy consists of an equal-weighted long-short position in the extreme O/S deciles. We implement the long-short strategy each week and compound the weekly returns within each calendar year. The unconditional long-short strategy (shown in grey) results in positive returns in 13 out of 15 years, with a mean return of 21.81% and a standard deviation of 20.64%.15 The second strategy takes long-short positions in extreme O/S deciles among firms in the bottom tercile of residual institutional

14The results are qualitatively similar when using volume-weighted average option leverage. 15A comparable strategy using ∆O/S results in average annual returns of 12.67%, and positive returns in 12 out of 15 sample years. Sorting by ΩO/S results in average annual returns of 7.87%, and positive returns in nine of 15 sample years (results untabulated). CHAPTER 2. OPTION TO STOCK VOLUME RATIO AND RETURNS 33

ownership (RI), corresponding to firms with the highest short-sale costs. This strat- egy (shown in black) produces positive returns in 13 out of 15 years of the sample, while increasing the mean of the annual cumulative returns to 40.37%.16 The third strategy corresponds to analogous long-short returns for firms in the lowest lever- age (LM) tercile. Conditional upon being in the lowest LM tercile, the long-short O/S strategy (shown in white) results in positive hedge returns in 13 of 15 years while again increasing the mean return relative to the unconditional O/S strategy to 27.46%. Across all three strategies, returns in the later sample years 2002–2010 are smaller than those in the early sample years 1996–2001 but still remain consistently positive and economically significant. Together, the results of Fig. 2.2 demonstrate a robust association between O/S and future returns throughout our sample period, and that this association is stronger when short-sale costs are high or option leverage is low. In addition to the above results pertaining to O/S, we also examine what the call to put volume ratio, C/P, tells us about future equity returns. Empirical Prediction 4 states that C/P is a positive predictor of future return skewness. The results of Table 2.6 confirm this prediction. For each firm-week, we compute C/P as:

VLCi,w C/Pi,w = , (2.11) VLPi,w

where VLCi,w is the total call volume for firm i in week w and VLPi,w is defined analogously for puts. Firms are sorted into deciles based on C/P, where the tenth (first) decile corresponds to high (low) levels of call volume relative to put volume.

16Because the long-short strategy results rely upon taking positions among equities with high short-sale costs, the reported results are not intended to reflect the actual returns achieved through implementation. CHAPTER 2. OPTION TO STOCK VOLUME RATIO AND RETURNS 34

For each calendar week, we calculate the cross-sectional skewness of the subsequent week’s returns, RET(1), for each decile portfolio of C/P, which results in a panel data set of approximately 7,330 observations. Table 2.6 contains the results of regressing cross-sectional skewness on the C/P decile rank. In column 1, the coefficient on the C/P decile rank is significantly positive (t-statistic = 5.89), indicating that C/P is positively associated with future return skewness. Column 2 demonstrates that this relation remains significant after controlling for the lagged skewness of a given C/P decile. The evidence in Table 2.6 is consistent with our model’s prediction that informed traders buy puts for extremely bad news, sell calls for moderate bad news, sell puts for moderate good news, and buy calls for extremely good news.

2.5 Additional analyses

Several existing studies examine the link between option market activity and earn- ings announcements. Skinner (1990) finds that the information content of earnings announcements declines following options listing, consistent with options facilitating informed trade prior to announcements. Amin and Lee (1997) find that open interest increases prior to announcements and possesses some predictive power for the sign of earnings news. RSS find that O/S significantly increases immediately prior to earnings announcements, suggesting that O/S reflects private information regarding earnings news. Consistent with this interpretation, they find that O/S positively pre- dicts the absolute magnitude of earnings news and that the effect is more pronounced when the earnings news is negative. Both findings are consistent with our prediction that option markets serve as an alternative venue for traders with negative private CHAPTER 2. OPTION TO STOCK VOLUME RATIO AND RETURNS 35

information seeking to avoid short-sale costs. Additionally, RSS find that the rela- tion between O/S and absolute announcement returns is less pronounced when there is a significant movement in equity prices prior to the announcement date, consis- tent with informed traders impounding private information into prices ahead of the announcement. In this section, we provide additional evidence that relative option volume reflects private information by examining whether prior week’s O/S provides predictive power for the sign and magnitude of quarterly earnings surprises. Our tests build upon RSS by examining the relation between O/S and signed earnings news and returns. We assemble a new data set from four sources. The OptionMetrics, Compustat Industrial Quarterly, CRSP daily stock, and Institutional Brokers’ Estimate System (I/B/E/S) consensus files provide information on option volume, quarterly firm at- tributes, equity prices, and earnings surprises, respectively. We apply the same sample restrictions outlined in Section 2.4. The intersection of these four databases results in a final sample consisting of 44,669 firm-quarter observations. To the extent that informed traders gravitate toward options ahead of negative news, we predict that O/S is negatively associated with the resulting earnings sur- prise. For each earnings announcement, we measure O/S in the calendar week that directly precedes it. For example, we measure O/S from Monday through Friday of each calendar week and examine the information content of earnings announcements occuring in the subsequent calendar week. This empirical design directly mimics the structure of our main analyses that use O/S in week w − 1 to predict returns in week w, except here we focus the analysis on the prediction of earnings news and earnings- announcement window returns revealed in week w. In Panel A of Table 2.7, we use CHAPTER 2. OPTION TO STOCK VOLUME RATIO AND RETURNS 36

three variables to capture the news released at earnings announcements. The first, SURPRISE, is the earnings surprise as measured by the firm’s actual earnings per share (EPS) minus the analyst consensus EPS forecast immediately prior to the an- nouncement, scaled by the beginning-of-quarter stock price. The second, standardized unexplained earnings (SUE), is an alternative measure of earnings surprise defined as the realized EPS minus EPS from four quarters prior, divided by the standard deviation of this difference over the prior eight quarters. The final, CAR(−1, +1), equals three-day cumulative market-adjusted returns during the earnings announce- ment window from t − 1 to t + 1, where day t is the earnings announcement date. Mirroring the construction of Table 2.3, Table 2.7 contains Fama-MacBeth regres- sion results, where standard errors are computed across quarterly coefficients. Panel A demonstrates that the prior calendar week’s O/S decile carries predictive power for future earnings surprises. The negative relation between relative option volume and earnings surprises (t-statistic = -2.36) is consistent with the negative O/S-return relation reflecting informed trade. We also find analogous results where SUE is the dependent variable. The coefficient on O/S is significanty negative (t-statistic = - 2.06), indicating that O/S is negatively associated with earnings innovations. The final column of Panel A presents the regression results when the announcement win- dow abnormal returns, CAR(−1, +1), is the dependent variable. The coefficient on O/S remains negative and statistically significant (t-statistic = -2.11) incremental to the firm’s momentum, size, book-to-market, and decile of equity volume, which is con- sistent with relative option volume reflecting private information about future asset values revealed in part by the earnings announcement. As an example of the economic significance, the lowest O/S decile outperforms the highest by 0.369% (= −0.041 × 9) CHAPTER 2. OPTION TO STOCK VOLUME RATIO AND RETURNS 37

in the three-day announcement window (all else equal), more than the return spread generated on average during an entire normal week (0.34%).17 Panel B of Table 2.7 examines the predictive power of O/S for returns follow- ing the announcement. We use five return windows: CAR(+2, +5), CAR(+2, +10), CAR(+2, +20), CAR(+2, +40), and CAR(+2, +60), where CAR(X,Y ), equals the cumulative market-adjusted return from t + X through t + Y . The O/S coefficient is insignificant across all of the return horizons, with t-statistics ranging from -0.52 to -1.53. These results indicate that most, if not all, of the private information in O/S in the week prior to the announcement is publicly revealed at the earnings an- nouncement, leaving no significant return predictability in the days following the announcement. Collectively, the results in Table 2.7 are consistent with O/S reflect- ing private information about future earnings which is impounded into prices during subsequent earnings announcements.

2.6 Conclusion

The central contribution of this paper is a mapping between observed transactions and the sign and magnitude of private information that does not require estimating order flow imbalances. Specifically, we examine the information content of option and equity volumes when agents are privately informed but trade direction is unobserved. We provide theoretical and empirical evidence that O/S, the amount of trading volume in option markets relative to equity markets, is a negative cross-sectional signal of

17In untabulated results, we find that O/S contains predictive power for earnings news as early as six weeks prior to the earnings announcement, indicating that informed traders’ anticipation of earnings news is reflected in the level of O/S several weeks prior to the announcement. Consistent with this interpretation, we find that ∆O/S and ΩO/S, which rely on weekly changes in O/S, fail to predict earnings announcement news. CHAPTER 2. OPTION TO STOCK VOLUME RATIO AND RETURNS 38

private information. Stocks in the lowest decile of O/S outperform the highest decile by 0.34% on a factor-adjusted basis in the week following portfolio formation. We offer a simple explanation for this finding, specifically that it results from how informed traders choose between trading in equity and option markets in the presence of short- sale costs. We model the capital allocation and price-setting processes in a multimarket set- ting and develop novel predictions regarding information transmission across markets. In equilibrium, short-sale costs cause informed traders to trade more frequently in option markets when in possession of a negative signal than when in possession of a positive signal, thus predicting that volume in options markets, relative to equity markets, is indicative of negative private information. By empirically documenting that O/S is a negative cross-sectional signal for future equity returns, our results are consistent with market frictions preventing equity prices from immediately reflecting the information content of O/S. Return predictability associated with O/S is rela- tively short-lived, decreasing sharply in the first few weeks and remaining statistically significant in the four weeks following portfolio formation, which suggests that it takes multiple weeks for the information in O/S to become fully reflected in equity prices. Our model also predicts that O/S is a stronger signal when short-sale costs are high or option leverage is low, and that volume differences across calls and puts predict future return skewness, all of which we confirm in the data. We measure short-sale costs using proprietary firm-specific data on institutional loan fees and loan supply from 2002–2009. We find mixed evidence that O/S alphas increase with short-sale costs and strong evidence that O/S alphas decrease with option leverage. Conditional on low average leverage traded in options, sorting stocks by deciles of O/S results in CHAPTER 2. OPTION TO STOCK VOLUME RATIO AND RETURNS 39

an average annual hedge return of 27%. Finally, we show that O/S predicts the sign and magnitude of earnings surprises and abnormal returns at quarterly earnings announcements, consistent with O/S reflecting traders’ private information. CHAPTER 2. OPTION TO STOCK VOLUME RATIO AND RETURNS 40

2.7 Tables and Figures Figure 2.1: Persistence of O/S-return relation. This figure presents the alphas associated with a portfolio that combines an equal-weighted long position in the lowest decile with an equal-weighted short position in the highest decile of each relative option volume signal. The top graph shows weekly alphas, where the surrounding error bars represent the 95% confidence interval. The bottom graph shows cumulative alphas. The portfolio is formed K weeks after the observation of the signal, where K ranges from 1 to 12. O/Si,w equals the ratio of option volume to equity volume of firm i in week w. ∆O/S equals the difference between O/S in the portfolio formation week and the average over the prior six months, scaled by this average. ΩO/S equals the percentile rank in the firm-specific time-series of O/S. Alphas are the intercept in a time-series regression of weekly strategy returns on contemporaneous weekly factor returns for the three Fama-French and momentum factors. The sample consists of 611,173 firm-weeks spanning 1996 through 2010. Alphas are shown as percentages. CHAPTER 2. OPTION TO STOCK VOLUME RATIO AND RETURNS 41

Figure 2.2: Cumulative hedge returns by year. This figure presents cumulative annual unadjusted returns to three strategies assuming weekly portfolio rebalancing for each year in the sample. The first strategy (shown in grey) consists of an equal-weighted long position in the lowest O/Si,w decile together with an equal-weighted short position in the highest O/Si,w decile. O/Si,w equals the ratio of option volume to equity volume of firm i in week w. In addition to O/Si,w deciles, firms are independently sorted into terciles of residual institutional ownership and option leverage. The second strategy (shown in black) consists of a long-short O/Si,w position for firms in the lowest tercile of residual institutional ownership (RI). RI is obtained from cross-sectional regressions as detailed in Nagel (2005). The third strategy (shown in white) consists of a long-short O/Si,w position for firms in the lowest leverage (LM) tercile, where LMi,w is the open-interest-weighted average λ of firm i in week w. The sample consists of 611,173 firm-weeks spanning 1996 through 2010. All returns are shown as percentages. CHAPTER 2. OPTION TO STOCK VOLUME RATIO AND RETURNS 42

Table 2.1: Descriptive statistics by year. Panel A provides sample size information and descriptive statistics of O/Si,w (shown as a percentage), where O/Si,w equals the ratio of option volume to equity volume of firm i in week w as outlined in Section 2.4. Panel B gives average firm characteristics by decile of O/S. The sample consists of 611,173 firm-weeks spanning 1996 through 2010. VLC (VLP) equals the total call (put) contract volume traded in a given week; each contract represents 100 shares. OPVOL equals the sum of VLC and VLP. EQVOL equals the total equity volume traded, in units of 100 shares. SIZE (LBM) equals the log of market capitalization (book-to-market) corresponding to firms’ most recent quarterly earnings announcement. MOMEN equals firms’ cumulative market-adjusted return measured over the six months prior to week w, in percent.

Panel A: Sample characteristics and O/S descriptive statistics by year Firms Firm-weeks MEAN P25 MEDIAN P75 SKEW 1996 1,020 12,006 6.260 2.163 4.359 8.494 5.822 1997 1,422 29,426 6.193 2.159 4.317 8.405 6.990 1998 1,655 32,970 5.636 1.866 3.768 7.381 5.803 1999 1,724 35,296 5.408 1.828 3.749 7.374 5.802 2000 1,733 40,696 5.024 1.873 3.738 7.057 68.868 2001 1,587 38,182 4.585 1.520 3.121 6.026 8.112 2002 1,533 36,087 3.835 1.283 2.765 5.619 4.507 2003 1,470 36,815 4.381 1.363 3.095 6.619 20.938 2004 1,590 41,062 5.425 1.615 3.782 8.280 9.801 2005 1,737 45,527 6.218 1.683 4.068 9.324 73.627 2006 1,848 52,299 7.329 1.941 4.786 10.775 26.834 2007 1,980 57,735 7.304 1.928 4.720 10.732 23.595 2008 1,914 57,035 6.249 1.523 3.722 8.648 8.684 2009 1,814 50,794 6.452 1.711 4.107 9.092 13.388 2010 1,870 45,243 6.322 1.542 3.792 8.643 9.521 ALL 611,173 5.775 1.733 3.859 8.165 19.486

Panel B: Firm characteristics by deciles of O/S VLC VLP OPVOL EQVOL SIZE LBM MOMEN 1 (Low) 228 124 353 74,095 7.734 0.375 0.461 2 479 249 728 66,965 7.555 0.359 2.147 3 827 439 1,266 74,415 7.542 0.350 2.972 4 1,342 727 2,069 85,184 7.594 0.342 3.754 5 2,080 1,160 3,240 97,036 7.671 0.332 4.300 6 3,390 1,924 5,314 116,808 7.788 0.323 5.189 7 5,264 3,103 8,368 136,539 7.933 0.312 5.751 8 8,072 4,993 13,064 156,023 8.091 0.301 6.878 9 12,145 7,791 19,936 164,785 8.190 0.293 7.942 10 (High) 25,197 15,807 41,003 148,488 8.128 0.273 11.058 High−low 24,968 15,683 40,651 74,393 0.394 -0.102 10.597 CHAPTER 2. OPTION TO STOCK VOLUME RATIO AND RETURNS 43 -statistics are shown in t equals the ratio of option volume to equity volume of i,w , where O/S i,w CAPM Three-factor Four-factor Table 2.2: Factor regression results by deciles of O/S, ∆O/S, and ΩO/S. INT MKTRF INT MKTRF SMB HML INT MKTRF SMB HML UMD (4.20) -(6.22)(3.45) -(6.13) (4.29) -(3.30) (3.43) (13.33) -(3.08) -(6.63) (14.46) (5.00) -(6.67) -(5.86) (4.19) (10.05) -(5.90) -(5.97) -(8.34) (11.00) -(5.99) -(9.10) (2.12) (35.30)(0.17) (37.79) (1.77)(0.72) (35.80) (37.13) -(0.09) (5.51) (37.33) (0.57) (4.86) (1.77) (36.69) (7.25) (2.37) -(0.22) (8.90) (33.02) (0.45) (2.45) (34.57) (1.01) (5.94) -(1.32) (33.86) -(8.21) (8.50) -(2.64) -(8.57) (9.87) -(6.82) -(0.31) (36.99)-(0.78) (38.21) -(0.40)-(0.43) (36.35) (36.86) -(0.81)-(0.31) -(2.33) (37.61) (38.22) (8.82) -(0.49)-(0.69) -(4.10) (36.45) (38.92) (8.72) -(0.27) (0.04)-(1.09) -(3.71) (37.84) (37.76) (33.53) (9.57) -(0.61) -(0.38)-(1.85) -(5.03) -(4.83) (38.86) (34.77) (38.89) (9.18) -(1.10) -(0.11) (9.87) -(6.74) -(6.56) (38.04) (33.63) -(7.25) (9.27) -(1.97) (9.78) (0.05) -(6.52) -(5.75) (39.29) -(7.29) (10.53) (35.02) (10.43) -(0.38) -(6.12) -(6.15) -(6.63) (36.14) (10.77) -(0.91) (9.85) -(7.64) (35.42) -(5.17) -(1.90) (9.69) -(7.25) (37.06) -(3.63) (10.87) -(6.06) -(3.13) (10.81) -(0.98) . Decile portfolios are formed at the conclusion of each week, ranging from 1 to 10 with the highest (lowest) values w + 1 and regressed on three sets of contemporaneous risk factors: the excess market return (MKTRF); three Fama-French 10 0.342 -0.180 0.302 -0.085 0.574 -0.318 0.338 -0.151 0.443 -0.275 -0.225 (9+10) 0.229 -0.144 0.193 -0.063 0.498 -0.255 0.224 -0.121 0.384 -0.219 -0.195 w 23 0.0144 1.135 0.0645 -0.028 1.1666 -0.007 1.183 -0.0727 1.115 0.048 1.253 -0.0428 0.089 -0.034 1.125 1.252 -0.029 0.404 1.1289 -0.070 -0.011 1.301 -0.065 -0.121 0.510 1.186 -0.043 0.035 0.511 1.287 -0.101 -0.217 1.182 -0.024 1.035 0.083 0.514 1.244 -0.202 0.004 1.224 -0.068 -0.052 1.059 0.580 0.455 -0.032 1.057 -0.273 1.202 -0.141 -0.092 -0.269 0.554 1.114 -0.261 0.552 -0.010 -0.350 1.157 -0.223 0.556 0.536 -0.360 1.119 -0.333 0.005 -0.240 0.560 0.598 -0.327 -0.032 1.170 -0.244 0.620 1.166 -0.379 -0.076 -0.214 0.588 -0.421 1.126 -0.181 0.559 -0.394 -0.123 0.618 -0.104 − − in week 1 i 1 (Low) 0.178 1.049 0.144 1.062 0.274 0.269 0.185 0.986 0.125 0.317 -0.257 10 (High) -0.164 1.229 -0.158 1.147 -0.300 0.587 -0.153 1.137 -0.318 0.593 -0.031 Panel A: Factor regressions results across deciles of O/S (1+2) parentheses. factors (MKTRF, SMB, and HML);is and the the three portfolio Fama-Frenchformation and alpha. momentum week factors and Panel (UMD). theΩO/S The B intercept average is is in over the this the defined regression percentile prior analogously (INT) rank six for in months, ∆O/S, the scaled where by firm-specific ∆O/S this time-series equals average. of the O/S. Panel difference C All between is returns O/S defined are analogously in shown for the as ΩO/S, portfolio percentages, where firm located in the 10th (1st)in decile. week The sample consists of 611,173 firm-weeks spanning 1996 through 2010. Portfolio returns are measured Panel A presents factor regression results across deciles of O/S CHAPTER 2. OPTION TO STOCK VOLUME RATIO AND RETURNS 44 O/S ∆ CAPM Three-factor Four-factor INT MKTRF INT MKTRF SMB HML INT MKTRF SMB HML UMD (0.19) (37.80) (0.11) (37.06) -(1.93) (8.32) (0.64) (34.28) -(4.85) (9.56) -(8.33) (1.37) (35.82) (1.27) (34.93) -(0.50) (7.32) (1.91) (32.22) -(3.73) (8.67) -(9.08) (3.38) (5.48) (3.68) (4.82) -(4.09) -(1.94) (4.38) (2.18) -(7.02) -(1.09) -(8.47) (2.45) (5.96) (2.68) (5.34) -(3.43) -(1.45) (3.49) (2.29) -(7.02) -(0.42) -(10.10) -(0.03) (38.03)-(0.19) (38.23) -(0.05)-(0.24) (37.18) (37.97) -(0.15)-(0.90) -(2.92) (37.68) (38.17) (7.98) -(0.15)-(1.00) -(4.66) (37.45) (38.70) (8.77) -(0.91) (0.36)-(0.68) -(5.68) (37.55) (34.33) (38.15) (8.52) -(1.02) (0.28)-(0.30) -(4.74) -(5.15) (38.24) (34.83) (39.19) (8.54) -(0.74) (8.88) (0.23)-(1.12) -(4.79) -(6.95) (37.94) -(6.57) (34.60) (37.05) (8.94) -(0.50) -(0.62) (9.74) -(3.99) -(7.56) (39.04) (34.77) -(6.88) (9.90) -(1.59) -(0.75) (9.32) -(0.91) -(6.18) (37.68) (35.46) -(5.96) (9.85) (9.14) -(0.50) -(6.08) (2.86) -(4.74) (35.22) (9.47) (10.40) -(0.31) -(5.16) -(4.35) (36.41) (10.37) -(1.38) -(1.91) -(3.93) (35.03) (10.17) (1.46) -(2.96) (10.80) -(3.47) 10 0.217 0.125 0.234 0.112 -0.159 -0.084 0.266 0.051 -0.279 -0.045 -0.206 (9+10) 0.129 0.112 0.140 0.102 -0.110 -0.052 0.172 0.043 -0.226 -0.014 -0.199 23 0.0184 -0.003 1.2525 1.270 -0.0186 0.010 1.281 -0.0227 -0.005 1.200 1.257 -0.083 1.2138 -0.013 -0.105 1.239 -0.089 0.503 -0.160 1.2099 -0.013 0.486 1.226 -0.059 -0.251 1.181 -0.078 0.055 0.525 1.186 -0.025 -0.301 0.031 1.170 1.116 -0.085 0.502 1.153 1.145 -0.248 0.024 -0.270 1.157 -0.060 0.497 0.556 -0.294 1.139 -0.243 0.019 1.120 -0.039 0.529 -0.284 0.505 -0.388 -0.052 1.121 -0.198 1.108 -0.229 0.570 0.546 1.123 -0.419 -0.062 -0.043 -0.236 0.540 -0.341 0.522 1.114 -0.040 -0.202 0.527 -0.327 1.083 -0.024 -0.160 0.532 -0.271 1.080 -0.143 0.570 -0.097 -0.127 0.539 -0.092 − − 1 1 (Low) 0.127 1.180 0.114 1.142 -0.027 0.447 0.163 1.050 -0.208 0.505 -0.310 10 (High) -0.090 1.055 -0.120 1.030 0.132 0.531 -0.103 0.999 0.071 0.551 -0.104 Panel B: Factor regressions results across deciles of (1+2) Table 2.2 [Continued]: Factor regression results by deciles of O/S, ∆O/S, and ΩO/S CHAPTER 2. OPTION TO STOCK VOLUME RATIO AND RETURNS 45 O/S Ω CAPM Three-factor Four-factor INT MKTRF INT MKTRF SMB HML INT MKTRF SMB HML UMD (0.76) (36.77) (0.87)(0.41) (35.48) (37.56) -(3.98) (6.07) (0.44) (36.58) (1.52) -(3.29) (32.82) (7.56) -(7.34)(0.34) (0.94) (7.44) (38.89) (33.77) -(9.54) -(5.93) (0.22) (8.65) (38.61) -(7.67) -(1.33) (9.60) (0.41) (36.00) -(2.31) (9.92) -(2.98) (0.10) (32.13) (0.22) (30.76) -(4.52) (5.83) (0.90) (28.12) -(8.29) (7.38) -(10.54) (1.19) (7.88) (1.53) (7.00) -(5.11) -(1.83) (2.45) (3.51) -(9.62) -(0.64) -(12.28) (1.40) (8.87) (1.77) (8.00) -(5.48) -(2.32) (2.82) (4.33) -(10.53) -(1.09) -(13.48) -(0.10) (37.43) -(0.13)-(0.17) (36.42) (37.45)-(1.01) -(2.63) (39.40) (7.41) -(0.21)-(0.40) (36.58) (38.69) -(1.10) (0.43)-(0.29) -(2.80) (38.65) (33.69) (40.65) (7.96) -(0.51) -(2.81) -(5.77) (38.28) (8.24) -(0.46) (0.15)-(1.22) (8.73) -(2.35) (40.67) (33.76) (38.68) (9.36) -(8.94) -(0.79) -(1.54) -(4.74) (35.80) (10.24) -(1.55) -(0.28) (8.73) -(4.55) (39.08) (35.57) -(0.28) -(5.79) (8.91) -(0.71) -(3.55) (37.98) (11.31) -(5.25) (9.80) -(2.45) -(1.46) (10.54) -(3.75) -(2.84) (36.75) -(1.17) (11.40) -(1.43) 10 0.111 0.260 0.140 0.234 -0.286 -0.114 0.205 0.112 -0.526 -0.037 -0.411 (9+10) 0.099 0.224 0.124 0.204 -0.234 -0.110 0.177 0.104 -0.431 -0.047 -0.338 23 0.0774 -0.009 1.3215 1.286 0.0396 0.085 -0.015 1.2677 -0.012 1.260 1.192 -0.085 1.2318 -0.237 0.040 1.183 -0.034 0.403 -0.1499 -0.018 1.209 0.468 1.172 -0.023 1.139 -0.088 -0.183 0.140 1.156 0.466 -0.146 0.038 0.028 1.131 1.156 -0.041 0.463 1.138 -0.138 1.130 1.118 -0.442 -0.034 0.082 0.450 -0.333 -0.115 0.013 0.469 1.107 1.130 0.017 0.511 0.527 -0.352 -0.062 1.081 -0.070 -0.339 1.084 -0.316 0.520 1.082 -0.259 -0.022 0.517 -0.063 0.499 -0.235 1.083 -0.269 -0.021 0.503 -0.194 0.481 -0.185 1.082 0.031 -0.166 0.533 -0.119 1.056 -0.119 0.536 -0.116 -0.085 0.520 -0.092 − − 1 1 (Low) 0.012 1.364 0.025 1.290 -0.318 0.456 0.096 1.156 -0.583 0.542 -0.454 10 (High) -0.099 1.104 -0.115 1.056 -0.032 0.570 -0.108 1.043 -0.057 0.578 -0.042 Panel C: Factor regressions results across deciles of (1+2) Table 2.2 [Continued]: Factor regression results by deciles of O/S, ∆O/S, and ΩO/S CHAPTER 2. OPTION TO STOCK VOLUME RATIO AND RETURNS 46 CHAPTER 2. OPTION TO STOCK VOLUME RATIO AND RETURNS 47

Table 2.3: Fama-MacBeth multivariate regressions. Panel A presents Fama-MacBeth regression results from regressing RET(1) on deciles of O/S, ∆OPVOL, and ∆EQVOL. The sample consists of 611,173 firm-weeks spanning 1996 through 2010. RET(1) is the firm’s return in the first week following the observation of O/Si,w, the ratio of option volume to equity volume of firm i in week w. OPVOLi,w equals the total option volume of firm i in week w. EQVOLi,w is defined analogously for equity volume. ∆OPVOL is the difference be- tween OPVOL in the observation week and the average OPVOL over the prior six months, scaled by this average. ∆EQVOL is defined analogously. Decile portfolios are formed at the conclusion of each week. Deciles range from 1 to 10 with the highest (lowest) values located in the 10th (1st) decile. RET(0) is the market-adjusted return in the portfolio formation week. MOMEN equals the cumulative market-adjusted returns measured over the prior six months. SIZE is the log of mar- ket capitalization of the firm and LBM is the log of the firm’s book-to-market ratio measured at the firm’s last quarterly announcement date. AMIHUD is the Amihud illiquidity ratio of firm i in week w. Panel B repeats this analysis using ∆O/S, the difference between O/S in the observation week and the average O/S over the prior six months, scaled by this average. In Panel C, ΩO/S is the percentile rank in the firm-specific time-series measured relative to the distribution of the firm’s O/S over the past six months. ΩOPVOL and ΩEQVOL are analogously defined for OPVOL and EQVOL. Standard errors are computed across weekly coefficient estimates, following Fama and MacBeth (1973). The resulting t-statistics are shown in parentheses. The notations ***, **, and * indicate the coefficient is significant at the 1%, 5%, and 10% level, respectively. All returns are shown as percentages.

Panel A: Fama-MacBeth regressions of RET(1) on O/S (1) (2) (3) (4) (5) (6) (7) Intercept -0.507 -0.297 -0.355 -0.331 -0.355 -0.368 -0.312 (-1.13) (-0.59) (-0.74) (-0.69) (-0.74) (-0.77) (-0.62) Decile(O/S) -0.026*** -0.025*** -0.022*** – -0.028*** -0.019*** -0.022*** (-3.99) (-3.64) (-3.32) – (-3.36) (-2.88) (-3.22) Decile(∆OPVOL) – – – -0.011** 0.013** – – – – – (-2.38) (1.98) – – Decile(∆EQVOL) – – – 0.030*** – 0.023*** 0.022*** – – – (4.99) – (3.99) (3.65) RET(0) – – -0.015*** -0.014*** -0.015*** -0.014*** – – – (-3.39) (-3.27) (-3.38) (-3.29) – MOMEN 0.003** 0.003** 0.003* 0.002* 0.002* 0.002* 0.003* (2.05) (2.14) (1.93) (1.74) (1.81) (1.71) (1.91) SIZE 0.037 0.023 0.026 0.011 0.024 0.019 0.016 (1.38) (0.77) (0.91) (0.41) (0.85) (0.65) (0.53) LBM 0.137 0.141 0.159 0.228 0.146 0.180 0.169 (0.79) (0.82) (0.97) (1.36) (0.90) (1.11) (1.00) AMIHUD -0.008* -0.007* -0.007* -0.007* -0.007* -0.008* (-1.93) (-1.71) (-1.87) (-1.77) (-1.67) (-1.85) Adj-R2 (%) 4.694 5.032 5.810 5.783 5.967 5.975 5.210 CHAPTER 2. OPTION TO STOCK VOLUME RATIO AND RETURNS 48

Table 2.3 [Continued]: Fama-MacBeth multivariate regressions.

Panel B: Fama-MacBeth regressions of RET(1) on ∆O/S (1) (2) (3) (4) (5) (6) (7) Intercept -0.458 -0.225 -0.294 -0.331 -0.254 -0.308 -0.241 (-0.99) (-0.45) (-0.61) (-0.69) (-0.53) (-0.64) (-0.47) Decile(∆O/S) -0.020*** -0.019*** -0.013*** – -0.042*** -0.011*** -0.017*** (-4.36) (-4.20) (-3.20) – (-4.51) (-2.71) (-3.77) Decile(∆OPVOL) – – – -0.011** 0.036*** – – – – – (-2.38) (3.53) – – Decile(∆EQVOL) – – – 0.030*** – 0.025*** 0.024*** – – – (4.99) – (4.18) (3.84) RET(0) – – -0.015*** -0.014*** -0.015*** -0.014*** – – – (-3.37) (-3.27) (-3.35) (-3.26) – MOMEN 0.003** 0.003** 0.003** 0.002* 0.003* 0.002* 0.003** (2.16) (2.23) (2.01) (1.74) (1.83) (1.77) (2.00) SIZE 0.030 0.015 0.019 0.011 0.014 0.011 0.008 (1.17) (0.54) (0.68) (0.41) (0.50) (0.41) (0.29) LBM 0.206 0.207 0.214 0.228 0.222 0.229 0.228 (1.16) (1.17) (1.27) (1.36) (1.32) (1.37) (1.31) AMIHUD – -0.008** -0.008* -0.007* -0.008* -0.007* -0.008** – (-2.11) (-1.90) (-1.87) (-1.91) (-1.86) (-2.02) Adj-R2(%) 4.506 4.822 5.606 5.783 5.778 5.780 5.009

Panel C: Fama-MacBeth regressions of RET(1) on ΩO/S (1) (2) (3) (4) (5) (6) (7) Intercept -0.424 -0.224 -0.274 -0.364 -0.243 -0.338 -0.264 (-0.90) (-0.44) (-0.56) (-0.75) (-0.49) (-0.69) (-0.51) Decile(ΩO/S) -0.016*** -0.015** -0.010* – -0.033*** -0.010* -0.016*** (-2.64) (-2.51) (-1.85) – (-3.09) (-1.76) (-2.73) Decile(ΩOPVOL) – – – -0.011* 0.029*** – – – – – (-1.76) (2.60) – – Decile(ΩEQVOL) – – – 0.027*** – 0.022*** 0.021*** – – – (4.06) – (3.36) (3.12) RET(0) – – -0.014*** -0.013*** -0.013*** -0.013*** – – – (-3.22) (-3.06) (-3.07) (-3.08) – MOMEN 0.003** 0.003** 0.003** 0.003* 0.003** 0.003* 0.003** (2.29) (2.36) (2.10) (1.88) (1.97) (1.93) (2.19) SIZE 0.028 0.015 0.017 0.012 0.011 0.011 0.008 (1.06) (0.52) (0.62) (0.42) (0.39) (0.41) (0.27) LBM 0.209 0.210 0.211 0.232 0.227 0.234 0.234 (1.20) (1.21) (1.27) (1.40) (1.38) (1.42) (1.35) AMIHUD – -0.008* -0.007* -0.006 -0.007* -0.006 -0.008* – (-1.90) (-1.74) (-1.55) (-1.66) (-1.54) (-1.91) Adj-R2 (%) 4.603 4.923 5.691 5.842 5.901 5.854 5.071 CHAPTER 2. OPTION TO STOCK VOLUME RATIO AND RETURNS 49 Table 2.4: Strategy alphas sorted by short-sale costs. This table presents alphas for portfolios double-sorted by one of three different short-sale cost mea- sures (RI, LF, and LS), and by one of three different relative option measures (O/S, ∆O/S, and ΩO/S, as defined in Table 2.2). RI (residual institutional ownership) is obtained from cross-sectional regressions as detailed in Nagel (2005). LF (loan fee) is the value-weighted average loan fee for in- stitutional loans in the month prior to portfolio formation, and LS (loan supply) is the quantity of shares available for lending scaled by shares outstanding at the end of the month prior to portfolio formation. In Panel A (B), firms are sorted each week into quintiles (deciles) of each relative option volume signal and quintiles (terciles) of each short-sale cost measure, and returns are measured the following week. Within each short-sale cost portfolio, strategy alphas are computed for a long-short position in the extreme O/S portfolios using a time-series regression on the three Fama-French and momentum factors (factor loadings not reported). The main sample consists of 611,173 firm-weeks spanning 1996 through 2010, however, the loan fee and supply data are only available from June 2002 through 2009. All returns are shown as percentages, t-statistics are in parentheses.

Panel A: Quintile alphas by quintiles of short-sale costs O/S ∆O/S ΩO/S RI(1): High short-sale costs 0.269 0.219 0.124 (2.24) (2.01) (0.99) RI(2) 0.319 0.174 0.313 (3.40) (1.75) (2.71) RI(3) 0.185 0.201 0.109 (2.39) (2.61) (1.17) RI(4) 0.227 0.242 0.154 (2.67) (2.89) (1.70) RI(5): Low short-sale costs 0.090 0.039 0.181 (1.17) (0.49) (1.98) High−low short-sale costs 0.179 0.180 -0.057 (1.34) (1.34) -(0.38) LF(1): Low short-sale costs 0.011 -0.151 0.098 (0.10) -(1.13) (0.72) LF(2) -0.070 -0.099 -0.105 -(0.64) -(0.91) -(0.87) LF(3) 0.156 0.088 0.066 (1.23) (0.68) (0.46) LF(4) 0.358 0.229 0.429 (2.85) (2.05) (2.92) LF(5): High short-sale costs 0.273 0.189 0.157 (2.06) (1.28) (0.90) High−low short-sale costs 0.262 0.340 0.059 (1.56) (1.87) (0.16) LS(1): High short-sale costs 0.258 0.553 0.431 (1.76) (3.76) (2.42) LS(2) 0.065 0.001 -0.013 (0.50) (0.00) -(0.09) LS(3) 0.130 0.119 0.144 (1.06) (1.09) (1.05) LS(4) -0.024 0.035 0.115 -(0.21) (0.31) (0.89) LS(5): Low short-sale costs -0.192 0.246 0.317 -(1.33) (2.19) (2.56) High−low short-sale costs 0.449 0.306 0.114 (2.53) (1.65) (0.63) CHAPTER 2. OPTION TO STOCK VOLUME RATIO AND RETURNS 50

Table 2.4 [Continued]: Strategy alphas sorted by short-sale costs.

Panel B: Decile alphas by terciles of short-sale costs O/S ∆O/S ΩO/S RI(1): High short-sale costs 0.471 0.366 0.271 (3.97) (3.34) (2.07) RI(2) 0.358 0.272 0.151 (3.90) (3.20) (1.41) RI(3): Low short-sale costs 0.183 0.209 0.259 (2.07) (2.32) (2.38) High−low short-sale costs 0.288 0.157 0.009 (2.10) (1.13) (0.06) LF(1): Low short-sale costs 0.038 -0.132 -0.029 -(0.34) -(1.11) -(0.20) LF(2) 0.123 0.213 0.218 (0.84) (1.46) (1.19) LF(3): High short-sale costs 0.309 0.456 0.378 (2.19) (2.98) (1.91) High−low short-sale costs 0.348 0.588 0.479 (1.98) (3.13) (2.08) LS(1): High short-sale costs 0.396 0.541 0.314 (2.52) (3.68) (1.68) LS(2) -0.010 0.170 0.176 -(0.07) (1.46) (1.19) LS(3): Low short-sale costs -0.184 0.258 0.323 -(1.17) (2.17) (2.15) High−low short-sale costs 0.580 0.283 0.014 (2.78) (1.55) (0.07) CHAPTER 2. OPTION TO STOCK VOLUME RATIO AND RETURNS 51

Table 2.5: Option volume alphas sorted by leverage. This table presents alphas for portfolios double-sorted by terciles of open-interest-weighted average leverage (LM) of firm i in week w, and by one of three different relative option measures (O/S, ∆O/S, and ΩO/S, as defined in Table 2.2). In Panel A (B), firms are sorted each week into quintiles (deciles) of each relative option volume signal and quintiles (terciles) of leverage, and returns are measured the following week. Within each LM portfolio, strategy alphas are computed for a long- short position in the extreme O/S portfolios using a time-series regression on the three Fama-French and momentum factors (factor loadings not reported). The sample consists of 611,173 firm-weeks spanning 1996 through 2010. All returns are shown as percentages, t-statistics are in parentheses.

Panel A: Quintile alphas by quintiles of leverage O/S ∆O/S ΩO/S LM(1): Low leverage 0.581 0.476 0.433 (4.67) (4.41) (3.59) LM(2) 0.247 0.021 0.100 (2.39) (0.22) (0.93) LM(3) 0.095 0.076 -0.053 (1.12) (0.95) -(0.61) LM(4) (0.10) (0.07) 0.143 (1.45) (1.25) (2.05) LM(5): High leverage -0.007 0.039 0.061 -(0.15) (0.86) (1.21) Low-high leverage 0.589 0.437 0.372 (4.42) (3.84) (2.89)

Panel B: Decile strategy alphas by terciles of leverage O/S ∆O/S ΩO/S LM(1): Low leverage 0.711 0.499 0.262 (5.53) (4.29) (1.83) LM(2) 0.259 0.236 0.207 (2.72) (2.80) (1.92) LM(3): High leverage 0.042 0.004 0.040 (0.71) (0.06) (0.60) Low-high leverage 0.669 0.496 0.221 (4.97) (3.81) (1.49) CHAPTER 2. OPTION TO STOCK VOLUME RATIO AND RETURNS 52

Table 2.6: Future return skewness by deciles of call-put volume ratio. The dependent variable in the table below is SKEW, defined as the cross-sectional skewness of weekly returns within a given portfolio in the week following portfolio formation. SKEW is calculated each calendar week and for each decile of C/Pi,w, where C/Pi,w equals the ratio of total call volume to total put volume of firm i in week w. Decile portfolios are formed at the conclusion of each week. Deciles range from 1 to 10 with the highest (lowest) values located in the 10th (1st) decile. The sample consists of 611,173 firm-weeks spanning 1996 through 2010, from which we compute 7,330 decile-weeks. Year fixed effects are included and standard errors are clustered at the weekly level. The resulting t-statistics are shown in parentheses. The notations ***, **, and * indicate the coefficient is significant at the 1%, 5%, and 10% level, respectively.

Dep. variable: SKEW (1) (2) Intercept 0.233*** 0.187*** (4.59) (3.68) Decile(C/P) 0.030*** 0.029*** (5.89) (5.69) Lag(SKEW) – 0.139*** – (10.91) Adj-R2 (%) 2.525 4.419 CHAPTER 2. OPTION TO STOCK VOLUME RATIO AND RETURNS 53

Table 2.7: Earnings surprises and earnings announcement returns. The sample for Table 2.7 consists of 44,669 quarterly earnings announcements during the 1996 through 2010 sample window. Each measure of earnings news is regressed on deciles of O/Si,w from the prior calendar week. O/Si,w equals the ratio of option volume to equity volume of firm i in week w. Deciles range from 1 to 10 with the highest (lowest) values located in the 10th (1st) decile. SURPRISE equals the firm’s actual EPS minus the consensus EPS forecasts immediately prior to the announcement, scaled by the beginning-of-quarter share price. SUE equals the standard unexplained earnings, calculated as realized EPS minus EPS from four-quarters prior, divided by its standard deviation over the prior eight quarters. CAR(X,Y ) is the cumulative market-adjusted return from day X to day Y relative to the earnings announcement. EQVOL equals the total equity volume traded, SIZE is the log of the firm’s market capitalization, and LBM is the log of the firm’s book-to-market ratio measured at the firm’s last quarterly announcement date. MOMEN equals the cumulative market-adjusted returns measured over the six months leading up to portfolio formation, and RET(0) is the cumulative market-adjusted return over the prior month. AMIHUD is the Amihud illiquidity ratio in the week prior to the announcement. All returns are calculated as percentages. Standard errors are computed across quarterly coefficient estimates, following Fama and MacBeth (1973). The resulting t-statistics are shown in parentheses. The notations ***, **, and * indicate the coefficient is significant at the 1%, 5%, and 10% level, respectively.

Panel A: Earnings announcement surprises Dep. variable: SURPRISE SUE CAR(-1,+1) Intercept 0.096*** 0.296*** 0.236 (2.74) (4.59) (0.49) Decile(O/S) -0.004** -0.008** -0.041** (-2.36) (-2.06) (-2.11) Decile(∆EQVOL) -0.003* 0.006 0.003 (-1.76) (1.47) (0.16) RET(0) 0.005*** 0.005** -0.063*** (7.19) (2.10) (-5.42) MOMEN 0.002*** 0.007*** 0.001 (9.62) (11.44) (0.51) SIZE 0.001 -0.023*** 0.028 (0.48) (-3.32) (0.56) LBM -0.167*** -0.822*** 0.318 (-3.98) (-7.89) (0.72) AMIHUD -0.006*** 0.003* -0.038** (-3.96) (1.67) (-2.49) Adj-R2 (%) 3.525 4.924 1.056 Chapter 3

Equity Risk Premia and the VIX Term Structure

3.1 Introduction

There is mounting theoretical and empirical evidence of time variation in equity mar- ket risk premia. These time variations are important both for economists studying asset pricing and for market participants making portfolio allocation decisions. How- ever, their use is limited without an ex-ante observable measure of expected returns. In many popular asset pricing models, risk premia can be inferred ex-ante as long as the variance of future returns conditional on all available information is observable, for example from options prices. In the intertemporal CAPM from Merton (1973), expected excess market returns are the product of conditional market variance and the coefficient of relative risk aversion for the representative agent. In both Campbell and Cochrane (1999) and Bansal and Yaron (2004) the risk premium are an approximately linear, time-invariant, function of conditional variance. All three models therefore predict that the time-series of equity risk premia and conditional equity variance are almost perfectly correlated, meaning that no other variable predicts future equity returns incremental to conditional variance. However, the data only provide weak support for this hypothesis. To quote Bollerslev, Tauchen, and Zhou (2009), “a significant time-invariant expected return- volatility tradeoff type relationship has largely proven elusive.” Conditional variance

54 CHAPTER 3. RISK PREMIA AND THE VIX TERM STRUCTURE 55

measures, whether constructed from option-implied volatilities or time-series models, have economically and statistically weak predictive power for future index returns1. In this paper, I argue that the expected return-conditional variance relation pre- dicted by theory is weak in the data because its nature changes every period. I show in a very general model that expected returns are indeed related to conditional variance linearly, but that the coefficients in this relation may change over time. In particular, the period-by-period shape of the variance-mean relation is determined by two economically meaningful quantities: the regression coefficient of that period’s equity return on the stochastic discount factor (SDF beta), and the variance of the part of returns orthogonal to the SDF (unpriced risk). The aforementioned asset pricing models assume these two quantities are essentially constant. However, when the market’s SDF beta and unpriced risk change over time, expected excess-returns are no longer a time-invariant linear function of conditional variance. Furthermore, I provide evidence that when risk premia are not easily inferred from conditional variance, an observer who is unaware of the marginal investor’s preferences can estimate equity risk premia using conditional variances at multiple horizons. In a stochastic volatility model where the SDF beta and unpriced risk are mean-reverting state variables, I show that the shape of the variance term structure allows an observer to decompose volatility into its constituent factors by relying on differences in their persistence. As a result, risk premia in the model are an affine function of the variance term structure. I apply this technique to the S&P 500 index, which has liquid options markets for many different strikes and times to expiration. Following the methodology used to calculate the CBOE’s volatility index (VIX), I compute model-free implied volatility estimates at many horizons, and show that the variance term structure can indeed dramatically improve index return predictability. I find that the VIX term structure predicts future S&P 500 returns, incrementally to any single VIX horizon, for holding periods from one month to one year. For example, the combined term structure predicts next-quarter excess log returns on the S&P 500 with a 5.2% adjusted R2 and joint significance at the 1% level. The return predictability is incremental to the

1Scruggs (1998) provides a good summary of the many papers estimating the relation between variance and future returns, and shows that this relation is stronger once long-term government bond returns are included as a second factor. I discuss more recent work in Section 3.2. CHAPTER 3. RISK PREMIA AND THE VIX TERM STRUCTURE 56

“volatility risk premia” factor (Bollerslev, Tauchen, and Zhou (2009) and Drechsler and Yaron (2011)), the aggregate dividend yield (e.g. Cochrane (2008) or Campbell and Shiller (1988)), the net payout yield (Boudoukh et al. (2007)), and the Cochrane Piazzessi factor from bond pricing (Cochrane and Piazzesi (2005)). The majority of the incremental return predictability comes from the fourth prin- cipal component of the VIX term structure, and not the first three (“level,” “slope,” and “curvature”) principal components. The factor has a “lightning bolt” shape, with loadings that slope downwards over the first three VIX horizons, jump up sharply, and then slope down again across the final three VIX horizons. It exhibits much less persistence than the VIX level, mitigating concerns over the small-sample predictive bias discussed in Stambaugh (1999). Finally, both the fourth principal component and risk premia fitted from the VIX term structure spike upwards around times of financial turmoil: the Asian financial crises (1997), the LTCM collapse (1998), and in the later portion of the recent financial crises (2008-2009). The exact linear combination of conditional variance horizons that reveals risk premia in my model depends on the model’s parametrization. Therefore, the principal component of the variance term structure that predicts future index returns also depends on the parametrization. I show that a reasonable calibration of my model matches the regression coefficients, as well as other relevant moments, observed in the data. Moreover, I show that as long as the different state variables have different persistences, the combined information in the VIX term structure provides an ex-ante measure of equity risk premia. I discuss the paper’s relation to prior research in Section 3.2, detail the model of priced and unpriced risks in the variance term structure in Section 3.3, show empirical results pertaining to the S&P 500 in Section 3.4, and conclude in Section 3.5.

3.2 Relation to prior research

My empirical results are closely related to those in Bakshi, Panayotov, and Skoulakis (2011), which pertain to forward variances inferred from option portfolios. Motivated Q R t+T by the bond pricing literature that focuses on Et {(− t rudu)}, where ru is the interest rate process, the authors compute the risk-neutral expectation of exponential CHAPTER 3. RISK PREMIA AND THE VIX TERM STRUCTURE 57

integrated variance:

  Z t+T  T −rT Q 2 Ht = e E exp − σudu t

T T2 T1 From these Ht , they compute the “forward” variances log(Ht )−log(Ht ), analogous to the forward interest rates studied in bond markets. The authors build a term structure of the four nearest-term variance forwards, and show that they combine to predict a slew of future macroeconomic indicators: growth in non-farm payroll, growth in industrial production, civilian unemployment, and others. Furthermore, they show that the first two forwards jointly predict future index returns in their 1998-2008 sample period. My results differ from those in Bakshi, Panayotov, and Skoulakis (2011) along multiple dimensions. The first is that I use the VIX2, which is a risk-neutral expec- tation of (non-exponential) integrated variance, or:

−rT Z t+T  2 e Q 2 VIXt,T = E σudu T t as a variance measure. The advantages of this measure are that it is widely used by both practitioners and academics, that it is positively related to σt, and that it T does not suffer from the Jensen’s inequality issue arising when taking the log of Ht . The disadvantage is that it is not analogous to the bond market term structure. The second difference is that I focus on the fundamental connection between conditional variances and expected returns, providing economic background for why the variance term structure predicts future equity returns. The third difference is that my empirical results extend the term structure to longer horizons (up to one year), and predict equity returns with higher R2 by using more variance horizons than the two used in Bakshi, Panayotov, and Skoulakis (2011). The theoretical relation between the time series of conditional variance and market risk premia originates with Merton (1973), which predicts that equity market returns rt+1 satisfy:

Et(rt+1) − rf = γvart(rt+1) where γ is the representative agent’s coefficient of relative risk aversion. As discussed CHAPTER 3. RISK PREMIA AND THE VIX TERM STRUCTURE 58

in detail below, risk premia are also a time-invariant linear (but not proportional) function of conditional variances in Campbell and Cochrane (1999) and Bansal and Yaron (2004). Backus and Gregory (1993) shows that for a representative agent model with power utility and fairly general asset return distribution, the shape of the con- ditional variance/expected return relation can have virtually any shape (increasing, decreasing, or nonmonotonic). I provide a simple formula relating conditional vari- ances to risk premia assuming only a stochastic discount factor; describe conditions under which the relation is linear; and argue that when the relation is non-linear, the term structure of conditional variances provides incremental information about expected returns. Empirically, as summarized in Scruggs (1998), the strength and even direction of the relation between conditional variance and future index returns depends on the model used for stochastic volatility, the sample period, and estimation methodol- ogy.2 Scruggs (1998) adds to this literature by showing the partial relation between conditional variance and future returns is positive once long-term government bond returns are added as a second factor. More recently, Ghysels, Santa-Clara, and Valka- nov (2005) finds a statistically and economically strong positive relation between conditional variance, calculated using an average of past daily squared returns with declining weights, and future index returns. Finally, Banerjee, Doran, and Peterson (2007) shows that option-implied variances predict future index returns positively once you control for the most recent innovation in option-implied variance. This pa- per argues that the relation is weaker in the data than implied by theory because of time variation in unpriced risk, and that the conditional variance term structure can provide a better estimate of expected returns than a single conditional variance. Two recent papers, Bollerslev, Tauchen, and Zhou (2009) and Drechsler and Yaron (2011), study the relation between variance risk premia and equity risk premia. Em- pirically, both show that the difference between end-of-month VIX2 and an estimate of statistical-measure variance positively predicts future index returns. Bollerslev, Tauchen, and Zhou (2009) uses past realized variance as a proxy for future statistical- measure variance, and Drechsler and Yaron (2011) uses predicted future variance from

2The results of French, Schwert, and Stambaugh (1987), Campbell (1987), Harvey (1989), Turner, Startz, and Nelson (1989), Baillie and DeGennaro (1990), and Glosten, Jagannathan, and Runkle (1993) are summarized well in Table 1 of Scruggs (1998). CHAPTER 3. RISK PREMIA AND THE VIX TERM STRUCTURE 59

a time-series model. In both papers, the evidence for return predictability is statisti- cally and economically strong from 1990-2007. I show that the return predictability continues in the additional 2008-2010 years in my sample. Both papers postulate that this return predictability arises because the time- Q P series of the variance risk premia (Vart (R) - Vart (R)) and the equity risk premia are positively correlated. Bollerslev, Tauchen, and Zhou (2009) derives this correla- tion in a discrete-time model with time variation in both consumption volatility and the volatility of consumption volatility. They show that the variance risk premium captures time variation in the volatility of consumption volatility, which is also cor- related with the time variation in risk premia. Drechsler and Yaron (2011) derives the positive correlation between equity and variance risk premia in broader setting that incorporates the long-run risk dynamics in Bansal and Yaron (2004) and so can simultaneously match the magnitudes of the equity risk premia and volatility risk premia we observe in the data, all while assuming a reasonable level of risk aver- sion. In their model, the variance risk premia comes from a drift difference in the consumption volatility process between Q and P, and from priced jump risk. Both the Bollerslev, Tauchen, and Zhou (2009) and the Drechsler and Yaron (2011) models predict no incremental role for the VIX term structure in capturing risk premia. The reason is that both models have two priced factors: the variance risk premia and the traditional risk-return relation (when conditional return volatility is high, expected returns are high). The first can be captured empirically by using the variance differences described above. The second is much simpler to implement empirically: a single VIX horizon should reveal the current conditional variance. For this reason, both Bollerslev, Tauchen, and Zhou (2009) and Drechsler and Yaron (2011) predict a much stronger VIX-return relation than the one observed in the data, and no role for multiple VIX horizons. In my multifactor volatility model, the nature of the variance-return relation varies across states, meaning there is a role for multiple VIX horizons in extracting risk premia. Mixon (2007) shows that the expectations hypothesis fails for the term structure of Black-Scholes implied volatility, namely that the “forward” implied volatilities em- bedded in the term structure are not one-for-one predictors of future realized “spot” implied volatilities. In untabulated results, I replicate the failure of the expectations CHAPTER 3. RISK PREMIA AND THE VIX TERM STRUCTURE 60

hypothesis for the new model-free VIX (as opposed to the old Black-Scholes based VXO used in Mixon (2007)) term structure. Prior research also shows theoretically and empirically that VIX term structure can be used to price VIX futures and options (e.g. Sepp (2008), and Zhu and Zhang (2007)). Duan and Yeh (2011) fits a fairly general single-factor stochastic volatility model to the observed S&P 500 index return and VIX term structure processes using a particle-filter based estimation. By contrast, I use a multifactor volatility model to study the relation between the VIX term structure and equity risk premium (rather than contemporaneous returns) and calibrate the model to match empirical moments rather than structurally estimating it. To my knowledge, the only multifactor model of index volatility is in Christof- fersen, Heston, and Jacobs (2009), which extends the Heston (1993) model by allowing the volatility process be the sum of two square-root processes. The two-factor volatil- ity model is able to capture the relative independence of the level and slope of the “smirk” observed in implied volatilities and therefore dramatically improve the upon the option pricing compared to Heston (1993). I augment their analysis by adding a time-varying SDF beta, looking at different times-to-expiration rather than different strike prices, and examining the economic relation between variance and expected returns. My central argument, that the VIX term structure allows an observer to distin- guish among components of the VIX that have different prices and persistences, has an analogue in the bond return predictability literature. If the short-rate process is composed of multiple factors with different pricing and persistences, like the volatility process in my model, the shape of the yield curve could be used to distinguish be- tween different short-rate factors and thereby better estimate bond risk premia. The bond risk premia literature (Dai and Singleton (2002), Cochrane and Piazzesi (2005), Cochrane and Piazzesi (2008), and the references therein) provides theoretical and empirical evidence that the shape yield of the curve reveals bond risk premia when there are multiple factors. There is a fundamental distinction, however, between the VIX term structure predicting equity index returns and the bond term structure predicting excess bond returns. In bond markets, the predicted returns are for the same asset whose yields CHAPTER 3. RISK PREMIA AND THE VIX TERM STRUCTURE 61

determine the term structure. In this paper, I use the term structure of option-implied variance to predict the returns of the underlying asset, relying on the economic con- nection between conditional variance and equity risk premia. The equivalent analysis for the treasury yield curve would examine the relation between bond risk premia and the term structure of bond return variance implied by bond options.

3.3 Model

The driving intuition in my model is that if the relation between conditional variance and risk premia is state-dependent, examining market expectations of volatility at multiple horizons allows an otherwise uninformed observer to distinguish between states and better estimate risk premia. I begin by illustrating the relation between variance and returns with two min- imal assumptions: no arbitrage, and the existence of a conditionally riskless asset. Together these assumptions imply that there exists a stochastic discount factor (SDF) ˜ ˜ Mt+1 that prices any traded asset i with gross return Ri,t+1 as follows:

˜ ˜ ˜ Et(Ri,t+1) − Rf,t = −covt(Ri,t+1, Mt+1)Rf,t (3.1)

where Rf,t is the gross risk-free rate at time t. I drop the subscript i hereafter for brevity. Now define αt, βt, and ˜t+1 so that:

˜ ˜ Rt+1 = αt − βtMt+1 + ˜t+1 (3.2) −cov (R˜ , M˜ ) β = t t+1 t+1 (3.3) t ˜ vart(Mt+1) ˜ covt(Mt+1, ˜t+1) = 0 (3.4)

With this sign convention, assets with a positive βt have a negative correlation with ˜ the SDF and are therefore subject to systematic risk. Note that αt − βtMt+1 is the ˜ ˜ projection of Rt+1 onto Mt+1, and that equation (3.2) is not a time-series regression because the coefficients αt and βt may be different in each period. ˜ Combining equations (3.1) and (3.3), and computing the variance of Rt+1 from CHAPTER 3. RISK PREMIA AND THE VIX TERM STRUCTURE 62

equation (3.2) gives:

˜ ˜ Et(Ri,t+1) − Rf,t = βtvart(Mt+1)Rf,t (3.5) ˜ 2 ˜ vart(Rt+1) = βt vart(Mt+1) + vart(˜t+1) (3.6)

The reason for manipulating the basic pricing equation into equations (3.5) and (3.6) is that these equations express both the conditional mean and variance of future returns as simple functions of three economically meaningful variables. The first,

SDF variance vart(m ˜ t+1), is an economy-wide variable, while the second, SDF beta βt, is asset-specific. Both are reflected in risk premia (equation (3.5)) as well as return variance (equation (3.6)). The final variable is the residual variance vart(˜t+1), which is also asset-specific and only impacts the return variance but not the mean of returns, and so I call it unpriced risk. To help interpret the SDF variance, SDF beta, and unpriced risk in what follows, I discuss the meaning of each in the context the asset pricing literature. The conditional ˜ SDF variance vart(Mt+1) is the variance of investor’s marginal utility of consumption in the next period. Changes in the volatility of marginal utility could be due to changes in the volatility of consumption growth (as in Bansal and Yaron (2004)), or to changes in the risk aversion of the marginal investor (as in Campbell and Cochrane (1999)). In this framework, both effects are absorbed into the SDF variance and impact expected returns and conditional variances for all traded assets.

The best analogies for the SDF beta βt and the unpriced variance vart(˜t+1) come from the CAPM. The asset-specific SDF beta βt serves the same role as the asset- specific “market beta” in the a period-by-period CAPM: it measures the exposure to priced risk for a specific asset. Similarly, what I call unpriced risk is analogous to “idiosyncratic volatility” in the CAPM. Both are orthogonal to what investor’s care ˜ about: market returns in the CAPM and the more general SDF Mt in this framework. If the CAPM holds period-by-period, the market portfolio is perfectly correlated with the SDF, implying that it has no unpriced risk. Other traded assets and portfolios, however, could have time variation in both βt and vart(˜t+1). Theorem 1 illustrates the general relation between conditional variance and the conditional risk premium. CHAPTER 3. RISK PREMIA AND THE VIX TERM STRUCTURE 63

˜ Theorem 1. Assuming no arbitrage, for any risky asset with returns Rt+1 and any ˜ ˜ SDF Mt+1 that prices Rt+1, we have:

˜ ˜ Et(Rt+1) − Rf,t = A0,t + A1,tvart(Rt+1) (3.7) where A = − vart(˜t+1))Rf,t , A = Rf,t , and the other variables are defined above. 0,t βt 1,t βt Proof. Manipulating equations (3.5) and (3.6) yields:

˜ ˜ 1 Et(Rt+1) − Rf,t = (vart(Rt+1) − vart(˜t+1)) Rf,t (3.8) βt

Equation (3.7) abbreviates this using the A0,t and A1,t notation. Note that Theorem 1 holds regardless of the time horizon, investor preferences, and for all types of assets. The only necessary assumptions are that there is a riskless asset, and that there is no arbitrage. In fact, equation (3.7) simplifies to the original pricing equation (3.1) with enough substitution. However, equation (3.7) is a useful representation because theoretical models often provide some structure for A0,t and

A1,t, from which we can use equation (3.7) to find ex-ante observable expected return estimates using ex-ante conditional variance estimates. The remainder of this section discusses different specifications for A0,t and A1,t. Theorem 1 examines a special case where both the SDF beta and unpriced risk are constant across time. This occurs when, while risk premia may be changing each ˜ ˜ period through changes in the SDF variance, the relation between Mt+1 and Rt+1 has the same slope and residual variance in each period. I show that this implies all time variation in risk premia can be captured by a time-invariant linear function of conditional variance. Moreover, I show the converse: if risk premia for any asset are a linear function of conditional variance, it implies that the asset has constant SDF beta and constant unpriced risk.

Corollary 1. Assuming the risk-free rate is fixed at Rf,t = Rf , and there is no arbitrage, the following two statements are equivalent:

2 1. Unpriced risk is fixed at vart(˜t+1)) = σ and the SDF beta is fixed at βt = β. ˜ ˜ 2. There exist constants A0 and A1 such that Et(Rt+1) − Rf = A0 + A1vart(Rt+1) in every period t. CHAPTER 3. RISK PREMIA AND THE VIX TERM STRUCTURE 64

2 Proof. Suppose vart(˜t+1)) = σ and βt = β. Plugging these values into A0,t and ˜ At,1 yields constant coefficients, meaning Theorem 1 implies Et(Rt+1) − Rf,t = A0 + 2 ˜ σ Rf Rf A1vart(Rt+1) in each period where A0 = − β A1 = β . ˜ Now suppose we have constants A0 and A1 such that Et(Rt+1) − Rf,t = A0 + ˜ A1vart(Rt+1) in every period. This implies that unpriced risk and SDF beta are constant over time, because if they were not, equation (3.7) would imply time-varying coefficients in the relation between risk premia and variance, a contradiction. The statements in Theorem (1) may seem quite restrictive, but they are met or nearly met in many modern asset pricing models. Both Campbell and Cochrane (1999) and Bansal and Yaron (2004), for example, predict that nearly all of the time-variation in equity risk premia is captured by a linear function of conditional return variance (see Appendix B.1 for details). The reason is that both assume equity markets provide a claim to dividends whose growth has constant correlation with consumption growth. This means that both the unpriced risk vart(˜t+1), and the SDF beta βt, are nearly constant across different states of the model. As a result, both models imply that a time invariant linear function conditional variance captures nearly all the variation in risk premia, meaning that conditional variance should predict future market returns and that nothing else should have significant predictive power incremental to conditional variance. A similar result applies in the models of variance risk premia in Bollerslev, Tauchen, and Zhou (2009) and Drechsler and Yaron (2011); both models predict a strong linear relation between conditional variance and future returns in addition to the relation between variance risk premia and future returns. Empirically, as detailed in Section 3.4, the relation between realized returns and conditional variance is weak. Moreover, there appears to be several other other vari- ables that capture risk premia better than conditional variances, namely the variance risk premia, the price-dividend ratio, and the variance term structure. These results suggest that the market’s SDF beta and unpriced variance change over time. In the next subsection, I impose some structure of the state variables that govern the variance-return relation, and show how the variance term structure can be combined to provide an observable measure of ex-ante risk premia when conditional variance alone is not enough. CHAPTER 3. RISK PREMIA AND THE VIX TERM STRUCTURE 65

A A three-factor model of variance and risk premia

I present a dynamic, discrete time, discrete state model of the relation between risk premia and the variance term structure. I focus here on the simplest possible frame- work that conveys my results. A more realistic continuous time, continuous state, version of the model in the Appendix B.3 produces qualitatively identical conclu- sions. ˜ In each period, the stochastic discount factor Mt+1 and the simple equity return ˜ Rt+1 each take one of two values:

( 1 1 ( 1 + σM,t w.p. µt + σR,t w.p. M˜ = Rf 2 R˜ = 2 t+1 1 1 t+1 1 − σM,t w.p. µt − σR,t w.p. Rf 2 2 where Rf is the risk-free rate, constant over time to focus my results on the term ˜ structure of variance and not interest rates, σM,t is the standard deviation of Mt+1,

µt is the expected equity return, and σR,t is the standard deviation of returns. The

SDF and return innovations have correlation ρt, which implies that:

1 (R˜ high |M˜ high) = (1 + ρ ) P t+1 t+1 2 t

2 There are three state variables in the model: SDF variance σM,t, SDF beta βt, 2 and unpriced risk σ,t. The distributional parameters depend on the state variables as follows:

2 2 2 2 σR,t = βt σM,t + σ,t (3.9) σM,t ρt = −βt (3.10) σR,t ˜ ˜ 2 µt − Rf = −Cov(Mt+1, Rt+1)Rf = βtσM,tRf (3.11)

The first two equations (3.9) and (3.10) are definitions, while equation (3.11) is the ba- sic pricing result. These definitions ensure that the general intuition developed above applies here. Equations (3.9) and (3.11) are exactly equations (3.5) and (3.6) above,

covt(R˜t+1,M˜ t+1) while equation (3.10) assures that βt is the regression coefficient − . As var(M˜ t+1) in the above, assets with negative SDF correlation (ρt < 0) have positive βt and CHAPTER 3. RISK PREMIA AND THE VIX TERM STRUCTURE 66

positive risk premia. 2 2 2 Each state variable has two possible values, σM,t ∈ {σM,L, σM,H }, βt ∈ {βL, βH }, 2 2 2 and σ,t ∈ {σ,L, σ,H }. Each also has a transition probability matrix that specifies its distribution over next-period states for each current-period state. These transition probability matrices are determined entirely by the three persistence parameters ρσm ,

ρβ, and ρσ . In each case, the persistence parameter is the period-to-period correlation in the state variable. For example, the transition probability matrix for βt can be summarized by:

1 (β = β ) = (1 + ρ ) P t+1 t 2 β 1 (β 6= β ) = (1 − ρ ) P t+1 t 2 β

The state transitions are uncorrelated with transitions in the other state variables, the asset’s return, and the SDF. Such correlations exist in the data, as evidenced by the fact that return variance tends to go up when markets go down (the so-called “leverage effect”). A version of this model with state innovations correlated with the SDF generates the leverage effect, but adds complexity without changing the relation between variance term structure and risk premia.

Observing risk premia in the model

While the marginal investor knows their preferences and the distribution of asset 2 2 values, and can therefore compute the three volatility states σM,t, βt, and σ,t,I study the problem of an outside observer hoping to infer the state using traded asset prices. For such an observer, the equity price reveals nothing about the volatility 2 state. Observing past returns could reveal the past volatility of returns σR,t−1 but 2 neither the state variables that compose σR,t−1 nor the current volatility of returns 2 σR,t. To assist the observer, I assume that variance swaps on the underlying asset are traded for multiple horizons. As shown in Carr and Madan (1998), in the absence of CHAPTER 3. RISK PREMIA AND THE VIX TERM STRUCTURE 67

jumps the strike of a variance swap IVt,T at time t expiring at time t + T is:

Z t+T  1 Q R IVt,T = Et vs ds (continuous time) T t T −1 1 X IV = EQ(σ2 ) (discrete time) (3.12) t,T T t R,t+s s=0

R The integrand vs is the quadratic variation of log(St) evaluated at t = s, and the expectation is under the risk-neutral measure Q. The second equation is the discrete analogue of the first, and can be computed easily in the model using the state transi- tion probabilities, together with return variance in each state as specified by equation (3.9). Because the state innovations are uncorrelated with the SDF, expected future variance is the same under the risk-neutral measure Q as it is under the statistical measure P. If the state innovation was correlated with the SDF innovation, the ex- pected future variance state could be computed using the state transition probabilities under Q. As discussed in more detail in Section 3.4, without data on variance swap pricing an observer can still compute IVt,T from the time t price of call and put options expiring at time t + T at enough different strike prices. The square of the CBOE’s VIX index is exactly such a computation, as are the model-free implied variance estimates discussed in Britten-Jones and Neuberger (2000), Jiang and Tian (2005), and elsewhere.

I show that the combined information in IVt,T at many different horizons T can 2 2 completely reveal the volatility state (σM,t, βt, σ,t), and therefore reveal any time variation in the equity risk premia. Intuitively, the reason is that when the different components of volatility have different persistences, many different volatility states could produce the same IVt,T for a single T , but each volatility state produces a unique shape of the IVt,T term structure. This intuition can be seen clearly in Figure 3.1, which shows the VIX term structure in each of the eight different states of the world3, as well as the annualized equity premia. Due to the array of possible SDF betas and unpriced risk, the equity risk premia is a non-linear, non-monotonic,

3Each of the three state variables can take one of two values, making a total of eight states for the model. CHAPTER 3. RISK PREMIA AND THE VIX TERM STRUCTURE 68

function of conditional variance for any single horizon. However, each state of the world is easily identifiable from the shape of the term structure. Theorem 2 shows more formally that risk premia are a linear function of the VIX term structure in the model.

Theorem 2. If each state variable has a different persistence (ρσM , ρβ, and ρσ are all different), the equity risk premium is an affine function of the implied variance term structure:

2 µt − Rf = βtσM,tRf = A0 + A1IVt,{T1,T2,T3,T4}

where IVt,{T1,T2,T3,T4} is a 4x1 vector of implied variances at four different horizons

{T1,T2,T3,T4} on date t, A0 is a constant, and A1 is 1x4 vector of constants. Proof. Define state vector:

0 h 2 2 2 2 2 i zt = σM,t βt σM,tβt σ,t

Note that that both the equity risk premium µt − Rf and the implied variance IVt,T are affine functions of this state vector:

µt − Rf = b0 + b1zt

IVt,T = c0,T + c1,T zt

⇒ IVt,{T1,T2,T3,T4} = C0 + C1zt

The constants b0 and c0,T , as well as the 1x4 vectors of constants b1 and c1,T , are functions of model parameters but not the state variables, as detailed in Appendix

B.2. The 4x1 vector C0 and the 4x4 matrix C1 are the four different c0,T and c1,T stacked on top of eachother.

Assuming C1 is invertible, we now have that:

−1
zt = C1 IVt,{T1,T2,T3,T4} − C0 −1
⇒ µt − Rf = b0 + b1C1 IVt,{T1,T2,T3,T4} − C0 (3.13)

Equation (3.13) is shows that the equity risk premia is an affine function of the CHAPTER 3. RISK PREMIA AND THE VIX TERM STRUCTURE 69

implied variance term structure whenever C1 is invertible, which I show in Appendix

B.2 holds exactly when the three persistences ρσM , ρβ, and ρσ are all different from each other.

Theorem 2 shows that an affine function of the VIX term structure reveals equity risk premia because each different VIX horizon loads differently on the underlying state variables, allowing the state variables to be inverted given enough different VIX horizons. The same procedure would work for any set of four assets whose prices were each a different affine function the state variables, allowing the matrix C1 to be inverted. Theorem 2 also predicts that all variation in expected returns are captured by an affine function of the VIX term structure, meaning that the VIX term structure (and nothing else) should predict future equity returns in a linear regression.

Calibrated Model

I present sample moments for a specific calibration of my model in order to illustrate that the regression coefficients relating the VIX term structure to future index returns, as described in Section 3.4, can arise in my relatively simple model. I calibrate the model to data on the S&P 500 by matching the moments listed in Table 3.1. The unit of time is one month, and I use the VIX term structure as described in Section 3.4 as a measure of the risk-neutral expected return variation. I show that the calibrated model is capable of generating weak return predictability for a single variance horizon and strong return predictability for multiple variance horizons, with coefficients closely matching those in the data. The first four moments in Table 3.1 are the unconditional mean of the one-month VIX and the standard deviations of the VIX at three different horizons. I calculate the unconditional mean VIX level under the statistical measure to match the data, but the model VIX itself is a risk-neutral expectation of future diffusion, as described above. The longer-horizon VIX are less volatile in both the model and the data because volatility mean reverts, meaning that the short-horizon VIX is normally farther from the long-run mean than the long-horizon VIX. The calibrated model matches these four moments fairly well, although the longer horizon VIX are more volatile in the data than the model. CHAPTER 3. RISK PREMIA AND THE VIX TERM STRUCTURE 70

The next two moments are the mean and variance of monthly excess returns. In the model, expected returns and statistical variance are given by equations (3.11) and (3.9), respectively. The calibrated model matches the unconditional risk premium quite closely. However, returns in the calibrated model are too volatile under the statistical measure. In the model, there is no difference between statistical and risk- neutral volatility, and so it cannot replicate the dramatic difference between average 2 VIX1 and realized return variance (variance risk premia) observed in the data. If the model were extended to allow correlation between the SDF and innovations in the volatility states, there would be a positive variance risk premia. The next moment is the correlation between the one-month VIX and the one- month VIX one month ago, which is is determined in the model by the combined persistence of the three different volatility measures. The calibrated model produces a 0.758 correlation, while the correlation in the data is 0.805. Moments labelled (8)-(12) in Table 3.1 are regression coefficients from a single regression of the simple excess return Rt+1 − Rf,t on a constant and VIX1,t; and a 2 2 2 2 multiple regression of Rt+1 − Rf,t on a constant, VIX1,t, VIX3,t, VIX6,t, and VIX12,t. The calibrated model matches the weakly positive single regression slope coefficient, as well as the “lightning bolt” shape of the multiple regression coefficients4. I match coefficients with only four of the six VIX horizons because, as shown in Theorem 2, any four VIX horizons completely reveals the risk premia in the model, meaning a regression with more than four horizons is not identified. Table 3.1 also presents the model parameters used in calibration. In order to match the remarkable volatility in the VIX index observed in my 1996-2010 sample, the spread between high and low parameter values is quite large. For example, the SDF variance is 0.06 in its high state and 0.01 in its low state. The persistence parameters are primarily responsible for the regression coefficients in the multi-VIX regression. A calibration with persistences 0.95 for SDF beta, 0.85 for SDF variance, and 0.55 for unpriced risk produces approximately the regression coefficients observed in the data. A different persistence combination would produce different regression coefficients, but as long as the state variables each have different persistences, the VIX term structure provides incremental return predictability beyond a single VIX

4The coefficients used in this calibration do not exactly match those in Table 3.3 because they pertain to regressions with simple returns, rather than log returns, on the left-hand side. CHAPTER 3. RISK PREMIA AND THE VIX TERM STRUCTURE 71

horizon.

3.4 Empirical results

In order to study the relation between conditional variances and risk premia, I need an empirical measure of conditional variance. I use an option-based measure, rather than a forecast from a time-series volatility model, because options markets are able to incorporate all public information about future variances. Time-series volatil- ity models, on the other hand, can only incorporate conditioning variables that are both observable and incorporated into the model, making them inherently backward- looking. The drawback of the option-implied variance measure is that it produces risk-neutral rather than statistical variances, an issue I return to later. I compute the VIX term structure by replicating the CBOE’s VIX calculation5, but with longer horizons than 30 days. The VIX calculation is based on the model- free implied volatility measure originating from Breeden and Litzenberger (1978). Assuming the availability of options at every strike price, the VIX is defined by:

rT Z Ft Z ∞  2 2e 1 1 VIX ≡ 2 putT (K)dK + 2 callT (K)dK (3.14) T 0 K Ft K where putT (K) and callT (K) are the prices at time 0 of puts and calls expiring at time T with strike price K. As shown in Neuberger (1994) and Carr and Madan (1998), if the S&P 500 follows a diffusion process dSt = rStdt + σtStdZt under the risk-neutral measure, VIX2 equals the risk-neutral expectation of average future instantaneous h i Q R T 2 variance E 0 σt dt . An alternate option-implied variance measure is the SVIX from Martin (2011), which is defined by:

rT Z Ft Z ∞  2 2e 1 1 SVIX ≡ 2 putT (K)dK + 2 callT (K)dK (3.15) T 0 S0 Ft S0

Equation (3.15) uses the same notation at equation (3.14) with the addition of S0, the price of the underlying at time 0. Martin (2011) shows that the SVIX measures the

5See www.cboe.com/micro/vix/vixwhite.pdf for more details. CHAPTER 3. RISK PREMIA AND THE VIX TERM STRUCTURE 72

risk-neutral variance of simple returns from time 0 to time T, and unlike the VIX its interpretation is robust to the presence of jumps. The SVIX2 and VIX2 turn out to be very highly correlated in the data, with the VIX2 exceeding the SVIX2 significantly only during periods of crisis like the fall of 2008. My empirical results employ the widely-followed VIX but are qualitatively identical when using SVIX. The standard approach to estimating the VIX equation (3.14) empirically, used by the CBOE to compute the VIX, discretizes the integral at the available strike prices and truncates it at the smallest and largest available strike prices. Jiang and Tian (2005) discusses each as a potential source for estimation error and, using simulated data, conclude that the number of strikes available for S&P 500 index options is sufficient to compute the one-month VIX. However, at longer horizons, options are less liquid and the range of plausible index values is larger. For that reason, Figure 3.2 presents the numeric integrals used to compute the twelve-month VIX on the first day of my sample, likely to be the worst day/horizon pair in my analysis. The twelve-month VIX is the linear interpolation of the VIX estimates on the two nearest expiration dates in the data, computed as the area under the two curves in Figure 3.2. While there is clearly some truncation error, especially for the range of strikes above 700, it looks to be only a small percentage of the total area under each curve. Critically, since equation (3.14) relies heavily on out-of-the-money put options with 1 low K and high K2 , liquidity in options markets tilts heavily towards exactly these options. The results in Bollerslev, Tauchen, and Zhou (2009) and Drechsler and Yaron (2011) indicate that past realized volatility, controlling for the forward-looking VIX, negatively predicts future index returns. To examine the incremental predictive power of the VIX term structure, I therefore need to control for realized volatility in my regressions. For a measure of realized volatility, I follow Drechsler and Yaron (2011) and use intraday S&P 500 futures data provided by tickdata.com. More specifically, I compute a time series of five-minute returns for S&P 500 futures and estimate the variation of log returns from t − 1 to t using:

n X h  i2 RVˆ = log R j t−1+ n j=1 CHAPTER 3. RISK PREMIA AND THE VIX TERM STRUCTURE 73

Intraday data at five-minute intervals has the advantage of more accurately estimating return variation than coarser daily data, but has the disadvantage that it requires accurate intraday prices. As discussed in Drechsler and Yaron (2011), the S&P 500 index level can often have stale prices in intraday data because it is the aggregation of 500 different prices, and not traded directly. S&P 500 futures, by contrast, are a single actively-traded asset, making stale prices much less likely. Using closing option quotes for S&P 500 index options and risk-free rates available from 1996 through 2010 via OptionMetrics, I compute the VIX term structure and a rolling estimate of past one-month return variation at the close of markets each day. Occasionally in my sample period, there were option expiries on the last trading day of the quarter in addition to the normal third Saturday of every month. I remove these observations because they generally did not have nearly the volume or range of strike prices offered by the normal expiration dates. I also remove a handful day/expiration date pairs which appear to have missing data in OptionMetrics, as well as a few individual option prices that are clearly data errors. Table 3.2 presents some descriptive statistics for the VIX term structure, including 2 VIXT for T = 1, 2, 3, 6, 9, and 12 months. These horizons were chosen to represent the approximate times-to-expiration available at any given time for index options. The first thing to note is that, on the median day, the term structure is upwards sloping from T = 1 to T = 6 and then about flat out until T = 12. There is, however, some variability in the shape of the term structure, for example the interquartile range 2 VIX6 of 2 is 0.95 to 1.31. Finally, the entire VIX term structure tends to be much higher VIX1 than past realized volatility, though in some rare cases this relation can reverse. Figure 3.3 plots the evolution of the term structure over my 1996-2010 sample. The primary thing to notice is that the term structure is downward sloping when 2 2 VIX1 is particularly high, and upward sloping when VIX1 is particularly low. This suggests that investors assume some mean reversion in volatility will bring it “towards normal” over the next year. Another salient feature of the VIX term structure visible in Figure 3.3 is that while the different VIX horizons are strongly correlated with one another, there are changes in the exact shape of the term structure beyond its level and slope. Table 3.3 illustrates the predictive power of the VIX term structure for future CHAPTER 3. RISK PREMIA AND THE VIX TERM STRUCTURE 74

returns. The dependent variable is future S&P 500 index returns, inclusive of divi- dends, starting at the close of markets one day after the observation of the VIX term structure. The one day gap is important because options markets close 15 minutes later than equity markets. The independent variables are the option-implied vari- 2 ances at different horizons, as measured by VIXT . These are scaled to be monthly variances in order to match the time period used for the model in Section 3.3. Panel A shows that, by itself, the one-month VIX has no predictive power for future one- and three-month returns, statistically insignificant power for future six- month returns, and some predictive power for one-year returns. Adding the full 2 2 VIX term structure (VIX2 ... VIX12) dramatically increases the return predictability, adding 1.99%, 5.16%, 6.60%, and 3.73% in incremental R2 for the one-, three-, six-, and twelve-month horizons, respectively. I reject the null hypothesis that the five longer horizon VIX all have coefficients equal to zero with p values 0.1%, 0.1%, and 0.4% for the three-, six-, and twelve-month horizons. I cannot reject the null for 2 2 next-month returns. The intermediate rows, which only include VIX1, VIX6, and 2 VIX12, demonstrate that the addition of two longer-horizon VIX provides much, but not all, of the incremental predictive power gained by the VIX term structure. Panel B repeats the analysis in Panel A but with past realized variance RV (de- fined above) as an additional regressor. As demonstrated in Bollerslev, Tauchen, and 2 Zhou (2009) and Drechsler and Yaron (2011), the difference between VIX1 and RV predicts future index returns quite well. However, at all horizons the predictive power 2 of the VIX term structure is incremental to the predictability afforded by VIX1 and 2 RV. I can reject the null hypothesis that the coefficients on VIXT are all zero for 2 T ≥ 2 after controlling for VIX1 and RV with p-values of 0.3%, 0.1%, and 0.3% for the three-, six-, and twelve-month horizons. As in Panel A, I cannot reject the null for one-month returns. At all horizons there is also a substantial increase in R2 upon the addition of the VIX term structure. Finally, Panel C shows that the VIX term structure predicts returns incrementally to three predictors established in prior research. The first is the dividend yield, measured as the log of the prior year’s total market-wide dividends scaled by the current total market capitalization, as in Cochrane (2008). The second is the net dividends+net repurchases payout yield, which is log(0.1 + market cap ), where net repurchases are the CHAPTER 3. RISK PREMIA AND THE VIX TERM STRUCTURE 75

total value of share repurchases less share issuances in the prior year, as in Boudoukh et al. (2007). The third and final is the bond risk premia factor from Cochrane and Piazzesi (2005), which predicts equity market returns as well as excess bond returns. The first two measures are available monthly on Michael Roberts’ website, and the I generate the third from the code available on John Cochrane’s website. The results in Panel C indicate that the VIX term structure predicts returns incrementally to these three extant predictors, with p values of 0.1%, 0.0%, and 0.6% for the three-, six-, and twelve-month horizons and a similar pattern of coefficients to Panel B. Each of the R2 are much higher in Panel C than Panels A and B, as expected given the high R2 associated with the net payout yield in Boudoukh et al. (2007). A natural question is whether results in Table 3.3 are driven by the use of a risk- neutral, rather than statistical, conditional variance measure. Section 3.3 argues that risk premia can be inferred from the term structure of statistical (not risk-neutral) variance. The VIX captures both the statistical variance, as desired in the model, and the variance risk premia. I cannot rule out the possibility that the term structure of the variance risk premia (and not the statistical variance) is what predicts returns. However, the variance risk premia models in Bollerslev, Tauchen, and Zhou (2009) and Drechsler and Yaron (2011) predict an incremental role for return predictability from a single horizon, not the full term structure, of the variance risk premia. The regressions in Table 3.3 indicate that something in the VIX term structure reflects time variation in risk premia, however interpreting the regression coefficients is difficult because of the colinearity of the right hand side variables. For this reason, I perform principal components analysis and assess the role of each orthogonal factor in predicting future index returns. Table 3.4 presents the results. As observed in bond markets, the first three principal components have patterns that can be described as the level, slope, and curvature of the term structure. Also similar to bond markets (see Cochrane and Piazzesi (2005)), I find that a significant portion of return predictability comes from principal components other than the level, slope, and curvature. The fourth principal component (PC4) provides return predictability consistently across return horizons from one month to twelve months. PC4 has a “lightning bolt” shape, sloping from positive down to negative over the first three expirations, then back to positive at the six-month expiration and sloping down again through the CHAPTER 3. RISK PREMIA AND THE VIX TERM STRUCTURE 76

twelve-month expiration. Table 3.5 shows that the return predictability associated with PC4 is statistically significant even if I remove the financial crisis of 2008-2009 from the sample6. Because the level factor was extremely high in late 2008, the rela- tion between the first principal component (“level”) and future returns substantially changes with the removal of the financial crises. The level factor predicts returns more strongly for shorter horizons but less strongly for longer horizons without the financial crisis in the sample. Figure 3.4 plots PC4 alongside the fitted next-quarter expected return from the regression in Panel A of Table 3.3 that includes all six VIX horizons. Each plot is a rolling average over the prior month to facilitate visual interpretation. The figure shows that PC4 is highly correlated with the fitted expected return, indicating that most of the incremental predictive for future returns demonstrated in Table 3.3 comes from PC4. Moreover, both fitted expected returns and the PC4 are quickly mean reverting7, suggesting that the VIX term structure captures shorter-run risk premia, for example premia arising from liquidity risk or short-term order imbalances. Figure 3.4 also highlights the periods in my sample with abnormally high un- certainty, as defined in Bloom (2009)8. Both PC4 and the fitted expected return spiked up at times in my sample intuitively associated with financial market turmoil. For example, the Asian crisis (November 1997), LTCM’s collapse (September-October 1998), and the later stages of the recent financial crisis (October 2008-June 2009) were all accompanied by upward moves in the PC4 and fitted expected returns. However, the uncertainty events not pertaining directly to the health of financial markets, the 9/11 terrorist attacks and the defaults of Worldcom and Enron, were not accompanied by spikes in PC4 of fitted expected returns. Panel B of Tables 3.4 and 3.5 show the principal components analysis repeated with past realized volatility as one of the time series. I find that RV fits right into the 2 level factor, loading positively along with all the VIXT horizons in the first principal component. The second PC reflects the difference between the VIX and RV, loading 2 negatively on RV and positively on the VIXT for T ≥ 2, with increasing weight on 6I remove days in my sample between October 1st, 2008 and July 1st, 2009, approximately the period in which the one-month VIX was extraordinarily high. 7PC4 has a monthly AR(1) coefficient of 0.39. 8I include each period from the list in Bloom (2009) occurring in my sample with the exception of the second Gulf War in 2003, which did not appear to have a big effect on index option markets. CHAPTER 3. RISK PREMIA AND THE VIX TERM STRUCTURE 77

2 the longer horizons. This PC is strongly correlated with VIX1-RV (correlation 76%, result untabulated), and improves upon its return predictability. However, the old fourth principal component remains essentially unchanged in the new fifth principal 2 component, meaning it is orthogonal to the improved VIXT -RV factor and predicts returns incrementally to it. As discussed in the Section 3.3, the incremental predictive power of small dif- ferences in the VIX term structure arises from the different factors that compose volatility having different persistences. When the VIX is high it could be that the SDF is particularly volatile, the index has an abnormally high SDF beta, unpriced risk is high, or some combination thereof. By looking at the exact timing of the mean reversion implied by the VIX term structure, an observer can distinguish among these possibilities, and therefore more accurately estimate the equity risk premium.

3.5 Conclusion

The tradeoff between risk and return is a fundamental concept in finance. However, quoting Bollerslev, Tauchen, and Zhou (2009): “a significant time-invariant expected return-volatility tradeoff type relationship has largely proven elusive.” I argue that even for broad market indices, the expected return-volatility tradeoff is complicated by time variation in the relation between returns and the stochastic discount factor. I show under minimal assumptions that assets with constant SDF beta and constant unpriced risk have a time-invariant linear relation between variance and returns. In a reduced-form model with a state-dependent variance-return relation, I show that the combined information in multiple VIX horizons can distinguish between priced and unpriced risk and, therefore, provide much better estimates of expected returns. Empirically, I show that the VIX term structure dramatically improves the pre- dictive power of the VIX and the variance risk premium for future S&P 500 returns. Adding the remainder of the VIX term structure to a regression of next-quarter S&P 2 2 500 returns on VIX1 and past realized variance increases the adjusted R from 4.8% to 8.9%. In the same regression, I reject the null hypothesis of zero coefficients on the entire VIX term structure with p-value 0.3%. Most of the return predictability comes from the fourth principal component of the term structure, which significantly CHAPTER 3. RISK PREMIA AND THE VIX TERM STRUCTURE 78

predicts returns at horizons from one month to one year throughout the 1996-2010 sample period. Collectively, the evidence in this paper indicates that multiple fac- tors, with different prices and persistences, combine to form the volatility of equity returns, and that an observer can distinguish between these factors using the VIX term structure.

3.6 Figures and Tables

Figure 3.1: Model Equity Risk Premia and the VIX Term Structure This figure shows the VIX term structure, and corresponding equity risk premia, in each possible state of the world in my calibrated model. The VIX term structure is presented as an annualized standard deviation. The equity risk premia, presented to the left of each curve, are also annualized. The model and its calibration are described in Section 3.4 and Table 3.1. CHAPTER 3. RISK PREMIA AND THE VIX TERM STRUCTURE 79

2 Figure 3.2: Functions integrated to compute VIX12 on January 2nd, 1996. This figure presents the functions integrated when computing the twelve-month VIX2 on the first day in my sample, January 2nd, 1996. Each function is the integrand of equation (3.14) in Section 3.4 of the paper. As described in the text, these functions are numerically integrated across the range of available strike prices, and interpolated to estimate the twelve-month VIX2. CHAPTER 3. RISK PREMIA AND THE VIX TERM STRUCTURE 80

Figure 3.3: Time Series of the Term Structure I compute the VIX term structure using the same procedure as the publicized VIX but with longer times-to-expiration than one month. The term structure includes VIX at one-, two-, three-, six-, nine-, and twelve-month horizons, and is computed daily throughout my 1996-2010 sample. This figure presents the rolling average over the prior month to ease visual interpretation. CHAPTER 3. RISK PREMIA AND THE VIX TERM STRUCTURE 81 horizons. Both time series are computed each day in my 1996-2010 sample, 2 Figure 3.4: Return predictability factors in the VIX term structure This figure presents two return-predicting(PC4) factors of that emerge the from term thequarter structure VIX excess as term log detailed structure. S&P inthen The 500 Table first smoothed index 3.4, is using returns and the the on the fourthidentified average the principal second in over component six is Bloom the VIX a prior (2009):terrorist fitted month attacks 1 value to (Sep from is 2001), ease the the 4 visual2009). regression is Asian interpretation. the in Financial Worldcom The Table Crisis and 3.3 shaded Enron (Nov of areas bankruptcies 1997), (Jul-Oct realized represent 2 2002) high next- and is volatility 5 the events is LTCM the crisis recent financial (Sep-Oct crisis 1998), (Oct 3 2008-Jun is the September 11th CHAPTER 3. RISK PREMIA AND THE VIX TERM STRUCTURE 82

Table 3.1: Data and Calibrated Model Moments This table presents model-calibrated and data estimated moments pertaining to the VIX term struc- ture and its relation to the equity risk premia, as well as the model parameters used in calibration. All moments are annualized. The VIX moments are computed in my 1996-2010 sample, while the mean and standard deviation of excess market returns are from a longer 1926-2010 sample. The 2 regression coefficient moments come from two different regressions. The “VIX1 alone” coefficient 2 comes from a regression of future excess equity returns on the 1-month VIX1, while the other four coefficients come from a regression of future excess equity returns on one-, three-, six-, and twelve- month VIX2. The model parameters, given on the right hand side, are defined in the text.

Moment Empirical Model (1) Mean(VIX1) 22.12% 23.12% Parameter Value (2) Std(VIX1) 8.76% 9.15% βH 0.35 (3) Std(VIX6) 7.07% 5.75% βL 0.02 (4) Std(VIX12) 6.45% 4.53% σ2 0.06 (5) Mean(R − Rf ) 7.56% 7.77% M,H 2 (6) Std(R − Rf ) 18.60% 23.12% σM,L 0.01 2 (7) corr(VIX1,t,VIX1,t+1) 0.805 0.758 σ,H 0.005 2 2 σ,L 0.001 (8) VIX1 alone coefficient 0.540 1.942 ρβ 0.95 (9) VIX2 coefficient -2.10 -3.69 1 ρVm 0.85 (10) VIX2 coefficient -7.02 -5.75 3 ρV 0.55 2  (11) VIX6 coefficient 21.26 29.79 2 (12) VIX12 coefficient -11.45 -18.86 CHAPTER 3. RISK PREMIA AND THE VIX TERM STRUCTURE 83

Table 3.2: Descriptive Statistics of Term Structure I compute the VIX term structure using the same procedure as the publicized VIX but with longer times-to-expiration than one month. For example, VIX2 is the interpolated two-month model-free implied volatility. RV is a rolling measure of past realized return variance, estimated using intraday S&P 500 futures returns at 5-minute intervals over the prior month. The sample is daily from 1996 through 2010. I each VIXT to get variances. Below are the time-series percentiles of the VIX term 2 structure, scaled by VIX1.

2 2 2 2 2 RV VIX2 VIX3 VIX6 VIX9 VIX12 2 2 2 2 2 2 VIX1 VIX1 VIX1 VIX1 VIX1 VIX1 Minimum 0.17 0.66 0.36 0.39 0.30 0.23 5th percentile 0.38 0.87 0.79 0.72 0.64 0.60 25th percentile 0.52 0.98 0.96 0.95 0.91 0.88 Median 0.65 1.04 1.07 1.12 1.10 1.11 75th percentile 0.81 1.10 1.18 1.31 1.35 1.41 95th percentile 1.24 1.20 1.34 1.58 1.70 1.80 Maximum 2.96 1.54 1.73 2.10 2.31 2.69 CHAPTER 3. RISK PREMIA AND THE VIX TERM STRUCTURE 84

Table 3.3: Return Prediction Regressions The dependent variable in each of these regressions is future S&P 500 index returns for one of four different horizons. Each return period begins 2 days after the regressors are observed to account for the non-synchronous market closing times for options and stocks. The regressors are the variance 2 2 term structure VIX1 ··· VIX12 as defined in Table 1, and a constant (not reported). Panel B also includes RV, past realized variance computed using log S&P 500 futures returns at 5-minute intervals over the prior month. Panel C also controls for three other return predictors from the literature: dividend yield, net payout yield, and the Cochrane Piazzesi factor (coefficients not reported). The last two columns show the adjusted R2 and a χ2 statistic for the joint significance of the variance term structure. Below the coefficients and χ2 statistics are p-values. Significance at the 10%, 5%, and 1% levels are denoted by *, **, and ***. The sample is daily from 1996-2010. Standard errors are adjusted using Newey-West with lags equal to one-and-a-half times the return horizon.

Panel A: Without Past Realized Variance Return 2 2 2 2 2 2 2 2 Horizon VIX1 VIX2 VIX3 VIX6 VIX9 VIX12 Adj. R χ 0.17 - - - - - 0.00% - (86.2%) ------4.42 - - 15.41* - -10.68* 1.50% 3.19 1 mo. (12.3%) - - (7.8%) - (9.8%) - (36.3%) 1.65 -9.64 -2.21 22.57* 1.28 -13.01** 1.99% 7.63 (79.7%) (28.0%) (72.5%) (5.7%) (89.3%) (3.7%) - (26.6%) 0.55 - - - - - 0.06% - (83.2%) ------11.83*** - - 36.54** - -22.32 4.01% 11.60*** 3 mo. (0.1%) - - (2.9%) - (13.1%) - (0.9%) 3.68 -25.79*** -0.61 38.90* 24.05 -36.69*** 5.22% 21.85*** (57.5%) (0.5%) (94.5%) (8.4%) (13.0%) (0.1%) - (0.1%) 4.04 - - - - - 1.94% - (10.1%) ------17.59*** - - 57.22** - -30.56 7.69% 21.59*** 6 mo. (0.0%) - - (1.3%) - (20.1%) - (0.0%) 2.93 -34.30** -1.01 63.03** 27.72 -47.83* 8.54% 23.21*** (76.6%) (3.5%) (93.2%) (4.8%) (22.9%) (6.3%) - (0.1%) 8.22** - - - - - 3.68% - (1.2%) ------13.85** - - 67.78** - -43.19 6.08% 13.64*** 12 mo. (1.4%) - - (1.0%) - (16.1%) - (0.3%) -0.57 -21.11 7.93 18.66 94.45** -86.38** 7.41% 19.17*** (97.3%) (49.0%) (65.2%) (60.3%) (1.5%) (3.7%) - (0.4%) CHAPTER 3. RISK PREMIA AND THE VIX TERM STRUCTURE 85

Table 3.3: Return Prediction Regressions, continued.

Panel B: With Past Realized Variance Return 2 2 2 2 2 2 2 2 Horizon RV VIX1 VIX2 VIX3 VIX6 VIX9 VIX12 Adj. R χ -2.88** 3.09** - - - - - 1.62% - (1.3%) (4.2%) ------2.81*** -1.21 - - 15.57* - -11.58* 2.99% 3.27 1 mo. (0.5%) (62.1%) - - (7.4%) - (7.4%) - (19.5%) -2.83*** 4.14 -8.05 -3.23 24.62** -2.08 -12.38** 3.46% 6.65 (0.5%) (51.7%) (41.4%) (60.8%) (4.0%) (82.5%) (4.3%) - (24.8%) -8.41*** 9.07*** - - - - - 4.84% - (0.0%) (0.6%) ------7.87*** -2.85 - - 36.97** - -24.84* 8.04% 7.24** 3 mo. (0.0%) (45.8%) - - (2.4%) - (8.4%) - (2.7%) -7.60*** 10.40 -21.50** -3.35 44.43** 15.00 -35.01*** 8.90% 17.65*** (0.0%) (15.4%) (2.5%) (69.2%) (4.6%) (37.8%) (0.2%) - (0.3%) -9.57*** 13.74*** - - - - - 4.60% - (0.0%) (0.0%) ------7.98*** -8.49* - - 57.66** - -33.11 9.46% 19.60*** 6 mo. (0.0%) (6.1%) - - (1.2%) - (16.3%) - (0.0%) -7.63*** 9.67 -30.00* -3.76 68.58** 18.64 -46.14* 10.12% 20.96*** (0.0%) (30.8%) (7.3%) (75.9%) (3.5%) (39.3%) (5.9%) - (0.1%) -10.70*** 19.06*** - - - - - 5.20% - (0.5%) (0.1%) ------9.77** -2.71 - - 68.32*** - -46.32 7.28% 10.22*** 12 mo. (1.2%) (73.5%) - - (1.0%) - (12.8%) - (0.6%) -8.50** 6.94 -16.32 4.86 24.84 84.34** -84.50** 8.30% 18.16*** (2.2%) (64.5%) (60.4%) (76.8%) (47.6%) (2.1%) (3.7%) - (0.3%) CHAPTER 3. RISK PREMIA AND THE VIX TERM STRUCTURE 86

Table 3.3: Return Prediction Regressions, continued.

Panel C: With Past Realized Variance, Dividend Yield, Net Payout Yield, and CP Factor Return 2 2 2 2 2 2 2 2 Horizon RV VIX1 VIX2 VIX3 VIX6 VIX9 VIX12 Adj. R χ -2.95** 2.65* - - - - - 3.33% - (1.9%) (8.4%) ------3.04*** -1.10 - - 15.52* - -13.64** 4.84% 4.80* 1 mo. (0.5%) (66.3%) - - (6.5%) - (3.5%) - (9.1%) -3.19*** 4.17 -7.83 -3.86 31.47*** -11.12 -10.90 5.70% 8.19 (0.4%) (54.9%) (45.8%) (53.9%) (0.8%) (27.0%) (10.9%) - (14.6%) -8.63*** 7.87** - - - - - 10.08% - (0.0%) (1.6%) ------8.56*** -2.34 - - 39.34** - -30.43** 13.04% 5.36* 3 mo. (0.0%) (56.4%) - - (2.3%) - (3.6%) - (6.8%) -8.59*** 10.88 -21.88** -4.46 61.15*** -7.20 -31.49** 14.02% 21.42*** (0.0%) (17.4%) (2.7%) (63.3%) (0.3%) (69.6%) (1.2%) - (0.1%) -9.87*** 11.18*** - - - - - 13.73% - (0.0%) (0.2%) ------9.11*** -7.82* - - 62.61** - -43.98* 17.35% 17.62*** 6 mo. (0.0%) (9.4%) - - (1.3%) - (8.2%) - (0.0%) -9.38*** 10.25 -30.65** -6.15 102.26*** -25.56 -39.05 18.40% 36.49*** (0.0%) (29.5%) (2.2%) (53.5%) (0.2%) (31.1%) (11.3%) - (0.0%) -8.53** 12.47** - - - - - 21.90% - (2.3%) (4.6%) ------9.10*** -6.00 - - 83.36*** - -69.55** 24.54% 10.08*** 12 mo. (0.6%) (27.3%) - - (0.8%) - (3.9%) - (0.7%) -9.15** 6.05 -22.14 -0.23 102.84*** -9.50 -68.93* 24.65% 16.50*** (0.6%) (66.6%) (39.6%) (98.7%) (0.8%) (73.8%) (6.3%) - (0.6%) CHAPTER 3. RISK PREMIA AND THE VIX TERM STRUCTURE 87

Table 3.4: Principal Components (Full Sample) Panel A presents principal components analysis for the VIX term structure without past realized variance, Panel B with past realized variance RV. The first block of both panels shows the coefficients defining each principal component. The second block gives the fraction of term structure variance explained by each principal component. The remaining blocks shows the coefficients, p-values, and incremental R2 from standardized regressions of future index returns on the principal components at different horizons. The sample is daily from 1996 through 2010. Standard errors are adjusted using Newey-West with lags equal to one-and-a-half times the return horizon.

Panel A: Without past realized volatility “Level” “Slope” “Curve” PC1 PC2 PC3 PC4 PC5 PC6 2 VIX1 0.51 0.60 -0.53 0.07 -0.27 -0.15 2 VIX2 0.47 0.24 0.25 -0.11 0.75 0.30 2 VIX3 0.43 0.00 0.67 -0.34 -0.43 -0.26 2 VIX6 0.37 -0.30 0.09 0.61 -0.30 0.55 2 VIX9 0.33 -0.44 -0.10 0.37 0.30 -0.68 2 VIX12 0.30 -0.54 -0.44 -0.60 -0.05 0.22 % of var: 95.47% 4.09% 0.25% 0.08% 0.06% 0.05% Next Month’s Return Regressions: Coeff 0.03 -0.05 0.03 0.12** -0.05 0.02 p-value (70.3%) (45.1%) (68.1%) (1.3%) (20.9%) (64.4%) Adj. R2 0.1% 0.3% 0.1% 1.4% 0.3% 0.1% Next Quarter’s Return Regressions: Coeff 0.06 -0.13** 0.05 0.17*** -0.06* -0.03 p-value (55.4%) (4.7%) (47.9%) (0.1%) (8.3%) (60.5%) Adj. R2 0.4% 1.7% 0.2% 2.7% 0.3% 0.1% Next Six Month’s Return Regressions: Coeff 0.18*** -0.16* 0.05 0.15** -0.06* -0.01 p-value (0.9%) (5.2%) (40.3%) (2.4%) (5.5%) (85.5%) Adj. R2 3.4% 2.5% 0.2% 2.3% 0.3% 0.0% Next Year’s Return Regressions: Coeff 0.21** -0.05 0.07 0.12* 0.01 -0.08** p-value (2.5%) (57.9%) (16.4%) (7.3%) (81.1%) (3.3%) Adj. R2 4.6% 0.3% 0.5% 1.5% 0.0% 0.6% CHAPTER 3. RISK PREMIA AND THE VIX TERM STRUCTURE 88

Table 3.4: Principal Components, continued.

Panel B: With past realized volatility “Level” “BTZ” “Slope B” “Curve” PC1 PC2 PC3 PC4 PC5 PC6 PC7 RV 0.48 -0.74 -0.46 -0.02 0.00 0.00 0.01 2 VIX1 0.46 -0.10 0.62 0.55 0.07 0.27 0.13 2 VIX2 0.41 0.07 0.32 -0.24 -0.11 -0.75 -0.30 2 VIX3 0.38 0.19 0.13 -0.66 -0.34 0.43 0.26 2 VIX6 0.32 0.31 -0.17 -0.10 0.61 0.30 -0.56 2 VIX9 0.28 0.37 -0.29 0.10 0.37 -0.29 0.69 2 VIX12 0.26 0.40 -0.41 0.43 -0.60 0.04 -0.22 % of var: 91.23% 6.80% 1.65% 0.19% 0.06% 0.04% 0.03% Next Month’s Return Regressions: Coeff 0.01 0.12 0.06 -0.01 0.12** 0.05 -0.04 p-value (87.6%) (11.9%) (12.8%) (81.7%) (1.5%) (22.0%) (44.1%) Adj. R2 0.0% 1.4% 0.4% 0.0% 1.4% 0.3% 0.2% Next Quarter’s Return Regressions: Coeff 0.03 0.23*** 0.07 -0.03 0.16*** 0.06* 0.00 p-value (70.2%) (0.0%) (14.1%) (69.1%) (0.0%) (5.3%) (99.7%) Adj. R2 0.1% 5.3% 0.6% 0.1% 2.7% 0.4% 0.0% Next Six Month’s Return Regressions: Coeff 0.16** 0.22*** 0.01 -0.03 0.15** 0.06** -0.01 p-value (1.3%) (0.1%) (86.6%) (54.0%) (2.1%) (3.9%) (82.8%) Adj. R2 2.4% 5.1% 0.0% 0.1% 2.3% 0.3% 0.0% Next Year’s Return Regressions: Coeff 0.20** 0.14* 0.06 -0.06 0.12* 0.00 0.07* p-value (3.9%) (9.1%) (41.6%) (21.4%) (7.0%) (88.2%) (5.0%) Adj. R2 3.8% 2.0% 0.4% 0.3% 1.6% 0.0% 0.4% CHAPTER 3. RISK PREMIA AND THE VIX TERM STRUCTURE 89

Table 3.5: Principal Components (No 2008-2009 Financial Crisis) Panel A presents principal components analysis for the VIX term structure without past realized variance, Panel B with past realized variance RV. The first block of both panels shows the coefficients defining each principal component. The second block gives the fraction of term structure variance explained by each principal component. The remaining blocks shows the coefficients, p-values, and incremental R2 from standardized regressions of future index returns on the principal components at different horizons. The sample is daily from 1996 through 2010, excluding the October 2008-June 2009 period of market turmoil. Standard errors are adjusted using Newey-West with lags equal to one-and-a-half times the return horizon. Panel A: Without past realized volatility “Level” “Slope” “Curve” PC1 PC2 PC3 PC4 PC5 PC6 2 VIX1 0.51 0.60 -0.53 0.07 -0.27 -0.15 2 VIX2 0.47 0.24 0.25 -0.11 0.75 0.30 2 VIX3 0.43 0.00 0.67 -0.34 -0.43 -0.26 2 VIX6 0.37 -0.30 0.09 0.61 -0.30 0.55 2 VIX9 0.33 -0.44 -0.10 0.37 0.30 -0.68 2 VIX12 0.30 -0.54 -0.44 -0.60 -0.05 0.22 % of var: 95.47% 4.09% 0.25% 0.08% 0.06% 0.05% Next Month’s Return Regressions: Coeff 0.09 -0.01 0.04 0.11** -0.01 0.04 p-value (22.5%) (87.2%) (64.4%) (3.8%) (73.8%) (52.7%) Adj. R2 0.9% 0.0% 0.1% 1.4% 0.1% 0.1% Next Quarter’s Return Regressions: Coeff 0.13* -0.05 0.10 0.15*** -0.02 -0.01 p-value (9.2%) (58.9%) (12.5%) (0.5%) (49.9%) (79.0%) Adj. R2 2.3% 0.1% 0.7% 2.3% 0.2% 0.0% Next Six Month’s Return Regressions: Coeff 0.07 -0.09 0.05 0.15* -0.06 0.00 p-value (42.6%) (35.0%) (39.3%) (5.9%) (11.1%) (95.1%) Adj. R2 1.2% 0.6% 0.1% 2.1% 0.4% 0.0% Next Year’s Return Regressions: Coeff 0.01 -0.04 0.08 0.12 -0.01 -0.07 p-value (95.7%) (71.5%) (13.3%) (11.2%) (80.0%) (14.4%) Adj. R2 0.0% 0.0% 0.5% 1.4% 0.0% 0.5% CHAPTER 3. RISK PREMIA AND THE VIX TERM STRUCTURE 90

Table 3.5: Principal Components (no 2008), continued.

Panel B: With past realized volatility “Level” “BTZ” “Slope B” “Curve” PC1 PC2 PC3 PC4 PC5 PC6 PC7 VOL2 0.48 -0.74 -0.46 -0.02 0.00 0.00 0.01 2 VIX1 0.46 -0.10 0.62 0.55 0.07 0.27 0.13 2 VIX2 0.41 0.07 0.32 -0.24 -0.11 -0.75 -0.30 2 VIX3 0.38 0.19 0.13 -0.66 -0.34 0.43 0.26 2 VIX6 0.32 0.31 -0.17 -0.10 0.61 0.30 -0.56 2 VIX9 0.28 0.37 -0.29 0.10 0.37 -0.29 0.69 2 VIX12 0.26 0.40 -0.41 0.43 -0.60 0.04 -0.22 % of var: 91.23% 6.80% 1.65% 0.19% 0.06% 0.04% 0.03% Next Month’s Return Regressions: Coeff 0.05 -0.08 0.05 -0.03 0.11** 0.02 -0.05 p-value (57.4%) (40.8%) (25.0%) (68.7%) (3.8%) (70.5%) (44.2%) Adj. R2 0.8% 0.5% 0.3% 0.1% 1.4% 0.1% 0.02% Next Quarter’s Return Regressions: Coeff 0.06 0.17** 0.10 -0.08 0.15*** 0.03 -0.01 p-value (50.4%) (4.8%) (11.6%) (16.0%) (0.0%) (40.2%) (88.3%) Adj. R2 1.7% 2.2% 1.0% 0.5% 2.3% 0.2% 0.0% Next Six Month’s Return Regressions: Coeff 0.01 0.18** 0.05 -0.04 0.15** 0.06* -0.02 p-value (93.3%) (3.0%) (49.0%) (46.0%) (4.7%) (5.8%) (62.7%) Adj. R2 0.8% 2.3% 0.2% 0.0% 2.1% 0.5% 0.0% Next Year’s Return Regressions: Coeff -0.07 0.16* 0.11 -0.06 0.12* 0.02 0.04 p-value (63.1%) (6.9%) (22.7%) (22.1%) (9.0%) (65.8%) (32.0%) Adj. R2 0.0% 1.1% 1.2% 0.4% 1.4% 0.0% 0.3% Appendix A

Appendices for Chapter 2

A.1 Simultaneous equations

The full set of simultaneous equations that characterize the equilibrium are:

α(φ( k5 ) − φ( k6 )) σ σ as = µ + σ (A.1) α(Φ( k6 ) − Φ( k5 )) + (1 − α)q σ σ 1 α(φ( k2 ) − φ( k1 )) σ σ bs = µ − σ (A.2) α(Φ( k2 ) − Φ( k1 )) + (1 − α)q σ σ 2 R ∞ p 2 2 α φ()C(, ση)d + (1 − α)q3φ(0) σ + σ k6  η ac = (A.3) α(1 − Φ( k6 )) + (1 − α)q σ 3 R k3 p 2 2 α φ()C(, ση)d + (1 − α)q4φ(0) σ + σ k2  η bc = (A.4) α(Φ( k3 ) − Φ( k2 )) + (1 − α)q σ σ 4 R k1 p 2 2 α −∞ φ()P (, ση)d + (1 − α)q5φ(0) σ + ση ap = (A.5) α(Φ( k1 )) + (1 − α)q σ 5 R k5 p 2 2 α φ()P (, ση)d + (1 − α)q6φ(0) σ + σ k4  η bp = (A.6) α(Φ( k5 ) − Φ( k4 )) + (1 − α)q σ σ 6

91 APPENDIX A. APPENDICES FOR CHAPTER 2 92

θ(P (k1, ση) − ap) = γ(bs(1 − ρ) − µ − k1)) (A.7)

γ(bs(1 − ρ) − µ − k2) = θ(bc − C(k2, ση)) (A.8)

θ(bc − C(k3, ση)) = 0 (A.9)

0 = θ(bp − P (k4, ση)) (A.10)

θ(bp − P (k5, ση)) = γ(µ + k5 − as) (A.11)

γ(µ + k6 − as) = θ(C(k6, ση) − ac) (A.12)

Eq. (A.1) – (A.6) are the zero profit conditions for the market maker. Eq. (A.1), for example, ensures that the ask price for a stock is exactly the expectation of V˜ conditional on a stock trade. Computing this conditional expectation in the case of options trades (Eq. (A.3) – (A.6)) requires integrating the value function for options C and P . These functions are the mean value of a call and put, respectively, conditional on the signal  and the standard deviation of η. Specifically,     ˜   C(, ση) ≡ E(C|˜ = ) = Φ  + φ ση (A.13) ση ση     ˜ −  P (, ση) ≡ E(P |˜ = ) = −Φ  + φ ση. (A.14) ση ση

Finally, Eq. (A.7) through (A.12) ensure informed traders are indifferent between the two neighboring portfolios at the cutoff points. For example, (A.7) ensures they are indifferent between buying puts and shorting stock given the signal k1. These equations cannot be solved in closed form due to the nonlinearity of C and P , however, we can prove some general results directly from the simultaneous equa- tions without needing a closed-form solution. Throughout, we assume the exogenous parameters are chosen so that there exists a set of equilibrium parameters satisfying (A.1) through (A.12) as well as k1 < k2 < k3 ≤ k4 < k5 < k6. For some parameters, no such equilibrium exists, typically because the informed trader never finds it op- timal to trade stock, implying k2 = k3; we focus on the case when informed traders use stock because our goal is to model the impact of short-sale costs, which are only relevant when informed traders use equity. We do consider parametrizations where the informed trader chooses to trade every signal, meaning k3 = k4. In this case, we have one fewer free parameter and we need to replace Eq. (A.9) and (A.10) with the single equation: θ(bc − C(k3, ση)) = θ(bp − P (k3, ση)). (A.15) APPENDIX A. APPENDICES FOR CHAPTER 2 93

A.2 Measure of leverage

Leverage λ in options markets is measured by the elasticity of the option pricing function C(S) with respect to S. For options priced according to Black-Scholes, we have:

2 ! log( S ) + (r + σ T S λ = Φ K √ 2 . (A.16) σ T C

The above elasticity represents the change in value of an option position with respect to the change in value of an option position, assuming that because the stock costs S S and the option C, the option position has C times as many contracts as the stock position has shares. In our model, order sizes are fixed exogenously, so we examine ∂C θ ∂C S λ = ∂S γ instead of λ = ∂S C :

∂C()     ∂C θ θ φ( ) + Φ( ) + 2σηφ( ) θ λ = = ∂ = ση ση ση ση . (A.17) ∂S γ ∂V γ 1 γ ∂ Since the options in our model are struck at µ, we measure λ when  = 0, giving us: 1 θ λ = . (A.18) JS 2 γ A.3 Proofs

Result 1. When uninformed demand satisfies q1 = q2 and q3 = q4 = q5 = q6, in equilibrium, E(V˜ |option trade) ≤ E(V˜ |equity trade). We obtain a strict inequality when ρ > 0.

Result 2. Given the same assumptions as Result 1, the difference in conditional means D ≡ E(V˜ |stock trade) − E(V˜ |option trade) is weakly increasing in the short- sale cost ρ.

˜ ˜ Proof. Define VO = E(V − µ|option trade) and VS = E(V − µ|stock trade). We show that D ≡ VO − VS is 0 for ρ = 0, strictly increasing in ρ at ρ = 0, and weakly increasing in ρ at all ρ > 0, which together implies both Result 1 and Result 2. Given the symmetry in uninformed trader demand and the normal distributions of V˜ , ˜, andη ˜, when ρ = 0 the entire problem is symmetric and therefore, we have k1 = −k6, k2 = −k5, k3 = −k4 in equilibrium. This, in turn, implies that VO = VS = 0 when ρ = 0. APPENDIX A. APPENDICES FOR CHAPTER 2 94

We first compute VO and VS as a function of the equilibrium cutoff points used by informed traders ki. α VS = (φ(k1) − φ(k2) + φ(k5) − φ(k6)) (A.19) pS

pS = (1 − α)(q1 + q2) + α(Φ(k2) − Φ(k1) + Φ(k6) − Φ(k5)) (A.20) α VO = (−φ(k1) + φ(k2) − φ(k3) + φ(k4) − φ(k5) + φ(k6)) (A.21) pO

pO = (1 − α)(q3 + q4 + q5 + q6) + α(Φ(k1) + Φ(k3) − Φ(k2) + Φ(k5) − Φ(k4) + 1 − Φ(k6)), (A.22) where pS and pO are the unconditional probabilities of a stock trade and an option trade occurring, respectively. We now consider the derivative of D with respect to ρ.

∂D V V = S − O ∂ρ ∂ρ ∂ρ 6   X VS VO ∂ki = − . (A.23) ∂k ∂k ∂ρ i=1 i i

∂ki The derivatives ∂ρ represent changes in equilibrium ki as ρ changes. The direct effect of ρ on ki is that, given unchanged prices, selling stock becomes less profitable than it was before. Of course, given the direct effect on ki, there is also the indirect effect that comes through prices: informed traders’ strategy changes, which changes prices, which in turn changes informed traders’ strategy. However, due to uninformed traders’ demand, these indirect effects dampen the direct effect but do not change its direction. We therefore focus on the first-order change in ki with respect to ρ. Since ρ does not appear in the indifference equations at k3, k4, k5, and k6 (Eq. ∂ki (A.9) – (A.12)), we have ∂ρ = 0 for all i ≥ 3. For k1, we work from the informed traders’ indifference Eq. (A.7).

γ(bs(1 − ρ) − µ − k1) = θ(P (k1, ση) − ap)   ∂k1 θ ∂P ∂k1 θ −k1 ∂k1 ⇒ −bs − = = − Φ ∂ρ γ ∂k ∂ρ γ ση ∂ρ ∂k1 bs ⇒ = −  . (A.24) ∂ρ 1 − θ Φ −k1 γ ση   Since θ ≥ 2 by assumption, and Φ −k1 > 0.5 because k < 0, we have ∂k1 > 0. γ ση 1 ∂ρ APPENDIX A. APPENDICES FOR CHAPTER 2 95

A similar calculation yields

∂k2 bs = −  . (A.25) ∂ρ 1 − θ Φ k2 γ ση

∂k2 In order to sign ∂ρ , we note that for signals slightly less than k2, the informed trader prefers selling stock, while for signals slightly more than k2, the informed trader prefers selling calls. This implies: ∂Profit from selling calls ∂Profit from selling stock (k ) > (k ) ∂k 2 ∂k 2 ∂ ∂ ⇒ θ(bc − C(k2, ση)) > γ(bs(1 − ρ) − µ − k2) ∂k2 ∂k2  k  ⇒ −θΦ 2 > −γ ση θ  k  ⇒ Φ 2 < 1 γ ση ∂k ⇒ 2 < 0. (A.26) ∂ρ

From Eq. (A.19) – (A.22), remembering that pS and pO are functions of ki, we compute:

∂VS φ(k1) = α (VS − k1) (A.27) ∂k1 pS

∂VS φ(k2) = α (k2 − VS) (A.28) ∂k2 pS

∂VO φ(k1) = α (k1 − VO) (A.29) ∂k1 pO

∂VO φ(k2) = α (VO − k2). (A.30) ∂k2 pO

As discussed above, VS = VO = 0 when ρ = 0. In this case, since k1 < k2 < 0, it is clear from (A.27)–(A.30) that ∂VS > 0, ∂VS < 0, ∂VO < 0, and ∂VO > 0. Furthermore, ∂k1 ∂k2 ∂k1 ∂k2 since these derivatives are all zero whenever VS > −k1, k2 > VS, k1 > VO, and VO −k2, respectively, the derivatives can never change signs. For example, as VS approaches k1, the derivative of VS approaches zero, meaning it stops changing and never crosses k1. Similar logic applies to the other thre derivatives, meaning that their sign when ρ = 0 applies for all ρ. APPENDIX A. APPENDICES FOR CHAPTER 2 96

Returning to Eq. (A.23), we have:

∂D  ∂V ∂V  ∂k  ∂V ∂V  ∂k = S − O 1 + S − O 2 (A.31) ∂ρ ∂k1 ∂k1 ∂ρ ∂k2 ∂k2 ∂ρ |{z} |{z} |{z} |{z} |{z} |{z} >0 <0 >0 <0 >0 <0 ∂D ⇒ > 0. ∂ρ

We have a strict inequality here because we assumed k1 < k2, meaning the informed trader shorts stocks for a non-empty set of signals. If the short-sale costs are suffi- ciently high that k1 = k2, further increases no longer have any impact on equilibrium ∂D and ⇒ ∂ρ = 0.

Result 3. The difference in conditional means D ≡ E(V˜ |option trade)−E(V˜ |equity trade) θ is decreasing in the leverage in options as measured by λ = 2γ . Proof. Since any solution to the simultaneous equations (A.1) – (A.12) for order sizes (γ,θ) is also a solution for order sizes (cγ,cθ) for all constants c, we assume without loss of generality that γ = 1. Following the notation from the proof of Result 1, we therefore want to show: 6   ∂D X VS VO ∂ki = − < 0 ∂θ ∂k ∂k ∂θ i=1 i i whenever ρ > 0. Since θ cancels out in the equations for k3 and k4, we have:

∂D  V V  ∂k  V V  ∂k = S − O 1 + S − O 2 + (A.32) ∂θ ∂k1 ∂k1 ∂θ ∂k2 ∂k2 ∂θ  V V  ∂k  V V  ∂k S − O 5 + S − O 6 . ∂k5 ∂k5 ∂θ ∂k6 ∂k6 ∂θ

∂ki We first focus on the partial derivatives ∂θ . Following the methodology used to APPENDIX A. APPENDICES FOR CHAPTER 2 97

∂ki compute ∂ρ in Result 1, we find:

∂k1 P (k1, ση) − ap =   > 0 (A.33) ∂θ θΦ −k1 − 1 ση

∂k2 bc − C(k2, ση) =   < 0 (A.34) ∂θ θΦ k2 − 1 ση

∂k5 bp − P (k5, ση) =   > 0 (A.35) ∂θ 1 − θΦ −k5 ση

∂k6 C(k6, ση) − ac =   < 0. (A.36) ∂θ 1 − θΦ k6 ση

The logic in Result 1 implies that bp > bc, −k2 > k5, and −k1 < k6 whenever ρ > 0. These facts, along with C(k, ση) = P (−k, ση), imply that:

∂k2 bc − C(k2, ση) bp − C(k2, ση) ∂k5 ∂k5 − =   <   = < (A.37) ∂θ k2 k2 ∂θ ∂θ 1 − θΦ 1 − θΦ k5=−k2 ση ση

∂k1 P (k1, ση) − ap P (k1, ση) − ac ∂k6 ∂k6 =   <   = − < − . (A.38) ∂θ −k1 −k1 ∂θ ∂θ θΦ − 1 θΦ − 1 k6=−k1 ση ση

∂ ∂ Adding together (A.37) and (A.38) and switching signs yields ∂θ (k2−k1) > ∂θ (k6− k5). Since both sides are negative, this implies that the “short stock” region k2 − k1 shrinks as θ increases, but not as fast as the “long stock” region k6 − k5 shrinks, implying that the sum in (A.32) is negative.

Result 4. Equity value has a higher skewness conditional on a call trade than con- α ditional on a put trade when qi > (1−α)139.2 . Proof. We show that the third centralized moments conditional on call and put trades satisfy:

˜ ˆ 3 ˜ ˆ 3 E((V − Vcall) |call trade) > 0 > E((V − Vput) |put trade), (A.39) ˆ ˜ where Vi is the expected value of V conditional on trade type i. Inequality (A.39) implies Result 4 because skewness is the third centralized moment scaled by a positive number. ˜ ˆ 3 We show here that E((V − Vcall) |call trade) > 0. The other half of inequality (A.39) follows from the same derivation applied to the put option. APPENDIX A. APPENDICES FOR CHAPTER 2 98

C C To simplify notation, we write E (·) as short-hand for E(·|call trade), and cm3 for the third centralized moment conditional on a call trade.

C C ˜ ˆ 3 cm3 = E ((V − Vcall) ) (A.40) = EC ((˜ − EC (˜) +η ˜)3).

Since ˜− EC (˜) andη ˜ are independent and both have zero mean conditional on a call trade, we have:

C C C 3 cm3 = E ((˜ − E (˜)) ) C C ˜ C ˜ 3 ⇒ cm3 ∝ E ((δ − E (δ)) ), (A.41) where δ˜ = ˜ and ∝ indicates that the two expressions have the same sign. σ Next we break up the expectation in (A.41) into two exhaustive cases: the trade was initiated by an informed trader and the trade was initiated by an uninformed trader. In each case, we expand (δ˜ − EC (δ˜))3, and in order to keep the expression as brief as possible, we write:

I i mi ≡ E(˜ |informed call trade) (A.42) U i mi ≡ E(˜ |uninformed call trade) (A.43) δˆ ≡ EC (δ˜) (A.44)

αC ≡ P (informed|call trade). (A.45)

After breaking up and expanding the expectation, we find:

C I I ˆ I ˆ2 ˆ3 U U ˆ U ˆ2 ˆ3 cm3 ∝ αC (m3 − 3m2δ + 3m1δ − δ ) + (1 − αC )(m3 − 3m2 δ + 3m1 δ − δ ) I I ˆ I ˆ2 ˆ3 ˆ ˆ3 = αC (m3 − 3m2δ + 3m1δ − δ ) + (1 − αC )(−3δ − δ ) I ˆ3 ˆ I = m3αC + 2δ − 3δ(1 + αC (m2 − 1)). (A.46)

ˆ I U I To arrive at Eq. (A.46) we use the fact that δ = αC m1 + (1 − αC )m1 = αC m1. From here, we prove three lemmas that together complete the proof under the following condition: α q > . (A.47) i (1 − α)139.2

This condition ensures that the number of uninformed traders in options markets does not approach zero, in which case markets begin to fail and the skewness result can 1 reverse. It is a condition easily satisfied for any normal parametrizations. If α > 10 , 1 1 we only require qi > 1250 and if qi > 84 , we only require α < 63%. APPENDIX A. APPENDICES FOR CHAPTER 2 99

I ˆ3 ˆ Lemma 1 shows that m3 > 0 when (A.47) holds. Lemma 2 shows that 2δ −3δ(1+ I I ˆ3 ˆ I αC (m2 −1)) > 0 when δ < 0. Lemma 3 shows that m3αC > −2δ +3δ(1+αC (m2 −1)) when δ > 0 and (A.47) holds. Put together with (A.46), these lemmas complete the proof.

˜ I Lemma 1. The third moment of δ conditional on an informed call trade, m3, is α positive whenever qi > (1−α)139.2 .

Proof. The lemma follows from informed traders’ equilibrium cutoff strategy, which assures that a call trade is either weakly bad news or extremely good news. We only need to rule out the possibility that uninformed traders are so scarce that the informed trader almost never buys calls, which would make the the distribution of δ˜ conditional on an informed trade similar to the distribution of δ˜ conditional on a call sell, which has a negative third moment. From the moments of the truncated normal distribution given in Jawitz (2004), we have:

2 2 2 I (j2 + 2)φ(j2) − (j3 + 2)φ(j3) + (j6 + 2)φ(j6) m3 = , (A.48) Φ(j3) − Φ(j2) + 1 − Φ(j6) ˜ where ji are the equilibrium cutoff points scaled down by σ so they are δ cutoffs rather than ˜ cutoffs. The function f(x) = (x2 + 2)φ(x) is positive, symmetric about x = 0, decreasing for x > 0, increasing for x < 0, and satisfies f(−¯j) + f(¯j) = f(0) for ¯j = 1.832. In equilibrium, we know that j2 ≤ j3 ≤ 0 ≤ j6 and |j3| < |j2| < |j6|, so I ¯ (A.48) tells us that m3 > 0 whenever j6 < j. Next we show that j6 < ¯j whenever (A.47) holds. Assume the contrary, that j6 ≥ ¯j. We consider only equilibria where the informed trader buys equity for some signals, so we know that at ˜ = ¯jσ the informed trader prefers equity to calls. Writing ˜ C(x, ση) for E(C|˜ = x), we have that:

j6 ≥ ¯j ⇒ γ(µ + σ¯j − as) > θ(C(¯jσ, ση) − ac). (A.49)

The right-hand side of (A.49) is increasing in ση, so if (A.49) holds when ση = 0, it holds for all ση. When ση = 0, we can solve for the equilibrium k6 directly from the simultaneous equations in Appendix A.1. In particular, we find that: µ − a + θa k = s c . (A.50) 6 θ − 1 APPENDIX A. APPENDICES FOR CHAPTER 2 100

So if k6 ≥ ¯jσ, we have: µ − a + θa ¯jσ ≥ s c  θ − 1

⇒ ¯jσ ≥ac αφ(¯j) ⇒ ¯jσ ≥ σ. (A.51) α(1 − Φ(¯j)) + (1 − α)q6

Solving (A.51) for q6, we find exactly the opposite of the condition (A.47), so we know ¯ I that (A.47) implies k6 < jσ and m3 > 0.

ˆ3 ˆ I Lemma 2. When δ < 0, we have that 2δ − 3δ(1 + αC (m2 − 1)) > 0.

Proof. This lemma holds because the quantity in question measures the differ- ence between non-centralized moments and centralized moments due to the change in mean. The lemma shows that when the mean of a variable is negative, the cen- tralized third moment is greater than the non-centralized third moment. To see this technically, first note that:

var(δ˜|call trade) = EC (δ˜2) − δˆ2 I ˆ2 = αC m2 + (1 − αC ) − δ I ˆ2 = 1 + αC (m2 − 1) − δ . (A.52)

And since variances are positive, we have:

I ˆ2 1 + αC (m2 − 1) − δ > 0 ˆ I ˆ3 ⇒ δ(1 + αC (m2 − 1)) < δ ˆ3 ˆ I ⇒ 2δ − 3δ(1 + αC (m2 − 1)) > 0. (A.53)

I ˆ3 ˆ Lemma 3. When δ > 0 and (A.47) holds, we have that m3αC > −2δ + 3δ(1 + I αC (m2 − 1)). Proof. The intuition for Lemma 3 is that when δˆ > 0, the centralized third moment is less than the non-centralized third moment, but the positive mean makes the third APPENDIX A. APPENDICES FOR CHAPTER 2 101

moment so large it is positive even after centralization. More rigorously, we have:

I ˆ3 ˆ I m3αC + 2δ − 3δ(1 + αC (m2 − 1)) I I 3 2 I I ∝ m3 + 2(m1) (αC ) − 3m1(1 + αC (m2 − 1)) I I I > m3 − 3m1(1 + αC (m2 − 1)). (A.54)

From Jawitz (2004), we have:

2 2 2 I (j2 + 2)φ(j2) − (j3 + 2)φ(j3) + (j6 + 2)φ(j6) m3 = (A.55) Φ(j3) − Φ(j2) + 1 − Φ(j6)

I (j2)φ(j2) − (j3)φ(j3) + (j6)φ(j6) m2 = (A.56) Φ(j3) − Φ(j2) + 1 − Φ(j6)

I φ(j2) − φ(j3) + φ(j6) m1 = . (A.57) Φ(j3) − Φ(j2) + 1 − Φ(j6)

Noting that any equilibrium satisfying (A.47) and δˆ > 0 in which the informed trader uses each asset with positive probability satisfies:

1. −¯j < j2 < j3 < 0 < j6 < ¯j.

2. |j3| < |j2| < |j6|.

3. φ(j2) − φ(j3) + φ(j6) > 0.

I I I We can substitute these conditions into (A.54) and find that m3 − 3m1(1 + αC (m2 − 1)) > 0, which in turn implies Lemma 3. Appendix B

Appendices for Chapter 3

B.1 Conditional means and variances in prior mod- els

In this appendix, I show that risk premia are an approximately linear function of conditional variance in two popular asset pricing models, Bansal and Yaron (2004) and Campbell and Cochrane (1999).

A Bansal and Yaron (2004) In the Appendix of Bansal and Yaron (2004), they provide the following expressions for the conditional means and variance of future log returns in their model in (equa- tions (A13) and (A14) in Bansal and Yaron (2004)):

2 2 2 2 2 vart(rm,t+1) = (βm,e + ϕd)σt + βm,wσw (B.1) 2 2 Et(rm,t+1 − rf,t) = (βm,eλm,e)σt + βm,wλm,wσw − 0.5vart(rm,t+1) (B.2)

Bansal and Yaron (2004) contains exact definitions and interpretation for each of these parameters, but the important point here is that there is only one variable on the right-hand sides that moves over time: the consumption growth volatility σt. Solving (B.1) for σt and then substituting into (B.2) yields:

Et(rm,t+1 − rf,t) = a0 + a1vart(rm,t+1) (B.3)   2 βm,eλm,eβm,w a0 ≡ βm,wσw λm,w − 2 2 βm,e + ϕd   βm,eλm,e 1 a1 ≡ 2 2 − βm,e + ϕd 2

102 APPENDIX B. APPENDICES FOR CHAPTER 3 103

B Campbell and Cochrane (1999) The only state variable in Campbell and Cochrane (1999) is the surplus consumption ratio (C − X)/C. Figures 5 and 6 in Campbell and Cochrane (1999), reproduced in Appendix Figure 1, show that the shape of the Sharpe Ratio as a function of surplus consumption is nearly identical to the shape of the conditional standard deviation. Dividing the conditional Sharpe Ratio by the conditional standard deviation yields the ratio of equity risk premia to conditional variance, so the similarity of the plots in Figures 5 and 6 suggest that this ratio is close to constant across states in the Campbell and Cochrane (1999) model. Appendix Figure 1 shows the relation between conditional variance and risk pre- mia in Campbell and Cochrane (1999) for both the consumption claim and the divi- dend claim. For both assets, the relation is nearly linear, indicating that equity risk premia are almost perfectly correlated with conditional variance in the model. Note that the dividend claim is to the right of the consumption claim in this plot because of the additional variance in dividend growth that is independent of the SDF, and therefore has no bearing on expected returns.

B.2 Details of Proofs

In this section, I provide a more detailed proof of the affine relation between the implied variance term structure and equity risk premia in my model (Theorem 2 in Section 3.3).

Theorem 2. If each state variable has a different persistence (ρσM , ρβ, and ρσ are all different), the equity risk premium is an affine function of the implied variance term structure:

2 µt − Rf = βtσM,tRf = A0 + A1IVt,{T1,T2,T3,T4}

where IVt,{T1,T2,T3,T4} is a 4x1 vector of implied variances at four different horizons {T1,T2,T3,T4} on date t, A0 is a constant, and A1 is 1x4 vector of constants. Proof. Define state vector:

 2 2 2 2 2 0 zt = σM,t βt σM,tβt σ,t APPENDIX B. APPENDICES FOR CHAPTER 3 104

Figure B.1: Risk premia and variances in Campbell and Cochrane (1999). APPENDIX B. APPENDICES FOR CHAPTER 3 105

The equity risk premium µt − Rf is an affine function of the state vector zt since:

2 µt − Rf = βtσM,tRf  2  βt + βH βL 2 = σM,tRf βH + βL

h Rf βhβL Rf i = 0 0 zt βH +βL βH +βL | {z } b1

2 βt +βH βL The second line relies on the discreteness of βt, which implies that βt = . This βH +βL equation does not hold for continuous βt. However, Theorem 2 is not an artifact of the discreteness. As discussed in Appendix B.3, the same result holds in the continuous case but requires 6 horizons of the VIX term structure rather than 4. The implied variance measure IVt,T is also an affine function of the state vector zt. Starting from the definition in Equation (3.12):

T −1 1 X IV = EQ(σ2 ) t,T T t R,t+s s=0 Since the variance states are independent of the SDF and eachother, we have that:

Q 2 2 2 2 Et (σR,t+s) = Et(βt+s)Et(σM,t+s) + Et(σ,t+s)

Since each state variable is an AR(1), we have:

E (σ2 ) = σ2 + ρs (σ2 − σ2 ) t M,t+s M σM M,t M 2 2 s 2 2 Et(βt+s) = β + ρβ(βt − β ) 2 2 s 2 2 Et(σ,t+s) = σ + ρσ (σ,t − σ )

2 2 2 where σM , β , and σ are unconditional means. Putting these together yields:

Q 2 Et (σR,t+s) = d0(s) + d1(s)zt d (s) = β2(1 − ρs )σ2 (1 − ρs ) + σ2(1 − ρ2 ) 0 β M σM  σ h i d (s) = β2ρs (1 − ρs ) σ2 ρs (1 − ρs ) ρs ρs ρs 1 σM β M β σM σM β σ T −1 T −1 ! 1 X 1 X ⇒ IV = d (s) + d (s) z = c (T ) + c (T )z t,T T 0 T 1 t 0 1 t s=0 s=0

PT −1 s These coefficients can be summed using the geometric series summation s=0 aρ = APPENDIX B. APPENDICES FOR CHAPTER 3 106

1−ρT a 1−ρ : ! ! 1 1 − ρT 1 − ρT 1 − ρT ρT  1 − ρT  2 2 σM β σM β 2 σ c0(T ) = β σM 1 − − + + σ 1 − T 1 − ρσM 1 − ρβ 1 − ρσM ρβ 1 − ρσ

1 h  1−ρT 1−ρT ρT   1−ρT 1−ρT ρT  1−ρT ρT T i 2 β σM β 2 β σM β σM β 1−ρσ c1(T ) = β 1−ρ − 1−ρ ρ σM 1−ρ − 1−ρ ρ 1−ρ ρ 1−ρ T β σM β β σM β σM β σ

The implied variance term structure IVt,{T1,T2,T3,T4} is also an affine function of zt:     c0(T1) c1(T1) c0(T2) c1(T2) IVt,{T1,T2,T3,T4} =   +   zt c0(T3) c1(T3) c0(T4) c1(T4) | {z } | {z } C0 C1

As long as the VIX term structure is composed of four different horizons, and the three persistences ρσM , ρβ, and ρσ are all different, C1 is invertible since each column of C1 changes at a different rate with respect to T , the only thing changing across rows of C1. Therefore, the risk premia can be computed from the implied variance term structure as follows:

−1
zt = C1 IVt,{T1,T2,T3,T4} − C0 −1
⇒ µt − Rf = b1C1 IVt,{T1,T2,T3,T4} − C0 (B.4)

B.3 Continuous version of the model

This appendix contains a continuous-time, continuous-state, analogue to the model in Section 3.3. The intuition and main results are identical, indicating that the stark discreteness in the main model is not driving any of the results. The following differential equation governs the stochastic discount factor (SDF) mt:

dmt p m m = −rtdt + vt dBt (B.5) mt where rt is the risk-free rate process, which I fix at r to keep the focus away from √ m the term structure of interest rates, and vt is the SDF’s diffusion process, which APPENDIX B. APPENDICES FOR CHAPTER 3 107

follows the square-root process used in Heston (1993) and Cox, Ingersoll, and Ross (1985):

m m p m vm dvt = κvm (θvm − vt )dt + σvm vt dBt (B.6)

m The parameter κvm represents the extent of mean reversion in vt , with higher val- ues resulting in more mean reversion and therefore less persistence. The parameters m θvm and σvm allow the unconditional mean and variance of vt to be any non-negative vm 1 number. The Brownian motion Bt has correlation ρvm,m with the SDF. In the data, we observe that innovations in the VIX and the equity market have a strong nega- tive correlation (the so-called leverage effect), which suggests a positive correlation

ρvm,m > 0 between innovations in m and vm. There is also a traded asset with value St, which is also correlated with innovations in mt, as dictated by the process: q dSt R  m q 2  = µtdt + vt ρS,tdBt + 1 − ρS,tdBt (B.7) St p m m p   = µtdt − βt vt dBt + vt dBt (B.8)

q R R vt  R 2 m Where vt is the diffusion of log(St), βt = −ρs,t m , and vt = vt − βt vt . I proceed vt √ m with (B.8) because it isolates the SDF diffusion vt , which is already specified by Equation (B.6). Since the returns of procyclical risky assets are negatively correlated with the SDF, ρt is negative for these assets and βt is positive. Since mt is a stochastic discount factor, no arbitrage implies that:

m µt = r + βtvt . (B.9)

m The three state variables have the following interpretations: vt is the variance of the  SDF, βt is the SDF beta, and vt the variance of the part of returns orthogonal to the SDF, or unpriced risk. These three quantities are discussed in more detail throughout the paper.  Both βt and vt also follow square-root processes:

p β dβt = κβ(θβ − βt)dt + σβ βtdBt (B.10)

  p  v dvt = κv (θv − vt )dt + σv vt dBt (B.11)

m These parallel vt with the exception that they are uncorrelated with the SDF. In reality, innovations in both the SDF beta and unpriced risk could be correlated with q 1Formally, dBvm = ρ dBm + 1 − ρ2 dB,vm where B,vm is an independent Brownian t vm,m t vm,m t t motion. APPENDIX B. APPENDICES FOR CHAPTER 3 108

Table B.1: Summary of Model Notation

State variables Exogenous Parameters

mt Level of SDF θvm Mean of vm St Level of asset value θβ Mean of β m vt Price of risk θv Mean of v

βt Quantity of priced risk κvm Mean reversion of vm  vt Quantity of unpriced risk κβ Mean reversion of β

κv Mean reversion of v

σvm Volatility of vm σβ Volatility of β

σv Volatility of v

ρvm,m Correlation of vm and m

the SDF, but by ignoring these correlations I can use the pairwise independence m  2 of (vt , βt, vt ) to compute many cross-moments like Et(βT vT ) that would otherwise be intractable. Additionally, while the correlation between the SDF and volatility generates the leverage effect and volatility risk premium in the model, it has little effect on the incremental value of the volatility term structure in identifying risk premia. Appendix Table 1 summarizes the ten exogenous parameters and five state vari- ables of the model. Two of the state variables, the SDF level mt and asset price level St, have no impact on the distribution of future log returns. The other three state variables are all stationary and Markov, allowing the computation of both conditional and unconditional moments.

A Risk and Return in the Model While the marginal investor knows their preferences and the distribution of asset m  values, and can therefore compute the three volatility states vt , βt, and vt , I study the problem of an outsider observer that tries to infer the state using traded asset prices. For such an observer, the current underlying price St reveals nothing about the volatility state. Observing the path of St prior to t in enough detail could, hypo- R 2 m  thetically, reveal the total return diffusion vt = βt vt + vt but could not discriminate between priced and unpriced risks. As in the text, I assist the observer by assuming there is a variance swap traded on the underlying asset. As shown in Martin (2011) and the references therein, in the absence of jumps the strike of a variance swap IVt,T at time t expiring at time t + T APPENDIX B. APPENDICES FOR CHAPTER 3 109

is the risk-neutral expected quadratic variation of log(ST ), or:

Z t+T  Z t+T Q R Q m  IVt,T = Et vs ds = Et [βsvs + vs] ds t t

I show that the combined information in IVt,T at many different horizons T can m  completely reveal the volatility state (vt , βt, vt ), and therefore reveal any time vari- ation in the equity risk premia. Intuitively, the reason is that if the different com- ponents of volatility have different persistences, many different volatility states could produce the same IVt,T for a single T , but each volatility state produces a unique shape of the IVt,T term structure. Mathematically, this arises because each IVt,T is an affine function of an augmented state vector Zt, where Zt is:

 m  2 m 2 m0 Zt = vt βt vt βt βtvt βt vt

The coefficients in the affine function mapping Zt to IVt,T vary with T , meaning that with enough different T , {IVt,T } can be inverted to reveal Zt. More specifically, given IVt,T at six different horizons, I can compute any other affine function of B · Zt as follows. First, note that IVt,T = A0(T ) + A1(T ) · Zt for 6 functions A0 : R → R and A1 : R → R . This implies:

wt = A0 + A1zt −1 ⇒ zt = A1 (wt − A0) −1 ⇒ B · zt = BA1 (wt − A0) where wt is a 6x1 column vector formed by stacking all six IVt,T , A0 is a 6x1 matrix of all the A0(T ), and A1 is a 6x6 matrix of all the A1(T ). The conditional moments take the form C0 + C1Zt for some constant C0 and vector of constants C1. The unconditional moments take the form C0 +C1E(Zt), where E(Zt) is the unconditional expected state. The instantaneous equity risk premium is the drift of the return process, namely m r + βtvt . Computing the expected simple return of the next T periods is tricky because the stochastic differential equation governing St has St in both the drift and diffusion terms. However, a simple application of Ito’s lemma reveals that:

m R
m m   d(log(St)) = r + βtvt − vt dt − βtvt dBt + vt dBt

R 2 m  where vt = βt vt +vt is the total return diffusion. From this, I compute the expected APPENDIX B. APPENDICES FOR CHAPTER 3 110

excess log return:    St+T Et log − rT = Et (log (St+T )) − log(St) − rT St Z t+T m R
= Et βsvs − vs ds t

m R The log risk premium is the integral of expected future drifts βsvs − vs conditional upon time t information. Like the other moments, the closed form for the expected log return is affine in Zt. The critical assumption that makes it possible to compute these moments in closed form is the independence of Vt and βt. By computing the moments in closed form, as opposed to simulating them, I am able to fit the model as described below.

B Matching Moments I calibrate the model to data on the S&P 500 by matching the moments listed in Appendix Table 2. The unit of time is one month, and use the VIX term structure as described in Section 3.4 as a proxy for the risk-neutral expected quadratic variation. I show that the unrestricted model is capable of generating weak return predictability for a single variance horizon and strong return predictability for multiple variance horizons with a single parametrization. However, if one of the three volatility com- ponents is restricted to being constant, the incremental predictive power of multiple variance horizons is dramatically reduced. The first four moments are unconditional means and standard deviations of the VIX at different horizons. These unconditional means are under the statistical mea- sure, so they can match the data, but the model VIX itself is a risk-neutral expectation of future diffusion. The longer-horizon VIX are less volatile because volatility mean reverts, meaning that the short horizon VIX is always farther from the long-run mean than the long horizon VIX. The calibrated model matches these four moments very closely. The next two moments are the mean and variation of monthly excess log returns. m In the model, the instantaneous drift is βtvt but the average one-period return is more complicated because of the mean reversion in both β and vm. As discussed above, for the St process described in Equation (B.5), the moments of log returns are substantially more tractable than the moments of simple returns. The variation of monthly excess log returns is defined as the statistical-measure analogue of the one APPENDIX B. APPENDICES FOR CHAPTER 3 111

Table B.2: Data and Calibrated Model Moments

Moment Empirical Calibrated Model 2 (1) Mean(VIX1) 4.72E-03 4.44E-03 2 (2) Std(VIX1) 4.69E-03 4.80E-03 2 (3) Std(VIX6) 3.32E-03 3.57E-03 2 (4) Std(VIX12) 2.87E-03 2.64E-03 (5) Mean(log(R1 − Rf )) 4.67E-03 4.03E-03 (6) Var(log(R1 − Rf )) 3.66E-03 4.43E-03 2 2 (7) corr(VIX1,t,VIX1,t+1) 0.783 0.866 (8) Single regression consant 0.005 0.001 2 (9) Single regression VIX1 coefficient 0.045 0.627 (10) Multiple regression constant -0.007 0.006 2 (11) Multiple regression VIX1 coefficient -11.80 -27.72 2 (12) Multiple regression VIX6 coefficient 36.31 39.77 2 (13) Multiple regression VIX12 coefficient -22.13 -12.34 period VIX:

Z 1  2 m  Var(R1) = E βt vt + vt dt (B.12) 0 which can be computed identically to IVt,T but without the adjustment to the drift m of vt necessary under the risk-neutral measure. Log returns in the calibrated model are too volatile under the statistical measure. 2 The model cannot replicate the dramatic difference between average VIX1 and realized 2 return variance observed in the data, and so it winds up with VIX1 slightly too small on average and Var(log(R − Rf )) much too high. The meager variance risk premia t there is in the model comes from the correlation between mt and vm. The Bollerslev, Tauchen, and Zhou (2009) and Drechsler and Yaron (2011) models that directly address the variance risk premium use a volatility of volatility process or asset price jumps to create a larger variance risk premium. The seventh moment is the correlation between the 1 month VIX and the 1 month VIX 1 month ago. This moment is determined in the model by the combined per- sistence of the three different volatility measures. The calibrated model that fits the other moments has slightly too much persistence (or, equivalently, not enough mean version). Moments (8)-(13) are regression coefficients from a single regression of log(Rt+1)- logRf,t+1) on a constant and VIX1,t; and a multiple regression of log(Rt+1)-log(Rf,t+1) APPENDIX B. APPENDICES FOR CHAPTER 3 112

on a constant, VIX1,t, VIX6,t, and VIX12,t. The calibrated model matches the nearly 0 intercepts, weakly positive single regression slope coefficient, as well as the tent shape of the multiple regression coefficients. However, the single regression slope coefficient is considerably higher in the calibrated model than the data. Bibliography

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