Abstract We propose a new measure of financial intermediary constraints based on how intermediaries manage their tail risk exposures. Using data for the trading activities in the market of deep out-of-the-money index put options, we identify periods when the variations in the net amount of trading between financial intermediaries and public investors are likely to be mainly driven by shocks to intermediary constraints. We then infer tightness of intermediary constraints from the quantities of option trading. A tightening of intermediary constraints according to our measure is associated with increasing option expensiveness, higher risk premia, deteriorating funding liquidity, and broker-dealer deleveraging. Received December 1, 2014; editorial decision May 19, 2017 by Editor Geert Bekaert. Authors have furnished an Internet Appendix, which is available on the Oxford University Press Web site next to the link to the final published paper online. In this paper, we present new evidence connecting financial intermediary constraints to asset prices. We propose to measure the tightness of intermediary constraints based on how the intermediaries manage their aggregate tail risk exposures. Using data on the trading activities between public investors and financial intermediaries in the market of deep out-of-the-money put options on the S&P 500 index (abbreviated as DOTM SPX puts), we exploit the price-quantity relations to identify periods when shocks to intermediary constraints are likely to be the main driver of the variations in the net amount of trading between public investors and financial intermediaries. This then enables us to infer tight intermediary constraints from the option trading quantities. We show that tight intermediary constraints under our measure are associated with increasing option expensiveness, rising risk premia for a wide range of financial assets, deteriorating funding liquidity, and deleveraging by broker-dealers. To construct the constraint measure, we start by computing the net amount of DOTM SPX puts that public investors in aggregate purchase each month (henceforth referred to as |$\textit{PNBO}$|), which also reflects the net amount of the same options that broker-dealers and market-makers sell in that month. While it is well known that financial intermediaries are net sellers of these types of options during normal times, we find that |$\textit{PNBO}$| varies significantly over time and tends to be negative during times of market distress. |$\textit{PNBO}$| could be low during periods of weak supply by intermediaries (because of tight constraints) or during periods of weak demand by public investors. One needs to separate these two effects in order to link |$\textit{PNBO}$| to intermediary constraints. We propose exploiting the relation between the quantities of trading, as measured by |$\textit{PNBO}$|, and prices (expensiveness of SPX options), as measured by the variance premium. Positive comovements in prices and quantities are consistent with the presence of demand shocks, whereas negative comovements are consistent with the presence of supply shocks.1 We summarize the daily price-quantity relations each month. On average, supply shocks are likely to be the main driver of the quantity of trading in months with a negative price-quantity relation. Then we take low |$\textit{PNBO}$| in a month with negative price-quantity relation as indicative of tight intermediary constraints. In monthly data from January 1991 to December 2012, |$\textit{PNBO}$| is significantly negatively related to option expensiveness, and this negative relation becomes stronger when jump risk in the market is higher. In daily data, the correlation between |$\textit{PNBO}$| and our measure of option expensiveness is negative in 159 out of 264 months. These results highlight the significant role that supply shocks play in the market of DOTM SPX puts. In periods with a negative price-quantity relation, |$\textit{PNBO}$| significantly predicts future market excess returns. A one-standard-deviation decrease in |$\textit{PNBO}$| (normalized |$\textit{PNBO}$|) in a month with negative price-quantity relation is on average associated with a 4.3% (3%) increase in the subsequent 3-month log market excess return. The |$R^2$| of the regression is 24.7% (12.3%). The predictive power of |$\textit{PNBO}$| is even stronger in months when market jump risk is above the median level (in addition to the negative price-quantity relation), but becomes much weaker in months when the price-quantity relations are positive. Besides equity, a lower |$\textit{PNBO}$| also predicts higher future excess returns for high-yield corporate bonds, an aggregate hedge fund portfolio, a carry trade portfolio, and a commodity index, and it predicts lower future excess returns on long-term Treasuries and (pay-fix) SPX variance swaps. The predictability results survive extensive robustness checks, which include different statistical methods for determining the significance of the predictive power, excluding the 2008–2009 financial crisis, extreme observations of |$\textit{PNBO}$|, different ways to define option moneyness, and an alternative quantity measure based on end-of-period open interest instead of trading volume, among others. In addition, we consider an alternative method for identifying periods of weak supply based on Rigobon (2003) (see also Sentana and Fiorentini 2001). Using the reduced-form econometric assumptions of this method, we extract supply shocks to intermediaries and confirm the ability of the inferred supply shocks to predict future stock returns. The return predictability results are consistent with the intermediary asset pricing theories, in which a reduction in the risk-sharing capacity of the financial intermediaries causes the aggregate risk premium in the economy to rise. An alternative explanation of the predictability results is that |$\textit{PNBO}$| is merely a proxy for standard macrofinancial factors that simultaneously drive the aggregate risk premium and intermediary constraints. If this alternative explanation is true, then the inclusion of proper risk factors in the predictability regression should drive away the predictive power of our constraint measure. We find that the predictive power of our measure is unaffected by the inclusion of a long list of return predictors in the literature, including various price ratios, consumption-wealth ratio, variance risk premium, default spread, term spread, and several tail risk measures. While these results do not lead to the rejection of the alternative explanation (there can always be omitted risk factors), they are at least consistent with intermediary constraints having a unique effect on the aggregate risk premium. Our intermediary constraint measure is significantly related to the funding condition measures of Fontaine and Garcia (2012) (extracted from the Treasury market) and Adrian, Etula, and Muir (2014) (based on the growth rate of broker-dealer leverage). At the same time, we also find that our constraint measure provides unique information about the aggregate risk premium not contained in the other funding liquidity measures. Our results suggest that when financial intermediaries switch from sellers of DOTM SPX puts to buyers (e.g., in the months following the Lehman Brothers bankruptcy in 2008), tight constraints likely forced intermediaries to aggressively hedge their tail risk exposures, rather than accommodate an increase in public investors’ demand to sell crash insurance. Examples of shocks to intermediary constraints include stricter regulatory requirements on banks’ tail risk exposures (e.g., because of the Dodd-Frank Act or Basel III) or losses incurred by intermediaries. To further examine the risk-sharing mechanism, we try to identify who among the public investors (retail or institutional) are the “liquidity providers” during times of distress: reducing the net amount of crash insurance acquired from financial intermediaries or even providing insurance to the latter group. We answer this question by comparing public investors’ demand in the markets of SPX verus SPY options. SPY options are options on the SPDR S&P 500 ETF Trust that have a significantly higher percentage of retail customers compared to SPX options. Our results suggest that institutional public investors are the liquidity providers during periods of distress. Our paper builds on and extends the work of Garleanu, Pedersen, and Poteshman (2009) (henceforth GPP) by incorporating the impact of supply shocks into the options market. In a partial equilibrium setting, GPP demonstrate how exogenous public demand shocks affect option prices when risk-averse dealers bear the inventory risks. In their model, the dealers’ intermediation capacity is fixed, and the model implies a positive relation between the public demand for options and the option premium. Unlike GPP, we consider shocks to the intermediary constraint and the endogenous relations among public demand for options, option pricing, and aggregate market risk premium.2 In the empirical analysis, we try to separate the effects of public demand shocks and shocks to intermediary constraints, and show that the latter is linked to the time-varying risk premia for a wide range of financial assets. Our empirical strategy based on the price-quantity dynamics is motivated by Cohen, Diether, and Malloy (2007), who use a similar strategy to identify demand and supply shocks in the equity shorting market. The recent financial crisis has highlighted the importance of understanding the potential impact of intermediary constraints on the financial markets and the real economy. Following the seminal contributions by Bernanke and Gertler (1989), Kiyotaki and Moore (1997), and Bernanke, Gertler, and Gilchrist (1999), other theoretical developments include Gromb and Vayanos (2002), Brunnermeier and Pedersen (2009), Geanakoplos (2009), He and Krishnamurthy (2013), Adrian and Boyarchenko (2012), Brunnermeier and Sannikov (2014), among others. In contrast to the fast-growing body of theoretical work, there is relatively little empirical work measuring intermediary constraints and studying their aggregate effects on asset prices. The notable exceptions include Adrian, Moench, and Shin (2010) and Adrian, Etula, and Muir (2014), who show that changes in aggregate broker-dealer leverage are linked to the time series and cross-section of asset returns. Our paper uses a new venue (the crash insurance market) to capture intermediary constraint variations and study their effects on asset prices. Moreover, compared to intermediary leverage changes, our measure has the advantage of being forward-looking and available at a higher (daily and monthly instead of quarterly) frequency. The ability of option volume to predict returns has been examined in other contexts. Pan and Poteshman (2006) show that option volume predicts near-future individual stock returns (up to 2 weeks). They find the source of this predictability to be the nonpublic information possessed by option traders. Our evidence of return predictability applies to the market index and to longer horizons (up to 4 months), and we argue that the source of this predictability is time-varying intermediary constraints. Finally, several studies have examined the role that derivatives markets play in the aggregate economy. Buraschi and Jiltsov (2006) study option pricing and trading volume when investors have incomplete and heterogeneous information. Bates (2008) shows how options can complete the markets in the presence of crash risk. Longstaff and Wang (2012) show that the credit market plays an important role in facilitating facilitates risk sharing among heterogeneous investors. Chen, Joslin, and Tran (2012) show that the market risk premium is highly sensitive to the amount of sharing of tail risks in at equilibrium. 1. Research Design Our goal is to measure the degree to which financial intermediaries are constrained through the ways they manage their exposures to aggregate tail risks. The market of DOTM SPX put options is well suited for this purpose. First, this market is large in terms of economic exposure to aggregate tail risks.3 Second, compared to other over-the-counter derivatives that also provide exposures to aggregate tail risks, the exchange-traded SPX options have the advantages of better liquidity and almost no counterparty risk (other than exchange failure). Third, the Options Clearing Corporation (OCC) classifies exchange option transactions by investor types, a classification that allows us to determine the net exposures of the financial intermediaries and is essential for constructing our measure. Specifically, the OCC classifies each option transaction as one of three categories based on who initiates the trade. They include public investors, firm investors, and market-makers. Transactions initiated by public investors include those initiated by retail investors and those by institutional investors such as hedge funds. Trades initiated by firm investors are those that securities broker-dealers (who are not designated market-makers) make for their own accounts or for another broker-dealer. Since we focus on financial intermediaries as a whole, it is natural to merge firm investors and market-makers into one group and observe how they trade against public investors. We classify DOTM puts as having a strike-to-price ratio |$K/S \leq 0.85$|. For robustness, we also consider different strike-to-price cutoffs, as well as cutoffs that adjust for option maturity and the volatility of the S&P 500 index (which is similar to cutoffs based on option delta). Another feature of option transaction is that an order can either be an open order (to open new positions) or be a closed order (to close existing positions). We will focus on open orders, because they are less likely to be mechanically influenced by existing positions (see Pan and Poteshman, 2006). We construct a measure of the public net buying-to-open volume for DOTM SPX puts (abbreviated as |$\textit{PNBO}$|). In period |$t$| (e.g., a day or a month), |$\textit{PNBO}_t$| is defined as \begin{align} \textit{PBNO}_t \equiv \mbox{public open-buy volume}_t - \mbox{public open-sell volume}_t. \label{eqn.pnbo} \end{align} (1) |$\textit{PNBO}$| represents the amount of new DOTM SPX puts bought (sold if negative) by public investors in a period. Because of the growth in size of the options market, there could be a time trend in the level or volatility of |$\textit{PNBO}$|. Thus, we also consider normalizing |$\textit{PNBO}$| by the average monthly volume of all SPX options traded by public investors over the past 3 months,4 \begin{align} \textit{PNBON}_t \equiv \frac{\textit{PNBO}_t} {\mbox{Average monthly public SPX volume over past 3 months}}. \label{eqn.pnbon} \end{align} (2) |$\textit{PNBON}$| helps address the potential issue with growth in the size of market, and |$\textit{PNBO}$| better captures the actual magnitude of the tail risk exposures being transferred between public investors and intermediaries. The latter matters for measuring the degree of intermediary constraints. Considering this trade-off, we conduct all of our main analyses using both |$\textit{PNBO}$| and |$\textit{PNBON}$|. It is well documented (see, e.g., Bollen and Whaley 2004) that, during normal times, public investors are net buyers of index puts, while financial intermediaries are net sellers. All else equal, when financial intermediaries become more constrained, their willingness to supply crash insurance to the market is reduced. It is thus tempting to infer the degree to which financial intermediaries are constrained based on the net amount of crash insurance they sell to public investors each period, as captured by |$\textit{PNBO}$|. However, besides weak supply from constrained intermediaries, weak public demand can also cause the equilibrium amount of crash insurance traded between public investors and intermediaries to be low. The challenge is to separate the effects of supply from demand. We address this problem in two ways: (1) by exploiting the price-quantity relation to identify periods of “supply environments,” when variations in |$\textit{PNBO}$| are likely to be mainly driven by shocks to intermediary constraints and (2) using the Rigobon (2003) method, which identifies demand and supply by exploiting the heteroscedasticity of shocks. Before presenting the identification methodology, we briefly explain our “price” measure, that is, the expensiveness of SPX options. One would ideally like to calculate the difference between the market price of an option and its hypothetical price without any market frictions. The latter is not observable and can only be approximated by adopting a specific pricing model. For simplicity and robustness, we use the variance premium (|$VP$|) in Bekaert and Hoerova (2014) as a proxy for overall expensiveness of SPX options, which is the difference between |$\rm VIX^2$| and the expected physical variance of the return of the S&P 500 index.5 1.1 Identification through the price-quantity relation Our first empirical strategy is motivated by Cohen, Diether, and Malloy (2007) (CDM), who identify shifts in demand versus supply in the securities shorting market by examining the relation between the changes in the loan fee (price) and the changes in the percentage of outstanding shares on loan (quantity). In their setting, a simultaneous increase (decrease) in the price and quantity indicates at least an increase (decrease) in shorting demand, whereas an increase (decrease) in price coupled with a decrease (increase) in quantity indicates at least a decrease (increase) in shorting supply. The same logic applies to the options market. The demand pressure theory of GPP predicts that a positive exogenous shock to the public demand for DOTM SPX puts forces risk-averse dealers to bear more inventory risks. As a result, the dealers will raise the price of the option (a move along the upward-sloping supply curve). Thus, demand shocks generate a positive relation between changes in prices and in quantities. Alternatively, if intermediation shocks tighten the constraints facing financial intermediaries (e.g., due to a loss of capital or higher capital requirements), they will become less willing to provide crash insurance to public investors. Then the premium for the DOTM SPX puts rises, while the equilibrium quantity of such options traded falls (a move along the downward-sloping demand curve). Different from CDM, we would like to identify periods of weak and strong supply (level), instead of negative or positive supply shocks (changes). For this reason, we cannot directly apply their identification strategy. However, the ability to identify supply and demand shocks is still useful for our setting. Consider a month with low |$\textit{PNBO}$|. If the price-quantity relations on a daily basis mainly suggest supply shocks in that month, then the low |$\textit{PNBO}$| is more likely to be driven by weak supply instead of weak demand. Based on this idea, we run the following regression using daily data in each month |$t$|: \begin{align} VP_{i(t)} &= a_{VP,t} + b_{VP,t} \ \textit{PNBO}_{i(t)} + d_{VP,t} \ J_{i(t)} + \epsilon^v_{i(t)}, \label{eqn.VP} \end{align} (3) where |$i(t)$| denotes day |$i$| in month |$t$|. The presence of jumps in the underlying stock index can affect |$VP$| even when markets are frictionless. Thus, when examining the relation between |$VP$| and |$PBNO$|, we control for the level of jump risk |$J$| in the S&P 500 index based on the measure of Andersen, Bollerslev, and Diebold (2007). A negative coefficient |$b_{VP,t} < 0$| in month |$t$| suggests that supply shocks are the dominant driver of price-quantity relations in that month, and we expect |$\textit{PNBO}$| to be informative about the variation in intermediary constraints during such times. |$b_{VP,t} < 0$| does not confirm the existence of supply shock in a particular month and does not rule out the presence of demand shocks in the same month. It does indicate that supply shocks are likely to be more significant relative to demand shocks. Similarly, |$b_{VP,t} > 0$| does not rule out the presence of supply shocks in a month, but demand shocks are likely to be more significant. During such periods, we do not expect |$\textit{PNBO}$| to be informative about supply conditions. Furthermore, we expect high jump risk to amplify the effect of shocks to intermediary constraints on the equilibrium quantity of options traded. This is because a main reason that intermediary constraint matters for their supply of DOTM index puts is the difficulty to hedge the market jump risk embedded in their inventory positions. If public demand does not become more volatile during such times, variations in |$\textit{PNBO}$| will be more informative about shocks to intermediary constraints when jump risk is high. The validity of this assumption is an empirical question. In summary, in a month when the price-quantity relation is on average negative (|$b_{VP,t} < 0$|), we expect small (or negative) value for |$\textit{PNBO}_t$| (cumulative net-buying by public investors for the month) to indicate tight intermediary constraints. 1.2 Identification through heteroscedasticity Besides the identification of “supply environments” based on the price-quantity relation, supply shocks can be identified through econometric identification. One method suited for our study is identification through heteroscedasticity of Rigobon (2003).6Rigobon (2003) considers a standard linear supply-demand relationship between prices (|$p_t$|) and quantities (|$q_t$|): \begin{align} p_t &= b + \beta q_t + \epsilon_t, \quad \textrm{(demand equation),} \label{eqn.generic demand} \\ \end{align} (4a) \begin{align} q_t &= a + \alpha p_t + \eta_t, \quad \textrm{(supply equation),} \label{eqn.generic supply} \end{align} (4b) where the volatilities of the supply and demand shocks are |$\sigma_\epsilon$| and |$\sigma_\eta$|, respectively. In general, the residuals will be correlated with the independent variables in each equation, and the parameters will not be identified. Rigobon (2003) solves this identification problem by considering the regime-dependent heteroscedasticity of |$(\epsilon, \eta)$|. Supposing that there are two regimes and the relative volatilities of the supply and demand shocks vary across the regimes, the supply and demand Equations can be identified. In parallel to our empirical strategy motivated by CDM, we also identify demand and supply shocks following the method of Rigobon (2003). There is suggestive evidence that in the market for DOTM SPX puts supply shocks are more volatile relative to demand shocks during the period of the U.S. financial crisis and the European sovereign debt crisis. We thus date the low- and high-supply volatility regimes accordingly and use the price and quantity measures discussed above to estimate the supply-demand system in Equations (4a) and (4b). The two identification methods presented in Sections 1.1 and 1.2 have their respective advantages. On the one hand, the Rigobon method has the advantage of being based on a clean set of parametric assumptions, a basis that helps with the precise identification of the supply equation and supply shocks. However, these assumptions might be restrictive and potentially inconsistent with the data (e.g., the assumption about the linear relations between prices and quantities, the number of volatility regimes, and whether other parameters might change across these regimes). On the other hand, the method based on price-quantity relation does not identify the supply environment or supply shocks as cleanly, but it imposes weaker assumptions on the demand and supply curve that likely makes the results more robust. After constructing our measures of intermediary constraints, we investigate how these measures are linked to asset prices. According to the theory of financial intermediary constraints (see, e.g., Gromb and Vayanos 2002; He and Krishnamurthy 2013), variations in the aggregate intermediary constraints not only affect option prices but also drive the risk premia of other financial assets. This theory implies that low |$\textit{PNBO}_t$|, when occurring in a period dominated by supply shocks, should imply high future expected excess returns on the market portfolio. That is, we expect |$b^-_r < 0$| in the following predictive regression: \begin{align} r_{t+j \rightarrow t+k} &= a_{r} + b^-_{r} \ I_{\{b_{VP,t} < 0\}} \, \textit{PNBO}_t + b^+_{r} \ I_{\{b_{VP,t} \geq 0\}} \, \textit{PNBO}_t \notag \\ &\quad+ c_{r} \ I_{\{b_{VP,t} < 0\}} + \epsilon_{t+j \rightarrow t+k} \label{eqn.r1} \end{align} (5) where |$r$| denotes log market excess return, and the notation |${t+j \rightarrow t+k}$| indicates the leading period from |$t+j$| to |$t+k$| (|$k>j \geq 0$|). Similarly, we expect |$\hat \eta_t$| extracted from the supply-demand system in Equations (4a and (4b) to predict future market excess returns as well, \begin{align} r_{t+j \rightarrow t+k} &= a_{s} + b_{s} \hat \eta_t + \epsilon_{t+j \rightarrow t+k}, \end{align} (6) where we expect |$b_s \leq 0$|. Besides the market portfolio, predictability should apply to other risky assets. Finally, the empirical strategy and testable hypotheses above are mainly based on economic intuition. In the Online Appendix, we present a dynamic general equilibrium model featuring time-varying intermediary constraints. The model not only helps formalize the main intuition but generates more rigorous predictions about how intermediary constraints affect the equilibrium price-quantity dynamics in the crash insurance market, the aggregate risk premium, and intermediary leverage. Moreover, we can use the calibrated model to examine the quantitative effects of intermediary constraints on asset prices. 2. Empirical Results We now present the empirical evidence connecting the option trading activities to the constraints of the financial intermediaries and the risk premia in financial markets. 2.1 Data Figure 1 plots the monthly time series of |$\textit{PNBO}$| and its normalized version |$\textit{PNBON}$|. Consistent with the findings of Pan and Poteshman (2006) and GPP, the net public purchase of DOTM SPX puts was positive in the majority of the months prior to the financial crisis in 2008, suggesting that broker-dealers and market-makers were mainly supplying crash insurance to public investors. A few notable exceptions include the period around the Asian financial crisis (December 1997), the period around the Russian default, the period around the financial crisis in Latin America (November 1998 to January 1999), the period around the Iraq War (April 2003), and 2 months in 2005 (March and November 2005).7 Figure 1 View largeDownload slide Time series of net public purchase for DOTM SPX puts |$\textit{PNBO}$| is the net amount of deep out-of-the-money (DOTM)(with |$K/S\leq0.85$|) SPX puts public investors buying-to-open each month. |$\textit{PNBON}$| is |$\textit{PNBO}$| normalized by average of previous 3-month total volume from public investors. “Asian” (Oct. 1997): period around the Asian financial crisis. “Russian” (Nov. 1998): period around Russian default. “Iraq” (Apr. 2003): start of the Iraq War. “Quant” (Aug. 2007): the crisis of quant-strategy hedge funds. “Bear Sterns” (Mar. 2008): acquisition of Bear Sterns by J. P. Morgan. “Lehman” (Sept. 2008): Lehman bankruptcy. “TARP” (Oct. 2008): establishment of TARP. “TALF1” (Nov. 2008): creation of TALF. “BoA” (Jan. 2009): Treasury, Fed, and FDIC assistance to Bank of America. “TALF2” (Feb. 2009): increase of TALF to |${\$}$|1 trillion. “Euro” (Dec. 2009): escalation of Greek debt crisis. “GB1” (Apr. 2010): Greece seeks financial support from euro and IMF. “EFSF” (May 2010): establishment of EFSM and EFSF; 110 billion bailout package to Greece agreed. “GB2” (Sept. 2010): a second Greek bailout installment. “Voluntary” (Jun. 2011): Merkel agrees to voluntary Greece bondholder role. “Referendum” (Oct. 2011): further escalation of Euro debt crisis with the call for a Greek referendum. Figure 1 View largeDownload slide Time series of net public purchase for DOTM SPX puts |$\textit{PNBO}$| is the net amount of deep out-of-the-money (DOTM)(with |$K/S\leq0.85$|) SPX puts public investors buying-to-open each month. |$\textit{PNBON}$| is |$\textit{PNBO}$| normalized by average of previous 3-month total volume from public investors. “Asian” (Oct. 1997): period around the Asian financial crisis. “Russian” (Nov. 1998): period around Russian default. “Iraq” (Apr. 2003): start of the Iraq War. “Quant” (Aug. 2007): the crisis of quant-strategy hedge funds. “Bear Sterns” (Mar. 2008): acquisition of Bear Sterns by J. P. Morgan. “Lehman” (Sept. 2008): Lehman bankruptcy. “TARP” (Oct. 2008): establishment of TARP. “TALF1” (Nov. 2008): creation of TALF. “BoA” (Jan. 2009): Treasury, Fed, and FDIC assistance to Bank of America. “TALF2” (Feb. 2009): increase of TALF to |${\$}$|1 trillion. “Euro” (Dec. 2009): escalation of Greek debt crisis. “GB1” (Apr. 2010): Greece seeks financial support from euro and IMF. “EFSF” (May 2010): establishment of EFSM and EFSF; 110 billion bailout package to Greece agreed. “GB2” (Sept. 2010): a second Greek bailout installment. “Voluntary” (Jun. 2011): Merkel agrees to voluntary Greece bondholder role. “Referendum” (Oct. 2011): further escalation of Euro debt crisis with the call for a Greek referendum. However, starting in 2007, |$\textit{PNBO}$| became significantly more volatile.8 It turned negative during the quant crisis in August 2007, when a host of quant-driven hedge funds experienced significant losses. It then rose significantly and peaked in October 2008, following the Lehman Brothers bankruptcy. As market conditions continued to deteriorate, |$\textit{PNBO}$| plunged rapidly and turned significantly negative in the following months. Following a series of government interventions, |$\textit{PNBO}$| bottomed out in April 2009, rebounded briefly, and then dropped again in December 2009, when the Greek debt crisis escalated. During the period from November 2008 to December 2012, public investors on average sold 44,000 DOTM SPX puts to open new positions each month. In contrast, they bought on average 17,000 DOTM SPX puts each month in the period from 1991 to 2007. One reason that the |$\textit{PNBO}$| series appears more volatile in the latter part of the sample is because the options market (e.g., in terms of total trading volume) had grown significantly over time. As the bottom panel of Figure 1 shows, after normalizing |$\textit{PNBO}$| with the total SPX volume (see the definition in Equation (2)), the |$\textit{PNBON}$| series no longer demonstrates visible trend in volatility. Table 1 reports the summary statistics of the option volume and pricing variables and their correlation coefficients. From January 1991 to December 2012, the public net buying-to-open volume of DOTM SPX puts (|$\textit{PNBO}$|) is close to 10,000 contracts per month on average (each contract has a notional size of 100 times the index). In comparison, the average total open interest for all DOTM SPX puts is around 0.9 million contracts during the period from January 1996 to December 2012. This figure highlights the significant difference between |$\textit{PNBO}$| and open interest. The option volume measures have relatively modest autocorrelations at monthly frequency (0.61 for |$\textit{PNBO}$| and 0.48 for |$\textit{PNBON}$|) compared to standard return predictors such as dividend yield and term spread. The correlation matrix in panel B shows that the various quantity measures are negatively related to variance premium (|$VP$|). In addition, both |$\textit{PNBO}$| and |$\textit{PNBON}$| are negatively correlated with the unemployment rate (see Table IA1 in the Online Appendix). Table 1 Summary statistics and correlation A. Summary statistics Mean Median SD AC(1) pp test |$\textit{PNBO}$| (|$10^3$| contracts) 10.00 9.67 51.12 0.61 0.00 |$\textit{PNBON}$| (|${\%}$|) 0.56 0.53 1.07 0.48 0.00 |$\textit{PNBO}_{ND}$| (|$10^3$| contracts) 138.08 117.10 113.64 0.76 0.01 |$\textit{PNOI}$| (|$10^3$| contracts) 28.79 19.85 63.21 0.74 0.00 |$\textit{PNOIN}$| (|${\%}$|) 0.18 0.15 0.36 0.70 0.00 |$J$| (%) 12.14 10.81 6.17 0.62 0.00 |$VP$| 19.37 14.56 21.66 0.54 0.00 A. Summary statistics Mean Median SD AC(1) pp test |$\textit{PNBO}$| (|$10^3$| contracts) 10.00 9.67 51.12 0.61 0.00 |$\textit{PNBON}$| (|${\%}$|) 0.56 0.53 1.07 0.48 0.00 |$\textit{PNBO}_{ND}$| (|$10^3$| contracts) 138.08 117.10 113.64 0.76 0.01 |$\textit{PNOI}$| (|$10^3$| contracts) 28.79 19.85 63.21 0.74 0.00 |$\textit{PNOIN}$| (|${\%}$|) 0.18 0.15 0.36 0.70 0.00 |$J$| (%) 12.14 10.81 6.17 0.62 0.00 |$VP$| 19.37 14.56 21.66 0.54 0.00 B. Correlation |$\textit{PNBO}$| |$\textit{PNBON}$| |$\textit{PNBO}_{\textit{ND}}$| |$\textit{PNOI}$| |$\textit{PNOIN}$| |$J$| |$\textit{PNBON}$| 0.71 |$\textit{PNBO}_{ND}$| 0.21 0.04 |$\textit{PNOI}$| 0.67 0.48 0.33 |$\textit{PNOIN}$| 0.58 0.59 0.23 0.90 |$\textit{J}$| 0.03 0.03 0.08 0.16 0.12 |$\textit{VP}$| –0.22 –0.15 –0.06 –0.07 –0.09 0.54 B. Correlation |$\textit{PNBO}$| |$\textit{PNBON}$| |$\textit{PNBO}_{\textit{ND}}$| |$\textit{PNOI}$| |$\textit{PNOIN}$| |$J$| |$\textit{PNBON}$| 0.71 |$\textit{PNBO}_{ND}$| 0.21 0.04 |$\textit{PNOI}$| 0.67 0.48 0.33 |$\textit{PNOIN}$| 0.58 0.59 0.23 0.90 |$\textit{J}$| 0.03 0.03 0.08 0.16 0.12 |$\textit{VP}$| –0.22 –0.15 –0.06 –0.07 –0.09 0.54 This table reports the summary statistics for the SPX options volume from public investors and pricing variables in the empirical analysis. |$\textit{PNBO}$|: public net open-buying volume of DOTM puts (|$K/S \leq 0.85$|). |$\textit{PNBON}$|: |$\textit{PNBO}$| normalized by average monthly public SPX volume over past 12 months (in million contracts). |$\textit{PNBO}_{ND}$|: public net open-buying volume of all SPX options excluding DOTM puts. |$\textit{PNOI}$|: public net open interest for DOTM SPX puts (in million contracts). |$\textit{PNOIN}$|: |$\textit{PNOI}$| normalized by the total public open interest of all options (long and short). |$J$|: monthly average of the daily physical jump risk measure by Andersen, Bollerslev, and Diebold (2007). |$VP$|: variance premium based on Bekaert and Hoerova (2014). AC(1) is the first-order autocorrelation for monthly time series; the pp test is the |$p$|-value for the Phillips-Perron test for unit root. The sample period is from January 1991 to December 2012. Table 1 Summary statistics and correlation A. Summary statistics Mean Median SD AC(1) pp test |$\textit{PNBO}$| (|$10^3$| contracts) 10.00 9.67 51.12 0.61 0.00 |$\textit{PNBON}$| (|${\%}$|) 0.56 0.53 1.07 0.48 0.00 |$\textit{PNBO}_{ND}$| (|$10^3$| contracts) 138.08 117.10 113.64 0.76 0.01 |$\textit{PNOI}$| (|$10^3$| contracts) 28.79 19.85 63.21 0.74 0.00 |$\textit{PNOIN}$| (|${\%}$|) 0.18 0.15 0.36 0.70 0.00 |$J$| (%) 12.14 10.81 6.17 0.62 0.00 |$VP$| 19.37 14.56 21.66 0.54 0.00 A. Summary statistics Mean Median SD AC(1) pp test |$\textit{PNBO}$| (|$10^3$| contracts) 10.00 9.67 51.12 0.61 0.00 |$\textit{PNBON}$| (|${\%}$|) 0.56 0.53 1.07 0.48 0.00 |$\textit{PNBO}_{ND}$| (|$10^3$| contracts) 138.08 117.10 113.64 0.76 0.01 |$\textit{PNOI}$| (|$10^3$| contracts) 28.79 19.85 63.21 0.74 0.00 |$\textit{PNOIN}$| (|${\%}$|) 0.18 0.15 0.36 0.70 0.00 |$J$| (%) 12.14 10.81 6.17 0.62 0.00 |$VP$| 19.37 14.56 21.66 0.54 0.00 B. Correlation |$\textit{PNBO}$| |$\textit{PNBON}$| |$\textit{PNBO}_{\textit{ND}}$| |$\textit{PNOI}$| |$\textit{PNOIN}$| |$J$| |$\textit{PNBON}$| 0.71 |$\textit{PNBO}_{ND}$| 0.21 0.04 |$\textit{PNOI}$| 0.67 0.48 0.33 |$\textit{PNOIN}$| 0.58 0.59 0.23 0.90 |$\textit{J}$| 0.03 0.03 0.08 0.16 0.12 |$\textit{VP}$| –0.22 –0.15 –0.06 –0.07 –0.09 0.54 B. Correlation |$\textit{PNBO}$| |$\textit{PNBON}$| |$\textit{PNBO}_{\textit{ND}}$| |$\textit{PNOI}$| |$\textit{PNOIN}$| |$J$| |$\textit{PNBON}$| 0.71 |$\textit{PNBO}_{ND}$| 0.21 0.04 |$\textit{PNOI}$| 0.67 0.48 0.33 |$\textit{PNOIN}$| 0.58 0.59 0.23 0.90 |$\textit{J}$| 0.03 0.03 0.08 0.16 0.12 |$\textit{VP}$| –0.22 –0.15 –0.06 –0.07 –0.09 0.54 This table reports the summary statistics for the SPX options volume from public investors and pricing variables in the empirical analysis. |$\textit{PNBO}$|: public net open-buying volume of DOTM puts (|$K/S \leq 0.85$|). |$\textit{PNBON}$|: |$\textit{PNBO}$| normalized by average monthly public SPX volume over past 12 months (in million contracts). |$\textit{PNBO}_{ND}$|: public net open-buying volume of all SPX options excluding DOTM puts. |$\textit{PNOI}$|: public net open interest for DOTM SPX puts (in million contracts). |$\textit{PNOIN}$|: |$\textit{PNOI}$| normalized by the total public open interest of all options (long and short). |$J$|: monthly average of the daily physical jump risk measure by Andersen, Bollerslev, and Diebold (2007). |$VP$|: variance premium based on Bekaert and Hoerova (2014). AC(1) is the first-order autocorrelation for monthly time series; the pp test is the |$p$|-value for the Phillips-Perron test for unit root. The sample period is from January 1991 to December 2012. Figure 2 provides information about the trading volume of SPX options at different moneyness. Over our entire sample, put options account for 63% of the total trading volume of SPX options. Among put options, out-of-the-money puts account for over 75% of the total trading volume; in particular, DOTM puts (with |$K/S < 0.85$|) account for 23% of the total volume. These statistics demonstrate the importance of the market for DOTM SPX puts. Figure 2 View largeDownload slide Percentage of total put and call volumes at different moneyness This figure plots the total fraction of volume for calls and puts at different levels of moneyness (measure by strike price, |$K$|, divided by spot price |$S$|). The height of each bar indicates the fraction of the total market volume at that moneyness level, while the colors indicate the breakdown within the strike between public (blue) and private (red) orders. Figure 2 View largeDownload slide Percentage of total put and call volumes at different moneyness This figure plots the total fraction of volume for calls and puts at different levels of moneyness (measure by strike price, |$K$|, divided by spot price |$S$|). The height of each bar indicates the fraction of the total market volume at that moneyness level, while the colors indicate the breakdown within the strike between public (blue) and private (red) orders. Although financial intermediaries can partially hedge the risks of their option inventories through dynamic hedging, the hedge is imperfect and costly. This is especially true for DOTM SPX puts, because they are highly sensitive to jump risk, which is difficult to hedge. To demonstrate this point, we regress put option returns on the returns of the corresponding hedging portfolios at both weekly and daily horizons. We consider both delta hedging (using the S&P 500 index) and delta-gamma hedging.9 The |$R^2$| of these regressions demonstrates the effectiveness of the hedging methods. As Table 2 shows, with daily (weekly) rebalancing, delta hedging can capture around 72% (76%) of the return variation of ATM SPX puts, but only 41% (34%) of the return variation of DOTM puts. With delta-gamma hedging, the |$R^2$| for ATM puts can exceed 90%, but it is still below 60% for DOTM puts. These results imply that when holding nonzero inventories of DOTM SPX puts, financial intermediaries will be exposed to significant inventory risks even after dynamically hedging these positions. It is because of such inventory risks that financial intermediaries become more reluctant to supply crash insurance to the public investors when they are more constrained. Table 2 Explaining options returns with hedging portfolios |$\frac{K}{S}<0.85$| |$0.85<\frac{K}{S}<0.95$| |$0.95<\frac{K}{S}<0.99$| |$0.99<\frac{K}{S}<1.01$| delta delt+gam delta delt+gam delta delt+gam delta delt+gam Weekly |$R^2$| 0.34 0.54 0.45 0.75 0.59 0.86 0.76 0.91 Daily |$R^2$| 0.41 0.59 0.46 0.74 0.56 0.82 0.72 0.87 |$\frac{K}{S}<0.85$| |$0.85<\frac{K}{S}<0.95$| |$0.95<\frac{K}{S}<0.99$| |$0.99<\frac{K}{S}<1.01$| delta delt+gam delta delt+gam delta delt+gam delta delt+gam Weekly |$R^2$| 0.34 0.54 0.45 0.75 0.59 0.86 0.76 0.91 Daily |$R^2$| 0.41 0.59 0.46 0.74 0.56 0.82 0.72 0.87 This table shows the |$R^2$| from regressing option returns on hedging portfolios returns. The dependent variables are the returns of put options with different moneyness. delta denotes the returns on the delta hedging portfolio for the corresponding put option. delt+gam denotes the returns on the delta-gamma hedging portfolio. The sample period is from 1996 to 2012. Table 2 Explaining options returns with hedging portfolios |$\frac{K}{S}<0.85$| |$0.85<\frac{K}{S}<0.95$| |$0.95<\frac{K}{S}<0.99$| |$0.99<\frac{K}{S}<1.01$| delta delt+gam delta delt+gam delta delt+gam delta delt+gam Weekly |$R^2$| 0.34 0.54 0.45 0.75 0.59 0.86 0.76 0.91 Daily |$R^2$| 0.41 0.59 0.46 0.74 0.56 0.82 0.72 0.87 |$\frac{K}{S}<0.85$| |$0.85<\frac{K}{S}<0.95$| |$0.95<\frac{K}{S}<0.99$| |$0.99<\frac{K}{S}<1.01$| delta delt+gam delta delt+gam delta delt+gam delta delt+gam Weekly |$R^2$| 0.34 0.54 0.45 0.75 0.59 0.86 0.76 0.91 Daily |$R^2$| 0.41 0.59 0.46 0.74 0.56 0.82 0.72 0.87 This table shows the |$R^2$| from regressing option returns on hedging portfolios returns. The dependent variables are the returns of put options with different moneyness. delta denotes the returns on the delta hedging portfolio for the corresponding put option. delt+gam denotes the returns on the delta-gamma hedging portfolio. The sample period is from 1996 to 2012. 2.2 Option volume and the expensiveness of SPX options We start by investigating the link between |$\textit{PNBO}$| and the expensiveness of SPX options as proxied by the variance premium (|$VP$|) in Bekaert and Hoerova (2014). Before constructing the measure |$b_{VP,t}$| in Equation (3) for the price-quantity relation based on daily data, we first examine the relation between |$\textit{PNBO}$| and |$VP$| at monthly frequency. Table 3 reports the results. In both the cases of |$\textit{PNBO}$| and |$\textit{PNBON}$|, the coefficient |$b_{\textit{VP}}$| is negative and statistically significant, consistent with the hypothesis that shocks to intermediary constraints generate a negative relation between the equilibrium quantities of DOTM SPX puts that public investors purchase and the expensiveness of SPX options. The coefficient (-94.60) in the univariate regression suggests that a one-standard-deviation decrease in |$\textit{PNBO}$| is associated with an increase in |$VP$| of 4.82, a 25% increase relative to the average variance premium. Similarly, a one-standard-deviation decrease in |$\textit{PNBON}$| is associated with a 3.15 unit increase in |$VP$|. Table 3 |$\textit{PNBO}$| and SPX option expensiveness |$\textit{VP}_{t} = a_{\textit{VP}} + b_{\textit{VP}} \ \textit{PNBO}_{t} + c_{\textit{VP}} \ J_{t} \times \textit{PNBO}_{t} + d_{\textit{VP}} \ J_{t} + \epsilon^v_{t}$| |$\textit{PNBO}$| |$\textit{PNBON}$| |$\textit{PNBO}_{\textit{ND}}$| |$a_{\textit{VP}}$| 20.32*** –2.83 –4.80 21.05*** –1.88 –5.92 20.96*** –1.13 –7.08 (2.60) (3.05) (3.17) (3.32) (3.35) (3.45) (3.35) (4.23) (7.40) |$b_{\textit{VP}}$| –94.60** –101.57*** 13.16 –2.94** –3.24*** 4.51 –11.48 –20.22 11.43 (40.18) (24.13) (27.09) (1.39) (1.40) (2.80) (18.16) (13.73) (29.18) |$c_{\textit{VP}}$| –6.83*** –0.57** –2.33 (1.81) (0.24) (2.85) |$d_{\textit{VP}}$| 1.91*** 2.05*** 1.90*** 2.20*** 1.92*** 2.38*** (0.31) (0.32) (0.35) (0.34) (0.38) (0.75) |$R^2$| 4.6 34.1 36.8 1.7 30.9 34.5 0.0 29.5 30.1 |$\textit{VP}_{t} = a_{\textit{VP}} + b_{\textit{VP}} \ \textit{PNBO}_{t} + c_{\textit{VP}} \ J_{t} \times \textit{PNBO}_{t} + d_{\textit{VP}} \ J_{t} + \epsilon^v_{t}$| |$\textit{PNBO}$| |$\textit{PNBON}$| |$\textit{PNBO}_{\textit{ND}}$| |$a_{\textit{VP}}$| 20.32*** –2.83 –4.80 21.05*** –1.88 –5.92 20.96*** –1.13 –7.08 (2.60) (3.05) (3.17) (3.32) (3.35) (3.45) (3.35) (4.23) (7.40) |$b_{\textit{VP}}$| –94.60** –101.57*** 13.16 &nda
