How To Find Out How A Financial Market Risk Premium Affects Option Demand And Risk Premium

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1 Demand for Crash Insurance, Intermediary Constraints, and Stock Returns Hui Chen Scott Joslin Sophie Ni September 4, 213 Abstract We show that the net public demand for deep out-of-the-money S&P 5 put options predicts future market returns. A one standard deviation decrease in the net public demand measure is associated with a 3.6% rise in the subsequent 3-month market return. The predictive power of option demand is distinct from that of the many financial and macro variables in the literature, is particularly strong during the recent financial crisis, and is unique for deep out-of-the-money puts. Moreover, nets public demand is negatively related to expensiveness of these options relative to the at-the-money options, and that periods of low net public demand for crash insurance are linked to deleveraging by security broker-dealers. To explain these findings, we build a dynamic general equilibrium model of the crash insurance market, where financial institutions play a key role in sharing tail risks. The financial constraints they face affect their risk-sharing capacity, which in turn affect the market risk premium. Our model shows that the equilibrium demand for disaster insurance by public investors provides information of dealer constraints. Compared to the standard measures such as the broker-dealer leverage, our demand measure has the advantage of being forward-looking and available at higher frequency. Chen: MIT Sloan and NBER (huichen@mit.edu). Joslin: USC Marshall (sjoslin@usc.edu). Ni: Hong Kong University of Science and Technology (sophieni@ust.hk). We thank David Bates, Gary Katz, Bryan Kelly, Andrew Lo, Dmitriy Muravyev, Jun Pan, Martin Schneider, Ken Singleton, Hao Zhou, and seminar participants at USC, MIT Sloan, HKUST, the AFA Meeting in San Diego, the Consortium of Systemic Risk Aanalytics Meeting, the China International Conference in Finance, and SITE for comments.

2 1 Introduction According to the Bank for International Settlements, the size of the global exchange-traded derivatives market is 56 trillion US dollars as of June 212 (measured by notional principal amount), and the total turnover in the second quarter of 212 alone is $43 trillion. The size of the over-the-counter derivatives market is even bigger, at $64 trillion. Impressive as these numbers are, the quantities of derivative trading do not play any significant role in the modern no-arbitrage based option pricing theory. 1 Since options are assumed to be redundant securities that can be dynamically replicated using stocks and bonds, the theory is silent on the interactions between the equilibrium quantities of option trading and option prices. Moreover, the theory is silent on whether the options market has any real effect on the aggregate economy, or it is just a side show. In this paper, we study the dynamic interactions between the option trading quantities, option prices, and the aggregate market risk premium in relation to financial intermediary constraints. Our analysis focuses on the deep out-of-the-money (DOTM) put options on the market index, which are insurance against market crashes. These derivatives are particularly interesting for two reasons. First, tail risks in financial markets can have disproportionately large effects on asset prices. Since the risk exposure of DOTM index puts is concentrated on tail risk, understanding this market will help us understanding how tail risks are traded and priced in equilibrium. Second, while the dealers can and do hedge the inventory risk via futures and the OTC markets, the hedge is imperfect and costly, especially in regards to tail risk. Thus, how constrained these financial intermediaries are will affect their risk sharing capacity, which in turn affects the option prices, the equilibrium amount of trading in tail risk, and the aggregate risk premium in the economy. Using data from the CBOE that measure the net demand of the public investors on the S&P5 index options (SPX), we find that the net public demand for DOTM put options on the market index are negatively related to the expensiveness of the DOTM puts relative to the at-the-money (ATM) options, as measured by the difference in their implied volatilities. Moreover, we show that the net public demand for DOTM puts significantly 1 See the seminal work of Black and Scholes (1973) and Merton (1973). 1

3 predicts future stock returns. A one standard deviation drop in the net public demand of DOTM puts is associated with 3.57% increase in the subsequent 3-month market excess returns, and the R 2 of for the predictability regression is 18.4%. This predictive power of option demand is distinct from that of the standard predictive variables in the literature, such as price ratios, consumption-wealth ratio, and variance premium, which suggests that the information about market risk premium contained in option demand is not captured by standard macro or financial variables. The predictability is stronger for open buy orders, and it is unique for DOTM index put options. Moreover, the predictability is especially strong during the recent financial crisis. To account for these empirical findings, we build a dynamic general equilibrium model of the crash insurance market. Financial intermediaries (dealers) are net providers of such insurance under normal conditions because they are better at managing crash risk than the public investors, or because they are less concerned with crash risk due to agency problems. 2 We capture this feature in reduced form by assuming that the financial intermediaries are more optimistic about crash risk. Over time, the risk sharing capacity of the intermediaries changes, both due to endogenous trading losses and exogenous shocks to the intermediation capacity. Intuitively, these shocks can be generated by changes in capital requirements or uncertainties about government guarantees. In the model, public investors equilibrium demand for disaster insurance depends on the level of crash risk in the economy, the wealth distribution between public investors and the financial intermediary, and shocks to the intermediation capacity. As the probability of a market crash rises, all else equal, public investors demand for disaster insurance tends to rise. However, if the financial intermediaries risk sharing capacity drops at the same time, the equilibrium amount of risk sharing can become smaller. Furthermore, because of limited risk sharing, public investors will now demand a higher premium for bearing crash risk, which leads to higher risk premium in all financial assets that are exposed to such risks. 2 Examples include government guarantees to large financial institutions and compensation schemes that encourages managers to take on tail risk. See e.g., Lo (21), Malliaris and Yan (21), Makarov and Plantin (211). 2

4 Our paper builds on the work of Garleanu, Pedersen, and Poteshman (29), who develop a partial equilibrium model to show how demand pressure affects option prices when risk-averse market makers have to bear inventory risk. 3 In their model, demand for options is exogenously given and the dealer s intermediation capacity is fixed. As a result, the model implies a positive relation between the equilibrium demand for disaster insurance and the option premium. We introduce shocks to dealers intermediation capacity and endogenize the price and quantity of disaster insurance jointly in general equilibrium. In such a setting, the relation between the equilibrium demand for disaster insurance and the option premium can be either positive or negative. More broadly, our empirical evidence is consistent with the theories of intermediary constraints in asset pricing, Adrian and Shin (21), He and Krishnamurhty (212). As Cochrane (211) observes, such constraints act as a way to change the effective risk aversion of the dealers. Rather than modeling the effects of the constraint using first principle, we directly model shocks to intermediation capacity by making the dealer have time-varying risk aversion, which makes the model tractable and easier to take to the data. Studies that examine the theoretical empirical implications for intermediary-based asset pricing include Longstaff and Wang (28), Adrian, Moench, and Shin (21), Adrian, Etula, and Muir (212), Cheng, Kirilenko, and Xiong (212), among others. Hong and Yogo (212) find that open interests in commodity futures are pro-cyclical and predict commodity returns positively. They argue that the positive link between market interest and future returns is due to under-reaction to good news. Our measure of net public demand is different from open interest. If public investors only trade among themselves (e.g., due to heterogeneous beliefs or background risks, etc), there will be large open interest and zero net public demand for options. Thus, any net positive public demand we observe in the market has to correspond to short positions taken by the dealers, which is not true for open interest. Moreover, the return predictability we find for net public demand is negative (opposite to that of open interest in Hong and Yogo (212)), and the predictability varies with the moneyness of the option, a characteristic that does not exist in the futures markets. 3 The general idea of demand pressure effects goes back to Keynes (1923) and Hicks (1939). 3

5 To explain our empirical results based on the theory of underreaction would require that people buy insurance after they have seen market volatility going up recently, either due to Bayesian updating or overreaction, and at the same time markets under-react to such rising fear. A problem with this explanation is that our net public demand variable is negatively correlated with excess volatility. This explanation cannot account for the collapse in public demand as crash risk rises. Also, we show that the results do not apply to SPY options. Several studies have highlighted the role that the derivatives markets play in the aggregate economy. For example, Liu and Pan (23) study the end-user s demand for derivatives to hedge jump risk in a portfolio choice problem. Bates (28) shows how options can be used to complete the markets in the presence of crash risk. Chen, Joslin, and Tran (212) show that the risk premium in the economy is highly sensitive to the trading among agents who disagree on the probability of disasters. Buraschi and Jiltsov (26) use options to complete the markets in an economy with incomplete and heterogeneous information. In this paper, we introduce time-varying risk aversion of the market makers, which induces changes in their intermediation capacity and provides novel predictions about the relation between options demand and risk premium. 2 Empirical Evidence In this section, we present the empirical evidence connecting the net public demand for deep out-of-the-money put options on the S&P 5 index (which we refer to as DOTM index puts) to the relative expensiveness of the DOTM index puts and the risk premium of the aggregate stock market. 2.1 Data and variables The data used to construct our option demand measures are from the Chicago Board Options Exchange (CBOE) and the International Securities Exchange (ISE). The Options Clearing Corporation classifies each option trade into one of three categories based on 4

6 who initiates the trade. They include public investors (or customers), firm investors, and market makers. Volumes initiated by public investors include the trades made by retail investors (via discount brokerage firms) and those by institutional investors such as hedge funds or mutual funds (via full-service brokerage firms). Volumes initiated by firm investors correspond to the trades that the securities broker/dealers (who are not designated market makers) make for their own accounts or for another broker/dealer. The SPX options volume data are available at daily frequency from the beginning of 1991 to the end of 211. The SPY options volume data cover the period from February 25 to December 211. Option pricing data are obtained from the OptionMetrics for the period of January 1996 to December 211. Market excess returns are constructed based on the CRSP value-weighted market index. Our main option demand variable is NetBuyOpen, which is defined as the monthly public investor open-buy orders of all the DOTM index puts (with K/S.85) minus their open-sell orders on the same set of options. Since options are in zero net supply, the amount of net open-buying by the public investors is equal to the amount of net open-selling by the firm investors and market makers. Since our focus is to connect option market trading activities to the constraints of securities broker/dealers, it is reasonable to group the firm investors together with the market makers. Besides NetBuyOpen, we also consider three alternative demand measures for the SPX options: (i) NetBuyOpenDetrend, which removes a linear time trend from NetBuyOpen; (ii) NetBuy, which is the net monthly purchase of DOTM index puts by public investors including both open orders and close orders; (iii) FirmNetBuy, which is the net monthly purchase of DOTM index puts by firm investors. Finally, for comparison, we also construct the net demand variable based on SPY options, which are options with the underlying being the SPDR S&P 5 exchange-traded fund, which is designed to track the S&P 5 index. Unlike the SPX options which trade only on the CBOE, the SPY options are cross-listed at several option markets. The majority of the trading volume for SPY options is at the CBOE and the ISE. Thus, our SPY demand variable aggregates the volume data from both exchanges. 5

7 3 x 15 No Bailout 2 Lehman Number of contracts 1 1 Russian Asian Enron 911 G2 G1 Quant G3 Bear Sterns 2 3 Euro Debt Jan1995 Jan2 Jan25 Jan21 Figure 1: Time Series of net public demand for DOTM index puts. NetBuyOpen is the volume of public open purchased DOTM (K/S <.85) puts minus public open sold DOTM puts. Asian refers to the Asian financial crisis in Nov Russian is the Russian default in Oct Quant indicates the event that a host of quant-driven hedge funds experienced losses on the back of the subprime crisis in Aug 27. Bear Sterns refers to its collapse in Mar 28. Lehman indicates the event that Lehman Brothers reports a $2.8 billion second quarter loss in Jun 28, which leads to its bankruptcy in Sept 28. No Bailout refers to the House rejection of the bailout package in Sept 28. G1, G2 and G3 indicate three government rescue plans; G1 is the Term Asset-Backed Securities Loan Facility (TALF) in Nov 28, G2 is the assistance to Bank of America from Federal Reserve and the FDIC in Jan 29, and G3 is the increase of TALF to $ 1 trillion by the Federal Reserve in Feb 29. Euro Debt indicates early stage of the Euro debt crisis with a wave of downgrading of government debt in some European states in Dec 29. Figure 1 plots the time series of the NetBuyOpen measure. Consistent with the finding of GPP, the net public demand for DOTM index puts is positive most of the time prior to the recent financial crisis in 28, suggesting that the broker-dealers and market-makers were net providers of market crash insurance while public investors were the net buyers of the insurance. NetBuyOpen has also seen significant growth over time. From the period of January 1991 to December 21 (the end of the GPP sample), public investors on average bought 8, 392 DOTM index puts to open in a month. In the period from January 22 to December 26, the average monthly net-buying volume by public investors rises to 29,

8 However, starting in 27, NetBuyOpen becomes significantly more volatile. It starts to increase after a host of quant-driven hedge funds experienced significant loss in August 27, and peaks in October 28, right after the bankruptcy of Lehman Brothers (in September 28). Then, as market conditions continue to deteriorate, NetBuyOpen plunges rapidly and turns negative in most months from then to the end of the sample. Following a series of government actions, NetBuyOpen bottomed in April 29, rebounding briefly, then plunging again following the downgrade of the government debt of Greece and other European countries in December 29. During this period from November 28 to December 211, public investors on average sold 47, 715 DOTM index puts to open each month. Moreover, NetBuyOpen drops visibly following almost all the crisis events that are associated with the turmoil in the financial markets during our sample period. Examples include the Asian financial crisis in 1997, the Russian Default and LTCM crisis in 1998, the Quant crisis in 27, the Lehman bankruptcy in 28, and the European debt crisis in 29. In contrast, there s no big variations in demand for crash insurance during non-financial crises such as 9-11 and Enron. To measure the relative expensiveness of the DOTM index puts, we compute the implied volatility slope as the difference between the implied volatility of the DOTM index puts and the ATM index puts with one month to maturity. We also consider a range of existing measures of macroeconomic risks that have been shown to be connected to the market risk premium. These measures include the price to earning ratio (P/E) and the dividend yield (D/P) of the market portfolio, the consumption-to-wealth ratio (CAY) by Lettau and Ludvigson (21) (available quarterly), and the variance risk premium (IVRV) by Bollerslev, Tauchen, and Zhou (29) (available from 1996). In addition, we use the year-over-year change in broker-dealer leverage (dlev) from the Flow of Funds, which Adrian, Moench, and Shin (21) show to be capable of predicting the returns of a broad set of equity and bond portfolios. Figure 2 plots the time series of NetBuyOpen (Panel A) along with the aggregate risk measures (Panels B, C, D), the relative expensiveness of the DOTM index puts (IVSlope 7

9 4 x A. NetBuyOpen B. D/P C. CAY D. Variance premium: IV RV E. IV Slope: IV DOTM IV ATM 4 F. Market excess return: Rm t t Figure 2: NetBuyOpen, aggregate risk measures, expensiveness of DOTM index puts, and leading market risk premium. in Panel E), and the leading 3-month market excess returns (Rm t t+3 in Panel F). The figure shows that the dividend yield rises significantly during the 28 financial crisis, as did the variance premium measure. Table 1 reports the summary statistics of the variables used in our analysis. On average, the net public open-purchase of the DOTM index puts is 12, 15.3 contracts per month. If both the open and close orders are included, the net public purchase is contracts per month. In contrast, the firm investors on average sell contracts of the DOTM index puts per month, which offsets 18% of the average public buy orders, with the rest taken on by the market-makers. NetBuyOpen and NetBuy have first-order auto-correlation of.62 and.41 at the monthly frequency, respectively, which are significantly lower than the autocorrelation for the price-earnings ratio (P/E, AC(1) =.96), the dividend yield 8

10 (D/P AC(1) =.98), and the consumption-wealth ratio (CAY, AC(1) =.93 quarterly). Their p-values of the Phillips-Perron unit root test are both essentially zero. Table 2 presents the pair-wise correlations between the demand measures and other market and macro variables. As indicated by Figure 2, the statistical properties of some of these variables change during the 28 financial crisis. For this reason, we report the correlations for both the whole sample period from 1991 to 211, and the period before the crisis (from 1991 to 27). During both periods, the public demand for DOTM index puts are pro-cyclical, as indicated by the positive correlation with industrial production growth (.16) and negative correlation with unemployment rate (.48). In addition, the correlation between NetBuyOpen and the implied volatility slope is.46 for the whole sample. Its correlation with the leading 3-month market risk premium is.43. These results are consistent with the interpretation that dealer constraints matter for option pricing and market risk premium, which is the focus of our empirical analysis in the next section. In addition, the public demand and the firm demand (FirmNetBuy) are negatively correlated, especially prior to the crisis, suggesting firm investors (broker-dealers) in aggregate tend to be trading on the opposite side of public investors. 2.2 Main results We first investigate the link between public demand for DOTM index puts and the relative expensiveness of these options. The basic regression specification is: IV Slope t = a + b NetBuyOpen t + ɛ t, (1) which is similar to the specification in GPP, except that our option expensiveness measure and the option demand measure are both for DOTM index puts, whereas they construct the measures for options with all moneyness. Table 3 reports the regression results both for the full sample ( ) and the sample period of the GPP study ( ). In the full sample, NetBuyOpen is negatively and statistically significantly related to IVSlope contemporaneously, which is consistent 9

11 with the theory of intermediary constraints in that when the financial intermediaries become more constrained, they reduce their supply of DOTM index puts and demand a higher premium on these options. The coefficient of 11. implies that a one-standard deviation drop in NetBuyOpen raises the implied volatility of one-month DOTM index puts by.6% relative to the ATM puts. In contrast, during the earlier sample period, the coefficient on NetBuyOpen is positive but insignificantly different from zero. Next, following GPP, we include the recent profits and losses (P&L) for the marketmakers and firm investors in the index options. 4 To the extent that the P&L in the option markets proxy for how constrained the financial intermediaries are, the theory of intermediary constraints predicts that the regression coefficient on the intermediary P&L term should be negative (DOTM puts becoming more expensive after the intermediaries have suffered recent losses), while the coefficient on the interaction term between NetBuy- Open and intermediary P&L should be positive (the sensitivity of option expensiveness to NetBuyOpen is stronger after intermediary losses). Both predictions are confirmed in the data. Moreover, these results are robust to the inclusion of control variables including the variance premium (IVRV), contemporaneous market returns, and the lagged IVSlope. During the period of 1996 to 21, GPP find that the aggregate public demand for all index options are significantly positively correlated with their measure of the expensiveness of ATM options based on the difference between the implied volatility and a reference volatility from Bates (26). This positive relation is consistent with their demand-based option pricing theory: when the dealers are risk-averse and cannot perfectly hedge away their inventory risk, an exogenous increase in public demand will increase option prices. The fact that we find a positive (albeit insignificant) coefficient on NetBuyOpen in this period is loosely consistent with their finding. However, our results show that the relation between the relative expensiveness of DOTM puts and the net public demand changed significantly since then. This change is possibly due to the fact that the time-variation in the intermediary constraints becomes significantly more relevant in the latter part of the sample, which simultaneously drives option expensiveness and the endogenous public demand for DOTM index puts. 4 The results are similar if we only use the market-maker P&L. 1

12 Having documented the relation between the demand and the expensiveness of DOTM index puts, we now examine the ability of public demand on DOTM index puts to predict market excess returns. The basic regression specification is: Rm t+j t+k = a + b NetBuyOpen t + ɛ t+j t+k, (2) where the notation t + j t + k indicates the leading period from t + j to t + k (j < k). This return predictability regression is again motivated by the theory of intermediary constraints. First, it is possible that the options dealers become more constrained when the market risk premium is high, which reduces their capacity to provide market crash insurance to public investors. The result is a low NetBuyOpen today should predict high future market returns. Second, it is also possible that intermediary constraints directly affect the aggregate market risk premium rather than simply reflecting it. If the financial intermediaries play an important risk sharing role in the economy, then when they become more constrained, the market risk premium rises, which again results in a negative relation between NetBuyOpen and future market returns. Table 4 reports the regression results. NetBuyOpen has strong predictive power for future market returns up to 4 months ahead. The coefficient estimates are all negative and statistically significant when the dependent variables are the 1st, 2nd, 3rd or 4th month market excess returns. The coefficient estimate for predicting one-month ahead market excess return is (t-stat = 3.42), with an R 2 of 7.2%. For 4-month ahead returns (Rm t+3 t+4, or simply Rm t+4 ), coefficient estimate is (t-stat = 2.16), with an R 2 of 3.6%. When we aggregate the effect for the cumulative market excess returns in the next 3 months, the coefficient estimate of 7.62 implies that a one-standard deviation decrease in NetBuyOpen raises the future 3-month market excess return by 3.57%. The R 2 is an impressive 18.4%. Table 4 also reports the coefficient estimates on a set of macro and financial variables that have been shown to predict market returns in the literature. They include the variance risk premium (IVRV) in Bollerslev, Tauchen, and Zhou (29), the price-to-earnings ratio (P/E), the dividend yield (D/P), the consumption-wealth ratio (CAY) in Lettau and 11

13 Ludvigson (21), and the yearly change in broker-dealer leverage (dlev) in Adrian and Shin (21) and Adrian, Moench, and Shin (21). In the case of CAY and dlev, the regression is at the quarterly frequency. Only the variance premium and the dividend yield (marginally) remain statistically significantly related to future market excess returns when NetBuyOpen is included in the regression. The coefficient on NetBuyOpen is always negative and statistically significant. The fact that the predictive power of NetBuyOpen remains statistically significant after controlling for other standard predictive variables is important because it suggests that the information contained in option demand about future market returns is not captured by the standard measures of market risk premium. This potentially allows us to disentangle the two alternative explanations of the negative relation between NetBuyOpen and future market returns as discussed above. It suggests that the time variation in intermediary constraints may indeed have extra impact on the aggregate risk premium. It is also interesting that NetBuyOpen renders the change in broker-dealer leverage (dlev) insignificant in the quarterly regression. In a univariate regression, dlev predicts future market excess returns negatively. Adrian and Shin (21) argue that this result reflects the aggregate consequences of financial intermediary balance sheet adjustments. Our NetBuyOpen measure is positively correlated with dlev, suggesting that financial intermediaries have reduced risk-sharing capacity when they are deleveraging. Compared to the change in leverage, our option-demand measure is likely a better measure of intermediary constraints due to the fact that it is forward-looking and it is available at high (daily) frequency. The time variation in the predictive power of NetBuyOpen is illustrated in Figure 3, which compares the realized 3-month excess returns on the market portfolio against those predicted by option demand. The improvement in the model performance is quite visible in the period starting in 25. In particular, the NetBuy-predicted market excess returns closely replicated the major swings in the realized returns in the period post Lehman bankruptcy. Figure 4 presents the out of sample predictability of NetBuyOpen. We report the R 2 12

14 3 predicted realized Predicted vs. Realized 3-Month Market Excess Returns Figure 3: Return predictability in the time series We compare the predicted vs. realized 3-month excess returns on the market portfolio. based on various sample split date because recent forecast literature suggests that sample splits themselves can be data-mined (Hansen and Timmermann (212)). To demonstrate the robustness of out-of-sample forecasts to various sample splits, we plot out-of-sample three month return predictive R 2 as early as 1996, which uses only five years of data as a training sample. The latest split we consider is 28, which uses a 17 year training sample. The figure shows NetBuyOpen consistently achieves an out-of-sample R 2 above 1%, and climbs up to 2% after 23. It outperforms volatility premium (IVRV) in predicting 3 month returns across all sample splits. Since 1999 sample split, the R 2 s generated from NetBuyOpen are two times of those from IVRV. Table 5 reports the results of robustness tests with various forms of NetBuyOpen. The options market has experienced dramatic growth during our sample period. It is worth to check whether the predictability exists if we use detrended NetBuyOpen. Panel A shows that all the coefficient estimates on NetBuyOpenDetrend are negative and statistically significant. Table 5 also examines the predictability of the NetBuyOpen in two different subsamples, the period before the 28 financial crisis, and the first half of the sample 13

15 .3 PutBuyOpen IVRV.25.2 R Figure 4: Out-of-Sample R 2 by Sample Split Date, 3 Month Returns. Forcasts are based on the demand for crash insurance (NetBuyOpen) and volatility premium (IVRV) in Bollerslev, Tauchen, and Zhou (29). (ending December 21). The predictive power of the NetBuyOpen is significantly stronger in the second half of the sample than in the first half. The coefficient on the NetBuyOpen variable is not statistically significant in the first half period. This result is consistent with the lack of a significant relation between NetBuyOpen and the expensiveness of DOTM index puts during the same period (see Table 3). It could be due to the fact that intermediary constraint is not as significant and variable in the first half of the sample as in the second half (especially the crisis period). It could be due to the fact that the options market were under-developed in the early periods, which makes NetBuyOpen a more noisy measure of intermediary constraints. NetBuyOpen reflects the newly established demand for crisis insurance. The supply of crisis insurance from market makers should not only include the newly established positions, but also the changes in existing positions. To measure this supply, we compute NetBuy as the sum of deep OTM puts net open and close volumes from public investors. Panel B of Table 5 reports the coefficient estimates on various forms of NetBuy. We can see that both the raw and detrended measures of NetBuy have predictability for the market returns. Similarly to the NetBuyOpen, it has stronger predictability if we include 14

16 28 financial crisis, and predict market returns in the first half sample period and the period before the crisis with significant t-statistics. In Panel C of Table 5, we examine the predictability of other SPX option trading variables, including deep OTM put net buy volume from firm investors (FirmNetBuy- Open and FirmNetBuy), and the combined net buy volume of public and firm investors (PubFirmNetBuyOpen and PubFirmNetBuy). We can see that the Firm investors net demand predict returns positively. This also leads to weaker, but still negatively significant predictability of the combined volume. This result is consistent with that some of the firm investors are option dealers, and have positions opposite to the public investors. Next, Table 6 examines the predictability of public demand variables based on different time-to-maturity and moneyness of the options. The option demand measures based on shorter maturity options have statistically more significant predictive power for future 3-month market excess returns. As for moneyness, Table 6 shows that the predictive power of option demand is unique for deep out-of-the-money SPX puts with K/S.85. An alternative explanation for our finding is that the demand for crash insurance is a proxy for jump risks. Kelly (212) shows that jump tail risks are positive associated with future market returns. It is possible that the demand for crash insurance is negatively related to jump risks. Table 7 show the predictability of NeBuyOpen after controlling various measure of jump risks. To measure jump risks, we use the normalized tail risk in Kelly (212), various measures of slope of implied volatility of SPX options. The results show that none of those jump risk measures can predict market returns in our sample period from 1991 to 211. Table 7 also shows that open interest of deep OTM put can not predict market returns either. 2.3 Determinants of the demand for DOTM puts In this section, we investigate the determinants of demand for crash insurance. In our model, the dealers are less willing to provide crash insurance when they become more averse to jump risk. Such rise in risk aversion is a proxy for the fact the dealers are becoming more constrained. In this section, we present empirical evidence that the net 15

17 2 x NetBuyOpen and Leverage 15 NetBuyOpen Leverage 1 NetBuyOpen 5 Leverage Q3 1999Q2 22Q3 28Q3 Figure 5: Time Series Plot NetBuyDetrend v/s Leverage NetBuyOpen is the volume of public open purchased deep OTM (K/S <.85) puts minus public open sold deep OTM puts. Leverage is the quartly leverage of the security brokers and dealers from. public demand for crash insurance is indeed connected to dealer constraints. We first examine the relation between demand for crash insurance and dealer leverage. Periods of leverage expansions by the dealers are likely periods when they are less constrained, whereas deleveraging typically occurs when dealers become highly constrained. We use the quarterly broker-dealer leverage data from the Flow of Funds. Figure 5 plots the time series of the dealer leverage and the demand for crash insurance, and shows that indeed market maker sell (publice demand) more crash insurance during high leverage periods than low leverage periods, especially during the financial crisis. Table 8 shows that demand for crash is significantly positively related to changes in dealer leverage. It also reports that the public demand for DOTM index puts is negatively related to the slope of implied volatility of SPX options, but without statistical significance. In addition, from its positive relation with industrial productivity and negative relation with unemployment rate, we can see that the demand for crash insurance is high when the economy in a good state. 16

18 3 A Dynamic Model In Section 2, we document a number of features of the market for crash insurance. In particular, there is time-varying equilibrium demand for crash insurance from public investors. The equilibrium demand is inversely related to the relative price of the out-ofthe-money put protection times in which the equilibrium demand is low are generally times when the protection is very expensive. The demand for crash insurance was also informative about future stock market returns over and above the information in option prices and macro-variables. We now examine an equilibrium model consistent with these empirical facts. As our model elaborates, the main mechanism we have in mind is a model whereby a public sector and intermediary face time-varying risk of a disaster. In general, the intermediary is more willing to bear the downside risk of the disaster. However, as the amount of risk rises, the intermediary becomes less willing (or less able) to share the risk of a crash. These ingredients allow us to capture our key empirical results. 3.1 A Simple Model with Intermediary Constraints We first consider a simple model that illustrates the main mechanism for the negative relationship between public demand for out-of-the-money options and subsequent market returns. The key feature of our model will be that as the amount of risk rises, the public demand curve will (necessarily) shift up, but the equilibrium demand will go down. This will follow because the dealer s supply curve will shift up more and the market clearly price will imply the lower equilibrium quantity. In our simple model, this will follow because the dealer faces a tighter capital constraints. This suggests that the tightness of dealer constraints can be understood in reduced form as time-variation in effective risk aversion, an observation made by Adrian and Shin (21), He and Krishnamurhty (212), Cochrane (211), and others. To illustrate the idea, consider a two-period model with two agents: a public investor and a dealer. We suppose that both agents have power utility over wealth in the second 17

19 Supply λ=1% Demand λ=1% Supply λ=3% Demand λ=3% Supply λ=3%, γ= quantity price Figure 6: Dealer constraint and derivative supply. This figure plots the equilibrium supply of the dealer and demand of the public investors when the dealer faces a CVaR constraint. The dealer is endowed with fixed wealth W = 2. The public investor is endowed with a wealth of either WH P = 4 or W L P = 2. The dealer s expected shortfall at the q = 1% level is capped at c = 5%. The other parameters of the model are λ = 1% (low crash risk) or 3% (high crash risk) and the relative risk aversion for the public and market makers are both γ = 3. The dotted line plots the supply of an unconstrained market maker with γ = 7.6, which gives rise to the same equilibrium as the constrained case with γ = 3. (final) period, with constant relative risk aversion γ. There are two possible states in the second period: a good state and a bad state, which occur with probability 1 λ and λ, respectively. The public investor receives a lower endowment in the bad state than in the good state (WH P > W L P ), while the dealer s endowment is riskless (W H D = W L D = W ). The heterogeneity in background risk generates the motive for trading in the form of the dealer writing insurance to the public investor against the bad state. Without loss of generality, we assume the insurance contract is an Arrow-Debreu security that pays off 1 unit of wealth in the bad state, and that the riskfree rate is normalized to. We also denote the number of insurance contracts the dealer sells by n, and the price of the contract by p. Next, we assume that the dealer faces an exogenous constraint on his total risk exposure. Specifically, the constraint is that the conditional Value-at-Risk (CVaR) at level q cannot exceed a fraction c of his wealth. The CVaR, which is also referred to as the expected 18

20 shortfall, is defined as the average value-at-risk (VaR) with confidence level from to q: ES q = E[loss being in worst q% tail] = 1 q q V ar α dα. (3) The resulting equilibrium is plotted in Figure 6, where we consider the cases of a low and high probability of disaster. We see that as the amount of risk rises, the demand curve for the public rises. However, at the same time the supply curve falls and the CVaR constraint begins to bind. The result is that as risk rises, the equilibrium quantity falls. Moreover, as indicated by the dashed black line, the same equilibrium quantity would be obtained if instead of the constraint binding more with higher levels of risk, the dealer was instead more risk averse as the amount of risk went up. This simple comparative static exercise shows that the relationship between the about of risk and the amount of trade depends crucially on how the risk-sharing capacity of the dealer changes with the level of crash risk in the economy. In reality, the dealers are large financial institutions, and many factors could changes their risk-sharing capacity, including losses in wealth from other investments, regulatory changes on capital requirement, and beliefs about government guarantees. Another observation from this example is that we can arrive at the same equilibrium if instead of imposing the CVaR constraint, we assume the dealer s risk aversion rises as the crash risk increases. It is worth noting that this mechanism differs from other studies that focus on market makers or arbitrageurs to share risk with public investors with exogenous stochastic demand. For example, in Garleanu, Pedersen, and Poteshman (29), demand is specified as an exogenous process. Vayanos and Vila (29) and Greenwood and Vayanos (212) also focus on public investors who have exogenous demand curves. In these papers, public investors are models as having flat demand curve (or alternatively as having exogenously specified equilibrium demand). Such approaches cannot immediately reconcile our empirical finding without introducing the possibility or correlation between the risk bearing capacity of the market makers with the public demand. 19

21 3.2 A Full Dynamic Model We now present a dynamic model for the market of disaster insurance. Our model builds on Chen, Joslin, and Tran (212) which is based on disagreement about a time-varying disaster probability. We extend their model by incorporating time-variation in the dealer s aversion to crash risk. Similar alternative models could be based purely on time-varying risk aversion or dealer constraints. We consider an aggregate endowment in the economy which follows a jump diffusion process where the endowment is subject to both a diffusive risk and a jump risk. In particular, sudden severe drops in the aggregate endowment are a source of disaster risk in this economy. There are two types of agents in the economy: small public investors and competitive dealers. We assume there exists a representative public investor, who is denoted by agent P, and a representative dealer, denoted by agent D. To induce the two types of agents to trade, we assume that they have different beliefs about the probability of disasters. As discussed earlier, such differences in beliefs capture in reduced form the advantages that dealers have in bearing disaster risk, whether it is due to differences in technology, agency problems, or behavioral biases. Specifically, we assume that both agents believe that the log aggregate endowment c t = log C t follows the process dc t = ḡdt + σ c dw c t d dn t (4) where ḡ and σ c are the expected growth rate and volatility of consumption without jumps, W c t is a standard Brownian motion under both agents beliefs, d is the constant size of consumption drop in a diaster 5. N t is a counting process whose jumps arrive with stochastic intensity λ t under the public investors beliefs, dλ t = κ( λ λ t )dt + σ λ λt dw λ t, (5) 5 As in Chen, Joslin, and Tran (212), one could generalize the model by allowing disaster size to have a time-invariant distribution. 2

22 where λ is the long-run average jump intensity under P s beliefs, and Wt λ is a standard Brownian motion independent of W c t. In general, the dealers are more willing to bear the disaster risk because (they act as if) they are more optimistic about disaster risk. We assume that they believe that the disaster intensity is given by ρλ t with ρ < 1. We summarize the public investors beliefs with the probability measure P P, and the dealers beliefs with the probability measure P D. Public investors have standard constant relative risk aversion (CRRA) utility: U P = E P [ ] C1 γ δt P,t e 1 γ dt, (6) where we focus on the cases where γ > 1. The superscript P reflects that the expectations are taken under the public investors beliefs. The utility function of the dealers are different: U D = E D [ ] C1 γ δt D,t Nt e 1 γ ea n=1 (λτ(n) λ) dt, (7) where the additional exponential term in the utility function captures the dealers timevarying risk aversion against market crashes. The specification generalizes the statedependent preferences proposed by Bates (28) in that it allows the dealers risk aversion against crashes to rise with the probability of disasters. Specifically, τ(n) is the time of the n th disaster since t =, τ(n) inf{s : N s = n}. 6 Thus, this crash-aversion term remains constant in between disasters. Suppose the dealer s log consumption drops by d D,τ(n) at the time of the n th disaster. Then, at the same time, the marginal utility of the dealer jumps up by e γd D,τ(n)+a(λ τ(n) λ) = e ( ) γ+ a(λ τ(n) λ) d d D,τ(n) D,τ(n), 6 The formulation using λ τ(n) as opposed to λ t ensures that the premium for bearing brownian risk will be unaffected by variation in the disaster intensity. 21

23 which implies that the dealer s effective relative risk aversion against the disaster is γ D,τ(n) = γ + a(λ τ(n) λ) d D,τ(n). (8) Thus, when a >, the dealers will have higher aversion to disaster risk than public investors when the conditional disaster intensity λ t is higher than average. The dealers effective risk aversion keeps rising as λ t becomes larger, and the parameter a controls how fast the risk aversion rises. For example, when the dealers owns all the wealth in the economy and the disaster intensity is 1% above its steady state mean, the dealers effective risk aversion against disasters will be γ +.1a/ d. The main motivation for dealers time-varying aversion to crash risk is the time-varying constraint faced by financial intermediaries. Rising crash risk in the economy raises the intermediaries capital/collateral requirements and tightens their constraints on tail risk exposures (e.g., Value-at-Risk constraints), which make them more reluctant to provide insurance against disaster risk. For example, see Adrian and Shin (21), He and Krishnamurhty (212). In this sense, the shocks to the disaster intensity in the model also serve the purpose of generating time variation in the intermediation capacity of the dealers. We can further generalize the specification by making the dealers aversion to crash risk driven by adding independent variations in the intermediation shocks. We also assume that markets are complete and agents are endowed with some fixed share of aggregate consumption (θ P, θ D = 1 θ P ). The equilibrium allocations can be characterized as the solution of the following planner s problem, specified under the probability measure P P, max C P t, CP t E P [ e δt (CP t ) 1 γ 1 γ + ζη te δt (CD t ) 1 γ e a Nt n=1 (λ τ(n) λ) 1 γ dt ], (9) subject to the resource constraint C P t + C D t = C t. Here, ζ is the the Pareto weight for the dealers and η t dp D dp P = ρ Nt e (1 ρ) t λsds. (1) 22

24 where ρ = λ D /λ, the relative likelihood of a jump under the two beliefs. From the first order condition and the resource constraint, we obtain the equilibrium consumption allocations Ct P = f P ( ζ t )C t and Ct D = (1 f P ( ζ t ))C t, where ζ t = ρ N t e (1 ρ) t λsds+α N t n=1 (λ τ(n) λ) ζ (11) and f P ( ζ) = ζ. (12) 1 γ The stochastic discount factor under P s beliefs, M P t, is given by M P t = e ρt (C P t ) γ = e δt f P ( ζ t ) γ C γ t. (13) We can solve for the Pareto weight ζ through the lifetime budget constraint for one of the agents (Cox and Huang (1989)), which is linked to the initial allocation of endowment. Our key focus will be on risk premiums and on the net public demand for disaster insurance which we relate to the market for deep out of the money puts in our empirical analysis. The risk premium for any security under each agent s beliefs is the difference between the expected return under P i and under the risk-neutral measure Q. E i t[r e ] = γσ c B P + (λ i t λ Q t )E d t [R], i = D, P, (14) where we use the shorthand that B P denotes the sensitivity of the security to Brownian shocks and Et d [R] is the expected return of the security conditional on a disaster. Since consumption will be relatively smooth in our calibration, the return of securities which are not highly levered on the brownian risk will be dominated by the jump risk term. Moreover, agents agree about the brownian risk and have the same risk aversion with respect to these shocks so there will be no variation in the Sharpe ratio for brownian risk. In light of these facts, we focus on the variation in the jump risk premium, as measured by λ Q /λ P P. The stochastic discount factor characterizes the unique risk neutral probability measure 23

25 Q (see, e.g., Duffie 21). The risk-neutral disaster intensity λ Q t Et d [Mt i ]/Mt i λ i t is determined by the expected jump size of the stochastic discount factor at the time of a disaster. When the risk-free rate and disaster intensity are close to zero, the risk-neutral disaster intensity λ Q t has the nice interpretation of (approximately) the value of a one-year disaster insurance contract that pays one at t + 1 when a disaster occurs between t and t + 1. In our setting, the risk-neutral jump intensity is given by λ Q t = e γ d ( 1 + (ρ ζt ) 1 γ ) γ (1 + ζ 1 γ t ) γ λ t (15) In order to define the market size, we must consider how the pareto efficient allocation is obtained. The equilibrium allocations can be implemented through competitive trading in a sequential-trade economy. Extending the analysis of Bates (28), we can consider four types of traded securities: (i) a risk-free money market account, (ii) a claim to aggregate consumption, and (iii) a disaster insurance contracts which pay green one dollar in the event of a disaster in exchange for a continuous premium. and (iv) a separate instrument sensitive only to shocks in the disaster intensity. As in Chen, Joslin, and Tran (212), since agents agree about the Brownian risk and have identical aversion to the risk, they will proportionally hold the risk according to their consumption share. With the instruments we have specified, this means they will proportionally hold the consumption claim. Thus the agents will hold proportional exposure to the disaster risk from their exposure to the consumption claim. Motivated by these facts, we define the net public demand for disaster insurance as the (scaled) difference between the consumption loss the public bears in equilibrium minus the consumption loss that the public would bear without insurance. That is, the public demand for insurance is the difference between e d(f P ( ζ t d ) f P ( ζ t )) (where ζ t d is the value of ζ t conditional on a disaster occurring at time t: ζ t d = ρe α(λ t λ) ζt ) and e d(f P ( ζ t d ) f P ( ζ t ). Thus we define the net public demand for insurance to be net public demand for disaster insurance = e d ( f P ( ζ t ρe α(λ t λ) ) f P ( ζ t ) ). (16) 24