The Supply and Demand of S&P 500 Put Options

The Supply and Demand of S&P 500 Put Options George Constantinides University of Chicago Lei Lian University of Massachusetts at Amherst October 28 2015 George Constantinides; Lei Lian The Supply and Demand of S&P Put Options October 28 2015 1 / 38

Overview We introduce endogenous supply shifts, in addition to demand shifts, in the market for S&P 500 put options The principal writers of index puts are risk neutral market makers who face a credit constraint, modeled as a VaR constraint The principal buyers of index puts are risk averse customers which buy the index to maximize their expected utility and hedge their exposure to downside risk by buying index puts Our model captures the scenario where market makers write overpriced index puts and portfolio managers buy them explains a novel sets of empirical evidence on the net buy and prices of put options George Constantinides; Lei Lian The Supply and Demand of S&P Put Options October 28 2015 2 / 38

Definitions Risk-Neutral (RN) Volatility: Britten-Jones and Neuberger (2000) Disaster Index: the difference between the RN realized variation and RN variance is a function of the disaster risk; as in Du and Kapadia (2012) ATM Puts: Model: K /S = 1 Data: 0.97 K /S < 1.03 OTM Puts: Model: K /S = 0.85 Data: 0.8 K /S < 0.9 George Constantinides; Lei Lian The Supply and Demand of S&P Put Options October 28 2015 3 / 38

Definitions, continued Net Buy: the average of daily executed total buy orders by public customers and firms to open new positions or close existing ones during the month minus their average daily executed total sell orders Implied Volatility Skew: the IV of 1-month (15-60 day) OTM puts minus the IV of 1-month ATM puts George Constantinides; Lei Lian The Supply and Demand of S&P Put Options October 28 2015 4 / 38

Motivation-1 Net buy of puts: increased before the financial crisis and sharply decreased during the crisis Demand pressure theory: opposite to the above facts Our model: explains the net buy by introducing a funding constraint and therefore supply shift The supply of puts by market makers decreased during the financial crisis because of the market makers tighter funding constraint The supply curve shifted to the left and the demand curve shifted to the right The supply shift turns out to be the driving factor in the decrease in the equilibrium net buy of puts George Constantinides; Lei Lian The Supply and Demand of S&P Put Options October 28 2015 5 / 38

Liabilities-to-Assets Ratio of Broker-Dealers Liability/Asset Ratio 1.010 1.020 1.030 02 1996 02 1998 02 2000 02 2002 02 2004 02 2006 02 2008 02 2010 02 2012 George Constantinides; Lei Lian The Supply and Demand of S&P Put Options October 28 2015 6 / 38

Motivation-2 Net buy of puts: decrease in the RN variance and disaster index The model implies when the RN variance and/or disaster index increase, public customers like to buy more puts as crash insurance but market makers become more credit-constrained The supply curve shifts to the left and the demand curve shifts to the right The supply shift turns out to be the driving factor in the decrease in the equilibrium net buy of puts George Constantinides; Lei Lian The Supply and Demand of S&P Put Options October 28 2015 7 / 38

Motivation-3 Net buy of OTM and ATM puts: decrease in their price The model implies both the supply and demand curves shift The supply shift turns out to be the dominant factor in the decrease in the equilibrium net buy and the put price increase George Constantinides; Lei Lian The Supply and Demand of S&P Put Options October 28 2015 8 / 38

Motivation-4 The IV skew of S&P 500 index puts: non-decrease in the disaster index and RN variance Widely used no-arbitrage models: the skew is decreasing in the disaster index and RN variance Our model recognizes as the disaster risk and variance increase, customers demand more puts as insurance while market makers become more credit-constrained in writing puts The resulting increase in the equilibrium price is more pronounced in OTM than in ATM puts because the credit constraint is more sensitive to OTM than ATM puts. The IV skew becomes steeper George Constantinides; Lei Lian The Supply and Demand of S&P Put Options October 28 2015 9 / 38

Motivation-4 Dealers and intermediaries credit constraints and funding liquidity Adrian and Shin (2014), Bates (2003), Brunnermeier and Pedersen (2009), Danielsson, Shin, and Zigrand (2004), Etula (2013), Gromb and Vayanos (2002), He and Krishnamurthy (2013), Shleifer and Vishny (1997), and Thurner, Farmer, and Gaenakoplos (2012) Put demand pressure: Gârleanu, Pedersen, and Poteshman (2009) Market makers risk aversion and credit constraints in reduced form: Chen, Joslin, and Ni (2014) George Constantinides; Lei Lian The Supply and Demand of S&P Put Options October 28 2015 10 / 38

A Model of the Supply and Demand for Index Put Options The stock price at the beginning of the month: exogenous and is normalized to one at the end of the month, the index price is: e µ+σɛ, ɛ (0, 1) in the no-disaster state e µ J +σ J ɛ J, ɛ J (0, 1) in the disaster state that occurs with prob. p. George Constantinides; Lei Lian The Supply and Demand of S&P Put Options October 28 2015 11 / 38

Customers Demand for Puts Customers (and firms) are risk averse. They invest in the index and riskless asset buy puts to hedge their down-side risk The customer has an initial wealth W 0, buys α shares of stock and β puts, invests α βp units of the numeraire in bonds max α,β E[U] where U = W 0 α βp + αs + β[k S] + A 2 (W 0 α βp + αs +β[k S] +) 2 A: the absolute risk aversion coefficient Their expected one-month utility maximization provides the endogenous demand for puts George Constantinides; Lei Lian The Supply and Demand of S&P Put Options October 28 2015 12 / 38

Market Makers Supply of Puts Market makers are risk neutral. They invest in the index and riskless asset write puts for profit face a VaR credit constraint The market maker has zero endowment, buy ˆα shares stock, buy ˆβ puts max E [ˆα(S 1) + ˆβ([K S] + P) ] ˆα, ˆβ subject to the exogenous VaR constraint prob {ˆα(S 1) + ˆβ ( [K S] + P ) < W } h Their expected one-month profit maximization provides the endogenous supply of puts George Constantinides; Lei Lian The Supply and Demand of S&P Put Options October 28 2015 13 / 38

Calibration The length of one period is 1 month Parameters: p: 0.04-0.16, implies 0.48-1.92 expected disasters p.a. σ: 0.02-0.14, implies volatility 0.07-0.48 p.a. µ: 0.005 implies equity premium 6% p.a. in the no-disaster state µ J : -0.04 σ J : 80/ 12, implies vol 80% p.a. of the equity premium in the disaster state For this range of parameters, the equity risk premium is 2.86% - 17.04% p.a. and the volatility is 7.38% - 45.01% Customer s initial wealth 500 and preference parameter 0.001 imply RRA=1 VaR threshold -20 (loss) and VaR probability 1% George Constantinides; Lei Lian The Supply and Demand of S&P Put Options October 28 2015 14 / 38

Supply and Demand σ = 0.04, p = 0.05 option option price price option option price price 0.018 0.018 0.020 option 0.020 option price 0.022 price 0.022 0.024 0.024 0.022 0.022 0.024 0.024 option 0.026 option price 0.026 price 0.028 0.028 0.030 0.030 0.018 0.018 0.020 0.020 0.022 0.022 0.024 0.024 Demand/Supply (ATM) (ATM) Demand/Supply (OTM) (OTM) Demand/Supply (ATM) (ATM) Demand/Supply (OTM) (OTM) Demand Demand Supply Supply Demand Demand Supply Supply 3000 3000 1000 10000 010001000 0.0018 0.0018 0.0020 option 0.0020 option price 0.0022 price 0.0022 Demand Demand Supply Supply Demand Demand Supply Supply 0 200 0 200 600 600 1000 1000 3000 3000 1000 10000 010001000 0 200 0 200 600 600 1000 1000 Quantity Quantity Quantity Quantity 0.022 0.022 0.024 0.024 0.026 0.026 0.028 0.028 0.030 0.030 Quantity Quantity Demand/Supply (ATM) (ATM) Demand/Supply (ATM) (ATM) Demand Demand Supply Supply Demand Demand Supply Supply 2500 2500 1500 1500 500 5000 500 0 500 2500 2500 1500 1500 500 5000 500 0 500 Quantity Quantity Quantity Quantity option option price price option option price price 0.0036 0.0036 0.0040 option 0.0040 option price 0.0044 price 0.0044 0.0018 0.0018 0.0020 0.0020 0.0022 0.0022 σ = 0.04, p = 0.10 0.0036 0.0036 0.0040 0.0040 0.0044 0.0044 Quantity Quantity Demand/Supply (OTM) (OTM) Demand/Supply (OTM) (OTM) Demand Demand Supply Supply Demand Demand Supply Supply 500 500 0 0 500 500 500 500 0 0 Quantity Quantity 500 500 Quantity Quantity George Constantinides; Lei Lian The Supply and Demand of S&P Put Options October 28 2015 15 / 38

Data Sources 1996-2012 CBOE: daily buy, and sell contracts for customers and firms small customer (<100 contracts), medium (100-200), and large (>200) CBOE: intra-day trades and bid-ask quotes of the S&P 500 options. select the last pair of bid-ask quotes at or before 14:45 CDT Tick Data Inc: Minute-level price of the S&P 500 index match the option quotes with the tick-level index price at the same minute. Federal Reserve s Flow of Funds database : broker-dealers liabilities-to-assets ratio George Constantinides; Lei Lian The Supply and Demand of S&P Put Options October 28 2015 16 / 38

Net Buy, RN Variance, Disaster Index NetBuy (OTM) 0.05 0.00 0.05 0.10 RN Variance 0.00 0.10 0.20 0.30 01 1996 01 1998 01 2000 01 2002 01 2004 01 2006 01 2008 01 2010 01 2012 01 1996 01 1998 01 2000 01 2002 01 2004 01 2006 01 2008 01 2010 01 2012 NetBuy (ATM) 0.00 0.05 0.10 0.15 0.20 Disaster Index 0.000 0.010 0.020 01 1996 01 1998 01 2000 01 2002 01 2004 01 2006 01 2008 01 2010 01 2012 01 1996 01 1998 01 2000 01 2002 01 2004 01 2006 01 2008 01 2010 01 2012 George Constantinides; Lei Lian The Supply and Demand of S&P Put Options October 28 2015 17 / 38

Model-Implied Net Buy of OTM Puts vs the Disaster 0.05 0.050.10 0.100.15 0.150.20 0.200.25 0.25 0.05 0.050.10 0.100.15 0.150.20 0.200.25 0.25 Index and RN RV Variance RV Variance RV Variance RV Variance 140 140 σ = 0.04 σ = 0.08 130 13 N N N N 130 13 Net Buy(OTM) 100 Net 300 Buy(OTM) 500 700 100 300 500 700 Net Buy(OTM) 150 Net Buy(OTM) 250 350 150 250 350 0.0010 0.0010 0.0020 0.0020 0.0030 0.0030 0.0010 0.0010 0.0015 0.0015 0.0020 0.0020 0.0025 0.0025 Disaster Disaster Index Index Disaster Disaster Index Index p = 0.06 p = 0.10 Net Buy(OTM) 140 Net 180 Buy(OTM) 220 140 180 220 Net Buy(OTM) 130 135 Net Buy(OTM) 140 145 150 130 135 140 145 150 0.05 0.050.10 0.10.15 0.150.20 0.20.25 0.25 0.05 0.050.10 0.10.15 0.150.20 0.20.25 0.25 RV Variance RV Variance RV Variance RV Variance George Constantinides; Lei Lian The Supply and Demand of S&P Put Options October 28 2015 18 / 38

Model-Implied Net Buy of ATM Puts vs the Disaster 0.05 0.050.10 0.100.15 0.150.20 0.200.25 0.25 0.05 0.050.10 0.100.15 0.150.20 0.200.25 0.25 Index and RN RV Variance RV Variance RV Variance RV Variance 85 85 σ = 0.04 σ = 0.08 76 N N N N 76 Net Buy(ATM) 70 80 Net 90 Buy(ATM) 100 120 70 80 90 100 120 Net Buy(ATM) 70 80 Net 90 Buy(ATM) 100 110 70 80 90 100 110 0.0010 0.0010 0.0020 0.0020 0.0030 0.0030 Disaster Disaster Index Index 0.0010 0.0010 0.0015 0.0015 0.0020 0.0020 0.0025 0.0025 Disaster Disaster Index Index p = 0.06 p = 0.10 Net Buy(ATM) 85 Net 90 Buy(ATM) 95 100 85 90 95 100 Net Buy(ATM) 76 Net 78 Buy(ATM) 80 82 76 78 80 82 0.05 0.050.10 0.100.15 0.150.20 0.200.25 0.25 RV Variance RV Variance 0.05 0.050.10 0.100.15 0.150.20 0.200.25 0.25 RV Variance RV Variance George Constantinides; Lei Lian The Supply and Demand of S&P Put Options October 28 2015 19 / 38

Observed Net Buy of OTM Puts versus the Disaster Index and the RN Variance George Constantinides; Lei Lian The Supply and Demand of S&P Put Options October 28 2015 20 / 38

Observed Net Buy of ATM Puts versus the Disaster Index and the RN Variance George Constantinides; Lei Lian The Supply and Demand of S&P Put Options October 28 2015 21 / 38

Discussion: The Results Are Consistent with the Model Net buy of OTM puts: significantly decreases in both the disaster index and the RN variance in the full period and subperiods Net buy of ATM puts: decreases in both the disaster index and the RN variance in the full period and subperiods but some regression coefficients are insignificant, which is also consistent with the model implications. George Constantinides; Lei Lian The Supply and Demand of S&P Put Options October 28 2015 22 / 38

Net Buy ( 140 Net 180 Buy ( Model-Implied Net Buy of OTM Puts vs the Price of Puts 0.40 0.40 0.45 0.45 0.50 0.50 0.40 0.40 0.44 0.44 0.48 0.48 0.52 0.52 140 180 Net Buy ( σ = 0.04 σ = 0.08 130 135 Net Buy 140 ( B S B S IV IV B S B S IV IV 130 135 140 Net Buy (OTM) 100 Net 300 Buy 500 (OTM) 700 100 300 500 700 Net Buy (OTM) 150 Net Buy 250 (OTM) 350 150 250 350 0.34 0.34 0.36 0.36 0.38 0.38 0.40 0.40 0.42 0.42 0.44 0.44 B S B S IV IV 0.36 0.36 0.38 0.38 0.40 0.40 0.42 0.42 0.44 0.44 B S B S IV IV p = 0.06 p = 0.10 Net Buy (OTM) Net 180 Buy (OTM) 220 140 140 180 220 Net Buy (OTM) 135 Net Buy 140 (OTM) 145 130 150 130 135 140 145 150 0.40 0.45 0.50 0.40 0.45 0.50 B S B S IV IV 0.40 0.44 0.48 0.52 0.40 0.44 0.48 0.52 B S B S IV IV George Constantinides; Lei Lian The Supply and Demand of S&P Put Options October 28 2015 23 / 38

Net Buy ( 85 Net Buy 90 ( Model-Implied Net Buy of ATM Puts vs the Price of Puts 0.1 0.1 0.2 0.2 0.3 0.3 0.4 0.4 0.5 0.5 0.2 0.2 0.3 0.3 0.4 0.4 0.5 0.5 Net Buy ( σ = 0.04 σ = 0.08 76 Net Buy 78 ( B S B S IV IV B S B S IV IV Net Buy (ATM) 70 80 Net 90 Buy 100 (ATM) 120 Net Buy (ATM) 70 80 Net Buy 90 (ATM) 100 110 85 90 76 78 70 80 90 100 120 70 80 90 100 110 0.16 0.160.18 0.180.20 0.200.22 0.220.24 0.24 B S B S IV IV 0.30 0.32 0.34 0.36 0.30 0.32 0.34 0.36 B S B S IV IV p = 0.06 p = 0.10 Net Buy (ATM) 85 Net Buy 90 (ATM) 95 100 85 90 95 100 Net Buy (ATM) Net Buy 78 (ATM) 80 76 82 76 78 80 82 0.1 0.2 0.3 0.4 0.5 0.1 0.2 0.3 0.4 0.5 B S B S IV IV 0.2 0.3 0.4 0.5 0.2 0.3 0.4 0.5 B S B S IV IV George Constantinides; Lei Lian The Supply and Demand of S&P Put Options October 28 2015 24 / 38

Observed Net Buy of OTM and ATM Puts vs the Price of Puts (in IV Units) George Constantinides; Lei Lian The Supply and Demand of S&P Put Options October 28 2015 25 / 38

Discussion: The Results Are Consistent with the Model Net buy of OTM puts: significantly decreases in their price Net buy of ATM puts: insignificantly decreases in their price George Constantinides; Lei Lian The Supply and Demand of S&P Put Options October 28 2015 26 / 38

The Net Buy of Puts versus the Market Makers Constraint σ = 0.04, p = 0.06 Net Net Buy Buy (OTM) 0 200 200 600 600 1000 1000 0 Net Net Buy Buy (ATM) 0 100 100 200 200 300 300 400 400 0 Net Net Buy Buy (OTM) 0 100 100 300 300 500 500 0 80 60 40 20 0 80 60 40 20 0 80 60 40 20 0 80 60 40 20 0 W* W* W* W* σ = 0.08, p = 0.10 Net Net Buy Buy (ATM) 50 150 150 250 250 0 50 350 350 0 80 60 40 20 0 80 60 40 20 0 W* W* 80 60 40 20 0 80 60 40 20 0 W* W* George Constantinides; Lei Lian The Supply and Demand of S&P Put Options October 28 2015 27 / 38

Discussion: The Results Are Consistent with the Model In a regression of the net buy of OTM puts on the L/A ratio, the regression coefficient is -1.512 (0.575) In a regression of the net buy of ATM puts on the L/A ratio, the regression coefficient is 0.696 (1.013) George Constantinides; Lei Lian The Supply and Demand of S&P Put Options October 28 2015 28 / 38

The Skew Response Puzzle A broad class of widely used no-arbitrage models that allows for stochastic volatility and jumps in the price and volatility implies that the skew is decreasing in the RN volatility and disaster index: Bates (2006): one-factor model Andersen, Fusari, and Todorov (2015): two-factor model This contradicts the empirical evidence George Constantinides; Lei Lian The Supply and Demand of S&P Put Options October 28 2015 29 / 38

The IV Skew Implied by the Bates (2006) No-Arbitrage Model IV Skew 0.05 0.06 0.07 0.08 0.09 0.10 0.11 0.12 IV Skew 0.05 0.06 0.07 0.08 0.09 0.10 0.11 0.12 0.05 0.10 0.15 0.20 0.25 0.30 0.35 RN Variance 0.000 0.002 0.004 0.006 0.008 0.010 Disaster Index George Constantinides; Lei Lian The Supply and Demand of S&P Put Options October 28 2015 30 / 38

Implied Volatility Skew versus the Risk-Neutral Variance 0.00 0.10 0.20 0.30 0.06 0.10 0.14 Full Period RN Variance IV Skew 0.02 0.06 0.10 0.14 0.06 0.10 0.14 Before Crisis RN Variance IV Skew 0.05 0.15 0.25 0.35 0.09 0.11 0.13 During Crisis RN Variance IV Skew 0.02 0.04 0.06 0.08 0.10 0.12 0.11 0.13 0.15 0.17 After Crisis RN Variance IV Skew George Constantinides; Lei Lian The Supply and Demand of S&P Put Options October 28 2015 31 / 38

Implied Volatility Skew versus the Risk-Neutral Disaster Index 0.000 0.010 0.020 0.06 0.10 0.14 Full Period Disaster Index IV Skew 0.000 0.002 0.004 0.006 0.008 0.06 0.10 0.14 Before Crisis Disaster Index IV Skew 0.000 0.005 0.010 0.015 0.020 0.025 0.09 0.11 0.13 During Crisis Disaster Index IV Skew 0.002 0.004 0.006 0.008 0.11 0.13 0.15 0.17 After Crisis Disaster Index IV Skew George Constantinides; Lei Lian The Supply and Demand of S&P Put Options October 28 2015 32 / 38

0.0 0.0 Model-Implied IV Skew 0.05 0.050.10 0.10.15 0.150.20 0.20.25 0.25 RN Variance RN Variance 0.0 0.0 0.05 0.050.10 0.100.15 0.150.20 0.200.25 0.25 RN Variance RN Variance σ = 0.04 σ = 0.08 IV Skew 0.170 0.180 IV Skew 0.190 0.200 0.170 0.180 0.190 0.200 IV Skew 0.06 IV 0.07 Skew 0.08 0.09 0.06 0.07 0.08 0.09 0.0010 0.0010 0.0020 0.0020 0.0030 0.0030 Disaster Disaster Index Index 0.0010 0.0010 0.0015 0.0015 0.0020 0.0020 0.0025 0.0025 Disaster Disaster Index Index p = 0.06 p = 0.10 IV Skew 0.05 IV Skew 0.15 0.25 0.05 0.15 0.25 IV Skew 0.05 IV Skew 0.15 0.25 0.05 0.15 0.25 0.05 0.050.10 0.10.15 0.150.20 0.20.25 0.25 RN Variance RN Variance 0.05 0.050.10 0.100.15 0.150.20 0.200.25 0.25 RN Variance RN Variance George Constantinides; Lei Lian The Supply and Demand of S&P Put Options October 28 2015 33 / 38 200 200

Model-Implied IV Skew The skew is increasing in the disaster index The skew is decreasing in the RN variance The net effect is confounded by the high correlation (90%) between the disaster index and RN variance George Constantinides; Lei Lian The Supply and Demand of S&P Put Options October 28 2015 34 / 38

The Observed IV Skew versus the Observed Disaster Index and RN Variance George Constantinides; Lei Lian The Supply and Demand of S&P Put Options October 28 2015 35 / 38

Discussion The skew is increasing in the disaster index in both the univariate and bivariate regressions but some coefficients are statistically insignificant The skew is sometimes decreasing or increasing in the RN variance The net effect is confounded by the high correlation (90%) between the disaster index and RN variance George Constantinides; Lei Lian The Supply and Demand of S&P Put Options October 28 2015 36 / 38

Conclusion The hedging of down-side risk by customers provides the endogenous demand for puts The profit-making by VaR-constrained market makers provides the endogenous supply of puts The intersection of the supply and demand provides the equilibrium put price and net buy by customers George Constantinides; Lei Lian The Supply and Demand of S&P Put Options October 28 2015 37 / 38

Conclusion, continued Consistent with the empirical evidence, the model predicts that the net buy of puts by customers is decreasing in the RN volatility and disaster index both for OTM and ATM puts decreasing in their price decreasing in the liabilities/assets ratio The model also potentially resolves the skew response puzzle George Constantinides; Lei Lian The Supply and Demand of S&P Put Options October 28 2015 38 / 38