The Supply and Demand of S&P 500 Put Options

Transcription

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

2 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 / 38

3 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 / 38

4 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 / 38

5 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 / 38

6 Liabilities-to-Assets Ratio of Broker-Dealers Liability/Asset Ratio George Constantinides; Lei Lian The Supply and Demand of S&P Put Options October / 38

7 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 / 38

8 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 / 38

9 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 / 38

10 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 / 38

11 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 / 38

12 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 / 38

13 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 / 38

14 Calibration The length of one period is 1 month Parameters: p: , implies expected disasters p.a. σ: , implies volatility p.a. µ: implies equity premium 6% p.a. in the no-disaster state µ J : 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% % p.a. and the volatility is 7.38% % Customer s initial wealth 500 and preference parameter 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 / 38

15 Supply and Demand σ = 0.04, p = 0.05 option option price price option option price price option option price price option option price price Demand/Supply (ATM) (ATM) Demand/Supply (OTM) (OTM) Demand/Supply (ATM) (ATM) Demand/Supply (OTM) (OTM) Demand Demand Supply Supply Demand Demand Supply Supply option option price price Demand Demand Supply Supply Demand Demand Supply Supply Quantity Quantity Quantity Quantity Quantity Quantity Demand/Supply (ATM) (ATM) Demand/Supply (ATM) (ATM) Demand Demand Supply Supply Demand Demand Supply Supply Quantity Quantity Quantity Quantity option option price price option option price price option option price price σ = 0.04, p = Quantity Quantity Demand/Supply (OTM) (OTM) Demand/Supply (OTM) (OTM) Demand Demand Supply Supply Demand Demand Supply Supply Quantity Quantity Quantity Quantity George Constantinides; Lei Lian The Supply and Demand of S&P Put Options October / 38

16 Data Sources CBOE: daily buy, and sell contracts for customers and firms small customer (<100 contracts), medium ( ), 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 / 38

17 Net Buy, RN Variance, Disaster Index NetBuy (OTM) RN Variance NetBuy (ATM) Disaster Index George Constantinides; Lei Lian The Supply and Demand of S&P Put Options October / 38

18 Model-Implied Net Buy of OTM Puts vs the Disaster Index and RN RV Variance RV Variance RV Variance RV Variance σ = 0.04 σ = N N N N Net Buy(OTM) 100 Net 300 Buy(OTM) Net Buy(OTM) 150 Net Buy(OTM) Disaster Disaster Index Index Disaster Disaster Index Index p = 0.06 p = 0.10 Net Buy(OTM) 140 Net 180 Buy(OTM) Net Buy(OTM) Net Buy(OTM) RV Variance RV Variance RV Variance RV Variance George Constantinides; Lei Lian The Supply and Demand of S&P Put Options October / 38

19 Model-Implied Net Buy of ATM Puts vs the Disaster Index and RN RV Variance RV Variance RV Variance RV Variance σ = 0.04 σ = N N N N 76 Net Buy(ATM) Net 90 Buy(ATM) Net Buy(ATM) Net 90 Buy(ATM) Disaster Disaster Index Index Disaster Disaster Index Index p = 0.06 p = 0.10 Net Buy(ATM) 85 Net 90 Buy(ATM) Net Buy(ATM) 76 Net 78 Buy(ATM) RV Variance RV Variance RV Variance RV Variance George Constantinides; Lei Lian The Supply and Demand of S&P Put Options October / 38

20 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 / 38

21 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 / 38

22 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 / 38

23 Net Buy ( 140 Net 180 Buy ( Model-Implied Net Buy of OTM Puts vs the Price of Puts Net Buy ( σ = 0.04 σ = Net Buy 140 ( B S B S IV IV B S B S IV IV Net Buy (OTM) 100 Net 300 Buy 500 (OTM) Net Buy (OTM) 150 Net Buy 250 (OTM) B S B S IV IV B S B S IV IV p = 0.06 p = 0.10 Net Buy (OTM) Net 180 Buy (OTM) Net Buy (OTM) 135 Net Buy 140 (OTM) B S B S IV IV B S B S IV IV George Constantinides; Lei Lian The Supply and Demand of S&P Put Options October / 38

24 Net Buy ( 85 Net Buy 90 ( Model-Implied Net Buy of ATM Puts vs the Price of Puts Net Buy ( σ = 0.04 σ = Net Buy 78 ( B S B S IV IV B S B S IV IV Net Buy (ATM) Net 90 Buy 100 (ATM) 120 Net Buy (ATM) Net Buy 90 (ATM) B S B S IV IV B S B S IV IV p = 0.06 p = 0.10 Net Buy (ATM) 85 Net Buy 90 (ATM) Net Buy (ATM) Net Buy 78 (ATM) B S B S IV IV B S B S IV IV George Constantinides; Lei Lian The Supply and Demand of S&P Put Options October / 38

25 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 / 38

26 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 / 38

27 The Net Buy of Puts versus the Market Makers Constraint σ = 0.04, p = 0.06 Net Net Buy Buy (OTM) Net Net Buy Buy (ATM) Net Net Buy Buy (OTM) W* W* W* W* σ = 0.08, p = 0.10 Net Net Buy Buy (ATM) W* W* W* W* George Constantinides; Lei Lian The Supply and Demand of S&P Put Options October / 38

28 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 (0.575) In a regression of the net buy of ATM puts on the L/A ratio, the regression coefficient is (1.013) George Constantinides; Lei Lian The Supply and Demand of S&P Put Options October / 38

29 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 / 38

30 The IV Skew Implied by the Bates (2006) No-Arbitrage Model IV Skew IV Skew RN Variance Disaster Index George Constantinides; Lei Lian The Supply and Demand of S&P Put Options October / 38

31 Implied Volatility Skew versus the Risk-Neutral Variance Full Period RN Variance IV Skew Before Crisis RN Variance IV Skew During Crisis RN Variance IV Skew After Crisis RN Variance IV Skew George Constantinides; Lei Lian The Supply and Demand of S&P Put Options October / 38

32 Implied Volatility Skew versus the Risk-Neutral Disaster Index Full Period Disaster Index IV Skew Before Crisis Disaster Index IV Skew During Crisis Disaster Index IV Skew After Crisis Disaster Index IV Skew George Constantinides; Lei Lian The Supply and Demand of S&P Put Options October / 38

33 Model-Implied IV Skew RN Variance RN Variance RN Variance RN Variance σ = 0.04 σ = 0.08 IV Skew IV Skew IV Skew 0.06 IV 0.07 Skew Disaster Disaster Index Index Disaster Disaster Index Index p = 0.06 p = 0.10 IV Skew 0.05 IV Skew IV Skew 0.05 IV Skew RN Variance RN Variance RN Variance RN Variance George Constantinides; Lei Lian The Supply and Demand of S&P Put Options October /

34 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 / 38

35 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 / 38

36 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 / 38

37 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 / 38

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 / 38