The Supply and Demand of S&P 500 Put
Options
George Constantinides
University of Chicago
Lei Lian
University of Massachusetts at Amherst
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
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
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)
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
Liabilities-to-Assets Ratio of Broker-Dealers
1.010 1.020 1.030 Liability/Asset Ratio 02−1996 02−1998 02−2000 02−2002 02−2004 02−2006 02−2008 02−2010 02−2012Motivation-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
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
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
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
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:
eµ+σe, e∼(0,1)in the no-disaster state
eµJ+σJeJ, e
J∼(0,1)in the disaster state that occurs with prob.p.
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 wealthW0, buysαshares of stock andβputs, invests−α−βP units of the numeraire in bonds max α,β E[U] where U = W0−α−βP+αS+β[K−S]+−A 2(W0−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
Market Makers’ Supply of Puts
Market makers are risk neutral. Theyinvest in the index and riskless asset write puts for profit
face a VaR credit constraint
The market maker has zero endowment, buyαˆ shares stock, buyβˆ puts
max ˆ α,βˆ E ˆ α(S−1) +βˆ([K−S]+−P)
subject to the exogenous VaR constraint
probαˆ(S−1) +βˆ [K−S]+−P<W∗ ≤h Their expected one-month profit maximization provides the endogenous supply of puts
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
µJ: -0.04
σJ: 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
Supply and Demand
σ=0.04,p=0.05 −3000 −1000 0 1000 0.018 0.020 0.022 0.024 Demand/Supply (ATM) Quantity option pr ice Demand Supply 0 200 600 1000 0.0018 0.0020 0.0022 Demand/Supply (OTM) Quantity option pr ice Demand Supply −2500 −1500 −500 0 500 0.022 0.024 0.026 0.028 0.030 Demand/Supply (ATM) Quantity option pr ice Demand Supply −500 0 500 0.0036 0.0040 0.0044 Demand/Supply (OTM) Quantity option pr ice Demand Supply −3000 −1000 0 1000 0.018 0.020 0.022 0.024 Demand/Supply (ATM) Quantity option pr ice Demand Supply 0 200 600 1000 0.0018 0.0020 0.0022 Demand/Supply (OTM) Quantity option pr ice Demand Supply −2500 −1500 −500 0 500 0.022 0.024 0.026 0.028 0.030 Demand/Supply (ATM) Quantity option pr ice Demand Supply −500 0 500 0.0036 0.0040 0.0044 Demand/Supply (OTM) Quantity option pr ice Demand Supply σ=0.04,p=0.10 −3000 −1000 0 1000 0.018 0.020 0.022 0.024 Demand/Supply (ATM) Quantity option pr ice Demand Supply 0 200 600 1000 0.0018 0.0020 0.0022 Demand/Supply (OTM) Quantity option pr ice Demand Supply −2500 −1500 −500 0 500 0.022 0.024 0.026 0.028 0.030 Demand/Supply (ATM) Quantity option pr ice Demand Supply −500 0 500 0.0036 0.0040 0.0044 Demand/Supply (OTM) Quantity option pr ice Demand Supply −3000 −1000 0 1000 0.018 0.020 0.022 0.024 Demand/Supply (ATM) Quantity option pr ice Demand Supply 0 200 600 1000 0.0018 0.0020 0.0022 Demand/Supply (OTM) Quantity option pr ice Demand Supply −2500 −1500 −500 0 500 0.022 0.024 0.026 0.028 0.030 Demand/Supply (ATM) Quantity option pr ice Demand Supply −500 0 500 0.0036 0.0040 0.0044 Demand/Supply (OTM) Quantity option pr ice Demand SupplyData 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
Net Buy, RN Variance, Disaster Index
−0.05 0.00 0.05 0.10 NetBuy (O TM) 01−1996 01−1998 01−2000 01−2002 01−2004 01−2006 01−2008 01−2010 01−2012 0.00 0.05 0.10 0.15 0.20 NetBuy (A TM) 01−1996 01−1998 01−2000 01−2002 01−2004 01−2006 01−2008 01−2010 01−2012 0.00 0.10 0.20 0.30 RN V ar iance 01−1996 01−1998 01−2000 01−2002 01−2004 01−2006 01−2008 01−2010 01−2012 0.000 0.010 0.020 Disaster Inde x 01−1996 01−1998 01−2000 01−2002 01−2004 01−2006 01−2008 01−2010 01−2012Model-Implied Net Buy of OTM Puts vs the Disaster
Index and RN Variance
σ=0.04 σ=0.08 ●●●●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.05 0.10 0.15 0.20 0.25 140 180 220 RV Variance Net Buy(O TM) ●●●●●● ●● ●● ● ● ●●●● ● ● ● ● ● ● ● ● ● 0.05 0.10 0.15 0.20 0.25 130 135 140 145 150 RV Variance Net Buy(O TM) ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.0010 0.0020 0.0030 100 300 500 700 Disaster Index Net Buy(O TM) ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.0010 0.0015 0.0020 0.0025 150 250 350 Disaster Index Net Buy(O TM) ●●●●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.05 0.10 0.15 0.20 0.25 140 180 220 RV Variance Net Buy(O TM) ●●●●●● ●● ●● ● ● ●●●● ● ● ● ● ● ● ● ● ● 0.05 0.10 0.15 0.20 0.25 130 135 140 145 150 RV Variance Net Buy(O TM) ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.0010 0.0020 0.0030 100 300 500 700 Disaster Index Net Buy(O TM) ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.0010 0.0015 0.0020 0.0025 150 250 350 Disaster Index Net Buy(O TM) p=0.06 p=0.10 ●●●●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.05 0.10 0.15 0.20 0.25 140 180 220 RV Variance Net Buy(O TM) ●●●●●● ●● ●● ●● ●●●● ● ● ● ● ● ● ● ● ● 0.05 0.10 0.15 0.20 0.25 130 135 140 145 150 RV Variance Net Buy(O TM) ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.0010 0.0020 0.0030 100 300 500 700 Net Buy(O TM) ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.0010 0.0015 0.0020 0.0025 150 250 350 Net Buy(O TM) ●●●●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.05 0.10 0.15 0.20 0.25 140 180 220 RV Variance Net Buy(O TM) ●●●●●● ●● ●● ●● ●●●● ● ● ● ● ● ● ● ● ● 0.05 0.10 0.15 0.20 0.25 130 135 140 145 150 RV Variance Net Buy(O TM) ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.0010 0.0020 0.0030 100 300 500 700 Net Buy(O TM) ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.0010 0.0015 0.0020 0.0025 150 250 350 Net Buy(O TM)
Model-Implied Net Buy of ATM Puts vs the Disaster
Index and RN Variance
σ=0.04 σ=0.08 ● ● ● ● ●● ● ● ●● ● ●●● ● ● ● ● ● ● ● ● ● ● ● 0.05 0.10 0.15 0.20 0.25 85 90 95 100 RV Variance Net Buy(A TM) ●● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●●● ● ● ● ● ● ● 0.05 0.10 0.15 0.20 0.25 76 78 80 82 RV Variance Net Buy(A TM) ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● 0.0010 0.0020 0.0030 70 80 90 100 120 Disaster Index Net Buy(A TM) ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.0010 0.0015 0.0020 0.0025 70 80 90 100 110 Disaster Index Net Buy(A TM) ● ● ● ● ●● ● ● ●● ● ●●● ● ● ● ● ● ● ● ● ● ● ● 0.05 0.10 0.15 0.20 0.25 85 90 95 100 RV Variance Net Buy(A TM) ●● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●●● ● ● ● ● ● ● 0.05 0.10 0.15 0.20 0.25 76 78 80 82 RV Variance Net Buy(A TM) ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● 0.0010 0.0020 0.0030 70 80 90 100 120 Disaster Index Net Buy(A TM) ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.0010 0.0015 0.0020 0.0025 70 80 90 100 110 Disaster Index Net Buy(A TM) p=0.06 p=0.10 ● ● ● ● ●● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● 0.05 0.10 0.15 0.20 0.25 85 90 95 100 RV Variance Net Buy(A TM) ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ● ● ● ● 0.05 0.10 0.15 0.20 0.25 76 78 80 82 RV Variance Net Buy(A TM) ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● 0.0010 0.0020 0.0030 70 80 90 100 120 Net Buy(A TM) ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.0010 0.0015 0.0020 0.0025 70 80 90 100 110 Net Buy(A TM) ● ● ● ● ●● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● 0.05 0.10 0.15 0.20 0.25 85 90 95 100 RV Variance Net Buy(A TM) ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ● ● ● ● 0.05 0.10 0.15 0.20 0.25 76 78 80 82 RV Variance Net Buy(A TM) ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● 0.0010 0.0020 0.0030 70 80 90 100 120 Net Buy(A TM) ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.0010 0.0015 0.0020 0.0025 70 80 90 100 110 Net Buy(A TM)
Observed Net Buy of OTM Puts versus the Disaster
Index and the RN Variance
Observed Net Buy of ATM Puts versus the Disaster
Index and the RN Variance
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.
Model-Implied Net Buy of OTM Puts vs the Price of
Puts
σ=0.04 σ=0.08 ● ● ● ● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.40 0.45 0.50 140 180 220 B−S IV Net Buy (O TM) ● ● ● ● ● ● ● ●●●● ● ● ●● ● ● ● ● ● ● ● ● ● ● 0.40 0.44 0.48 0.52 130 135 140 145 150 B−S IV Net Buy (O TM) ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.34 0.36 0.38 0.40 0.42 0.44 100 300 500 700 B−S IV Net Buy (O TM) ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ●● ●● ●● ●● ●● ●● ●● ● ● ● ● ● ● ● ● ● ● 0.36 0.38 0.40 0.42 0.44 150 250 350 B−S IV Net Buy (O TM) ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.40 0.45 0.50 140 180 220 B−S IV Net Buy (O TM) ● ● ● ● ● ● ● ● ●●● ● ● ●● ● ● ● ● ● ● ● ● ● ● 0.40 0.44 0.48 0.52 130 135 140 145 150 B−S IV Net Buy (O TM) ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.34 0.36 0.38 0.40 0.42 0.44 100 300 500 700 B−S IV Net Buy (O TM) ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ●● ●● ●● ●● ●● ●● ●● ● ● ● ● ● ● ● ● ● ● 0.36 0.38 0.40 0.42 0.44 150 250 350 B−S IV Net Buy (O TM) p=0.06 p=0.10 ● ● ● ● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.40 0.45 0.50 140 180 220 B−S IV Net Buy (O TM) ● ● ● ● ● ● ● ●●●●● ● ●● ● ● ● ● ● ● ● ● ● ● 0.40 0.44 0.48 0.52 130 135 140 145 150 B−S IV Net Buy (O TM) ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.34 0.36 0.38 0.40 0.42 0.44 100 300 500 700 Net Buy (O TM) ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ●● ●● ●● ●● ●● ●● ●● ● ● ● ● ● ● ● ● ● ● 0.36 0.38 0.40 0.42 0.44 150 250 350 Net Buy (O TM) ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.40 0.45 0.50 140 180 220 B−S IV Net Buy (O TM) ● ● ● ● ● ● ● ● ●●●● ● ●● ● ● ● ● ● ● ● ● ● ● 0.40 0.44 0.48 0.52 130 135 140 145 150 B−S IV Net Buy (O TM) ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.34 0.36 0.38 0.40 0.42 0.44 100 300 500 700 Net Buy (O TM) ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ●● ●● ●● ●● ●● ●● ●● ● ● ● ● ● ● ● ● ● ● 0.36 0.38 0.40 0.42 0.44 150 250 350 Net Buy (O TM)Model-Implied Net Buy of ATM Puts vs the Price of
Puts
σ=0.04 σ=0.08 ● ●● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● 0.1 0.2 0.3 0.4 0.5 85 90 95 100 B−S IV Net Buy (A TM) ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ● ● ● ● ● ● 0.2 0.3 0.4 0.5 76 78 80 82 B−S IV Net Buy (A TM) ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.16 0.18 0.20 0.22 0.24 70 80 90 100 120 B−S IV Net Buy (A TM) ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.30 0.32 0.34 0.36 70 80 90 100 110 B−S IV Net Buy (A TM) ● ●● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● 0.1 0.2 0.3 0.4 0.5 85 90 95 100 B−S IV Net Buy (A TM) ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ● ● ● ● ● ● 0.2 0.3 0.4 0.5 76 78 80 82 B−S IV Net Buy (A TM) ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.16 0.18 0.20 0.22 0.24 70 80 90 100 120 B−S IV Net Buy (A TM) ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.30 0.32 0.34 0.36 70 80 90 100 110 B−S IV Net Buy (A TM) p=0.06 p=0.10 ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● 0.1 0.2 0.3 0.4 0.5 85 90 95 100 B−S IV Net Buy (A TM) ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ● ● ● ● ● ● 0.2 0.3 0.4 0.5 76 78 80 82 B−S IV Net Buy (A TM) ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.16 0.18 0.20 0.22 0.24 70 80 90 100 120 Net Buy (A TM) ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.30 0.32 0.34 0.36 70 80 90 100 110 Net Buy (A TM) ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● 0.1 0.2 0.3 0.4 0.5 85 90 95 100 B−S IV Net Buy (A TM) ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ● ● ● ● ● ● 0.2 0.3 0.4 0.5 76 78 80 82 B−S IV Net Buy (A TM) ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.16 0.18 0.20 0.22 0.24 70 80 90 100 120 Net Buy (A TM) ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.30 0.32 0.34 0.36 70 80 90 100 110 Net Buy (A TM)Observed Net Buy of OTM and ATM Puts vs the Price
of Puts (in IV Units)
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
The Net Buy of Puts versus the Market Makers’
Constraint
σ=0.04,p=0.06 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● −80 −60 −40 −20 0 0 200 600 1000 W* Net Buy (O TM) ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● −80 −60 −40 −20 0 0 100 200 300 400 W* Net Buy (A TM) ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● −80 −60 −40 −20 0 0 100 300 500 W* Net Buy (O TM) ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● −80 −60 −40 −20 0 0 50 150 250 350 W* Net Buy (A TM) σ=0.08,p=0.10 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● −80 −60 −40 −20 0 0 200 600 1000 W* Net Buy (O TM) ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● −80 −60 −40 −20 0 0 100 200 300 400 W* Net Buy (A TM) ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● −80 −60 −40 −20 0 0 100 300 500 W* Net Buy (O TM) ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● −80 −60 −40 −20 0 0 50 150 250 350 W* Net Buy (A TM)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)
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
The IV Skew Implied by the Bates (2006) No-Arbitrage
Model
0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.05 0.06 0.07 0.08 0.09 0.10 0.11 0.12 RN Variance IV Sk e w 0.000 0.002 0.004 0.006 0.008 0.010 0.05 0.06 0.07 0.08 0.09 0.10 0.11 0.12 Disaster Index IV Sk e wImplied 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 Sk e w ● ● ● ● ● ● ● ● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ●●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ●● ●● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ● ● ● ● 0.02 0.06 0.10 0.14 0.06 0.10 0.14 Before Crisis RN Variance IV Sk e w ●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● 0.05 0.15 0.25 0.35 0.09 0.11 0.13 During Crisis RN Variance IV Sk e w ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 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 Sk e wImplied 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 Sk e w ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ●● ● ● 0.000 0.002 0.004 0.006 0.008 0.06 0.10 0.14 Before Crisis Disaster Index IV Sk e w ●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● 0.000 0.005 0.010 0.015 0.020 0.025 0.09 0.11 0.13 During Crisis Disaster Index IV Sk e w ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.002 0.004 0.006 0.008 0.11 0.13 0.15 0.17 After Crisis Disaster Index IV Sk e wModel-Implied IV Skew
σ=0.04 σ=0.08 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●●●●● ● 0.05 0.10 0.15 0.20 0.25 0.05 0.15 0.25 RN Variance IV Sk e w ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●●●●● 0.05 0.10 0.15 0.20 0.25 0.05 0.15 0.25 RN Variance IV Sk e w ● ● ● ● ● ● ● ● ● ● ● ●●● ●● ●● ● ●● ● ● ● ● 0.0010 0.0020 0.0030 0.170 0.180 0.190 0.200 Disaster Index IV Sk e w ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●●●●●● ●● ●● 0.0010 0.0015 0.0020 0.0025 0.06 0.07 0.08 0.09 Disaster Index IV Sk e w ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●●●●●● 0.05 0.10 0.15 0.20 0.25 0.05 0.15 0.25 RN Variance IV Sk e w ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●●●●● 0.05 0.10 0.15 0.20 0.25 0.05 0.15 0.25 RN Variance IV Sk e w ● ● ● ● ● ● ● ● ● ● ● ●●● ●● ●● ● ●● ● ● ● ● 0.0010 0.0020 0.0030 0.170 0.180 0.190 0.200 Disaster Index IV Sk e w ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●●●●●● ●● ●● 0.0010 0.0015 0.0020 0.0025 0.06 0.07 0.08 0.09 Disaster Index IV Sk e w p=0.06 p =0.10 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●●●●●● ● 0.05 0.10 0.15 0.20 0.25 0.05 0.15 0.25 RN Variance IV Sk e w ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●●●●●● 0.05 0.10 0.15 0.20 0.25 0.05 0.15 0.25 RN Variance IV Sk e w ● ● ● ● ● ● ● ● ● ●● ● ●● ●● ● ● ● ●● ● ● ● ● 0.0010 0.0020 0.0030 0.170 0.180 0.190 0.200 IV Sk e w ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●●●●● ●●● ● ● 0.0010 0.0015 0.0020 0.0025 0.06 0.07 0.08 0.09 IV Sk e w ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●●●●●●● 0.05 0.10 0.15 0.20 0.25 0.05 0.15 0.25 RN Variance IV Sk e w ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●●●●●● 0.05 0.10 0.15 0.20 0.25 0.05 0.15 0.25 RN Variance IV Sk e w ● ● ● ● ● ● ● ● ● ●● ● ●● ●● ● ● ● ●● ● ● ● ● 0.0010 0.0020 0.0030 0.170 0.180 0.190 0.200 IV Sk e w ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●●●●● ●●● ● ● 0.0010 0.0015 0.0020 0.0025 0.06 0.07 0.08 0.09 IV Sk e wModel-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
The Observed IV Skew versus the Observed Disaster
Index and RN Variance
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
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
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
