Stock Price Pinning near Option Expiration Dates

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1 Stock Price Piig ear Optio Expiratio Dates Marco Avellaeda, New York Uiversity Geady Kasya, New York Uiversity Michael D. Lipki, Katama Tradig & Columbia Uiversity George Papaicolaou Coferece Paris, December st

2 Summary Empirical evidece of stock piig (Ni-Pearso-Poteshma, 3) Liear price-impact model (Avellaeda & Lipki ) No-liear price-impact model (Ave., Kasya & Lipki, 7) Numerical simulatio of piig for differet price-impact fuctios Phase trasitio at p=/ Rigorous mathematical proofs for the differet regimes. Bibliography

, Kasya & Lipki, 7) Numerical simulatio of piig for differet price-impact

3 Share Price Piig o Optio Expiratio Dates $5. B. Stock B pied Stock A did ot A. $.5 Time $. Optio expiratio Friday, (3 rd Friday of the moth).

4 Price KO: Sep 8 to Oct Day /7 Close=$45.5 Expiratio

5 Statistical Evidece of Piig Stock Price Clusterig o Optio Expiratio Dates, Preprit, Jue 6, 3 Authors: Sophie Xiaoya Ni, Neil Pearso ad Alle M. Poteshma (U. of Illiois Urbaa-Champaig) Data. Ivy DB (OptioMetrics) Ja 996, Sep : All stocks traded i US exchages All optios traded i US exchages Ed of day bid-ask quotes, volume, ope iterest. CBOE Statistics Ope iterest ad tradig volume, Ja 996 to Dec 4 Ivestor Categories: Market Makers, Firm Prop Traders, Large Firm Cliets, Discout Firm Cliets

Ivy DB (OptioMetrics) Ja 996, Sep : All stocks traded i US exchages All optios traded i US exchages Ed of day bid-ask

6 The U. Illiois Urbaa study At least 8 expiratio dates 4,395 optioable stocks o at least oe date 84,449 optioable stock-expiratio pairs, o-optioable stocks o at least oe date 47,7 o-optioable stock-expiratio pairs

7 Percetage of o-optioable stocks closig withi $.5 of a iteger multiple of $5 % 3. 5 Expiratio Friday r e la t i v e t r a d i g d a t e f r o m o p t i o e x p i r a t i o d a t e Relative Tradig Date from Optio Expiratio Date (Courtesy: Ni, Pearso & Poteshma)

5 - - 9-8 - 7-6 - 5-4 - 3 - - 3 4 5 6 7 8 9 r e la t i v e t r a d i g d a

8 Percetage of optioable stocks closig withi $.5 of a iteger multiple of $5 3 % r e la t i v e t r a d i g d a t e f r o m o p t i o e x p i r a t i o d a t e Relative Tradig Date from Optio Expiratio Date (Courtesy: Ni, Pearso & Poteshma)

5 - - 9-8 - 7-6 - 5-4 - 3 - - 3 4 5 6 7 8 9 r e la t i v e t r a d i g

9 Percetage of optioable stocks closig withi $.5 of a strike price % r e la t i v e t r a d i g d a t e f r o m o p t i o e x p i r a t i o d a t e Relative Tradig Date from Optio Expiratio Date (Courtesy: Ni, Pearso & Poteshma)

10 Percetage of o-optioable stocks closig withi $.5 of a iteger multiple of $5 % r e la t i v e t r a d i g d a t e f r o m o p t io e x p ir a t i o d a t e Relative Tradig Date from Optio Expiratio Date (Courtesy: Ni, Pearso & Poteshma)

5 6 - - 9-8 - 7-6 - 5-4 - 3 - - 3 4 5 6 7 8 9 r e la t i v e t r a d i g

11 Percetage of optioable stocks closig withi $.5 of a iteger multiple of $5 % r e la t i v e t r a d i g d a t e f r o m o p t i o e x p i r a t i o d a t e Relative Tradig Date from Optio Expiratio Date (Courtesy: Ni, Pearso & Poteshma)

12 Percetage of optioable stocks closig withi $.5 of a strike price % r e la t i v e t r a d i g d a t e f r o m o p t i o e x p i r a t i o d a t e Relative Tradig Date from Optio Expiratio Date (Courtesy: Ni, Pearso & Poteshma)

13 Percetage of o-optioable stocks closig withi $.5 of a iteger multiple of $5 % r e la t i v e t r a d i g d a t e f r o m o p t io e x p ir a t i o d a t e Relative Tradig Date from Optio Expiratio Date (Courtesy: Ni, Pearso & Poteshma)

14 I search for a explaatio JDEC i March /6/ // // // /3/ /6/ Large sale of optios o this day /7/ /8/ /3/ /3/ 5/3/ 6/3/ 7/3/ 8/3/ 9/3/ /3/ 3/3/ 3/4/ 3/5/ 3/6/

15 Cotracts 6 JDEC Mar Put & Call Ope Iterest /9/ // Average traded vol i stocks = MM shares /3/ /5/ /7/ /9/ // /3/ /5/ /7/ Date 3// 3/3/ 3/5/ 3/7/ 3/9/ 3// 3/3/ 3/5/ Notioal umber of shares correspodig to OI = 5.6 MM shares

16 Our Model: Feedback Due to Demad for Deltas Assumptio. Ope Iterest is uusually large Assumptio. Market-makers professioal delta-hedgers are et very log optios Proposed mechaism for piig: Hedgers are et log optios, hece log Gamma. They sell stock whe it rises ad buy stock whe it falls. Sice the aggregate amout of stock required is large compared to typical daily tradig volume, this drives the stock to the strike price

et log optios, hece log Gamma. They sell stock whe it rises ad buy stock whe it falls.

17 Takig ito accout demad for stock: Price-Impact Fuctios ds S E D V p D V p=. p=.5 Farmer, Lillo, Matega X. Gabaix p= liear model, (A. & Lipki) p=.5 covex model (Bouchaud, ) Choice of p is a fudametal questio i Ecoophysics.

18 Liear Model ( A & Lipki, ) Price-Demad Elasticity Eq. S S E D D Price-respose due to demad for deltas S S E. OI D B. -S. Delta for oe optio

19 Estimatig the Demad for Deltas usig Black-Scholes,, l l ), (, 3/ a y e a y t a K S y K S d d N t T dt t From Black-Scholes

20 Dyamics for Stock Price dw dt e t T t T a y D E OI dy K S y dw dt t D E OI S ds t T t T a y 3/. l. restorig force `couplig costat bouded support oise

21 Mote Carlo Simulatio Time scale=-log(t-t)

22 probability Cumulative PDF for price at expiratio date (Beta=.) Z()

23 Dimesioless Variables., l,, T D E OI T a K S T T y T t s T y dw ds e s s d s s ) ( ) ( 3/ ) ( ) (

24 Dimesioless Model (alpha=) for Liear Price-Impact Fuctio d ( s) 3/ e ( s) ds dw Liear restorig force with icreasig couplig with time ad compact support.

25 The Potetial Well =l(s/k)/(sigma*sqrt(tau)) Price experieces a force that becomes stroger, more localied ear expiratio s=. s=.5 d ds ( s) 3/ e s (, ) s

26 Solvig the liear respose model (p=) s F e F s F, 3/ Assume Alpha= Forward Fokker-Plack equatio: Look for solutio of the form: s F ukow,, exp,

27 ODE for the `Phase Fuctio (WKB) s e s s F c c O c e e O e 3/ ' ' 3/ exp, ' ' ' - ' ' ' ' ' Eikoal Equatio Exact solutio of the FFP Equatio!

28 A Formula for the Piig Probability Prob ) (, lim ), ( lim : Satisfies exp ), ( e s s s e s P s P e s s P

29 prob Piig Probability: Depedece o Beta Piig Probability Beta Zo=.5 Zo= E. OI D T - Icreases with OI - Decreases with volat, expiratio - Decreases with the distace to strike

30 probability Piig probability: depedece o _ % 8% 6% 4% % % 8% 6% 4% % % Alpha = Beta =. =l(s/k)/(vol*sqrt(t))

31 OK, but does this story explai stock piig? We kow that stock piig at optio expiratio exists for optioable stocks Our model makes two assumptios to justify piig - Large umber of deltas relative to total volume - Market-makers are log optios Use CBOE data to calculate piig statistics coditioal o market-makers positio ear expiratio

32 Observatios with market-makers et log (~$.5)

33 Market-makers + firm proprietary traders et log

34 Market-makers et short

35 Market-makers + firm proprietary traders et short

36 Lipki & Stato: Validatio of the Depedece of Volatility/OI P pi OI V P pi OI Source of Data: IVY/OptioMetrics

37 Piig Probability vs. Ope Iterest OI Ratio Mike Lipki, Sasha Stato, Columbia empirical fiace course

38 Dimesioless Model for Power-Law Price-Impact Fuctio (p>) Price chage= Price impact+ oise ds S cost. t p sig dt t dw d ( sig 3 p / s) p e p ( s) ds dw Dimesioless eq. without irrelevat drift terms (alpha=).

39 Piig uder o-liear priceimpact models (i) If p<=/, there is o piig, i.e. P[()= ()=]=, for all. (ii) If p>/ piig occurs with fiite probability (<) ad l P(() () ) C, p P pi e C p, p /

40 Calculatio of Piig Probabilities by MC Simulatio (Geady Kasya) Smooth fit ear p=.5

41 Absece of Piig for p</ (I) d p pr p r r sgre dt dw Novikov coditio: If we have E W e drift dt the measure iduced by the -process i path-space is absolutely cotiuous with respect to Wieer Measure.

42 Absece of Piig for p</ (II) Verify that Novikov coditio holds: / if, ) ( drift p e E e E e E p L t p t dt W dt W dt W

43 Self-similarity approach (RG) Kasya (7) : exploit the self-similarity properties of the model ' ' a a t' ( a ) a t Parabolic re-scalig trasformatio a d' ' p p ' dt' dw ' ' a p Re-scalig give the same model with a differet couplig costat (RG)

44 Absece of piig at p=/. Chapma-Kolmogorov P ',.5P.5 ' d'. Reormaliatio group P.5 ' P ' P ',.5 P ' ',.5 ' d P P d' This gives a cotradictio. The piig probability must be ero. Note: we used mootoicity i (maximum priciple).

45 Vaishig of Piig Probability at p=.5 (log scale)

46 Log probability vs. /(p-/)

47 Z Sketch of proof of l P ~ - C/(p-) (G. Kasya, 7) Tau

48 / 4 3/ if / if,,... ; / ;... / ;... / ;,,..., 4 / p p P P P P P p Estimatig the probability of remaiig iside the parabola Probabilities of trasitioig betwee time ad 3/4 with differet values of the couplig coefficiet.

49 Z Large Deviatios Estimate Exit from parabolic regio Tau -.5 P 3/ 4) / () ( e, C

50 Z Most likely exit path Tau Follow the flow (to =) for time Go agaist (large) flow for time / /

51 Large Deviatios Estimate for exp because exp exp, exp / / / 3 C ds d ds C ds e s ds d ds s v ds d P s p p p `Actio asymptotics (Varadha, Aecott, etc.)

52 Lower boud for piig probability () l exp / 4 3/ () 3 )) l ( ( l p p C e C e C P e C e C e P P p C C C C e C p p p C p p Re-scaled betas! Large deviatios estimate

53 Coclusios Piig of stock prices at strikes ear expiratio dates ca be explaied i terms of price-impact due to demad for deltas Liear model is exactly solvable. Agrees very well with data. Power-law price-impact fuctios such as those used i Ecoophysics gives rise to SDEs that geeralie the liear model The price impact fuctio dp / P p Q produces piig at the strike if ad oly if p>/. Models of price-impact with p>.5 are therefore more likely to be observed i practice, sice they are cosistet with the observable pheomeo of piig.

54 Refereces Krisha, Hari I. Nelke The effect of stock piig o optio prices, RISK, December Avellaeda, M. ad M.D. Lipki A market-iduced mechaism for stock piig, Quatitative Fiace, vol 3, pp 47-45, Dec. 3 (sub. Mar 3) Ni, S.X., N. Pearso ad A. M. Poteshma Stock Price Clusterig o Optio Expiratio Dates, Workig Paper, U. Illiois at Urbaa-Champaig, Jue 3 Jeai, Marc, Iori, Giulia ad Samuel, David Modelig Stock Piig, Available at SSRN: Kasya, Geady, Avellaeda, M. ad Lipki M., forthcomig Kasya, Geady, Ph. D. Thesis, NYU, forthcomig

Power-law Price-impact Models and Stock Pinning near Option Expiration Dates. Marco Avellaneda Gennady Kasyan Michael D. Lipkin

Power-law Price-impact Models and Stock Pinning near Option Expiration Dates Marco Avellaneda Gennady Kasyan Michael D. Lipkin PDE in Finance, Stockholm, August 7 Summary Empirical evidence of stock pinning