A Market-Induced Mechanism For Stock Pinning

Transcription

1 IMPA, Rio d Janiro Tnth Annual Workshop on Drivativ Scuritis and Risk Managmnt Cntr For Applid Probability, Columbia Univrsity A Markt-Inducd Mchanism For Stock Pinning Marco Avllanda Courant Institut of Mathmatical Scincs Nw York Univrsity Michal D. Lipkin Katama Trading LLC, Amrican Stock Exchang Pinning on Option Expiration Dats $5. B. Stock B pinnd Shar Pric A. $.5 Stock A did not Tim $.

2 Pric KO: Sp 8 to Oct Day /7 Clos$45.5 Expiration KO: Sp 8 to Oct 7 3 Volum,, 9,, 8,, 7,, 6,, 5,, 4,, 3,,,,,, Day Avg. daily vol. (yrs) 5 M shars

3 KO : Oct 5,6, Pric Tick Statistical Evidnc of Pinning Stock Pric Clustring on Option Expiration Dats, Prprint, Jun 6, 3 Authors: Sophi Xiaoyan Ni, Nil Parson and Alln M. Potshman (U. of Illinois Urbana-Champaign) Data. Ivy DB (OptionMtrics) Jan 996, Sp : All stocks tradd in US xchangs All options tradd in US xchangs End of day bid-ask quots, volum, opn intrst. CBOE Statistics Opn intrst and trading volum, Jan 996 to Dc 4 Invstor Catgoris: Markt Makrs, Firm Prop Tradrs, Larg Firm Clints, Discount Firm Clints 3

4 UI Urbana Study: Optionabl vs. Non-Optionabl Stocks At last 8 xpiration dats 4,395 optionabl stocks on at last on dat 84,449 optionabl stock-xpiration pairs, non-optionabl stocks on at last on dat 47,7 non-optionabl stock-xpiration pairs Prcntag of non-optionabl stocks closing within $.5 of an intgr multipl of $5 % 3. 5 Expiration Friday Rlativ Trading Dat from Option Expiration Dat (Courtsy: Ni, Parson & Potshman) 4

5 Prcntag of optionabl stocks closing within $.5 of an intgr multipl of $5 3 % Rlativ Trading Dat from Option Expiration Dat (Courtsy: Ni, Parson & Potshman) Prcntag of optionabl stocks closing within $.5 of a strik pric % Rlativ Trading Dat from Option Expiration Dat (Courtsy: Ni, Parson & Potshman) 5

6 Prcntag of non-optionabl stocks closing within $.5 of an intgr multipl of $5 % Rlativ Trading Dat from Option Expiration Dat (Courtsy: Ni, Parson & Potshman) Prcntag of optionabl stocks closing within $.5 of an intgr multipl of $5 % Rlativ Trading Dat from Option Expiration Dat (Courtsy: Ni, Parson & Potshman) 6

7 Prcntag of optionabl stocks closing within $.5 of a strik pric % Rlativ Trading Dat from Option Expiration Dat (Courtsy: Ni, Parson & Potshman) Non-optionabl stocks that latr bcam optionabl closing within $.5 of an intgr multipl of $.5 % Rlativ Trading Dat from Option Expiration Dat 7

8 Optionabl stocks that wr prviously non-optionabl closing within $.5 of an intgr multipl of $.5 % rlativ trading dat from option xpiration dat Optionabl stocks that latr bcam non-optionabl closing within $.5 of an intgr multipl of $.5 % Rlativ Trading Dat from Option Expiration Dat 8

9 Non-optionabl stocks that wr prviously optionabl closing within $.5 of an intgr multipl of $.5 % Rlativ Trading Dat from Option Expiration Dat In sarch for an xplanation.5 JDEC in March /6/ // // // /3/ /6/ Larg sal of options on 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/ 9

10 6 JDEC Mar Put & Call Opn Intrst Contracts /9/ // /3/ Avrag tradd vol in stocks MM shars /5/ /7/ /9/ // /3/ /5/ /7/ Dat 3// 3/3/ 3/5/ 3/7/ 3/9/ 3// 3/3/ 3/5/ Notional numbr of shars corrsponding to OI 5.6 MM shars Our Modl: Fdback Du to Dmand for Dltas Assumption. Opn Intrst is unusually larg Assumption. Markt-makrs profssional dlta-hdgrs ar nt vry long options Proposd mchanism for pinning: Hdgrs ar nt long options, hnc long Gamma. Thy sll stock whn it riss and buy stock whn it falls. Sinc th aggrgat amount of stock rquird is larg compard to daily trading volum (supply), this drivs th stock to th strik pric

11 Accounting for Pric Impact of Hdgrs D D E S S Pric-Dmand Elasticity Eq. Pric-rspons du to dmand for dltas Dlta for on option - S. B.. δ δ D E OI S S Estimating th Dmand for Dltas ( ),, ln ln ), (, 3/ σ σ π δ σ µ σ σ µ σ δ δ δ a y a y t a K S y K S d d N t T dt t From Black-Schols

12 Dynamics for Stock Pric Add nois: (xognous stuff) ds S E. OI D δ dt + σdw t S y ln K dy E. OI y D π σ a( T t) 3/ ( T t) ( y+ a( T t )) σ ( T t) dt + σdw Dynamics for Stock Pric ds S E. OI D δ dt + σdw t S y ln K dy E. OI y D π σ a( T t) 3/ ( T t) ( y+ a( T t )) σ ( T t ) dt + σdw `coupling constant rstoring forc boundd support nois

13 Dimnsionlss Variabls y σ T, s t T, y σ T S ln σ T K a T α, σ β E. OI D πσ T β d ( α( s) ) ( s) 3/ ( + α (s )) ( s) ds + dw Th Potntial Wll ln(s/k)/(sigma*sqrt(tau)) Pric xprincs a forc that bcoms strongr, mor localid nar xpiration s. s.5 d ds ( s) 3/ ( s ) ( α, β ) s 3

14 Mont Carlo Simulation of SDE y (ln S/K) Paths Bta ~., Alpha t (days to xpiration) Pinning Probability: Dpndnc on Bta Pinning Probability prob Bta Zo.5 Zo E. OI β D πσ T - Incrass with OI - Dcrass with volat, xpiration - Dcrass with th distanc to strik 4

15 Pinning Probability: Dpndnc of Z % 8% 6% 4% probability % % 8% 6% 4% % % Bta ln(s/k)/(vol*sqrt(t)) Cumulativ PDF for pric at xpiration dat (Bta.)..8 probability Z() 5

16 6 Solving th modl s F F s F +, 3/ β Assum Alpha Forward Fokkr-Planck quation: Look for solution of th form: ( ) ( ) ς ς φ φ s F unknown,, xp, ODE for th `Phas Function (WKB) ( ) ( ) ( ) ( ) ( ) ( ) ( ) s s s F c c O c O 3/ ' ' 3/ xp, ' ' ' - ' ' ' ' ' β φ ςφ φ β ς φ βςφ φ βςφ φ φ ςφ φ ς ς ς Eikonal Equation Exact solution of th FFP Equation!

17 A Formula for th Pinning Probability P(, s) xp Satisfis : lim P(, s) s + lim P(, s) s + β s ( s ) Prob ( ( ) ( ) ) β Comparison btwn simulation and xact rsult: P(,) % 8% 6% 4% probability % % 8% 6% 4% % % Alpha Bta ln(s/k)/(vol*sqrt(t))

18 Z BETA PINNING PROBABILITY SIMULATION 8.35% 7.35% 5.5% 3.55% 37.8% 43.95% 49.% 53.68% 58.4% 6.45% 65.78% 68.8% 7.6% 74.5% 76.3% THEORETICAL 9.5% 7.47% 5.% 3.88% 38.% 43.78% 48.9% 53.6% 57.85% 6.7% 65.% 68.39% 7.8% 73.9% 76.3% Non-ro Alpha: asymptotics µ + σ Thorm : Lt α. For ach ε < /, thr xists σ T a constant C indpndn t of α, β, and, such that P ( ) ( ( ) ( ) ) t t xp ε α α Cβ α + xp ε β t ( ε ) ( ) / ε t n / Proof: WKB xpansion in up to trms of ordr / 8

19 OK, but dos this story xplain stock pinning? W know that stock pinning at option xpiration xists for optionabl stocks Our modl maks two assumptions to justify pinning - Larg numbr of dltas rlativ to total volum - Markt-makrs ar long options Obsrvations with markt-makrs nt long (~$.5) 9

20 Markt-makrs + firm propritary tradrs nt long Markt-makrs nt short

21 Markt-makrs + firm propritary tradrs nt short Pinning vs. `Dpltion Invrt sign of th coupling constant: gt ``dpltion Trminal CDF Bta +.5 Trminal CDF Bta Data dos not indicat `dpltion for MM nt short Instad it indicats a vry slight pinning (unxplaind) Howvr, most pinning taks plac whn MM ar nt long ( consistnt with modl)

22 Effct on Front-Month Option Prics 3-day call prics Valu Strik Pric Bta.5 Bta. Compar B-S with xpctd valu of payoff with rspct to nw procss 3-day implid volatilitis Volatility 45% 4% 35% 3% 5% % 5% % 5% % Strik pric Bta.5 Bta. Effct on Scond-Month Option Prics 6-day call prics Valu ($) Bta.5 Bta Strik ($) 6-Day Implid Vols 45% 4% Volatility (%) 35% 3% 5% Bta.5 Bta % 5% Strik ($)

23 Conclusions & Furthr Rsarch Pinning of optionabl stocks on option xpiration dats was statistically stablishd in Ni, Parson and Potshman (3, prprint). W proposd a modl that provids a markt-drivn mchanism for pinning basd on pric-impact du to dlta-hdging. Assumptions: - Larg opn Intrst/ (avg. stock volum) - Markt-makrs (hdgrs) ar nt long Our modl: a Langvin quation with a forc that bcoms singular at xpiration and has shrinking domain of influnc Modl is analytically tractabl using WKB and xactly solvabl in a spcial cas. Conditioning th data on MM nt long / short givs rsults which ar consistnt with th proposd mchanism (x post) Estimating pinning probability conditional on stock pric, volatility and tim-tomaturity is possibl -- mor work rmains to b don Rfrncs Krishnan, Hari I. Nlkn Th ffct of stock pinning on option prics, RISK, Dcmbr Avllanda, M. and M.D. Lipkin A markt-inducd mchanism for stock pinning, Quantitativ Financ, vol 3, pp 47-45, 3 (sub. Mar 3) Ni, S.X., N. Parson and A. M. Potshman Stock Pric Clustring on Option Expiration Dats, Working Papr, U. Illinois at Urbana-Champaign, Jun 3 3