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

Transcription

1 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

2 Summary Empirical evidence of stock pinning (Ni-Pearson-Poteshman, 3) Linear price-impact model (Avellaneda & Lipkin ) Non-linear price-impact model (Ave., Kasyan & Lipkin, 7) Numerical simulation of pinning for different price-impact functions Phase transition at p/ Rigorous mathematical proofs for the different regimes. Bibliography

3 Pinning on Option Expiration Dates $5. B. Stock B pinned Share Price A. $.5 Stock A did not Time $. Option expiration Friday, (3 rd Friday of the month).

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

5 Statistical Evidence of Pinning Stock Price Clustering on Option Expiration Dates, Preprint, June 6, 3 Authors: Sophie Xiaoyan Ni, Neil Pearson and Allen M. Poteshman (U. of Illinois Urbana-Champaign) Data. Ivy DB (OptionMetrics) Jan 996, Sep : All stocks traded in US exchanges All options traded in US exchanges End of day bid-ask quotes, volume, open interest. CBOE Statistics Open interest and trading volume, Jan 996 to Dec 4 Investor Categories: Market Makers, Firm Prop Traders, Large Firm Clients, Discount Firm Clients

6 The U. Illinois Urbana study At least 8 expiration dates 4,395 optionable stocks on at least one date 84,449 optionable stock-expiration pairs, non-optionable stocks on at least one date 47,7 non-optionable stock-expiration pairs

7 Percentage of non-optionable stocks closing within $.5 of an integer multiple of $5 % 3. 5 Expiration Friday Relative Trading Date from Option Expiration Date (Courtesy: Ni, Pearson & Poteshman)

8 Percentage of optionable stocks closing within $.5 of an integer multiple of $5 3 % Relative Trading Date from Option Expiration Date (Courtesy: Ni, Pearson & Poteshman)

9 Percentage of optionable stocks closing within $.5 of a strike price % Relative Trading Date from Option Expiration Date (Courtesy: Ni, Pearson & Poteshman)

10 Percentage of non-optionable stocks closing within $.5 of an integer multiple of $5 % Relative Trading Date from Option Expiration Date (Courtesy: Ni, Pearson & Poteshman)

11 Percentage of optionable stocks closing within $.5 of an integer multiple of $5 % Relative Trading Date from Option Expiration Date (Courtesy: Ni, Pearson & Poteshman)

12 Percentage of optionable stocks closing within $.5 of a strike price % Relative Trading Date from Option Expiration Date (Courtesy: Ni, Pearson & Poteshman)

13 Our Model: Feedback Due to Demand for Deltas Assumption. Open Interest is unusually large Assumption. Market-makers professional delta-hedgers are net very long options Proposed mechanism for pinning: Hedgers are net long options, hence long Gamma. They sell stock when it rises and buy stock when it falls. Since the aggregate amount of stock required is large compared to typical daily trading volume, this drives the stock to the strike price

14 JDEC Mar Put & Call Open Interest Contracts /9/ // /3/ /5/ /7/ /9/ // /3/ /5/ /7/ 3// 3/3/ 3/5/ 3/7/ 3/9/ 3// 3/3/ 3/5/ Date Average traded vol in stocks MM shares Notional number of shares corresponding to OI 5.6 MM shares

15 3/5/ 3/6/ 3/4/ In search for an explanation JDEC in March /6/ // // // /3/ /6/ /7/ /8/ /3/ /3/ 5/3/ 6/3/ 7/3/ 8/3/ 9/3/ /3/ 3/3/ Large sale of options on this day

16 Taking into account demand for stock: Price-Impact Functions ds S E < D V > p < D V > >> p. p.5 Farmer, Lillo, Mantegna X. Gabaix p linear model, (A. & Lipkin) p.5 convex model (Bouchaud, ) Choice of p is a fundamental question in Econophysics.

17 Linear Model ( A & Lipkin, ) Price-Demand Elasticity Eq. S S E D D Price-response due to demand for deltas S S E. OI D δ δ B. - S. Delta for one option

18 Estimating the Demand for Deltas using Black-Scholes ( ),, ln ln ), (, 3/ τ σ τ στ τ π δ σ µ σ τ σ µ τ σ δ τ δ δ a y e a y t a K S y K S d d N t T dt t From Black-Scholes

19 Dynamics for Stock Price ds S E. OI D δ dt t + σdw y ln S K dy E. OI D π y σ a( T t) 3/ ( T t) e ( y+ a( T t )) σ ( T t ) dt + σdw `coupling constant restoring force bounded support noise

20 Dimensionless Variables., ln,, T D E OI T a K S T T y T t s T y πσ σ α σ σ σ ( ) ( ) dw ds e s s d s s + + ) ( ) ( 3/ ) ( ) ( α α

21 Dimensionless Model (alpha) for Linear Price-Impact Function d ( s) 3/ e ( s) ds + dw Linear restoring force with increasing coupling with time and compact support.

22 The Potential Well Price experiences a force that becomes stronger, more localied near expiration s. s.5 ln(s/k)/(sigma*sqrt(tau)) d ds ( s) 3/ e ( s ) ( α, ) s

23 Solution of the Avellaneda-Lipkin linear response model (p) s F e F s F +, 3/ τ τ τ Assume Alpha Forward Fokker-Planck equation: Look for solution of the form: ( ) ( ) τ ς ς φ τ φ τ s F unknown,, exp,

24 ODE for the `Phase Function (WKB) ( ) ( ) ( ) ( ) ( ) ( ) ( ) s e s s F c c O c e e O e 3/ ' ' 3/ exp, ' ' ' - ' ' ' ' ' φ ςφ φ τ ς φ ςφ φ τ τ ςφ φ τ φ ςφ φ ς ς ς Eikonal Equation Exact solution of the FFP Equation!

25 A Formula for the Pinning Probability P(, s) exp Satisfies : lim P(, s) s + lim P(, s) s + s e ( s ) Prob ( ( ) ( ) ) e e

26 Pinning Probability: Dependence on Beta Pinning Probability pro b Beta Zo.5 Zo E. OI D πσ T - Increases with OI - Decreases with volat, expiration - Decreases with the distance to strike

27 Pinning probability: dependence on _ % 8% 6% 4% % % 8% 6% 4% % % probability ln(s/k)/(vol*sqrt(t)) Alpha Beta.

28 Dimensionless Model for Power-Law Price-Impact Function (p>) Price change Price impact+ noise ds S δ const. t p δ sign dt t + σdw d ( sign 3 p / s) p ( ) e p ( s) ds + dw Dimensionless eq. without irrelevant drift terms (alpha).

29 Pinning under non-linear priceimpact models Proposition: (i) If p</, there is no pinning, i.e. P[() ()], for all. (ii) If p>/ pinning occurs with finite probability (<) and ln P(() () ) (, ) C p P pin e C p, p > /

30 Calculation of Pinning Probabilities by MC Simulation (Gennady Kasyan) Smooth fit near p.5

31 Absence of Pinning for p</ (I) d Ω Ω p τ dt + τ pr p ( r) r sg( r) e dw Novikov condition: If we have E W e ( drift ) dt < the measure induced by the -process in path-space is absolutely continuous with respect to Wiener Measure.

32 Absence of Pinning for p</ (II) Verify that Novikov condition holds: ( ) ( ) / if, ) ( drift < < < Ω Ω p e E e E e E p L t p t dt W dt W dt W ζ τ

33 Self-similarity approach (RG) Kasyan (7) : exploit the self-similarity properties of the model ' τ ' a a τ t' ( a ) + a t Parabolic re-scaling transformation d' a ( τ ') p p Ω ' dt' + dw τ ' ' a p Re-scaling give the same model with a different coupling constant (RG)

34 P Absence of pinning at p/. Chapman-Kolmogorov P [ ( ) ( ) ] π ( ',.5) P [ ( ) (.5) ' ]. Renormaliation group P [ ' ] [ ( ) (.5) ' ] P ( ) ( ) [ ' ] ( ( ) ) π ',.5 d' P [ ( ) ( ) ] [ ( ) ( ) ] π ( ',.5) P ( ) ( ) < P [ ( ) ( ) ] d' d' This gives a contradiction. The pinning probability must be ero. Note: we used monotonicity in (maximum principle).

35 Vanishing of Pinning Probability at p.5 (log scale)

36 Log probability vs. /(p-/)

37 Sketch of proof of ln P ~ - C/(p-) (G. Kasyan, 7).5.5 Z Tau

38 τ n P Estimating the probability of remaining / 4 n, [ ( ) ] P ( ) < ; ( τ) < / ;... ( τ n ) { n P ( τ ) < / ; n,,... } n n n n,,... { n < / ;...} inf inf p inside the parabola < < n n+ P P { ( ) < / ( ) } n τ τ ( p ) ( n( p) ) n n n { ( 3/ 4) < / ( ) } Probabilities of transitioning between time and 3/4 with different values of the coupling coefficient.

39 Large Deviations Estimate Exit from parabolic region.5.5 Z Tau P { 3/ 4) > / () < } C ( e, >>

40 Most likely exit path.5.5 Z Tau Follow the flow (to ) for time Go against (large) flow for time τ / /

41 Large Deviations Estimate P τ ( γ ) exp v( γ, s) dγ ds ds `Action asymptotics (Varadhan, Aencott, etc.) exp τ τ / dγ + ds γ p ( s) 3 p/ e pγ ( s ) ds exp C τ τ / ds because dγ ds exp { C} for >>

42 Lower bound for pinning probability [ ] ( ) ( ) ( ) { } ( ) ( ) ( ) ( ) [ ] ( ) ( ) () ln exp / 4 3/ () 3 )) ln ( ( ln << < < + p p C e C e C P e C e C e P P n p n C n C n C C e n C n p n p n n p n C p n p n Re-scaled betas! Large deviations estimate

43 Conclusions Pinning of stock prices at strikes near expiration dates can be explained in terms of price-impact due to demand for deltas by hedgers Modeling the demand for deltas using power-law price-impact functions such as those used in Econophysics gives rise to SDEs that generalie the Avellaneda-Lipkin linear model The price impact function dp / P Q p produces pinning at the strike if and only if p>/. Models of price-impact with p>.5 are therefore more likely to be observed in practice, since they are consistent with the observable phenomenon of pinning.

44 References Krishnan, Hari I. Nelken The effect of stock pinning on option prices, RISK, December Avellaneda, M. and M.D. Lipkin A market-induced mechanism for stock pinning, Quantitative Finance, vol 3, pp 47-45, Dec. 3 (sub. Mar 3) Ni, S.X., N. Pearson and A. M. Poteshman Stock Price Clustering on Option Expiration Dates, Working Paper, U. Illinois at Urbana-Champaign, June 3 Jeannin, Marc, Iori, Giulia and Samuel, David Modeling Stock Pinning, Available at SSRN: Kasyan, Gennady, Avellaneda, M. and Lipkin M., forthcoming Kasyan, Gennady, Ph. D. Thesis, NYU, forthcoming