Evnine-Vaughan Associates, Inc. A multi-timescale statistical feedback model of volatility: Stylized facts and implications for option pricing Lisa Borland October, 2005
Acknowledgements: Jeremy Evnine Roberto Osorio Jean-Philippe Bouchaud Benoit Pochart
Layout Stylized facts of markets - Why we need a new model The non-gaussian model -Properties -Applications: Options and Credit The multi-time scale model -Capturing the stylized facts Work in progress and conclusions
Properties of Financial Time-Series Power Law distributions, persistent over very many timescales: minutes to weeks Cumulative distribution power law tail -3 Gopikrishnan,Plerou,Nunes Amaral,Meyer,Stanley (1999)
Properties of Financial Time-Series Power Law distributions, persistent Slow decay to Gaussian, as τ 0.25 Volatility clustering and correlation Volatility relaxation (Omori law) Close-to log-normal distribution of volatility Returns normally diffusive over time-scales Leverage effect (Skew: Negative returns higher volatility) Time-Reversal asymmetry
o Empirical --- Gaussian q=1.43 Tsallis Distribution
Consequences Risk control: under-estimate rare events Derivative markets: (options, credit) wrong model of underlying leads to wrong pricing, wrong hedging
Challenge A model that can reproduce the stylized facts A model that can reproduce option prices, credit etc. A model that captures the correct dynamical features Desirable Intuition Parsimony Analytic tractability
Popular models Stochastic volatility (Heston 1993) Levy noise GARCH Multifractal models (Bacry,Delour,Muzy, 2001) Problems Typically converge too quickly to Gaussian Less parsimonious Do not reproduce time reversal assymmetry
The Standard Stock Price Model Y= ln S dy = µ dt + σdω < < µ : rate of σ : volatility dω > = 0 return dω(t)dω( t') >= δ ( t t')
The Standard Stock Price Model Y= ln S dy = µ dt + σdω µ : rate of σ : volatility < ω > = 0 return < ω(t) ω( t') >= δ ( t t') dp = dt 1 2 2 d P 2 dω Fokker-Planck Equation P( ω) = 1 2πt 2 ω exp( 2t ) Gaussian Distribution
The Generalized Returns Model Borland L, Phys. Rev.Lett 89 (2002) Borland L, Quantitative Finance 2 (2002) dy = µ dt + σdω
The Generalized Returns Model Borland L, Phys. Rev.Lett 89 (2002) Borland L, Quantitative Finance 2 (2002) dy = µ dt + σdω dω = P( Ω) 1 q 2 dω
The Generalized Returns Model Borland L, Phys. Rev.Lett 89 (2002) Borland L, Quantitative Finance 2 (2002) dy = µ dt + σdω dω = P( Ω) 1 q 2 dω dp dt = 1 2 2 d P dω 2 q 2 Nonlinear Fokker-Planck P = 1 Z( t) (1 (1 q) β ( t) Ω 1 2 1 q ) Tsallis Distribution
In other words: State dependent deterministic model dω t = [ a + ( q 1) b Ω t t 1 2 2 t ] dω t Work with Ω = Ω(S) as a computational tool allowing us to find the solution
Extensions to Model: dy i = 1 q 2 σ[ Pq ( Ωi Ω)] dω i Current Model: Ω = 0 More realistic model: Ω eg moving average
Extensions to Model: dy i = 1 q 2 σ[ Pq ( Ωi Ω)] dω i Current Model: Ω = 0 More realistic model: Ω eg moving average Or: dy i = i σ 1 j= 0 [ P q ( Ω i Ω j )] 1 q 2 dω i (see later in this talk)
Not a perfect model of returns: Well-defined starting price and time Nevertheless: Reproduces fat-tails and volatility clustering Closed form option-pricing formulae Success for options and credit (CDS) pricing
Example European Call Stock Price rt c = e max[ S( T ) K,0] Q S( T ) 2 σ = S(0) exp Ω + T rt 2 T 0 P( Ω t ) 1 q dt
Example European Call Stock Price rt c = e max[ S( T ) K,0] Q S( T ) 2 σ = S(0) exp Ω + T rt 2 T 0 P( Ω t ) 1 q dt Integrate using generalized Feynman-Kac γ ( T ) Ω 2 T
Example European Call Stock Price rt c = e max[ S( T ) K,0] Q S( T ) = S(0) exp Ω T + rt 2 σ 2 γ ( T ) (1 q) g( T ) Ω 2 T Payoff if { S ( T ) > K } d1 ΩT d 2
Example European Call rt c = e max[ S( T ) K,0] Q c = e rt d 2 d 1 ( S( T ) K) P q ( Ω T ) dω Τ = S rt ( 0) M q,σ e KNq,σ q = 1: P is Gaussian q >1 : P is fat tailed Tsallis dist.
q=1.5
q=1.5 q=1 K=50, T=0.4, sigma=0.3, r=.06
Volatility Smiles o Empirical implied vols q=1.43 implied vols
Implied Volatility JY Futures 16 May 2002 T=17 days 14 13 Vol 12 11 10 9 72 76 80 84 Strike
Implied Volatility JY Futures 16 May 2002 T=37 days 14 13 Vol 12 11 10 9 72 76 80 84 Strike
Implied Volatility JY Futures 16 May 2002 T=62 days 14 13 Vol 12 11 10 9 72 76 80 84 Strike
Implied Volatility JY Futures 16 May 2002 T=82 days 14 13 Vol 12 11 10 9 72 76 80 84 Strike
Implied Volatility JY Futures 16 May 2002 T=147 days 14 13 Vol 12 11 10 9 72 76 80 84 Strike
Example Currency Futures: (500 options) q Mean square relative pricing error 1. 0.16 1.4 0.008 Benefits of a more parsimonious model: 1) Better pricing - arbitrage opportunities 2) Better hedging
(with Jean-Philippe Bouchaud) The Generalized Model with Skew ds dω = α = µ Sdt + σs dω P( Ω) 1 q 2 dω Volatility ds S S α 1 Leverage Correlation
1.5 1.0 Return [Hourly] 0.5 0.0-0.5-1.0-1.5 0 1 2 3 4 Time [Years]
Example European Call rt c = e max[ S( T ) K,0] Q = e rt d 2 d 1 ( S( T ) K) P q ( Ω T ) dω Τ c = S rt ( 0) M q, α, σ e KNq, α, σ Borland L, Bouchaud J-P, Quantitative Finance (2004)
q=1.5 α = 0.5
BS Implied Volatility Strike
SP500 OX q=1.5, alpha = -1. T=.03 T=0.1 T=0.2 T=0.3 T=0.55 Strike K Strike K
alpha ~ 0.3
Call Bid/Ask Theoretical value 15 TOL 10 cbid 5 0 4OCT30A 4OCT35A 4OCT40A 4OCT45A 4OCT50A Series
Call Bid/Ask Theoretical value 12 10 TOL 8 cbid 6 4 2 0 4NOV35A 4NOV40A 4NOV45A 4NOV50A V12
Call Bid/Ask Theoretical value 20 TOL 15 cbid 10 5 0 4DEC25A 4DEC30A 4DEC35A 4DEC40A 4DEC45A 4DEC50A 4DEC50A 4DEC50A V56
Call Bid/Ask Theoretical value 30 TOL cbid 20 10 0 5JAN10A 5JAN15A 5JAN20A 5JAN25A 5JAN30A 5JAN35A 5JAN40A 5JAN45A 5JAN50A V23
Call Bid/Ask Theoretical value 15 TOL cbid 10 5 0 5MAR30A 5MAR35A 5MAR40A 5MAR45A 5MAR50A V34
cbid Call Bid/Ask Theoretical value 31 TOL 21 11 1 6JAN15A 6JAN20A 6JAN25A 6JAN30A 6JAN35A 6JAN40A 6JAN45A 6JAN50A V45
Volatility ds V = = V S 2 ( q, α, σ )
Volatility ds V = = V S 2 ( q, α, σ ) q-alpha-sigma Volatility vs. VIX 70.00 60.00 50.00 Sqa ISD VIX Close 40.00 30.00 20.00 10.00 0.00 10/28/1995 3/11/1997 7/24/1998 12/6/1999 4/19/2001 9/1/2002 1/14/2004
Options look good. What about pricing credit? Borland L, Evnine J, Pochart B, cond-mat/0505359 (2005) Chirayathumadom R, et al, Investment Practice Report Project, Stanford University (2004)
Equity is a call option on underlying assets of firm Assets = Debt + Equity Merton Model (1974) A T < D : Bond holders receive AT Stock holders receive 0 A T > D : Stock holders receive A T D Bond holders receive D
Key assumptions - Underlying assets follow stochastic log normal process - Debt in terms of single zero coupon bond - Black-scholes valuation for European call option Asset Process: da = µa dt + σadz
Key assumptions - Underlying assets follow stochastic log normal process - Debt in terms of single zero coupon bond - Black-scholes valuation for European call option Asset Process: da = µa dt + σadz Generalized Process: α da= µa dt + σa dω
Merton Model and Credit Spread Equity S =< max[ A,0] 0 T D > S ds da 0 σ S = A0 σ A Debt D 0 = A0 S0
Merton Model and Credit Spread Equity S =< max[ A,0] 0 T D > S ds da 0 σ S = A0 σ A Debt D 0 = A0 S0 De yt = D 0 Credit Spread 1 = T D log De 0 y r rt y=risky yield
Merton Model and Credit Spread Equity S =< max[ A,0] 0 T D > S = A M DN rt 0 0 q,α q,α e S ds da 0 σ S = A0 σ A Debt D 0 = A0 S0 De yt = D 0 Credit Spread 1 = T D log De 0 y r rt y=risky yield
Analysis Sectors 1 through 7 are Aerospace, Communication, Construction, Energy, High tech equipment, Financial services and Retail Q values Industry sectors Q q across industries q 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 aerospace, auto manufacturing, airlines Communication, IT electrical, construction machinery energy Equipments(medical and electronic) Financial services retail, personal products, food processing 1.6 1.5 1.4 1.3 1.2 1.1 1 1 Companies Q
Credit Implied volatility 0.37 0.32 Reality q=1, alpha=0 q=1.2, alpha=1 q=1.4, alpha=1 q=1.4, alpha=0.5 q=1.4, alpha=0 0.27 0.22 0.17 Standard model 0.12 0.4 0.6 0.8 1.0 1.2 1.4 1.6 d D/A(0)
Results - q mainly in range 1.2 1.5 and α = 0.3 - Extremely good model prediction
Summary Non-Gaussian model well describes many features of: Stock Markets Option Markets Debt and Credit Markets Now : Extending model of underlying
A multi-time scale non-gaussian model of stock returns [Borland L., cond-mat/0412526 2004] dy = W i σ 1 j= w ij [ P q ( y i y j )] 1 q d ω i P 1 q q = 1 (1 (1 q) β ( ) 2 ij yi y j ij Z ) Motivation: Traders act on all different time horizons
A multi-time scale non-gaussian model of stock returns [Borland L., cond-mat/0412526 2004] dy = W i σ 1 j= w ij [ P q ( y i y j )] 1 q d ω i P 1 q q = 1 (1 (1 q) β ( ) 2 ij yi y j ij Z ) Motivation: Traders act on all different time horizons Single-time model: wij = δ j0
More General A multi-timescale model for volatility [Borland and Bouchaud,(2005)] ω τ y = σ i σ 2 1 i 1 = W j = w ij z[( y i y j )] ARCH-like z = z z + 2 ( y y ) 2 0 i j ( i j) τ
A multi-timescale model for volatility [Borland and Bouchaud,(2005)] ω τ y = σ i )] [( 1 1 2 j i i j ij y y z w W = = σ ARCH-like ( ) 2 2 1 0 ) ( ) ( ) ( j i j i y y j i z y y j i z z z + + = τ τ
A multi-timescale model for volatility [Borland and Bouchaud,(2005)] y = σω τ i w ij = ( i j) a kurtosis decay σ 2 1 i 1 = W j = w ij z[( y i y j )] ARCH-like skew tails z = z z 1 + ( y y ) + ( y y ) 0 i j ( ) ( ) i j i j i j τ τ z 2 elementary timescale 2
Parameters: g z 2 controls the tails α controls the memory τ the elementary time scale z 0 base volatility z 1 skew
Calibration U n i v e r s a l z 2 = 0.85 α = 1.15 τ = 1/ 300 = 7 min controls the tails controls the memory the elementary time scale z 0 base volatility z 1 skew
Volatility
Distribution of volatility
Volatility-Volatility correlation variogram 2 α l
(Build-up ) and decay of kurtosis Elementary timescale tau =1/300 day Signature of jumps in real data?
Multifractal scaling M n ( l ) = x + x i l i n = A n l ζ n
Matches results for SP500 [Sornette, Malevergne, Muzy, 2003)] Volatility relaxation 2 Evolution of conditioned on an initial volatility σ σ 2 e 2s
Also time-reversal assymmetry Similar to long-memory HARCH(Zumbach &Lynch,2003)
Regression gives z 0. 9 2 X i = l= 1 ( y i y l i l 1+ α 2 )
Summary Analytic results Borland,Bouchaud 2005 Volatility-volatility correlations: decay as Model well-defined with power-law tails for 2 α l z2 < 1, α > 1 Volatility normally diffusive ( y) 2 = z0 1 z 2 t Numerical results Tsallis distribution excellent description on all time-scales Distribution of volatility Multifractal scaling Volatility relaxation Time-reversal assymmetry Tested premise of model on real data
Implications of model to market Soft-calibration (due to long relaxation) Model operating close to an instability Past price changes do influence future investor behavior Jumps (news) in addition to feedback effects
Implications for option pricing Price depends on past path history : Low vol period different price than high vol period Returns: Tsallis-Student distributions on all time-scales q 1 in a predictable way Approximation: Use single-time qασ model with q(t).
Conclusions - Simple multi-time scale model -Captures many statistical properties of real returns -Closed form solution for single time case: options,credit -Current and future work: General analytic solution
References: Borland L, Phys. Rev.Lett 89 (2002) Borland L, Quantitative Finance 2 (2002) Borland L, Bouchaud J-P, Quantitative Finance(2004) Chirayathumadom R, et al, Investment Practice Report Project, Stanford University (2004) Borland L, cond-mat/04122526 (2004) Borland L, Evnine J, Pochart B, cond-mat/0505359 (2005) Borland L, Bouchaud J-P, Muzy J-F, Zumbach G, Wilmott Magazine, (March 2005) Borland L, Bouchaud J-P,arXiv:physics/0507073 (2005)