Lisa Borland. A multi-timescale statistical feedback model of volatility: Stylized facts and implications for option pricing

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1 Evnine-Vaughan Associates, Inc. A multi-timescale statistical feedback model of volatility: Stylized facts and implications for option pricing Lisa Borland October, 2005

2 Acknowledgements: Jeremy Evnine Roberto Osorio Jean-Philippe Bouchaud Benoit Pochart

3 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

4

5 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)

6 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

7 o Empirical --- Gaussian q=1.43 Tsallis Distribution

8

9 Consequences Risk control: under-estimate rare events Derivative markets: (options, credit) wrong model of underlying leads to wrong pricing, wrong hedging

10 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

11 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

12

13 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')

14 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 d P 2 dω Fokker-Planck Equation P( ω) = 1 2πt 2 ω exp( 2t ) Gaussian Distribution

15 The Generalized Returns Model Borland L, Phys. Rev.Lett 89 (2002) Borland L, Quantitative Finance 2 (2002) dy = µ dt + σdω

16 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

17 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 = d P dω 2 q 2 Nonlinear Fokker-Planck P = 1 Z( t) (1 (1 q) β ( t) Ω q ) Tsallis Distribution

18 In other words: State dependent deterministic model dω t = [ a + ( q 1) b Ω t t t ] dω t Work with Ω = Ω(S) as a computational tool allowing us to find the solution

19 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

20 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)

21

22 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

23 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

24 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

25 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

26 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.

27 q=1.5

28 q=1.5 q=1 K=50, T=0.4, sigma=0.3, r=.06

29 Volatility Smiles o Empirical implied vols q=1.43 implied vols

30 Implied Volatility JY Futures 16 May 2002 T=17 days Vol Strike

31 Implied Volatility JY Futures 16 May 2002 T=37 days Vol Strike

32 Implied Volatility JY Futures 16 May 2002 T=62 days Vol Strike

33 Implied Volatility JY Futures 16 May 2002 T=82 days Vol Strike

34 Implied Volatility JY Futures 16 May 2002 T=147 days Vol Strike

35 Example Currency Futures: (500 options) q Mean square relative pricing error Benefits of a more parsimonious model: 1) Better pricing - arbitrage opportunities 2) Better hedging

36 (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

37 Return [Hourly] Time [Years]

38 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)

39 q=1.5 α = 0.5

40 BS Implied Volatility Strike

41 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

42 alpha ~ 0.3

43

44 Call Bid/Ask Theoretical value 15 TOL 10 cbid 5 0 4OCT30A 4OCT35A 4OCT40A 4OCT45A 4OCT50A Series

45 Call Bid/Ask Theoretical value TOL 8 cbid NOV35A 4NOV40A 4NOV45A 4NOV50A V12

46 Call Bid/Ask Theoretical value 20 TOL 15 cbid DEC25A 4DEC30A 4DEC35A 4DEC40A 4DEC45A 4DEC50A 4DEC50A 4DEC50A V56

47 Call Bid/Ask Theoretical value 30 TOL cbid JAN10A 5JAN15A 5JAN20A 5JAN25A 5JAN30A 5JAN35A 5JAN40A 5JAN45A 5JAN50A V23

48 Call Bid/Ask Theoretical value 15 TOL cbid MAR30A 5MAR35A 5MAR40A 5MAR45A 5MAR50A V34

49 cbid Call Bid/Ask Theoretical value 31 TOL JAN15A 6JAN20A 6JAN25A 6JAN30A 6JAN35A 6JAN40A 6JAN45A 6JAN50A V45

50 Volatility ds V = = V S 2 ( q, α, σ )

51 Volatility ds V = = V S 2 ( q, α, σ ) q-alpha-sigma Volatility vs. VIX Sqa ISD VIX Close /28/1995 3/11/1997 7/24/ /6/1999 4/19/2001 9/1/2002 1/14/2004

52 Options look good. What about pricing credit? Borland L, Evnine J, Pochart B, cond-mat/ (2005) Chirayathumadom R, et al, Investment Practice Report Project, Stanford University (2004)

53 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

54 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

55 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

56 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

57 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

58 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

59 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 aerospace, auto manufacturing, airlines Communication, IT electrical, construction machinery energy Equipments(medical and electronic) Financial services retail, personal products, food processing Companies Q

60 Credit Implied volatility 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= Standard model d D/A(0)

61 Results - q mainly in range and α = Extremely good model prediction

62 Summary Non-Gaussian model well describes many features of: Stock Markets Option Markets Debt and Credit Markets Now : Extending model of underlying

63 A multi-time scale non-gaussian model of stock returns [Borland L., cond-mat/ ] 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

64 A multi-time scale non-gaussian model of stock returns [Borland L., cond-mat/ ] 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

65 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

66 A multi-timescale model for volatility [Borland and Bouchaud,(2005)] ω τ y = σ i )] [( j i i j ij y y z w W = = σ ARCH-like ( ) ) ( ) ( ) ( j i j i y y j i z y y j i z z z + + = τ τ

67 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

68 Parameters: g z 2 controls the tails α controls the memory τ the elementary time scale z 0 base volatility z 1 skew

69 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

70

71 Volatility

72 Distribution of volatility

73 Volatility-Volatility correlation variogram 2 α l

74 (Build-up ) and decay of kurtosis Elementary timescale tau =1/300 day Signature of jumps in real data?

75 Multifractal scaling M n ( l ) = x + x i l i n = A n l ζ n

76 Matches results for SP500 [Sornette, Malevergne, Muzy, 2003)] Volatility relaxation 2 Evolution of conditioned on an initial volatility σ σ 2 e 2s

77 Also time-reversal assymmetry Similar to long-memory HARCH(Zumbach &Lynch,2003)

78 Regression gives z X i = l= 1 ( y i y l i l 1+ α 2 )

79 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

80 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

81 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).

82 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

83 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/ (2004) Borland L, Evnine J, Pochart B, cond-mat/ (2005) Borland L, Bouchaud J-P, Muzy J-F, Zumbach G, Wilmott Magazine, (March 2005) Borland L, Bouchaud J-P,arXiv:physics/ (2005)

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