Stylized facts and implications for option pricingieor.columbia.edu/.../pdf-files/Borland_Lisa.pdfEvnine-Vaughan Associates, Inc. A multi-timescale statistical feedback model of volatility:

Evnine-Vaughan Associates, Inc.

A multi-timescale statistical feedback model of volatility:

Stylized facts and

implications for option pricing

Lisa BorlandOctober, 2005

Acknowledgements:

Jeremy Evnine Roberto Osorio

Jean-Philippe BouchaudBenoit 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 -3Gopikrishnan,Plerou,Nunes Amaral,Meyer,Stanley (1999)

Properties of Financial Time-Series

• Power Law distributions, persistent• Slow decay to Gaussian, as • 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

25.0−τ

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

• Stochastic volatility (Heston 1993)

• Levy noise• GARCH • Multifractal models (Bacry,Delour,Muzy, 2001)

Popular models

Problems• Typically converge too quickly to Gaussian• Less parsimonious• Do not reproduce time reversal assymmetry

The Standard Stock Price Model

SY ln=

)'()'(0

ttt(t)dd dω

−>=<=><

δωω

ty volatili:returnofrate

σµ :

ωσµ ddtdY + =

The Standard Stock Price Model

)'()'(0

ttt(t) ω

−>=<=><

δωω

ty volatili:returnofrate

σµ :

Gaussian Distribution

Fokker-Planck Equation2

2

21

ωdPd

dtdP

=

)2

exp(21)(

2

ttP ω

πω −=

ωσµ ddtdY + =

SY ln=

Ω+ = ddtdY σµ

The Generalized Returns Model Borland L, Phys. Rev.Lett 89 (2002)

Borland L, Quantitative Finance 2 (2002)

ωdPdq

21

)(−

Ω=Ω

Ω+ = ddtdY σµ

The Generalized Returns Model The Generalized Returns Model Borland L, Phys. Rev.Lett 89 (2002)


The Generalized Returns Model The Generalized Returns Model Borland L, Phys. Rev.Lett 89 (2002)


Tsallis Distribution

Nonlinear Fokker-Planck2

22

21

Ω=

−

dPd

dtdP q

qtqtZ

P −Ω−−= 11

2 ))()1(1()(

1 β

ωdPdq

21

)(−

Ω=Ω

Ω+ = ddtdY σµ

In other words:

State dependent deterministic model

ttttt dbqad ω21

2 ])1([ Ω−+=Ω

Work with

)(SΩ=Ω

as a computational tool allowing us to find the solution

Extensions to Model:

i

q

iqi dPdY ωσ 21

)]([−

Ω−Ω=

0=ΩCurrent Model:

eg moving averageMore realistic model: Ω

Extensions to Model:

i

q

iqi dPdY ωσ 21

)]([−

Ω−Ω=

0=ΩCurrent Model:

eg moving averageMore realistic model: Ω

Or:

i

i

j

q

jiqi dPdY ωσ ∑−

=

−

ΩΩ=1

0

21

)]|([

(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 CallQ

rT KTSec ]0,)(max[ −= −

Stock Price

⎭⎬⎫

⎩⎨⎧

Ω2

− + Ω= ∫ −2 T q

tT dtPrTSTS0

1)(exp)0()( σ



Stock Price

⎭⎬⎫

⎩⎨⎧

Ω2

− + Ω= ∫ −2 T q

tT dtPrTSTS0

1)(exp)0()( σ

Integrate using generalized Feynman-Kac

2)(T

T Ω∝ γ



Stock Price

⎭⎬⎫

⎩⎨⎧

Ω−−2

− + Ω=2

2)()1()(exp)0()(TT TgqTrTSTS γσ

21 dd T ≤Ω≤)( KTS >Payoff if



∫ Τ− ΩΩ− =

2

1

)())((d

dTq

rT dPKTSec

σσ ,,)0( qrT

q KNeMS −−=

q = 1: P is Gaussian q >1 : P is fat tailed Tsallis dist.

q=1.5

K=50, T=0.4, sigma=0.3, r=.06

q=1.5q=1

Volatility Smiles

o Empirical implied vols__ q=1.43 implied vols

72 76 80 84Strike

9

10

11

12

13

14

Vol

Implied Volatility JY Futures 16 May 2002 T=17 days

72 76 80 84Strike

9

10

11

12

13

14

Vol


72 76 80 84Strike

9

10

11

12

13

14

Vol


72 76 80 84Strike

9

10

11

12

13

14

Vol


72 76 80 84Strike

9

10

11

12

13

14

Vol


Example Currency Futures: (500 options)

1. 0.16

1.4 0.008

q Mean square relative pricing error

Benefits of a more parsimonious model:

1) Better pricing - arbitrage opportunities

2) Better hedging

The Generalized Model with Skew

1−∝ αSSdS

Ω+ = dSSdtdS ασµ

ωdPdq

21

)(−

Ω=Ω

Volatility

Leverage Correlation

(with Jean-Philippe Bouchaud)

0 1 2 3 4

Time [Years]

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

Ret

urn

[Hou

rly]



∫ Τ− ΩΩ− =

2

1

)())((d

dTq

rT dPKTSe

σασα ,,,,)0( qrT

q KNeMSc −−=

Borland L, Bouchaud J-P, Quantitative Finance (2004)

5.0−=αq=1.5

Strike

BS Implied Volatility

Strike K Strike K

T=.03

T=0.1

T=0.2

T=0.3

T=0.55

SP500 OX q=1.5, alpha = -1.

alpha ~ 0.3

4OCT30A 4OCT35A 4OCT40A 4OCT45A 4OCT50ASeries

0

5

10

15

cBid

Call Bid/Ask Theoretical value

TOL

4NOV35A 4NOV40A 4NOV45A 4NOV50AV12

0

2

4

6

8

10

12cB

id


TOL

0

5

10

15

20

cBid

4DEC25A 4DEC30A 4DEC35A 4DEC40A 4DEC45A 4DEC50A 4DEC50A 4DEC50AV56


TOL

5JAN10A 5JAN15A 5JAN20A 5JAN25A 5JAN30A 5JAN35A 5JAN40A 5JAN45A 5JAN50AV23

0

10

20

30

cBid


TOL

5MAR30A 5MAR35A 5MAR40A 5MAR45A 5MAR50AV34

0

5

10

15

cBid


TOL

6JAN15A 6JAN20A 6JAN25A 6JAN30A 6JAN35A 6JAN40A 6JAN45A 6JAN50AV45

1

11

21

31

cBid


TOL

),,(2

σαqVS

dSV =⎟⎠⎞

⎜⎝⎛=Volatility

),,(2

σαqVS

dSV =⎟⎠⎞

⎜⎝⎛=Volatility

q-alpha-sigma Volatility vs. VIX

0.00

10.00

20.00

30.00

40.00

50.00

60.00

70.00

10/28/1995 3/11/1997 7/24/1998 12/6/1999 4/19/2001 9/1/2002 1/14/2004

Sqa ISD VIX Close

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)

Merton Model (1974)

• Equity is a call option on underlying assets of firm

Assets = Debt + Equity

:DAT < Bond holders receive TA

Stock holders receive 0

:DAT > Stock holders receive DAT −

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:

dA= µA dt + σA Ωdα

Merton Model and Credit Spread

>−=< ]0,max[ 0 DAS TEquity

000 SAD −=

AS AdAdSS σσ 00 =

Debt



000 SAD −=

AS AdAdSS σσ 00 =

Debt0DDe yT =−

⎟⎠⎞

⎜⎝⎛−=− −rTDe

DT

ry 0log1Credit Spread y=risky yield



000 SAD −=

AS AdAdSS σσ 00 =αα ,,00 q

rTq DNeMAS −−=

Debt0DDe yT =−

⎟⎠⎞

⎜⎝⎛−=− −rTDe

DT

ry 0log1Credit Spread y=risky yield

AnalysisSectors 1 through 7 are Aerospace, Communication, Construction, Energy, High tech equipment, Financial services and Retail

Q

00.20.40.60.8

11.21.41.6

aero

spac

e, a

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man

ufac

turin

g,ai

rline

s

Com

mun

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ion,

IT

elec

trica

l,co

nstru

ctio

nm

achi

nery

ener

gy

Equ

ipm

ents

(med

ical

and

elec

troni

c)

Fina

ncia

l ser

vice

s

reta

il, p

erso

nal

prod

ucts

, foo

dpr

oces

sing

Q values across Industry sectors

1

1.1

1.2

1.3

1.4

1.5

1.6

1

Companies

Q

q across industries q

0.4 0.6 0.8 1.0 1.2 1.4 1.6

d

0.12

0.17

0.22

0.27

0.32

0.37q=1, alpha=0q=1.2, alpha=1q=1.4, alpha=1q=1.4, alpha=0.5q=1.4, alpha=0

Standard model

“Reality”

Credit Implied volatility

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 MarketsOption MarketsDebt and Credit Markets

Now :

Extending model of underlying

A multi-time scale non-Gaussian model of stock returns[Borland L., cond-mat/0412526 2004]

i

i

j

qjiqij dyyPw

Wdy ωσ ∑

−

−∞=

−=1

1)]|([

))()1(1(1 21jiij

ij

qq yyq

ZP −−−=− β

Motivation: Traders act on all different time horizons

A multi-time scale non-Gaussian model of stock returns[Borland L., cond-mat/0412526 2004]

i

i

j

qjiqij dyyPw

Wdy ωσ ∑

−

−∞=

−=1

1)]|([

))()1(1(1 21jiij

ij

qq yyq

ZP −−−=− β

Motivation: Traders act on all different time horizons

0jijw δ=Single-time model:

More GeneralA multi-timescale model for volatility [Borland and Bouchaud,(2005)]

τωσ iy =∆

)][(1 12

ji

i

jij yyzw

W−= ∑

−

−∞=

σ

ARCH-like

( )22

0 )( ji yyji

zzz −−

+=τ

A multi-timescale model for volatility [Borland and Bouchaud,(2005)]

τωσ iy =∆

)][(1 12

ji

i

jij yyzw

W−= ∑

−

−∞=

σ

ARCH-like

( )221

0 )()()( jiji yy

jizyy

jizzz −

−+−

−+=

ττ

A multi-timescale model for volatility [Borland and Bouchaud,(2005)]

kurtosis decay

)][(1 12

ji

i

jij yyzw

W−= ∑

−

−∞=

σ

τωσ iy =∆ aij jiw −−= )(

tailsskew

elementary timescale

( )221

0 )()()( jiji yy

jizyy

jizzz −

−+−

−+=

ττ

ARCH-like

1

0

z

z

g

τ

α

2zParameters: controls the tails

controls the memory

the elementary time scale

base volatility

skew

Calibration Universal

1

0

2

min7300/1

15.1

85.0

z

z

z

==

=

=

τ

α

controls the tails

controls the memory

the elementary time scale

base volatility

skew

Volatility

Distribution of volatility

Volatility-Volatility correlation variogram

α−2l

(Build-up ) and decay of kurtosis

Elementary timescale tau =1/300 day

Signature of jumps in real data?

Multifractal scaling

nlAxxlM nn

ilinζ=−= + ||)(

Evolution of 2σ conditioned on an initial volatility se22σVolatility relaxation

Matches results for SP500 [Sornette, Malevergne, Muzy, 2003)]

Also time-reversal assymmetry

Similar to long-memory HARCH(Zumbach &Lynch,2003)

∑∞

=+

−−=

11

2)(l

liii l

yyX α

Regression gives 9.02 ≈z

Analytic results Borland,Bouchaud 2005

• Volatility-volatility correlations: decay as

• Model well-defined with power-law tails for

• Volatility normally diffusive

α−2l

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

tz

zy ∆−

=∆2

02

1)(

1,12 >< αz

Summary

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-scalesq 1 in a predictable way

• Approximation: Use single-time model with q(T).

ασq

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, 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)

Documents

Stylized facts and implications for option pricingieor.columbia.edu/.../pdf-files/Borland_Lisa.pdfEvnine-Vaughan Associates, Inc. A multi-timescale statistical feedback model of volatility: