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

Evnine-Vaughan Associates, Inc.

A multi-timescale statistical feedback model of volatility:

Stylized facts

and

implications for option pricing

Lisa Borland

Acknowledgements: Jeremy Evnine

Roberto Osorio Jean-Philippe Bouchaud

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

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 Æ higher volatility)

• Time-Reversal asymmetry

τ

o Empirical --- Gaussian

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

• Stochastic volatility

• Levy noise

• GARCH

• Multifractal models

Popular models

Problems

• Typically converge too quickly to Gaussian

• Less parsimonious

The Standard Stock Price Model

S

Y ln

=

σ

µ

ω

σ

µ

dt

d

dY

=

+

The Standard Stock Price Model

σ

µ

2

1

ω

d

P

d

dt

dP =

)

2

exp(

2

1

)

(

t

t

P

ω

π

ω

=

−

ω

σ

µ

dt

d

dY

=

+

S

Y ln

=

Ω

+

=

dt

d

dY

µ

σ

The Generalized Returns Model

ω

d

P

d

)

(

Ω

=

Ω

Ω

+

=

dt

d

dY

µ

σ

The Generalized Returns Model

The Generalized Returns Model

The Generalized Returns Model

The Generalized Returns Model

Tsallis Distribution Nonlinear Fokker-Planck 2 2 2

2

1

Ω

=

d

P

d

dt

dP

_β

ω

d

P

d

)

(

Ω

=

Ω

Ω

+

=

dt

d

dY

µ

σ

In other words:

State dependent deterministic model

t t t t t

a

q

b

d

d

ω

_]

)

1

(

[

+

−

Ω

=

Ω

)

(S

Ω

=

Ω

Extensions to Model: i q i q i

P

d

dY

σ

ω

)]

(

[

Ω

−

Ω

=

0

=

Ω

Extensions to Model: i q i q i

P

d

dY

σ

ω

)]

(

[

Ω

−

Ω

=

0

=

Ω

Ω

P

d

dY

σ

∑

ω

Ω

Ω

=

)]

|

(

[

Not a perfect model of returns:

Well-defined starting price and time

Nevertheless:

Reproduces fat-tails and volatility clustering Closed form option-pricing formulae

Example European Call Q rT _{S T K} e c = − max[ ( )− ,0] Stock Price

⎭

⎬

⎫

⎩

⎨

⎧

Ω

2

−

+

Ω

=

∫

rT

P

dt

S

T

S

)

(

exp

)

0

(

)

(

σ

Example European Call Q rT _{S T K} e c = − max[ ( )− ,0] Stock Price

⎭

⎬

⎫

⎩

⎨

⎧

Ω

2

−

+

Ω

=

∫

rT

P

dt

S

T

S

)

(

exp

)

0

(

)

(

σ

Integrate using generalized Feynman-Kac 2

)

(

T

Ω

∝

γ

Example European Call Q rT _{S T K} e c = − max[ ( )− ,0] Stock Price

⎭

⎬

⎫

⎩

⎨

⎧

Ω

−

−

2

−

+

Ω

=

₍

₀

₎

_exp

₍

₎

₍

₁

₎

₍

₎

)

(

rT

T

q

g

T

S

T

S

σ

γ

d

d

≤

Ω

≤

}

)

(

{

S

T

>

K

_Æ

Example European Call Q rT _{S T K} e c = − max[ ( )− ,0]

∫

₋

_Ω

_Ω

=

)

(

)

)

(

(

_S

_T

_K

_P

_d

e

c

)

0

(

M

e

KN

S

−

=

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

q=1.5

Volatility Smiles

o Empirical implied vols __ q=1.43 implied vols

72 76 80 84 Strike 9 10 11 12 13 14 Vo l

72 76 80 84 Strike 9 10 11 12 13 14 Vo l

72 76 80 84 Strike 9 10 11 12 13 14 Vo l

72 76 80 84 Strike 9 10 11 12 13 14 Vo l

72 76 80 84 Strike 9 10 11 12 13 14 Vo l

Example Currency Futures:

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

∝

S

S

dS

Ω

+

=

Sdt

S

d

dS

µ

σ

ω

d

P

d

)

(

Ω

=

Ω

0 1 2 3 4 Time [Years] -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 Return [Hourl y ]

Example European Call Q rT _{S T K} e c = − max[ ( )− ,0]

∫

₋

_Ω

_Ω

=

)

(

)

)

(

(

_S

_T

_K

_P

_d

e

)

0

(

M

e

KN

S

c

=

−

5

.

0

−

=

α

q=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.

4OCT30A 4OCT35A 4OCT40A 4OCT45A 4OCT50A Series 0 5 10 15 cB id

Call Bid/Ask Theoretical value

4NOV35A 4NOV40A 4NOV45A 4NOV50A V12 0 2 4 6 8 10 12 cB id

Call Bid/Ask Theoretical value

0 5 10 15 20 cB id

4DEC25A 4DEC30A 4DEC35A 4DEC40A 4DEC45A 4DEC50A 4DEC50A 4DEC50A V56

Call Bid/Ask Theoretical value

5JAN10A 5JAN15A 5JAN20A 5JAN25A 5JAN30A 5JAN35A 5JAN40A 5JAN45A 5JAN50A V23 0 10 20 30 cB id

Call Bid/Ask Theoretical value

5MAR30A 5MAR35A 5MAR40A 5MAR45A 5MAR50A V34 0 5 10 15 cB id

Call Bid/Ask Theoretical value

6JAN15A 6JAN20A 6JAN25A 6JAN30A 6JAN35A 6JAN40A 6JAN45A 6JAN50A V45 1 11 21 31 cBid

Call Bid/Ask Theoretical value

)

,

,

(

σ

α

q

V

S

dS

V

_⎟

=

⎠

⎞

⎜

⎝

⎛

=

Volatility

)

,

,

(

σ

α

q

V

S

dS

V

_⎟

=

⎠

⎞

⎜

⎝

⎛

=

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

:

D

A

<

A

:

D

A

>

_A

_D

T

−

• 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:

• 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:

Merton Model and Credit Spread

>

−

=<

max[

,

0

]

A

D

S

A

S

D

=

−

σ

σ

Merton Model and Credit Spread

>

−

=<

max[

,

0

]

A

D

S

A

S

D

=

−

σ

σ

D

De

=

⎟

⎠

⎞

⎜

⎝

⎛

−

=

−

De

D

T

r

y

1

_log

Merton Model and Credit Spread

>

−

=<

max[

,

0

]

A

D

S

A

S

D

=

−

σ

σ

e

DN

M

A

S

=

−

D

De

=

⎟

⎠

⎞

⎜

⎝

⎛

−

=

−

De

D

T

r

y

1

_log

Analysis

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

Q 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 aer os pac e, aut o m a nuf ac tu ri ng, a ir lines C o m m un ic at ion, I T e lec tr ic al , c ons tr uc ti on m a c h iner y ener gy E qui pm en ts (m edi c a l and el ec tr oni c ) F inan c ial s e rv ic es re ta il , per s onal pr oduc ts , f ood pr oc e s s in g

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 0.37 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 Standard model “Reality”

Credit Implied volatility

Results

- q mainly in range 1.2 – 1.5 and α = 0.3

Summary

Non-Gaussian model well describes many features of:

Stock Markets Option Markets

Debt and Credit Markets

Now :

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

P

y

y

d

w

W

dy

σ

∑

ω

=

_[

₍

_|

_)]

)

)

(

)

1

(

1

(

1

q

y

y

Z

P

=

−

−

β

−

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

P

y

y

d

w

W

dy

σ

∑

ω

=

_[

₍

_|

_)]

)

)

(

)

1

(

1

(

1

q

y

y

Z

P

=

−

−

β

−

Motivation: Traders act on all different time horizons

0

j ij

w

=

δ

More General

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

τ

ω

σ

y

=

∆

)]

[(

1

z

y

y

w

W

−

=

∑

σ

ARCH-like

(

)

(

y

y

)

j

i

z

z

z

−

−

+

=

τ

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

τ

ω

σ

y

=

∆

)]

[(

1

z

y

y

w

W

−

=

∑

σ

ARCH-like

(

)

(

)

(

)

)

(

i

j

y

y

z

y

y

j

i

z

z

z

−

−

+

−

−

+

=

τ

τ

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

)]

[(

1

z

y

y

w

W

−

=

∑

σ

τ

ω

σ

y

=

∆

i

j

w

=

(

−

)

(

)

(

)

(

)

)

(

i

j

y

y

z

y

y

j

i

z

z

z

−

−

+

−

−

+

=

τ

τ

ARCH-like

1 0 z z g

τ

α

z

Parameters: _{controls the tails}

controls the memory

the elementary time scale

base volatility

Calibration U n i v e r s a l 1 0 2 min 7 300 / 1 15 . 1 85 . 0 z z z = = = =

τ

α

controls the tails

controls the memory

the elementary time scale

base volatility

Volatility-Volatility correlation variogram

α − 2

(Build-up ) and decay of kurtosis

Elementary timescale tau =1/300 day Signature of jumps in real data?

Multifractal scaling

l

A

x

x

l

M

(

)

=

|

−

|

=

Evolution of

σ

σ

e

Volatility relaxation

Also time-reversal assymmetry

∑

0

.

9

≈

z

Analytic results

• Volatility-volatility correlations: decay as • Model well-defined with power-law tails for • Volatility normally diffusive

α

− 2

l

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

t z z y ∆ − = ∆ 2 0 2 1 ) (

1

,

1

<

α

>

z

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

Implications for option pricing

• Price depends on past path history :

Low vol period Æ different price than high vol period

• Returns:

Tsallis-Student distributions on all time-scales

q Æ 1 in a predictable way

• Approximation:

Use single-time model with

q(T).

ασ

Conclusions

- Simple multi-time scale model

-Captures many statistical properties of real returns

-Closed form solution for single time case: options,credit

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)