An empirical An empirical characterization of the characterization of the

market processmarket process

(T(The fractional volatility model)he fractional volatility model)

Contents1. Market prices are non-differentiable2. Geometric Brownian motion ?3. Volatility as a process. Volatility models4. The induced volatility process5. Time scales and pdf’sAppendix: Derivation of the Black-Scholes formula6. Option pricing. Risk-neutral approachMathExc1 – Stochastic integration with respect to fBm7. An option pricing equation using fBm stochastic integrationMathExc2 – An introduction to fractional calculus8. Leverage, fBm representation and fractional calculus interpretation

1. Market prices are non-differentiable

Detrending is needed for stationarity

The returns process

r(t)=log(S(t+1))-log(S(t))

Automatically detrended

)())(log( )(1 tStS dt

dtSdt

d =

2. Geometric Brownian motion ?

?(a basis for most mathematical finance studies –Black-Scholes, etc.)Consequences:

(i)

Price changes would be lognormal

(ii)

Self-similar (Law(Xat)=Law(aH Xt)) with Hurst coefficient = 1/2

E S t S tS t

H( ) ( )( )

+ − − ≈Δ Δ Δμ

))(2

))(2

(lnexp(

)(21)(ln 2

22

2 tT

tTSS

tTSSp t

T

t

T

−

⎥⎦

⎤⎢⎣

⎡−−−

−−

=σ

σμ

πσ

dWdtSdS

t

t σμ +=

2. Geometric Brownian motion ?Empirical tests :

P(r1) is not lognormal

r1=log(S(t+1)/S(t))

Deviations from scaling

Larger deviations for high-frequency data

σ is not constantReturn

Conclusion : Nor do returns follow geometric Brownian motionnor is σ constant (not even a smooth function of S and t)

Δ≈Δ−=Δ Δ− dtdS

SrSSr t

ttttt

1)()log()log()(

“Stilized” experimental facts(i) Returns have nearly no autocorrelation(ii) The autocorrelations of |rt|d decline slowly withincreasing lag. Long memory effect(iii) Leptokurtosis : asset returns have distributionswith fat tails and excess peakedness at the mean(iv) Autocorrelations of sign rt are insignificant(v) Volatility clustering : tendency of large changes to follow large changes and small changes to follow smallchanges. Volatility occurs in bursts.(vi) Volatility is mean-reversing and the distribution isclose to lognormal or inverse gamma(vii) Leverage effect : volatility tends to rise more following a large price fall than following a price rise(viii) Why volatility is important : Uncertainty and riskare the driving factors for investors’ behavior

3. Volatility as a processWhen the future is uncertain investors are less likely to invest. Therefore uncertainty (volatility) would have to be changing over time.“... build a forecasting model for variance and make it a well-definedprocess ...” (Robert Engle – 1982)Structural model

Conditional variance

Homoscedasticity = variance of errors is constantHeteroscedasticity = variance of errors is not constant

uxxy ++++= ...33221 βββ

termerrorufactors...,, 321 βββ

[ ],..., 2122

−−= tttt uuuEσ

2. Volatility modelsARCH(q) (Autoregressive conditionally heteroscedastic)

GARCH (1,1) (Generalized ARCH)

IGARCH (Integrated GARCH)Leverage :GJR (Glosten, Jagannathan, Runkle)

EGARCH (exponential GARCH)

11 =+βα

2222

2110

2 ... qtqttt uuu −−− ++++= αααασ

21

2110

2−− ++= ttt u βσαασ

otherwiseuifIuI

tt

tttt

0;01)(

11

21

21110

2

=<=+++=

−−

−−− βσγαασ

⎥⎥⎦

⎤

⎢⎢⎣

⎡−+++=

−

−

−

−− πσ

ασ

γσβωσ 2)ln()ln(2

1

1

21

121

2

t

t

t

ttt

uu

Stochastic volatility modelsIn GARCH models, the conditional volatility is a deterministicfunction of past quantities. In Stochastic Volatility models it isitself a random process.Heston model

Two-time scales model (Perello, Masoliver)

Comte, Renault

W’ is fractional Brownian motion

')()(

0')(20

22 dWdtd

dWdWdWdtSdS

ttt

ttt

γσσσσ

ρσμ

+−Ω−=

<=+=

'')(0''')(

0')(

000000

0

dWdtddWdWdWdtd

dWdWdWedtSdS

tt

ttt

ttt

γξξξγξξξ

ρμ ξ

+−Ω−==+−Ω−=

<=+=

')ln()(ln)(

dWdtkddWdtSdS

tt

ttt

γσθσσμ

+−=+=

4. The induced volatility processLet logSt be a stochastic process defined on a tensor product probability space Ω⊗Ω’logSt(ω,ω’) with Ω being Wiener space(M1) Then, if logSt(ω,ω’) is square integrable in Ω

for fixed ω’

σt(ω, ω’) is called the “Induced volatility”

(E1) Obtained from the dataσt

2 (·, ω’)≈ var(log St)/(T0 - T1)

(μ=0)

tt

t dBdtS

dS )',()',( ωσμω •+=•

What does the data suggest for σt ?σt is not self similar

However Rσ(t) is Σ log σ(nδ) = β t + Rσ(t)H ≈ 0.8 - 0.9

E t tt

Hσ σσ

( ) ( )( )

+ − ≠Δ Δ

E R t R tR t

Hσ σ

σ

( ) ( )( )

+ − =Δ Δ

What does the data suggest for σt ?Recall:If a process Xt has finite variance, stationary increments and is self-similar, thenCov(Xs,, Xt)=(|s|2H+|t| 2H-|s-t| 2H)E(X1

2)(M2) The simplest such process is a zero-mean Gaussian process, Fractional Brownian motion BH

twith long-range dependence for H>1/2Conclusion :log σt = β + (k/δ) ( BH

t - BHt - δ )

σt modeled by a stochastic exponential of fractional noise

The fractional volatility model (FVM)Two coupled processes :

dSt = μ St dt + σt St dBt

log σt = β + (k/δ) ( BHt - BH

t - δ )

log σt driven by fractional noise, not by fractional Brownian motion

5. Time scales and pdf’sFrom

log σt is a Gaussian process with mean β and covariance

Then

( ))()(ln δδ

βσ −−+= tBtBkHHt

( )⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛−−−= H

tktBtBk 2

2

21)()(exp δ

δδ

δθσ

{ }HHH ussuuskus 2222

2

22

),( −−+−++−= δδδ

ψ

( )⎭⎬⎫

⎩⎨⎧ −−= −− 222

2

1 2logexp

21)( HH kk

pδ

βσδσπ

σδ

5. Time scales and pdf’sand for the returns

with

The probability distribution of the returns depends on the observation time scale δ

⎪⎪

⎭

⎪⎪

⎬

⎫

⎪⎪

⎩

⎪⎪

⎨

⎧

Δ

⎟⎟⎠

⎞⎜⎜⎝

⎛Δ⎟⎟⎠

⎞⎜⎜⎝

⎛−−

Δ=

Δ+

Δ+2

22

2 22

logexp

21)(log

σ

σμ

πσσ

t

t

t

tS

S

SSp

∫∞

Δ+Δ+ ≅0

)(log)()(logt

t

t

t

SSppd

SSP σδδ σσ

5. Time scales and pdf’s (NYSE 1973-2000)H=0.83 k=0.59 β= - 5 δ=1 Δ=1

6. Time scales and pdf’s (NYSE 1973-2000)

6. Time scales and pdf’s (USD-Euro 05-06 2001)

6. Time scales and pdf’s (Scaling ??)

6. Time scales and pdf’s (Scaling ??)

Closed form and return asymptoticsFrom

one obtains

Asymptotic behavior :

( )2

20222

2)(8θ

λδθ β

Δ−Δ=== − rrkCe H

∫∞

Δ=Δ0

))(()())(( rppdrP σδδ σσ

21

log1

1)(1

41))((

2

=

⎟⎠⎞

⎜⎝⎛ −−

−Γ

Δ=Δ

z

dzd

CH

zek

rPλ

δ λδθπ

( )2log111))((λ

δ λCerP

−

Δ≈Δ

Appendix:Derivation of the Black-Scholes formula

Assumptions: 1) The price of the underlying instrument St follows a geometric Brownian motion defined by

where Wt is a Wiener process with constant drift μ and volatility σ. 2) It is possible to short sell the underlying stock. 3) There are no arbitrage opportunities. 4) Trading in the stock is continuous. 5) There are no transaction costs or taxes. 6) All securities are perfectly divisible (i.e. it is possible to buy any fraction of a share). 7)It is possible to borrow and lend cash at a constant risk-free interest rate. 8) The stock does not pay a dividend

Let V(S,σ) be the value of a call option. By Itō's lemma we have

Now consider a trading strategy under which one holds one optionand continuously trades in the stock in order to hold

shares. At time t, the value of these holdings will be

The composition of this portfolio, called the delta-hedge portfolio, will vary from time-step to time-step. Let R denote the accumulated profit or loss from following this strategy. Then over the time period [t, t + dt], the instantaneous profit or loss is

Substituting dV and dS from the equations above we are left with

This last equation contains no dW term. That is, it is entirely riskless (delta neutral). Thus, given that there is no arbitrage, the rate of return on this portfolio must be equal to the rate of return on any other riskless instrument. Assuming the risk-free rate of return to be r we must have over the time period [t, t + dt]

If we now insert the expression for П and divide through by dt we obtain the Black–Scholes PDE:

This is the law of evolution of the value of the option. With the assumptions of the Black–Scholes model, this partial differential equation holds whenever V is twice differentiable with respect to Sand once with respect to t.

Solution of the Black-Scholes equationFor a call option the PDE above has the boundary condition

Introduce the change-of-variables

Then the Black–Scholes PDE becomes a diffusion equation

After some algebra we obtain

where

and Φ is the standard normal cumulative distribution function.The formula for the price of a put option follows from this via put-call parity

)()(),()( tStPTtBKtV +=⋅+

The Greeks under Black–Scholes:

)()( tVtC = )()( tPtC =

Option pricing in FVM. “Risk-neutral approach”

Option pricing in FVM. “Risk-neutral approach”

Option pricing in FVM. “Risk-neutral approach”

Option pricing in FVM. “Risk-neutral approach”

The option pricing equation in FVM

The option pricing equation in FVM

The option pricing equation in FVM

Option pricing in FVM. Numerical solutions

Option pricing in FVM. Numerical solutions

The option pricing in FVM. Analytical solution

10/23/2008

8. Leverage and the fractional calculus interpretation

10/23/2008

10/23/2008

10/23/2008

Fractional Brownian motion and fractional calculus

10/23/2008

10/23/2008

10/23/2008

The fractional calculus interpretation of the FVM

10/23/2008

ReferencesR V M and M. J. Oliveira; A data-reconstructedfractional volatility model, Economics, discussionpaper 2008-22R V M and M. J. Oliveira; Fractional volatility andoption pricing, arxiv:cond-mat/0404684R. V. M.; The fractional volatility model: An agent-based interpretation, Physica A: Stat. Mech. and Applic. 387 (2008) 3987-3994R. V. M.; A fractional calculus interpretation of the fractional volatility model, Nonlinear Dynamics, doi:10.1007/s11071-008-9372-0