Comte, Renault - Long Memory in Continuous-Time Stochastic Volatility Models.pdf

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Mathematical Finance, Vol. 8, No. 4 (October 1998), 291323

LONG MEMORY IN CONTINUOUS-TIME STOCHASTIC

VOLATILITY MODELS

FABIENNECOMTE

URA 1321 and LSTA, University Paris 6 and CREST-ENSAE

ERICRENAULT

GREMAQ-IDEI, University of Toulouse and Institut Universitaire de France

This paper studies a classical extension of the Black and Scholes model for option pricing, oftenknown as the Hull and White model. Our specification is that the volatility process is assumed not onlyto be stochastic, but also to have long-memory features and properties. We study here the implicationsof this continuous-time long-memory model, both for the volatility process itself as well as for the

global asset price process. We also compare our model with some discrete time approximations. Thenthe issue of option pricing is addressed by looking at theoretical formulas and properties of the implicitvolatilities as well as statistical inference tractability. Lastly, we provide a few simulation experimentsto illustrate our results.

KEYWORDS: continuous-time option pricing model, stochastic volatility, volatility smile, volatilitypersistence, long memory

1. INTRODUCTION

If option prices in the market were conformable with the BlackScholes (1973) formula,all the BlackScholes implied volatilities corresponding to various options written on the

same asset would coincide with the volatility parameter of the underlying asset. In reality

this is not the case, and the BlackScholes (BS) implied volatility imp

t,T heavily depends

on the calendar timet, the time to maturity T t, and the moneyness of the option. Thismay produce various biases in option pricing or hedging when BS implied volatilities are

used to evaluate new options or hedging ratios. These price distortions, well-known to

practitioners, are usually documented in the empirical literature under the terminology of

the smile effect, where the so-called smile refers to the U-shaped pattern of implied

volatilities across different strike prices.

It is widely believed that volatility smiles can be explained to a great extent by a modeling

of stochastic volatility, which could take into account not only the so-called volatility

clustering (i.e., bunching of high and low volatility episodes) but also the volatility effects

of exogenous arrivals of information. This is why Hull and White (1987), Scott (1987), and

Melino and Turnbull (1990) have proposed an option pricing model in which the volatility

of the underlying asset appears not only time-varying but also associated with a specific

A previous version of this paper has benefitted from helpful comments from S. Pliska, L. C. G. Rogers, M. Taqqu,and two anonymous referees. All remaining errors are ours.

Initial manuscript received December 1995; final revision received September 1997.

Address correspondence to F. Comte, ISUP Boite 157, Universite Paris 6, 4 Place Jussieu, 75 252 Paris cedex05 France; e-mail: [email protected].

c 1998 Blackwell Publishers, 350 Main St., Malden, MA 02148, USA, and 108 Cowley Road, Oxford,OX4 1JF, UK.

291


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292 FABIENNE COMTE AND ERIC RENAULT

risk according to the stochastic volatility (SV) paradigm

d S(t)

S(t)=(t, S(t))dt+ (t)dw1(t)

d(ln (t))

=k(

ln (t))dt

+dw2(t),

(1.1)

whereS(t) denotes the price of the underlying asset, (t) is its instantaneous volatility, and

(w1(t), w2(t))is a nondegenerate bivariate Brownian process. The nondegenerate feature

of(w1, w2) is characteristic of the SV paradigm, in contrast to continuous-time ARCH-

type models where the volatility process is a deterministic function of past values of the

underlying asset price.

The logarithm of the volatility is assumed to follow an OrnsteinUhlenbeck process,

which ensures that the instantaneous volatility process is stationary, a natural way to gener-

alize the constant-volatility Black and Scholes model. Indeed, any positive-valued station-

ary process could be used as a model of the stochastic instantaneous volatility (see Ghysels,Harvey and Renault (1996) for a review). Of course, the choice of a given statistical model

for the volatility process heavily influences the deduced option pricing formula. More pre-

cisely, Hull and White (1987) show that, under specific assumptions, the price at time tof

a European option of exercise date Tis the expectation of the Black and Scholes option

pricing formula where the constant volatility is replaced by its quadratic average over the

period:

2t,T= 1

T

t

T

t

2(u) du,(1.2)

and where the expectation is computed with respect to the conditional probability distri-

bution of2t,T given (t). In other words, the square of implied BlackScholes volatility

imp

t,T appears to be a forecast of the temporal aggregation 2

t,Tof the instantaneous volatility

viewed as a flow variable.

It is now well known that such a model is able to reproduce some empirical stylized

facts regarding derivative securities and implied volatilities. A symmetric smile is well

explained by this option pricing model with the additional assumption of independence

betweenw1 andw2 (see Renault and Touzi (1996)). Skewness may explain the correlation

of the volatility process with the price process innovations, the so-called leverage effect

(see Hull and White 1987). Moreover, a striking empirical regularity that emerges from

numerous studies is the decreasing amplitude of the smile being a function of time to

maturity; for short maturities the smile effect is very pronounced (BS implied volatilities

for synchronous option prices may vary between 15% and 25%), but it almost completely

disappears for longer maturities. This is conformable to a formula like (1.2) because it

shows that, when time to maturity is increased, temporal aggregation of volatilities erases

conditional heteroskedasticity, which decreases the smile phenomenon.

The main goal of the present paper is to extend the SV option pricing model in order

to capture well-documented evidence ofvolatility persistenceand particularly occurrence

of fairly pronounced smile effects even for rather long maturity options. In practice, thedecrease of the smile amplitude when time to maturity increases turns out to be much

slower than it goes according to the standard SV option pricing model in the setting (1.1).

This evidence is clearly related to the so-called volatility persistence, which implies that

temporal aggregation (1.2) is not able to fully erase conditional heteroskedasticity.

Generally speaking, there is widespread evidence that volatility is highly persistent.


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LONG MEMORY IN CONTINUOUS-TIME STOCHASTIC VOLATILITY MODELS 293

Particularly for high frequency data one finds evidence of near unit root behavior of the

conditional variance process. In the ARCH literature, numerous estimates of GARCH

models for stock market, commodities, foreign exchange, and other asset price series are

consistent with an IGARCH specification. Likewise, estimation of stochastic volatility

models show similar patterns of persistence (see, e.g., Jacquier, Polson and Rossi 1994).

These findings have led to a debate regarding modeling persistence in the conditional

variance process either via a unit root or a long memory-process. The latter approach has

been suggested both for ARCH and SV models; see Baillie, Bollerslev, and Mikkelsen

(1996), Breidt, Crato, and De Lima (1993), and Harvey (1993). This allows one to consider

mean-reverting processes of stochastic volatility rather than the extreme behavior of the

IGARCH process which, as noticed by Baillie et al. (1996), has low attractiveness for asset

pricing since the occurence of a shock to the IGARCH volatility process will persist for

an infinite prediction horizon.

The main contribution of the present paper is to introduce long-memory mean reverting

volatility processes in the continuous time Hull and White setting. This is particularlyattractive for option pricing and hedging through the so-called term structure of BS implied

volatilities (see Heynen, Kemna, and Vorst 1994). More precisely, the long-memory feature

allows one to capture the well-documented evidence of persistence of the stochastic feature

of BS implied volatilities, when time to maturity increases. Since, according to (1.2), BS

implied volatilities are seen as an average of expected instantaneous volatilities in the same

way that long-term interest rates are seen as average of expected short rates, the type of

phenomenon we study here is analogous to the studies by Backus and Zin (1993) and Comte

and Renault (1996) who capture persistence of the stochastic feature of long-term interest

rates by using long-memory models of short-term interest rates.

Indeed, we are able to extend Hull and White option pricing to a continuous-time long-

memory model of stochastic volatility by replacing the Wiener process w2 in (1.1) by

a fractional Brownian motion w2 , with restricted to 0 < 12 (instead of|| 0, 0< 0 r(h) r(0)h

h0

0,

which could be interpreted as a near-integrated behavior

r(h) r(0)h

= h 1

hh

0

ln

1

0

if (t) is considered as a continuous-time AR(1) process with a correlation coefficient

near 1.

3Two processes are called equivalent if they coincide almost surely.


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This analogy between a unit root hypothesis and its fractional alternatives has already

been used for unit root tests by Robinson (1993). Robinsons methodology could be a useful

tool for testing integrated volatility against long memory in stochastic volatility behavior.

The concept of persistence that we advance thanks to the fractional framework is that

of long memory instead of indefinite persistence of shocks as in the IGARCH framework.

Indeed, we can prove the following result:

PROPOSITION 2.2. In the context of Proposition 2.1, we have

(i) r(h)is of order O(|h|21)for h +.(ii) lim0 2 f()=cR+, where f()=

R

r(h)ei h dh is the spectral density

of.

Proposition 2.2 illustrates that the volatility process itself (and not only its logarithm) does

entail the long-memory properties (generally summarized as in (i) and (ii) by the behavior

of the covariance function near infinity and of the spectral density near zero) we could

expect in the FSV model.

3. DISCRETE APPROXIMATIONS OF THE FSV MODEL

3.1. The Volatility Process

The volatility process dynamics are characterized by the fact that x(t)= ln (t) is asolution of the fractional SDE (2.1). So we know two integral expressions for x(t)(with

the notations of Section 2.1):

x (t)= t

0

(t s)(1 + ) d x

()(s)= t

0

a(t s) dw2(s),

wherea(t s)is given by (2.2).A discrete time approximation of the volatility process is a formula to numerically eval-

uate these integrals using only the values of the involved processes x ()(s) andw2(s) on

a discrete partition of [0, t]: j/n, j = 0, 1, . . . , [nt].4 A natural way to obtain suchapproximations (see Comte 1996) is to approximate the integrands by step functions:

xn,1(t)= t

0

t [ns]

n

(1 + ) d x

()(s) and xn,2(t)= t

0

a

t [ns ]

n

dw2(s),(3.1)

which gives, neglecting the last terms for large values ofn ,

xn (t)=[nt]

j=1

t j1n

(1

+)

x () j

n and(3.2)xn (t)=

[nt]j=1

a

t j 1

n

w2

j

n

,

4[z] is the integerksuch thatkz


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where we use the following notations: x () ( j

n)= x () ( j

n) x ()( j1

n ) and w2(

j

n)=

w2( j

n) w2( j1

n ).

Indeed, all these approximations converge toward thexprocess in distribution in the sense

of convergence in distribution for stochastic processes as defined in Billingsley (1968); this

convergence is denoted by D

. This result is proved in Comte (1996).

PROPOSITION 3.1. xn,1D x , xn,2 D x, xn D x, and xn D x when n goes to

infinity.

The proxyxn is the most useful for comparing our FSV model with the standard discretetime models of conditional heteroskedasticity, whereas the most tractable for mathematical

work isxn .

3.2. FSV versus FIGARCH

Expression (3.2) provides a proxy xn of x in function of the process x () ( jn ), j =0, 1, . . . , [nt], which is an AR(1) process associated with an innovation processu(

j

n), j=

0, 1, . . . , [nt]. Let us denote by

(1 nL n)x ()

j

n

=u

j

n

(3.3)

the representation of this process, whereL nis the lag operator corresponding to the sampling

scheme j

n, j= 0, 1, . . ., Ln Y( jn )=Y( j1n ),and n= ek/n is the correlation coefficient

for the time interval 1n

.

Since the process x () is asymptotically stationary, we can assume without loss of gen-

erality that its initial value is zero, x ()( j

n) = 0 for j 0, which of course implies

u(j

n)=0 for j 0.Then we can write

x

n jn=j

i=1 (j i+ 1)

n (1 + ) x () in x () i 1n =

j1i=0

(i+ 1) i n (1 + ) L

in

x ()

j

n

.

Thus,

xn (j

n)=

j1

i=0(i+ 1) i n (1

+)

L in

(1 nLn )1u

j

n.(3.4)

Expression (3.4) gives a parameterization of the volatility dynamics in two parts: a long-

memory part that corresponds to the filter+

i=0 aiLin /n

with ai= ((i +1) i )/(1+)and a short-memory part that is characterized by the AR(1) process: (1 nLn)1u( jn ).

We can show that the long-memory filter is long-term equivalent to the usual discrete

time long-memory filter(1 L ) = +i=0 biL i , wherebi= (i+ )/((i+ 1)()),


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in the sense that there is a long-term relationship (a cointegration relation) between the

two types of processes. Indeed, we can show (see Comte 1996) that the two long-memory

processes, Yt=+

i=0 ai u ti and Zt=+

i=0 bi u ti , where aiand bi are defined previouslyand u tis any short-term memory stationary process, are cointegrated: Yt Zt is shortmemory and +i=0|ai bi |


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And by a remark ofthe same type as (3.2), we can also considerYn(t)=[nt]

j=1 ( j1

n )w1(

j

n).

It can be proved that:

LEMMA3.1. YnDY andYn DY , when n grows to infinity.

But from a practical viewpoint, the discretizationsYn andYn are not very useful becausethey are based on the values of the process , which cannot be computed without some other

errors of discretization. Thus we are more interested in the following joint discretization:

n(t)=exp

[nt]j=1

a

t j 1

n

w2

j

n

,(3.6)

Yn(t)=[nt]

j=1 n j 1

n w1

j

n .We can then prove the following proposition.

PROPOSITION 3.2.

Ynn

D

Y

and thus

Sn= lnYnn

D

S

when n .

Another parameterization can be obtained by using n (t) = exp( xn (t)) rather thann(t)= exp( xn (t)); the previous section has shown how this parameterization is givenby and n.

We have something like a discrete time stochastic variance model a la Harvey et al. (1994)

which converges toward our FSV model when the sampling interval 1n

converges toward

zero. The only difference is that, when =0, lnn (t)is not an AR(1) process but a long-memory stationary process. Such a generalization has in fact been considered in discrete

time by Harvey (1993) in a recent working paper. He works withyt= tt, t I I D(0, 1),t= 1, . . . , T, 2t = 2 exp(ht), (1 L)dht= t, t I I D(0, 2 ), 0 d 1. Theanalogy with (3.6) is then obvious, with the remaining problem being the choice of the right

approximation of the fractional derivation studied in the previous subsection. Moreover,our case is a little different from the one studied by Harvey in that we have in mind a

volatility process of the type ARFIMA (1, , 0) where he has an ARFIMA(0, d, 0). But

such discrete time models may be also useful for statistical inference.

4. OPTION PRICING AND IMPLIED VOLATILITIES

4.1. Option Pricing

The maintained assumption of our option pricing model is characterized by the price

model (1.2), where (w1(t), w2(t)) is a standard Brownian motion. Let (, F,P) bethe fundamental probability space. (Ft)t[0,T] denotes the P-augmentation of the filtra-tion generated by (w1(),w2( )), t. It coincides with the filtration generated by(S(),()), tor( S(),x ()()), t, withx (t)=ln (t).

We look here for the call option premium Ct, which is the price at time t T of aEuropean call option on the financial asset of price St at t, with strike Kand maturing at


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timeT. The asset is assumed not to pay dividends, and there are no transaction costs.

Let us assume that the instantaneous interest rate at timet, r(t), is deterministic, so that

the price at timetof a zero coupon bond of maturity T is B(t, T)=exp( Tt

r(u) du).

We know from Harrison and Kreps (1981) that the no free lunch assumption is equivalent

to the existence of a probability distributionQon(,F), equivalent to P, under which the

discountedprice processesare martingales. We emphasize that no change of probability of

the Girsanov type could have transformed the volatility process into a martingale, but there

is no such problem for the price process S(t). This stresses the interest of such models

where the nonstandard fractional properties are set on (t)and not directly on S(t). This

avoids any of the possible problems of stochastic integration with respect to a fractional

process, which does not admit any standard decomposition. Indeed, the process appears

only as a predictible and even L 2 continuous integrand.

Then we can use the standard arguments. An equivalent measure Q is characterized by a

continuous version of the density process ofQ with respect to P (see Karatzas and Shreve

1991, p. 184):

M(t)=exp t

0

(u)d W(u) 12

t0

(u)(u) du

,

where W= (w1, w2) and= (1, 2) is adapted to{Ft} and satisfies the integrabilitycondition

T0

(u)(u) du


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whereEQ(.| Ft) is the conditional expectation operator, givenFt, when the price dynamicsis governed by Q. Sincew1 and are independent under Q, the Q distribution of ln(ST/St)given by dlnSt= (r(t)( (t)2/2)dt+ (t)dw1(t) conditionally on bothFtand the wholevolatility path ( (t))t[0,T]is Gaussian with mean

T

t r(u)du 1

2 T

t (u)2duand variance

Tt (u)2du. Therefore, computing the expectation (4.1) conditionally on the volatilitypath gives:

Ct= S(t)E

Qt

mt

Ut,T+ Ut,T

2

Ft emtEQt m tUt,T Ut,T2Ft ,(4.2)

wherem t= ln

S(t)K B(t,T)

, Ut,T=

Tt

(u)2 du, and(u)= 12

ue

t2/2 dt.The dynamics ofare now given by

ln (t)

(0)=k t

0

ln (u) du t

0

(t s)(1 + ) 2(s) ds

+ w2 (t),

where

w2(t)= t

0

(t s)(1 + ) dw

2(s).

Then differentiatingx (t)=ln (t)with fractional order gives:

d x ()(t)=(kx () (t) + 2(t))dt+ dw2(t),(4.3)

where

x ()(t)= ddt

t0

(t s)(1 )x(s) ds

is the derivative of (fractional) order ofx .

We can give the general solution of (4.3):

x () (t)=

c + t

0

eks 2(s)ds+ t

0

eks dw2(s)

ekt

and deducex by fractional integration.

As usual, when one wants to perform statistical inference using arbitrage pricing models,

two approaches can be imagined: either specify a given parametric form of the risk premium

or assume that the associated risk is not compensated. When trading of volatility is observed

it might be relevant to assume a risk premium on it. But we choose here, for the sake of

simplicity (see, e.g., Engle and Mustafa 1992 or Pastorello et al. 1993 for similar strategiesin short-memory settings) to assume that the volatility risk is not compensated, i.e., that

2=0. Under this simplifying assumption, which has some microeconomics foundations(see Pham and Touzi 1996), the probability distributions ofUt,Tare the same under P and

underQ. In other words the expectation operator in the option pricing formula (4.2) can be

considered with respect to P.


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4.2. Implied Volatilities

Practitioners are used to computing the so-called BlackScholes implicit volatility by

inversion of the BlackScholes option pricing formula on the observed option prices. If

we assume that these option prices are given by (4.2) and that the volatility risk is not

compensated, the BlackScholes implicit volatility appears to be a forecast of the averagevolatility t,Ton the lifetime of the option (

2t,T= (T t)1U2t,T). If we consider the proxy

of theoptionprice (4.2) deduced from a first-order Taylor expansion (around (Tt)1EU2t,T)of the BlackScholes formula considered as a function of2t,T, the BlackScholes implicit

volatility dynamics would be directly related to the dynamics of

2i mp ,T(t)= 1

T t T

t

E

2(u)| Ft

du.

To describe the dynamics of this implicit volatility we start by analyzing the conditional

laws and moments of:

E( (t+ h)| Ft)=exp

g(t+ h) + t

0

a(t+ h s) dw2(s) + 12

h0

a2(x ) d x

forx(t)= ln (t)= g(t) + t0

a(t s) dw2(s), g(t)= x(0) + (1 ekt), and a (x )asusual. Or, if we work with the stationary version of:

E( (t+ h)| Ft)=exp t

a(t+ h s) dw(2)(s) + 1

2

h0

a2(x) d x

.

To have an idea of the behavior of the implicit volatility, we can prove:

PROPOSITION 4.1. yt= E(2(t+ 1)| Ft) is a long-memory process in the sense thatcov(yt,yt+h )is of order O (|h|21)for h + and ]0, 1/2[.

Var(E( (t+ h)| Ft))is of order O (|h|21)for h + if ]0, 1/2[and of ordere

k

|h

|if=0.Proposition 4.1 shows that, thanks to the long-memory property of the instantaneous volatil-

ity process, the stochastic feature of forecasted volatility does not vanish at the very high

exponential rate but at the lower hyperbolic rate. This rate of convergence explains the

stochastic feature of implicit volatilities, even for fairly long maturity options.

SinceT > t, we can set T= t+ . We take= 1 for simplicity and study the long-memory properties of the stationary (if we work with the stationary version of) process

which is now defined by

2imp(t)= 1

0

E

2(t+ u)| Ft du.PROPOSITION 4.2. zt :=2imp(t) is a long-memory process in the sense thatcov(zt,zt+h )

is of order O (|h|21)for h + and ]0, 1/2[.


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We have already documented (see Section 6.5) some empirical evidence to confirm the

theoretical result of Proposition 4.2. Indeed, when we use daily data on CAC40 and option

prices on CAC40 (of the Paris Stock Exchange) and we try to estimate a long-memory

parameter by regression on the log-periodogram (see Robinson 1996), we find that the

stock price process Sis a short-memory process and the B.S. implicit volatility process is

a long-memory one.

Finally, the dynamics of conditional heteroskedasticity of the stock price process Scan

be described through the marginal kurtosis. We are not only able to prove a convergence

property like Corollary 3.2 of Drost and Werker (1996) but also to measure the effect of the

long-memory parameter on the speed of convergence:

PROPOSITION 4.3. Let(h)= E|Y(h)EY(h)|4 = EZ(h)4 denote the fourth centeredmoment of the rate of return Y(h)=ln S(h)

S(0) on[0, h], with Z(t)=

t

0 (u) dw1(u). Then

(h)/ h2 is bounded on R.

Moreover, let kurtY(h)=(h)/(V a rY (h))2 denote the kurtosis coefficient of Y(h). Then

limh0

kurtY(h)=3 E(4)

(E(2))2 >3, for

0,

1

2

at rate h2+1 (continuity in = 0),6 and limh+kurtY(h)= 3for [0, 12 [ at rateh21 if]0, 1

2[,7 and at rate e(k/2)h if=0.

The discontinuity in 0 of the speed of convergence of limh+kurtY(h) with respect to is additional evidence of the persistence in volatility introduced by the parameter. When

there is long memory( >0)the leptokurtic feature due to conditional heteroskedasticity

vanishes with temporal aggregation at a slow hyperbolic rate, while with a usual short-

memory volatility process it vanishes at an exponential rate.

Note that the limit forh going to 0 ofkurtY(h)is close to 3 (and thus the log-returnY is

close to Gaussian) if and only if Var 2 is close to 0, that is, ifis close to deterministic

(small value of the diffusion coefficient ); this leads us back to the standard BlackScholes

world.

5. STATISTICAL INFERENCE IN THE FSV MODEL

5.1. Statistical Inference from Stock Prices

Several methods are provided in Comte and Renault (1996) and Comte (1996) to estimate

the parameters of an OrnsteinUhlenbeck long-memory process, which here is the set

of parameters(, k, , )implied by the first-order equation fulfilled by the log-volatility

process. Those methods of course are all based on a discrete time sample of observations

of one path of ln . Such a path is not available here.

The idea then is to find approximations of the path deduced from the observed S(ti )and to replace the true observations usually used by their approximations in the estimation

procedure. Let us recall briefly that those procedures are as follows:

6That is,kurtY(h) 3[E(4)/(E(2))2] is of orderh 2+1 forh0.7That is,kurtY(h) 3 is of orderh 21 forh +.


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either we find by log-periodogram regression using the semiparametric resultsof Robinson (1996) and (k, , ) by estimating an AR(1) process after fractional

differentiation at the estimated order,

or all parameters are estimated by minimizing the Whittle-type criterium approximat-ing the likelihood in the frequency domain, as studied by Fox and Taqqu (1986) and

Dahlhaus (1989).

The natural idea for approximating is then based on the quadratic covariation ofY(t)=ln(S(t)). Indeed,Yt=

t0

2(s) ds and, if{t1, . . . , tm}is a partition of [0, t] andt0= 0,then

limstep0

m

k=1(Ytk Ytk1 )2 = Ytin probability, where step= Max

1im{|ti ti1|}.

Then as(Yt Yth )/ h h0

2(t)a.s. and provided that high-frequency data are avail-

able, we can think of cumulating the two limits by considering a partition of the partition

to obtain estimates of the derivative of the quadratic variation.

Let [0, T] be the interval of observation, let tk= kT/N,N= np, be the dates ofobservations, and let Ytk, k= 0, . . . ,N, be the sample of observations of the log-prices.Then we have n blocks of length p and we set:

Y(N)t =

[(t N)/T]=[nt]k=0 (Ytk Ytk1 )2 so

that(Y(N)

t

Y(N)

t

h )/ h is computed from the underlying blocks with h

=T/n. In other

words,

2n,p(t)= n

T

[ t NT

]k=[ t N

T ]p+1

(Ytk Ytk1 )2

because[((t (T/n)N)/T]=[t N/ T p]. Then we have:

PROPOSITION 5.1. Let Y(t)=

t

0 (s) dw1(s) and (t)= (t) withgiven by formula

(2.7). Then >0,

limn+p+

supt[0,T]

p1E

2n,p(t) 2(t)2 =0.

Thus p must be as large as possible for the rate of convergence to be optimal. On the

other hand we are interested in large sizes n of the sample of deduced volatilities. This is

the reason there is a trade-off between n and p, taking into account the constraint N= n p.A possible choice could be to choosen and pof order

N.Then we have to estimate , supposed to be a constant, and we notice that the finite

variation terms that have been omitted in Yare known to have no weight in the quadratic

covariation. The estimate of can be chosen here as usual (see Renault and Touzi 1996):

enp t = 1np

npk=1

StkStk1

, t= Tnp

, tk= kTnp , or sometimes: np= npT 1npnp

k=1StkStk1

Stk1,

which completes the estimation procedure.


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5.2. Statistical Inference from Option Prices

Another way to estimate the volatility parameters could be the use of the informational

content of option prices and associated implied volatilities in the spirit of Engle and Mustafa

(1992) or Pastorello et al. (1993) (assuming that the volatility risk is not compensated).

Unfortunately, the non-Markovian feature of the long-memory process implies that the Hulland White option pricing formula is not so simple to invert to recover latent instantaneous

volatilities as in the usual case. Nevertheless, if sufficiently high frequency data are available

to approximate integrals by finite sums, we are able to generalize the Pastorello et al. (1993)

procedure thanks to a first-stage estimate of the long-memory parameter . To see this point,

let us assume for instance that we observe at timesti , i= 0, 1, . . . , n, option pricesCti foroptions of exercise dates ti+ (for a fixed ), that are at the money in the generalizedsense: Sti= KtiB(ti , ti+ ),where Kti is the exercise price of an option traded at dateti .In this case, we know from (4.2) that:

Cti= Sti

2EP

Uti ,ti +

2

1

Fti .(5.1)The information set Fti in the above expectation is defined as the sigma-field generated

by (w1(),(), ti ). But since the two processes w1 and are independent andUti ,ti + is depending on only, the information provided byw

1(), ti is irrelevant inthe expectation (5.1). Moreover, thanks to (2.4) and (2.3), we know that the sigma-field

generated byx ( )

=ln (),

ti , coincides with the sigma-field generated by the short-

memory process x ()(), ti . On the other hand, thanks to (2.3), Uti ,ti + appears like acomplicated function (see Appendix B) ofx () (), ti+ .

In other words, (5.1) gives the option price as a function of:

first, the past values x () (), ti , which define the deterministic part ofUti ,ti +, second, the OrnsteinUhlenbeck parameters(k, , ), which characterize the condi-

tional probability distribution ofx () (), > ti , given the available information Ftisummarized byx () (ti ),

third, the long-memory parameter , which defines the functional relationship be-

tweenUti ,ti + and the processx ().

The BlackScholes implicit volatilityB Simp(ti )is by definition related to the option price

Cti in a one-to-one fashion by

Cti= Sti

2

B Simp(ti )

2

1

.(5.2)

The comparison of (5.1) and (5.2) shows that the dynamics ofB Simp(ti )are determined not

only by the dynamics of the OrnsteinUhlenbeck process x () but also by the complicated

functional relationship between Uti ,ti + and the past values of x(). This is why the BS

implicit volatility is itself a long-memory process whose dynamics cannot analytically be

related to the dynamics of the instantaneous latent volatility. Nevertheless, the relationship


17/33


betweenUti ,ti + and x() can be approximated by (see Appendix B):

U2ti ,ti+

= ti +

ti

exp2


18/33


FIGURE6.1. Simulated path of log-stock price in the long-memory FSV model;N= 1000,h=0.02,(, k, )=(0.3, 1, 0.01).

6.2. An Apparent Unit Root

Another comparison can be made with Baillie et al.s (1996) work. Indeed, they argue

that their discrete time fractional model gives another representation of persistence that can

remain stationary, contrary to usual unit roots models.

Here, we want to show that our model may exhibit an apparent unit root if a wrong

parameterization is assumed for estimation. For that purpose, we look at what is obtained

if the model is estimated as if it were a GARCH(1,1) process:

t= lnSt= tzt, Et1zt= 0, Vart1zt= 12t= + a2t1+ b2t1.

In other words,(1 L)2t= + (1 bL)t where=a+ band is a white noise.We estimate the parameter (,, b) through minimizing l(,1, . . . , T)=

Tt=1(ln

2t+

2t2

t ).The results are reported in Table 6.1 and Figure 6.2. One hundred forty samples

have been generated, starting with 5000 points (with a step h h

=0.1)and with one point

out of ten (i.e., 500 points), kept for the estimation procedure with a step h = 1 and(, k, )= (0.3, 2, 1). We find also an apparent unit root for , and the empirical dis-tribution of clearly appears to be centered at 1. We can see that is the more stable

of the estimated coefficients and is always very near 1. Other tries have been made with

other parameters for the continuous time model with the same kinds of results. Thus,

continuous-time fractional models are good representations of apparent persistence.


19/33


TABLE6.1

An Apparent Unit Root for . A GARCH(1,1) Is Estimated Instead of the True FSV

Simulated. 140 Samples Generated

Empirical mean Empirical std. dev. 1.5843 1.3145

1.0453 0.1109

b 0.1935 0.3821

FIGURE 6.2. Empirical distribution of when the FSV model is estimated as a GARCH

(1,1) process;t= tzt,2t= + a2t1+ b2t1,=a+ b; 140 samples generated.

6.3. Comparison of the Filters

Now we give an illustration of the quality of the continuous-time filter defined by ai=((i+ 1) i )(see Section 3.2) as compared with the usual discrete time one (1 L ) .

We generated Nobservations at step hh= 0.01 of the AR(1) process x () as given informula (3.3) with (, k, )=(0.3, 3, 1), which gave an N-sample ofx as given by (3.2).Then we kept n= N/10 observations of the true AR(1), x () , and of the x process. Weapplied the continuous-time filter at step h= 10hh= 0.1 to then-samplex , which gaveobservations of a process x ()1 ; we applied the discrete time filter at steph= 10hh to thesame n-samplex , which gave observations of a process x

()2 . The paths ofx

(), x()1 , and

x()

2 can be compared. It appears that the continuous-time filter is better than the discrete

time one.

We generated 100 such samples ofx (),x()

1 , andx()

2 and computed L1 andL 2 distances


20/33


TABLE6.2

L1 andL2 Distances between the Original Paths and the Filtered Paths with the Two

Filters. Filter 1 Is the Continuous-Time Filter, Filter 2 the Discrete Time One

x (),x ()1 x () ,x ()2dL1 0.1048 0.2135

dL2 0.1359 0.3607

betweenx () andx()i ,i= 1, 2; that is,

dL 1 (x (),x ()i )= 1nn

j=1|x ()(j ) x ()i (j )|,

dL 2 (x(),x

()i )=

1

n

nj=1

(x ()(j ) x ()i (j ))2, i= 1, 2.

The results are reported in Table 6.2. Even if the numbers do not have any meaning in

themselves, the comparison leads clearly to the conclusion that the first filter is significantly

better. For a convincing comparison of the twelve first partial autocorrelations of the three

samples, see Comte (1996).

6.4. Estimation of by Log-Periodogram Regression in Three Models

Lastly, we compared the estimations ofobtained by regression of lnI() on ln , where

I()is the periodogram (see Geweke and Porter-Hudak 1983 for the idea, Robinson 1996

for the proof of the convergence and asymptotic normality of the estimator, and Comte 1996

to check the assumptions given by Robinson).

We used 100 samples with length 400, where 4000 points were generated for the

continuous-time models with a step 0.1 and one point out of ten was kept for the esti-

mation. We had(, k, )=(0.3, 3, 1), in particular=0.3 in all cases.But we compared two ways of estimating : either working directly on the log-period-

ogram of the processx (t)=l n (t) (which exactly corresponds to our fractional OrnsteinUhlenbeck model) or working on (t)=e x p(x(t)), since it fulfills the same long-memoryproperties (see Proposition 2.2).

As a benchmark for this estimation of, we considered a third estimation through the

following procedure. Assuming that the observed path would be associated withx(t)=(1L)x ()(t) (with a sampling frequencyh=1) instead ofx (t), we could then estimate by a log-periodogram regression on the path

x(t), which is referred to below as the

ARFIMA method. The three methods should provide consistent estimators of the samevalue for . The results are reported in Table 6.3: they are better with x than with an

ARFIMA model or with exp x , and the recommendation is to work withx instead of expx .

Let us nevertheless notice that the bad result for the ARFIMA model could be explained by

the fact that the discrete time filter(1 L ) had been applied to the low frequency pathx () (t)with steph=1.


21/33


TABLE6.3

Estimation of with Log-Periodogram Regression for Three Models. 100 Samples,

Length 400. True Value: =0.3

Empirical mean Empirical std. dev.x 0.2877 0.0629

expx 0.2568 0.0823

ARFIMA 0.3447 0.0864

6.5. Estimation with Real Data

We carried out log-periodogram regressions on real stock prices and implicit volatilitiesassociated with options on CAC40 of the Paris Stock Exchange. First, for the log-prices,

we found from our 775 daily data that price= 0.0035. This is near = 0 and confirmsthe short-memory feature of prices. On the contrary, we found for a sequence|t+1 t|that volat= 0.2505, which illustrates the long-memory feature of the volatilities. Weemphasize that we have to take absolute values of the increments of the implicit volatilities

(see Ding, Granger, and Engle 1993), since we otherwise have:

ln (t

+1/t) t

+1

t t (t

+1

t)

2

0.30493 0.37589 0.67034 0.21490

so that squaring is also possible. Missing values are replaced by the global mean.

Two preliminary conclusions can be derived from the previous empirical evidence. First,

it appears that the volatility is not stationary and must be differenced. Secondly, the long-

memory phenomenom is stronger when we consider absolute values (or squared values) of

the differenced volatility. This seems to indicate some asymmetric feature in the volatility

dynamics, as observed in asset prices by Ding et al. (1993).

These two points lead us to modify our long-memory diffusion equation on (t). This

work is still in progress. Nevertheless, the previous empirical evidence has to be interpreted

cautiously, because if we take into account small sample biases, it is clear that an autore-

gressive operator(1 L)with close to one is empirically difficult to identify against afractional differentiation(1 L ) .

APPENDIX A: PROOFS

Proof of Proposition 2.1. We are going to use the result given in equation (2.6) (see Comte

1996) which gives forx= ln( ) andh0: rx (h)=rx (0) + .|h|2+1 + o(|h|2+1) with,forh0:

V( x (t+ h) x(t))=2(rx (0) rx (h))=2+

0

a2(x ) d x

0

a(x)a(x+ h) d x

,


22/33


and the fact that if X N(0, s2) then E(expX)= es2/2. Then, still for h 0 (andr(h)=r(h)):

r(h)

=E(exp(

x (t

+h)

exp(

x(t))

E(exp(

x(t

+h))

E(exp(

x (t))

r(h)= E

exp

t

(a(t+ h s) + a(t s)) dw2(s)

exp t+h

t

a(t+ h s) dw2(s)

Eexp

t+h

a(t

+h

s) dw2(s) Eexp

t

a(t

s) dw2(s)

This yields, forh0, with the second point, to

r(h)=exp+

0

a2(x) d x

exp

+0

a(x+ h)a(x ) d x

1

.

Then, forh0:

r(h) r(0)= exp+

0

a2(x ) d x

exp

+0

a(x+ h)a(x) d x

exp +

0

a2(x) d x

,

and factorizing the first right-hand term again:

r(h) r(0)=exp2 +0

a2(x ) d x

(exp (rx (h) rx (0)) 1) .

Then forh0, K= exp(+0

a2(x) d x), we have:

r(h) r(0)=K2

exp(.|h|2+1 + o(|h|2+1)) 1= K2 |h|2+1 + o(|h|2+1)which gives the announced result.

Proof of Proposition 2.2. (i) The previous computations give, with the same K as

above: r(h)= K(exp(rx (h)) 1)and it has been proved in Comte (1996) that rx (h)=|h|21 + o(|h|21)forh +, whereis a constant. This implies straightforwardlythatr(h)= K|h|21 + o(|h|21), which gives (i).


23/33


(ii)+

0 r(h) cos(h)dh =

A0

r(h) cos(h)dh+ 1+

A r(

u

) cos udu. Now for

A chosen great enough, the development of r near + implies 2 f() =2

A0

r(h) cos(h)dh++

A u21 cos ud u+o(1), and consequentlylim0 2 f()=

+

0 u21 cos udu where the integral is convergent near 0 because 2 >0 and near+

because: +1 u21 cos ud u=[u21 sin u]+1 (2 1) +1 u22 sin ud u where allterms are obviously finite.

Proof of Lemma 3.1. For the proof of the first convergence: YnDYon a compact set

[0, T], we check the L 2 pointwise convergence ofYn (t)towardY(t), and then a tightness

criterion as given by Billingsley (1968, Th. 12.3): E|Yn (t2) Yn(t1)|p C.|t2 t1|q withp >0, q >1, andCa constant.

TheL2 convergence is ensured by computing:

E(Yn (t) Y(t))2 = E t0

[ns ]n (s) dw1(s)2

= E t

0

[ns ]

n

(s)

2ds

= t

0

E

[ns ]

n

(s)

2ds with Fubini.

Then theL

2

convergence is obviously given by an inequality:E

( (t2) (t1))2

C.|t2 t1| for a positiveand a constant C.As usual, let x (t)= t

0a(t s) dw2(s)and lett1t2.

E( (t2) (t1))2 = E(exp(x ((t2)) exp(x (t1)))2 = E

e2x (t2) + e2x (t1) 2ex (t1)+x(t2)= e2

t10

a2(x )d x + e2t2

0a2(x )d x

2e 12t1

0a2(x )d x+ 1

2

t20

a2(x )d x+t1

0a(x )a(t2t1+x )d x

= e

2t2

0a2(x )d x

1 + e2t2t1 a2(x)d x 2e 32 t2t1 a2(x)d xt10 a(x )(a(x )a(t2t1+x ))d x 2e2

t20

a2(x )d x

1 e

32

t2t1

a2(x )d xt1

0a(x)(a(x)a(t2t1+x ))d x

.

The term inside the last parentheses being necessarily nonnegative, the term in the last

great exponential is nonpositive. Moreover| t2t1

a2(x ) d x| M21 |t2 t1| with M1 =supx[0,T] |a(x)|, and sincea is -Holder,

t10

a(x )(a(x) a(t2 t1+ x )) d xC|t2 t1| t1

0

|a(x)| d x C|t2 t1|M1T,

which implies|t2t1

a2(x ) d x+ t10

a(x)(a(x)a(t2t1+x )) d x| M2|t2t1| , with


24/33


M2= M2(T). Then using thatu 0, 01 eu |u|,we have E( (t2) (t1))2 2K2M2|t2 t1| , ]0, 12 [ where K = exp(

+0

a2(x)d x) as previously. Then,

E( ([ns ]/n) (s))2 2 K2M2( 1n ) gives:

E(Yn (t) Y(t))2 2K2M2Tn

t [0, T],

which ensures the L2 convergence.

We use a straightforward version of Burkholders inequality (see Protter 1992, p. 174),

E|Mt|p CpEMp/2t , where Cp is a constant and Mta continuous local martingale,M0=0, to write (with an immediate adaptation of the proof on [t1, t2] instead of [0, t]):

E|Yn(t2) Yn (t1)|p

= E t2t1 [ns ]n dw1(s)p

CpE t2t1 2 [ns ]n dsp/2

.

Let us choose p=4:

E|Yn (t2) Yn(t1)|4 C4E t2

t1

2

[ns ]

n

ds

2= C4

[t1,t2]2E

2

[nu ]

n 2

[nv]

n du dv

C4 [t1,t2]2

E4 E4 du dv (given by (2.7))

= C4E4(t2 t1)2, E4 =exp

8

+0

a2(x)ds

.

This gives the tightness and thus the convergence.

The second convergence is deduced from the first one, the decomposition: Yn(t)=Yn(t) + un (t), with un(t) = ( [nt]n )(w1(t) w1( [nt]n )), and Theorem 4.1 ofBillingsley (1968): (Xn

D

X and (Xn,Zn) P

0) (ZnD

X), where (x ,y)=supt[0,T] |x(t)y(t)|. Here (Yn,Yn )=sup |un(t)| and un (t)= M([nt]/n) is a martingaleso that Doobs inequality (see Protter 1992, p. 12, Th. 20) gives:

E

supt[0,T]

|un (t)|2

4 supt[0,T]

E(un (t)2).

Then,

Eun (t)2 = E

2

[nt]

n

w1(t) w1

[nt]

n

2

= E

2

[nt]

n

E

w1(t) w1

[nt]

n

2


25/33


=

t [nt]n

E

2

[nt]

n

1

nE(2),

which achieves the proof.

Proof of Proposition 3.2.

First we prove the following implication:

Yn DY or tightn

D or tightand

Yn(t)

n(t)

L 2

Y(t)

(t)

imply

Ynn

D

Y

.

Indeed, the functional convergences of both sequences imply their tightness and thus

the tightness of the joint process. This can be seen from the very definition of tightness

as given in Billingsley (1968), that can be written for Yn :,Kn (compact set) sothat P(Yn Kn) >1 (/2)and then, for this, we have forn:Kn (compact set)so that P(n Kn ) >1 2 . Then:

P((Yn,n ) KnKn)= 1 P(Yn / Kn orn / Kn) 1 P(Yn / Kn) P(n / Kn)

= P(

Yn

Kn )

+P(

n

Kn)

1

1 .

Now, the tightness and the pointwise L 2 convergence of the couple imply the conver-

gence of the joint process.

Let us check the pointwiseL 2 convergence ofYn :

E(Yn(t) Y(t))2 = E [nt]/n

0 n

[ns ]

n (s)

dw1(s) +

t

[nt]

n

(s) dw1(s)

2

= [nt]/n

0

E

n

[ns ]

n

(s)

2ds+

t[nt]/n

E

2(s)

ds.

The last right-hand term is less than 1nE(2) and goes to zero when ngrows to infinity

and the first right-hand term can be written, for the part under the integral, as

E n [ns ]

n (s))2= Eexp [nt]/n

0

a t [ns ]

n dw2(s) exp

t0

a(t s) dw2(s)2

.

Now, forXnandXfollowingN(0,EX2n ) andN(0,EX

2) respectively,E(eXn eX)2 =


26/33


e2EX2n +e2EX2 2eE(Xn+X)2/2, which goes to zero when ngrows to infinity ifEX2n

n+EX2 and E(Xn+X)2

n+4EX2.

This can be checked quite straightforwardly here with X=

t

0a(t s) dw2(s)and

Xn= [nt]/n0 a(t [ns]n ) dw2(s), so that:EX2n=

[nt]/n0

a2

t [ns ]n

ds

n+

t0

a2(t s) ds= EX2,

E(Xn+X)2 = [nt]/n

0

a2

t [ns ]n

ds+

t0

a2(t s) ds

+ 2 [nt]/n0

a t [ns ]n a(t s) ds

n+4EX2.

This result gives in fact both L2 convergences ofYn(t)and ofn(t). The tightness ofn is then known from Comte (1996) and the tightness ofYn can

be deduced from the proof of Lemma 3.1 with E4n (t)instead ofE4, which is still

bounded.

Proof of Proposition 4.1. We work with (t)

= exp(

t

a(t

s)dw2(s)), but the

results would obviously still be valid with the only asymptotically stationary version of.We use here and for the proof of Proposition (4.2), the following result:

0, limh+

h12+

a(x)a(x+ h) d x

=C,(A.1)

where C is a constant. This result can be straighforwardly deduced from Comte and

Renault (1996) through rewriting the proof of the result about the long-memory property

of the autocovariance function (extended here to the case =

0).

We know that yt= E(2(t+1)| Ft)= exp(21

0 a2(x ) d x) exp(2

ta(t+1

s) dw2(s)). Then:

cov(yt+h ,yt)= E(yt+hyt) E(yt+h )E(yt)

= exp

4

10

a2(x) d x

Eexp2 t+h

a(t+ h+ 1 s) dw2

(s)

+ 2 t

a(t+ h s) dw2(s)

exp

4

10

a2(x ) d x

exp

4

+0

a2(x+ 1) d x


27/33


= exp

4

+0

a2(x) d x

exp

4

+1

a(x+ h)a(x ) d x

1

.

This proves the stationarity of the y process, and, with (A.1), which gives the order

of the term inside the exponential, implies the announced order h 21. E( (t+ h)| Ft), still with the stationary version of, is given by

exp

t

a(t+ h s) dw2(s)

exp

1

2

t+ht

a2(t+ h s) ds

.

Then, as(E(E( (t+ h| Ft)))2 =(E (t+ h))2, we have:

Var (E( (t+ h)| Ft))= exp h

0

a2(x ) d x

exp

2+

0

a2(x+ h) d x

exp+

0

a2(x) d x

= exp

+0

a2(x) d x

exp

+h

a2(x) d x

1

.

As a(x )=

x

a(x )

= x

1.x

a(x)

= O(x

1) for x

+since we know that

limx+x a(x )=a. Then, forh +,

+h

(x 1)2(x a(x))2 d x= O

a2

+h

x 22 d x

= O(h21).

Developing again the exponential of this term for great h gives the orderh 21, andeven the limit of the variance divided by h21 for h +, which isK(a2/12)with K

=exp(+0 a2(x ) d x).

For=0, a(x )=ek x gives obviously for the variance an order C ekh .

Proof of Proposition 4.2. We have to compute cov(zt,zt+h ).

E(zt+hzt)= E 1

0

E(2(t+ h+ u)| Ft+h ) du 1

0

E(2(t+ v)| Ft) dv

= 1

0 1

0 Ee(2u

0a2(x ) d x)

e

(2t+h

a(t

+h

+u

s) dw2(s))

e

(2v

0a2(x ) d x )

e

(2t

a(t

+v

s) dw2(s)) du dv

= 1

0

10

e

(2u

0a2(x ) d x+2

v0

a2(x ) d x)e

(2t

(a(t+h+us)+a(t+vs))2 ds )

e(2t+h

ta2(t+h+us) ds )

du dv

= 1

0

10

e(4+

0a2(x ) d x+4

+0

a(x+h+u)a(x) d x )du dv.


28/33


Moreover, with the same kind of computations we haveE(zt+h )E(zt)=exp(4+

0 a2(x) d x)

so that:

cov(zt,zt+

h )

= exp4

+

0

a2(x) d x 1

0

10

exp

4

+v

a(x+ h+ u v)a(x) d x

1

du dv

.

Thenz is stationary and another use of (A.1) gives the order h 21 forh +.

Proof of Proposition 4.3. LetZ(t)= t0

(u)dw1(u). Then we know (see Protter 1992

p. 174, for p= 4) that EZ(t)4 = 4(41)2 E

t

0 Z2(s)dZs . AsZs=

s

0 2(u) du, we

find: EZ(t)4

= 6E(t0 Z2(s)2(s) ds)= 6 t0 E(Z2(s)2(s)) ds, with Fubinis theorem.Then E(Z2(t)2(t))= E(t

0( (t) (u)) dw1(u))2 = E t

0(2(t)2(u)) du ( andw1 are

independent). This yields

EZ(t)4 =6 t

0

s0

(r2 (|s u|) + (E2)2) du ds,

whereris the autocovariance function, and, lastly, (h)=3h2(E2)2 + 3 [0,h]2

r2 (|su|) du ds.

Near zero, the autocovariance function of2 is of the same kind as the one of, witha replaced by 2a, since= exp(x). Then we know from Proposition 2.1 that, forh0: r2 (h)=r2 (0) + C h2+1 + o(h2+1), whereCis a constant and ]0, 12 [.Then replacing in (h)gives

(h)=3h2((E2)2 + r2 (0)) + 3C

(2+ 2)(2+ 3) h2+3 + o(h2+3).

For= 0,a(x)= exp(kx )givesr2 (h)= e2/ kexp( 2kekh ) 1which leads to:r2 (h)=r2 (0) 2e4/ kh+ o(h),forh0. This implies the continuity for=0.Now, E(Y(h) EY(h))2 = E(h

0 (u) dw1(u))2 = E h

0 2(u) du= hE2 implies

that

limh0

kurtY(h)=3 E4

(E2)2 >3.

From Proposition 2.2, we know that, for ]0, 12

[ andh +,

[0,h]2

r2 (|s u|) du ds= O [1,h]2

u21 du ds= O(h2+1),

but for = 0, r2 (h) = e2/ k

2k

ekh + o(ekh ). This gives the result and theexponential rate for=0.


29/33


Proof of Proposition 5.1. Letm=[N t/T], then

E2n,p(t)= n

T

m

k=mp+1E

tk

tk

1

(s) dw1(s)

2

= nT

m

k=mp+1E

tk

tk

1

2(s) ds

= n

TE

tmtmp

2(s) ds

= n

TE(2)(tm tmp)=

n

Tp T

npE2, = E2,

where E2 =exp( 12

+0

a2(x ) d x). This ensures the L1 convergence of the sequence,

uniformly int.

Before computing the mean square, let:

f(z)= f(|z|)= E2(u)2(u+ |z|)= exp

4

+0

a2(x )d x+ 4+

0

a(x )a(x+ |z|) d x

.

Then

E[2n,p(t) 2(t)]2 = En

T

m

k=mp+1(Ytk Ytk1 )2 2(t)

2

= n2

T2E

m

k=mp+1(Ytk Ytk1 )2

2

+ E4(t) 2 nTE

m

k=mp+12(t)(Ytk Ytk1 )2

2.

We consider separately the different terms.

E

2(t)(Ytk Ytk1 )22 = E tk

tk1 (t) (s) dw1(s)

2= E

tktk1

2(t)2(s) ds

= tk

tk1f(t s) ds

as andw1 are independent. As in a previous proof, we have:

E

tktk1

(s) dw1(s)

4=3

[tk1,tk]2

f(u v) du dv,

and for j= k: E[(Ytk Ytk1 )2(Ytj Ytj1 )2]= E(tk

tk12(s) ds tj

tj12(s) ds). This


30/33


gives:

E

m

k=mp+1(Ytk Ytk1 )2

2

= 2m

k=mp+1 [tk1,tk]2f(u v) du dv

+

[tmp ,tm ]2f(u v) du dv.

Now with all the terms:

E[2n,p(t) 2(t)]= n2

T2

2

mk=mp+1

[tk1,tk]2

f(u v) du dv

+ [tmp ,tm ]2

f(u v) du dv+ E4(t) 2n

T

[tmp ,tm ]

f(t s) ds

= 2n2

T2

mk=mp+1

[tk1,tk]2

(f(u v) f(0)) du dv

+ n2

T2 [tmp ,tm ]2 (f(u v) f(0)) du dv 2n

T

[tmp ,tm ]

(f(t s) f(0)) ds+ 2E4

p.

Let K1= exp(8+

0 a2(s) ds). Then f(h)= K1r(h)whereris as in Proposition 2.1.

Proposition 2.1 then implies| f(h) f(0)| K1C | h |2+1, where C is a positiveconstant.

This implies that: >0, >0, so that |v u| < |f(v u) f(0)| < .Letthen

=1/p,then

=()is fixed and

E[2n,p(t)2(t)]2 2n2

T2p

T

np

21

p+n

2

T2

T

n

21

p+ 2n

Tp T

np1

p+2E

2

p

if|tm tmp| = Tn < , which implies |tk tk1| = Tnp < . Then

E[2n,p(t) 2(t)]2 2

p+ 1 + 2 + 2E2

1

p=

2

p+ 3 + 2E2

1

p.

Then a >0, n >T/ p1a.E 2n,p(t) 2(t)2 Cpa , whereC= 5 + 2E2.Thestationarity implies that theresult is uniform in t, sothat lim

n,p+sup

t[0,T]p1aE[2n,p(t)

2(t)]2 =0.


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APPENDIX B

We suppose herer= 0 and2=0, so that x ()(t)=(ln )() (t)can be written:

x ()(t)=ektx () (0) + t0

eks dw2(s) .ThenU2t,T=

Tt

2(u)du can be written:

U2t,T= T

t

exp

2

t0

(u s)(1 + ) d x

() (s)

exp

2

ut

(u s)(1 + ) d x

()(s)

du.

Then the first part, exp(2 t0 (us)(1+) d x ()(s)), is deterministic knowingFt. For the secondpart, since we have for s > t that x ()(s)=ek(st)x (t) + s

t ek(sx )dw2(x ),we find

that

x () (s)=ek(ts)x ()(t) + t

s

ek(sx )dw2(x),

u

t

(u s)

(1 + )d x ()(s)

= x ()(t)

k u

t

(u s)

(1 + )ek(st) ds

k u

t

(u s)(1 + )

st

ek(sx)dw2(x)

+ u

t

(u s)(1 + ) dw

2(s).

This term depends then only on (x () (t)) and on future increments of the Brownian

motionw2; those increments are independent ofFt. This is the reason we can write

exp

2 u

t

(u s)(1 + ) d x

()(s)

= f(x ()(t); Z(u, t, )).

At time t= ti , this gives the announced formula, with ln[f(x ()(t);Z(u, t,))]=x ()(t)(t, u) +Z(u, t, );(t, u)is a deterministic function, Z(u, t, ) is a process inde-pendent ofFt:

Z(t, u, )= k ut (u s)

(1 + ) st ek(sx )dw2(x) + u

t

(u

s)

(1 + ) dw2

(s).

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