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Volatility Modulated Volterra Processes

Ole E. Barndorff-Nielsen Thiele Centre Department of Mathematical Sciences University of Aarhus

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Synopsis

Volatility Modulated Volterra Processes,

page 2 of 72

.

Intro: Turbulence and Finance; MultipowerVariation Volterra processes Volatility modulated Volterra Processes (VMVP) Ambit processes 1-dim MA BM setting: Y = g σ B Concrete model type Realised Variation Ratio

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Introduction

Volatility Modulated Volterra Processes,

page 3 of 72

Modelling framework: in Finance The basic framework for stochastic volatility modeling in nance is that of Brownian semimartingales Yt = Y0 +

Z t

0 σsdBs +

Z t

0 asds

where σ and a are cadlag processes and B is Brownian motion, with σ expressing the volatility. In general, Y, σ, B and a will be multidimensional.

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Introduction

Volatility Modulated Volterra Processes,

page 4 of 72

Modelling framework: Turbulence (Phenomenological approach) Whereas Brownian semimartingales are 'cumulative' in nature, for free turbulence it is physically natural to model timewise velocity dy- namics by stationary processes: At time t and at a xed position x in the turbulent eld, the velocity vector is specied as Vt = µ + Yt with

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Introduction

Volatility Modulated Volterra Processes,

page 5 of 72

Yt (x) =

Z t

∞

Z

R3 g(t s, x ξ)σs (ξ) W (dξds)

+

Z t

∞

Z

R3 q(t s, x ξ)as (ξ) dξds.

where W is white noise, with σ expressing the intermittency (= volatility). In general, Y, g, σ, W, q, and a will be multidimensional.

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Introduction

Volatility Modulated Volterra Processes,

page 6 of 72

Multipower Variations For any stochastic process Y = fYtgt0 (or Y = fYtgt2R) the quadratic variation (QV) process [Y] and the bipower variation (BV) process fYg are, respectively, the lim- its in probability, when they exist, of the realised quadratic variation (RQV) [Yδ] and the realised bipower variation (RBV) fYδg. To dene RVR and RBP , for any δ > 0 let Yδ denote the δ-discretisation

• f Y, i.e. (Yδ)t = Ybt/δcδ, and recall that for a standard normal vari-

able u we have µ1 = E fjujg =

p

2/π. Furthermore, for positive integers n and δ = n1, let ∆n

j Y = Yjδ Y(j1)δ.

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Introduction

Volatility Modulated Volterra Processes,

page 7 of 72

Then RVR and RBP are given, respectively, by

[Yδ]t =

bntc

∑

j=1

• ∆n

j Y

2 and

fYδgt = π

2 [Yδ][1,1]

t

with

[Yδ][1,1]

t

=

bt/nc

∑

j=2

• ∆n

j1Y

• ∆n

j Y

• .

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Introduction

Volatility Modulated Volterra Processes,

page 8 of 72

General multipower: n Y[r]

δ

• t = cr [Yδ][r]

t

where

[Yδ][r]

t

=

bntc

∑

j=k+1

• ∆n

jkY

• rk
• ∆n

j Y

• r0 .

More generally,

bntc

∑

j=k+1

f1

• ∆n

jkY

• fk
• ∆n

j Y

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Introduction

Volatility Modulated Volterra Processes,

page 9 of 72

Applications In Finance δ1

2

• [Yδ]t σ2+

t , fYδgt σ2+ t

Lstably

!

N2

• (0, 0) , 2
• 1

1 1 1 + ϑ

• σ4+

t

• where ϑ = π2/4 + π 5 ( .

= 0.609).

Feasible results.

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Introduction

Volatility Modulated Volterra Processes,

page 10 of 72

Applications in Turbulence

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Volterra processes

Volatility Modulated Volterra Processes,

page 11 of 72

Brownian Volterra processes (BVP): Yt =

Z ∞

∞ Kt (s) dBs +

Z ∞

∞ Qt (s) ds,

Here K and Q are deterministic functions, sufciently regular to give suitable meaning to the integrals. Backward type: Yt =

Z t

∞ Kt (s) dBs +

Z t

∞ Qt (s) ds.

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Volterra processes

Volatility Modulated Volterra Processes,

page 12 of 72

Lévy Volterra processes (LVP): Yt =

Z ∞

∞ Kt (s) dLs +

Z ∞

∞ Qt (s) ds

Here L denotes a Lévy process on R and K and Q are deterministic kernels, satisfying certain regularity conditions. Backward type: Yt =

Z t

∞ Kt (s) dLs +

Z t

∞ Qt (s) ds.

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Volterra processes

Volatility Modulated Volterra Processes,

page 13 of 72

Stochastic integration in this kind of setting is discussed for BVP in [Hu03], [Dec05], [DecSa06] and for LVP in [BeMar07]. When is Y a semimartingale? In that case what is the character

• f its spectral representation?

Andreas Basse [Bas07a], [Bas07b], [Bas07c], for Brownian case.

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Volterra processes

Volatility Modulated Volterra Processes,

page 14 of 72

Tempo-spatial Volterra processes: Yt (x) =

Z ∞

∞

Z

Ξ Kt (ξ, s; x) L# (dξds) +

Z ∞

∞

Z

Ξ Qt (ξ, s; x) dξds

Here K and Q are deterministic functions, Ξ is a region in Rd and L# is a homogeneous Lévy basis on Ξ R. Backward type: Yt (x) =

Z t

∞

Z

Ξ Kt (ξ, s; x) L# (dξds) +

Z t

∞

Z

Ξ Qt (ξ, s; x) dξds

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Volatility modulated Volterra processes

Volatility Modulated Volterra Processes,

page 15 of 72

Volatility modulated Volterra Processes (VMVP): Yt (x) =

Z ∞

∞

Z

Ξ Kt (ξ, s; x) σs (ξ) L# (dξds) +

Z ∞

∞

Z

Ξ Qt (ξ, s; x) as (ξ) dξds

where σ is a positive stochastic process, embodying the volatility or

• intermittency. (K and Q deterministic, σ and a stochastic.)

Backwards moving average type: Yt (x) =

Z t

∞

Z

Ξ g (ξ x, t s) σs (ξ) L# (dξds)

+

Z t

∞

Z

Ξ q (ξ x, t s) as (ξ) dξds

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Inference on the volatility

Volatility Modulated Volterra Processes,

page 16 of 72

A central issue in these settings is how to draw inference on the volatility process σ. In cases where the processes are semimartingales, the theory of multipower variations provides effective tools for this. ([BNGJPS07], [BNGJS06] and references given there) However, VMVP processes are generally not of semimartingale type and the question of how to proceed then is largely unsolved and poses mathematically challenging problems.

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Inference on the volatility

Volatility Modulated Volterra Processes,

page 17 of 72

It is further of interest to consider cases where processes express- ing possible jumps or noise in the dynamics are added. Some of these problems are presently under study in joint work with Jose-Manuel Corcuera, Neil Shephard, Jürgen Schmiegel and Mark Podolski.

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Ambit processes

Volatility Modulated Volterra Processes,

page 18 of 72

Ambit processes: ([BNSch07a]) Yt (x) = µ +

Z

At(σ) g (t s, jξ xj) σs (ξ) W (dξ, ds)

+

Z

Dt(σ) q (t s, jξ xj) as (ξ) dξds

Here At (σ) and Dt (σ) are termed ambit sets.

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Ambit processes

Volatility Modulated Volterra Processes,

page 19 of 72

(t(w), σ(w)) Xw At(w)(σ(w))

❅

• Ambit processes

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Ambit processes

Volatility Modulated Volterra Processes,

page 20 of 72

t t′ σ

• σ′
• ✲

✻

Two overlappng ambit sets

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1-dim. BM MA setting

Volatility Modulated Volterra Processes,

page 21 of 72

Recall: Modelling time series by stochastic processes of the form V = µ + Y with Yt =

Z t

∞ g (t u) σudBu +

Z t

∞ h(t u)asdu.

(1) Here B is Brownian motion, the kernels h and g are determinis- tic, positive and square integrable functions on (0, ∞), presumed known, and σ is a stationary process which expresses the time- dependent variation or volatility of the process Y. Moreover, a and σ are stochastic processes satisfying the same as- sumptions as are usual for Brownian semimartingales; in particular, σ is square integrable.

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1-dim. BM MA setting

Volatility Modulated Volterra Processes,

page 22 of 72

Concretely we (BN+Schmiegel) think of this as a modelling frame- work for the time-wise behaviour of the main component of the ve- locity vector (i.e. the component in the mean direction of the uid motion) in a turbulent eld. Question: To what extent is the integral of the squared volatility

• ver the interval [0, t], i.e.

σ2+

t

=

Z t

0 σ2 udu,

consistently estimable by a suitably normalised version of the re- alised quadratic variation of Y when the limiting scheme consid- ered is that Y is observed at the time points jδ, j = 1, ..., n, where δ = t/n, and n ! ∞ with t xed?

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1-dim. BM MA setting

Volatility Modulated Volterra Processes,

page 23 of 72

Conjecture: (of work in progress by BN+Corcuera+Podolskij) The theory of multipower variation can be extended to processes of the form (1) under conditions on g of which the essential one is that the function R (t) =

Z ∞

g (t + u) g (u) du satises the following (given on next slide) three assumptions (A1)- (A3) where ¯ R = 2

• kgk2 R
• and 0 < γ < 5

4:

Note The conjecture holds true for power variation when σ is a constant, as follows from [GuyLe89].

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1-dim. BM MA setting

Volatility Modulated Volterra Processes,

page 24 of 72

(A1) ¯ R (t) = tγL0 (t) for some slowly varying (at 0) function L0, which is continuous on (0, ∞). (A2) ¯ R00 (t) = tγ2L2 (t) for some slowly varying (at 0) function L2, which is continuous on (0, ∞). (A3) There exists a b 2 (0, 1) with lim sup

x!0

sup

y2[x,xb]

• L2 (y)

L0 (x)

• < ∞.

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Conjecture: Some rst considerations

Volatility Modulated Volterra Processes,

page 25 of 72

Recall: Yt =

Z t

∞ g (t u) σudBu +

Z t

∞ h(t u)asdu.

The inuence of the `drift term', that is the second integral, will dis- appear under the limiting procedure we have in mind, so henceforth that term is assumed not to be present, and we write the expression for Y briey as Y = g σ B.

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Conjecture: Some rst considerations

Volatility Modulated Volterra Processes,

page 26 of 72

To ensure that Y is well dened we assume that g (t u) σu is square integrable with respect to u on (∞, t], for all t 2 R. Furthermore, we suppose that g is differentiable on (0, ∞) and that for any ε > 0 and any t the integral R tε

∞ ˙

g2(t u)σ2

udu exists and g

is Lipschitz of order 2 on [ε, ∞).

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Conjecture: Some rst considerations

Volatility Modulated Volterra Processes,

page 27 of 72

Suppose for the moment that σ = 1 identically. Then Y = g B and this process has autocovariance and autocorrelation functions R (t) =

Z ∞

g(t + u)g (u) du and r (t) =

Z ∞

¯ g(t + u) ¯ g (u) du where ¯ g = g/ kgk and

kgk2 =

Z ∞

g2 (u) du. We let ¯ r (t) = 1 r (t) and ¯ R (t) = 2 kgk2 ¯ r (t) .

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Quadratic variation of Y = g σ B

Volatility Modulated Volterra Processes,

page 28 of 72

Let ∆n

j Y = Yjδ Y(j1)δ and for any q > 0, let V(Y, q)n t be the

realised q-th order power variation of Y, i.e. V(Y, q)n

t = nq/21 n

∑

j=1

• ∆n

j Y

• q

. For q = 2 this is the realised quadratic variation, which will be the basis for estimating σ2+

t .

We let ¯ V(Y, 2)n

t =

δ 2 kgk2 ¯ r (δ) V(Y, 2)n

t .

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Quadratic variation of Y = g σ B

Volatility Modulated Volterra Processes,

page 29 of 72

Special restrictive setting: We suppose that the process σ is independent of the Brownian motion B, and we will argue condi- tionally on σ. Remark: Under (A1)-(A3) the variance of ¯ V(Y, 2)n

t will go to 0 as

δ ! 0. What remains in order to establish consistency is then that E f ¯ V(Y, 2)n

t jσg p

! σ2+

t

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Quadratic variation of Y = g σ B

Volatility Modulated Volterra Processes,

page 30 of 72

Behaviour of E fV(Y, 2)n

t jσg

Note that Yt+δ Yt =

Z t+δ

t

g (δ + t u) σudBu

+

Z t

∞ (g (δ + t u) g (t u)) σudBu.

Hence, for arbitrary ε > 0, δE fV(Y, 2)n

t jσg =

Z δ

0 δ n

∑

j=1

σ2

jδvg2 (v) dv

+

Z ∞

δ

n

∑

j=1

σ2

(j1)δv (g (δ + v) g (v))2 dv

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Quadratic variation of Y = g σ B

Volatility Modulated Volterra Processes,

page 31 of 72

After some calculation we nd (key relation) E f ¯ V(Y, 2)n

t jσg = σ2+ t

+ ¯

R (δ)1 A(δ) where A (δ) = A0 (δ) + A1 (δ; ε) + A2 (δ; ε) with

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Quadratic variation of Y = g σ B

Volatility Modulated Volterra Processes,

page 32 of 72

A0 (δ) =

Z δ

@δ

n

∑

j=1

σ2

jδv σ2+ t

1 A g2 (v) dv and, for any ε > 0, A1 (δ; ε) =

Z ε

@δ

n

∑

j=1

σ2

(j1)δv σ2+

t

1 A (g (δ + v) g (v))2 dv A2 (δ; ε) =

Z ∞

ε

@δ

n

∑

j=1

σ2

(j1)δv σ2+

t

1 A (g (δ + v) g (v))2 dv.

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Quadratic variation of Y = g σ B

Volatility Modulated Volterra Processes,

page 33 of 72

Let c0 (δ) =

Z δ

0 g2 (v) dv

and c (δ) =

Z ∞

(g (δ + v) g (v))2 dv.

and note that c0 (δ) + c (δ) = ¯ R (δ). Furthermore, let ˆ σ2+

sjt = δ n

∑

j=1

σ2

(j1)δs

and note that ˆ σ2+

sjt !

Z ts

s

σ2

udu.

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Quadratic variation of Y = g σ B

Volatility Modulated Volterra Processes,

page 34 of 72

It follows that for any ε > 0

jE f ¯

V(Y, 2)n

t jσg σ2+ t j

sup

0vδ

jˆ

σ2+

vjt σ2+ t jc0 (δ)

¯ R (δ)

+ sup

0vε

jˆ

σ2+

vjt σ2+ t j c (δ)

¯ R (δ)

+ sup

ε<v<∞

jˆ

σ2+

vjt σ2+ t jC (ε)

δ2 ¯ R (δ)

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Conclusion

Volatility Modulated Volterra Processes,

page 35 of 72

The upshot of these considerations is that if c0 (δ) and c (δ) are of the same asymptotic order as δ ! 0, with this common order being smaller than that of δ2, then ¯ V(Y, 2)n

t p

! σ2+

t .

More boldly, one may surmise that it will be possible to derive a feasible asymptotic normal limit result for inference on σ2+

t

under some additional assumption on the behaviour of g at 0.

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A class of moving average models

Volatility Modulated Volterra Processes,

page 36 of 72

Particular case: Suppose that σ = 1 and g (t) = tν1eαt (3) with ν > 1

2 and α > 0.

Remark The derivative ˙ g of g is not square integrable if 1

2 < ν < 1

• r 1 < ν 3

2; hence, in these cases Y is not a semimartingale.

For ν = 1 the process Y is a semimartingale, in fact a modulated version of the Gaussian Ornstein-Uhlenbeck process. Note also that when ν > 3

2 then Y is of nite variation and hence, trivially, a

semimartingale.

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A class of moving average models

Volatility Modulated Volterra Processes,

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Remark Suppose that the volatility process is constant, σt = σ. In this case ([GuyLe89]) ¯ V(Y, 2)n

t p

! t σ2.

In fact, considerably more is true: [GuyLe89] derived associated (nonfeasible) limit law results It follows from those results that the limit distribution is normal if

1 2 < ν < 5 4, with rate δ3/2¯

r (δ), while it belongs to the second order Wiener chaos, with rate δ2ν3, for 5

4 < ν < 3 2.

Extension to the power variations V(Y, q)n

t , q > 0, are also given in

[GuyLe89].

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Volatility Modulated Volterra Processes,

page 38 of 72

The following analysis uses a number of, mostly well known, prop- erties of modied Bessel functions of the third type Kν (not given explicitly here). Steps in analysis:

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A class of moving average models

Volatility Modulated Volterra Processes,

page 39 of 72

(i) Properties of the autocorrelation function r of Y = g B: Exact formulae for the autocorrelation function r and its derivatives. (ii) Asymptotic properties of ¯ r (t) = 1 r (t) for t ! 0. (iii) Verication that (A1)-(A3) are satised (iv) Asymptotics of c0 (δ) and c (δ) for δ ! 0 (v) Example illustrating the asymptotics of E fV(Y, 2)n

t g for a very

special choice of σt that allows explicit calculations.

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Volatility Modulated Volterra Processes,

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Formulae for r and its derivatives The autocorrelation function r of Y = g B has the form r (t) = (2α)2ν1 Γ (2ν 1)eαt

Z ∞

(t + u)ν1 uν1e2αudt.

By formulae for the Bessel functions of type K we nd r (t) = ˇ Kν1

2 (αt) .

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Volatility Modulated Volterra Processes,

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Suppose for notational simplicity that α = 1, and let c (ν) = 2ν+1Γ (ν)1 . Then, we nd, for ν 2

• 1

2, 3 2

• we nd

¯ r0 (t) = c

• ν 1

2

• c

3

2 ν

t2ν2 ˇ K3

2ν (t)

¯ r00 (t) = c

• ν 1

2

• t2ν3 n

¯ K5

2ν (t) ¯

K3

2ν (t)

• ¯

r000 (t) = c

• ν 1

2

• t2ν4 n

¯ K7

2ν (t) 3 ¯

K5

2ν(t)

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Volatility Modulated Volterra Processes,

page 42 of 72

Behaviour of ¯ r = 1 r near 0 Using formulae for the Bessel functions of type K we nd that for t ! 0 the complementarry autocorrelation function ¯ r (t) = 1 r (t) behaves as 22ν+1Γ(3

2ν)

Γ(ν+1

2) (αt)2ν1 + O

• t2

for

1 2 < ν < 3 2

¯ r (t)

1 2 (αt)2 j log tj

for ν = 3

2 1 4(ν3

2) (αt)2 + O

• t2ν1

for

3 2 < ν

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Volatility Modulated Volterra Processes,

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Verication of assumptions (A1)-(A3): Conditions (A1)-(A3) are satised (with γ = 2ν 1 and ν 2

• 1

2, 3 2

• , i.e.

γ 2 (0, 2))

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Volatility Modulated Volterra Processes,

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On (A1): The complementary autocorrelation function ¯ r is of the form ¯ r(t) = t2ν1L0 (t) with L0 (t) = t2ν+1 1 ˇ Kν1

2 (αt)

• and

L0 (t) ! 22ν+1 Γ 3

2 ν

• Γ
• ν + 1

2

• for

t ! 0. It follows that L0 is slowly varying at 0, and hence assumption (A1) is met.

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Volatility Modulated Volterra Processes,

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On (A2): Note that ¯ r00(t) = t2ν3L2(t) with L2 (t) = c

• ν 1

2 n ¯ K5

2ν (t) ¯

K3

2ν (t)

• ,

where L2 is slowly varying at 0 with L2 (t) ! 23 (ν 1) Γ 3

2 ν

• Γ
• ν 1

2

• for

t ! 0.

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Volatility Modulated Volterra Processes,

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The latter follows from the rewrite ¯ r00 (t) = t2ν3c

• ν 1

2

• c

3 2 ν 1 n

(3 2ν) ˇ

K5

2ν (t) ˇ

K3

2ν (t)

• = t2ν322 Γ

3

2 ν

• Γ
• ν 1

2

• n

(3 2ν) ˇ

K5

2ν (t) ˇ

K3

2ν (t)

• .

Thus (A2) holds.

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Volatility Modulated Volterra Processes,

page 47 of 72

On (A3): Finally, we nd L0

2 (t) = c

• ν 1

2 n ¯ K0

5 2ν (t) ¯

K0

3 2ν (t)

• = c
• ν 1

2

• t

n ¯ K1

2ν (t) ¯

K3

2ν (t)

• = c
• ν 1

2

• t

n t2ν+1 ¯ Kν1

2 (t) ¯

K3

2ν (t)

• = c
• ν 1

2

• t2ν+2
• c(ν 1

2)1t2ν+1 ˇ Kν1

2 (t) c(ν 3

2)1t2ν1 ˇ K3

2ν (t)

• Hence (for ν 2
• 1

2, 3 2

• ) L2 (t) is increasing near 0. Consequently

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Volatility Modulated Volterra Processes,

page 48 of 72

lim sup

x!0

sup

y2[x,xb]

• L2 (y)

L0 (x)

• lim sup

x!0

• L2
• xb

L0 (x)

• ;

Here, as x ! 0, L0 (x) ! 22ν+1 Γ 3

2 ν

• Γ
• ν + 1

2

. while L2

• xb

! c

• ν 1

2 ( c 5 2 ν 1

c

3 2 ν 1) . Therefore also condition (A3) is satised.

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Volatility Modulated Volterra Processes,

page 49 of 72

Asymptotic behaviour of c0 (δ) and c (δ), taking α = 1, c0 (δ) = 1 2ν 1δ2ν1 + O

• δ2ν+n1

1 2ν1

• 22(ν1)Γ(ν)Γ(3

2ν)

Γ(1

2)

1

• δ2ν1 + O
• δ2

for

1 2 < ν < 3 2

c (δ)

1 2δ2j log δj

for ν = 3

2

22νΓ(2ν1)

ν3

2

δ2 + O

• δ2ν1

for

3 2 < ν

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Volatility Modulated Volterra Processes,

page 50 of 72

Key Example Consider now the special case where σ is given by σu = e(ψ1)u. This particular choice allows explicit calculation of E fV(Y, 2)n

t g. Af-

ter some calculation one nds

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δ 2 kgk2 ¯ r (δ) E fV(Y, 2)n

t g =

@δ

bt/δc

∑

j=1

e2(1ψ)jδ 1 A ψ(2ν1)A (δ)

ψ(2ν1)A (δ) σ2+

t

where A (δ) = e(1ψ)δ ¯ r (ψδ) ¯ r (δ) +

• 1 e(1ψ)δ2

¯ r (δ) .

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When 1

2 < ν 3 2 we have

A (δ) ψ2ν1 + O

• δ2

and hence δ 2 kgk2 ¯ r (δ) E fV(Y, 2)n

t g ! σ2+ t .

On the other hand, if ν > 3

2 we obtain

δ 2 kgk2 ¯ r (δ) E fV(Y, 2)n

t g ! ψ2

• ψ2ν+1 + 4
• ν 3

2

• (1 ψ)2
• σ2+

t .

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Remark For the concrete model considered here, i.e. Yt =

Z t

∞ (t u)ν1 eα(tu)σudBu,

let Xt =

Z t

∞ eαs (t s)1ν Ysds.

Then (Fubini!?), for 1

2 < ν < 3 2,

Xt =

Z t

∞ eαs (t s)1ν

Z s

∞ (s u)ν1 eα(su)σudBuds

=

Z t

∞ eαuσudBu

Z t

u (t s)1ν (s u)ν1 ds

= B (1 ν, ν)

Z t

∞ eαuσudBu.

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page 54 of 72

Recall that RVR and RBP are given, respectively, by

[Yδ]t =

bntc

∑

j=1

• ∆n

j Y

2 and fYδgt = π

2 [Yδ][1,1] t

with

[Yδ][1,1]

t

=

bt/nc

∑

j=2

• ∆n

j1Y

• ∆n

j Y

• .

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Let

[Yδ]0 =

n

∑

j=2

• ∆n

j1Y

2 , and

[Yδ]00 =

n

∑

j=2

• ∆n

j Y

2 . and dene the realised variation ratio (RVR) by

fYδ] =

fYδg [Yδ]0+[Yδ]00

2

. The probability limit of this ratio, when it exists, is the variation ratio VR), denoted fY], i.e.

fY] = p- limfYδ].

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The RVR as a diagnostic tool for model checking.

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Note The variation ratio may well exist even in cases where the quadratic variation and the bipower variation are both innite or both

• zero. This is the case, in particular, for Y = g σ B with g (t) =

tν1eαt, innite occurring for 1

2 < ν < 1 and zero for 1 < ν < 3 2.

(Another simple example of this is Yt = t or Yt = 1 for then fYg =

[Y] = 0 while fY] = π

2.)

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We have

fYδg = π

4

• [Yδ]0 + [Yδ]00

π

4

n

∑

j=2

• ∆n

j Y

• ∆n

j1Y

• 2

. From this equation it follows that 0 fYδ] π 2 and hence 0 fY] π 2 .

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It also follows that RVR is close to π

2 if the correlation between

cor n

• ∆n

j1Y

• ,
• ∆n

j Y

• is close to 1 for all j. This, in turn, holds if

cor n ∆n

j1Y, ∆n j Y

• is close to 1 or 1 for all j.

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To see the latter, recall that for arbitrary standard normal variables u and v with correlation coefcient ρ E fjuvjg = 2 π

• ρ arcsin ρ +

q 1 ρ2

• .

It follows that cor fjuj , jvjg = 2 π

• ρ arcsin ρ +

q 1 ρ2 1

• .

(4) Note that cor fjuj , jvjg does not depend on the sign of ρ and that it is an increasing function of jρj.

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Under certain conditions we will have that, for δ ! 0,

fYδ]

E ffYδgg E n[Yδ]0+[Yδ]00

2

• .

Suppose in particular that Y = g σ B with g (t) = tν1eαt. Then

fYδ] ρ (δ) arcsin ρ (δ) +

q 1 ρ (δ)2 with ρ (δ) = ¯ r (2δ) 2¯ r (δ) 1.

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Example Suppose that ¯ r is of the form ¯ r (t) = cγtγ + o (tγ) for t ! 0 and some γ 6= 1, and where cγ is a positive constants (i.e. as is the case for g (t) = tν1eαt with 1

2 < ν < 3 2 and γ = 2ν 1).

Then

fYδ] cγ2γδγ + o (tγ)

2cγδγ + o (tγ) 1

= 2γ1 1 + o (1) .

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References

Volatility Modulated Volterra Processes,

page 64 of 72

[BNCP07] Barndorff-Nielsen, O.E., Corcuera, J.M. and Podolskij, M. (2007): Power variation for Gaussian processes with sta- tionary increments. (Submitted.) [BNGJPS07] Barndorff-Nielsen, O.E., Graversen, S.E., Jacod, J., Podol- skij, M. and Shephard, N. (2006): A central limit theorem for realised power and bipower variations of continuous

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[BNSch07a] Barndorff-Nielsen, O.E. and Schmiegel, J. (2007): Am- bit processes; with applications to turbulence and cancer

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[BNSch08] Barndorff-Nielsen, O.E. and Schmiegel, J. (2007): Time change, volatility and turbulence. To appear in Proceedings

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[BNS06] Barndorff-Nielsen, O.E. and Shephard, N. (2006): Multi- power variation and stochastic volatility. In M. do Rosário Grossinho, A.N. Shiryaev, M.L. Esquível and P .E. Oliveira: Stochastic Finance. New York: Springer. Pp. 73-82. [BNS08] Barndorff-Nielsen, O.E. and Shephard, N. (2008): Financial Volatility in Continous Time. (2007). Cambridge University

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