Volatility density estimation by multiplicative deconvolution Sergio Brenner Miguela aInstitut f ur angewandte Mathematik und Interdisciplinary Center for Scientic Computing

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Volatility density estimation by multiplicative deconvolution

Sergio Brenner Miguela

aInstitut f¨ur angewandte Mathematik und Interdisciplinary Center for Scientiﬁc Computing

(IWR), Im Neuenheimer Feld 205, Heidelberg University, Germany

ARTICLE HISTORY

Compiled October 4, 2022

ABSTRACT

We study the non-parametric estimation of an unknown stationary density fV of an

unobserved strictly stationary volatility process (Vt)t≥0on R2

+:= (0,∞)2based on

discrete-time observations in a stochastic volatility model. We identify the under-

lying multiplicative measurement error model and build an estimator based on the

estimation of the Mellin transform of the scaled, integrated volatility process and a

spectral cut-oﬀ regularisation of the inverse of the Mellin transform. We prove that

the proposed estimator leads to a consistent estimation strategy. A fully data-driven

choice of k∈R2

+is proposed and upper bounds for the mean integrated squared risk

are provided. Throughout our study, regularity properties of the volatility process

are necessary for the analsysis of the estimator. These assumptions are fulﬁlled by

several examples of volatility processes which are listed and used in a simulation

study to illustrate a reasonable behaviour of the proposed estimator.

KEYWORDS

MSC2010 Primary 62G05; secondary 62G07, 62M05 ;

Stochastic volatility model, Non-parametric statistics, Multiplicative measurement

errors, Mellin transform, Adaptivity

1. Introduction

In this work, we are interested in estimating the unknown stationary den-

sity fV:R2

+→R+of an unobserved, strictly stationary volatility process

(Vt)t≥0,Vt= (Vt,1, Vt,2)Tin a stochastic volatility model with discrete-time observa-

tions. More precisely, we assume that we have access to the discrete-time observations

Z∆,...,Z∆n, n ∈N,∆∈(0,1), of the solution (Zt)t≥0of the stochastic diﬀerential

equation

dZt=ΣtdWt,Σt:= pVt,10

0pVt,2,Z0:= 0

0,(1.1)

where (Wt)t≥0,Wt= (Wt,1, Wt,2)Tis a standard Brownian motion on R2, stochasti-

cally independent of (Vt)t≥0.

In the non-parametric literature, the stochastic volatility model has been intensively

studied in the earlier 2000s. Introduced by [19] as a natural expansion of the constant

arXiv:2210.00054v1 [math.ST] 30 Sep 2022

volatility model studied by [4], the interpretation of the volatility as a stochastic pro-

cess itself enabled the theory to explain in-practice-observed phenomenons, as pointed

out by [22].

The stochastic volatility model has been intensively studied by the authors of [14], [15]

and [16] developing limit theorems of the empirical distribution, studying parameter

estimation and including the model in a hidden markov model framework.

Later on, non-parametric estimators have been studied for instance by [9] and [26]

where [9] considered a regression-type estimation problem while [26] considered the

point-wise estimation of the stationary density of the volatility process. Both, [26] and

[9] studied kernel estimators and univariate volatility processes. The generalisation

of [26] for multivariate volatility processes was done by [25] with an isotropic choice

of the bandwidth, while a diﬀerent structure of multivariate volatility processes had

been considered in [26]. A penalised projection estimator of the stationary density

was studied in [10]. Assuming that the volatility process is an diﬀusion process [11]

proposed a penalised projection estimator for the volatility and drift coeﬃcients in a

stochastic volatility model.

Frequently, the mentioned authors built their non-parameteric estimators on a log-

transformation of the data in order to rewrite the estimation problem into an adddi-

tive deconvolution problem and use standard deconvolution estimators. This was a

common strategy in the non-parametric literature to adress multiplicative errors. In

contrary to this strategy, [3] studied the mutliplicative measurement error model di-

rectly by using the Mellin transform to solve the underlying multiplicative convolution.

[3] proposed a kernel density estimator and studied its pointwise risk. Based on this

work, [7] constructed a spectral cut-oﬀ estimator in the multiplicative measurement

error model with global risk. [5] then generalised the results of [7], which are stated

for univariate variables, for multivariate density estimation under multiplicative mea-

surement errors with anisotropic choice of the smoothing parameter.

Based on the results of [5], we will consider a multivariate stochastic volatility model,

similar to the one considered in [25], and propose an anisotropic non-parametric esti-

mator of the stationary density exploiting the rich theory of Mellin transforms.

Our approach diﬀers in the following way from the existing literature. Instead of using

a log-transformation of the data, we adress the multiplicative deconvolution problem

directly. Despite the fact that this seems to be more natural, we are additionally able

to identify and study the underlying inverse problem in a more convenient way, as

done in [5] and state more general results. Indeed, our results include the log transfor-

mation approach as a special case, as pointed out by [3] and [7]. In contrary to [25],

we study an anisotropic choice of the smoothing parameter which in general leads to

a more ﬂexible estimator, compare [12] and [5].

The paper is structured as follows. In Section 1, we introduce the bivariate stochas-

tic volatility model, identify the underlying multiplicative deconvolution problem and

collect the regularity assumptions on the volatility process (Vt)t≥0. In Section 2, we

introduce the Mellin transform and build an estimator based on the estimation of

the Mellin transform of the scaled, integrated volatility process and a spectral cut-oﬀ

regularisation of the inverse Mellin transform. We measure the performance of our

estimator in terms of the mean integrated squared error and provide upper bounds for

arbitrary choices of k∈R+. We then propose a fully data-driven choice of k∈R2

based only on the observations Z∆,...,Zn∆and bound the risk of the resulting data-

driven density estimator. Several examples of volatility processes are then studied in

Section 3.1 and used in a simulation study to show reasonable the performance of

the proposed estimation strategy. More general results for the density estimation in a

multiplicative measurement error model with stationary data are stated in Section 4,

which are needed in the proofs of the results of Section 2. The proof ofs Section 1, 2

and 4 are collected in the Appendix 5.

Stochastic volatility model In this paper, we consider the following version of

a multivariate stochastic volatility model, motivated by [13], which has also been

considered by [25].

For a strictly stationary unobserved Markov process (Vt)t≥0, we consider the solution

(Zt)t≥0of the stochastic diﬀerential equation (1.1) where (Wt)t≥0is a standard 2-

dimensional Brownian motion, stochastically independent of the process (Vt)t≥0. Then

motivated by the work [17], respectively [10], we study the scaled increments of our

discrete-time sample (Z∆j)j∈JnKfor ∆ ∈(0,1) and JnK:= [1, n]∩N.

More precisely, let Dj:= ∆−1/2(Z∆j−Z∆(j−1)), understood componentwise for j∈

JnK, then conditioned on (Vt)t≥0we have

Dj=1

√∆ Rj∆

(j−1)∆ Vt,1dWt,1

Rj∆

(j−1)∆ Vt,2dWt,2!∼N(0,ΣVj),ΣVj:= Vj,10

0Vj,2

exploiting the independence of (Vt)t≥0and (Wt)t≥0, where Vj,` := ∆−1Rj∆

(j−1)∆ Vs,`ds,

`∈ {1,2}. As a direct consequence, we write

Yj:= Yj,1

Yj,2:= D2

j,1

j,2=Vj,1Uj,1

Vj,2Uj,2=: Xj,1Uj,1

Xj,2Uj,2=: XjUj(1.2)

where Xjand Uja stochastically independent and (Uj)j∈JnKis an i.i.d. (independent,

identically distributed) sequence with U1,1, U1,2i.i.d.

∼χ2

1= Γ(1/2,1/2). In other words,

the stochastic volatility model can be expressed as a multiplicative measurement error

model with χ-squared, respectively Gamma distributed noise. While the authors from

[17], [10], [26] and [25] used a log-transformation of the data, we will instead exploit

the theory of multivariate Mellin transform and their use in non-parametric density

estimation introduced in [5] to build a multiplicative deconvolution density estimator.

Assumption on the volatility process (Vt)t≥0Throughout this paper, we will

need to assume some regularity of the volatility process (Vt)t≥0to ensure the well-

deﬁnedness of the upcoming objects and to deduce consistency of our proposed estima-

tion strategy. As usual in non-parametric approaches, we aim to consider an ensemble

of assumptions which can be proven for a wide class of examples of volatility processes.

To motivate that these assumptions are not restrictive, we will show in Section 3.1 a

number of examples of frequently studied volatility processes.

Now let us assume that the discrete-time sample (Zj∆)j∈JnKis drawn from a process

(Zt)t≥0solving (1.1) where

(A0) (Wt)t≥0is a two-dimensional Brownian motion, independent of the process

(Vt)t≥0on R2

(A1) (Vt)t≥0is a time-homogeneous Markov process, with continuous sample paths,

strictly stationary and ergodic. The stationary distribution of (Vt)t≥0admits a

density fVwith respect to the Lebesgue measure on R2

(A2) (Vt)t≥0is β-mixing, with RR+βV(s)ds < ∞, where

βV(s) = TV(P(V0,Vs),PV0⊗PVs), s ∈R+,

where TV is the total variation distance.

For the estimation we will be in need of the following additional assumption

(A3) There exists a constant c>0 such that E(|log(X1,1)−log(V0,1)|+|log(X1,2)−

log(V0,2)|)≤c√∆.

In Section 3.1, we will deliver examples of volatility processes (Vt)t≥0which satisfy

(A0)-(A3). While assumptions (A0)- (A2) are widely considered in the literature and

proven for several diﬀusion processes, assumption (A3) is of rather technical nature.

A practical proposition in the univariate case was proposed by [10]. Here, we want to

state a bivariate counterpart. The proof of Proposition 1.1 can be found in Section 5.2.

Here, we denote for a∈R2the Euclidean norm by |a|2

R2:= a2

1+a2

2and for a matrix

A∈R(2,2) the Frobenius norm by |A|2

F:= A2

1,1+A2

1,2+A2

2,1+A2

2,2.Furthermore, for

p∈N0we denote by Cp(D) the set of all p-times continuously diﬀerentiable functions

on D ⊆ R2

Proposition 1.1. Suppose the volatility process (Vt)t≥0satisﬁes (either) one of the

following conditions

(i) Vt= (exp(Zt,1),exp(Zt,2))T, t ≥0,where (Zt)t≥0is a strictly stationary and

ergodic diﬀusion process on R2satisfying dZt=e

b(Zt) + e

a(Zt)df

Wt,(f

Wt)t≥0

standard Brownian motion on R2such that there exists e

L > 0with

a(x)kF+|e

b(x)|R2≤e

L(1 + |x|R2)

for all x∈R2,e

bi,e

ai,j ∈ C0(R2)∩C1(R2

+)for i, j ∈J2Kand E(|Z0|2

R2)<∞or

(ii) or (Vt)t≥0is a strictly stationary and ergodic diﬀusion process on R2

+satisfying

dVt=b(Vt) + a(Vt)df

Wtsuch that there exists L > 0with

ka(x)kF+|b(x)|R2≤e

L(1 + |x|R2)

for all x∈R2,bi,ai,j ∈ C0(R2)∩C1(R2

+)for i, j ∈J2Kand additionally let

E(supt∈[0,∆](Vt,i)−2),E(|V0|2

R2)<∞for i∈J2Khold true.

Then (Vt)t≥0satisﬁes (A3).

After this brief introduction to the stochastic volatility model, let us propose a non-

parametric density estimator based on a multiplicative deconvolution.

2. Stochastic volatility density estimation

In this section we introduce the Mellin transform and start to collect some of its major

properties, which are stated in [5]. We then propose our estimator.

Notations and deﬁnitions of the Mellin transform For two vectors u=

(u1, u2)T,v= (v1, v2)T∈R2and a scalar λ∈Rwe deﬁne the componentwise multipli-

cation uv := u·v:= (u1v1, u2v2)Tand denote by λuthe usual scalar multiplication.

Further, if v1, v26= 0 we deﬁne the multivariate power by vu:= vu1

1vv2

2. Addition-

ally, we deﬁne the componentwise division by u/v:= (u1/v1, u2/v2)TWe denote

the usual Euclidean scalar product and norm on R2by hu,viR2:= Pi∈J2Kuiviand

|u|R2:= phu,uiR2. Moreover, we set 1:= (1,1)T∈R+, respectively 0:= (0,0)T.

For a positive random vector Zwith E(Zc−1) = E(Zc1−1

1Zc2−1

2)<∞,c∈R2, we

deﬁne the Mellin transform Mc[Z] of Zas the function

Mc[Z] : R2→C,t7→ Mc[Z](t) := E(Zc−1+it).

As a consequence the convolution theorem for the Mellin transform holds true, that

is for UV independent with E((UV )c−1)<∞,

Mc[UV ](t) = Mc[U](t)Mc[V](t),t∈R2.

If Zemits a Lebesgue density h:R2

+→R+, then we can write Mc[Z](t) =

RR2

+xc−1+ith(x)dx,t∈R2. Motivated by this, we deﬁne the set L1(R2

+,xc−1) :=

{h:R2

+→C:khkL1(R2

+,xc−1):= RR2

+|h(x)|xc−1dx<∞}. Then we can generalise the

notion of the Mellin transform for L1(R2

+,xc−1) function. Indeed, for h∈L1(R2

+,xc−1)

we deﬁne the Mellin transform of hat the development point c∈R2as the function

Mc[h] : R2→Cby

Mc[h](t) := ZR2

xc−1+ith(x)dx,t∈R2.(2.1)

In analogy to the Fourier transform, one can deﬁne the Mellin transform for square in-

tegrable functions. We deﬁne the weighted norm by khk2

x2c−1:= RR2

+|h(x)|2x2c−1dxfor

a measurable function h:R2

+→Cand denote by L2(R2

+,x2c−1) the set of all complex-

valued, measurable functions with ﬁnite k.kx2c−1-norm and by hh1, h2ix2c−1:=

RR2

+h1(x)h2(x)x2c−1dxfor h1, h2∈L2(R2

+,x2c−1) the corresponding weighted scalar

product. Similarly, we deﬁne L2(R2) := {H:R2→Cmeasurable : kHk2

R2:=

RR2H(x)H(x)dx<∞}.

We are then able to deﬁne the Mellin transform as the isomorphism Mc:

L2(R2

+,x2c−1)→L2(R2).For a precise deﬁnition of the multivariate Mellin trans-

form and its connection to the Fourier transform, we refer to [5]. Nevertheless, if

h∈L1(R2

+,xc−1)∩L2(R2

+,x2c−1) both notions coincide. By abuse of notation we will

denote by Mc[h] both notions, for h∈L1(R2

+,xc−1), respectively h∈L2(R2

+,x2c−1)

of the Mellin transform. For a more detailed collection of the properties of the Mellin

transform we refer to Section 4, respectively [5].

Estimation strategy For k∈R2

+we deﬁne the hypercuboid [−k,k] := [−k1, k1]×

[−k2, k2]. Then based on the work of [5], we deﬁne for any c∈R2with E(Yc−1

1)<∞

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VolatilitydensityestimationbymultiplicativedeconvolutionSergioBrennerMiguelaaInstitutfurangewandteMathematikundInterdisciplinaryCenterforScienticComputing(IWR),ImNeuenheimerFeld205,HeidelbergUniversity,GermanyARTICLEHISTORYCompiledOctober4,2022ABSTRACTWestudythenon-parametricestimationofanunknowns...

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Volatility density estimation by multiplicative deconvolution Sergio Brenner Miguela aInstitut f ur angewandte Mathematik und Interdisciplinary Center for Scientic Computing

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