Estimation of the volatility persistence in a discretly observed diffusion model

Mathieu Rosenbaum

Description

Estimation of the volatility persistence in a discretly observed diffusion model Mathieu Rosenbaum LS-CREST, Laboratoire de Statistique, Timbre J340, 3 avenue Pierre Larousse, 92240 Malakoff, France. Laboratoire d'Analyse et de Mathematiques Appliquees, CNRS UMR 8050 et Universite de Marne-la-Vallee, France. Abstract We consider the stochastic volatility model dYt = ?t dBt, with B a Brownian motion and ? of the form ?t = ? (∫ t 0 a(u)dWHu ) , where WH is a fractional Brownian motion, independent of the driving Brownian motion B, with Hurst parameter H ≥ 1/2. This model allows for persistence in the volatility ?. The parameter of interest is H and the functions ? and a are treated as nuisance parameters. For a fixed objective time T, we construct from discrete data Yi/n, i = 0, . . . , nT, a wavelet based estimator of H, inspired by adaptive estimation of quadratic functionals. We show that the accuracy of our estimator is n?1/(4H+2) and that this rate is optimal in a minimax sense. Resume On considere le modele a volatilite stochastique defini par les equations precedentes, ou B est un mouvement brownien et WH un mouvement brownien fractionnaire, independant de B, de parametre de Hurst H ≥ 1/2.

• fractional brownian


• stochastic volatility


• stochastique defini par les equations precedentes


• proprietes de persistance dans la volatilite ?


• rate vn


• objective time

Sujets

• Renault
• Rosenbaum
• Stochastic volatility
• maths

Informations

• Publié par: pefav
• Nombre de lectures: 13
• Langue: English

Extrait

Estimation of the volatility persistence in discretly observed diffusion model

Mathieu Rosenbaum

LS-CREST, Laboratoire de Statistique, Timbre J340, 3 avenue Pierre Larousse, 92240 Malakoff, France. Laboratoired’AnalyseetdeMath´ematiquesAppliqu´ees, CNRSUMR8050etUniversite´deMarne-la-Vall´ee,France.

Abstract

a

We consider the stochastic volatility model dY t =σ t dB t , withBa Brownian motion andσof the form t σ t = Φ  Z a(u)dW uH  , 0 whereW H is a fractional Brownian motion, independent of the driving Brownian motionB, with Hurst parameterH≥1/2. This model allows for persistence in the volatilityσ. The parameter of interest isHand the functions Φ andaare treated as nuisance parameters. For a fixed objective time T, we construct from discrete data Y i/n , i= 0, . . . , nT ,a wavelet based estimator ofH, inspired by adaptive estimation of quadratic functionals. We show that the accuracy of our estimator isn −1/(4H+2) and that this rate is optimal in a minimax sense.

Resume ´ ´

Onconsid`erelemod`ele`avolatilit´estochastiqued´efiniparlese´quationspre´ce´dentes, ou`Best un mouvement brownien etW H un mouvement brownien fractionnaire, ind´ependantdeBsttredeHurperamae`d,H≥1/eperme2t.dCemod`elrierepeorud despropri´ete´sdepersistancedanslavolatilit´eσpara.Lereeˆni´ter’d`mtestteHet les fonctions Φ etatiarttnose´seocmmdeseaparm`etresdenuisancoP.enurupmets objectiffix´eTtia`tsurriedaptrn´eesdoncr`esdisset,oonncY i/n , i= 0, . . . , nT ,un estimateur par ondelettes deHs’leditsepsnie´ri,oinnleeledfsnotcaptativemationad quadratiques. On montre que l ´ ision de notre estimateur estn −1/(4H+2) et que a prec celle-ci est optimale au sens minimax.

Key words:Stochastic volatility models; High frequency data; Fractional Brownian motion; Adaptive estimation of quadratic functionals; Wavelet methods. 1991 MSC:60G18; 60G99; 60H05; 60H40; 60G15; 62F12; 62F99; 62M09; 62P05.

Email address:[e-mail](Mathieu Rosenbaum).

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Introduction

1.1 Stochastic volatility and volatility persistence

Since the celebrated model of Black and Scholes, the behaviour of financial assets is modeled by processes of type

dS t =µ t dt+σ t dB t , whereSis the price of the asset,Ba Brownian motion andµa drift process. The volatility coefficientσrepresents the fluctuations ofSand plays a crucial role in trading, option pricing and hedging. It is well known that stochastic volatility models, where the volatility is a random process, are a way to deal with the endemic time-varying volatility and to reproduce various stylised facts observed on the markets, see Shephard [30], Barndorff-Nielsen, Nico- lato and Shephard [3]. Among these stylised facts, there are many arguings about volatility persistence. This presence of memory in the volatility has in particular consequences for option pricing, see Taylor [31], Comte, Coutin and Renault [8]. Hence continuous time dynamics have been introduced to capture this phenomenon, see Comte and Renault [9], Comte, Coutin and Renault [8] or Barndorff-Nielsen and Shephard [4]. Paradoxically, in statistical finance, the question of volatility persistence has been mostly treated with discrete time models, see among others Breidt, Crato and De Lima [6], Harvey [16], AndersenandBollerslev[1],Robinson[29],HurvichandSoulier[20],Teyssie`re [33]. Concurrently, statistical methods to detect this volatility persistence have been specifically developed for these models, see Hurvich, Moulines and Soulier [18], Deo, Hurvich and Lu [12], Hurvich and Ray [19], Lee [23], Jensen [22]. In this paper, our objective is to build for continuous time models a statistical program allowing to recover information about volatility persistence.

1.2 A diffusion model with fractional stochastic volatility

We consider a class of diffusion models whose volatility is a non-linear trans- formation of a stochastic integral with respect to fractional Brownian motion. Recall that a fractional Brownian motion (W tH , t≥0), with Hurst parameter H∈[0,1] is a self-similar centered Gaussian process with covariance function

E[W t H W H s ( 2 1 =]|s| 2H +|t| 2H − |t−s| 2H ).

As soon asH >1/2, the use of fractional Brownian motion (fbm for short) is a way to allow for persistence. Indeed, its increments are positively correlated

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and the value of the Hurst parameter quantifies the presence of so-called long- memory in the dynamic, see Mandelbrot and Van Ness [25], Taqqu [13]. We define on a rich enough probability space (Ω,A,P) a Brownian motionBand a fractional Brownian motionW H , independent ofB, with unknown Hurst parameterH∈(1/2,1). We fix an objective timeT >0 and we consider the 1-dimensional stochastic processYdefined by Y t =y 0 + Z 0 t σ s dB s , y 0 ∈R, t∈[0, T],(1) whereσis another 1-dimensional stochastic process of the form t σ t = Φ  Z a(u)dW H u  .(2) 0 The functions Φ andaare deterministic and unknown. The stochastic integral with respect to fractional Brownian motion is defined as the limit in L 2 (P) of the associated Riemann sums (for details and properties, we refer to Lin [24]).

This framework is an extension of the model introduced in mathematical finance by Comte and Renault [9]. Remark also that forH= 1/2, under smoothness assumptions on Φ, letting a= 1, f= (Φ 2 ) 0 ◦Φ −1 andg= (Φ 2 ) 00 ◦Φ −1 ,

we equivalently have dσ t 2 =g(σ t 2 )dt+f(σ t 2 )dW t . Thus, we (partially) retrieve the standard stochastic volatility diffusion frame- work, see for example Hull and White [17], Melino and Turnbull [26] or Musiela and Rutkowski [27] for a more exhaustive study.

Finally, the assumptions onaand Φ in the model (1)-(2) are the following:

Assumption A.

(i)t→a(t) is continuously differentiable, (ii) There exist 0≤α < β≤Tsuch that inf t∈[α,β] a 2 (t)>0. Assumption B.LetIdenote the indicator function,

(i)x→Φ(x) is twice continuously differentiable, (ii) For somec 1 >0,c 2 >0 andγ≥0,|(Φ 2 ) 0 (x)| ≥c 1 |x| γ I |x|∈[0,1] +c 2 I |x|>1 , (iii) For somec 3 >0 andm≥0,|(Φ 2 ) 00 (x)| ≤c 3 (1 +|x| m ).

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