Stochastic volatility with long-range dependence

I. Casas, J. Gao

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Abstract

Since Merton (1969), the description of a contingent claim as a Brownian motion is commonly accepted. Thus an option price, a future price, a share price, a bond price, interest rates etc., can be modelled with a Brownian motion. In summary, any financial series which present value depends on only a few previous values, may be modelled with a continuous - time diffusion - type process. The general diffusion equation is given by, dX(t) = μ(X(t))dt + σ(X(t))dB(t), (1) where μ(X(t)) is the drift function and σ(X(t)) is the volatility function of the process. There is a vast list of references related to developments on the short-term interest rate as a stochastic diffusion. For instance: a) Vacisek (1977) studied equation (1) for a mean-reverting drift function and a constant volatility; b) Cox, Ingersoll and Ross (1985) posed the CIR model which contains the square root of X(t) as part of the diffusion function. There exist financial data with long-range dependence (LRD). The typical data with this property is obtained as aggregation of several processes of type (1). For instance, portfolios or indexes such as the S&P 500, Nikkei, FTSE, etc. It is important to exploit this property as information imbedded in the data can be used to find arbitrage opportunities. Processes with LRD are modelled by, dX(t) = μ(X(t))dt + σ(X(t))dB β(t). (2) Equation (2) differs from equation (1) in the diffusion term: the classical Brownian motion is substituted by a fractional Brownian motion of the form Bβ(t) = ∫0 t(t-s)β/Γ(1+β)dB(s), where B(t) is the standard Brownian motion and Γ(x) is the usual Γ function. The fractional Brownian motion has dependent increments, in essence the fractional Brownian motion displays LRD which is measured by β. Classically the Hurst index, H ∈ (1/2 , 1), indicates the process displays LRD. The parameter β is related to H as β = H - 1/2 (see Beran 1994, p.52-53). If 0 < β < 1/2 then the process is said to have LRD, if -1/2 < β < 0 then the process is said to have intermediate-range dependence (IRD) and if β = 0 then the process is of type (1). As a particular case of (2), we are interested in the form dX(t) = -σX(t)dt + σdBβ(t), (3) where the α > 0 is the drift parameter and the the diffusion function is given by the parameter σ < 0. This particular expression is chosen because it has a solution (Comte and Renault, 1996) that is equivalent to a stationary process. Thirty years ago, Black and Scholes (1973) assumed a constant volatility to derive their famous option pricing equation. The implied volatility values obtained from this equation show skewness, suggesting that the assumption of constant volatility is not feasible. In fact, the volatility shows an intermittent behaviour with periods of high values and periods of low values. In addition, the asset volatility cannot be directly observed. The stochastic volatility (SV) model deals with these two facts. Hull and White (1987) amongst others study the logarithm of SV as an Ornstein-Uhlenbeck process. Andersen and Lund (1997) extend the CIR model to associate the spot interest rate with stochastic volatility process through estimating the parameters with the efficient method of moments. Comte and Renault (1998) specified the fractional stochastic volatility model (FSV), as an extension of the SV with stochastic volatility displaying LRD. This paper proposes a new methodology to estimate the volatility parameters from the returns. Our research shows that the parameter estimation of the FSV model can be transformed to the parameter estimation of a process without LRD. A priori, the assumption of long memory on the stochastic volatility might suggest LRD on the underlier and returns. It is shown that the returns do not display LRD independently of long memory in the volatility values.

Original languageEnglish
Title of host publicationMODSIM05 - International Congress on Modelling and Simulation
Subtitle of host publicationAdvances and Applications for Management and Decision Making, Proceedings
EditorsAndre Zerger, Robert M. Argent
PublisherModelling and Simulation Society of Australia and New Zealand Inc. (MSSANZ)
Pages802-806
Number of pages5
ISBN (Electronic)0975840029, 9780975840023
ISBN (Print)0975840002, 9780975840009
Publication statusPublished - 1 Dec 2005
Externally publishedYes
EventInternational Congress on Modelling and Simulation 2005: Advances and Applications for Management and Decision Making - Melbourne, Australia
Duration: 12 Dec 200515 Dec 2005

Publication series

NameMODSIM05 - International Congress on Modelling and Simulation: Advances and Applications for Management and Decision Making, Proceedings

Conference

ConferenceInternational Congress on Modelling and Simulation 2005
Abbreviated titleMODSIM 2005
CountryAustralia
CityMelbourne
Period12/12/0515/12/05

Keywords

  • Continuous-time model
  • Fractional volatility process
  • Stochastic differential equations

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