# Stochastic volatility with long-range dependence

I. Casas, J. Gao

Research output: Chapter in Book/Report/Conference proceedingConference PaperOtherpeer-review

## 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 language English MODSIM05 - International Congress on Modelling and Simulation Advances and Applications for Management and Decision Making, Proceedings Andre Zerger, Robert M. Argent Modelling and Simulation Society of Australia and New Zealand Inc. (MSSANZ) 802-806 5 0975840029, 9780975840023 0975840002, 9780975840009 Published - 1 Dec 2005 Yes International Congress on Modelling and Simulation 2005: Advances and Applications for Management and Decision Making - Melbourne, AustraliaDuration: 12 Dec 2005 → 15 Dec 2005

### Publication series

Name MODSIM05 - International Congress on Modelling and Simulation: Advances and Applications for Management and Decision Making, Proceedings

### Conference

Conference International Congress on Modelling and Simulation 2005 MODSIM 2005 Australia Melbourne 12/12/05 → 15/12/05

## Keywords

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