Long memory in continuous‐time stochastic volatility models

Abstract

This paper studies a classical extension of the Black and Scholes model for option pricing, often known as the Hull and White model. Our specification is that the volatility process is assumed not only to be stochastic, but also to have long-memory features and properties. We study here the implications of this continuous-time long-memory model, both for the volatility process itself as well as for the global asset price process. We also compare our model with some discrete time approximations. Then the issue of option pricing is addressed by looking at theoretical formulas and properties of the implicit volatilities as well as statistical inference tractability. Lastly, we provide a few simulation experiments to illustrate our results. Copyright Blackwell Publishers Inc 1998.

Estimating the Volatility Occupation Time via Regularized Laplace Inversion

“…Li et al (2013) provide a two-step estimation method for estimating the VOT from a high-frequency record of X by first nonparametrically estimating the spot variance process over [0, T ] and then constructing a direct plug-in estimator corresponding to (2). Their estimation method is based on a thresholding technique (Mancini (2001)) to separate volatility from jumps and forming blocks of asymptotically decreasing length to account for the time variation of volatility (Foster and Nelson (1996), Comte and Renault (1998)). …”

“…Thus, in this paper we impose neither the existence of invariant distribution of the volatility process nor mixing-type conditions. While such conditions may be reasonable for analyzing data from a long sample period, they are unlikely to "kick in" sufficiently fast in short samples in view of the high persistence of the volatility process (Comte and Renault (1998)). In our setup, we allow the volatility process to be nonstationary and strongly serially dependent.…”

Asymmetry and Leverage in Realized Volatility

“…The asymmetric SV model of Harvey and Shephard (1996) is considered to be an Euler-Maruyama approximation of the continuous-time model (3), with negative correlation. Three major extensions of such diffusion-based SV models incorporate jumps to volatility process (Eraker, Johannes and Polson (2003)), model volatility as a function of a number of factors (Chernov et al (2003)), and allow the log-volatility to follow a long memory process (Comte and Renault (1998)). …”

No-arbitrage, leverage and completeness in a fractional volatility model

“…Notice that this empirically based model is different from the usual stochastic volatility models which assume the volatility to follow an arithmetic or geometric Brownian process. Also in Comte and Renault [9] and Hu [10], it is fractional Brownian motion that drives the volatility, not its derivative (fractional noise). δ is the observation scale of the process.…”