This paper introduces the concept of stochastic volatility of volatility in continuous time and, hence, extends standard stochastic volatility (SV) models to allow for an additional source of randomness associated with greater variability in the data. We discuss how stochastic volatility of volatility can be defined both non–parametrically, where we link it to the quadratic variation of the sto...

Empirical evidence from equity markets clearly shows that the volatility of asset returns varies randomly in time. Typically, this randomness is referred to as stochastic volatility. In this article, we review how stochastic volatility can be modeled, and the use of asymptotic analysis to quantify (i) how the presence of stochastic volatility affects option prices, and (ii) how stochastic volat...

This paper introduces a new class of stochastic volatility models which allows for stochastic volatility of volatility. Such models are given by volatility modulated non–Gaussian Ornstein Uhlenbeck processes. We study the probabilistic properties of such models both under the physical and under the risk neutral probability measure, where we focus in particular on the role of the volatility of v...

An estimation method is developed for extracting the latent stochastic volatility from VIX, a volatility index for the S&P 500 index return produced by the Chicago Board Options Exchange (CBOE) using the so-called model-free volatility construction. Our model specification encompasses all mean-reverting stochastic volatility option pricing models with a constant-elasticity of variance and those...

In this chapter we use particle filtering methods to estimate volatility and examine volatility dynamics for three financial time series during the early part of the current credit crisis. We compare estimates from a pure stochastic volatility model, a stochastic volatility model with jumps and a Garch model to each other and to the market volatilities implied by actual option prices. Our three...

In this paper we examine the effect of stochastic volatility on optimal portfolio choice in both partial and general equilibrium settings. In a partial equilibrium setting we derive an analog of the classic Samuelson–Merton optimal portfolio result and define volatility-adjusted risk aversion as the effective risk aversion of an individual investing in an asset with stochastic volatility. We ex...

This paper proposes a class of nonlinear stochastic volatility models. The proposed class encompasses many parametric stochastic volatility models that have appeared in the literature, including the well known lognormal stochastic volatility model. The new class is based on the Box-Cox transformation and offers an alternative to the one introduced in Andersen (1994). An advantage of our propose...