Almut E. D. Veraart

Modelling Energy Spot Prices by Volatility Modulated Lévy-Driven Volterra Processes

This paper introduces the class of volatility modulated Levy-driven Volterra (VMLV) processes and their important subclass of Levy semistationary (LSS) processes as a new framework for modelling energy spot prices. The main modelling idea consists of four principles: First, deseasonalised spot prices can be modelled directly in stationarity. Second, stochastic volatility is regarded as a key factor for modelling energy spot prices. Third, the model allows for the possibility of jumps and extreme spikes and, lastly, it features great flexibility in terms of modelling the autocorrelation structure and the Samuelson effect. We provide a detailed analysis of the probabilistic properties of VMLV processes and show how they can capture many stylised facts of energy markets. Further, we derive forward prices based on our new spot price models and discuss option pricing. An empirical example based on electricity spot prices from the European Energy Exchange confirms the practical relevance of our new modelling framework.

Inference for the Jump Part of Quadratic Variation of Itô Semimartingales

Recent research has focused on modelling asset prices by Ito semimartingales. In such a modelling framework, the quadratic variation consists of a continuous and a jump component. This paper is about inference on the jump part of the quadratic variation, which can be estimated by the difference of realised variance and realised multipower variation. The main contribution of this paper is twofold. First, it provides a bivariate asymptotic limit theory for realised variance and realised multipower variation in the presence of jumps. Second, this paper presents new, consistent estimators for the jump part of the asymptotic variance of the estimation bias. Eventually, this leads to a feasible asymptotic theory which is applicable in practice. Finally, Monte Carlo studies reveal a good finite sample performance of the proposed feasible limit theory.

Modelling Electricity Day-Ahead Prices by Multivariate Lévy Semistationary Processes

This paper presents a new modelling framework for day-ahead electricity prices based on multivariate Levy semistationary (\(\mathcal{M}\mathcal{L}\mathcal{S}\mathcal{S}\)) processes. Day-ahead prices specify the prices for electricity delivered over certain time windows on the next day and are determined in a daily auction. Since there are several delivery periods per day, we use a multivariate model to describe the different day-ahead prices for the different delivery periods on the next day. We extend the work by [4] on univariate Levy semistationary processes to a multivariate setting and discuss the probabilistic properties of the new class of stochastic processes. Furthermore, we provide a detailed empirical study using data from the European energy exchange (EEX) and give new insights into the intra-daily correlation structure of electricity day-ahead prices in the EEX market. The flexible structure of \(\mathcal{M}\mathcal{L}\mathcal{S}\mathcal{S}\) processes is able to reproduce the stylized facts of such data rather well. Furthermore, these processes can be used to model negative prices in electricity markets which started to occur recently and cannot be described by many classical models.

Ambit Processes and Stochastic Partial Differential Equations

Ambit processes are general stochastic processes based on stochastic integrals with respect to Levy bases. Due to their flexible structure, they have great potential for providing realistic models for various applications such as in turbulence and finance. This papers studies the connection between ambit processes and solutions to stochastic partial differential equations. We investigate this relationship from two angles: from the Walsh theory of martingale measures and from the viewpoint of the Levy noise analysis.

Stochastic Volatility of Volatility and Variance Risk Premia

This article introduces a new class of stochastic volatility models which allows for stochastic volatility of volatility (SVV): Volatility modulated non-Gaussian Ornstein--Uhlenbeck (VMOU) processes. Various probabilistic properties of (integrated) VMOU processes are presented. Further we study the effect of the SVV on the leverage effect and on the presence of long memory. One of the key results in the article is that we can quantify the impact of the SVV on the (stochastic) dynamics of the variance risk premium (VRP). Moreover, provided the physical and the risk-neutral probability measures are related through a structure-preserving change of measure, we obtain an explicit formula for the VRP. Copyright The Author, 2012. Published by Oxford University Press. All rights reserved. For Permissions, please email: journals.permissions@oxfordjournals.org , Oxford University Press.

Modelling Electricity Forward Markets by Ambit Fields

This paper proposes a new modelling framework for electricity forward markets, which is based on ambit fields. The new model can capture many of the stylised facts observed in energy markets. One of the main differences to the traditional models lies in the fact that we do not model the dynamics, but the forward price directly, where we focus on models which are stationary in time. We give a detailed account on the probabilistic properties of the new model and we discuss martingale conditions and change of measure within the new model class. Also, we derive a model for the spot price which is obtained from the forward model through a limiting argument.

Approximating Levy Semistationary Processes via Fourier Methods in the Context of Power Markets

The present paper discusses simulation of Levy semistationary (LSS) processes in the context of power markets. A disadvantage of applying numerical integration to obtain trajectories of LSS processes is that such a scheme is not iterative. We address this problem by introducing and analyzing a Fourier simulation scheme for obtaining trajectories of these processes in an iterative manner. Furthermore, we demonstrate that our proposed scheme is well suited for simulation of a wide range of LSS processes, including, in particular, LSS processes indexed by a kernel function which is steep close to the origin. Finally, we put our simulation scheme to work for simulating the price of path-dependent options to demonstrate the advantages of the proposed Fourier simulation scheme.

Modelling electricity futures by ambit fields

In this paper we propose a new modelling framework for electricity futures markets based on so-called ambit fields. The new model can capture many of the stylised facts observed in electricity futures and is highly analytically tractable. We discuss martingale conditions, option pricing, and change of measure within the new model class. Also, we study the corresponding model for the spot price, which is implied by the new futures model, and show that, under certain regularity conditions, the implied spot price can be represented in law as a volatility modulated Volterra process.

Modelling Energy Spot Prices by Lévy Semistationary Processes

This paper introduces a new modelling framework for energy spot prices based on Levy semistationary processes. Levy semistationary processes are special cases of the general class of ambit processes. We provide a detailed analysis of the probabilistic properties of such models and we show how they are able to capture many of the stylised facts observed in energy markets. Furthermore, we derive forward prices based on our spot price model. As it turns out, many of the classical spot models can be embedded into our novel modelling framework.

Stochastic Volatility of Volatility in Continuous Time

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 stochastic variance process, and parametrically, where we propose two new SV models which allow for stochastic volatility of volatility. In addition, we show that volatility of volatility can be estimated by a novel estimator called pre–estimated spot variance based realised variance.