arXiv subjects

Fred Espen Benth

Publications and source records attributed to Fred Espen Benth.

At least 55 records · Page 3Linked to original sources

arXiv2017

Cointegration in continuous time for factor models

We develop cointegration for multivariate continuous-time stochastic processes, both in finite and infinite dimension. Our definition and analysis are based on factor processes and operators mapping to the space of prices and cointegration. The focus is on commodity markets, where both spot and forward prices are analysed in the context of cointegration. We provide many examples which include the most used continuous-time pricing models, including forward curve models in the Heath-Jarrow-Morton paradigm in Hilbert space.

math.PR↗

arXiv2017

The Heston stochastic volatility model in Hilbert space

We extend the Heston stochastic volatility model to a Hilbert space framework. The tensor Heston stochastic variance process is defined as a tensor product of a Hilbert-valued Ornstein-Uhlenbeck process with itself. The volatility process is then defined by a Cholesky decomposition of the variance process. We define a Hilbert-valued Ornstein-Uhlenbeck process with Wiener noise perturbed by this stochastic volatility, and compute the characteristic functional and covariance operator of this process. This process is then applied to the modelling of forward curves in energy markets. Finally, we compute the dynamics of the tensor Heston volatility model when the generator is bounded, and study its projection down to the real line for comparison with the classical Heston dynamics.

math.PR↗

arXiv2017

Continuous-time autoregressive moving-average processes in Hilbert space

We introduce the class of continuous-time autoregressive moving-average (CARMA) processes in Hilbert spaces. As driving noises of these processes we consider Levy processes in Hilbert space. We provide the basic definitions, show relevant properties of these processes and establish the equivalents of CARMA processes on the real line. Finally, CARMA processes in Hilbert space are linked to the stochastic wave equation and functional autoregressive processes.

math.PR↗

arXiv2016

Integration theory for infinite dimensional volatility modulated Volterra processes

We treat a stochastic integration theory for a class of Hilbert-valued, volatility-modulated, conditionally Gaussian Volterra processes. We apply techniques from Malliavin calculus to define this stochastic integration as a sum of a Skorohod integral, where the integrand is obtained by applying an operator to the original integrand, and a correction term involving the Malliavin derivative of the same altered integrand, integrated against the Lebesgue measure. The resulting integral satisfies many of the expected properties of a stochastic integral, including an Itô formula. Moreover, we derive an alternative definition using a random-field approach and relate both concepts. We present examples related to fundamental solutions to partial differential equations.

math.PR↗

arXiv2016

Simulation of volatility modulated Volterra processes using hyperbolic stochastic partial differential equations

We propose a finite difference scheme to simulate solutions to a certain type of hyperbolic stochastic partial differential equation (HSPDE). These solutions can in turn estimate so called volatility modulated Volterra (VMV) processes and Lévy semistationary (LSS) processes, which is a class of processes that have been employed to model turbulence, tumor growth and electricity forward and spot prices. We will see that our finite difference scheme converges to the solution of the HSPDE as we take finer and finer partitions for our finite difference scheme in both time and space. Finally, we demonstrate our method with an example from the energy finance literature.

math.ST↗

arXiv2015

Approximation of forward curve models in commodity markets with arbitrage-free finite dimensional models

In this paper we show how to approximate a Heath-Jarrow-Morton dynamics for the forward prices in commodity markets with arbitrage-free models which have a finite dimensional state space. Moreover, we recover a closed form representation of the forward price dynamics in the approximation models and derive the rate of convergence uniformly over an interval of time to maturity to the true dynamics under certain additional smoothness conditions. In the Markovian case we can strengthen the convergence to be uniform over time as well. Our results are based on the construction of a convenient Riesz basis on the state space of the term structure dynamics.

q-fin.MF↗

arXiv2015

Representation and approximation of ambit fields in Hilbert space

We lift ambit fields as introduced by Barndorff-Nielsen and Schmiegel to a class of Hilbert space-valued volatility modulated Volterra processes. We name this class Hambit fields, and show that they can be expressed as a countable sum of weighted real-valued volatility modulated Volterra processes. Moreover, Hambit fields can be interpreted as the boundary of the mild solution of a certain first order stochastic partial differential equation. This stochastic partial differential equation is formulated on a suitable Hilbert space of functions on the positive real line with values in the state space of the Hambit field. We provide an explicit construction of such a space. Finally, we apply this interpretation of Hambit fields to develop a finite difference scheme, for which we prove convergence under some Lipschitz conditions.

math.PR↗

arXiv2015

Ornstein-Uhlenbeck processes in Hilbert space with non-Gaussian stochastic volatility

We propose a non-Gaussian operator-valued extension of the Barndorff-Nielsen and Shephard stochastic volatility dynamics, defined as the square-root of an operator-valued Ornstein-Uhlenbeck process with Levy noise and bounded drift. We derive conditions for the positive definiteness of the Ornstein-Uhlenbeck process, where in particular we must restrict to operator-valued Levy processes with "non-decreasing paths". It turns out that the volatility model allows for an explicit calculation of its characteristic function, showing an affine structure. We introduce another Hilbert space-valued Ornstein-Uhlenbeck process with Wiener noise perturbed by this class of stochastic volatility dynamics. Under a strong commutativity condition between the covariance operator of the Wiener process and the stochastic volatility, we can derive an analytical expression for the characteristic functional of the Ornstein-Uhlenbeck process perturbed by stochastic volatility if the noises are independent. The case of operator-valued compound Poisson processes as driving noise in the volatility is discussed as a particular example of interest. We apply our results to futures prices in commodity markets, where we discuss our proposed stochastic volatility model in light of ambit fields.

math.PR↗

arXiv2014

Derivatives pricing in energy markets: an infinite dimensional approach

Based on forward curves modelled as Hilbert-space valued processes, we analyse the pricing of various options relevant in energy markets. In particular, we connect empirical evidence about energy forward prices known from the literature to propose stochastic models. Forward prices can be represented as linear functions on a Hilbert space, and options can thus be viewed as derivatives on the whole curve. The value of these options are computed under various specifications, in addition to their deltas. In a second part, cross-commodity models are investigated, leading to a study of square integrable random variables with values in a "two-dimensional" Hilbert space. We analyse the covariance operator and representations of such variables, as well as presenting applications to pricing of spread and energy quanto options.

q-fin.MF↗

arXiv2014

Pricing and hedging of energy spread options and volatility modulated Volterra processes

We derive the price of a spread option based on two assets which follow a bivariate volatility modulated Volterra process dynamics. Such a price dynamics is particularly relevant in energy markets, modelling for example the spot price of power and gas. Volatility modulated Volterra processes are in general not semimartingales, but contain several special cases of interest in energy markets like for example continuous-time autoregressive moving average processes. Based on a change of measure, we obtain a pricing expression based on a univariate Fourier transform of the payoff function and the characteristic function of the price dynamics. Moreover, the spread option price can be expressed in terms of the forward prices on the underlying dynamics assets. We compute a linear system of equations for the quadratic hedge for the spread option in terms of a portfolio of underlying forward contracts.

q-fin.PR↗

arXiv2014

A change of measure preserving the affine structure in the BNS model for commodity markets

For a commodity spot price dynamics given by an Ornstein-Uhlenbeck process with Barndorff-Nielsen and Shephard stochastic volatility, we price forwards using a class of pricing measures that simultaneously allow for change of level and speed in the mean reversion of both the price and the volatility. The risk premium is derived in the case of arithmetic and geometric spot price processes, and it is demonstrated that we can provide flexible shapes that is typically observed in energy markets. In particular, our pricing measure preserves the affine model structure and decomposes into a price and volatility risk premium, and in the geometric spot price model we need to resort to a detailed analysis of a system of Riccati equations, for which we show existence and uniqueness of solution and asymptotic properties that explains the possible risk premium profiles. Among the typical shapes, the risk premium allows for a stochastic change of sign, and can attain positive values in the short end of the forward market and negative in the long end.

q-fin.PR↗

arXiv2014

Representation of infinite dimensional forward price models in commodity markets

We study the forward price dynamics in commodity markets realized as a process with values in a Hilbert space of absolutely continuous functions defined by Filipović. The forward dynamics are defined as the mild solution of a certain stochastic partial differential equation driven by an infinite dimensional Lévy process. It is shown that the associated spot price dynamics can be expressed as a sum of Ornstein-Uhlenbeck processes, or more generally, as a sum of certain stationary processes. These results link the possibly infinite dimensional forward dynamics to classical commodity spot models. We continue with a detailed analysis of multiplication and integral operators on the Hilbert spaces and show that Hilbert-Schmidt operators are essentially integral operators. The covariance operator of the Lévy process driving the forward dynamics and the diffusion term can both be specified in terms of such operators, and we analyse in several examples the consequences on model dynamics and their probabilistic properties. Also, we represent the forward price for contracts delivering over a period in terms of an integral operator, a case being relevant for power and gas markets. In several examples we reduce our general model to existing commodity spot and forward dynamics.

q-fin.PR↗

arXiv2013

A pricing measure to explain the risk premium in power markets

In electricity markets, it is sensible to use a two-factor model with mean reversion for spot prices. One of the factors is an Ornstein-Uhlenbeck (OU) process driven by a Brownian motion and accounts for the small variations. The other factor is an OU process driven by a pure jump Lévy process and models the characteristic spikes observed in such markets. When it comes to pricing, a popular choice of pricing measure is given by the Esscher transform that preserves the probabilistic structure of the driving Lévy processes, while changing the levels of mean reversion. Using this choice one can generate stochastic risk premiums (in geometric spot models) but with (deterministically) changing sign. In this paper we introduce a pricing change of measure, which is an extension of the Esscher transform. With this new change of measure we also can slow down the speed of mean reversion and generate stochastic risk premiums with stochastic non constant sign, even in arithmetic spot models. In particular, we can generate risk profiles with positive values in the short end of the forward curve and negative values in the long end. Finally, our pricing measure allows us to have a stationary spot dynamics while still having randomly fluctuating forward prices for contracts far from maturity.

q-fin.PR↗

arXiv2013

Modelling energy spot prices by volatility modulated Lévy-driven Volterra processes

This paper introduces the class of volatility modulated Lévy-driven Volterra (VMLV) processes and their important subclass of Lévy 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.

q-fin.PR↗

arXiv2013

On stochastic integration for volatility modulated Brownian-driven Volterra processes via white noise analysis

This paper generalizes the integration theory for volatility modulated Brownian-driven Volterra processes onto the space G* of Potthoff-Timpel distributions. Sufficient conditions for integrability of generalized processes are given, regularity results and properties of the integral are discussed. We introduce a new volatility modulation method through the Wick product and discuss its relation to the pointwise-multiplied volatility model.

math.PR↗

arXiv2012

Recent advances in ambit stochastics with a view towards tempo-spatial stochastic volatility/intermittency

Ambit stochastics is the name for the theory and applications of ambit fields and ambit processes and constitutes a new research area in stochastics for tempo-spatial phenomena. This paper gives an overview of the main findings in ambit stochastics up to date and establishes new results on general properties of ambit fields. Moreover, it develops the concept of tempo-spatial stochastic volatility/intermittency within ambit fields. Various types of volatility modulation ranging from stochastic scaling of the amplitude, to stochastic time change and extended subordination of random measures and to probability and Lévy mixing of volatility/intensity parameters will be developed. Important examples for concrete model specifications within the class of ambit fields are given.

math.PR↗

arXiv2012

On stochastic integration for volatility modulated Lévy-driven Volterra processes

This papers develops a stochastic integration theory with respect to volatility modulated Lévy-driven Volterra (VMLV) processes. It extends recent results in the literature to allow for stochastic volatility and pure jump processes in the integrator. The new integration operator is based on Malliavin calculus and describes an anticipative integral. Fundamental properties of the integral are derived and important applications are given.

math.PR↗

arXiv2012

Optimal portfolios in commodity futures markets

We consider portfolio optimization in futures markets. We model the entire futures price curve at once as a solution of a stochastic partial differential equation. The agents objective is to maximize her utility from the final wealth when investing in futures contracts. We study a class of futures price curve models which admit a finite-dimensional realization. Using this, we recast the portfolio optimization problem as a finite-dimensional control problem and study its solvability.

q-fin.PM↗