Fred Espen Benth

A Non‐Gaussian Ornstein–Uhlenbeck Process for Electricity Spot Price Modeling and Derivatives Pricing

A mean‐reverting model is proposed for the spot price dynamics of electricity which includes seasonality of the prices and spikes. The dynamics is a sum of non‐Gaussian Ornstein–Uhlenbeck processes with jump processes giving the normal variations and spike behaviour of the prices. The amplitude and frequency of jumps may be seasonally dependent. The proposed dynamics ensures that spot prices are positive, and that the dynamics is simple enough to allow for analytical pricing of electricity forward and futures contracts. Electricity forward and futures contracts have the distinctive feature of delivery over a period rather than at a fixed point in time, which leads to quite complicated expressions when using the more traditional multiplicative models for spot price dynamics. In a simulation example it is demonstrated that the model seems to be sufficiently flexible to capture the observed dynamics of electricity spot prices. The pricing of European call and put options written on electricity forward contracts is also discussed.

The volatility of temperature and pricing of weather derivatives

We propose an Ornstein–Uhlenbeck process with seasonal volatility to model the time dynamics of daily average temperatures. The model is fitted to approximately 45 years of daily observations recorded in Stockholm, one of the European cities for which there is a trade in weather futures and options on the Chicago Mercantile Exchange. Explicit pricing dynamics for futures contracts written on the number of heating/cooling degree-days (so-called HDD/CDD futures) and the cumulative average daily temperature (so-called CAT futures) are calculated, along with a discussion on how to evaluate call and put options with these futures as underlying.

Stochastic modelling of temperature variations with a view towards weather derivatives

Daily average temperature variations are modelled with a mean‐reverting Ornstein–Uhlenbeck process driven by a generalized hyperbolic Lévy process and having seasonal mean and volatility. It is empirically demonstrated that the proposed dynamics fits Norwegian temperature data quite successfully, and in particular explains the seasonality, heavy tails and skewness observed in the data. The stability of mean‐reversion and the question of fractionality of the temperature data are discussed. The model is applied to derive explicit prices for some standardized futures contracts based on temperature indices and options on these traded on the Chicago Mercantile Exchange (CME).

Optimal portfolio selection with consumption and nonlinear integro-differential equations with gradient constraint: A viscosity solution approach

Abstract. We study a problem of optimal consumption and portfolio selection in a market where the logreturns of the uncertain assets are not necessarily normally distributed. The natural models then involve pure-jump Lévy processes as driving noise instead of Brownian motion like in the Black and Scholes model. The state constrained optimization problem involves the notion of local substitution and is of singular type. The associated Hamilton-Jacobi-Bellman equation is a nonlinear first order integro-differential equation subject to gradient and state constraints. We characterize the value function of the singular stochastic control problem as the unique constrained viscosity solution of the associated Hamilton-Jacobi-Bellman equation. This characterization is obtained in two main steps. First, we prove that the value function is a constrained viscosity solution of an integro-differential variational inequality. Second, to ensure that the characterization of the value function is unique, we prove a new comparison (uniqueness) result for the state constraint problem for a class of integro-differential variational inequalities. In the case of HARA utility, it is possible to determine an explicit solution of our portfolio-consumption problem when the Lévy process posseses only negative jumps. This is, however, the topic of a companion paper [7].

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.

Extracting and Applying Smooth Forward Curves From Average-Based Commodity Contracts with Seasonal Variation

Several important new classes of derivative instruments, notably those related to weather and to electricity, have payoffs based on the average value of the underlying over some period of time. For electricity in the Nord Pool market, for example, actively traded contracts exist for a variety of partly overlapping averaging periods out several years into the future. Current market prices for such contracts inherently depend on the markets projection of the future spot price of electricity over the averaging periods, i.e., instantaneous forward prices for the relevant time periods. But since these forward prices are not observable directly, and they can be expected to exhibit the seasonality embedded in the spot market, extracting a smooth implied instantaneous forward curve from market data is a challenging task. This article offers a procedure for doing just that, by modeling the forwards to include both a seasonal component and a noise component, and requiring the noise to have maximum smoothness over the term structure of forward prices.

On arbitrage-free pricing of weather derivatives based on fractional Brownian motion

We derive an arbitrage‐free pricing dynamics for claims on temperature, where the temperature follows a fractional Ornstein–Uhlenbeck process. Using a fractional white noise calculus, one can express the dynamics as a special type of conditional expectation not coinciding with the classical one. Using a Fourier transformation technique, explicit expressions are derived for claims of European and average type, and it is shown that these pricing formulas are solutions of certain Black and Scholes partial differential equations. Our results partly confirm a conjecture made by Brody, Syroka and Zervos.

OPTIMAL PORTFOLIO MANAGEMENT RULES IN A NON-GAUSSIAN MARKET WITH DURABILITY AND INTERTEMPORAL SUBSTITUTION

Abstract. We consider an optimal portfolio-consumption problem which incorporates the notions of durability and intertemporal substitution. The logreturns of the uncertain assets are not necessarily normally distributed. The natural models then involve Lévy processes as driving noise instead of the more frequently used Brownian motion. The optimization problem is a singular stochastic control problem and the associated Hamilton-Jacobi-Bellman equation is a nonlinear second order degenerate elliptic integro-differential equation subject to gradient and state constraints. For utility functions of HARA type, we calculate the optimal investment and consumption policies together with an explicit expression for the value function when the Lévy process has only negative jumps. For the classical Merton problem, which is a special case of our optimization problem, we provide explicit policies for general Lévy processes having both positive and negative jumps. Instead of following the classical approach of using a verification theorem, we validate our solution candidates within a viscosity solution framework. To this end, the value function of our singular control problem is characterized as the unique constrained viscosity solution of the Hamilton-Jacobi-Bellman equation in the case of general utilities and general Lévy processes.

The information premium for non-storable commodities

For non-storable commodities forward looking information about market conditions is not necessarily incorporated in today’s prices, and the standard assumption that the information filtration is generated by the asset is fundamentally wrong. Electricity and weather are the typical markets we have in mind. We discuss pricing of forward contracts on non-storable commodities based on an enlargement of the information filtration. The method is able to incorporate future information of the spot, which is not accounted for in the present spot price behaviour. The notions of information drift and premium are introduced, and we argue that significant parts of the supposedly irregular market price of risk observed in electricity markets is in reality due to information miss-specification in the model. Some examples based on Brownian motion and Lévy processes and the theory of initial enlargement of filtrations are considered, where we are able to shed some insight into the nature of the information drift and premium being relevant for the electricity markets. The examples include cases where we take temperature forecasts and CO2 emission costs into account when pricing electricity forwards.

A remark on the Equivalence between Poisson and Gaussian Stochastic Partial Differential Equations

We discuss the connection between Gaussian and Poisson noise Wick-type stochastic partial differential equations.