We study the forward price dynamics in commodity markets realised as a process with values in a Hilbert space of absolutely continuous functions defined by Filipović (Consistency problems for Heath-Jarrow-Morton interest rate models, 2001). 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 OrnsteinUhlenbeck 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.

ABSTRACT. This paper aims at transferring the philosophy behind Heath-Jarrow-Morton to the modelling of call options with all strikes and maturities. Contrary to the approach by Carmona and Nadtochiy [8] and related to the recent contribution [10] by the same authors, the key parametrisation of our approach involves time-inhomogeneous Lévy processes instead of local volatility models. We provide necessary and sufficient conditions for absence of arbitrage. Moreover we discuss the construction of arbitrage-free models. Specifically, we prove their existence and uniqueness given basic building blocks.

ABSTRACT. 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, crosscommodity 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.

The increase in longevity, the ultra-low interest rates and the guarantees associated to pension benefits have put significant strain on the pension industry. Consequently, insurers need to be in a financially sound position while offering satisfactory benefits to participants. In this paper, we propose a pension design that goes beyond the idea of annuity pools and unit-linked insurance products. The purpose is to replace traditional guarantees with low volatility, mainly achieved by collective smoothing algorithms and an adequate asset management. With the aim of offering security to the insured, we discuss the optimisation of some key variables of the proposed pension product to target both a satisfactory level of the initial pension and stable pension payments over time. By combining such well-known products as unit-linked and annuities, we show that it is possible to design a pension product with both high-expected return and low risk for the policyholder. However, differently than in the classical unit-linked framework, we do not allow the individuals to choose the underlying funds. Instead, the funds are under the surveillance of an insurance company's professional risk management, which induces better informed decisions.

We consider an insurance entity endowed with an initial capital and a surplus process modelled as a Brownian motion with drift. It is assumed that the company seeks to maximise the cumulated value of expected discounted dividends, which are declared or paid in a foreign currency. The currency fluctuation is modelled as a Lévy process. We consider both cases: restricted and unrestricted dividend payments. It turns out that the value function and the optimal strategy can be calculated explicitly.