A note on arbitrage‐free pricing of forward contracts in energy markets

Arbitrage theory is used to price forward (futures) contracts in energy markets, where the underlying assets are non‐tradeable. The method is based on the so‐called ‘fitting of the yield curve’ technique from interest rate theory. The spot price dynamics of Schwartz is generalized to multidimensional correlated stochastic processes with Wiener and Lévy noise. Findings are illustrated with examples from oil and electricity markets.     

Portfolio optimization in a Lévy market with intertemporal substitution and transaction costs

We investigate an infinite horizon investment-consumption model in which a single agent consumes and distributes her wealth between a risk-free asset (bank account) and several risky assets (stocks) whose prices are governed by Lévy (jump-diffusion) processes. We suppose that transactions between the assets incur a transaction cost proportional to the size of the transaction. The problem is to maximize the total utility of consumption under Hindy-Huang-Kreps intertemporal preferences. This portfolio optimisation problem is formulated as a singular stochastic control problem and is solved using dynamic programming and the theory of viscosity solutions. The associated dynamic programming equation is a second order degenerate elliptic integro-differential variational inequality subject to a state constraint boundary condition. The main result is a characterization of the value function as the unique constrained viscosity solution of the dynamic programming equation. Emphasis is put on providing a framework that allows for a general class of Lévy processes. Owing to the complexity of our investment-consumption model, it is not possible to derive closed form solutions for the value function. Hence, the optimal policies cannot be obtained in closed form from the first order conditions for the dynamic programming equation. Therefore, we have to resort to numerical methods for computing the value function as well as the associated optimal policies. In view of the viscosity solution theory, the analysis found in this paper will ensure the convergence of a large class of numerical methods for the investment-consumption model in question.     

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.     

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 modeling of forward curves in energy and commodity 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.     

The Minimal Entropy Martingale Measure and Numerical Option Pricing for the Barndorff–Nielsen–Shephard Stochastic Volatility Model

Abstract

We develop and apply a numerical scheme for pricing options in the stochastic volatility model proposed by Barndorff–Nielsen and Shephard. This non-Gaussian Ornstein–Uhlenbeck type of volatility model gives rise to an incomplete market, and we consider the option prices under the minimal entropy martingale measure. To numerically price options with respect to this risk neutral measure, one needs to consider a Black and Scholes type of partial differential equation, with an integro-term arising from the volatility process. We suggest finite difference schemes to solve this parabolic integro-partial differential equation, and derive appropriate boundary conditions for the finite difference method. As an application of our algorithm, we consider price deviations from the Black and Scholes formula for call options, and the implications of the stochastic volatility on the shape of the volatility smile.     

Pricing of Spread Options on a Bivariate Jump Market and Stability to Model Risk

Abstract

We study the pricing of spread options and we obtain a Margrabe-type formula for a bivariate jump-diffusion model. Moreover, we study the robustness of the price to model risk, in the sense that we consider two types of bivariate jump-diffusion models: one allowing for infinite activity small jumps and one not. In the second model, an adequate continuous component describes the small variation of prices. We illustrate our computations by several examples.     

Hedging of Spatial Temperature Risk with Market-Traded Futures

Abstract

The main objective of this work is to construct optimal temperature futures from available market-traded contracts to hedge spatial risk. Temperature dynamics are modelled by a stochastic differential equation with spatial dependence. Optimal positions in market-traded futures minimizing the variance are calculated. Examples with numerical simulations based on a fast algorithm for the generation of random fields are presented.     

Nonlinear Evolution Equations with Gradient Coupled Noise

We show existence and uniqueness of solutions for a class of nonlinear evolution equations with gradient coupled noise. Our results are obtained by using a simple transformation relating the equation under consideration to an underlying deterministic partial differential equation. Both the Itô and the Stratonovich conventions are treated. Several examples show that the properties of solutions for Itô equations can differ significantly from those of Stratonovich equations.     

Anticipative calculus for Lévy processes and stochastic differential equations*

We develop an anticipative calculus for Lévy processes with finite second moment for analysing anticipating stochastic differential equations. The calculus is based on the chaos expansion of square-integrable random variables in terms of iterated integrals with respect to the compensated Poisson random measure. We define a space of smooth and generalized random variables in terms of such chaos expansions, and present anticipative stochastic integration, the Wick product and the so-called 𝒮-transform. These concepts serve as tools for studying general Wick type stochastic differential equations with anticipative initial conditions. We apply the 𝒮-transform to find the unique solutions to a class of linear stochastic differential equations. The solutions can be expressed in terms of the Wick product.