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

Pricing of spread options in energy markets

The objective of this thesis is to value spread options with payoff function on the form max(S_1(T)-hS_2(T)-K,0), where S_1(T) and S_2(T) are the spot prices of two energy commodities at maturity time T, h is the heatrate and K is the strike price. We model (X_1(t),X_2(t))=(log(S_1(t)),log(S_2(t)) as a bivariate Ornstein-Uhlenbeck Lévy process. First, we consider an Ornstein-Uhlenbeck process driven by a bivariate Brownian motion, then we extend the model to an Ornstein-Uhlenbeck process driven by a bivariate Lévy process with jumps. We compute the characteristic function of (X_1(t),X_2(t)) in both models, and study the stationary properties of the distribution of (X_1(t),X_2(t)). Then we we derive a closed form formula for the option price in the continuous model for the case K=0. In the model with jumps, we use a Fourier transform method to express the price as an integral of the characteristic function of (X_1(T),X_2(T)) times the Fourier transform of the payoff function. When K!=0, we use a first order Taylor-expansion to approximate the option price. We find a closed form formula for the approximated price in the continuous model, and use simulations to check how good the approximation is for different values of K and for different values of the correlation between the two underlying price processes

Metatimes, random measures and cylindrical random variables

Metatimes constitute an extension of time-change to general measurable spaces, defined as mappings between two σ-algebras. Equipping the image σ-algebra of a metatime with a measure and defining the composition measure given by the metatime on the domain σ-algebra, we identify metatimes with bounded linear operators between spaces of square integrable functions. We also analyse the possibility to define a metatime from a given bounded linear operator between Hilbert spaces, which we show is possible for invertible operators. Next we establish a link between orthogonal random measures and cylindrical random variables following a classical construction. This enables us to view metatime-changed orthogonal random measures as cylindrical random variables composed with linear operators, where the linear operators are induced by metatimes. In the paper we also provide several results on the basic properties of metatimes as well as some applications towards trawl processes