Essays in mathematical finance modeling the futures price

Abstract: This thesis consists of four papers dealing with the futures price process.In the first paper, we propose a two-factor futures volatility model designed for the US natural gas market, but applicable to any futures market where volatility decreases with maturity and varies with the seasons. A closed form analytical expression for European call options is derived within the model and used to calibrate the model to implied market volatilities. The result is used to price swaptions and calendar spread options on the futures curve.In the second paper, a financial market is specified where the underlying asset is driven by a d-dimensional Wiener process and an M dimensional Markov process. On this market, we provide necessary and, in the time homogenous case, sufficient conditions for the futures price to possess a semi-affine term structure. Next, the case when the Markov process is unobservable is considered. We show that the pricing problem in this setting can be viewed as a filtering problem, and we present explicit solutions for futures. Finally, we present explicit solutions for options on futures both in the observable and unobservable case.The third paper is an empirical study of the SABR model, one of the latest contributions to the field of stochastic volatility models. By Monte Carlo simulation we test the accuracy of the approximation the model relies on, and we investigate the stability of the parameters involved. Further, the model is calibrated to market implied volatility, and its dynamic performance is tested.In the fourth paper, co-authored with Tomas Björk and Camilla Landén, we consider HJM type models for the term structure of futures prices, where the volatility is allowed to be an arbitrary smooth functional of the present futures price curve. Using a Lie algebraic approach we investigate when the infinite dimensional futures price process can be realized by a finite dimensional Markovian state space model, and we give general necessary and sufficient conditions, in terms of the volatility structure, for the existence of a finite dimensional realization. We study a number of concrete applications including the model developed in the first paper of this thesis. In particular, we provide necessary and sufficient conditions for when the induced spot price is a Markov process. We prove that the only HJM type futures price models with spot price dependent volatility structures, generically possessing a spot price realization, are the affine ones. These models are thus the only generic spot price models from a futures price term structure point of view.