Advisor: John Lowengrub

Advisor:  Dr. Martin Zeman

Advisor: Professor Knut Solna

Abstract: We introduce some popular models in the energy markets. Then we
propose to incorporate a stochastic volatility feature to an existing
multi-factor deterministic volatility model in order to take into account
the observed implied volatility skews for each of the commodities in the
simulation of monthly forward prices. As examples we consider natural gas,
crude oil and heating oil price and option data. Our objective is to
explore the role of stochastic volatility modeling for calibration and
simulation of price paths and scenario analysis. The linkage between
price, option data and modeling is captured by the so called "Vs" in our
approach. These are the effective group market parameters that capture the
main impact of an uncertain and fluctuating volatility, in particular how
these affect prices. To explore the significance of incorporating this
link we carry out an initial calibration test to explore the role of the
"Vs" in the commodity price distribution. We find that indeed the
distribution of the commodity prices are significantly affected by
incorporating the leading correction that accounts for the effect of
uncertain volatility parameters which manifests itself in the data via
strong "skew" effect in the option pricing data. An added benefit of this
modeling framework is that it enables us to use observations around and
not only at the money in a consistent way, thus, providing robustness and
stability in calibration also at the order one level.

Advisor:  Professor Hongkai Zhao

Advisor: Professor Chuu-Lian Terng

In this thesis defense, I will explain

(1) properties of a classical soliton equation----the KdV equation,

(2) the symmetry and Hamiltonian properties of the Matrix modified Constrained KP hierarchy,

(3) an integrable curve flow on the affine n-space.