UC Santa Barbara

In this thesis, we discuss three new topics in Financial Mathematics using partial differential equation (PDE): uncertain volatility with stochastic bounds, Ross recovery with multivariate driving states and mean field games on the Erdos Renyi random graph, in three chapters respectively.

In Chapter 1, we study a class of uncertain volatility models with stochastic bounds, over which volatility stays between two bounds, but instead of using two deterministic bounds, the uncertain volatility fluctuates between two stochastic bounds generated by its inherent stochastic volatility process. We then apply a regular perturbation analysis upon the worst-case scenario price, and derive the first order approximation in the regime of slowly varying stochastic bounds. The original problem which involves solving a fully nonlinear PDE in dimension two for the worst-case scenario price, is reduced to solving a nonlinear PDE in dimension one and a linear PDE with source, which gives a tremendous computational advantage.

In Chapter 2, we address the problem of recovering the real world probability distribution from observed option prices by avoiding the intensively debated transition independence, through placing structure on the dynamics of the numeraire portfolio in a preference-free manner. We firstly utilize the Ito and Feynman-Kac theorem to derive a a uniformly elliptic operator, whose inverse is a compact linear operator, based on boundary conditions, and then apply the Krein-Rutman theorem to guarantee the uniqueness of the positive eigenfunction, which happens to generate the physical transition probability.

In Chapter 3, we analyze a model of inter-bank lending and borrowing, by means of mean field games on the Erdos Renyi random graph. An open-loop Nash equilibrium is obtained using a system of fully coupled forward backward stochastic differential equations (FBSDEs), whose unique solution leads to the master equation. We explore the approximation to the finite player game equilibrium through a decoupled system of diffusion equations generated by the master equation under frozen graph, and through a weakly interacting particle system on random graph generated by the master equation under random graph, respectively.