Georgia Institute of Technology College of Sciences

Applicability of Pearson's correlation as a measure of explained variance is by now well understood. One of its limitations is that it does not account for asymmetry in explained variance. Aiming to obtain broad applicable correlation measures, we use a pair of r-squares of generalized regression to deal with asymmetries in explained variances, and linear or nonlinear relations between random variables. We call the pair of r-squares of generalized regression generalized measures of correlation (GMC). We present examples under which the paired measures are identical, and they become a symmetric correlation measure which is the same as the squared Pearson's correlation coefficient. As a result, Pearson's correlation is a special case of GMC. Theoretical properties of GMC show that GMC can be applicable in numerous applications and can lead to more meaningful conclusions and decision making. In statistical inferences, the joint asymptotics of the kernel based estimators for GMC are derived and are used to test whether or not two random variables are symmetric in explaining variances. The testing results give important guidance in practical model selection problems. In real data analysis, this talk presents ideas of using GMCs as an indicator of suitability of asset pricing models, and hence new pricing models may be motivated from this indicator.

We prove stochastic representation formulae for solutions to elliptic boundary value and obstacle problems associated with a degenerate Markov diffusion process on the half-plane. The degeneracy in the diffusion coefficient is proportional to the \alpha-power of the distance to the boundary of the half-plane, where 0 < \alpha < 1 . This generalizes the well-known Heston stochastic volatility process, which is widely used as an asset price model in mathematical finance and a paradigm for a degenerate diffusion process. The generator of this degenerate diffusion process with killing, is a second-order, degenerate-elliptic partial differential operator where the degeneracy in the operator symbol is proportional to the 2\alpha-power of the distance to the boundary of the half-plane. Our stochastic representation formulae provides the unique solution to the degenerate partial differential equation with partial Dirichlet condition, when we seek solutions which are suitably smooth up to the boundary portion \Gamma_0 contained in the boundary of the half-plane. In the case when the full Dirichlet condition is given, our stochastic representation formulae provides the solutions which are not guaranteed to be any more than continuous up to the boundary portion \Gamma_0 .

The relation between robust utility maximization problems and quadratic backward stochastic differential equations will be explored in this talk. Motivated by the solution of the dual formulation of the robust hedging problem for semi-martingales, when the model adopted is a diffusion it is possible to describe more completely the solution using the dynamic programming intuition, as well as some results of BSDEs.

It is commonly accepted that certain financial data exhibit long-range dependence. A continuous time stochastic volatility model is considered in which the stock price is geometric Brownian motion with volatility described by a fractional Ornstein-Uhlenbeck process. Two discrete time models are also studied: a discretization of the continuous model via an Euler scheme and a discrete model in which the returns are a zero mean iid sequence where the volatility is a fractional ARIMA process. A particle filtering algorithm is implemented to estimate the empirical distribution of the unobserved volatility, which we then use in the construction of a multinomial recombining tree for option pricing. We also discuss appropriate parameter estimation techniques for each model. For the long-memory parameter, we compute an implied value by calibrating the model with real data. We compare the performance of the three models using simulated data and we price options on the S&P 500 index. This is joint work with Prof. Alexandra Chronopoulou, which appeared in Quantitative Finance, vol 12, 2012.