I am interested in Mathematical Finance. Specific areas that I have worked in are Portfolio Optimization, Equilibrium, Optimal Stopping and Sequential Analysis.
We investigate the impact of imbalanced derivative markets - markets in which not all agents hedge - on the underlying stock market. The availability of a closed-form representation for the equilibrium stock prices in the context of a complete (imbalanced) market with terminal consumption allows us to study how this equilibrium outcome is affected by the risk aversion of agents and the degree of imbalance. In particular, it is shown that the derivative imbalance leads to significant changes in the equilibrium stock price processes: volatility changes from constant to local, while risk premia increase or decrease depending on the replicated contingent claim, and become stochastic processes. Moreover, the model produces implied volatility skew consistent with empirical observations.
We study the quickest change-point detection problems for the correlated components of a multidimensional Wiener process changing their drift rates at certain random times. These problems seek to determine the times of alarm which are as close as possible to the unknown change-point (disorder) times at which some of the components have changed their drift rates. The optimal times of alarm are shown to be the first times at which the appropriate posterior probability processes exit certain regions restricted by the stopping boundaries. We characterise the value functions and optimal boundaries as unique solutions of the associated free boundary problems for partial differential equations. We provide estimates for the value functions and boundaries which are solutions to the appropriately constructed ordinary differential free boundary problems.
We compute the generalised Laplace transforms of the first times at which two- dimensional jump-diffusion processes exit from certain regions formed by constant boundaries. The method of proof is based on the solutions of the associated integro-differential boundary value problems for the corresponding value functions. We consider two- dimensional processes driven by constantly correlated drifted Brownian motions and compound Poisson processes with exponential jumps and common jump components. The results are illustrated on some two-dimensional jump-diffusion models including the non- affine pure jump analogues of Stein and Stein and Heston models of stochastic volatility.
We study the sequential hypothesis testing and quickest change-point (disorder) detection problems with linear delay penalty costs for certain observable time-inhomogeneous Gaussian diffusions and fractional Brownian motions. The method of proof consists of the reduction of the initial problems into the associated optimal stopping problems for one- dimensional time-inhomogeneous diffusion processes and the analysis of the associated free boundary problems for partial differential operators. We derive explicit estimates for the Bayesian risk functions and optimal stopping boundaries for the associated weighted likelihood ratios and obtain their exact asymptotic growth rates under large time values.
We describe a method for construction of jump analogues of certain one-dimensional diffusion processes. The method is based on the reduction of the stochastic differential equations satisfied by the processes to the ones with linear diffusion coefficients, which are reducible to the associated ordinary differential equations, by using the appropriate integrating factor processes. The analogues are constructed by means of adding the jump components linearly into the reduced stochastic differential equations. We illustrate the method on the construction of jump analogues of several diffusion processes and expand the notion of market price of risk to the resulting non-affine jump-diffusion models.
We compute the Laplace transforms of the first exit times for certain one-dimensional jump-diffusion processes from connected regions formed by constant boundaries. The method of proof is based on the solutions of the associated integro-differential boundary value problems for the corresponding value functions. We consider jump-diffusion processes solving stochastic differential equations driven by Brownian motions and compound Poisson processes with multi-exponential jumps. The results are illustrated on the non-affine pure jump analogues of certain mean-reverting or diverting diffusion processes which represent closed-form solutions of the appropriate stochastic differential equations.