My research interests lie in the area of Financial Mathematics. More precisely, I work on the problems arising in Derivatives Pricing and Hedging, Optimal Investment and Systemic Risk modeling, applying the methods from Probability Theory, Partial Differential Equations and Functional Analysis.
One of my main subjects of interest is the construction of the so-called "Market-Based Models" for derivatives prices. The idea of this approach is to model the future dynamics of derivatives (e.g. options) prices directly, as the prices of primary assets (e.g. stocks). It allows to make a model consistent with the currently observed derivatives prices, while making full use of the information contained in their historical values. On the mathematical side, it amounts to developing a tractable representation of a family of martingales with given (nonlinear) functional dependencies at expiry times.
Static Hedging is another project I am currently working on. The problem is to find a relation between the complicated (exotic) financial instruments and the more simple (liquid) ones. One can, then, exploit this relation to offset the risks associated with the exotic product by trading in the liquid ones. In particular, I have been developing the static hedging strategies (involving, at most, two trades) for barrier options with the European-type ones. This leads to rather interesting inverse problems which can be solved by means of Probability Theory and Functional Analysis.
Optimal Investment is concerned with the optimal allocation of capital among available financial assets. In this context, the criterion of optimality is of high importance. The classical criterion of optimality is given by utility function. Proving existence and uniqueness of the solution to the re- sulting optimization problem, as well as analyzing its properties and developing the approximation schemes, still generates a lot of challenging mathematical problems in the areas of Probability Theory and Partial Differential Equations. In addition, there are many practical reasons to consider alternative criterions of optimality, which yield brand new optimization problems. One of such alternative criterions is introduced by the "Forward Performance" theory, and, in the simplest cases, it leads to partial differential equations which cannot be analyzed by standard methods but can be solved by means of Potential Theory.