Speaker: Camelia Pop, Department of Mathematics, University of Pennsylvania
ABSTRACT
The fractional Laplacian operator is the infinitesimal generator of the symmetric $2s$-stable Markov process. Variations of such processes, such as normal and generalized tempered stable processes, are widely used in mathematical finance in models of asset prices with discontinuous paths. A problem of great interest, and which is still not well understood, is to establish that prices of American-style options on assets with discontinuous paths are solutions to evolutionary obstacle problems defined by suitable nonlocal operators, and to prove regularity of such solutions and of the free boundary, that is, the set of points where the value function of the American option coincides with its intrinsic value. We develop tools necessary to obtain such results when the infinitesimal generator of the underlying asset price process is the fractional Laplacian with drift with parameter $sin (1/2,1)$. We develop a new monotonicity formula, which is used together with perturbation methods in H"older spaces, to obtain the optimal regularity of solutions. We apply this result to study the regularity of the free boundary in a neighborhood of regular points, and to obtain the stochastic representation of the solutions. The latter result allows us to interpret solutions to stationary obstacle problems defined by the fractional Laplacian with drift as prices of perpetual American options when the underlying asset price process is driven by a symmetric stable process with drift. The talk will be based on joint work with Charles Epstein and Arshak Petrosyan.