Valuation and hedging in a market driven by discontinuous processes.

Summary This thesis studies markets whose risk assets are discontinuous processes. The interest in such modelizations is justified by numerous statistical studies, which show discontinuities in the observed price trajectories. The first two chapters are devoted to valuation problems in an incomplete market with a risk-free asset and a risk asset. We assume that the risk asset is a jump spread. In this incomplete market, we determine the range of viable prices (i.e., not creating arbitrage opportunities) of European contingent assets, and we study some properties of these prices. We then analyze the case of Asian and American finite-maturity options, as well as American perpetual put options. The third chapter examines the completion of such markets on the one hand, and on the other hand presents some examples of complete markets driven by a single jumping asset. The following chapters are devoted to the hedging problem. We consider an agent whose strategy is specified by a utility function and we study the maximization of the expected utility of his terminal wealth. When the market is incomplete and driven by a mixed diffusion, we establish the existence of an optimal strategy that we characterize. We prove that the use of utility functions brings a significant reduction in the range of viable prices. Finally, we consider the hedging failure in markets driven by mixed diffusion.