Complete markets with discontinuous security price
A classical model for continuous time asset pricing models is a geometric Brownian motion, namely, the standard model. An important feature of the standard model is that markets are ``complete'': That is, there will always exist a self-financing trading strategy that replicates the contingent claim. However as has been long known, empirical studies show that stock prices often have jumps (that is, are not continuous). Various models incorporating discontinuities have been proposed. The most common is the ``jump diffusion'' model, where Poisson jumps have been added to Brownian (or ``white'') noise, modeled by a Wiener process. These models do have equivalent martingale measures, and they do have market completeness. In this paper authors are interested in models that have strict market completeness and jumps that are intrinsic to the stock price with a unique equivalent martingale measure, and no arbitrage. For this framework the authors produce a family of distinct semimartingales, indexed by a parameter \(\beta\) (\(-2\leq\beta\leq 0\)), that gives rise to strict market completeness. For each value of the parameter \(\beta\) the model is just as rich as the standard model using white noise and a drift; as \(\beta\) increases to zero the model converges weakly to the standard model. A hedging result analogous to the one of \textit{I. Karatzas, D. Okone} and \textit{Jinju Li} [Stochastics Stochastics Rep. 37, No. 3, 127-131 (1991; Zbl 0745.60056)] is presented.
scientific article; zbMATH DE number 1336683