Continuous-time Security Markets

Abstract

This chapter presents basic results associated with continuous-time financial modelling. The first section deals with a continuous-time model, which is based on the notion of the Itô stochastic integral with respect to a semi-martingale. Such a model of financial market, in which the arbitrage-free property hinges on the chosen class of admissible trading strategies, is termed the standard market model hereafter. The relevance of a judicious choice of a numeraire process is also discussed. On a more theoretical side, we briefly comment on the class of results — informally referred to as a fundamental theorem of asset pricing — which say, roughly, that the absence of arbitrage opportunities is equivalent to the existence of a martingale measure. Let us emphasize that the theory developed in this chapter applies both to stock markets and bond markets. Therefore, it can also be seen as a theoretical background to the second part of this text. For simplicity, we restrict ourselves as usual to the case of processes with continuous sample paths. Putting aside a somewhat higher level of technical complexity, models of discontinuous prices can be dealt with along the same lines. Let us observe that in a typical jump-diffusion model, price discontinuities are introduced through a Poisson component (in this regard, we refer to Cox and Ross (1975), Merton (1976), Ahn and Thompson (1988), Aase (1988), Madan et al. (1989), Shirakawa (1991), Ahn (1992), Dengler (1993), Cutland et al. (1993a), Mercurio and Runggaldier (1993), Björk (1995), Lando (1995) or Mulinacci (1996)).