Continuous-time Security Markets

This chapter furnishes a summary of basic results associated with continuous-time financial modelling. The first section deals with a continuous-time model, which is based on the Itô stochastic integral with respect to a semimartingale. 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. We discuss the relevance of a judicious choice of a numeraire asset. 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. The theory developed in this chapter applies both to stock markets and bond markets. It can thus be seen as a theoretical background to the second part of this text.