The Continuous-time Trading Model

Abstract

This chapter extends the model described in Chapter 3 to the continuous-time trading case. It aims at deriving valuations of contingent claims in a model, where agents may rearrange their portfolios at every time t ∈ [O,T]. For that purpose continuous-time selffinancing trading strategies are introduced in Section 4.1. Continuous-time selffinancing trading strategies allow for continuously rearranging the basic securities without requiring nor generating funds between the initial date zero and the final date T. A continuous-time selffinancing trading strategy is defined as the limit of a sequence of simple selffinancing trading strategies with respect to a certain norm introduced on the space of all continuous-time trading strategies. This norm takes into account both gains or losses due to price changes and funds to be invested (which might of course be negative thus representing withdrawals) that are associated with a trading strategy. Continuous-time selffinancing trading strategies are characterized and the properties of the space of all continuous-time selffinancing trading strategies are determined. Furthermore it is examined whether contingent claims that are generated by continuous-time selffinancing trading strategies can be approximated by simply attainable contingent claims. Section 4.2 characterizes contingent claims that are generated by continuous-time selffinancing trading strategies. The characterization rests essentially on the concept of a stable subspace. The stable subspace considered is the space of all stochastic integrals with respect to the given security price processes. Under suitable regularity conditions, it is shown that a contingent claim, which is generated by a continuous-time selffinancing trading strategy, can be valued independently of preferences by the initial investment associated with the selffinancing generating trading strategy. Section 4.3 considers specific security price processes. For those security price processes the classes of contingent claims that can be generated by continuous-time selffinancing trading strategies are determined and valuation formulas are obtained. Section 4.4 bridges the gap between the valuation method proposed in this monograph and the valuation method to be found in the literature based on difference-differential equations. Complete models are dealt with in Section 4.5. A model is called complete, if every contingent claim can be generated by a selffinancing trading strategy. Section 4.6 shows by means of counterexamples which assumptions are indispensable for the results obtained in Chapter 4.