s(0,k,0) =£16

We are now in a position to introduce the concept of trading strategy. A trading strategy is a time-dependent (K + l)-dimensional vector ©, of holdings of the various securities on the different trading dates. As an example, a possible strategy at time 0 is shown in Table B.l . Here, at time 0, 2 units (long position) are held of security 0, 0 units of security 1, —20 units (short position) of security 2,..., 12 units of security K. The vector © is therefore made up of 0 = {<9(0), (9(1), 6(2),6(K)}. The initial value of the resulting portfolio is given by the scalar product ©S, i.e. by (2*£2.4 + 0*£32 + ... + 12*£16) = £196.78.

We are going to be particularly interested in strategies where the changes in holdings over time obey important restrictions. Let us start with a vector of holdings which are known with certainty at time 0. These initial holdings are determined at this initial time on the basis of the information set So, i.e. on the basis of the knowledge of the price vector S(0) and of the knowledge of what securities prices will be possible at all later times (see Section A.3). Let us then stipulate that we are not going to adjust these holdings until the securities prices at time 1 are revealed. In order to emphasise this fact, we shall denote the vector of holdings from time 0 to time 1 as ©(1). After the time-1 prices are known, i.e. after the filtration has been augmented from So to fo, and part of the initial uncertainty has been removed, we shall then modify the holdings in the light of the new price vector (or, more generally, of the new information set Jj). These new holdings, that will be left unchanged until the information set is further augmented to Si by revelation of the time-2 price vector, are denoted by ©(2). This procedure will then be repeated over the subsequent time steps all the way up to the penultimate (7—1) time step, where the allocation ©(/) will be determined. By the way it has been constructed, the vector of holdings is therefore not only a random vector process, but, in addition, it is previsible (predictable), i.e. the trading strategies that conform to this requirement are St-\-measurable (see Sections A.2 and A.3). Intuitively, the previsibility condition means that only trading strategies will be considered where no 'peeking ahead' is allowed; this condition is essential if we want to create a process © that is not only mathematically well defined, but also realisable in practice.

Remembering that the scalar product of two vectors A and B, of respective elements {a,} and {6;}, is given by

A ■ B = ^^ a¡b¡, one can immediately see that the value Vt at time t of the trading strategy is given by the scalar product of the vector of holdings © and the vector S containing the prices of the securities:

0 0

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