Trading Strategies

The Unified Model describes prices and cash flows.

A trading strategy is an increasing sequence of stopping times τ_j and functions Γ_j\colon{\mathcal{A}}_{τ_j}\to\boldsymbol{R}^I indicating how much of each instrument to purchase at τ_j.

Consider the strategy of purchasing a stock when it goes below some level L then selling the stock when it goes above some level H > L. Let (S < L) = \{(t,ω)\in [0,\infty)\times Ω| S_t(ω) < L\} and (S < H) = \{(t,ω)\in [0,\infty)\times Ω| S_t(ω)\} where S_t is the stock price at time t and ω is the outcome determining the stock trajectory. The first trading time is τ_0(ω) = \inf\{t|S_t(ω) < L\} when we buy one share. The second trading time is τ_1(ω) = \inf\{t > τ_0(ω)|S_t(ω) > H\} when we sell one share.

If we have an increasing sequence of numbers t_0 < t_1 < t_2 < \cdots let I = \{t_j\}. We can recover the sequence by defining *I = \min\{t\in I\} and +I = I>*I = \{t\in I\mid t > *I\}. Clearly t_0 = *I, t_1 = *(+I), t_2 = *(+(+I)), etc.

This can be generalized from a sequence of numbers to subsets of a totally ordered set. If I is a subset of a totally ordered set define *I = \inf\{t\in I\} and +I = I > *I. If I has no accumulation points this determines a sequence t_j = *(+^jI), j\ge0.

Recall the disjoint union of I_0 and I_1 is (\{0\}\times I_0)\cup(\{1\}\times I_1). It has elements of the form (j,i) where j\in\{0,1\} and i\in I_0 if j = 0 and i\in I_1 if j = 1. If i\in I_0\cup I_1 then we don’t know if i belongs to I_0 or to I_1, or both. The disjoint union preserves this information.

Let I = (j, I_0\sqcup I_1), j\in\{0,1\}. Define *I = (j, *I_j) and +(j, I_0\sqcup I_1) = (j + 1\mod 2, (I_0 > *I_j)\sqcup (I_1 > *I_j)).

Getting back to our trading example, we can define {(S < L), (S > H) = (0, (S < L)\sqcup(S > H))}. The first trade is determined by (*(S < L), (S > H))(ω) = (0, τ_0(ω)). The second trade is determined by (*(+(S < L), (S > H)))(ω) = (1, τ_1(ω)).