Equity Valuation and Portfolio Management

GENERAL MODELS

Let us generalize the idea of statistical arbitrage. Imagine that there are two price processes (realistic processes will be stochastic), A(t, x_t, z_z) and B(t, y_t, z_z). The arguments can be thought of as time (t), a unique factor to A and B (x and y, respectively), and a common factor (z). If you let w_A,t and w_B,t represent the number of shares invested in each, and these numbers need not be constant, then a statistical arbitrage opportunity exists if there is a zero cost position that never (i.e., for all t) becomes negative.

The initial position, at time zero, is given by w_A,0A(0,x₀,z₀) + w_B,0B(0,y₀,z₀). This must be equal to zero in order to be a zero cost position to establish.

At any time after the position has been established, the value of the position must be greater than or equal to zero, represented by w_A,tA(t,x_t,z_t) + w_B,tB(t,y_t,z_t).

The portfolio positions, w_A,t and w_B,t, evolve according to the relationship that the strategy is “self-funding,” or “self-financing.” This simply means that when a position is established at time t, the portfolio is adjusted at t + 1 only with the proceeds from having held the portfolio positions from time t. This can be represented by the equation

At the terminal time, T, the portfolio value must be greater than zero to yield a profit, . The reason we use w_A,T−1and w_B,T−1 in the terminal portfolio value is that the position ...