Nonlinear constraints and spin glasses

Finally, we outline the main idea of how portfolios and spin glasses can be related [3]. This shows that nonlinear constraints can lead to many solutions for the optimal portfolio. Consider a portfolio with futures, for which a margin is required for both sides. Limiting such margins requires an additional constraint

$$\gamma \sum_i \vert p_i\vert x_i .$$ (46)

Hence, we may define an optimal portfolio as the minimum of

$$\frac{1}{2} \sum_{ij}p_iC_{ij}p_j -\zeta \sum_i p_i m_i -\gamma\sum_i x_i\vert p_i\vert .$$ (47)

This yields,

$$0 = \sum_j C_{ij}p_j -\zeta m_i -\gamma x_i S_i ,$$ (48)

i.e.,

$$p_i = \zeta \sum_j C^{-1}_{ij} m_j + \gamma \sum_j C^{-1}_{ij} x_j S_j ,$$ (49)

where $S_i$ = sign$(p_i)$. Setting $x_j$ = 1, and taking the sign yields

$$S_i = {\rm sign} \left[ h_i + \sum_j^N J_{ij}S_j \right] ,$$ (50)

with $h_i$ = $\zeta \sum_j C^{-1}_{ij} m_j$ and $J_{ij}$ = $\gamma C^{-1}_{ij} x_j$. This is the equation, for a state which is (locally) stable under the discrete synchronous Hopfield dynamic, $S_i^\prime$ = ${\rm sign} \left[ h_i + \sum_j J_{ij}S_j \right]$, of a spin glass-like Hamiltonian

$$H = \sum_{ij}^N S_i J_{ij}S_j + \sum_i^N h_i S_i .$$ (51)

(For example, in a Hopfield model, $J_{ij}$ = $\sum_a \xi_i^{(a)}\xi_j^{(a)}$, the $\xi_i^{(a)}$ representing patterns to be stored. In an EA-model (Edwards, Anderson) the $J_{ij}$ are Gaussian random variables with distance dependent variance $\sigma_{J_{ij}}$ = $\Delta(\vert i-j\vert)$. In a SK-model (Sherrington, Kirkpatrick) the $J_{ij}$ are Gaussian random variables with distance independent variance $\sigma_{J_{ij}}$ = $J^2/N$, For portfolio theory one can choose a covariance ${\bf C}$ = ${\bf M}^T{\bf M}+{\bf D}$ built from random matrices ${\bf M}$, ${\bf D}$.) As one knows that the ground state of spin glasses can be highly degenerated one can expect a similar effect for such portfolios. For a random matrix treatment see [3].