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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
|
(46) |
Hence, we may define an optimal portfolio as the minimum of
|
(47) |
This yields,
|
(48) |
i.e.,
|
(49) |
where = sign.
Setting = 1, and taking the sign yields
|
(50) |
with =
and =
.
This is the equation,
for a state which is (locally) stable
under the discrete synchronous Hopfield dynamic,
=
,
of a spin glass-like Hamiltonian
|
(51) |
(For example, in a Hopfield model,
=
,
the representing patterns to be stored.
In an EA-model (Edwards, Anderson)
the are Gaussian random variables
with distance dependent variance
= .
In a SK-model (Sherrington, Kirkpatrick)
the are Gaussian random variables
with distance independent variance
= ,
For portfolio theory one can choose
a covariance =
built from random matrices , .)
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].
Next: Bibliography
Up: Econophysics WS1999/2000: Some Notes
Previous: Linear regression
Joerg_Lemm
2000-02-25