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Advantage of Filtering for Portfolio Optimization in Financial Markets with Partial Information
(2016)
In a financial market we consider three types of investors trading with a finite
time horizon with access to a bank account as well as multliple stocks: the
fully informed investor, the partially informed investor whose only source of
information are the stock prices and an investor who does not use this infor-
mation. The drift is modeled either as following linear Gaussian dynamics
or as being a continuous time Markov chain with finite state space. The
optimization problem is to maximize expected utility of terminal wealth.
The case of partial information is based on the use of filtering techniques.
Conditions to ensure boundedness of the expected value of the filters are
developed, in the Markov case also for positivity. For the Markov modulated
drift, boundedness of the expected value of the filter relates strongly to port-
folio optimization: effects are studied and quantified. The derivation of an
equivalent, less dimensional market is presented next. It is a type of Mutual
Fund Theorem that is shown here.
Gains and losses eminating from the use of filtering are then discussed in
detail for different market parameters: For infrequent trading we find that
both filters need to comply with the boundedness conditions to be an advan-
tage for the investor. Losses are minimal in case the filters are advantageous.
At an increasing number of stocks, again boundedness conditions need to be
met. Losses in this case depend strongly on the added stocks. The relation
of boundedness and portfolio optimization in the Markov model leads here to
increasing losses for the investor if the boundedness condition is to hold for
all numbers of stocks. In the Markov case, the losses for different numbers
of states are negligible in case more states are assumed then were originally
present. Assuming less states leads to high losses. Again for the Markov
model, a simplification of the complex optimal trading strategy for power
utility in the partial information setting is shown to cause only minor losses.
If the market parameters are such that shortselling and borrowing constraints
are in effect, these constraints may lead to big losses depending on how much
effect the constraints have. They can though also be an advantage for the
investor in case the expected value of the filters does not meet the conditions
for boundedness.
All results are implemented and illustrated with the corresponding numerical
findings.
Continuous-time regime-switching models are a very popular class of models for financial applications. In this work the so-called signal-to-noise matrix is introduced for hidden Markov models where the switching is driven by an unobservable Markov chain. Its relations to filtering, i.e. state estimation of the chain given the available observations, and portfolio optimization are investigated. A convergence result for the filter is derived: The filter converges to its invariant distribution if the eigenvalues of the signal-to-noise matrix converge to zero. This matrix is then also used to prove a mutual fund representation for regime-switching models and a corresponding market reduction which is consistent with filtering and portfolio optimization. Two canonical cases for the reduction are analyzed in more detail, the first based on the market regimes and the second depending on the eigenvalues. These considerations are presented both for observable and unobservable Markov chains. The results are illustrated by numerical simulations.