6 research outputs found
Efficient implementation of linear programming decoding
While linear programming (LP) decoding provides more flexibility for
finite-length performance analysis than iterative message-passing (IMP)
decoding, it is computationally more complex to implement in its original form,
due to both the large size of the relaxed LP problem, and the inefficiency of
using general-purpose LP solvers. This paper explores ideas for fast LP
decoding of low-density parity-check (LDPC) codes. We first prove, by modifying
the previously reported Adaptive LP decoding scheme to allow removal of
unnecessary constraints, that LP decoding can be performed by solving a number
of LP problems that contain at most one linear constraint derived from each of
the parity-check constraints. By exploiting this property, we study a sparse
interior-point implementation for solving this sequence of linear programs.
Since the most complex part of each iteration of the interior-point algorithm
is the solution of a (usually ill-conditioned) system of linear equations for
finding the step direction, we propose a preconditioning algorithm to
facilitate iterative solution of such systems. The proposed preconditioning
algorithm is similar to the encoding procedure of LDPC codes, and we
demonstrate its effectiveness via both analytical methods and computer
simulation results.Comment: 44 pages, submitted to IEEE Transactions on Information Theory, Dec.
200
Numerical Optimisation Problems in Finance
This thesis consists of four projects regarding numerical optimisation and financial
derivative pricing.
The first project deals with the calibration of the Heston stochastic volatility
model. A method using the Levenberg-Marquardt algorithm with the analytical
gradient is developed. It is so far the fastest Heston model calibrator and meets the
speed requirement of practical trading.
In the second project, a triply-nested iterative method for the implementation of
interior-point methods for linear programs is proposed. It is the first time that an
interior-point method entirely based on iterative solvers succeeds in solving a fairly
large number of linear programming instances from benchmark libraries under the
standard stopping criteria.
The third project extends the Black-Scholes valuation to a complex volatility
parameter and presents its singularities at zero and infinity. Fractals that describe
the chaotic nature of the Newton-Raphson calculation of the implied volatility are
shown for different moneyness values. Among other things, these fractals visualise
dramatically the effect of an existing modification for improving the stability and
convergence of the search. The project studies scientifically an interesting problem
widespread in the financial industry, while revealing artistic values stemming from
mathematics.
The fourth project investigates the consistency of a class of stochastic volatility
models under spot rate inversion, and hence their suitability in the foreign exchange
market. The general formula of the model parameters for the inversion rate is given,
which provides basis for further investigation. The result is further extended to the
affine stochastic volatility model. The Heston model, among the other members
in the stochastic volatility family, is the only one that we found to be consistent
under the spot inversion. The conclusion on the Heston model verifies the arbitrage
opportunity in the variance swap
A study of preconditioners for network interior point methods
ABSTRACT. We study and compare preconditioners available for network interior point methods. We derive upper bounds for the condition number of the preconditioned matrices used in the solution of systems of linear equations defining the algorithm search directions. The preconditioners are tested using PDNET, a state-of-the-art interior point code for the minimum cost network flow problem. A computational comparison using a set of standard problems improves the understanding of the effectiveness of preconditioners in network interior point methods. 1