On the Convergence of Discrete Time Hedging Schemes
Mats Brodén
Centre for Mathematical Sciences
Mathematical Statistics
Lund Institute of Technology,
Lund University,
2008
ISSN 1404028X

Abstract:

In the first part of this thesis discrete time hedging is considered. In
paper A an adaptive hedging scheme where the hedge portfolio is rebalanced
when the hedge ratios differ by some amount, here denoted ?, is investigated.
An expression of the normalized expected mean squared hedging error as ?
tends to zero is derived. The result is compared to an earlier result on
discrete time hedging on an equidistant time grid. For a reasonable setting
it is shown that the adaptive hedging scheme is more efficient. In paper
B discrete time hedging on an equidistant time grid using two hedge instruments
is investigated. It is shown that this hedging scheme improves the order
of convergence of the mean squared hedging error considerably compared to
the case when one hedge instrument is used.


The second part of the thesis concerns parameter estimation of option pricing
models. A framework based on a statespace formulation of the option pricing
model is introduced. Introducing a measurement error of observed market prices
the measurements are treated in a statistically consistent way. This will
reduce the effect of noisy measurements. Also, by introducing stochastic
dynamics for the parameters the statistical framework is made adaptive. In
a simulation study it is shown that the filtering framework is capable of
tracking parameters as well as latent processes. We compare estimates from
S&P 500 option data using Extended Kalman Filters as well as Iterated
Extended Kalman Filters with estimates using the standard methods weighted
least squares and penalized weighted least squares. It is shown that the
filter estimates are the most accurate.




