On the Convergence of Discrete Time Hedging Schemes

Mats Brodén


Centre for Mathematical Sciences
Mathematical Statistics
Lund Institute of Technology,
Lund University,
2008

ISSN 1404-028X
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 re-balanced 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 state-space 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.