# Asymptotic Analysis of Hedging Errors Induced by Discrete Time Hedging

## Mats Brodén

### Centre for Mathematical Sciences Mathematical Statistics Lund University 2010

##### ISBN 978-91-7473-024-1 LUTFMS-1037-2010

Abstract:
The firsts part of this thesis deals with approximations of stochastic integrals and discrete time hedging of derivative contracts; two closely related subjects. Paper A considers the problem of approximating the value of a Wiener process. The discretization points are placed at times when absolute difference between the value of the process and the approximation reaches a threshold level. It is shown that the difference between the process and the approximation normalized by the threshold level tends to a random variable that is triangularly distributed as the threshold level tends to zero. In Paper B the result from Paper A is generalized to Wiener driven SDEs, and then applied to adaptive discrete time hedging. The hedge portfolio is rebalanced when the abosolute difference between the delta of the hedge portfolio and the derivative contract reaches a threshold level. The rate of convergence of the expected squared hedging error as the threshold level approaches zero is analyzed. In Paper C 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. In Paper D we analyze the errors arising from discrete readjustmendt of the hedge portfolio when hedging options in exponential Lévy models, and establish the rate at which the expected squared error goes to zero when the readjustment frequency increases.
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 models 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.
Key words:
Discrete time hedging, discretization error, L2 convergence, Calibration, Non-linear Kalman filters