Asymptotic Analysis of Hedging Errors Induced by Discrete Time Hedging
Centre for Mathematical Sciences
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.
Discrete time hedging, discretization error, L2 convergence,
Calibration, Non-linear Kalman filters