This thesis proposes an optimization formulation to ensure
accuracy and stability in the local volatility function calibration.
The unknown local volatility function is represented by kernel
splines. The proposed optimization formulation minimizes calibration
error and an L1 norm of the vector of coefficients for the
kernel splines. The L1 norm regularization forces some
coefficients to be zero at the termination of optimization. The
complexity of local volatility function model is determined by the
number of nonzero coefficients. Thus by using a regularization
parameter, the proposed formulation balances the calibration
accuracy with the model complexity. In the context of the support
vector regression for function based on finite observations, this
corresponds to balance the generalization error with the number of
support vectors. In this thesis we also propose a trust region
method to determine the coefficient vector in the proposed
optimization formulation. In this algorithm, the main computation of
each iteration is reduced to solving a standard trust region
subproblem. To deal with the non-differentiable L1 norm in the
formulation, a line search technique which allows crossing
nondifferentiable hyperplanes is introduced to find the minimum
objective value along a direction within a trust region. With the
trust region algorithm, we numerically illustrate the ability of
proposed approach to reconstruct the local volatility in a synthetic
local volatility market. Based on S&P 500 market index option data,
we demonstrate that the calibrated local volatility surface is
smooth and resembles in shape the observed implied volatility
surface. Stability is illustrated by considering calibration using
market option data from nearby dates.
| Identifer | oai:union.ndltd.org:WATERLOO/oai:uwspace.uwaterloo.ca:10012/4080 |
| Date | 19 September 2008 |
| Creators | Wang, Cheng |
| Source Sets | University of Waterloo Electronic Theses Repository |
| Language | English |
| Detected Language | English |
| Type | Thesis or Dissertation |
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