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Financial mathematical models provide useful tools for option pricing.
These
physical models give us a good first order approximation to the
underlying dynamics in the financial market. Their power in option
pricing can be significantly enhanced when they are combined
with statistical approaches, which empirically learn and correct pricing errors
through estimating the state price densities. In this talk,
two new semiparametric techniques are proposed for estimating state price
densities and pricing financial derivatives. Our empirical studies
based on the options of SP500 index show our methods outperform
the ad hoc Black-Scholes method and significantly so
when the latter method has large pricing errors,
based on over 100,000 tests.
The second part of the talk will focus on estimation high-dimensional
covariance matrix for portfolio allocation and risk management.
Motivated by the Capital Asset Pricing Model, we propose to use
a factor model to reduce the dimensionality and to estimate the
covariance matrix. The performance is compared with the sample
covariance matrix. We demonstrate and identify the situations under which
the factor approach can gain substantially the performance
and the cases where the gains are only marginal.
Furthermore, the impacts of the covariance matrix
estimation on portfolio allocation and risk management are studied.
The theoretical results are convincingly supported by a
thorough simulation study.
