''Numerical Solutions of Stochastic Differential Equations
with Jumps in
Finance ''
Abstract.
In financial and actuarial modelling and other areas of
application,
stochastic differential equations with jumps have been
employed to describe
the dynamics of various state variables. The numerical
solution of such equations
is more complex than that of those only driven by Brownian
motions. The aim
of this lecture is to present various numerical methods used
in quantitative finance
for models involving stochastic differential equations with
jumps. It emphasises
mathematical concepts, techniques and intuition crucial for
modern numerical methods
in derivative pricing and risk management. Questions of
numerical stability and
convergence will be discussed. Several recent results will be
presented on higher-order
methods for scenario and Monte Carlo simulation, including
implicit, predictor corrector
and extrapolation methods.
Abstract. This lecture introduces into the benchmark approach,
which provides a
generalized framework for financial market modelling. It
allows for a unified
treatment of derivative pricing, portfolio optimization and
risk management.
It extends beyond the classical asset pricing theories, with
significant differences
emerging for extreme maturity contracts and risk measures
relevant to pensions
and insurance. The Law of the Minimal Price will be presented
for derivative pricing.
A Naïve Diversification Theorem allows forming a proxy for the
numeraire
portfolio. The richer modelling framework of the benchmark
approach leads to the
derivation of tractable, realistic models under the real world
probability measure.
It will be explained how the approach differs from the
classical risk neutral approach.
Examples on long term and extreme maturity derivatives
demonstrate the important
fact that a range of contracts can be less expensively priced
and hedged than suggested
by classical theory.
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