The role of adaptivity in a numerical method for the Cox-Ingersoll-Ross model
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Date
2022-07-04
Authors
Maulana, Heru
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Publisher
University College Cork
Published Version
Abstract
We demonstrate the effectiveness of an adaptive explicit Euler method for the
approximate solution of the Cox-Ingersoll-Ross model. This relies on a class of
path-bounded timestepping strategies which work by reducing the stepsize as
solutions approach a neighbourhood of zero. The method is hybrid in the sense
that a convergent backstop method is invoked if the timestep becomes too small,
or to prevent solutions from overshooting zero and becoming negative.
Under parameter constraints that imply Feller’s condition, we prove that such
a scheme is strongly convergent, of order at least 1/2. Control of the strong error
is important for multi-level Monte Carlo techniques. Under Feller’s condition we
also prove that the probability of ever needing the backstop method to prevent
a negative value can be made arbitrarily small. Numerically, we compare this
adaptive method to fixed step implicit and explicit schemes, and a novel semi-implicit adaptive variant. We observe that the adaptive approach leads to methods that are competitive
in a domain that extends beyond Feller’s condition, indicating suitability for the
modelling of stochastic volatility in Heston-type asset models.
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Keywords
Cox–Ingersoll–Ross model , Adaptive timestepping , Explicit Euler–Maruyama method , Strong convergence , Positivity
Citation
Maulana, H. 2022. The role of adaptivity in a numerical method for the Cox-Ingersoll-Ross model. PhD Thesis, University College Cork.