THESIS
2023
1 online resource (xi, 120 pages) : color illustrations
Abstract
In this thesis, we discuss two types of Monte Carlo simulation methods and their applications
in quantitative finance. In the first project, we investigate analytical solvability
of models with affine stochastic volatility (SV) and Lévy jumps by deriving a unified formula
for the conditional moment generating function of the log-asset price and providing
the condition under which this new formula is explicit. We then combine our theoretical
results, the Hilbert transform method, various interpolation techniques, with the dimension
reduction technique to propose unified simulation schemes for solvable models with
affine SV and Lévy jumps. Our approach is applicable to a broad class of models, maintains
good accuracy, and enables efficient pricing of discretely monitored path-dependent...[
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In this thesis, we discuss two types of Monte Carlo simulation methods and their applications
in quantitative finance. In the first project, we investigate analytical solvability
of models with affine stochastic volatility (SV) and Lévy jumps by deriving a unified formula
for the conditional moment generating function of the log-asset price and providing
the condition under which this new formula is explicit. We then combine our theoretical
results, the Hilbert transform method, various interpolation techniques, with the dimension
reduction technique to propose unified simulation schemes for solvable models with
affine SV and Lévy jumps. Our approach is applicable to a broad class of models, maintains
good accuracy, and enables efficient pricing of discretely monitored path-dependent
derivatives. We analyze various sources of errors arising from the simulation approach
and present error bounds. Extensive numerical results demonstrate that our method is
highly accurate, efficient, simple to implement, and widely applicable. In the second
project, we explore various enhancements in rare event simulation algorithms to achieve
better computational efficiency and accuracy for computing marginal risk contributions of
the popular Gaussian copula model and t-copula model. The Glasserman-Li exponential
twisting method is seen to have a close link with the saddlepoint approximation method.
We develop the hybrid saddlepoint approximation method that inherits the merit of the
nice analytical tractability exhibited by the saddlepoint approximation framework. We
enhance the efficiency of calculating the saddlepoint via an interpolation procedure. We
also propose new formulas to enhance the cross entropy method for the estimation of the
optimal parameters in the t-copula model. Extensive numerical tests on computing risk
contributions were performed on various copula models with multiple risk factors. Our
enhanced exponential twisting method, cross entropy method, and hybrid saddlepoint
approximation method are seen to exhibit a high level of efficiency, accuracy, and reliability
when compared with existing importance sampling algorithms in computing risk
measures and marginal risk contributions in copula credit models.
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