This thesis investigates numerical methods relating to the Laplace inversion in financial engineering. In the first essay, we propose a novel Laplace transform-based method that includes a two-sided Laplace inversion algorithm working on a uniform grid and an extended algorithm performing at unequally spaced points. We provide a new angle to balance the accuracy and efficiency theoretically: we first seek to ensure that the two algorithms can achieve any desired accuracy and then optimize their efficiencies in large-scale computation based on the guaran-teed accuracy. Numerical schemes for the implementation of both algorithms are provided. Our Laplace transform-based method has many important financial applications, including an exact simulation to price assets, model estimation and mo...[ Read more ]

This thesis investigates numerical methods relating to the Laplace inversion in financial engineering. In the first essay, we propose a novel Laplace transform-based method that includes a two-sided Laplace inversion algorithm working on a uniform grid and an extended algorithm performing at unequally spaced points. We provide a new angle to balance the accuracy and efficiency theoretically: we first seek to ensure that the two algorithms can achieve any desired accuracy and then optimize their efficiencies in large-scale computation based on the guaran-teed accuracy. Numerical schemes for the implementation of both algorithms are provided. Our Laplace transform-based method has many important financial applications, including an exact simulation to price assets, model estimation and model calibration. These applications show our method is accurate, fast and easy to implement. In the second essay, we study the estimation of financial stochastic models via the method of maximum likelihood, considering extreme samples. When there are extreme samples, the accuracy of log-likelihood approximations may not be guaranteed, which further leads to potential errors in the parameter estimation. Extreme events do appear. For example, the U.S. stock index once collapsed several times in 2020 due to COVID-19. We propose a Laplace inversion method with error control of the log-likelihood approximation to solve the problem caused by the extreme samples. The maximum likelihood estimator (MLE) computed by our method enjoys the same consistency and asymptotic normality as the true yet uncomputable MLE. Two novel numerical schemes, including the naive and asymptotic optimal allocations of error tolerance, are also provided for the implementation of our method. Numerical examples show the method is robust, accurate and efficient.