The Quadratic Exponential (QE) model is a market standard simulation method for the Heston stochastic volatility model. We identify certain... Show moreThe Quadratic Exponential (QE) model is a market standard simulation method for the Heston stochastic volatility model. We identify certain numerical problems with the standard discretization and modify the original method to correct these problems. We implement our modified QE scheme for the Heston model in the Guaranteed Automatic Integration Library (GAIL)|a suite of algorithms that includes Monte Carlo and quasi-Monte Carlo methods for multidimensional integration and computation of means. GAIL computes answers to satisfy user-defined error tolerances. We also implement variance reduction techniques for our modified QE scheme in GAIL. The numerical results show that our modified scheme is fast and accurate, and satisfies the user-defined error tolerances. M.S. in Applied Mathematics, May 2017 Show less
In some definite integral problems the analytical solution is either unknown or hard to compute. As an alternative, one can approximate the... Show moreIn some definite integral problems the analytical solution is either unknown or hard to compute. As an alternative, one can approximate the solution with numerical methods that estimate the value of the integral. However, for high dimensional integrals many techniques suffer from the curse of dimensionality. This can be solved if we use quasi-Monte Carlo methods which do not suffer from this phenomenon. Section 2.2 describes digital sequences and rank-1 lattice node sequences, two of the most common points used in quasi-Monte Carlo. If one uses quasi-Monte Carlo, there is still another problem to address: how many points are needed to estimate the integral within a particular absolute error tolerance. In this dissertation, we propose two automatic cubatures based on digital sequences and rank-1 lattice node sequences that estimate high dimensional problems. These new algorithms are constructed in Chapter 3 and the user-specified absolute error tolerance is guaranteed to be satisfied for a specific set of integrands. In Chapter 4 we define a new estimator that satisfies a generalized tolerance function and includes a relative error tolerance option. An important property of quasi-Monte Carlo methods is that they are effective when the function has low effective dimension. In [1], Sobol’ defined the global sensitivity indices, which measure what part of the variance is explained by each dimension. We can use these indices to measure the effective dimensionality of a function. In Chapter 5 we extend our digital sequences cubature to estimate first order and total effect Sobol’ indices. Ph.D. in Applied Mathematics, December 2016 Show less
