Pricing Multi-asset Path-dependent Options Through Monte Carlo Simulations Open Access

Xia, Hanqiu (2015)

Permanent URL: https://etd.library.emory.edu/concern/etds/x059c789f?locale=en
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Abstract

Monte Carlo simulation (MC) is an approach that is widely used in high-dimensional numerical integration and one of its main nancial applications is option pricing. The aim of this thesis is to evaluate the price of an Asian option using standard Monte Carlo method and Quasi-Monte Carlo method (QMC) respectively. Since QMC's convergence rate is determined by nominal problem dimension, the convergence rate of QMC increases as the problem dimension increases, which limits the performance of QMC in high dimensions. Hence, in this thesis, we also consider several techniques which are proposed to capture the eective dimensions and improve the efficiency of QMC in high-dimensional situations. The techniques include principal component analysis (PCA) and Kronecker product approximation (KPA) and they are applied for both constant and time-dependent volatilities. Finally, we conduct numerical experiments and compare the precision and computational time between Quasi-Monte Carlo and Monte Carlo methods.

Table of Contents

1 Introduction 1
1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Mathematical Model . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2 Monte Carlo Methods 8
2.1 Standard Monte Carlo Methods . . . . . . . . . . . . . . . . . . 8
2.2 Quasi-Monte Carlo Methods . . . . . . . . . . . . . . . . . . . . 13
3 Eective Dimensions Selection 17
3.1 Principal Component Analysis . . . . . . . . . . . . . . . . . . .18
3.1.1 Mathematical Background of PCA . . . . . . . . . . . . . .18
3.1.2 New Covariance Matrix after PCA . . . . . . . . . . . . . .20
3.1.3 Improvement of PCA . . . . . . . . . . . . . . . . . . . . . . 21
3.2 Kronecker Product Approximation . . . . . . . . . . . . . . . . 22
3.2.1 One of Two Matrices is Fixed . . . . . . . . . . . . . . . . . 23
3.2.2 Neither Matrices are Fixed . . . . . . . . . . . . . . . . . . 25
4 Numerical Experiments 28
4.1 Computational Time Comparison . . . . . . . . . . . . . . . . 30
4.2 Estimation of Option Price . . . . . . . . . . . . . . . . . . . . . 32
5 Conclusions 34

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