Guan, Peiqiu. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2009. / Includes bibliographical references (p. 67-71). / Abstract also in Chinese. / Chapter 1 --- Introduction --- p.1 / Chapter 2 --- Literature Review --- p.6 / Chapter 2.1 --- Characteristic Function --- p.6 / Chapter 2.1.1 --- Definition --- p.6 / Chapter 2.1.2 --- Inverse Fourier Transform --- p.8 / Chapter 2.1.3 --- Fast Fourier Transform (FFT) --- p.9 / Chapter 2.2 --- Levy Processes --- p.13 / Chapter 2.2.1 --- Definition --- p.13 / Chapter 2.2.2 --- Levy-Khinchine Formula --- p.15 / Chapter 2.2.3 --- Levy Processes in Finance --- p.17 / Chapter 2.3 --- Exotic Options --- p.17 / Chapter 2.3.1 --- Barrier Options --- p.18 / Chapter 2.3.2 --- Lookback Options --- p.19 / Chapter 2.3.3 --- Asian Options --- p.20 / Chapter 3 --- FFT Network Model --- p.23 / Chapter 3.1 --- Weaknesses of Traditional Tree Approaches --- p.24 / Chapter 3.2 --- FFT Network Model --- p.30 / Chapter 3.3 --- Basic Transition Probability Matrix --- p.31 / Chapter 3.4 --- Basic FFT Network Pricing Algorithm --- p.35 / Chapter 3.4.1 --- Plain Vanilla Options --- p.35 / Chapter 4 --- FFT Network for Exotic Options --- p.38 / Chapter 4.1 --- Barrier Option Pricing --- p.38 / Chapter 4.2 --- Forward Shooting Grid --- p.41 / Chapter 4.3 --- FSG in FFT Network --- p.43 / Chapter 4.4 --- Lookback and Knock-in Options --- p.45 / Chapter 4.4.1 --- American Lookback Option Pricing Algorithm --- p.48 / Chapter 4.4.2 --- Knock-in American Option Pricing Algorithm --- p.50 / Chapter 4.5 --- Asian Option Pricing --- p.51 / Chapter 4.5.1 --- Asian Option Pricing Algorithm --- p.54 / Chapter 5 --- Numerical Implementation --- p.57 / Chapter 5.1 --- Numerical Scheme --- p.57 / Chapter 5.2 --- Numerical Result --- p.60 / Chapter 6 --- Conclusion --- p.65 / Bibliography --- p.67

Tang, Sze Ki = 計算美式賣權的最優提早履約邊界及期權金 / 鄧思麒. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2010. / Includes bibliographical references (leaves 96-102). / Abstracts in English and Chinese. / Tang, Sze Ki = Ji suan Mei shi mai quan de zui you ti zao lu yue bian jie ji qi quan jin / Deng Siqi. / Chapter 1 --- Introduction --- p.1 / Chapter 1.1 --- The Black-Scholes Option Pricing Model --- p.1 / Chapter 1.1.1 --- Geometric Brownian Motion --- p.1 / Chapter 1.1.2 --- The Black-Scholes Equation --- p.3 / Chapter 1.1.3 --- The European Put Option --- p.5 / Chapter 1.1.4 --- The American Put Option --- p.7 / Chapter 1.1.5 --- Perpetual American Option --- p.9 / Chapter 1.2 --- Literature Review --- p.9 / Chapter 1.2.1 --- Direct Numerical Method --- p.10 / Chapter 1.2.2 --- Analytical Approximation --- p.11 / Chapter 1.2.3 --- Analytical Representation --- p.12 / Chapter 1.2.4 --- Mean-Reverting Lognormal Process --- p.13 / Chapter 1.2.5 --- Constant Elasticity of Variance Process --- p.15 / Chapter 1.2.6 --- Model Parameters with Time Dependence --- p.17 / Chapter 1.3 --- Overview --- p.18 / Chapter 2 --- Mean-Reverting Lognormal Model --- p.21 / Chapter 2.1 --- Moving Barrier Rebate Options under GBM --- p.21 / Chapter 2.2 --- Simulating American Puts under GBM --- p.25 / Chapter 2.3 --- Special Case: Time Independent Parameters --- p.26 / Chapter 2.3.1 --- Reduction to Ingersoll's Approximations --- p.26 / Chapter 2.3.2 --- Perpetual American Put Option --- p.28 / Chapter 2.4 --- Moving Barrier Rebate Options under MRL Process --- p.29 / Chapter 2.4.1 --- Reduction to Black-Scholes Model --- p.30 / Chapter 2.5 --- Simulating the American Put under MRL Process --- p.32 / Chapter 3 --- Constant Elasticity of Variance Model --- p.34 / Chapter 3.1 --- Transformations --- p.35 / Chapter 3.2 --- Homogeneous Solution on a Semi-Infinite Domain --- p.37 / Chapter 3.3 --- Particular Solution on a Semi-Infinite Domain --- p.38 / Chapter 3.4 --- Moving Barrier Options with Rebates --- p.39 / Chapter 3.5 --- Simulating the American Options --- p.40 / Chapter 3.6 --- Implication from the Special Case L = 0 --- p.41 / Chapter 4 --- Optimization for the Approximation --- p.43 / Chapter 4.1 --- Introduction --- p.43 / Chapter 4.2 --- The Optimization Scheme --- p.44 / Chapter 4.2.1 --- Illustrative Examples --- p.44 / Chapter 4.3 --- Discussion --- p.45 / Chapter 4.3.1 --- Upper Bound of the Exact Early Exercise Price --- p.45 / Chapter 4.3.2 --- Tightest Lower Bound of the American Put Option Price --- p.48 / Chapter 4.3.3 --- Ingersoll's Early Exercise Decision Rule --- p.51 / Chapter 4.3.4 --- Connection between Ingersoll's Rule and Samuelson's Smooth Paste Condition --- p.51 / Chapter 4.3.5 --- Computation Efficiency --- p.52 / Chapter 4.4 --- Robustness Analysis --- p.53 / Chapter 4.4.1 --- MRL Model --- p.53 / Chapter 4.4.2 --- CEV Model --- p.55 / Chapter 4.5 --- Conclusion --- p.57 / Chapter 5 --- Multi-stage Approximation Scheme --- p.59 / Chapter 5.1 --- Introduction --- p.59 / Chapter 5.2 --- Multistage Approximation Scheme for American Put Options --- p.60 / Chapter 5.3 --- Black-Scholes GBM Model --- p.61 / Chapter 5.3.1 --- "Stage 1: Time interval [0, t1]" --- p.61 / Chapter 5.3.2 --- "Stage 2: Time interval [t1, T]" --- p.62 / Chapter 5.4 --- Mean Reverting Lognormal Model --- p.63 / Chapter 5.4.1 --- "Stage 1: Time interval [0, t1]" --- p.63 / Chapter 5.4.2 --- "Stage 2: Time interval [t1, T]" --- p.64 / Chapter 5.5 --- Constant Elasticity of Variance Model --- p.66 / Chapter 5.5.1 --- "Stage 1: Time interval [0, t1]" --- p.66 / Chapter 5.5.2 --- "Stage 2: Time interval [t1, T]" --- p.67 / Chapter 5.6 --- Duration of Time Intervals --- p.69 / Chapter 5.7 --- Discussion --- p.72 / Chapter 5.7.1 --- Upper Bounds for the Optimal Early Exercise Prices --- p.73 / Chapter 5.7.2 --- Error Analysis --- p.74 / Chapter 5.8 --- Conclusion --- p.77 / Chapter 6 --- Numerical Analysis --- p.79 / Chapter 6.1 --- Sensitivity Analysis of American Put Options in MRL Model --- p.79 / Chapter 6.1.1 --- Volatility --- p.79 / Chapter 6.1.2 --- Risk-free Interest Rate and Dividend Yield --- p.80 / Chapter 6.1.3 --- Speed of Mean Reversion --- p.81 / Chapter 6.1.4 --- Mean Underlying Asset Price --- p.83 / Chapter 6.2 --- Sensitivity Analysis of American Put Options in CEV Model --- p.85 / Chapter 6.2.1 --- Elasticity Factor --- p.87 / Chapter 6.3 --- American Options with time-dependent Volatility --- p.87 / Chapter 6.3.1 --- MRL American Options --- p.89 / Chapter 6.3.2 --- CEV American Options --- p.90 / Chapter 6.3.3 --- Discussion --- p.91 / Chapter 7 --- Conclusion --- p.94 / Bibliography --- p.96 / Chapter A --- Derivation of The Duhamel Superposition Integral --- p.101 / Chapter A.1 --- Time Independent Inhomogeneous Boundary Value Problem --- p.101 / Chapter A.2 --- Time Dependent Inhomogeneous Boundary Value Problem --- p.102

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This document demonstrates the use of the C++ programming language as a simulation tool in the efficient pricing of exotic European options. Extensions to the basic problem of simulation pricing are undertaken including variance reduction by conditional expectation, control and antithetic variates. Ultimately we were able to produce a modularized, easily extend-able program which effectively makes use of Monte Carlo simulation techniques to price lookback, Asian and barrier exotic options. Theories of variance reduction were validated except in cases where we used control variates in combination with the other variance reduction techniques in which case we observed increased variance. Again, the main aim of this half thesis was to produce a C++ program which would produce stable pricings of exotic options.

In this thesis, we study the FX option pricing problem and put forward a 4-factor hybrid stochastic-local volatility model. The model, which describes the dynamics of an exchange rate, its volatility and the domestic and foreign short rates, allows for a perfect calibration to European options and has a good hedging performance. Due to the high-dimensionality of the problem, we propose a Monte Carlo simulation scheme that combines the full truncation Euler scheme for the stochastic volatility component and the stochastic short rates with the log-Euler scheme for the exchange rate. We analyze exponential integrability properties of Euler discretizations for the square-root process driving the stochastic volatility and the short rates, properties which play a key role in establishing the finiteness of moments and the strong convergence of numerical approximations for a large class of stochastic differential equations in finance, including the ones studied in this thesis. Hence, we prove the strong convergence of the exchange rate approximations and the convergence of Monte Carlo estimators for a number of vanilla and exotic options. Then, we calibrate the model to market data and discuss its fitness for pricing FX options. Next, due to the relatively slow convergence of the Monte Carlo method in the number of simulations, we examine a variance reduction technique obtained by mixing Monte Carlo and finite difference methods via conditioning. We consider a purely stochastic version of the model and price vanilla and exotic options by simulating the paths of the volatility and the short rates, and then evaluating the "inner" Black-Scholes-type expectation by means of a partial differential equation. We prove the convergence of numerical approximations and carry out a theoretical variance reduction analysis. Finally, we illustrate the efficiency of the method through a detailed quantitative assessment.

University of Macau / Faculty of Science and Technology / Department of Mathematics

University of Macau / Faculty of Science and Technology / Department of Mathematics

University of Macau / Faculty of Science and Technology / Department of Mathematics

Nicotinic acid adenine dinucleotide phosphate (NAADP) is a potent Ca²⁺ mobilising messenger in mammalian and non-mammalian cells. Studies on a variety of cell types suggest that NAADP evokes Ca²⁺ release from a lysosome-related store and via activation of a receptor distinct from either ryanodine receptors (RyR) or inositol 1,4,5-trisphosphate (IP₃) receptors (IP₃R). However, the identity of the NAADP receptor has, until now, remained elusive. In this thesis I have shown that NAADP-evoked Ca²⁺ release from lysosomes is underpinned by two-pore channels (TPCs), of which there are 3 subtypes, TPC1, TPC2 and TPC3. When stably over-expressed in HEK293 cells, TPC2 was found to be specifically targeted to lysosomes, while TPC1 and TPC3 were targeted to endosomes. Initial Ca²⁺ signals via TPC2, but not those via TPC1, were amplified into global Ca²⁺ waves by Ca²⁺-induced Ca²⁺ release (CICR) from the endoplasmic reticulum (ER) via IP₃Rs. I have shown that, consistent with a role for TPCs in NAADP-mediated Ca²⁺ release, TPC2 is expressed in pulmonary arterial smooth muscle cells (PASMCs), is likely targeted to lysosomal membranes, and that TPCs also underpin NAADP-evoked Ca²⁺ signalling in this cell type. However, and in contrast to HEK293 cells, in PASMCs NAADP evokes spatially restricted Ca²⁺ bursts that are amplified into global Ca²⁺ waves by CICR from the sarcoplasmic reticulum (SR) via a subpopulation of RyRs, but not via IP₃Rs. I have demonstrated that lysosomes preferentially co-localise with RyR subtype 3 (RyR3) in the perinuclear region of PASMCs to comprise a “trigger zone” for Ca²⁺ signalling by NAADP, away from which a propagating Ca²⁺ wave may be carried by subsequent recruitment of RyR2. The identification of TPCs as a family of NAADP receptors may further our understanding of the mechanisms that confer the versatility of Ca²⁺ signalling which is required to regulate such diverse cellular functions as gene expression, fertilization, cell growth, and ultimately cell death.

published_or_final_version / Business Administration / Master / Master of Business Administration