實證證據顯示商品價格有均值回歸和跳躍的特性。由於一些商品期權收益涉及歷史商品價格的算術平均，因此我們求出算術亞式期權在均值回歸跳躍擴散過程下的分析解。比分析解是對資產價格最終值和實際平均值的聯合特徽函數進行快速傅立葉變換獲得。我們通過數值模擬研究來檢驗此建議方法的準確度和計算效率。 / Empirical evidence indicates that commodity prices are mean reverting and exhibit jumps. As some commodity option payoff involves the arithmetic average of historical commodity prices, we derive an analytical solution to arithmetic Asian options under a mean reverting jump diffusion process. The analytical solution is implemented with the fast Fourier transform based on the joint characteristic function of the terminal asset price and the realized average value. We also examine the accuracy and computational efficiency of the proposed method through numerical studies. / Detailed summary in vernacular field only. / Chung, Shing Fung. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2013. / Includes bibliographical references (leaves 40-42). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstracts also in Chinese. / Chapter 1 --- Introduction --- p.1 / Chapter 2 --- Model with constant parameters --- p.5 / Chapter 2.1 --- Model specification --- p.6 / Chapter 2.2 --- Joint characteristic function --- p.8 / Chapter 3 --- Model with time-dependent parameters --- p.12 / Chapter 3.1 --- Model specification --- p.13 / Chapter 3.2 --- Joint characteristic function --- p.13 / Chapter 4 --- Fast Fourier transform on Asian option prices --- p.18 / Chapter 5 --- Numerical results --- p.20 / Chapter 5.1 --- Comparison of the analytical solution and Monte Carlo simulation . --- p.20 / Chapter 5.2 --- Price sensitivity and model parameters --- p.26 / Chapter 5.3 --- Price sensitivity and payoff structure --- p.26 / Chapter 6 --- Conclusion --- p.33 / Chapter A --- Normally distributed jump size --- p.34 / Bibliography --- p.40

針對Lévy過程下的二維期權定價問題，本文提出了一種基於快速傅利葉變換(FFT)的解決方案，稱之為二維快速傅利葉變換網絡。不論是時間從屬還是線性組合，此方法適用於所有能取得聯合特徵函數的二維Lévy構建。快速傅利葉變換的種種優點使得比數值方法在不影響結果精確性的前提下，大大降低了所需計算時間。理論上，更高維的Lévy期權定價問題也可以通過擴展數值網絡解決。除此之外，我們還探究了資產波動性亦服從Lévy過程的單資產期權定價。這種資產價值和波動性由一組相關Lévy過程驅動的模型被稱為時間轉換Lévy過程。最後，關於美式及奇異期權定價的數值算例驗證了文中方法的準確性和有效性。 / We propose a two-dimensional network to retrieve the price of two-asset option under Lévy processes by using the fast Fourier transform (FFT). It can be applied to different multivariate Lévy constructions such as subordination and linear combination provided that the joint characteristic function is obtainable. With the prevalent implementation of FFT, the network approach results in significant computational time reduction while maintaining satisfactory accuracy. In general, multi-dimensional option pricing problems are also solvable by extending this network. Furthermore, we investigate option pricing on a single asset where the asset return and its volatility are driven by a pair of dependent Lévy processes. Such a model is also called the random time-changed Lévy process. Numerical examples are given to demonstrate the efficiency and accuracy of FFT-network applied to exotic and American-style options. / Detailed summary in vernacular field only. / Wang, Weiyin. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2013. / Includes bibliographical references (leaves 41-43). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstracts also in Chinese. / List of Tables --- p.ii / List of Figures --- p.iii / Chapter 1 --- Introduction --- p.1 / Chapter 2 --- Literature Review --- p.4 / Chapter 2.1 --- Lévy Process --- p.4 / Chapter 2.1.1 --- Definition and Properties --- p.4 / Chapter 2.1.2 --- Multivariate Lévy Construction --- p.6 / Chapter 2.2 --- Fast Fourier Transform (FFT) in Option Pricing --- p.9 / Chapter 2.2.1 --- European Option on One Asset --- p.9 / Chapter 2.2.2 --- European Option on Two Assets --- p.11 / Chapter 3 --- Two-dimensional FFT-network Model --- p.13 / Chapter 3.1 --- Two-dimensional FFT-network --- p.15 / Chapter 3.2 --- Two-asset Option Pricing --- p.22 / Chapter 3.2.1 --- General Model --- p.22 / Chapter 3.2.2 --- Specific Models --- p.23 / Chapter 3.3 --- Random Time-changed Lévy Process --- p.25 / Chapter 3.3.1 --- Model --- p.26 / Chapter 3.3.2 --- Correlation Adjustment --- p.28 / Chapter 4 --- Numerical Examples --- p.31 / Chapter 4.1 --- Two-asset Option --- p.31 / Chapter 4.1.1 --- Spread Option Pricing --- p.31 / Chapter 4.1.2 --- Pricing under Diffierent Multivariate Lévy Constructions --- p.36 / Chapter 4.2 --- One-asset Option under Random Time-changed Lévy Process --- p.37 / Chapter 5 --- Conclusion --- p.40 / Bibliography --- p.41

有各式各樣的參數與非參數期貨定價模型被廣泛應用於金融領域。其中一些模型的組合能顯著提升期貨定價的準確性。更具體的說，可以先通過參數模型擬合數據，再使用非參數模型學習並修正誤差估價誤差。本論文作為范和Mancini(2009) 結果的延伸，將市場交易的歐式期權價格作為輸入數據，運用「有參數模型指導的非參數定價方法」對障礙期權進行估價。「自動誤差修正估價法」運用非參數方法對由參數估價法產生的誤差進行修正，使得障礙期權的非參數定價模型可以被視為一系列的歐式期權定價的組合。在整個障礙期權的估價過程中，本論文同時提供了一種分數階快速傅裡葉變換的應用，可通過由非參數方法獲得的標的資產對數的存活函數計算標的資產對數最大值分佈的特徵函數。 / There are a variety of parametric and nonparametric option pricing models commonly used in Finance. A combination of them can enhance the pricing performance significantly. Specifically, one proposes to fit the data with a parametric method and then correct the pricing errors empirically with a nonparametric learning approach. This thesis extends Fan and Mancini's (2009) model-guided nonparametric method to barrier option pricing using market traded European option data. Adopting automatic correction of errors (ACE) method to estimate the risk neutral conditional survivor function, by which the pricing error of the initial parametric estimates is captured nonparametrically, enables the nonparametric pricing procedure to value a barrier option as a sum of sequence of European options. As a byproduct from the valuation process, this thesis also provides a modified fractional fast Fourier transform technique compute the characteristic function of the running maximum log-price of the underlying asset nonparametrically through the calibrated survivor functions. / Detailed summary in vernacular field only. / Chi, Chengzhan. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2013. / Includes bibliographical references (leaves 38-39). / Abstracts also in Chinese. / Chapter 1 --- Introduction --- p.1 / Chapter 2 --- Nonparametric Local Regression Modelling --- p.4 / Chapter 2.1 --- Function Estimation by Local Constant --- p.4 / Chapter 2.2 --- Function Estimation by Local Linear Regression --- p.5 / Chapter 3 --- Nonparametric ACE European Option Pricing --- p.7 / Chapter 3.1 --- European Option Prices and Risk Neutral Survivor Functions --- p.7 / Chapter 3.2 --- Estimation of Risk Neutral Survivor Functions --- p.10 / Chapter 3.2.1 --- Risk Neutral Survivor Functions and Traded Options --- p.10 / Chapter 3.2.2 --- Survivor Function Estimation with Nonparametric ACE Method --- p.11 / Chapter 3.3 --- Representation of European Option Prices at Log-asset Level and Numerical Example --- p.15 / Chapter 4 --- Nonparametric ACE Barrier Option Pricing Framework --- p.20 / Chapter 4.1 --- Continuous-time Barrier Option --- p.20 / Chapter 4.2 --- Discrete Approximation and Backward Induction --- p.21 / Chapter 4.3 --- Decomposed Problems --- p.25 / Chapter 5 --- Nonparametric Estimation of Cumulative Distribution Function of M{U+2C7C}(R{U+209C}) --- p.28 / Chapter 5.1 --- Survivor Functions and Maxima Probabilities --- p.28 / Chapter 5.2 --- Characteristic Functions of Maxima --- p.30 / Chapter 5.2.1 --- Algorithm --- p.30 / Chapter 5.2.2 --- Preparation --- p.31 / Chapter 5.2.3 --- Fast Fourier Transform (FFT) --- p.31 / Chapter 5.2.4 --- Fractional Fast Fourier Transform (FRFT) --- p.33 / Chapter 5.2.5 --- Derivation of ΦR{U+209C} --- p.34 / Chapter 5.3 --- Numerical Experiments --- p.35 / Chapter 6 --- Conclusion --- p.37 / Bibliography --- p.38

在現今的社會，協整技術已被廣泛應用於金融和計量經濟領域，特別用於構建股票市場的統計套利策略。在這一篇論文中，我們主要考察在衍生品市場中，基於協整技術的套利交易策略，這一策略的主要研究對象是隱含波動率。利用隱性波動率的線性組合的均值回歸的特性，通過配對兩隻帶有正利差(如theta) 的短期平價歐式跨式期權來獲利。同時，構建實際波動率的模型和預測未來實際波動率的模型將會用於補充這一交易策略的不足，隱性一實際條件和Gamma-Vega條件被引入來提高交易策略的效率。這一策略的績效分析是基於三年的歷史外匯期權數據。從實證數據中，基於協整技術的策略能賺取利潤，而且Vega在利潤中起著重要的作用，並且無論是隱性一實際條件還是Gamma-Vega條件都是有效的。 / The notion of cointegration has been widely used in finance and econometrics, in particular in constructing statistical arbitrage strategies in the stock market. In this thesis, an arbitrage trading strategy for derivatives based on cointegration is studied to account for the volatility factor. Pairs of short dated at-the-money straddles of European options with positive net carry (i.e. theta) are used to capture the mean-reverting property of the linear combinations of implied volatilities. Furthermore, modeling and forecasting realized volatility are also considered as a supplement to the trading strategy. Implied-Realized Criertion and Gamma-Vega Criterion are introduced to improve the trading strategy. A performance analysis is conducted with a 3-year historical data of Foreign Exchange Options. From the empirical results, the portfolio based on the cointegration strategy makes a profit, where Vega plays a dominant role, and either the Implied-Realized Criertion or the Gamma-Vega Criterion is effective. / Detailed summary in vernacular field only. / Pun, Lai Fan. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2013. / Includes bibliographical references (leaves 43-45). / Abstracts also in Chinese. / List of Tables --- p.v / List of Figures --- p.vi / Chapter 1 --- Introduction --- p.1 / Chapter 2 --- Basic Ideas --- p.4 / Chapter 2.1 --- Cointegration and Johansen’s Methodology --- p.4 / Chapter 2.1.1 --- Cointegration --- p.4 / Chapter 2.1.2 --- Johansen’s Methodology --- p.5 / Chapter 2.2 --- Cointegration Pairs Trading Strategy --- p.6 / Chapter 2.3 --- Modelling and Forecasting Realized Volatility --- p.8 / Chapter 3 --- Cointegration Pairs Trading Strategy On Derivatives --- p.10 / Chapter 3.1 --- Trading On Implied Volatility --- p.10 / Chapter 3.2 --- Cointegration Trading Strategy --- p.12 / Chapter 3.3 --- Greek Letters --- p.13 / Chapter 3.3.1 --- Requirements of the Trade --- p.13 / Chapter 3.3.2 --- Approximation of the Expected P/L --- p.15 / Chapter 3.4 --- Foreign Exchange Options --- p.18 / Chapter 3.4.1 --- Cointegration Pairs --- p.19 / Chapter 3.4.2 --- Trading Process --- p.21 / Chapter 3.4.3 --- More Examples --- p.22 / Chapter 4 --- Further Trading Strategies --- p.26 / Chapter 4.1 --- Estimation of Realized Volatility --- p.26 / Chapter 4.2 --- Implied-Realized Criterion --- p.27 / Chapter 4.3 --- Gamma-Vega Criterion --- p.29 / Chapter 4.4 --- Summary --- p.32 / Chapter 5 --- Conclusion and Further Discussion --- p.37 / A --- p.39 / B --- p.41 / Bibliography --- p.43

Abstract not available

Research conducted in mathematical finance focuses on the quantitative modeling of financial markets. It allows one to solve financial problems by using mathematical methods and provides understanding and prediction of the complicated financial behaviors. In this thesis, efforts are devoted to derive and extend stochastic optimization models in financial economics and establish practical algorithms for representing and solving problems in mathematical finance. An option gives the holder the right, but not the obligation, to buy or sell an underlying asset at a specified strike price on or before a specified date. In this thesis, a valuation model for a perpetual convertible bond is developed when the price dynamics of the underlying share are governed by Markovian regime-switching models. By making use of the relationship between the convertible bond and an American option, the valuation of a perpetual convertible bond can be transformed into an optimal stopping problem. A novel approach is also proposed to discuss an optimal inventory level of a retail product from a real option perspective in this thesis. The expected present value of the net profit from selling the product which is the objective function of the optimal inventory problem can be given by the actuarial value of a real option. Hence, option pricing techniques are adopted to solve the optimal inventory problem in this thesis. The goal of risk management is to eliminate or minimize the level of risk associated with a business operation. In the risk measurement literature, there is relatively little amount of work focusing on the risk measurement and management of interest rate instruments. This thesis concerns about building a risk measurement framework based on some modern risk measures, such as Value-at-Risk (VaR) and Expected Shortfall (ES), for describing and quantifying the risk of interest rate sensitive instruments. From the lessons of the recent financial turmoils, it is understood that maximizing profits is not the only objective that needs to be taken into account. The consideration for risk control is of primal importance. Hence, an optimal submission problem of bid and ask quotes in the presence of risk constraints is studied in this thesis. The optimal submission problem of bid and ask quotes is formulated as a stochastic optimal control problem. Portfolio management is a professional management of various securities and assets in order to match investment objectives and balance risk against performance. Different choices of time series models for asset price may lead to different portfolio management strategies. In this thesis, a discrete-time dynamic programming approach which is flexible enough to deal with the optimal asset allocation problem under a general stochastic dynamical system is explored. It’s also interesting to analyze the implications of the heteroscedastic effect described by a continuous-time stochastic volatility model for evaluating risk of a cash management problem. In this thesis, a continuous-time dynamic programming approach is employed to investigate the cash management problem under stochastic volatility model and constant volatility model respectively. / published_or_final_version / Mathematics / Doctoral / Doctor of Philosophy

We present a tree algorithm, called the willow tree, for financial derivative pricing. The setup of the tree uses a fixed number of spatial nodes at each time step. The transition probabilities are determine by solving linear programming problems. The willow tree method is radically superior in numerical performance when compared to the binomial tree method.

Most recent empirical option valuation studies build on the affine square root (SQR) stochastic volatility model. The SQR model is a convenient choice, because it yields closed-form solutions for option prices. However, relatively little is known about the empirical shortcomings of this model. In the first essay, we investigate alternatives to the SQR model, by comparing its empirical performance with that of five different but equally parsimonious stochastic volatility models. We provide empirical evidence from three different sources. We first use realized volatilities to assess the properties of the SQR model and to guide us in the search for alternative specifications. We then estimate the models using maximum likelihood on a long sample of S& P500 returns. Finally, we employ nonlinear least squares on a time series of cross sections of option data. In the estimations on returns and options data, we use the particle filtering technique to retrieve the spot volatility path. The three sources of data we employ all point to the same conclusion: the SQR model is misspecified. Overall, the best of alternative volatility specifications is a model we refer to as the VAR model, which is of the GARCH diffusion type. / In the second essay, we estimate the Constant Elasticity of Variance (CEV) model in order to study the level of nonlinearity in the volatility dynamic. We also estimate a CEV process combined with a jump process (CEVJ) and analyze the effects of the jump component on the nonlinearity coefficient. Estimation is performed using the particle filtering technique on a long series of S&P500 returns and on options data. We find that both returns data and returns-and-options data favor nonlinear specifications for the volatility dynamic, suggesting that the extensive use of linear models is not supported empirically. We also find that the inclusion of jumps does not affect the level of nonlinearity and does not improve the CEV model fit. / The third essay provides an empirical comparison of two classes of option valuation models: continuous-time models and discrete-time models. The literature provides some theoretical limit results for these types of dynamics, and researchers have used these limit results to argue that the performance of certain discrete-time and continuous-time models ought to be very similar. This interpretation is somewhat contentious, because a given discrete-time model can have several continuous-time limits, and a given continuous-time model can be the limit for more than one discrete-time model. Therefore, it is imperative to investigate whether there exist similarities between these specifications from an empirical perspective. Using data on S&P500 returns and call options, we find that the discrete-time models investigated in this paper have the same performance in fitting the data as selected continuous-time models both in and out-of-sample.

We present a tree algorithm, called the willow tree, for financial derivative pricing. The setup of the tree uses a fixed number of spatial nodes at each time step. The transition probabilities are determine by solving linear programming problems. The willow tree method is radically superior in numerical performance when compared to the binomial tree method. / Science, Faculty of / Mathematics, Department of / Graduate

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