The exotic options that are examined in this thesis have a combination of non-standard characteristics which can be found in shout, multi-callable, pathdependent and Bermudan options. These options are called reset options. A reset option is an option which allows the holder to reset, one or more times, certain terms of the contract based on pre-specified rules during the life of the option. Overall in this thesis, an attempt has been made to tackle the modeling challenges that arise from the exotic properties of the reset option embedded in segregated funds. Initially, the relevant literature was reviewed and the lack of published work, advanced enough to deal with the complexities of the reset option, was identified. Hence, there appears to be a clear and urgent need to have more sophisticated approaches which will model the reset option. The reset option on the maturity guarantee of segregated funds is formulated as a non-stationary finite horizon Markov Decision Process. The returns from the underlying asset are modeled using a discrete time approximation of the lognormal model. An Optimal Exercise Boundary of the reset option is derived where a threshold value is depicted such that if the value of the underlying asset price exceeds it then it is optimal for the policyholder to reset his maturity guarantee. Otherwise, it is optimal for the policyholder to rollover his maturity guarantee. It is noteworthy that the model is able to depict the Optimal Exercise Boundary of not just the first but of all the segregated fund contracts which can be issued throughout the planning horizon of the policyholder. The main finding of the model is that as the segregated fund contract approaches its maturity, the threshold value in the Optimal Exercise Boundary increases. However, in the last period before the maturity of the segregated fund, the threshold value decreases. The reason for this is that if the reset option is not exercised it will expire worthless. The model is then extended to re ect on the characteristics of the range of products which are traded in the market. Firstly, the issuer of the segregated fund contract is allowed to charge a management fee to the policyholder. The effect from incorporating this fee is that the policyholder requires a higher return in order to optimally reset his maturity guarantee while the total value of the segregated fund is diminished. Secondly, the maturity guarantee becomes a function of the number of times that the reset option has been exercised. The effect is that the policyholder requires a higher return in order to choose to reset his maturity guarantee while the total value of the segregated fund is diminished. Thirdly, the policyholder is allowed to reset the maturity guarantee at any point in time within each year from the start of the planning horizon, but only once. The effect is that the total value of the segregated fund is increased since the policyholder may lock in higher market gains as he has more reset decision points. In response to the well documented deficiencies of the lognormal model to capture the jumps experienced by stock markets, extensions were built which incorporate such jumps in the original model. The effect from incorporating such jumps is that the policyholder requires a higher return in order to choose to reset his maturity guarantee while the total value of the segregated fund is diminished due to the adverse effect of the negative jumps on the value of the underlying asset.

The purpose of the current study is to apply delay and probability discounting in areas of treatment options, problem behaviors, and restraint usage with staff members. There was a total of 31 participants that completed three probability and delay discounting surveys either on the computer or by paper/pencil. Before the three surveys, they completed a demographic questionnaire. Participants had to choose from two choice, one that was immediate and the other had a delay in time. Survey one gave a scenario for treatment options, survey two had a scenario for problem behavior management, and then survey three’s scenario was about engaging in restraints. For each of the surveys, the results showed that staff did not engage in discounting. Survey one the AUC scores ranged from 0 to 0.99 (M= 0.77, SD=0.31) with R² value of 0.4156, survey two’s AUC score ranged from 0 to 0.99 (M= 0.54, SD=0.38) with R² value of 0.4356 and survey three’s AUC scores ranged from 0 to 0.99 (M= 0.53, SD=0.40) and R² value of 0.3498. Three different functions were used to show the best fit for the discounting curve, exponential, logarithmic, and polynomial. Overall, the three surveys showed that the participants had a lower level of impulsivity.

L’objet de cette thèse est l’étude de problèmes d’évaluation d’options dans les modèles à volatilité stochastique. La première partie est centrée sur les options américaines dans le modèle de Heston. Nous donnons d’abord une caractérisation analytique de la fonction de valeur d’une option américaine comme l’unique solution du problème d’obstacle parabolique dégénéré associé. Notre approche est basée sur des inéquations variationelles dans des espaces de Sobolev avec poids étendant les résultats récents de Daskalopoulos et Feehan (2011, 2016) et Feehan et Pop (2015). On étudie aussi les propriétés de la fonction de valeur d’une option américaine. En particulier, nous prouvons que, sous des hypothèses convenables sur le payoff, la fonction de valeur est décroissante par rapport à la volatilité. Ensuite nous nous concentrons sur le put américaine et nous étendons quelques résultats qui sont bien connus dans le monde Black-Scholes. En particulier nous prouvons la convexité stricte de la fonction de valeur dans la région de continuation, quelques propriétés de la frontière libre, la formule de Prime d’Exercice Anticipée et une forme faible de la propriété du smooth fit. Les techniques utilisées sont de type probabiliste. Dans la deuxième partie nous abordons le problème du calcul numérique du prix des options européennes et américaines dans des modèles à volatilité stochastiques et avec sauts. Nous étudions d’abord le modèle de Bates-Hull-White, c’est-à-dire le modèle de Bates avec un taux d’intérêt stochastique. On considère un algorithme hybride rétrograde qui utilise une approximation par chaîne de Markov (notamment un arbre “avec sauts multiples”) dans la direction de la volatilité et du taux d’intérêt et une approche (déterministe) par différence finie pour traiter le processus de prix d’actif. De plus, nous fournissons une procédure de simulation pour des évaluations Monte Carlo. Les résultats numériques montrent la fiabilité et l’efficacité de ces méthodes. Finalement, nous analysons le taux de convergence de l’algorithme hybride appliqué à des modèles généraux de diffusion avec sauts. Nous étudions d’abord la convergence faible au premier ordre de chaînes de Markov vers la diffusion sous des hypothèses assez générales. Ensuite nous prouvons la convergence de l’algorithme: nous étudions la stabilité et la consistance de la méthode hybride par une technique qui exploite les caractéristiques probabilistes de l’approximation par chaîne de Markov / We study option pricing problems in stochastic volatility models. In the first part of this thesis we focus on American options in the Heston model. We first give an analytical characterization of the value function of an American option as the unique solution of the associated (degenerate) parabolic obstacle problem. Our approach is based on variational inequalities in suitable weighted Sobolev spaces and extends recent results of Daskalopoulos and Feehan (2011, 2016) and Feehan and Pop (2015). We also investigate the properties of the American value function. In particular, we prove that, under suitable assumptions on the payoff, the value function is nondecreasing with respect to the volatility variable. Then, we focus on an American put option and we extend some results which are well known in the Black and Scholes world. In particular, we prove the strict convexity of the value function in the continuation region, some properties of the free boundary function, the Early Exercise Price formula and a weak form of the smooth fit principle. This is done mostly by using probabilistic techniques.In the second part we deal with the numerical computation of European and American option prices in jump-diffusion stochastic volatility models. We first focus on the Bates-Hull-White model, i.e. the Bates model with a stochastic interest rate. We consider a backward hybrid algorithm which uses a Markov chain approximation (in particular, a “multiple jumps” tree) in the direction of the volatility and the interest rate and a (deterministic) finite-difference approach in order to handle the underlying asset price process. Moreover, we provide a simulation scheme to be used for Monte Carlo evaluations. Numerical results show the reliability and the efficiency of the proposed methods.Finally, we analyze the rate of convergence of the hybrid algorithm applied to general jump-diffusion models. We study first order weak convergence of Markov chains to diffusions under quite general assumptions. Then, we prove the convergence of the algorithm, by studying the stability and the consistency of the hybrid scheme, in a sense that allows us to exploit the probabilistic features of the Markov chain approximation

Presale of residential units refers to putting the units on sale before they are completed. The value of presale to the developer comes from the flexibility of timing the presale launch so as to optimize the expected payoff. We model the developer's optimal launch timing as a real option, and the purchaser's series of presale payments with the flexibility to default as compound options. By assuming a stochastic property price process, we derive model frameworks that a risk-averse developer should adopt in launching the presale under single and multiple payment schemes. The frameworks solve the optimal conditions, contract structures, and prices for the launch. We then extend the model to optimize developers' payoffs in monopolistic and imperfect market equilibria. Finally, by assuming a jump-diffusion demand shock process and based on game theoretic approach, we derive sub-game Nash equilibrium optimal strategies that determine when and at what price developers should launch for presale with stochastic or deterministic rare market events. All the models thus derived are subject to probabilities of purchaser defaults, which will happen if the contract prices are too high when compared to market prices. Our model frameworks confirm that the launch option values increase with increases in price growth rates and variances, but decrease in risk-free rates. Furthermore, developers tend to delay the launch when good events are anticipated, while launching presale earlier at lower prices in times of expected bad events. The equilibrium strategies also provide an alternative explanation to oversupply in property markets. We further illustrate effects of rare events on presale launching strategies through government intervention (particularly public housing and housing subsidies) and output flow uncertainty in competitive equilibrium. Our general optimal strategic models are robust in a few aspects. First, we include the time factor that is crucial for some real options. Second, only slight adjustments are required to cope with market changes, or jumps. Finally, the strategies thus derived can be extensively and flexibly applied to other real options which incur multi-stage contingent payoffs, and whose price processes are characterized by stochastic jump-diffusion process. / Lai Neng. / "October 2000." / Source: Dissertation Abstracts International, Volume: 62-01, Section: A, page: 0270. / Supervisor: Ko Wang. / Thesis (Ph.D.)--Chinese University of Hong Kong, 2000. / Includes bibliographical references (p. 184-192). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Electronic reproduction. Ann Arbor, MI : ProQuest dissertations and theses, [200-] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / School code: 1307.

Choi Tsz Wang. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2004. / Includes bibliographical references (leaves 43-45). / Abstracts in English and Chinese. / Chapter 1 --- Introduction --- p.1 / Chapter 2 --- Review of Structural Models --- p.5 / Chapter 2.1 --- The Merton model --- p.5 / Chapter 2.2 --- The default barrier model of Black and Cox --- p.7 / Chapter 3 --- Estimating the Merton model --- p.10 / Chapter 3.1 --- The Variance Restriction (VR) method --- p.10 / Chapter 3.2 --- The Maximum Likelihood estimation (ML) method --- p.12 / Chapter 3.3 --- Comparison between VR and ML methods --- p.13 / Chapter 4 --- Implications of Using the Proxy in Default Barrier Estimation --- p.15 / Chapter 4.1 --- Rejection of SC framework --- p.16 / Chapter 4.2 --- Positive barrier implication --- p.17 / Chapter 4.3 --- Barier over debt implication --- p.17 / Chapter 4.4 --- Numerical illustration --- p.19 / Chapter 5 --- The Proposed Framework --- p.22 / Chapter 5.1 --- Maximum likelihood estimation --- p.23 / Chapter 5.2 --- Barrier-to-debt ratio specification --- p.25 / Chapter 5.3 --- Simulation checks --- p.26 / Chapter 5.4 --- Comments on the performance of α --- p.29 / Chapter 6 --- Estimation with Empirical Data --- p.33 / Chapter 6.1 --- Description of data --- p.33 / Chapter 6.2 --- Empirical results --- p.35 / Chapter 7 --- Conclusion --- p.41 / References --- p.43

Chan Chun Man. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2005. / Includes bibliographical references (leaves 63-66). / Abstracts in English and Chinese. / Chapter 1 --- Introduction --- p.1 / Chapter 2 --- Volatility Smile and Stochastic Volatility Models --- p.6 / Chapter 2.1 --- Volatility Smile --- p.6 / Chapter 2.2 --- Stochastic Volatility Model --- p.9 / Chapter 2.3 --- Multiscale Stochastic Volatility Model --- p.12 / Chapter 3 --- Lookback Options --- p.14 / Chapter 3.1 --- Lookback Options --- p.14 / Chapter 3.2 --- Lookback Spread Option --- p.15 / Chapter 3.3 --- Dynamic Fund Protection --- p.16 / Chapter 3.4 --- Floating Strike Lookback Options under Black-Scholes Model --- p.17 / Chapter 4 --- Floating Strike Lookback Options under Multiscale Stochastic Volatility Model --- p.21 / Chapter 4.1 --- Multiscale Stochastic Volatility Model --- p.22 / Chapter 4.1.1 --- Model Settings --- p.22 / Chapter 4.1.2 --- Partial Differential Equation for Lookbacks --- p.24 / Chapter 4.2 --- Pricing Lookbacks in Multiscale Asymtoeics --- p.26 / Chapter 4.2.1 --- Fast Tirnescale Asymtotics --- p.28 / Chapter 4.2.2 --- Slow Tirnescale Asymtotics --- p.31 / Chapter 4.2.3 --- Price Approximation --- p.33 / Chapter 4.2.4 --- Estimation of Approximation Errors --- p.36 / Chapter 4.3 --- Floating Strike Lookback Options --- p.37 / Chapter 4.3.1 --- Accuracy for the Price Approximation --- p.39 / Chapter 4.4 --- Calibration --- p.40 / Chapter 5 --- Other Lookback Products --- p.43 / Chapter 5.1 --- Fixed Strike Lookback Options --- p.43 / Chapter 5.2 --- Lookback Spread Option --- p.44 / Chapter 5.3 --- Dynamic Fund Protection --- p.45 / Chapter 6 --- Numerical Results --- p.49 / Chapter 7 --- Conclusion --- p.53 / Appendix --- p.55 / Chapter A --- Verifications --- p.55 / Chapter A.1 --- Formula (4.12) --- p.55 / Chapter A.2 --- Formula (4.22) --- p.56 / Chapter B --- Proof of Proposition --- p.57 / Chapter B.1 --- Proof of Proposition (4.2.2) --- p.57 / Chapter C --- Black-Scholes Greeks for Lookback Options --- p.60 / Bibliography --- p.63

Gao, Fei. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2011. / Includes bibliographical references (leaves 52-53). / Abstracts in English and Chinese. / Chapter 1 --- Introduction --- p.1 / Chapter 1.1 --- Option Pricing with GARCH --- p.1 / Chapter 1.2 --- Mean Reversion in GARCH --- p.3 / Chapter 1.3 --- Thesis Setting --- p.4 / Chapter 2 --- Literature Review --- p.5 / Chapter 2.1 --- GARCH Model --- p.5 / Chapter 2.2 --- Locally Risk-Neutral Valuation --- p.8 / Chapter 2.3 --- Conditional Esscher Transform --- p.9 / Chapter 3 --- The Model --- p.12 / Chapter 3.1 --- The Mean-Reverting GARCH Model --- p.12 / Chapter 3.2 --- The Characteristic Functions --- p.15 / Chapter 3.3 --- Identification of Pricing Measures --- p.21 / Chapter 3.3.1 --- Conditional Esscher Transform --- p.21 / Chapter 3.3.2 --- Our Proposed Change of Measure --- p.25 / Chapter 4 --- Option Pricing --- p.30 / Chapter 4.1 --- Fast Fourier Transform --- p.30 / Chapter 4.2 --- Option on Futures : --- p.32 / Chapter 4.3 --- Numerical Analysis --- p.35 / Chapter 5 --- Empirical Analysis - Application to the crude oil market --- p.37 / Chapter 5.1 --- Description of data --- p.37 / Chapter 5.2 --- Estimation --- p.38 / Chapter 5.3 --- Comparisons --- p.40 / Chapter 6 --- Summary and Future work --- p.42 / Chapter 7 --- Appendix --- p.43 / Bibliography --- p.52

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Bao, Zhenhua. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2008. / Abstracts in English and Chinese. / Chapter 1 --- Introduction --- p.1 / Chapter 2 --- Literature Review --- p.5 / Chapter 2.1 --- Review of the Models --- p.6 / Chapter 2.1.1 --- Black-Scholes-Merton Model --- p.6 / Chapter 2.1.2 --- Merton's Jump Diffusion Model --- p.8 / Chapter 2.1.3 --- Stochastic Volatility Jump Diffusion Model --- p.10 / Chapter 2.1.4 --- Constant Elasticity of Variance (CEV) Model --- p.13 / Chapter 2.2 --- Kou´ةs Double Exponential Jump Diffusion Model --- p.16 / Chapter 2.2.1 --- The Model Formulation --- p.16 / Chapter 2.2.2 --- The Merits of the Model --- p.17 / Chapter 2.2.3 --- Preliminary Results --- p.20 / Chapter 2.2.4 --- Extant Results on Option Pricing under the Model --- p.21 / Chapter 2.3 --- The Laplace Transform and Its Inversion --- p.24 / Chapter 2.3.1 --- The Laplace Transform --- p.24 / Chapter 2.3.2 --- One-dimensional Euler Laplace Transform Inversion Algorithm --- p.25 / Chapter 2.3.3 --- Two-dimensional Euler Laplace Transform Inversion Algorithm --- p.28 / Chapter 2.4 --- Monte Carlo Simulation for Double Exponential Jump Diffusion --- p.32 / Chapter 3 --- Pricing Double Barrier Option via Laplace Transform --- p.34 / Chapter 3.1 --- Double Barrier Option and the First Passage Time --- p.35 / Chapter 3.2 --- Preliminary Results --- p.35 / Chapter 3.3 --- Laplace Transform of the First Passage Time --- p.38 / Chapter 3.4 --- Pricing Double Barrier Option via Laplace Transform --- p.50 / Chapter 4 --- Numerical Results --- p.54 / Chapter 5 --- Conclusion --- p.57

Zheng, Fangbing. / "September 2010." / Thesis (M.Phil.)--Chinese University of Hong Kong, 2010. / Includes bibliographical references (leaves 32-33). / Abstracts in English and Chinese.