The valuation of many types of financial contracts and contingent claim agreements is complicated by the possibility that one party will default on their contractual obligations. This dissertation develops a general model that prices Black-Scholes options subject to intertemporal default risk using stochastic barrier option pricing theory. The explicit closed-form solution is obtained by generalizing the reflection principle to k-space to determine the appropriate transition density function. The European analytical valuation formula has a straightforward economic interpretation and preserves much of the intuitive appeal of the traditional Black-Scholes model. The hedging properties of this model are compared and contrasted with the default-free model. The model is extended to include partial recoveries. In one situation, the option holder is assumed to recover α (a constant) percent of the value of the writer’s assets at the time of default. This version of the partial recovery option leads to an analytical valuation formula for a first passage option - an option with a random payoff at a random time. / Ph. D.

Thesis (MBA)--Stellenbosch University, 2005. / ENGLISH ABSTRACT: This study project focuses on the use of real options valuation in a tactical financing setting. The objective is to identify real option values in financial restructuring situations. These options are generated by the use of hybrid financial instruments such as warrants, preferred stocks and convertibles. In the analysis, it will be demonstrated that the binomial approach, a method commonly used in real options analysis, can be applied to draw a monetary value from specific financial transactions (e.g., leverage buyouts). When used optimally, the binomial approach provides a forceful insight into the dynamics of the transaction. The study recognises the possible impact of capital structure decisions in the analysis, but understates it to avoid complexity. The real options perspective encourages a conscious search for monetary benefits and thus improves the decision-making of managers involved in financial restructuring operations. / AFRIKAANSE OPSOMMING: Hierde werkstuk fokus op die gebruik van rieëIe opsie teorie om taktiese finansieringsbesluitneming te evalueer. Opsies word gegenereer deur die gebruik van hibridiese finansiele instrumente soos bestuursopsie-orders, voorkeuraandele en omskepbare instrumente. In hierdie studie word 'n oorsig oor die teorie soos dit in literatuur verskyn gegee, asook voorbeelde van finansiele herstrukturering om die waarde van die toepassing daarvan te illustreer. In hierdie studie word erkenning gegee aan die moontlike impak wat kapitaalstruktuur-besluitneming op die ontleding mag hê. Die impak hiervan word egter weens die kompleksiteit daarvan ignoreer. Nieteenstaande hierdie beperking, word besluitneming rakende finansiele herstrukturering verbeter deur die perspektief wat deur die rieëIe opsie-benadering verkry word, soos in hierdie werkstuk uitgewys word.

Thesis (MSc)--Stellenbosch University, 2015 / AFRIKAANSE OPSOMMING : Die groeiende belangstelling in kalibrering en modelrisiko is ’n redelik resente ontwikkeling in finansiële wiskunde. Hierdie proefskrif fokusseer op hierdie sake, veral in verband met die prysbepaling van vanielje-en eksotiese opsies, en vergelyk die prestasie van verskeie Lévy modelle. ’n Nuwe metode om modelrisiko te meet word ook voorgestel (hoofstuk 6). Ons kalibreer eers verskeie Lévy modelle aan die log-opbrengs van die S&P500 indeks. Statistiese toetse en grafieke voorstellings toon albei aan dat suiwer sprongmodelle (VG, NIG en CGMY) die verdeling van die opbrengs beter beskryf as die Black-Scholes model. Daarna kalibreer ons hierdie vier modelle aan S&P500 indeks opsie data en ook aan "CGMY-wˆ ereld" data (’n gesimuleerde wÃłreld wat beskryf word deur die CGMY-model) met behulp van die wortel van gemiddelde kwadraat fout. Die CGMY model vaar beter as die VG, NIG en Black-Scholes modelle. Ons waarneem ook ’n effense verskil tussen die nuwe parameters van CGMY model en sy wisselende parameters, ten spyte van die feit dat CGMY model gekalibreer is aan die "CGMYwêreld" data. Versperrings-en terugblik opsies word daarna geprys, deur gebruik te maak van die gekalibreerde parameters vir ons modelle. Hierdie pryse word dan vergelyk met die "ware" pryse (bereken met die ware parameters van die "CGMY-wêreld), en ’n beduidende verskil tussen die modelpryse en die "ware" pryse word waargeneem. Ons eindig met ’n poging om hierdie modelrisiko te kwantiseer / ENGLISH ABSTRACT : The growing interest in calibration and model risk is a fairly recent development in financial mathematics. This thesis focussing on these issues, particularly in relation to the pricing of vanilla and exotic options, and compare the performance of various Lévy models. A new method to measure model risk is also proposed (Chapter 6). We calibrate only several Lévy models to the log-return of S&P500 index data. Statistical tests and graphs representations both show that pure jump models (VG, NIG and CGMY) the distribution of the proceeds better described as the Black-Scholes model. Then we calibrate these four models to the S&P500 index option data and also to "CGMY-world" data (a simulated world described by the CGMY model) using the root mean square error. Which CGMY model outperform VG, NIG and Black-Scholes models. We observe also a slight difference between the new parameters of CGMY model and its varying parameters, despite the fact that CGMY model is calibrated to the "CGMY-world" data. Barriers and lookback options are then priced, making use of the calibrated parameters for our models. These prices are then compared with the "real" prices (calculated with the true parameters of the "CGMY world), and a significant difference between the model prices and the "real" rates are observed. We end with an attempt to quantization this model risk.

Made available in DSpace on 2015-03-04T18:57:53Z (GMT). No. of bitstreams: 1 Estevao_MSc2013.pdf: 798745 bytes, checksum: 3a737b306d09e12a7beaa6d3f396c369 (MD5) Previous issue date: 2013-06-17 / Conselho Nacional de Desenvolvimento Cientifico e Tecnologico / We address the theory of no-arbitrage pricing of derivatives and hedging strategies. The continuous-time model of the underlying stock price that we consider is the Geometric Brownian Motion, which parameters (mean rate of return and volatility) are initially set as stochastic processes and, in the sequel, specified as deterministic functions of time or constant values. With a view to providing self-sufficiency for the text, we have included the necessary fundamental theory and derived the Partial Differential Equation (PDE) for the price of derivatives with payoffs that are functions of the time and of the stock price at maturity, while the stock is now governed by the local volatility model (in which the parameters are functions of time and of the stock price at each moment). Focusing the particular niche where parameters are, except for very mild constraints, arbitrary deterministic functions of time, we develop explicit formulae for both the price and the hedging strategy for an European call option, as well as the particular shape of the associated PDEs. The generalization of the above scenario corresponds to the main result of this thesis which, to the best of our knowledge, is new: we assume (as above) the model where parameters are arbitrary deterministic functions of time and an European call option with a moving barrier of a special sort - which we name discounted barrier. Still, we obtain explicit formulas for both the exact price and hedging strategy. The shape of the barrier option under consideration is attractive from the point of view of the dealer, since it is in fact constant if tested against the discounted risky asset price. Moreover, the riskless asset - which accounts for discounting - is the dealers reference for profit evaluation. Some tools employed in this work are the risk-neutral (or martingale) measure and an extension of the Reflection Principle for Brownian Motion. / Nós abordamos a teoria de preços livres de arbitragem de derivativos e estratégias de hedging. O modelo a tempo contínuo que consideramos para o preço das ações é o Movimento Browniano Geométrico, cujos parâmetros (taxa média de retorno e volatilidade) são inicialmente definidos como processos estocásticos, para daí serem especificados por funções determinísticas do tempo ou valores constantes. Com vistas a dar um cunho autossuficiente à dissertação, desenvolvemos a teoria de base e a Equação Diferencial Parcial (EDP) para o preço de derivativos cujos payoffs são funções do tempo e do preço da ação, ambos na expiração, enquanto que a ação é governada pelo modelo de volatilidade local (no qual os parâmetros são funções do tempo e do preço da ação a cada instante). No caso particular onde os parâmetros são, salvo restrições brandas, funções determinísticas arbitrárias do tempo, desenvolvemos fórmulas explícitas para o preço e para a estratégia de hedging para uma opção de compra Europeia, bem como a forma particular das EDPs associadas. A generalização do cenário acima constitui o resultado principal desta dissertação, novo na literatura: assumimos (como acima) o modelo onde os parâmetros são funções determinísticas arbitrárias do tempo e uma opção de compra Europeia com uma barreira móvel de um tipo específico - a qual chamamos barreira descontada. Ainda assim, obtemos fórmulas explícitas tanto para o preço quanto para a estratégia de hedging. O formato da barreira móvel considerada é atrativo do ponto de vista prático de mercado, uma vez que é, de fato, constante se testada contra o preço descontado do ativo de risco. Ademais, é em relação ao ativo sem risco - que dita o desconto - que os dealers aferem seus lucros. Algumas ferramentas empregadas neste trabalho são a medida risco-neutro (medida martingale) e uma extensão do Princípio da Reflexão para o Movimento Browniano.

by Law Kwok Fu, Frank. / Thesis (M.B.A.)--Chinese University of Hong Kong, 1992. / Includes bibliographical references (leaf 34). / ACKNOWLEDGEMENT --- p.2 / Chapter 1 . --- INTRODUCTION --- p.3 / Chapter 2. --- SPOT CONTRACT --- p.7 / Chapter 3. --- FORWARD CONTRACT --- p.8 / Chapter 4. --- CURRENCY FUTURES --- p.10 / Chapter 5. --- CURRENCY OPTIONS --- p.14 / Chapter 6. --- CURRENCY WARRANTS --- p.17 / Chapter 7. --- CONCLUSION. --- p.32 / BIBLIOGRAPHY / APPENDIX / Chapter 1. --- Some of currency futures and options listed in overseas exchanges --- p.35 / Chapter 2. --- Details of currency warrants available in the market --- p.37 / Chapter 3 . --- Raw data --- p.38 / Chapter 4. --- Graphs of the DM spot rate and the daily price movements of 3 warrants --- p.41 / Chapter 5-7. --- The relative daily change in DM spot rate in % against the daily change in price of the 3 DM warrants --- p.45

Chu Chun-fai, Carlin. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2001. / Includes bibliographical references (leaves 69-73). / Abstracts in English and Chinese. / Chapter Part I: --- INTRODUCTION --- p.1 / Chapter Part II: --- LITERATURE REVIEW --- p.4 / Chapter - --- Financial option-pricing theory / Chapter - --- Real option-pricing theory / Chapter - --- Real option-pricing theory in Management Information System Area / Chapter Part III: --- CASE BACKGROUND --- p.14 / Chapter - --- Case Background / Chapter - --- Availability of online banking services in Hong Kong / Chapter - --- Online banking investment in the Hong Kong Chinese Bank / Chapter Part IV: --- RESEARCH MODEL --- p.19 / Chapter - --- Research model / Chapter - --- Modelling of the optimal timing problem of HKCB / Chapter - --- Justification of geometric Brownian motion assumption for using Black-Scholes formula / Chapter Part V : --- DATA COLLECTION --- p.30 / Chapter Part VI: --- ANALYSIS RESULT --- p.35 / Chapter - --- Analysis result / Chapter - --- Sensitivity analysis on the selected parameters / Chapter - --- Suggested investment timing / Chapter Part VII: --- DISCUSSIONS AND IMPLICATIONS --- p.44 / Chapter - --- Result discussion / Chapter - --- Implications for researchers / Chapter - --- Implications for practitioners / Chapter Part VIII: --- LIMITATIONS AND CONTRIBUTIONS --- p.48 / Chapter - --- Limitation on data collection process / Chapter - --- Limitations on Black-Scholes model / Chapter - --- Contributions / APPENDIX / Appendix A -Limitation of traditional Discounted Cash Flow analysis --- p.51 / Appendix B -Banks services available to the customers --- p.54 / Appendix C -Sample path of a Geometric Brownian Motion --- p.56 / Appendix D -Discounted Cash Flows analysis of immediate entry of online banking investment --- p.57 / Appendix E -Black-Scholes formula and its interpretation for non-traded --- p.61 / Appendix F -Questionnaire for Online banking investment --- p.64 / Appendix G -Availability of online banking services in May 2001 --- p.67 / Appendix H -Sensitivity analysis on the number of initial usage --- p.68 / Appendix I -Reference List --- p.69

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.

This thesis is concerned with the small-time asymptotics and expansions of call option prices, when the log-return processes of the underlying stock prices follow several Levy-based models. To be specific, we derive the time-to-maturity asymptotic behavior for both at-the-money (ATM), out-of-the-money (OTM) and in-the-money (ITM) call-option prices under several jump-diffusion models and stochastic volatility models with Levy jumps. In the OTM and ITM cases, we consider a general stochastic volatility model with independent Levy jumps, while in the ATM case, we consider the pure-jump CGMY model with or without an independent Brownian component. An accurate modeling of the option market and asset prices requires a mixture of a continuous diffusive component and a jump component. In this thesis, we first model the log-return process of a risk asset with a jump diffusion model by combining a stochastic volatility model with an independent pure-jump Levy process. By assuming smoothness conditions on the Levy density away from the origin and a small-time large deviation principle on the stochastic volatility model, we derive the small-time expansions, of arbitrary polynomial order, in time-t, for the tail distribution of the log-return process, and for the call-option price which is not at-the-money. Moreover, our approach allows for a unified treatment of more general payoff functions. As a consequence of our tail expansions, the polynomial expansion in t of the transition density is also obtained under mild conditions. The asymptotic behavior of the ATM call-option prices is more complicated to obtain, and, in general, is given by fractional powers of t, which depends on different choices of the underlying log-return models. Here, we focus on the CGMY model, one of the most popular tempered stable models used in financial modeling. A novel second-order approximation for ATM option prices under the pure-jump CGMY Levy model is derived, and then extended to a model with an additional independent Brownian component. The third-order asymptotic behavior of the ATM option prices as well as the asymptotic behavior of the corresponding Black-Scholes implied volatilities are also addressed.

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

This dissertation contains two topics in operations research. The first topic is to design a distribution network to facilitate the repeated movement of shipments from many origins to many destinations. A method is developed to estimate transportation costs as a function of the number of terminals and moreover to determine the best number of terminals. The second topic is to study dynamics of a buyer's behavior when the buyer can buy goods through both option contracts and a spot market and the buyer attempts to learn the probability distribution of the spot price. The buyer estimates the spot price distribution as though it is exogenous. However, the spot price distribution is not exogenous but is endogenous because it is affected by the buyer's decision regarding option purchases.