General diffusion processes (GDP), or Ito's processes, are potential candidates for the modeling of asset prices, interest rates and other financial quantities to cope with empirical evidence. This thesis considers the applications of general diffusions in finance and potential extensions. In particular, we focus on financial problems involving (optimal) stopping times. A typical example is the valuation of American options. We investigate the use of Laplace-Carson transform (LCT) in valuing American options, and discuss its strengthen and weaknesses. Homotopy analysis from topology is then introduced to derive closed-form American option pricing formulas under GDP. Another example is taken from optimal dividend policies with bankruptcy procedures, which is closely related to excursion time and occupation time of a general diffusion. With the aid of Fourier transform, we further extend the analysis to the case of multi-dimensional GDP by considering the currency option pricing with mean reversion and multi-scale stochastic volatility. / Zhao, Jing. / Adviser: Hoi-Ying Wong. / Source: Dissertation Abstracts International, Volume: 72-04, Section: B, page: . / Thesis (Ph.D.)--Chinese University of Hong Kong, 2010. / Includes bibliographical references (leaves 97-105). / 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 Information and Learning Company, [200-] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstract also in Chinese.

When pricing American-style options on d assets by Monte Carlo methods, one usually stores the simulated asset prices at all time steps on all paths in order to determine when to exercise the options. If N time steps and M paths are used; then the storage requirement is d · M · N. In this thesis, we give two simulation methods to price multi-asset American-style options, where the storage requirement only grows like (d + 1)M + N. The only additional computational cost is that we have to generate each random number twice instead of once. For machines with limited memory, we can now use larger values of M and N to improve the accuracy in pricing the options. / by Wong Chi Yan. / Adviser: Raymond H. Chan. / Source: Dissertation Abstracts International, Volume: 70-03, Section: B, page: 1708. / Thesis (Ph.D.)--Chinese University of Hong Kong, 2008. / Includes bibliographical references (leaves 79-82). / 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 Information and Learning, [200-] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstracts in English and Chinese. / School code: 1307.

by Veronica Ho Pui Kwan. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1997. / Includes bibliographical references (leaves 99-103). / Prologue --- p.1 / Chapter Essay I: --- Examination of the GARCH Option Pricing Model in the case of Hang Seng Index Option / Chapter 1. --- Introduction --- p.4 / Chapter 2. --- Holes' in the Black-Scholes Model --- p.7 / Chapter 3. --- A Big 'Hole' -- Varying Volatility --- p.14 / Chapter 4. --- A Remedy : the GARCH Option Pricing Model --- p.31 / Chapter 5. --- Research Methodology and Data --- p.38 / Chapter 6. --- Empirical Results --- p.50 / Chapter 7. --- Conclusion --- p.67 / Chapter Essay II: --- Barrier Options / Chapter 1. --- Introduction on Barrier Option --- p.70 / Chapter 2. --- Pricing Models --- p.74 / Chapter 3. --- Hedging of Barrier Option --- p.81 / Chapter 4. --- Examination of a Down-and-Out Put Option --- p.88 / References --- p.99

Thesis (MSc)--Stellenbosch University, 2013. / ENGLISH ABSTRACT: We study Monte Carlo methods for pricing American options where the stock price dynamics follow exponential pure jump L évy models. Only stock price dynamics for a single underlying are considered. The thesis begins with a general introduction to American Monte Carlo methods. We then consider two classes of these methods. The fi rst class involves regression - we briefly consider the regression method of Tsitsiklis and Van Roy [2001] and analyse in detail the least squares Monte Carlo method of Longsta and Schwartz [2001]. The variance reduction techniques of Rasmussen [2005] applicable to the least squares Monte Carlo method, are also considered. The stochastic mesh method of Broadie and Glasserman [2004] falls into the second class we study. Furthermore, we consider the dual method, independently studied by Andersen and Broadie [2004], Rogers [2002] and Haugh and Kogan [March 2004] which generates a high bias estimate from a stopping rule. The rules we consider are estimates of the boundary between the continuation and exercise regions of the option. We analyse in detail how to obtain such an estimate in the least squares Monte Carlo and stochastic mesh methods. These models are implemented using both a pseudo-random number generator, and the preferred choice of a quasi-random number generator with bridge sampling. As a base case, these methods are implemented where the stock price process follows geometric Brownian motion. However the focus of the thesis is to implement the Monte Carlo methods for two pure jump L évy models, namely the variance gamma and the normal inverse Gaussian models. We first provide a broad discussion on some of the properties of L évy processes, followed by a study of the variance gamma model of Madan et al. [1998] and the normal inverse Gaussian model of Barndor -Nielsen [1995]. We also provide an implementation of a variation of the calibration procedure of Cont and Tankov [2004b] for these models. We conclude with an analysis of results obtained from pricing American options using these models. / AFRIKAANSE OPSOMMING: Ons bestudeer Monte Carlo metodes wat Amerikaanse opsies, waar die aandeleprys dinamika die patroon van die eksponensiële suiwer sprong L évy modelle volg, prys. Ons neem slegs aandeleprys dinamika vir 'n enkele aandeel in ag. Die tesis begin met 'n algemene inleiding tot Amerikaanse Monte Carlo metodes. Daarna bestudeer ons twee klasse metodes. Die eerste behels regressie - ons bestudeer die regressiemetode van Tsitsiklis and Van Roy [2001] vlugtig en analiseer die least squares Monte Carlo metode van Longsta and Schwartz [2001] in detail. Ons gee ook aandag aan die variansie reduksie tegnieke van Rasmussen [2005] wat van toepassing is op die least squares Monte Carlo metodes. Die stochastic mesh metode van Broadie and Glasserman [2004] val in die tweede klas wat ons onder oë neem. Ons sal ook aandag gee aan die dual metode, wat 'n hoë bias skatting van 'n stop reël skep, en afsonderlik deur Andersen and Broadie [2004], Rogers [2002] and Haugh and Kogan [March 2004] bestudeer is. Die reëls wat ons bestudeer is skattings van die grense tussen die voortsettings- en oefenareas van die opsie. Ons analiseer in detail hoe om so 'n benadering in die least squares Monte Carlo en stochastic mesh metodes te verkry. Hierdie modelle word geï mplementeer deur beide die pseudo kansgetalgenerator en die verkose beste quasi kansgetalgenerator met brug steekproefneming te gebruik. As 'n basisgeval word hierdie metodes geï mplimenteer wanneer die aandeleprysproses 'n geometriese Browniese beweging volg. Die fokus van die tesis is om die Monte Carlo metodes vir twee suiwer sprong L évy modelle, naamlik die variance gamma en die normal inverse Gaussian modelle, te implimenteer. Eers bespreek ons in breë trekke sommige van die eienskappe van L évy prossesse en vervolgens bestudeer ons die variance gamma model soos in Madan et al. [1998] en die normal inverse Gaussian model soos in Barndor -Nielsen [1995]. Ons gee ook 'n implimentering van 'n variasie van die kalibreringsprosedure deur Cont and Tankov [2004b] vir hierdie modelle. Ons sluit af met die resultate wat verkry is, deur Amerikaanse opsies met behulp van hierdie modelle te prys.

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

Thesis (MBA)--Stellenbosch University, 2002. / Some digitised pages may appear illegible due to the condition of the original hard copy / ENGLISH ABSTRACT: With the trends of business moving away from large, corporate companies to small, flexible and innovative alternatives, the need to value new companies are becoming important. A new company generally does not have substantial historical data available and it is therefore difficult to determine potential revenue streams and hence accurate valuations. The focus of this study is to find an appropriate method to attempt the valuation of a new company and this is explained by means of a case study. Three basic approaches exist to value companies. The Discounted Cash Flow (DCF) method analyses risk and return to estimate a discount rate and presents the value of the company as a Net Present Value (NPV). Relative Valuation methods compare the fundamentals of a company to that of other companies. Contingent Claim Valuation methods base the value of a company on the fact that decisions may be deferred into the future until more information is evident. The basis of this valuation technique is that of Option Pricing Theory in which the Black-Scholes technique and binomial models are used .: This method is normally used on assets that have optionlike features e.g. equity in a company, natural resource rights, product patents or any decision that may be deferred into the future. Decisions (options) deferred may be identified as growth-, staged-, flexibility-, exit-, learning- and expanding options. This is also known as the Real Option Pricing Theory. According to this model the investment proposal may be mapped as a series of call options (Luehrman, 1998a). The amount of money expended in the project corresponds to the option's exercise price (X), the present value of the asset built or acquired corresponds to the stock price (S), the length of time the company can defer the investment decision corresponds to the option's time to expiration (t) and the uncertainty about the future value of the project's cashflow corresponds to the standard deviation of return on the stock (c). Seven steps are used to obtain the value of the call option and the value is reflected by two option-value metries namely the value-to-cost (NPVq) and cumulative volatility (cr--Jt).The two metries are plotteá on a graph (defined as Options Space) in order to visualize and interpret the results. Mushroom Biomedical Systems developed three highly novel and patented products. The company was valued using the conventional OeF method and valued as a staged investment using the Real Option Pricing Theory according to Luehrman's model (1998a). The values of two products are similar using the OeF and Real Options methods. Most of the investment capital was required during the first phases of these products resulting in the investment of the second phases not holding high risks or value. The value of the third product is significantly higher using the Real Options method compared to the OeF. This is ascribed to the forced delay of phase one. The value of this future decision is worth more than the current decision due to expected new information that might arise. By "creating an option" value is added by forcing management to actively make two decisions about the continuation of the project at a future date. Applying Real Option Pricing Theory suggests inherent value in uncertainty when there is freedom to choose different courses of action in the face of different market conditions. With the OeF analysis the impact of risk is seen as depressing the value of the investment. By contrast, real options show that risk can be influenced through managerial flexibility, which becomes a central instrument to create value. / AFRIKAANSE OPSOMMING: Die beweging van die besigheidswêreld vanaf groot korporatiewe maatskappye na kleiner, buigsame en innoverende alternatiewe het 'n behoefte geskep om die waarde van sulke nuwe maatskappye te kan bepaal. 'n Nuwe maatskappy het tipies nie historiese data beskikbaar nie wat die vooruitskatting van potensiële inkomste strome en dus akkurate waardasies moeilik maak. Die fokus van hierdie studie is die bepaling van 'n toepaslike metode om die waarde van 'n nuwe maatskappy te bepaal en dit word deur middel van 'n gevalle studie verduidelik. Drie basiese metodes bestaan om maatskappye te waardeer. Die Verdiskonteerde Kontantvloei Stroom (VKS) metode gebruik risiko en opbrengs om 'n verdiskonteringskoers te bepaal en reflekteer die waarde van die maatskappy as die Netto Teenswoordige Waarde (NTW). Relatiewe Waardasie metodes vergelyk die fundamentele eienskappe van 'n maatskappy met die van ander maatskappye. Die Gebeurlikheids Waardasie metode koppel waarde aan die feit dat besluite uitgestel kan word totdat meer informasie beskikbaar is. Die basis van hierdie tegniek is Opsie Teorie waarin die Black-Scholes tegniek en binomiaal model gebruik word. Hierdie metode word gewoonlik gebruik waar bates "opsie-tipe" eienskappe besit soos aandeelhouding in 'n maatskappy, natuurlike mynregte; produk patente of enige besluit wat uitgestel kan word na 'n datum in die toekoms. Besluite (opsies) wat uitgestel word kan geïdentifiseer word as groei-, stap-vir-stap-, buigbaarheids-, uittree-, lerings- en uitbreidingsopsies. Hierdie metode staan ook bekend as die Ware Opsie Prysings Teorie. Volgens hierdie metode kan 'n beleggingsgeleentheid voorgestel word as 'n reeks koopopsies (Luehrman, 1998a). Die totale uitgawe word voorgestel deur die uitoefeningsprys (X), die teenswoordige waarde van die bate word voorgestel deur die aandeel waarde (S), die tydperk wat die besluit uitgestel kan word, word voorgestel deur die opsie vervaltyd (t), en die onsekerheid van die bate se kontantvloeistroom word voorgestel deur die standaardafwyking van die opbrengs van die bate (c). Sewe stappe word geneem om die waarde van die koopopsie te bepaal wat uitgedruk word deur twee opsiewaarde komponente naamlik waarde-tot-koste (NPVq) en kummulatiewe volatiliteit ((1'Jt). Die twee komponente word grafies voorgestel (genoem Opsie Spasie) om resultate te visualiseer en te interpreteer. Mushroom Biomedical Systems het drie unieke en gepatenteerde produkte ontwikkel. Die maatskappy is met die konvensionele VKS metode gewaardeer en volgens Luehrman (1998a) se Ware Opsie Prysings model as 'n stap-vir-stap opsie gewaardeer. Die waardes van twee van die produkte is dieselfde met die VKS metode en die Opsie Teorie metode. Die meeste van die kapitaal is tydens die eerste fases van die twee produkte benodig met die gevolg dat die tweede fases nie veel risiko of waarde inhou nie. Die waarde van die derde produk is aansienlik meer met die Opsie Teorie metode in vergelyking met die VKS metode. Dit word toegeskryf aan die gedwonge vertraging van fase een. Die waarde gekoppel daaraan om die besluit in die toekoms te neem is meer werd as om die besluit nou te neem a.g.v. verwagte nuwe informasie. Deur hierdie opsie "te skep" word waarde toegevoeg omdat bestuur gedwing word om aktief twee besluite in die toekoms te neem rakende die voortsetting van die projek. Die gebruik van Ware Opsie Prysings Teorie skep 'n inherente waarde wanneer daar verskillende besluite geneem kan word soos mark kondisies verander. Met die VKS metode word risiko gesien as 'n faktor wat waarde laat afneem. In teenstelling hiermee dui die Ware Opsie Teorie dat risiko beïnvloed kan word deur bestuur se vermoëns, wat 'n belangrike instrument is vir waardeskepping.

published_or_final_version / Statistics and Actuarial Science / Doctoral / Doctor of Philosophy

published_or_final_version / abstract / Real Estate and Construction / Doctoral / Doctor of Philosophy

常方差彈性(CEV) 模型能夠刻畫波動率微笑的優點使之成為期權定價中的實用工具，然而它在應用到美式衍生工具時面臨分析上及計算上的挑戰。現行的解析方法是對代表著期權價格函數和其最佳履約曲線的自由邊界問題進行拉普拉斯卡森變換(LCT) ，繼而獲得在此變換下的解析解，可是此解含有合流超線幾何函數，使得它的數值計算在某些參數下顯得不穩定及低效。本文運用漸近法徹底解決美式期權在常方差彈性模型下的定價問題，並用永久性和限時性的美式看跌期權作為例子闡述所提出的方法。 / The constant elasticity of variance (CEV) model is a practical approach to option pricing by fitting to the implied volatility skew. Its application to American-style derivatives, however, poses analytical and numerical challenges. By taking the Laplace Carson transform (LCT) to the free-boundary value problem characterizing the option value function and the early exercise boundary, the analytical result involves confluent hyper-geometric functions. Thus, the numerical computation could be unstable and inefficient for certain set of parameter values. We solve this problem by an asymptotic approach to the American option pricing problem under the CEV model. We demonstrate the use of the proposed approach using perpetual and finite-time American puts. / Detailed summary in vernacular field only. / Pun, Chi Seng. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2013. / Includes bibliographical references (leaves 39-40). / Abstracts also in Chinese. / Chapter 1 --- Introduction --- p.1 / Chapter 2 --- Problem Formulation --- p.4 / Chapter 2.1 --- The CEV model --- p.4 / Chapter 2.2 --- The free-boundary value problem --- p.5 / Chapter 2.2.1 --- Perpetual American put --- p.5 / Chapter 2.2.2 --- Finite-time American put --- p.6 / Chapter 3 --- Asymptotic expansion of American put --- p.8 / Chapter 3.1 --- Perpetual American put --- p.8 / Chapter 3.2 --- Finite-time American put --- p.16 / Chapter 4 --- Numerical examples --- p.24 / Chapter 4.1 --- Perpetual American put --- p.24 / Chapter 4.2 --- Finite-time American put --- p.26 / Chapter 5 --- Conclusion --- p.29 / Chapter A --- Proof of Lemma 3.1 --- p.30 / Chapter B --- Property of ak --- p.32 / Chapter C --- Explicit formulas for u₂(S) --- p.34 / Chapter D --- Closed-form solutions --- p.37 / Bibliography --- p.40

Ng Chi-Tim. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2004. / Includes bibliographical references (leaves 110-114). / Abstracts in English and Chinese. / Chapter Chapter 1 --- Introduction --- p.4 / Chapter Chapter 2 --- Mathematical Background --- p.7 / Chapter 2.1 --- Fractional Stochastic Integral --- p.8 / Chapter 2.2 --- Wick's Calculus --- p.9 / Chapter 2.3 --- Malliavin Calculus --- p.19 / Chapter 2.4 --- Fractional Ito's Lemma --- p.27 / Chapter Chapter 3 --- The Fractional Black Scholes Model --- p.34 / Chapter 3.1 --- Fractional Geometric Brownian Motion --- p.35 / Chapter 3.2 --- Arbitrage Opportunities --- p.38 / Chapter 3.3 --- Fractional Black Scholes Equation --- p.40 / Chapter Chapter 4 --- Generalization --- p.43 / Chapter 4.1 --- Stochastic Gradients of Fractional Diffusion Processes --- p.44 / Chapter 4.2 --- An Example : Fractional Black Scholes Mdel with Varying Trend and Volatility --- p.46 / Chapter 4.3 --- Generalization of Fractional Black Scholes PDE --- p.48 / Chapter 4.4 --- Option Pricing Problem for Fractional Black Scholes Model with Varying Trend and Volatility --- p.55 / Chapter Chapter 5 --- Alternative Fractional Models --- p.59 / Chapter 5.1 --- Fractional Constant Elasticity Volatility (CEV) Models --- p.60 / Chapter 5.2 --- Pricing an European Call Option --- p.61 / Chapter Chapter 6 --- Problems in Fractional Models --- p.66 / Chapter Chapter 7 --- Arbitrage Opportunities --- p.68 / Chapter 7.1 --- Two Equivalent Expressions for Geometric Brownian Motions --- p.69 / Chapter 7.2 --- Self-financing Strategies --- p.70 / Chapter Chapter 8 --- Conclusions --- p.72 / Chapter Appendix A --- Fractional Stochastic Integral for Deterministic Integrand --- p.75 / Chapter A.1 --- Mapping from Inner-Product Space to a Set of Random Variables --- p.76 / Chapter A.2 --- Fractional Calculus --- p.77 / Chapter A.3 --- Spaces for Deterministic Functions --- p.79 / Chapter Appendix B --- Three Approaches of Stochastic Integration --- p.82 / Chapter B.1 --- S-Transformation Approach --- p.84 / Chapter B.2 --- Relationship between Three Types of Stochastic Integral --- p.89 / Reference --- p.90