Assignment (MComm)--University of Stellenbosch, 2007. / Bibliography. Also available via the Internet.

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

An Asian option is an example of exotic options. Its payoff depends on the average of the underlying asset prices. The average may be over the entire time period between initiation and expiration or may be over some period of time that begins later than the initiation of the option and ends with the options expiration. The average may be from continuous sampling or may be from discrete sampling. The primary reason to base an option payoff on an average asset price is to make it more difficult for anyone to significantly affect the payoff by manipulation of the underlying asset price. The price of Asian options is not known in closed form, in general, if the arithmetic average is taken into effect. In this dissertation, we shall investigate the pricing theory for Asian options. After a brief introduction to the Black-Scholes theory, we derive the partial differential equations for the value process of an Asian option to satisfy. We do this in several approaches, including the usual extension to Asian options of the Black-Scholes, and the sophisticated martingale approach. Both fixed and floating strike are considered. In the case of the geometric average, we derive a closed form solution for the Asian option. Moreover, we investigate the Asian option price theory under stochastic volatility which is a recent trend in the study of path-dependent option theory. / Thesis (M.Sc.)-University of KwaZulu-Natal, Westville, 2007.

Thesis (Ph. D.)--Hong Kong University of Science and Technology, 2004. / Includes bibliographical references (leaves 120-126). Also available in electronic version. Access restricted to campus users.

Exotic options are derivatives which have features that makes them more complex than vanilla traded products. Thus, finding their fair value is not always an easy task. We look at a particular example of the exotic options - the power option - whose payoffs are nonlinear functions of the underlying asset price. Previous analyses of the power option have only obtained solutions using probability methods for the case of the constant stock volatility and interest rate. Using Lie symmetry analysis we obtain an optimal system of the Lie point symmetries of the power option PDE and demonstrate an algorithmic method for finding solutions to the equation. In addition, we find a new analytical solution to the asymmetric type of the power option. We also focus on the more practical and realistic case of time dependent parameters: volatility and interest rate. Utilizing Lie symmetries, we are able to provide a new exact solution for the terminal pay off case. We also consider the power parameter of the option as a time dependent factor. A new solution is once again obtained for this scenario. / Thesis (Ph.D.)-University of KwaZulu-Natal, Durban, 2013.

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

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.

Thesis (MComm (Statistics and Actuarial Science))--University of Stellenbosch, 2007. / The use of options on various stock markets over the world has introduced a unique opportunity for investors to hedge, speculate, create synthetic financial instruments and reduce funding and other costs in their trading strategies. The power of options lies in their versatility. They enable an investor to adapt or adjust her position according to any situation that arises. Another benefit of using options is that they provide leverage. Since options cost less than stock, they provide a high-leverage approach to trading that can significantly limit the overall risk of a trade, or provide additional income. This versatility and leverage, however, come at a price. Options are complex securities and can be extremely risky. In this document several aspects of trading and valuing some exotic options are investigated. The aim is to give insight into their uses and the risks involved in their trading. Two volatility-dependent derivatives, namely compound and chooser options; two path-dependent derivatives, namely barrier and Asian options; and lastly binary options, are discussed in detail. The purpose of this study is to provide a reference that contains both the mathematical derivations and detail in valuating these exotic options, as well as an overview of their applicability and use for students and other interested parties.