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

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

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

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

Thesis (PhD)--Macquarie University, Division of Economic and Financial Studies, Dept. of Statistics, 2008. / Bibliography: p. 73-77. / Introduction -- Lévy processes used in option pricing -- Option pricing for Lévy processes -- Option pricing based on empirical characteristic functions -- Performance of the five models on historical data -- Conclusions -- References -- Appendix A. Proofs -- Appendix B. Supplements -- Appendix C. Matlab programs. / Pricing problems of financial derivatives are among the most important ones in Quantitative Finance. Since 1973 when a Nobel prize winning model was introduced by Black, Merton and Scholes the Brownian Motion (BM) process gained huge attention of professionals professionals. It is now known, however, that stock market log-returns do not follow the very popular BM process. Derivative pricing models which are based on more general Lévy processes tend to perform better. --Carr & Madan (1999) and Lewis (2001) (CML) developed a method for vanilla options valuation based on a characteristic function of asset log-returns assuming that they follow a Lévy process. Assuming that at least part of the problem is in adequate modeling of the distribution of log-returns of the underlying price process, we use instead a nonparametric approach in the CML formula and replaced the unknown characteristic function with its empirical version, the Empirical Characteristic Functions (ECF). We consider four modifications of this model based on the ECF. The first modification requires only historical log-returns of the underlying price process. The other three modifications of the model need, in addition, a calibration based on historical option prices. We compare their performance based on the historical data of the DAX index and on ODAX options written on the index between the 1st of June 2006 and the 17th of May 2007. The resulting pricing errors show that one of our models performs, at least in the cases considered in the project, better than the Carr & Madan (1999) model based on calibration of a parametric Lévy model, called a VG model. --Our study seems to confirm a necessity of using implied parameters, apart from an adequate modeling of the probability distribution of the asset log-returns. It indicates that to precisely reproduce behaviour of the real option prices yet other factors like stochastic volatility need to be included in the option pricing model. Fortunately the discrepancies between our model and real option prices are reduced by introducing the implied parameters which seem to be easily modeled and forecasted using a mixture of regression and time series models. Such approach is computationaly less expensive than the explicit modeling of the stochastic volatility like in the Heston (1993) model and its modifications. / Mode of access: World Wide Web. / x, 111 p. ill., charts

Thesis (MBA)--Stellenbosch University, 2003. / ENGLISH ABSTRACT: This study project aims to use "Real Option Analysis" as a tool to translate financial hedging strategies into business strategies that can be used to hedge business projects against their associated risks. Financial investments are often hedged by means of further investment in financial option structures. These option structures give the investor the option (and sometimes the obligation) to change the constituents of his original investment, depending on changes in the external environment. A well engineered option structure will protect the investor against downside risk, while maximizing profits from upside risk. The objective of this study project is then to adapt some of the standard structures to such an extent that they can be used with similar success in the real business environment. This adaptation is done by means of Real Option Analysis - a relatively new theory whereby business uncertainty and managerial flexibility can be evaluated and quantified in a way similar to financial options. It will be seen that a careful application of Real Option Analysis allows one to take a certain business situation, identify the risks inherent to it, find a suitable option structure to hedge against those risks, and modify this option structure so that it can be implemented as a pure business strategy. This analysis is supported by a detailed derivation of a popular Real Option Analysis model, and an in depth discussion of the differences between Real- and financial options as well as difficulties associated with the implementation of Real Option-based strategies. Several examples of specific business situations are analyzed and it is concluded that Real Option Analysis can provide useful, practical and competitive strategies. Above all, the thought process leading to said strategies is deemed to provide powerful insight into the dynamics of the business/project under evaluation. / AFRIKAANSE OPSOMMING: Hierdie studie projek poog om "Real Option Analysis" te gebruik om finansiele immuniserings strategiee om te skakel in besigheids strategiee wat gebruik kan word om besigheids projekte te beskerm teen hul inherente risikos. Finansiele beleggings word dikwels geimmuniseer deur middel van verdere beleggings in finansiele opsie strukture. Hierdie strukture gee aan die belegger die opsie (en soms die verpligting) om die samestelling van sy oorspronklike belegging aan te pas na gelang van veranderinge in die omgewing. 'n Goed ontwerpte struktuur sal die belegger toelaat om sy winste te maksimeer terwyl verliese as gevolg van negatiewe risiko beperk word. Die doel van die studie projek is dan om sommige van hierdie standaard opsie strukture aan te pas sodat dit nie net in die beleggings wereld nie, maar ook in die besigheids wereld toegepas kan word. Hierdie aanpassing word gedoen met behulp van "Real Option Analysis" - 'n relatief nuwe teorie waarvolgens besigheids onsekerhede and bestuurs aanpasbaarhede geevalueer en gekwantifiseer kan word op 'n soortgelyke wyse as finansiele opsies. Dit sal gesien word dat 'n deeglike toepassing van "Real Option Analysis" die gebruiker toelaat om 'n besigheids situasie te evalueer, die risikos daaran verbonde te identifiseer, 'n toepaslike opsie struktuur te vind wat beskerming sal bied teen hierdie risikos, en dan hierdie struktuur aan te pas sodat dit as 'n besigheid strategie toegepas kan word. Hierdie analise word ondersteun deur die afleiding van 'n populere "Real Option Analysis" model, 'n bespreking van die verskille tussen Rieele- en finansiele opsies, sowel as komplikasies wat verwag kan word tydens die implimentasie van 'n strategie gebasseer op Rieele Opsies. Verskeie voorbeelde van spesifieke besigheids situasies word geanaliseer en dit gee aanleiding tot die gevolgtrekking dat "Real Option Analysis" wel sinvolle, bruikbare en kompeterende strategiee kan voorsien. Verder word daar aangedui dat die denk proses wat lei tot hierdie strategiee, 'n kragtige bron van insig in die besigheid/projek dinamika kan gee.

This thesis contains a discussion of four problems arising from the application of stochastic differential equations and real option theory to investment decision problems in a continuous-time framework. It is based on four papers written jointly with the author’s supervisor. In the first problem, we study an evolutionary stock market model in a continuous-time framework where uncertainty in dividends is produced by a single Wiener process. The model is an adaptation to a continuous-time framework of a discrete evolutionary stock market model developed by Evstigneev, Hens and Schenk-Hoppé (2006). We consider the case of fix-mix strategies and derive the stochastic differential equations which determine the evolution of the wealth processes of the various market players. The wealth dynamics for various initial set-ups of the market are simulated. In the second problem, we apply an entry-exit model in real option theory to study concessionary agreements between a private company and a state government to run a privatised business or project. The private company can choose the time to enter into the agreement and can also choose the time to exit the agreement if the project becomes unprofitable. An early termination of the agreement by the company might mean that it has to pay a penalty fee to the government. Optimal times for the company to enter and exit the agreement are calculated. The dynamics of the project are assumed to follow either a geometric mean reversion process or geometric Brownian motion. A comparative analysis is provided. Particular emphasis is given to the role of uncertainty and how uncertainty affects the average time that the concessionary agreement is active. The effect of uncertainty is studied by using Monte Carlo simulation. In the third problem, we study numerical methods for solving stochastic optimal control problems which are linear in the control. In particular, we investigate methods based on spline functions for solving the two-point boundary value problems that arise from the method of dynamic programming. In the general case, where only the value function and its first derivative are guaranteed to be continuous, piecewise quadratic polynomials are used in the solution. However, under certain conditions, the continuity of the second derivative is also guaranteed. In this case, piecewise cubic polynomials are used in the solution. We show how the computational time and memory requirements of the solution algorithm can be improved by effectively reducing the dimension of the problem. Numerical examples which demonstrate the effectiveness of our method are provided. Lastly, we study the situation where, by partial privatisation, a government gives a private company the opportunity to invest in a government-owned business. After payment of an initial instalment cost, the private company’s investments are assumed to be flexible within a range [0, k] while the investment in the business continues. We model the problem in a real option framework and use a geometric mean reversion process to describe the dynamics of the business. We use the method of dynamic programming to determine the optimal time for the private company to enter and pay the initial instalment cost as well as the optimal dynamic investment strategy that it follows afterwards. Since an analytic solution cannot be obtained for the dynamic programming equations, we use quadratic splines to obtain a numerical solution. Finally we determine the optimal degree of privatisation in our model from the perspective of the government.

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

published_or_final_version / Business Administration / Master / Master of Business Administration

published_or_final_version / Business Administration / Master / Master of Business Administration