ANALYTICAL STUDY AND GENERALISATION OF SELECTED STOCK OPTION VALUATION MODELS

In this work, the classical Black-Scholes model for stock option valuation on the basis of some stochastic dynamics was considered. As a result, a stock option val- uation model with a non-�xed constant drift coe�cient was derived. The classical Black-Scholes model was generalised via the application of the Constant Elasticity of Variance Model (CEVM) with regard to two cases: case one was without a dividend yield parameter while case two was with a dividend yield parameter. In both cases, the volatility of the stock price was shown to be a non-constant power function of the underlying stock price and the elasticity parameter unlike the constant volatility assumption of the classical Black-Scholes model. The It^o's theorem was applied to the associated Stochastic Di�erential Equations (SDEs) for conversion to Partial Dif- ferential Equations (PDEs), while two approximate-analytical methods: the Modi�ed Di�erential Transformation Method (MDTM) and the He's Polynomials Technique (HPT) were applied to the Black-Scholes model for stock option valuation; in both cases the integer and time-fractional orders were considered, and the results obtained proved the latter as an extension of the former. In addition, a nonlinear option pric- ing model was obtained when the constant volatility assumption of the classical linear Black-Scholes option pricing model was relaxed through the inclusion of transaction cost (Bakstein and Howison model). Thereafter, this nonlinear option pricing model was extended to a time-fractional ordered form, and its approximate-analytical solu- tions were obtained via the proposed solution technique. For e�ciency and reliability of the method, two cases with �ve examples were considered: Case 1 with two ex- amples for time-integer order, and Case 2 with three examples for time-fractional order, and the results obtained show that the time-fractional order form generalises the time-integer order form. Thus, the Black-Scholes and the Bakstein and Howison models for stock option valuation were generalised and extended to time-fractional order, and analytical solutions of these generalised models were provided

Innovations in Quantitative Risk Management

Quantitative Finance; Game Theory, Economics, Social and Behav. Sciences; Finance/Investment/Banking; Actuarial Science

Econometric Modeling and Evaluation of Fiscal-Monetary Policy Interactions

Thesis (Ph.D.) - Indiana University, Economics, 2015How do fiscal and monetary policies interact to determine inflation? The conventional view rests on the Taylor principle, that central banks can control inflation by raising nominal interest rate more than one-for-one with inflation. This principle embeds an implicit assumption that the government always adjusts taxes or spending to assure fiscal solvency. But when the required fiscal adjustments are not assured, as may occur during periods of fiscal stress, monetary policy may no longer be able to determine inflation. Under this alternative view, policy roles are reversed, with fiscal policy determining the price level and monetary policy acting to stabilize debt. Because these two policy regimes imply starkly different policy advice, identifying the prevailing regime is a prerequisite to understanding the macro economy and to making good policy choices. This dissertation employs econometric modeling and evaluation techniques to examine the empirical implications of the dynamic interactions between post-war U.S. fiscal and monetary policies. Chapter One compares two econometric interpretations of a dynamic macro model designed to study U.S. policy interactions. Two main findings emerge. First, the data overwhelmingly support the conventional view of inflation determination under the prevailing, "strong" econometric interpretation that takes literally all of the model's implications for the data. But this result is susceptible to any potential model misspecification. Second, according to the alternative, "minimal" econometric interpretation that is immune to the difficulties with the strong interpretation, the two views of inflation determination can explain the data about equally well. These findings imply that the apparent statistical support in favor of the conventional view over the alternative in the literature stems largely from the strong interpretation rather than from compelling empirical evidence. Therefore, a prudent policymaker should broaden her perspective beyond any single view on the inflation process. Chapter Two, joint with Todd B. Walker, develops an analytic function approach to solving generalized multivariate linear rational expectations models. This solution method is shown to provide important insights into equilibrium dynamics of well-known models. Chapter Three further demonstrates the usefulness of this method via a conventional new Keynesian model

Current Topics on Risk Analysis: ICRA6 and RISK2015 Conference

Peer ReviewedPostprint (published version

Innovations in Quantitative Risk Management

Quantitative Finance; Game Theory, Economics, Social and Behav. Sciences; Finance/Investment/Banking; Actuarial Science

Current Topics on Risk Analysis: ICRA6 and RISK2015 Conference

Artículos presentados en la International Conference on Risk Analysis ICRA 6/RISK 2015, celebrada en Barcelona del 26 al 29 de mayo de 2015.Peer ReviewedPostprint (published version

Stochastic programs and their value over deterministic programs

A dissertation submitted to the Faculty of Arts, University of the Witwatersrand, Johannesburg, in fulfilment of the requirements for the degree of Master of Arts.Real-life decision-making problems can often be modelled by mathematical programs (or optimization models). It is common for there to be uncertainty about the parameters of such optimization models. Usually, this uncertainty is ignored and a simplified deterministic program is obtained. Stochastic programs take account of this uncertainty by including a probabilistic description of the uncertain parameters in the model. Stochastic programs are therefore more appropriate or valuable than deterministic programs in many situations, and this is emphasized throughout the dissertation. The dissertation contains a development of the theory of stochastic programming, and a number of illustrative examples are formulated and solved. As a real-life application, a stochastic model for the unit commitment problem facing Eskom (one of the world's largest producers of electricity) is formulated and solved, and the solution is compared with that of the current strategy employed by Eskom.AC 201

5th International Probabilistic Workshop: 28-29 November 2007, Ghent, Belgium

These are the proceedings of the 5th International Probabilistic Workshop. Even though the 5th anniversary of a conference might not be of such importance, it is quite interesting to note the development of this probabilistic conference. Originally, the series started as the 1st and 2nd Dresdner Probabilistic Symposium, which were launched to present research and applications mainly dealt with at Dresden University of Technology. Since then, the conference has grown to an internationally recognised conference dealing with research on and applications of probabilistic techniques, mainly in the field of structural engineering. Other topics have also been dealt with such as ship safety and natural hazards. Whereas the first conferences in Dresden included about 12 presentations each, the conference in Ghent has attracted nearly 30 presentations. Moving from Dresden to Vienna (University of Natural Resources and Applied Life Sciences) to Berlin (Federal Institute for Material Research and Testing) and then finally to Ghent, the conference has constantly evolved towards a truly international level. This can be seen by the language used. The first two conferences were entirely in the German language. During the conference in Berlin however, the change from the German to English language was especially apparent as some presentations were conducted in German and others in English. Now in Ghent all papers will be presented in English. Participants now, not only come from Europe, but also from other continents. Although the conference will move back to Germany again next year (2008) in Darmstadt, the international concept will remain, since so much work in the field of probabilistic safety evaluations is carried out internationally. In two years (2009) the conference will move to Delft, The Netherlands and probably in 2010 the conference will be held in Szczecin, Poland. Coming back to the present: the editors wish all participants a successful conference in Ghent

Applications of Laplace transform for evaluating occupation time options and other derivatives

The present thesis provides an analysis of possible applications of the Laplace Transform (LT) technique to several pricing problems. In Finance this technique has received very little attention and for this reason, in the first chapter we illustrate with several examples why the use of the LT can considerably simplify the pricing problem. Observed that the analytical inversion is very often difficult or requires the computation of very complicated expressions, we illustrate also how the numerical inversion is remarkably easy to understand and perform and can be done with high accuracy and at very low computational cost. In the second and third chapters we investigate the problem of pricing corridor derivatives, i.e. exotic contracts for which the payoff at maturity depends on the time of permanence of an index inside a band (corridor) or below a given level (hurdle). The index is usually an exchange or interest rate. This kind of bond has evidenced a good popularity in recent years as alternative instruments to common bonds for short term investment and as opportunity for investors believing in stable markets (corridor bonds) or in non appreciating markets (hurdle bonds). In the second chapter, assuming a Geometric Brownian dynamics for the underlying asset and solving the relevant Feynman-Kac equation, we obtain an expression for the Laplace transform of the characteristic function of the occupation time. We then show how to use a multidimensional numerical inversion for obtaining the density function. In the third chapter, we investigate the effect of discrete monitoring on the price of corridor derivatives and, as already observed in the literature for barrier options and for lookback options, we observe substantial differences between discrete and continuous monitoring. The pricing problem with discrete monitoring is based on an appropriate numerical scheme of the system of PDE's. In the fourth chapter we propose a new approximation for pricing Asian options based on the logarithmic moments of the price average