Dissertations / Theses on the topic 'Applied mathematics|Finance'

This thesis consists of two parts. The first part addresses the parameter estimation and calibration of the Generalized Hyperbolic (GH) distributions. In this part we review the classical expectation maximization (EM) algorithm and factor analysis for the GH distribution. We also propose a simple shrinkage estimator driven from the penalized maximum likelihood. In addition an on-line EM algorithm is implemented to the GH distribution; and its regret for general exponential family can be represented as a mixture of Kullback-Leibler divergence. We compute the Hellinger distance of the joint GH distribution to measure the performances of all the estimators numerically. Empirical studies for long-term and short-term predictions are also performed to evaluate the algorithms.

In the second part we applied the GH distribution to portfolio optimization and risk allocation. We show that the mean-risk portfolio optimization problem of a certain type of normal mixture distributions including the GH distribution can be reduced to a two dimensional problem by fixing the location parameter and the skewness parameter. In addition, we show that the efficient frontier of the mean-risk optimization problem can be extended to the three dimensional space. We also proposed a simple algorithm to deal with the transaction costs. The first and second derivatives of the CVaR are computed analytically when the underlying distribution is GH. With these results we are able to extend the effective number of bets (ENB) to general risk measures with the GH distribution. By diagonalizing the Hessian matrix of a risk measure we are able to extract locally independent marginal contributions to the risk. The minimal torsion approach can still be applied to get the local coordinators of the marginal contributions.

This thesis presents three applications of Mathematics to Finance, from the empirical to the analytic level. The first part shows how the CBOE's market volatility index (VIX) has seasonal movements, using several statistical and econometrical tools. These tools complement each other. The application is shown in a way that illustrates how dangerous it is to apply only ordinary least squares methods to look for seasonality. The second part shows interesting patterns emerging from empirical distributions of S&P100 index (OEX) returns over some horizons conditioned to the VIX level. The third part shows the main features of the distributions of option's payoff obtained using mainly analytical tools. However, a computer is needed to get a general picture of the distribution useful for speculators and traders in isolation.

Modern general equilibria under uncertainty are modeled based on the recognition that all risks cannot be eliminated, perfect hedging is not possible, and some risk exposures must be tolerated. Therefore, we need to define the set of acceptable risks as a primitive of the financial economy. This set will be a cone, hence the word conic. Such a conic perspective challenges classical economics by introducing finance into the economic models and enables us to rewrite major chapters of classical micro- and macro-economics textbooks.

Includes bibliographical references (leaves 39-40).
In this study we apply Markov Chain Monte Carlo methods in the Bayesian framework to estimate Stochastic Volatility models using South African financial market data. A single move Gibbs sampler is used to sample parameters from the posterior distribution. Volatility is used as measure of an asset's risk. It is particularly important in risk management, derivatives pricing, and portfolio selection. When pricing derivatives it is important to quote the correct volatility trading in the market, hence there is need for good estimates of volatility. To capture the stylised facts about asset returns we used the model extended for fat tails and correlated errors. To support this model against the basic model of Taylor (1986), we computed Bayes Factors of Jacquier, Polson and Ross (2004). The extended model was found to be far superior to the basic model.

Thesis (MSc (Mathematical Sciences. Applied Mathematics))--University of Stellenbosch, 2009.
In this thesis, we investigate two numerical methods to price nancial options. We look at two types of options, namely European options and Asian options. The numerical methods we use are the nite di erence method and numerical inversion of the Laplace transform. We apply nite di erence methods to partial di erential equations with both uniform and non-uniform spatial grids. The Laplace inversion method we use is due to Talbot. It is based on the midpoint-type approximation of the Bromwich integral on a deformed contour. When applied to Asian options, we have the problem of computing the hypergeometric function of the rst kind. We propose a new method for numerically calculating the hypergeometric function. This method too is based on using Talbot contours. Throughout the thesis, we use the Black-Scholes equation as our benchmark problem.

High Frequency Trading (HFT) represents an ever growing proportion of all financial transactions as most markets have now switched to electronic order book systems. This dissertation proposes a novel methodology to analyze idiosyncrasies of the high frequency market microstructure and embed them in classical continuous time models.

The main technical result is the derivation of continuous time equations which generalize the self-financing relationships of frictionless markets to electronic markets with limit order books. We use NASDAQ ITCH data to identify significant empirical features such as price impact and recovery, rough paths of inventories and vanishing bid-ask spreads. Starting from these features, we identify microscopic identities holding on the trade clock, and through a diffusion limit argument, derive continuous time equations which provide a macroscopic description of properties of the order book.

These equations naturally differentiate between trading via limit and market orders. We give several applications to illustrate their impact and how they can be used to the benefit of Low Frequency Traders (LFTs). In particular, option pricing and market making models are proposed and solved, leading to new insights as to the impact of limit orders and market orders on trading strategies.

Low substantiation rates are pervasive across the federal Office of Inspector General (OIG) community due to high levels of uncertainty and limited data availability at the time of case selection. OIG management often selects cases based on intuition and past experience. Intuitive project selection has proven unsuccessful because the methods are often subjective, prone to bias, and lead to error. The high uncertainty surrounding case selection and the current selection method employed by OIG management teams results in a significant loss of investigative resources spent on unsubstantiated cases. This research presents a novel approach to predict OIG investigative case substantiation using natural language processing techniques and multinomial naïve Bayes to retrieve information from complaint intakes. It aims to improve OIG substantiation rates and reduce the cost associated with unsubstantiated cases. The model developed in this study significantly outperformed OIG management and was 20% more accurate in the prediction of substantiated and unsubstantiated cases. This model will augment investigative case selection and improve investigative targeting, increase impact of investigative work, and improve OIG investigative resource allocation. Its application will result in a significant savings by reducing the resources dedicated to cases with a low probability of substantiation.

In this thesis, we will develop the fundamental properties of financial mathematics, with a focus on establishing meaningful connections between martingale theory, stochastic calculus, and measure-theoretic probability. We first consider a simple binomial model in discrete time, and assume the impossibility of earning a riskless profit, known as arbitrage. Under this no-arbitrage assumption alone, we stumble upon a strange new probability measure Q, according to which every risky asset is expected to grow as though it were a bond. As it turns out, this measure Q also gives the arbitrage-free pricing formula for every asset on our market. In considering a slightly more complicated model over a finite probability space, we see that Q once again makes its appearance. Finally, in the context of continuous time, we build a framework of stochastic calculus to model the trajectories of asset prices on a finite time interval. Under the absence of arbitrage once more, we see that Q makes its return as a Radon-Nikodym derivative of our initial probability measure. Finally, we use the properties of Q and a stochastic differential equation that models the dynamics of the assets of our market, known as the Ito formula, in order to derive the classic Black-Scholes Equation.

The point of this thesis is to compare classic asset pricing models using historic UK data. It looks at three of the most commonly used asset pricing models in Finance and tests the suitability of each for the UK market. The models considered are the Capital Asset Pricing Model (1964, 65 and 66) (CAPM), the Fama-French 3-Factor Model (1993) (FF3F) and the Carhart 4-Factor Model (1997) (C4F). The models are analysed using a 34 year sample period (1980-2014). The sample data follows the structure explained in Gregory et al (2013) and is compiled of stocks from the London Stock Exchange (LSE). The stocks are grouped into portfolios arranged by market capitalisation, book-to-market ratio, past 2-12 month stock return and past 12 month standard deviation of stock return. Statistical analysis is performed and the suitability of the models is tested using the methods of Black, Jensen \& Scholes (1972), Fama \& MacBeth (1973) and Gibbons, Ross \& Shanken (1989). The results compare descriptive and test statistics across the range of risk factors and test portfolios for the each testing method on all three models. They show that although the UK market has some noticeable factor anomalies, none of the models clearly explains the 1980-2014 stock returns. However, of the three models, C4F shows the highest explanatory power in predicting stock returns.

The purpose of this paper is threefold. First, the paper builds on the work done previously done in the area of option implied probability distribution functions (PDFs) by extending the methods described by Breeden and Litzenberger (1978) to individual equity options. Second, it describes a closed-form, onto mapping from a two-dimensional volatility surface to the risk-neutral PDF. Lastly the paper performs an event study on the implied risk-neutral PDFs of companies which are the target of corporate takeover. While there was not sufficient data to determine any statistical relationship, there is observational evidence that option market implied PDFs may be predictive of future takeovers.

I en företagstransaktion anlitas vanligen en investmentbank för att bistå med värdering av bolaget samt agera rådgivare. Investmentbanker agerar som en slag företagsmäklare som är antingen på köp- eller säljsidan av transaktionen. När bolagsvärdet presenteras i en pitch till säljarna brukar de använda sig av flera metoder för att beräkna värdet av företaget. För att öka precisionen i värdering av ett nordiskt industribolag ställdes frågan om multipel regressionsanalys kunde användas som ett komplement i en bolagsvärdering och hur den stod sig gentemot en klassisk värderingsmetod som Precedent Transactions Analysis. Dessa frågor kom att analyseras och besvarades genom att skapa en regressionsmodell som byggde på data hämtad från företagens finansiella rapporter. Den insamlade datan byggde på 132 transaktioner av nordiska industribolag under perioden 2009-2019. Regressionsmodellen kom sedan att jämföras mot en PTA-värdering som byggde på tidigare företagstransaktioner av bolag med liknande finansiell och affärsmässig bakgrund som bolaget i fråga skulle värderas. Denna studie visar på att regressionsanalys kan användas som en komplement till de olika värderingsmetoderna men bör ej användas för att värdera nordiska industribolag med avhandlingens val av variabler och skall inte ersätta någon av de klassiska värderingsmetoderna.
Prior to a company being sold or acquired they usually contact an investment bank to support with the valuation of the company, execute the sale and act as advisors for the actors that wish to buy or sell. Investment banks acts as a kind of company broker which is either on the buy or the sell side. When the company value is presented, they usually utilize several methods to calculate the value of the company. During the last decade the frequency of transactions on the Nordic industry market have increased significantly. To increase the precision in the valuation of a Nordic industrial company, the question was asked if multiple regression analysis could be used as a valuation method? Also, how did it compare itself against a classical valuation method like Precedent Transaction Analysis? These questions came to be analyzed and answered by creating a regression modell built of data gathered from financial reports. The regression model then came to be compared to the PTA-valuation which built on previous company transactions with companies that were alike in financial background. This study shows that regression analysis could be used as a complement to the different valuation methods. However the model should not be used to evaluate Nordic industrial companies with the choice of variables in the thesis, since the reliability of the model is unpredictable. Regression analysis as a stand-alone valuation method should be taken with great caution and not replace neither of the classical valuation methods.

Newsvendor Models with Monte Carlo Sampling by Ijeoma Winifred Ekwegh The newsvendor model is used in solving inventory problems in which demand is random. In this thesis, we will focus on a method of using Monte Carlo sampling to estimate the order quantity that will either maximizes revenue or minimizes cost given that demand is uncertain. Given data, the Monte Carlo approach will be used in sampling data over scenarios and also estimating the probability density function. A bootstrapping process yields an empirical distribution for the order quantity that will maximize the expected profit. Finally, this method will be used on a newsvendor example to show that it works in maximizing profit.

Valuing a company is an important task in finance, especially before a potential merger or acquisition of a company. It is then of great importance for both parties in a deal to make an accurate estimate of the value of the company. The goal of this paper is to investigate how well regression analysis can be applied in this matter and if it can perform at par or better than more frequently used methods in the industry today. The study was conducted within the utilities sector in the US and Europe, with data collected from historic public transactions dating back to 2009. The study concludes that a regression model as a valuation tool can generate several advantages as it identifies key value drivers and is based on core mathematical concepts. However, the model created in this thesis underperforms compared to the prominent methods in place today. For further research this thesis may provide useful insight into different areas to consider when creating a valuation model.
Att värdera ett företag är en viktig uppgift inom finanssektorn, särskilt innan en potentiell sammanslagning eller förvärv av ett företag. Det är då av stor vikt för båda parter i en affär att göra en exakt uppskattning av företagets värde. Målet med denna studie är att undersöka hur väl regressionsanalys kan tillämpas i denna fråga och om den kan generera samma eller bättre resultat än mer använda värderingsmetoder inom branschen idag. Studien genomfördes inom el-, gas- och vattensektorn i USA och Europa, med data som samlats in från historiska offentliga transaktioner som går tillbaka till 2009. Studien drar slutsatsen att en regressionsmodell som ett värderingsverktyg kan generera flera fördelar eftersom den identifierar viktiga faktorer som driver en värdering och baseras på grundläggande matematiska begrepp. Modellen som skapats i denna avhandling underpresterar dock jämfört med de framstående metoderna som finns idag. För ytterligare forskning kan denna studie ge användbar insikt i olika områden att beakta när man skapar en värderingsmodell.

In this work, we study and develop certain aspects of the analytics of asymmetry for univariate and multivariate data. Accordingly, the above work consists of three separate parts.

In the first part of our work, we introduce a new approach to measure the univariate and multivariate skewness based on quantiles and the properties of odd and even functions. We illustrate through numerous examples and simulations that in the multivariate case the Mardia’s measure of skewness fails to provide consistent and meaningful interpretations. However, our new measure appears to provide an index which is more reasonable.

In the second part of our work, our emphasis is to moderate or eliminate asymmetry of multivariate data when the interest is in the study of dependence. Copula transformation has been used as an all-purpose transformation to introduce multivariate normality. Using this approach, even though information about marginal distributions is lost, we are still able to study dependence based modeling problems for asymmetric data using the technique developed for multivariate normal data. We illustrate a variety of applications in areas such as multiple regression, principal component, factor analysis, partial least squares and structural equation models. The results are promising in that our approach shows improvement over results obtained when asymmetry is ignored.

The last part of this work is based on the applications of our copula transformation to financial data. Specifically, we consider the problem of estimation of “beta risk” associated with a particular financial asset. Taking S&P500 index as a proxy for market, we suggest three versions of “beta estimates” which are useful in situations when the returns of the assets and market proxy do not have the most ideal probability distribution, namely, bivariate normal or when data may contain some very extreme (high or low) returns. Using the copula based methods, developed earlier in this dissertation, and winsorization, we obtain the estimates which in high skewness scenarios perform better than the traditional least square estimate of market beta.

Le but de cette thèse est d'apporter une contribution à la problématique de valorisation de produits dérivés en marchés incomplets. Nous considérons tout d'abord les cibles stochastiques introduites par Soner et Touzi (2002) afin de traiter le problème de sur-réplication, et récemment étendues afin de traiter des approches plus générales par Bouchard, Elie et Touzi (2009). Nous généralisons le travail de Bouchard {\sl et al} à un cadre plus général où les diffusions sont sujettes à des sauts. Nous devons considérer dans ce cas des contrôles qui prennent la forme de fonctions non bornées, ce qui impacte de façon non triviale la dérivation des EDP correspondantes. Notre deuxième contribution consiste à établir une version des cibles stochastiques qui soit robuste à l'incertitude de modèle. Dans un cadre abstrait, nous établissons une version faible du principe de programmation dynamique géométrique de Soner et Touzi (2002), et nous dérivons, dans un cas d'EDS controllées, l'équation aux dérivées partielles correspondantes, au sens des viscosités. Nous nous intéressons ensuite à un exemple de couverture partielle sous incertitude de Knightian. Finalement, nous nous concentrons sur le problème de valorisation de produits dérivées {\sl hybrides} (produits dérivés combinant finance de marché et assurance). Nous cherchons plus particulièrement à établir une condition suffisante sous laquelle une règle de valorisation (populaire dans l'industrie), consistant à combiner l'approches actuarielle de mutualisation avec une approche d'arbitrage, soit valable.

Derivative pricing, model calibration, and sensitivity analysis are the three main problems in financial modeling. The purpose of this study is to present an algorithm to improve the pricing process, the calibration process, and the sensitivity analysis of the double Heston model, in the sense of accuracy and efficiency. Using the optimized caching technique, our study reduces the pricing computation time by about 15%. Another contribution of this thesis is: a novel application of the Automatic Differentiation (AD) algorithms in order to achieve a more stable, more accurate, and faster sensitivity analysis for the double Heston model (compared to the classical finite difference methods). This thesis also presents a novel hybrid model by combing the heuristic method Differentiation Evolution, and the gradient method Levenberg--Marquardt algorithm. Our new hybrid model significantly accelerates the calibration process.

Pensionsmyndigheten's (the Swedish Pensions Agency) current calculation of the internal rate of return for 7.7 million premium pension savers is both time and resource consuming. This rate of return mirrors the overall performance of the funded part of the pension system and is analyzed internally, but also reported to the public monthly and yearly based on differently sized data samples. This thesis aims to investigate the possibility of utilizing other approaches in order to improve the performance of these calculations. Further, the study aims to verify the results stemming from said calculations and investigate their robustness. In order to investigate competitive matrix methods, a sample of approaches are compared to the more classical numerical methods. The approaches are compared in different scenarios aimed to mirror real practice. The robustness of the results are then analyzed by a stochastic modeling approach, where a small error term is introduced aimed to mimic possible errors which could arise in data management. It is concluded that a combination of Halley's method and the Jacobi-Davidson algorithm is the most robust and high performing method. The proposed method combines the speed and robustness from numerical and matrix methods, respectively. The result show a performance improvement of 550% in time, while maintaining the accuracy of the current server computations. The analysis of error propagation suggests the output error to be less than 0.12 percentage points in 99 percent of the cases, considering an introduced error term of large proportions. In this extreme case, the modeled expected number of individuals with an error exceeding 1 percentage point is estimated to be 212 out of the whole population.
Pensionsmyndighetens nuvarande beräkning av internräntan för 7,7 miljoner pensionssparare är både tid- och resurskrävande. Denna avkastning ger en översikt av hur väl den fonderade delen av pensionssystemet fungerar. Detta analyseras internt men rapporteras även till allmänheten varje månad samt årligen baserat på olika urval av data. Denna uppsats avser att undersöka möjligheten att använda andra tillvägagångssätt för att förbättra prestanda för denna typ av beräkningar. Vidare syftar studien till att verifiera resultaten som härrör från dessa beräkningar och undersöka deras stabilitet. För att undersöka om det finns konkurrerande matrismetoder jämförs ett urval av tillvägagångssätt med de mer klassiska numeriska metoderna. Metoderna jämförs i flera olika scenarier som syftar till att spegla verklig praxis. Stabiliteten i resultaten analyseras med en stokastisk modellering där en felterm införs för att efterlikna möjliga fel som kan uppstå i datahantering. Man drar slutsatsen att en kombination av Halleys metod och Jacobi-Davidson-algoritmen är den mest robusta och högpresterande metoden. Den föreslagna metoden kombinerar hastigheten från numeriska metoder och tillförlitlighet från matrismetoder. Resultatet visar en prestandaförbättring på 550 % i tid, samtidigt som samma noggrannhet som ses i de befintliga serverberäkningarna bibehålls. Analysen av felutbredning föreslår att felet i 99 procent av fallen är mindre än 0,12 procentenheter i det fall där införd felterm har stora proportioner. I detta extrema fall uppskattas det förväntade antalet individer med ett fel som överstiger 1 procentenhet vara 212 av hela befolkningen.

Although options have been traded for many centuries, it has remained a relatively thinly traded financial instrument. Paradoxically, the theory of option pricing has been studied extensively. This is due to the fact that many of the financial instruments that are traded in the market place have an option-like structure, and thus the development of a methodology for option-pricing may lead to a general methodology for the pricing of these derivative-assets. This thesis will focus on the development of the theory of option pricing. Initially, a fundamental principle that underlies the theory of option valuation will be given. This will be followed by a discussion of the different types of option pricing models that are prevalent in the literature. Special attention will then be given to a detailed derivation of both the Black-Scholes and the Binomial Option pricing models, which will be followed by a proof of the convergence of the Binomial pricing model to the Black-Scholes model. The Black-Scholes model will be adapted to take into account the payment of dividends, the possibility of a changing inter est rate and the possibility of a stochastic variance for the rate of return on the underlying as set. Several applications of the Black-Scholes model will finally be presented.
Thesis (M.Sc.)-University of Natal, 1992.

Numerical methods such as Monte Carlo method (MCM), finite difference method (FDM) and finite element method (FEM) have been successfully implemented to solve financial partial differential equations (PDEs). Sophisticated computational algorithms are strongly desired to further improve accuracy and efficiency.

The relatively new spectral element method (SEM) combines the exponential convergence of spectral method and the geometric flexibility of FEM. This dissertation carefully investigates SEM on the pricing of European options and their Greeks (Delta, Gamma and Theta). The essential techniques, Gauss quadrature rules, are thoroughly discussed and developed. The spectral element method and its error analysis are briefly introduced first and expanded in details afterwards.

Multi-element spectral element method (ME-SEM) for the Black-Scholes PDE is derived on European put options with and without dividend and on a condor option with a more complicated payoff. Under the same Crank-Nicolson approach for the time integration, the SEM shows significant accuracy increase and time cost reduction over the FDM. A novel discontinuous payoff spectral element method (DP-SEM) is invented and numerically validated on a European binary put option. The SEM is also applied to the constant elasticity of variance (CEV) model and verified with the MCM and the valuation formula. The Stochastic Alpha Beta Rho (SABR) model is solved with multi-dimensional spectral element method (MD-SEM) on a European put option. Error convergence for option prices and Greeks with respect to the number of grid points and the time step is analyzed and illustrated.

Dissertation

A robust computational framework is presented for the risk-neutral valuation of capital guarantees written on discretely-reallocated portfolios following the Constant Proportion Portfolio Insurance (CPPI) strategy. Aiming to address the (arguably more realistic) cases where analytical results are unavailable, this framework accommodates risky-asset jumps, volatility surfaces, borrowing restrictions, nonuniform reallocation schedules and autonomous CPPI floor trajectories. The two-asset state space representation developed herein facilitates visualising the CPPI strategy, which in turn provides insight into grid design and interpolation. It is demonstrated that given a deterministic process for the risk-free rate, the pricing problem can be cast as solving cascading systems of 1D partial integro-differential equations (PIDEs). This formulation’s stability and monotonicity are studied. In addition to making more sense financially, the limited borrowing variant of the CPPI strategy is found to be better suited than the classical (unlimited borrowing) counterpart for bounded-domain calculations. Consequently, it is demonstrated how the unlimited borrowing problem can be approximated by imposing an artificial borrowing limit. For implementation validation, analytical solutions to special cases are derived. Numerical tests are presented to demonstrate the versatility of this framework.