The family of stable distributions represents an important generalization of the Gaussian family; stable random variables obey a generalized central limit theorem where the assumption of finite variance is replaced with one of power law decay in the tails. Possessing heavy tails, asymmetry, and infinite variance, non-Gaussian stable distributions can be suitable for inference in settings featuring impulsive, possibly skewed noise. A general lack of analytical form for the densities and distributions of stable laws has prompted research into computational methods of estimation. This report introduces stable distributions through a discussion of their basic properties and definitions in chapter 1. Chapter 2 surveys applications, and chapter 3 discusses a number of procedures for inference, with particular attention to time series models in the ARMA setting. Further details and an application can be found in the appendices. / text

In this dissertation, I investigate three related topics on asset pricing: the consumption-based asset pricing under long-run risks and fat tails, the pricing of VIX (CBOE Volatility Index) options and the market price of risk embedded in stock returns and stock options. These three topics are fully explored in Chapter II through IV. Chapter V summarizes the main conclusions. In Chapter II, I explore the effects of fat tails on the equilibrium implications of the long run risks model of asset pricing by introducing innovations with dampened power law to consumption and dividends growth processes. I estimate the structural parameters of the proposed model by maximum likelihood. I find that the stochastic volatility model with fat tails can, without resorting to high risk aversion, generate implied risk premium, expected risk free rate and their volatilities comparable to the magnitudes observed in data. In Chapter III, I examine the pricing performance of VIX option models. The contention that simpler-is-better is supported by the empirical evidence using actual VIX option market data. I find that no model has small pricing errors over the entire range of strike prices and times to expiration. In general, Whaley’s Black-like option model produces the best overall results, supporting the simpler-is-better contention. However, the Whaley model does under/overprice out-of-the-money call/put VIX options, which is contrary to the behavior of stock index option pricing models. In Chapter IV, I explore risk pricing through a model of time-changed Lévy processes based on the joint evidence from individual stock options and underlying stocks. I specify a pricing kernel that prices idiosyncratic and systematic risks. This approach to examining risk premia on stocks deviates from existing studies. The empirical results show that the market pays positive premia for idiosyncratic and market jump-diffusion risk, and idiosyncratic volatility risk. However, there is no consensus on the premium for market volatility risk. It can be positive or negative. The positive premium on idiosyncratic risk runs contrary to the implications of traditional capital asset pricing theory.

While Monte Carlo methods for bootstrapping are typically easy to implement, they can be quite time intensive. This work aims to extend an established convolutional method of bootstrapping to work when convolutions in two or more dimensions are required. The convolutional method relies on efficient computational tools rather than Monte Carlo simulation which can greatly reduce the computation time. The proposed method is particularly well suited for the analysis of degradation data when the data are not collected on time intervals of equal length. The convolutional bootstrapping method is typically much faster than the Monte Carlo bootstrap and can be used to produce exact results in some simple cases. Even in more complicated applications, where it is not feasible to find exact results, mathematical bounds can be placed on the resulting distribution. With these benefits of the convolutional method, this bootstrapping approach has been shown to be a useful alternative to the traditional Monte Carlo bootstrap.

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.

Esta dissertação revisa a literatura acadêmica existente sobre a teoria de opções utilizando os modelos de precificação com saltos. Os conceitos foram equalizados, a nomenclatura foi padronizada, sendo gerado um material de referência sobre o assunto. O pressuposto de lognormalidade com volatilidade constante não é aceito pelo mercado financeiro. É freqüente, no meio acadêmico, a busca de modelos que reproduzam os fenômenos observados de leptocurtose ou assimetria dos log-retornos financeiros e que possuam a mesma robustez e facilidade para manipulação analítica do consagrado modelo de Black-Scholes. Os modelos com saltos são uma alternativa para esse problema. Avaliou-se o modelo de Kou no mercado acionário brasileiro composto por um componente de difusão que segue um movimento browniano geométrico e um componente de saltos que segue um processo de Poisson com intensidade do salto descrito por uma distribuição duplamente exponencial. A simulação histórica do modelo aponta, em geral, uma superioridade preditiva do modelo, porém as dificuldades de calibração dos parâmetros e de hedge em mercados incompletos são as principais deficiências para o uso dos modelos com saltos. / This master dissertation reviews the academic literature about option pricing and hedging with jumps. The theory was equalized and the notation was standardized, becoming this document a reference document about this subject. The log-normality with constant volatility is not accepted by the market. Academics search consistent models with the same analytical capabilities like Black-Scholes? model which can support the observed leptokurtosis or asymmetry of the financial daily log-returns behavior. The jump models are an alternative to these issues. The Kou?s model was evaluated and this one consists of two parts: the first part being continuous and following a geometric Brownian motion and the second being a jump process with its jump intensity defined by a double exponential distribution. The model backtesting showed a better predictive performance of the Kou´s model against other models. However, there are some handicaps regarding to the parameters calibration and hedging.

<p>This work consists of four articles concerning Gaussian probability laws with stochastic means and variances. The first paper introduces a new way of approximating the probability distribution of a function of random variables. This is done with a Gaussian probability law with stochastic mean and variance. In the second paper an extension of the Generalized Hyperbolic class of probability distributions is presented. The third paper introduces, using a Gaussian probability law with stochastic mean and variance, a GARCH type stochastic process with skewed innovations. </p><p>In the fourth paper a Lévy process with second order stochastic volatility is presented, option pricing under such a process is also considered.</p>

This work consists of four articles concerning Gaussian probability laws with stochastic means and variances. The first paper introduces a new way of approximating the probability distribution of a function of random variables. This is done with a Gaussian probability law with stochastic mean and variance. In the second paper an extension of the Generalized Hyperbolic class of probability distributions is presented. The third paper introduces, using a Gaussian probability law with stochastic mean and variance, a GARCH type stochastic process with skewed innovations. In the fourth paper a Lévy process with second order stochastic volatility is presented, option pricing under such a process is also considered.

Esta dissertação revisa a literatura acadêmica existente sobre a teoria de opções utilizando os modelos de precificação com saltos. Os conceitos foram equalizados, a nomenclatura foi padronizada, sendo gerado um material de referência sobre o assunto. O pressuposto de lognormalidade com volatilidade constante não é aceito pelo mercado financeiro. É freqüente, no meio acadêmico, a busca de modelos que reproduzam os fenômenos observados de leptocurtose ou assimetria dos log-retornos financeiros e que possuam a mesma robustez e facilidade para manipulação analítica do consagrado modelo de Black-Scholes. Os modelos com saltos são uma alternativa para esse problema. Avaliou-se o modelo de Kou no mercado acionário brasileiro composto por um componente de difusão que segue um movimento browniano geométrico e um componente de saltos que segue um processo de Poisson com intensidade do salto descrito por uma distribuição duplamente exponencial. A simulação histórica do modelo aponta, em geral, uma superioridade preditiva do modelo, porém as dificuldades de calibração dos parâmetros e de hedge em mercados incompletos são as principais deficiências para o uso dos modelos com saltos. / This master dissertation reviews the academic literature about option pricing and hedging with jumps. The theory was equalized and the notation was standardized, becoming this document a reference document about this subject. The log-normality with constant volatility is not accepted by the market. Academics search consistent models with the same analytical capabilities like Black-Scholes? model which can support the observed leptokurtosis or asymmetry of the financial daily log-returns behavior. The jump models are an alternative to these issues. The Kou?s model was evaluated and this one consists of two parts: the first part being continuous and following a geometric Brownian motion and the second being a jump process with its jump intensity defined by a double exponential distribution. The model backtesting showed a better predictive performance of the Kou´s model against other models. However, there are some handicaps regarding to the parameters calibration and hedging.

過去實證研究發現，資產的動態過程存在不連續的跳躍與大波動伴隨大波動的波動度叢聚現象而造成資產報酬分配呈現出厚尾與高狹峰的情況，然而，此現象並不能完全被傳統所使用幾何布朗運動模型與跳躍擴散模型給解釋。因此，本文設定資產模型服從Lévy 過程中Generalized Hyperbolic (GH)的normal inverse Gaussian(NIG) 和 variance gamma (VG)兩個模型，然而，Lévy 過程是一個跳躍過程，是屬於一個不完備的市場，這將使得平賭測度並非唯一，因此，本文將採用Gerber 和 Shiu (1994)所提的Esscher 轉換來求得平賭測度。關於美式選擇權將採用LongStaff and Schwartz (2001)所提的最小平方蒙地卡羅模擬法來評價美式選擇權。實證結果發現VG有較好的評價績效，此外，進一步探討流動性與價內外的情況對於評價誤差的影響，亦發現部分流動性高的樣本就較小的評價誤差；此外，價外的選擇權其評價誤差最大。另一方面從交易的觀點來看，次貸風暴後交易對手信用風險愈來愈受到重視，此外，近年來由於巨災事件的頻傳，使得傳統保險公司風險移轉的方式，漸漸透過資本市場發行衍生性商品來進行籌資，以彌補其在巨災發生時所承擔的損失。因此，透過發行衍生性商品來進行籌資，必須考量交易對手的信用風險，否則交易對手違約，就無法獲得額外的資金挹注，因此，本文評價巨災權益賣權，並考量交易對手信用風險對於其價格的影響。 / In the traditional models such as geometric Brownian motion model or the Merton jump diffusion model can’t fully depict the distributions of return for financial securities and the those return always have heavy tail and leptokurtic phenomena due to the price jump or volatilities of return changing over time. Hence, the first article uses two time-changed Lévy models: (1) normal inverse Gaussian model and (2) variance gamma model to capture the dynamics of asset for pricing American option. In order to deal with the early-exercised problem of the American option, we use the LSM to determine the optimal striking point until maturity. In the empirical analyses, we can find the VG model have better performance than the other three models in some cases. In addition, with the comparison the pricing performance under different liquidity and moneyness conditions, we also find in some samples increasing the liquidity really can reduce the pricing errors, at the same time, the maximum pricing errors appears in the OTM samples in all cases. The global subprime crisis during 2008 and 2009 arouses much more attention of the counterparty risk and the number of default varies with economic condition. Hence, we investigate the counterparty risk impact on the price of the catastrophe equity put with a Markov-modulated default intensity model in the second study. At the same time, we also extend the stochastic interest rate setting in Jaimungal and Wang (2006) and relax some restrictive assumption of Black-Scholes model by taking the regime-switching effects of the economic status, then use the Markov-modulated processes to model the dynamics of the underlying asset and interest rate. In the numerical analyses, we illustrate the impact of the recovery rate, time to maturity, jump intensity of the equity and default intensity of counterparty on the CatEPut price.

With the acceptance of the new Basel II banking regulation (implemented in South Africa in January 2008) the search for improved ways of modeling the most important banking activities has become very topical. Since the notion of Levy-process was introduced, it has emerged as an important tool for modeling economic variables in a Basel II framework. In this study, we investigate the stochastic dynamics of banking items that are driven by such processes. In particular, we discuss bank provisioning for loan losses and deposit withdrawals. The first type of provisioning is related to the earnings that the bank sets aside in order to cover loan defaults. In this case, we apply principles from robustness to a situation where the decision maker is a bank owner and the decision rule determines the optimal provisioning strategy for loan losses. In this regard, we formulate a dynamic banking loan loss model involving a provisioning portfolio consisting of provisions for expected losses and loan loss reserves for unexpected losses. Here, unexpected loan losses and provisioning for expected losses are modeled via a compound Poisson process and an exponential Levy process, respectively. We use historical evidence from OECD (Organization for Economic Corporation and Development) countries to support the fact that the provisions for loan losses-to-total assets ratio is negatively correlated with aggregate asset prices and the private credit-to-GDP ratio. Secondly, we construct models for provisioning for deposit withdrawals. In particular, we build stochastic dynamic models which enable us to analyze the interplay between deposit withdrawals and the provisioning for these withdrawals via Treasuries and reserves. Further insight is gained by considering a numerical problem and a simulation of the trajectory of the stochastic dynamics of the sum of the Treasuries and reserves. Since managing the risk that depositors will exercise their withdrawal option is an important aspect of this thesis, we consider the idea of a hedging provisioning strategy for deposit withdrawals in an incomplete market setting. In this spirit, we discuss an optimal risk management problem for a commercial bank whose main activity is to obtain funds through deposits from the public and use the Treasuries and reserves to cater for the resulting withdrawals. Finally, we provide a brief analysis of some of the issues arising from the dynamic models of the banking items derived. / Thesis (Ph.D. (Computer, Statistical and Mathematical Sciences))--North-West University, Potchefstroom Campus, 2008.