Stochastic volatility (SV) models provide a natural framework for a representation of time series for financial asset returns. As a result, they have become increasingly popular in the finance literature, although they have also been applied in other fields such as signal processing, telecommunications, engineering, biology, and other areas. In working with the SV models, an important issue arises as how to estimate their parameters efficiently and to assess how well they fit real data. In the literature, commonly used estimation methods for the SV models include general methods of moments, simulated maximum likelihood methods, quasi Maximum likelihood method, and Markov Chain Monte Carlo (MCMC) methods. Among these approaches, MCMC methods are most flexible in dealing with complicated structure of the models. However, due to the difficulty in the selection of the proposal distribution for Metropolis-Hastings methods, in general they are not easy to implement and in some cases we may also encounter convergence problems in the implementation stage. In the light of these concerns, we propose in this thesis new estimation methods for univariate and multivariate SV models. In the simulation of latent states of the heavy-tailed SV models, we recommend the slice sampler algorithm as the main tool to sample the proposal distribution when the Metropolis-Hastings method is applied. For the SV models without heavy tails, a simple Metropolis-Hastings method is developed for simulating the latent states. Since the slice sampler can adapt to the analytical structure of the underlying density, it is more efficient. A sample point can be obtained from the target distribution with a few iterations of the sampler, whereas in the original Metropolis-Hastings method many sampled values often need to be discarded. In the analysis of multivariate time series, multivariate SV models with more general specifications have been proposed to capture the correlations between the innovations of the asset returns and those of the latent volatility processes. Due to some restrictions on the variance-covariance matrix of the innovation vectors, the estimation of the multivariate SV (MSV) model is challenging. To tackle this issue, for a very general setting of a MSV model we propose a straightforward MCMC method in which a Metropolis-Hastings method is employed to sample the constrained variance-covariance matrix, where the proposal distribution is an inverse Wishart distribution. Again, the log volatilities of the asset returns can then be simulated via a single-move slice sampler. Recently, factor SV models have been proposed to extract hidden market changes. Geweke and Zhou (1996) propose a factor SV model based on factor analysis to measure pricing errors in the context of the arbitrage pricing theory by letting the factors follow the univariate standard normal distribution. Some modification of this model have been proposed, among others, by Pitt and Shephard (1999a) and Jacquier et al. (1999). The main feature of the factor SV models is that the factors follow a univariate SV process, where the loading matrix is a lower triangular matrix with unit entries on the main diagonal. Although the factor SV models have been successful in practice, it has been recognized that the order of the component may affect the sample likelihood and the selection of the factors. Therefore, in applications, the component order has to be considered carefully. For instance, the factor SV model should be fitted to several permutated data to check whether the ordering affects the estimation results. In the thesis, a new factor SV model is proposed. Instead of setting the loading matrix to be lower triangular, we set it to be column-orthogonal and assume that each column has unit length. Our method removes the permutation problem, since when the order is changed then the model does not need to be refitted. Since a strong assumption is imposed on the loading matrix, the estimation seems even harder than the previous factor models. For example, we have to sample columns of the loading matrix while keeping them to be orthonormal. To tackle this issue, we use the Metropolis-Hastings method to sample the loading matrix one column at a time, while the orthonormality between the columns is maintained using the technique proposed by Hoff (2007). A von Mises-Fisher distribution is sampled and the generated vector is accepted through the Metropolis-Hastings algorithm. Simulation studies and applications to real data are conducted to examine our inference methods and test the fit of our model. Empirical evidence illustrates that our slice sampler within MCMC methods works well in terms of parameter estimation and volatility forecast. Examples using financial asset return data are provided to demonstrate that the proposed factor SV model is able to characterize the hidden market factors that mainly govern the financial time series. The Kolmogorov-Smirnov tests conducted on the estimated models indicate that the models do a reasonable job in terms of describing real data.

El objetivo de esta tesis es modelar y predecir la volatilidad de las series financieras con modelos de volatilidad en tiempo discreto y continuo.En mi primer capítulo, intento modelar las principales características de las series financieras, como a persistencia y curtosis. Los modelos de volatilidad estocástica estimados son extensiones directas de los modelos de Gallant y Tauchen (2001), donde incluyo un elemento de retro-alimentación. Este elemento es de extrema importancia porque permite captar el hecho de que períodos de alta volatilidad están, en general, seguidos de periodos de gran volatilidad y viceversa. En este capítulo, como en toda la tesis, uso el método de estimación eficiente de momentos de Gallant y Tauchen (1996). De la estimación surgen dos modelos posibles de describir los datos, el modelo logarítmico con factor de volatilidad y retroalimentación y el modelo logarítmico con dos factores de volatilidad. Como no es posible elegir entre ellos basados en los tests efectuados en la fase de la estimación, tendremos que usar el método de reprogección para obtener mas herramientas de comparación. El modelo con un factor de volatilidad se comporta muy bien y es capaz de captar la "quiebra" de los mercados financieros de 1987.En el segundo capítulo, hago la evaluación del modelo con dos factores de volatilidad en términos de predicción y comparo esa predicción con las obtenidas con los modelos GARCH y ARFIMA. La evaluación de la predicción para los tres modelos es hecha con la ayuda del R2 de las regresiones individuales de la volatilidad "realizada" en una constante y en las predicciones. Los resultados empíricos indican un mejor comportamiento del modelo en tiempo continuo. Es más, los modelos GARCH y ARFIMA parecen tener problemas en seguir la marcha de la volatilidad "realizada". Finalmente, en el tercer capítulo hago una extensión del modelo de volatilidad estocástica de memoria larga de Harvey (2003). O sea, introduzco un factor de volatilidad de corto plazo. Este factor extra aumenta la curtosis y ayuda a captar la persistencia (que es captada con un proceso integrado fraccional, como en Harvey (1993)). Los resultados son probados y el modelo implementado empíricamente. / The purpose of my thesis is to model and forecast the volatility of the financial series of returns by using both continuous and discrete time stochastic volatility models.In my first chapter I try to fit the main characteristics of the financial series of returns such as: volatility persistence, volatility clustering and fat tails of the distribution of the returns.The estimated logarithmic stochastic volatility models are direct extensions of the Gallant and Tauchen's (2001) by including the feedback feature. This feature is of extreme importance because it allows to capture the low variability of the volatility factor when the factor is itself low (volatility clustering) and it also captures the increase in volatility persistence that occurs when there is an apparent change in the pattern of volatility at the very end of the sample. In this chapter, as well as in all the thesis, I use Efficient Method of Moments of Gallant and Tauchen (1996) as an estimation method. From the estimation step, two models come out, the logarithmic model with one factor of volatility and feedback (L1F) and the logarithmic model with two factors of volatility (L2). Since it is not possible to choose between them based on the diagnostics computed at the estimation step, I use the reprojection step to obtain more tools for comparing models. The L1F is able to reproject volatility quite well without even missing the crash of 1987.In the second chapter I fit the continuous time model with two factors of volatility of Gallant and Tauchen (2001) for the return of a Microsoft share. The aim of this chapter is to evaluate the volatility forecasting performance of the continuous time stochastic volatility model comparatively to the ones obtained with the traditional GARCH and ARFIMA models. In order to inquire into this, I estimate using the Efficient Method of Moments (EMM) of Gallant and Tauchen (1996) a continuous time stochastic volatility model for the logarithm of asset price and I filter the underlying volatility using the reprojection technique of Gallant and Tauchen (1998). Under the assumption that the model is correctly specified, I obtain a consistent estimator of the integrated volatility by fitting a continuous time stochastic volatility model to the data. The forecasting evaluation for the three estimated models is going to be done with the help of the R2 of the individual regressions of realized volatility on the volatility forecasts obtained from the estimated models. The empirical results indicate the better performance of the continuous time model in the out-of-sample periods compared to the ones of the traditional GARCH and ARFIMA models. Further, these two last models show difficulties in tracking the growth pattern of the realized volatility. This probably is due to the change of pattern in volatility in this last part of the sample. Finally, in the third chapter I come back to the model specification and I extend the long memory stochastic volatility model of Harvey (1993) by introducing a short run volatility factor. This extra factor increases kurtosis and helps the model capturing volatility persistence (that it is captured by a fractionally integrated process as in Harvey (1993) ). Futhermore, considering some restrictions of the parameters it is possible to fit the empirical fact of small first order autocorrelation of squared returns. All these results are proved theoretically and the model is implemented empirically using the S&P 500 composite index returns. The empirical results show the superiority of the model in fitting the main empirical facts of the financial series of returns.

In this paper we present a finite volume method for a two-dimensional Black-Scholes equation with stochastic volatility governing European option pricing. In this work, we first formulate the Black-Scholes equation with a tensor (or matrix) diffusion coefficient into a conversative form. We then present a finite volume method for the resulting equation, based on a fitting technique proposed for a one-dimensional Black-Scholes equation. We show that the method is monotone by proving that the system matrix of the discretized equation is an M-matrix. Numerical experiments, performed to demonstrate the usefulness of the method, will be presented.

This dissertation presents four essays. The first chapter builds a real-options, term structure model of the firm incorporating stochastic volatility and endogenous default to shed new light on the value premium, financial distress, momentum, and credit spread puzzles. The paper uses recently developed methodologies based on asymptotic expansions to solve the model. The second chapter, coauthored with Adam Guren, presents a model that shows how foreclosures can exacerbate a housing bust and delay the housing market's recovery. Quantitatively, the model successfully explains aggregate and retail price declines, the foreclosure share of volume, and the number of foreclosures both nationwide and across MSAs. The third and fourth chapters, coauthored with Stephen W. Salant and Jason Winfree, discuss the economics of untraceable experience goods in a variety of settings. The third chapter drops the "small country" assumption in the trade literature on collective reputation and shows how large exporters like China can address severe problems assuring the quality of its exports. The fourth chapter demonstrates how regulations in the formal sector of developing countries can lead to a quality gap between formal and informal sector goods. It moreover investigates how changes in regulation affect quality, price, aggregate production, and the number of firms in each sector. / Economics

Problem: The standard Black-Scholes framework cannot incorporate the volatility smiles usually observed in the markets. Instead, one must consider alternative stochastic volatility models such as the SABR. Little research about the suitability of the SABR model for Swedish market (swaption) data has been found. Purpose: The purpose of this paper is to account for and to calibrate the SABR model for swaptions trading on the Swedish market. We intend to alter the calibration techniques and parameter values to examine which method is the most consistent with the market. Method: In MATLAB, we investigate the model using two different minimization techniques to estimate the model’s parameters. For both techniques, we also implement refinements of the original SABR model. Results and Conclusion: The quality of the fit relies heavily on the underlying data. For the data used, we find superior fit for many different swaption smiles. In addition, little discrepancy in the quality of the fit between methods employed is found. We conclude that estimating the α parameter from at-the-money volatility produces slightly smaller errors than using minimization techniques to estimate all parameters. Using refinement techniques marginally increase the quality of the fit.

In this Master thesis, we use a singular and regular perturbation theory to derive an analytic approximation formula for the expected discounted penalty function. Our model is an extension of Cramer–Lundberg extended classical model because we consider a more general insurance risk model in which the compound Poisson risk process is perturbed by a Brownian motion multiplied by a stochastic volatility driven by two factors- which have mean reversion models. Moreover, unlike the classical model, our model allows a ruin to be caused either by claims or by surplus’ fluctuation. We compute explicitly the first terms of the asymptotic expansion and we show that they satisfy either an integro-differential equation or a Poisson equation. In addition, we derive the existence and uniqueness conditions of the risk model with two stochastic volatilities factors.

Stochastic volatility (SV) models provide a natural framework for a representation of time series for financial asset returns. As a result, they have become increasingly popular in the finance literature, although they have also been applied in other fields such as signal processing, telecommunications, engineering, biology, and other areas. In working with the SV models, an important issue arises as how to estimate their parameters efficiently and to assess how well they fit real data. In the literature, commonly used estimation methods for the SV models include general methods of moments, simulated maximum likelihood methods, quasi Maximum likelihood method, and Markov Chain Monte Carlo (MCMC) methods. Among these approaches, MCMC methods are most flexible in dealing with complicated structure of the models. However, due to the difficulty in the selection of the proposal distribution for Metropolis-Hastings methods, in general they are not easy to implement and in some cases we may also encounter convergence problems in the implementation stage. In the light of these concerns, we propose in this thesis new estimation methods for univariate and multivariate SV models. In the simulation of latent states of the heavy-tailed SV models, we recommend the slice sampler algorithm as the main tool to sample the proposal distribution when the Metropolis-Hastings method is applied. For the SV models without heavy tails, a simple Metropolis-Hastings method is developed for simulating the latent states. Since the slice sampler can adapt to the analytical structure of the underlying density, it is more efficient. A sample point can be obtained from the target distribution with a few iterations of the sampler, whereas in the original Metropolis-Hastings method many sampled values often need to be discarded. In the analysis of multivariate time series, multivariate SV models with more general specifications have been proposed to capture the correlations between the innovations of the asset returns and those of the latent volatility processes. Due to some restrictions on the variance-covariance matrix of the innovation vectors, the estimation of the multivariate SV (MSV) model is challenging. To tackle this issue, for a very general setting of a MSV model we propose a straightforward MCMC method in which a Metropolis-Hastings method is employed to sample the constrained variance-covariance matrix, where the proposal distribution is an inverse Wishart distribution. Again, the log volatilities of the asset returns can then be simulated via a single-move slice sampler. Recently, factor SV models have been proposed to extract hidden market changes. Geweke and Zhou (1996) propose a factor SV model based on factor analysis to measure pricing errors in the context of the arbitrage pricing theory by letting the factors follow the univariate standard normal distribution. Some modification of this model have been proposed, among others, by Pitt and Shephard (1999a) and Jacquier et al. (1999). The main feature of the factor SV models is that the factors follow a univariate SV process, where the loading matrix is a lower triangular matrix with unit entries on the main diagonal. Although the factor SV models have been successful in practice, it has been recognized that the order of the component may affect the sample likelihood and the selection of the factors. Therefore, in applications, the component order has to be considered carefully. For instance, the factor SV model should be fitted to several permutated data to check whether the ordering affects the estimation results. In the thesis, a new factor SV model is proposed. Instead of setting the loading matrix to be lower triangular, we set it to be column-orthogonal and assume that each column has unit length. Our method removes the permutation problem, since when the order is changed then the model does not need to be refitted. Since a strong assumption is imposed on the loading matrix, the estimation seems even harder than the previous factor models. For example, we have to sample columns of the loading matrix while keeping them to be orthonormal. To tackle this issue, we use the Metropolis-Hastings method to sample the loading matrix one column at a time, while the orthonormality between the columns is maintained using the technique proposed by Hoff (2007). A von Mises-Fisher distribution is sampled and the generated vector is accepted through the Metropolis-Hastings algorithm. Simulation studies and applications to real data are conducted to examine our inference methods and test the fit of our model. Empirical evidence illustrates that our slice sampler within MCMC methods works well in terms of parameter estimation and volatility forecast. Examples using financial asset return data are provided to demonstrate that the proposed factor SV model is able to characterize the hidden market factors that mainly govern the financial time series. The Kolmogorov-Smirnov tests conducted on the estimated models indicate that the models do a reasonable job in terms of describing real data.

This thesis makes a number of contributions in the derivative pricing and risk management literature and to the growing literature that exploits information embedded in option prices. First, it develops an effective numerical scheme for importance sampling scheme of Fouque and Tullie (2002) based on a 2-dimensional lookup table of stock price and time to maturity that dramatically improves the speed of this importance sampling scheme. Second, the thesis presents an application of this importance sampling scheme in a Multi-Level Monte Carlo simulation. Such combination yields greater variance reduction compared to Multi-Level Monte Carlo or importance sampling alone. Third, it demonstrates how the Greeks can be computed using the Likelihood Ratio Method based on characteristic function, and how combining it with importance sampling leads to a significant variance reduction for the Greeks. Finally, it documents the positive relationship between the risk-neutral skewness (RNS) and future realized stock returns that is driven by the underperformance of highly negative RNS portfolio. The results provide strong evidence that the underperformance of stocks with the lowest RNS is driven by those stocks that are associated with a higher hedging demand, relative overvaluation and are also too costly or too risky to sell short. Moreover, by decomposing RNS into its systematic and idiosyncratic components, this thesis shows that the latter drives the positive relationship with future realized stock returns.

<div>We develop a growth option and asset pricing model that incorporates uncertain cash flow volatility by way of a bounded quadratic diffusion. Using different measures of risk uncertainty, we study the combined effects of risk and its associated uncertainty on project values, firm investment, and the resulting returns. Uncertain cash flow volatility is modeled by a Jacobi process, and our main interest is the effect of the max uncertainty arising from the diffusion term. For comparison, we also model the volatility by a CIR process. In regards to the Jacobi process, we consider upper and lower bounds on cash flow volatility as measures of uncertainty. For the max uncertainty and upper bound, we find that higher uncertainty leads to less investment, higher returns, and lower project values. In the case of the lower bound, we find that higher uncertainty leads to more investment, lower returns, and higher project values. Comparatively, using a CIR process in place of the Jacobi process yields differences in returns and growth option values, showing the importance of the diffusion term in the volatility process. Finally, we have reduced the computational complexity of the simulation. This allows the user to generate long time series and run cross sectional regressions with many firms.</div>

In this dissertation, the price of variance swaps under stochastic volatility models based on the work done by Barndorff-Nielsen and Shepard (2001) and Heston (1993) is discussed. The choice of these models is as a result of properties they possess which position them as an improvement to the traditional Black-Scholes (1973) model. Furthermore, the popularity of these models in literature makes them particularly attractive. A lot of work has been done in the area of pricing variance swaps since their inception in the late 1990’s. The growth in the number of variance contracts written came as a result of investors’ increasing need to be hedged against exposure to future variance fluctuations. The task at the core of this dissertation is to derive closed or semi-closed form expressions of the fair price of variance swaps under the two stochastic models. Although various researchers have shown that stochastic models produce close to market results, it is more desirable to obtain the fair price of variance derivatives using models under which no assumptions about the dynamics of the underlying asset are made. This is the work of a useful analytical formula derived by Demeterfi, Derman, Kamal and Zou (1999) in which the price of variance swaps is hedged through a finite portfolio of European call and put options of different strike prices. This scheme is practically explored in an example. Lastly, conclusions on pricing using each of the methodologies are given. / Dissertation (MSc)--University of Pretoria, 2019. / Mathematics and Applied Mathematics / MSc (Financial Engineering) / Unrestricted