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Bayesian Inference for Stochastic Volatility ModelsMen, Zhongxian January 1012 (has links)
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.
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Modelling and forecasting stochastic volatilityLopes Moreira de Veiga, Maria Helena 19 April 2004 (has links)
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.
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A Multidimensional Fitted Finite Volume Method for the Black-Scholes Equation Governing Option PricingHung, Chen-Hui 05 July 2004 (has links)
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.
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Essays in Financial and Housing EconomicsMcQuade, Timothy 24 June 2014 (has links)
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
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The SABR Model : Calibrated for Swaption's Volatility Smile / SABR Modellen : Kalibrerad för Swaptioner med VolatilitetsleendeTran, Nguyen, Weigardh, Anton January 2014 (has links)
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.
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Asymptotic expansion of the expected discounted penalty function in a two-scalestochastic volatility risk model.Ouoba, Mahamadi January 2014 (has links)
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.
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Bayesian Inference for Stochastic Volatility ModelsMen, Zhongxian January 1012 (has links)
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.
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Numerical and empirical studies of option pricingStilger, Przemyslaw January 2014 (has links)
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.
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Uncertain Growth Options and Asset PricingBrian G Hogle (11059854) 22 July 2021 (has links)
<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>
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Pricing and hedging variance swaps using stochastic volatility modelsBopoto, Kudakwashe January 2019 (has links)
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
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