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Andersson, Kristina. "Stochastic Volatility". Thesis, Uppsala University, Department of Mathematics, 2003. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-121722.
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Galiotos, Vassilis. "Stochastic Volatility and the Volatility Smile". Thesis, Uppsala University, Department of Mathematics, 2008. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-120151.
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Le, Truc. "Stochastic volatility models". Monash University, School of Mathematical Sciences, 2005. http://arrow.monash.edu.au/hdl/1959.1/5181.
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Zeytun, Serkan. "Stochastic Volatility, A New Approach For Vasicek Model With Stochastic Volatility". Master's thesis, METU, 2005. http://etd.lib.metu.edu.tr/upload/3/12606561/index.pdf.
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In the original Vasicek model interest rates are calculated
assuming that volatility remains constant over the period of
analysis. In this study, we constructed a stochastic volatility
model for interest rates. In our model we assumed not only that interest rate process but also the volatility process for interest rates follows the mean-reverting Vasicek model. We derived the density function for the stochastic element of the interest rate process and reduced this density function to a series form. The parameters of our model were estimated by using the method of moments. Finally, we tested the performance of our model using the data of interest rates in Turkey.
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Cap, Thi Diu. "Implied volatility with HJM–type Stochastic Volatility model". Thesis, Mälardalens högskola, Akademin för utbildning, kultur och kommunikation, 2021. http://urn.kb.se/resolve?urn=urn:nbn:se:mdh:diva-54938.
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In this thesis, we propose a new and simple approach of extending the single-factor Heston stochastic volatility model to a more flexible one in solving option pricing problems. In this approach, the volatility process for the underlying asset dynamics depends on the time to maturity of the option. As this idea is inspired by the Heath-Jarrow-Morton framework which models the evolution of the full dynamics of forward rate curves for various maturities, we name this approach as the HJM-type stochastic volatility (HJM-SV) model. We conduct an empirical analysis by calibrating this model to real-market option data for underlying assets including an equity (ABB stock) and a market index (EURO STOXX 50), for two separated time spans from Jan 2017 to Dec 2017 (before the COVID-19 pandemic) and from Nov 2019 to Nov 2020 (after the start of COVID-19 pandemic). We investigate the optimal way of dividing the set of option maturities into three classes, namely, the short-maturity, middle-maturity, and long-maturity classes. We calibrate our HJM-SV model to the data in the following way, for each class a single-factor Heston stochastic volatility model is calibrated to the corresponding market data. We address the question that how well the new HJM-SV model captures the feature of implied volatility surface given by the market data.
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Jacquier, Antoine. "Implied volatility asymptotics under affine stochastic volatility models". Thesis, Imperial College London, 2010. http://hdl.handle.net/10044/1/6142.
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This thesis is concerned with the calibration of affine stochastic volatility models with jumps. This class of models encompasses most models used in practice and captures some of the common features of market data such as jumps and heavy tail distributions of returns. Two questions arise when one wants to calibrate such a model: (a) How to check its theoretical consistency with the relevant market characteristics? (b) How to calibrate it rigorously to market data, in particular to the so-called implied volatility, which is a normalised measure of option prices? These two questions form the backbone of this thesis, since they led to the following idea: instead of calibrating a model using a computer-intensive global optimisation algorithm, it should be more efficient to use a less robust—hence faster—algorithm, but with an accurate starting point. Henceforth deriving closed-form approximation formulae for the implied-volatility should provide a way to obtain such accurate initial points, thus ensuring a faster calibration. In this thesis we propose such a calibration approach based on the time-asymptotics of affine stochastic volatility models with jumps. Mathematically since this class of models is defined via its Laplace transform, the tools we naturally use are large deviations theory as well as complex saddle-point methods. Large deviations enable us to obtain the limiting behaviour (in small or large time) of the implied volatility, and saddle-point methods are needed to obtain more accurate results on the speed of convergence. We also provide numerical evidence in order to highlight the accuracy of the closed-form approximations thus obtained, and compare them to standard pricing methods based on real calibrated data.
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Ozkan, Pelin. "Analysis Of Stochastic And Non-stochastic Volatility Models". Master's thesis, METU, 2004. http://etd.lib.metu.edu.tr/upload/3/12605421/index.pdf.
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Changing in variance or volatility with time can be modeled as deterministic by using autoregressive conditional heteroscedastic (ARCH) type models, or as stochastic by using stochastic volatility (SV) models. This study compares these two kinds of models which are estimated on Turkish / USA exchange rate data. First, a GARCH(1,1) model is fitted to the data by using the package E-views and then a Bayesian estimation procedure is used for estimating an appropriate SV model with the help of Ox code. In order to compare these models, the LR test statistic calculated for non-nested hypotheses is obtained.
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Vavruška, Marek. "Realised stochastic volatility in practice". Master's thesis, Vysoká škola ekonomická v Praze, 2012. http://www.nusl.cz/ntk/nusl-165381.
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Realised Stochastic Volatility model of Koopman and Scharth (2011) is applied to the five stocks listed on NYSE in this thesis. Aim of this thesis is to investigate the effect of speeding up the trade data processing by skipping the cleaning rule requiring the quote data. The framework of the Realised Stochastic Volatility model allows the realised measures to be biased estimates of the integrated volatility, which further supports this approach. The number of errors in recorded trades has decreased significantly during the past years. Different sample lengths were used to construct one day-ahead forecasts of realised measures to examine the forecast precision sensitivity to the rolling window length. Use of the longest window length does not lead to the lowest mean square error. The dominance of the Realised Stochastic Volatility model in terms of the lowest mean square errors of one day-ahead out-of-sample forecasts has been confirmed.
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Hrbek, Filip. "Metody předvídání volatility". Master's thesis, Vysoká škola ekonomická v Praze, 2015. http://www.nusl.cz/ntk/nusl-264689.
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In this masterthesis I have rewied basic approaches to volatility estimating. These approaches are based on classical and Bayesian statistics. I have applied the volatility models for the purpose of volatility forecasting of a different foreign exchange (EURUSD, GBPUSD and CZKEUR) in the different period (from a second period to a day period). I formulate the models EWMA, GARCH, EGARCH, IGARCH, GJRGARCH, jump diffuison with constant volatility and jump diffusion model with stochastic volatility. I also proposed an MCMC algorithm in order to estimate the Bayesian models. All the models we estimated as univariate models. I compared the models according to Mincer Zarnowitz regression. The most successfull model is the jump diffusion model with a stochastic volatility. On the second place they were the GJR- GARCH model and the jump diffusion model with a constant volatility. But the jump diffusion model with a constat volatilit provided much more overvalued results.The rest of the models were even worse. From the rest the IGARCH model is the best but provided undervalued results. All these findings correspond with R squared coefficient.
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Lopes, Moreira de Veiga Maria Helena. "Modelling and forecasting stochastic volatility". Doctoral thesis, Universitat Autònoma de Barcelona, 2004. http://hdl.handle.net/10803/4046.
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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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Tsang, Wai-yin, i 曾慧賢. "Aspects of modelling stochastic volatility". Thesis, The University of Hong Kong (Pokfulam, Hong Kong), 2000. http://hub.hku.hk/bib/B31223515.
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Kovachev, Yavor. "Calibration of stochastic volatility models". Thesis, Uppsala universitet, Analys och sannolikhetsteori, 2014. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-227502.
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Tsiotas, Georgios K. "Nonlinearities in stochastic volatility models". Thesis, University of Essex, 2002. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.394112.
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PEREIRA, RICARDO VELA DE BRITTO. "VOLATILITY: A HIDDEN STOCHASTIC PROCESS". PONTIFÍCIA UNIVERSIDADE CATÓLICA DO RIO DE JANEIRO, 2010. http://www.maxwell.vrac.puc-rio.br/Busca_etds.php?strSecao=resultado&nrSeq=16816@1.
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CONSELHO NACIONAL DE DESENVOLVIMENTO CIENTÍFICO E TECNOLÓGICOA volatilidade é um parâmetro importante de modelagem do mercado financeiro. Ela controla a medida de risco associado à dinâmica estocástica de preço do título financeiro, afetando também o preço racional dos derivativos.Existe evidência empírica que a volatilidade é por sua vez também um processo estocástico, subjacente ao dos preços. Assim, a volatilidade não pode ser observada diretamente e tem que ser estimada, constituindo-se de um processo estocástico escondido.Nesta dissertação, consideramos um estimador para a volatilidade diária do índice da BOVESPA, baseado em banco de dados intradiários. Fazemos uma análise estatística descritiva da série temporal obtida, obtendo-se a função densidade de probabilidade, os momentos e as correlações. Comparamos os resultados empíricos com as previsões teóricas de vários modelos de volatilidade estocástica. Consideramos a classe de equações de Itô-Langevin formada por um processo de reversão à média e um processo difusivo de Wiener generalizado, com componentes de ruído multiplicativo e/ou aditivo. A partir dessa análise, é sugerido um modelo para descrever as flutuações de volatilidade dos preços do mercado acionário brasileiro.
Volatility is a key model parameter of the financial market. It controls the risk associated to the stochastic dynamics of the asset prices and also affects the rational price of derivative products. There are empirical evidences that the volatility is also a stochastic process, underlined to the price one. Therefore, the volatility is not directly observed and must be estimated, constituting a hidden stochastic process. In this work, we consider an estimate for the daily volatility of the BOVESPA index, computed from the intraday database. We perform a descriptive statistical analysis of the resulting time series, obtaining the probability density function, moments and correlations. We compare the empirical outcomes with the theoretical forecasts of many stochastic volatility models. We consider the class of Itô-Langevin equations composed by a mean reverting process and a generalized diffusive Wiener process with multiplicative and/or additive noise components. From this analysis, we propose a model that describes the volatility fluctuations of the Brazilian stock market.
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Chen, Jilong. "Pricing derivatives with stochastic volatility". Thesis, University of Glasgow, 2016. http://theses.gla.ac.uk/7703/.
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This Ph.D. thesis contains 4 essays in mathematical finance with a focus on pricing Asian option (Chapter 4), pricing futures and futures option (Chapter 5 and Chapter 6) and time dependent volatility in futures option (Chapter 7). In Chapter 4, the applicability of the Albrecher et al.(2005)'s comonotonicity approach was investigated in the context of various benchmark models for equities and com- modities. Instead of classical Levy models as in Albrecher et al.(2005), the focus is the Heston stochastic volatility model, the constant elasticity of variance (CEV) model and the Schwartz (1997) two-factor model. It is shown that the method delivers rather tight upper bounds for the prices of Asian Options in these models and as a by-product delivers super-hedging strategies which can be easily implemented. In Chapter 5, two types of three-factor models were studied to give the value of com- modities futures contracts, which allow volatility to be stochastic. Both these two models have closed-form solutions for futures contracts price. However, it is shown that Model 2 is better than Model 1 theoretically and also performs very well empiri- cally. Moreover, Model 2 can easily be implemented in practice. In comparison to the Schwartz (1997) two-factor model, it is shown that Model 2 has its unique advantages; hence, it is also a good choice to price the value of commodity futures contracts. Fur- thermore, if these two models are used at the same time, a more accurate price for commodity futures contracts can be obtained in most situations. In Chapter 6, the applicability of the asymptotic approach developed in Fouque et al.(2000b) was investigated for pricing commodity futures options in a Schwartz (1997) multi-factor model, featuring both stochastic convenience yield and stochastic volatility. It is shown that the zero-order term in the expansion coincides with the Schwartz (1997) two-factor term, with averaged volatility, and an explicit expression for the first-order correction term is provided. With empirical data from the natural gas futures market, it is also demonstrated that a significantly better calibration can be achieved by using the correction term as compared to the standard Schwartz (1997) two-factor expression, at virtually no extra effort. In Chapter 7, a new pricing formula is derived for futures options in the Schwartz (1997) two-factor model with time dependent spot volatility. The pricing formula can also be used to find the result of the time dependent spot volatility with futures options prices in the market. Furthermore, the limitations of the method that is used to find the time dependent spot volatility will be explained, and it is also shown how to make sure of its accuracy.
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Venter, Rudolf Gerrit. "Pricing options under stochastic volatility". Diss., Pretoria : [s.n.], 2003. http://upetd.up.ac.za/thesis/available/etd09052005-120952.
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Tsang, Wai-yin. "Aspects of modelling stochastic volatility /". Hong Kong : University of Hong Kong, 2000. http://sunzi.lib.hku.hk/hkuto/record.jsp?B22078952.
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Covaciu, Livia Andreea <1991>. "Stochastic volatility with big data". Master's Degree Thesis, Università Ca' Foscari Venezia, 2015. http://hdl.handle.net/10579/6933.
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The thesis aims to discuss stochastic volatility when a big amount of data is involved. Therefore I follow Windle and Carvalho (2015) and Casarin (2015) papers where a state-space model for observations and latent variables in the space of positive symmetric matrices is introduced. Moreover, I use Gibbs sample and MCMC method in order to discuss the Bayesian inference. One-step ahead and multi-step-ahead forecasting are evaluated because of their importance in economics and business. Since this model can have important applications in finance, one can use realized covariance matrices as data to predict latent time-varying covariance matrices. I present factor-like models, GARCH-like model and univariate stochastic volatility models to give an alternative to the model from the mentioned papers.
It is known that financial markets data often expose volatility clustering, where time series have periods of high volatility and periods of low
volatility. As a matter of fact, time-varying volatility appears more than constant volatility, and accurate modelling of time-varying volatility is of great importance, considering economic and financial data. In our case working with a nonlinear model by using MCMC posterior approximation can be a quite challenging issue. Computational time in Monte Carlo simulations is reduced by implementing a parallel algorithm in Matlab which is able to split our database and run the blocks in the same time.
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Abi, Jaber Eduardo. "Stochastic Invariance and Stochastic Volterra Equations". Thesis, Paris Sciences et Lettres (ComUE), 2018. http://www.theses.fr/2018PSLED025/document.
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La présente thèse traite de la théorie des équations stochastiques en dimension finie. Dans la première partie, nous dérivons des conditions géométriques nécessaires et suffisantes sur les coefficients d’une équation différentielle stochastique pour l’existence d’une solution contrainte à rester dans un domaine fermé, sous de faibles conditions de régularité sur les coefficients.Dans la seconde partie, nous abordons des problèmes d’existence et d’unicité d’équations de Volterra stochastiques de type convolutif. Ces équations sont en général non-Markoviennes. Nous établissons leur correspondance avec des équations en dimension infinie ce qui nous permet de les approximer par des équations différentielles stochastiques Markoviennes en dimension finie.Enfin, nous illustrons nos résultats par une application en finance mathématique, à savoir la modélisation de la volatilité rugueuse. En particulier, nous proposons un modèle à volatilité stochastique assurant un bon compromis entre flexibilité et tractabilitéThe present thesis deals with the theory of finite dimensional stochastic equations.In the first part, we derive necessary and sufficient geometric conditions on the coefficients of a stochastic differential equation for the existence of a constrained solution, under weak regularity on the coefficients. In the second part, we tackle existence and uniqueness problems of stochastic Volterra equations of convolution type. These equations are in general non-Markovian. We establish their correspondence with infinite dimensional equations which allows us to approximate them by finite dimensional stochastic differential equations of Markovian type. Finally, we illustrate our findings with an application to mathematical finance, namely rough volatility modeling. We design a stochastic volatility model with an appealing trade-off between flexibility and tractability
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Broodryk, Ryan. "The Lifted Heston Stochastic Volatility Model". Master's thesis, Faculty of Commerce, 2021. http://hdl.handle.net/11427/32614.
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Can we capture the explosive nature of volatility skew observed in the market, without resorting to non-Markovian models? We show that, in terms of skew, the Heston model cannot match the market at both long and short maturities simultaneously. We introduce Abi Jaber (2019)'s Lifted Heston model and explain how to price options with it using both the cosine method and standard Monte-Carlo techniques. This allows us to back out implied volatilities and compute skew for both models, confirming that the Lifted Heston nests the standard Heston model. We then produce and analyze the skew for Lifted Heston models with a varying number N of mean reverting terms, and give an empirical study into the time complexity of increasing N. We observe a weak increase in convergence speed in the cosine method for increased N, and comment on the number of factors to implement for practical use.
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Choi, Chiu Yee. "A multivariate threshold stochastic volatility model /". View abstract or full-text, 2005. http://library.ust.hk/cgi/db/thesis.pl?MATH%202005%20CHOI.
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Kalavrezos, Michail, i Michael Wennermo. "Stochastic Volatility Models in Option Pricing". Thesis, Mälardalen University, Department of Mathematics and Physics, 2008. http://urn.kb.se/resolve?urn=urn:nbn:se:mdh:diva-538.
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In this thesis we have created a computer program in Java language which calculates European call- and put options with four different models based on the article The Pricing of Options on Assets with Stochastic Volatilities by John Hull and Alan White. Two of the models use stochastic volatility as an input. The paper describes the foundations of stochastic volatility option pricing and compares the output of the models. The model which better estimates the real option price is dependent on further research of the model parameters involved.
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Aldberg, Henrik. "Bond Pricing in Stochastic Volatility Models". Thesis, Uppsala University, Department of Mathematics, 2008. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-120524.
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Bjarnason, Thorir. "Stochastic volatility, convex prices and bubbles". Thesis, Uppsala University, Department of Mathematics, 2007. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-120913.
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Malaikah, Honaida Muhammed S. "Stochastic volatility models and memory effect". Thesis, University of Manchester, 2011. https://www.research.manchester.ac.uk/portal/en/theses/stochastic-volatility-models-and-mempry-effect(424f6c71-a0e7-44ba-afbb-cc5f74ae075c).html.
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Sandmann, Gleb. "Stochastic volatility : estimation and empirical validity". Thesis, London School of Economics and Political Science (University of London), 1997. http://etheses.lse.ac.uk/1456/.
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Estimation of stochastic volatility (SV) models is a formidable task because the presence of the latent variable makes the likelihood function difficult to construct. The model can be transformed to a linear state space with non-Gaussian disturbances. Durbin and Koopman (1997) have shown that the likelihood function of the general non-Gaussian state space model can be approximated arbitrarily accurately by decomposing it into a Gaussian part (constructed by the Kalman filter) and a remainder function (whose expectation is evaluated by simulation). This general methodology is specialised to the estimation of SV models. A finite sample simulation experiment illustrates that the resulting Monte Carlo likelihood estimator achieves full efficiency with minimal computational effort. Accurate values of the likelihood function allow inference within the model to be performed by means of likelihood ratio tests. This enables tests for the presence of a unit root in the volatility process to be constructed which are shown to be more powerful than the conventional unit root tests. The second part of the thesis consists of two empirical applications of the SV model. First, the informational content of implied volatility is examined. It is shown that the in- sample evolution of DEM/USD exchange rate volatility can be accurately captured by implied volatility of options. However, better forecasts of ex post volatility can be constructed from the basic SV model. This suggests that options implied volatility may not be market's best forecast of the future asset volatility, as is often assumed. Second, the regulatory claim of a destabilising effect of futures market trading on stock market volatility is critically assessed. It is shown how volume-volatility relationships can be accurately modelled in the SV framework. The variables which approximate the activity in the FT100 index futures market are found to have no influence on the volatility of the underlying stock market index.
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Guo, Chuan. "The stochastic volatility Markov-functional model". Thesis, University of Warwick, 2016. http://wrap.warwick.ac.uk/91418/.
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In this thesis we study low-dimensional stochastic volatility interest rate models for pricing and hedging exotic derivatives. In particular we develop a stochastic volatility Markov-functional model. In order to implement the model numerically, we further propose a general algorithm by working with basis functions and conditional moments of the driving Markov process. Motivated by a data driven study, we choose a SABR type model as a driving process. With this choice we specify a pre-model and develop an approximation to evaluate conditional moments of the SABR driver which serve as building blocks for the practical algorithm. Having discussed how to set up a stochastic volatility Markov-functional model next we study the calibration of a LIBOR based version of the model with the SABR type driving process. We consider a link between separable SABR LIBOR market models and stochastic volatility LIBOR Markov-functional models. Based on the link we propose a calibration routine to feed in SABR marginals by calibrating to the market vanilla options. Moreover we choose the parameters of the SABR driver by fitting to the market correlation structure. We compare the stochastic volatility Markov-functional model developed in the thesis with one-dimensional (non-stochastic-volatility) swap Markov-functional models in terms of pricing and hedging Bermudan type products. By doing so we investigate effects of correlation structure, implied volatility smiles and the introduction of stochastic volatility on Bermudan type products. Finally we compare Quasi-Gaussian models with Markov-functional models in terms of specification and calibration. In particular we study Quasi-Gaussian models formulated in the Markov-functional model framework to make clear the relationship between the two models.
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Pham, Duy. "Markov-functional and stochastic volatility modelling". Thesis, University of Warwick, 2012. http://wrap.warwick.ac.uk/55161/.
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In this thesis, we study two practical problems in applied mathematical fi nance. The first topic discusses the issue of pricing and hedging Bermudan swaptions within a one factor Markov-functional model. We focus on the implications for hedging of the choice of instantaneous volatility for the one-dimensional driving Markov process of the model. We find that there is a strong evidence in favour of what we term \parametrization by time" as opposed to \parametrization by expiry". We further propose a new parametrization by time for the driving process which takes as inputs into the model the market correlations of relevant swap rates. We show that the new driving process enables a very effective vega-delta hedge with a much more stable gamma profile for the hedging portfolio compared with the existing ones. The second part of the thesis mainly addresses the topic of pricing European options within the popular stochastic volatility SABR model and its extension with mean reversion. We investigate some effcient approximations for these models to be used in real time. We first derive a probabilistic approximation for three different versions of the SABR model: Normal, Log-Normal and a displaced diffusion version for the general constant elasticity of variance case. Specifically, we focus on capturing the terminal distribution of the underlying process (conditional on the terminal volatility) to arrive at the implied volatilities of the corresponding European options for all strikes and maturities. Our resulting method allows us to work with a variety of parameters which cover long dated options and highly stress market condition. This is a different feature from other current approaches which rely on the assumption of very small total volatility and usually fail for longer than 10 years maturity or large volatility of volatility. A similar study is done for the extension of the SABR model with mean reversion (SABR-MR). We first compare the SABR model with this extended model in terms of forward volatility to point out the fundamental difference in the dynamics of the two models. This is done through a numerical example of pricing forward start options. We then derive an effcient probabilistic approximation for the SABRMR model to price European options in a similar fashion to the one for the SABR model. The numerical results are shown to be still satisfactory for a wide range of market conditions.
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Murara, Jean-Paul. "Asset Pricing Models with Stochastic Volatility". Licentiate thesis, Mälardalens högskola, Utbildningsvetenskap och Matematik, 2016. http://urn.kb.se/resolve?urn=urn:nbn:se:mdh:diva-31576.
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Asset pricing modeling is a wide range area of research in Financial Engineering. In this thesis, which consists of an introduction, three papers and appendices; we deal with asset pricing models with stochastic volatility. Here stochastic volatility modeling includes diffusion models and regime-switching models. Stochastic volatility models appear as a response to the weakness of the constant volatility models. In Paper A , we present a survey on popular diffusion models where the volatility is itself a random process and we present the techniques of pricing European options under each model. Comparing single factor stochastic volatility models to constant factor volatility models it seems evident that the stochastic volatility models represent nicely the movement of the asset price and its relations with changes in the risk. However, these models fail to explain the large independent fluctuations in the volatility levels and slope. We consider Chiarella and Ziveyi model, which is a subclass of the model presented in Christoffersen and in paper A, we also explain a multi-factor stochastic volatility model presented in Chiarella and Ziveyi. We review the first-order asymptotic expansion method for determining European option price in such model. Multiscale stochastic volatilities models can capture the smile and skew of volatilities and therefore describe more accurately the movements of the trading prices. In paper B, we provide experimental and numerical studies on investigating the accuracy of the approximation formulae given by this asymptotic expansion. We present also a procedure for calibrating the parameters produced by our first-order asymptotic approximation formulae. Our approximated option prices will be compared to the approximation obtained by Chiarella and Ziveyi. In paper C, we implement and analyze the Regime-Switching GARCH model using real NordPool Electricity spot data. We allow the model parameters to switch between a regular regime and a non-regular regime, which is justified by the so-called structural break behaviour of electricity price series. In splitting the two regimes we consider three criteria, namely the intercountry price di_erence criterion, the capacity/flow difference criterion and the spikes-in-Finland criterion. We study the correlation relationships among these criteria using the mean-square contingency coe_cient and the co-occurrence measure. We also estimate our model parameters and present empirical validity of the model.
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Chen, Ke. "Essays on stochastic volatility and jumps". Thesis, University of Manchester, 2013. https://www.research.manchester.ac.uk/portal/en/theses/essays-on-stochastic-volatility-and-jumps(7ce79e77-2806-443e-84c1-8b3ec922cc9f).html.
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This thesis studies a few different finance topics on the application and modelling of jump and stochastic volatility process. First, the thesis proposed a non-parametric method to estimate the impact of jump dependence, which is important for portfolio selection problem. Comparing with existing literature, the new approach requires much less restricted assumption on the jump process, and estimation results suggest that the economical significance of jumps is largely mis-estimated in portfolio optimization problem. Second, this thesis investigates the time varying variance risk premium, in a framework of stochastic volatility with stochastic jump intensity. The proposed model considers jump intensity as an extra factor which is driven by realized jumps, in addition to a stochastic volatility model. The results provide strong evidence of multiple factors in the market and show how they drive the variance risk premium. Thirdly, the thesis uses the proposed models to price options on equity and VIX consistently. Based on calibrated model parameters, the thesis shows how to calculate the unconditional correlation of VIX future between different maturities.
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Yoon, Jungyeon Ji Chuanshu. "Option pricing with stochastic volatility models". Chapel Hill, N.C. : University of North Carolina at Chapel Hill, 2008. http://dc.lib.unc.edu/u?/etd,1964.
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Thesis (Ph. D.)--University of North Carolina at Chapel Hill, 2008.Title from electronic title page (viewed Dec. 11, 2008). "... in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the Department of Statistics and Operations Research Statistics." Discipline: Statistics and Operations Research; Department/School: Statistics and Operations Research.
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Chen, Huaizhi. "Estimating Stochastic Volatility Using Particle Filters". Cleveland, Ohio : Case Western Reserve University, 2009. http://rave.ohiolink.edu/etdc/view?acc_num=case1247125250.
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Thesis (M.S.)--Case Western Reserve University, 2009Title from PDF (viewed on 19 August 2009) Department of Mathematics Includes abstract Includes bibliographical references Available online via the OhioLINK ETD Center
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Terenzi, Giulia. "Option prices in stochastic volatility models". Thesis, Paris Est, 2018. http://www.theses.fr/2018PESC1132/document.
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L’objet de cette thèse est l’étude de problèmes d’évaluation d’options dans les modèles à volatilité stochastique. La première partie est centrée sur les options américaines dans le modèle de Heston. Nous donnons d’abord une caractérisation analytique de la fonction de valeur d’une option américaine comme l’unique solution du problème d’obstacle parabolique dégénéré associé. Notre approche est basée sur des inéquations variationelles dans des espaces de Sobolev avec poids étendant les résultats récents de Daskalopoulos et Feehan (2011, 2016) et Feehan et Pop (2015). On étudie aussi les propriétés de la fonction de valeur d’une option américaine. En particulier, nous prouvons que, sous des hypothèses convenables sur le payoff, la fonction de valeur est décroissante par rapport à la volatilité. Ensuite nous nous concentrons sur le put américaine et nous étendons quelques résultats qui sont bien connus dans le monde Black-Scholes. En particulier nous prouvons la convexité stricte de la fonction de valeur dans la région de continuation, quelques propriétés de la frontière libre, la formule de Prime d’Exercice Anticipée et une forme faible de la propriété du smooth fit. Les techniques utilisées sont de type probabiliste. Dans la deuxième partie nous abordons le problème du calcul numérique du prix des options européennes et américaines dans des modèles à volatilité stochastiques et avec sauts. Nous étudions d’abord le modèle de Bates-Hull-White, c’est-à-dire le modèle de Bates avec un taux d’intérêt stochastique. On considère un algorithme hybride rétrograde qui utilise une approximation par chaîne de Markov (notamment un arbre “avec sauts multiples”) dans la direction de la volatilité et du taux d’intérêt et une approche (déterministe) par différence finie pour traiter le processus de prix d’actif. De plus, nous fournissons une procédure de simulation pour des évaluations Monte Carlo. Les résultats numériques montrent la fiabilité et l’efficacité de ces méthodes. Finalement, nous analysons le taux de convergence de l’algorithme hybride appliqué à des modèles généraux de diffusion avec sauts. Nous étudions d’abord la convergence faible au premier ordre de chaînes de Markov vers la diffusion sous des hypothèses assez générales. Ensuite nous prouvons la convergence de l’algorithme: nous étudions la stabilité et la consistance de la méthode hybride par une technique qui exploite les caractéristiques probabilistes de l’approximation par chaîne de MarkovWe study option pricing problems in stochastic volatility models. In the first part of this thesis we focus on American options in the Heston model. We first give an analytical characterization of the value function of an American option as the unique solution of the associated (degenerate) parabolic obstacle problem. Our approach is based on variational inequalities in suitable weighted Sobolev spaces and extends recent results of Daskalopoulos and Feehan (2011, 2016) and Feehan and Pop (2015). We also investigate the properties of the American value function. In particular, we prove that, under suitable assumptions on the payoff, the value function is nondecreasing with respect to the volatility variable. Then, we focus on an American put option and we extend some results which are well known in the Black and Scholes world. In particular, we prove the strict convexity of the value function in the continuation region, some properties of the free boundary function, the Early Exercise Price formula and a weak form of the smooth fit principle. This is done mostly by using probabilistic techniques.In the second part we deal with the numerical computation of European and American option prices in jump-diffusion stochastic volatility models. We first focus on the Bates-Hull-White model, i.e. the Bates model with a stochastic interest rate. We consider a backward hybrid algorithm which uses a Markov chain approximation (in particular, a “multiple jumps” tree) in the direction of the volatility and the interest rate and a (deterministic) finite-difference approach in order to handle the underlying asset price process. Moreover, we provide a simulation scheme to be used for Monte Carlo evaluations. Numerical results show the reliability and the efficiency of the proposed methods.Finally, we analyze the rate of convergence of the hybrid algorithm applied to general jump-diffusion models. We study first order weak convergence of Markov chains to diffusions under quite general assumptions. Then, we prove the convergence of the algorithm, by studying the stability and the consistency of the hybrid scheme, in a sense that allows us to exploit the probabilistic features of the Markov chain approximation
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Salikhova, Alsu <1982>. "Stochastic Volatility Analysis for Hedge Funds". Master's Degree Thesis, Università Ca' Foscari Venezia, 2013. http://hdl.handle.net/10579/3351.
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Ahy, Nathaniel, i Mikael Sierra. "Implied Volatility Surface Approximation under a Two-Factor Stochastic Volatility Model". Thesis, Mälardalens högskola, Akademin för utbildning, kultur och kommunikation, 2018. http://urn.kb.se/resolve?urn=urn:nbn:se:mdh:diva-40039.
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Due to recent research disproving old claims in financial mathematics such as constant volatility in option prices, new approaches have been incurred to analyze the implied volatility, namely stochastic volatility models. The use of stochastic volatility in option pricing is a relatively new and unexplored field of research with a lot of unknowns, where new answers are of great interest to anyone practicing valuation of derivative instruments such as options. With both single and two-factor stochastic volatility models containing various correlation structures with respect to the asset price and differing mean-reversions of variance the question arises as to how these values change their more observable counterpart: the implied volatility. Using the semi-analytical formula derived by Chiarella and Ziveyi, we compute European call option prices. Then, through the Black–Scholes formula, we solve for the implied volatility by applying the bisection method. The implied volatilities obtained are then approximated using various models of regression where the models’ coefficients are determined through the Moore–Penrose pseudo-inverse to produce implied volatility surfaces for each selected pair of correlations and mean-reversion rates. Through these methods we discover that for different mean-reversions and correlations the overall implied volatility varies significantly and the relationship between the strike price, time to maturity, implied volatility are transformed.
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Duben, Josef. "Oceňování opcí se stochastickou volatilitou". Master's thesis, Vysoká škola ekonomická v Praze, 2011. http://www.nusl.cz/ntk/nusl-72010.
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The thesis is dealing with option pricing. The basic Black-Scholes model is described, along with the reasons that led to the development of stochastic volatility models. SABR model and Heston model are described in detail. These models are then applied to equity options in the times of high volatility. The models and their application are then evaluated.
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Yuksel, Ayhan. "Credit Risk Modeling With Stochastic Volatility, Jumps And Stochastic Interest Rates". Master's thesis, METU, 2007. http://etd.lib.metu.edu.tr/upload/2/12609206/index.pdf.
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This thesis presents the modeling of credit risk by using structural approach. Three fundamental questions of credit risk literature are analyzed throughout the research: modeling single firm credit risk, modeling portfolio credit risk and credit risk pricing. First we analyze these questions under the assumptions that firm value follows a geometric Brownian motion and the interest rates are constant. We discuss the weaknesses of the geometric brownian motion assumption in explaining empirical properties of real data. Then we propose a new extended model in which asset value, volatility and interest rates follow affine jump diffusion processes. In our extended model volatility is stochastic, asset value and volatility has correlated jumps and interest rates are stochastic and have jumps. Finally, we analyze the modeling of single firm credit risk and credit risk pricing by using our extended model and show how our model can be used as a solution for the problems we encounter with simple models.
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Meng, Yu. "Bayesian Analysis of a Stochastic Volatility Model". Thesis, Uppsala University, Department of Mathematics, 2009. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-119972.
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Hafner, Reinhold. "Stochastic implied volatility : a factor-based model /". Berlin [u.a.] : Springer, 2004. http://www.loc.gov/catdir/enhancements/fy0813/2004109369-d.html.
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Shi, Fangwei. "Asymptotic analysis of new stochastic volatility models". Thesis, Imperial College London, 2017. http://hdl.handle.net/10044/1/60648.
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A good options pricing model should be able to fit the market volatility surface with high accuracy. While the standard continuous stochastic volatility models can generate volatility smiles consistent with market data for relatively larger maturities, these models cannot reproduce market smiles for small maturities, which have the well-observed 'small-time explosion' feature. In this thesis we propose three new types of stochastic volatility models, and we focus on the small-time asymptotic behaviour of the implied volatility in these models. We show that these models can generate implied volatilities with explosion, hence they can theoretically provide a better fit to the market data. The thesis is organised as follows. Chapter 0 is the introduction. We briefly discuss the development and performance of standard continuous stochastic volatility models, and raise the small-time fitness issue of these traditional models. In Chapter 1 we propose the randomised Heston model and analyse its small and large time asymptotic behaviours. In particular, we show that any small-time explosion rate in between of [0, 1/2] for the implied variance can be captured by a suitable choice of the initial randomisation. In Chapter 2 we propose a fractional version of the Heston model and detail the small-time asymptotic behaviour of the implied volatility in this setting. We precise the link between the explosion rate and the Hurst parameter. Finally, in Chapter 3 we propose a new stochastic volatility model based on the recent work by Conus and Wildman in which the stock price can have past dependency. We show that in the case of a CIR variance process this model has similar behaviours to a fractional Heston environment.
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Shi, Lishan. "Stochastic volatility in mean option pricing models". Thesis, University of Cambridge, 2006. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.614015.
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Monge, Adriana Ocejo. "Time-change and control of stochastic volatility". Thesis, University of Warwick, 2014. http://wrap.warwick.ac.uk/62030/.
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The central theme of this thesis is the behavior of the value function of general optimal stopping problems under a stochastic volatility model when varying the volatility dynamics. We first use a combination of time-change and coupling techniques to show regularity properties of the value function. We consider a large class of terminal payoffs: when the first component of the model is a stochastic differential equation without drift we allow for general measurable functions, and when it has a drift we impose a mild condition which includes possibly unbounded and discontinuous functions. We also consider a running cost which can be any non-negative and bounded Borel function. Moreover, we derive the solution of a zero-sum game of stopping and control, which arises when considering some parameter uncertainty in the volatility dynamics. In both finite and infinite horizon, we exhibit the existence of a saddle point using stochastic control and martingale arguments as well as the probabilistic representation of solutions to free-boundary problems. Overall, our approach in mainly theoretical, however we impose only verifiable conditions. We then discuss some examples arising in American option pricing where our results are applicable. In particular, we are able to compare American option prices under different volatility models in a variety of settings and we establish that the optimal exercise boundary for the associated option is a monotone function of the volatility.
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Rafiou, AS. "Foreign Exchange Option Valuation under Stochastic Volatility". University of the Western Cape, 2009. http://hdl.handle.net/11394/7777.
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>Magister Scientiae - MScThe case of pricing options under constant volatility has been common practise for decades. Yet market data proves that the volatility is a stochastic phenomenon, this is evident in longer duration instruments in which the volatility of underlying asset is dynamic and unpredictable. The methods of valuing options under stochastic volatility that have been extensively published focus mainly on stock markets and on options written on a single reference asset. This work probes the effect of valuing European call option written on a basket of currencies, under constant volatility and under stochastic volatility models. We apply a family of the stochastic models to investigate the relative performance of option prices. For the valuation of option under constant volatility, we derive a closed form analytic solution which relaxes some of the assumptions in the Black-Scholes model. The problem of two-dimensional random diffusion of exchange rates and volatilities is treated with present value scheme, mean reversion and non-mean reversion stochastic volatility models. A multi-factor Gaussian distribution function is applied on lognormal asset dynamics sampled from a normal distribution which we generate by the Box-Muller method and make inter dependent by Cholesky factor matrix decomposition. Furthermore, a Monte Carlo simulation method is adopted to approximate a general form of numeric solution The historic data considered dates from 31 December 1997 to 30 June 2008. The basket contains ZAR as base currency, USD, GBP, EUR and JPY are foreign currencies.
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Cullinan, Cian. "Implementation of Bivariate Unspanned Stochastic Volatility Models". Master's thesis, University of Cape Town, 2018. http://hdl.handle.net/11427/29266.
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Unspanned stochastic volatility term structure models have gained popularity in the literature. This dissertation focuses on the challenges of implementing the simplest case – bivariate unspanned stochastic volatility models, where there is one state variable controlling the term structure, and one scaling the volatility. Specifically, we consider the Log-Affine Double Quadratic (1,1) model of Backwell (2017). In the class of affine term structure models, state variables are virtually always spanned and can therefore be inferred from bond yields. When fitting unspanned models, it is necessary to include option data, which adds further challenges. Because there are no analytical solutions in the LADQ (1,1) model, we show how options can be priced using an Alternating Direction Implicit finite difference scheme. We then implement an Unscented Kalman filter — a non-linear extension of the Kalman filter, which is a popular method for inferring state variable values — to recover the latent state variables from market observable data
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Wort, Joshua. "Pricing with Bivariate Unspanned Stochastic Volatility Models". Master's thesis, Faculty of Commerce, 2019. http://hdl.handle.net/11427/31323.
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Unspanned stochastic volatility (USV) models have gained popularity in the literature. USV models contain at least one source of volatility-related risk that cannot be hedged with bonds, referred to as the unspanned volatility factor(s). Bivariate USV models are the simplest case, comprising of one state variable controlling the term structure and the other controlling unspanned volatility. This dissertation focuses on pricing with two particular bivariate USV models: the Log-Affine Double Quadratic (1,1) – or LADQ(1,1) – model of Backwell (2017) and the LinearRational Square Root (1,1) – or LRSQ(1,1) – model of Filipovic´ et al. (2017). For the LADQ(1,1) model, we fully outline how an Alternating Directional Implicit finite difference scheme can be used to price options and implement the scheme to price caplets. For the LRSQ(1,1) model, we illustrate a semi-analytical Fourierbased method originally designed by Filipovic´ et al. (2017) for pricing swaptions, but adjust it to price caplets. Using the above numerical methods, we calibrate each (1,1) model to both the British-pound yield curve and caps market. Although we cannot achieve a close fit to the implied volatility surface, we find that the parameters in the LADQ(1,1) model have direct control over the qualitative features of the volatility skew, unlike the parameters within the LRSQ(1,1) model.
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Cowen, Nicholas. "Local Stochastic Volatility—The Hyp-Hyp Model". Master's thesis, Faculty of Commerce, 2021. http://hdl.handle.net/11427/32556.
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Volatility modelling is used predominantly in order to explain the volatility smile observed in the market. Stochastic volatility models are mainly used to capture the curvature of a volatility smile while local volatility models generally model the skew. Jackel and Kahl ¨ (2008) present a hyperbolic-local hyperbolic-stochastic volatility (Hyp-Hyp) model which aims to improve upon existing local and stochastic volatility models such as the stochastic alpha, beta, rho (SABR) and constant elasticity of variance (CEV) models. The advantageous features of the Hyp-Hyp model are corroborated by implementing the model. Jackel and Kahl ¨ (2008) investigate the accuracy of a scaled analytical approximation for implied volatility, based on approximations presented by Watanabe (1987) and Fouque et al. (2000), for the Hyp-Hyp model. They use the approximation to derive an expression for the delta of an option. This dissertation analyses the Hyp-Hyp model, as well as the approximation, by deriving expressions for other sensitivities and by investigating the effect of the Hyp-Hyp model parameters on the volatility smile. The accuracy of the analytical approximation for functional forms other than those defined by the Hyp-Hyp model is explored. A derivation of the approximation is undertaken, presenting corrections to the expressions introduced by Kahl (2007) and used by Jackel and Kahl ¨ (2008).
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Häfner, Reinhold. "Stochastic implied volatility : a factor-based model /". Berlin ; New York : Springer, 2004. http://www.loc.gov/catdir/enhancements/fy0813/2004109369-d.html.
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Zanchini, Giulia. "Stochastic local volatility model for fx markets". Master's thesis, Alma Mater Studiorum - Università di Bologna, 2014. http://amslaurea.unibo.it/7685/.
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Questa tesi verte sullo studio di un modello a volatilità stocastica e locale, utilizzato per valutare opzioni esotiche nei mercati dei cambio. La difficoltà nell'implementare un modello di tal tipo risiede nella calibrazione della leverage surface e uno degli scopi principali di questo lavoro è quello di mostrarne la procedura.
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Kövamees, Gustav. "Particle-based Stochastic Volatility in Mean model". Thesis, KTH, Matematisk statistik, 2019. http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-257505.
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This thesis present a Stochastic Volatility in Mean (SVM) model which is estimated using sequential Monte Carlo methods. The SVM model was first introduced by Koopman and provides an opportunity to study the intertemporal relationship between stock returns and their volatility through inclusion of volatility itself as an explanatory variable in the mean-equation. Using sequential Monte Carlo methods allows us to consider a non-linear estimation procedure at cost of introducing extra computational complexity. The recently developed PaRIS-algorithm, introduced by Olsson and Westerborn, drastically decrease the computational complexity of smoothing relative to previous algorithms and allows for efficient estimation of parameters. The main purpose of this thesis is to investigate the volatility feedback effect, i.e. the relation between expected return and unexpected volatility in an empirical study. The results shows that unanticipated shocks to the return process do not explain expected returns.Detta examensarbete presenterar en stokastisk volatilitets medelvärdes (SVM) modell som estimeras genom sekventiella Monte Carlo metoder. SVM-modellen introducerades av Koopman och ger en möjlighet att studera den samtida relationen mellan aktiers avkastning och deras volatilitet genom att inkludera volatilitet som en förklarande variabel i medelvärdes-ekvationen. Sekventiella Monte Carlo metoder tillåter oss att använda icke-linjära estimerings procedurer till en kostnad av extra beräkningskomplexitet. Den nyligen utvecklad PaRIS-algoritmen, introducerad av Olsson och Westerborn, minskar drastiskt beräkningskomplexiteten jämfört med tidigare algoritmer och tillåter en effektiv uppskattning av parametrar. Huvudsyftet med detta arbete är att undersöka volatilitets-återkopplings-teorin d.v.s. relationen mellan förväntad avkastning och oväntad volatilitet i en empirisk studie. Resultatet visar på att oväntade chockar i avkastningsprocessen inte har förklarande förmåga över förväntad avkastning.
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Zhao, Ze. "Stochastic volatility models with applications in finance". Diss., University of Iowa, 2016. https://ir.uiowa.edu/etd/2306.
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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.