Dissertations / Theses on the topic 'Variance Gamma Model'

Doctor of Philosophy (PhD)
This thesis mainly builds on the Variance Gamma (VG) model for financial assets over time of Madan & Seneta (1990) and Madan, Carr & Chang (1998), although the model based on the t distribution championed in Heyde & Leonenko (2005) is also given attention. The primary contribution of the thesis is the development of VG models, and the extension of t models, which accommodate a dependence structure in asset price returns. In particular it has become increasingly clear that while returns (log price increments) of historical financial asset time series appear as a reasonable approximation of independent and identically distributed data, squared and absolute returns do not. In fact squared and absolute returns show evidence of being long range dependent through time, with autocorrelation functions that are still significant after 50 to 100 lags. Given this evidence against the assumption of independent returns, it is important that models for financial assets be able to accommodate a dependence structure.

The submitted work deals with option pricing. Mathematical approach is immediately followed by an economic interpretation. The main problem is to model the underlying uncertainities driving the stock price. Using two well-known valuation models, binomial model and Black-Scholes model, we explain basic principles, especially risk neutral pricing. Due to the empirical biases new models have been developped, based on pure jump process. Variance gamma process and its special symmetric case are presented.

Mestrado em Mathematical Finance
O resultado principal desta dissertação é a demonstração da fórmula de serie de soma tripla para o preço de uma opção Europeia induzido por um processo Variance Gamma. Com esta intenção, apresentamos certas propriedades e noções sobre processos de Lévy e análise complexa multidimensional, dando ênfase à aplicação do cálculo de resíduos ao integral Mellin-Barnes. Subsequentemente, iremos construir a representação na forma do integral Mellin-Barnes, em C^3, para o preço de uma opção e, apoiados pelo anteriormente mencionado cálculo de resíduos, deduziremos a representação em serie de soma tripla para o preço de uma opção Europeia e os seus correspondentes gregos. Para terminar, dando uso à nova formula, serão computados e discutidos alguns valores para um caso de estudo particular.
The main result of this dissertation is the proof of the triple sum series formula for the price of an European call option driven by the Variance Gamma process. With this intention, we present some notions and properties of Lévy processes and multidimensional complex analysis, with emphasis on the application of residue calculus to the Mellin-Barnes integral. Subsequently, we construct the Mellin-Barnes integral representation, in C^3, for the price of the option and, buttressed with the aforementioned residue calculus, we deduce the triple sum series representation for the price of the European option and its corresponding greeks. Finally, with the use of the new formula, some values for a particular case study are computed and discussed.
info:eu-repo/semantics/publishedVersion

We utilise several asset pricing models that allow for discontinuities in the returns and volatility time series in order to obtain estimates of Value-at-Risk (VaR). The first class of model that we use mixes a continuous diffusion process with discrete jumps at random points in time (Poisson Jump Diffusion Model). We also apply a purely discontinuous model that does not contain any continuous component at all in the underlying distribution (Variance Gamma Model). These models have been shown to have some success in capturing certain characteristics of return distributions, a few being leptokurtosis and skewness. Calibrating these models onto the returns of an index of Australian stocks (All Ordinaries Index), we then use the resulting parameters to obtain daily estimates of VaR. In order to obtain the VaR estimates for the Poisson Jump Diffusion Model and the Variance Gamma Model, we introduce the use of an innovation from option pricing techniques, which concentrates on the more tractable characteristic functions of the models. Having then obtained a series of VaR estimates, we then apply a variety of criteria to assess how each model performs and also evaluate these models against the traditional approaches to calculating VaR, such as that suggested by J.P. Morgan???s RiskMetrics. Our results show that whilst the Poisson Jump Diffusion model proved the most accurate at the 95% VaR level, neither the Poisson Jump Diffusion or Variance Gamma models were dominant in the other performance criteria examined. Overall, no model was clearly superior according to all the performance criteria analysed, and it seems that the extra computational time required to calibrate the Poisson Jump Diffusion and Variance Gamma models for the purposes of VaR estimation do not provide sufficient reward for the additional effort than that currently employed by Riskmetrics.

In this thesis, the structural model in credit risk and the credit derivatives is studied under both Black-Scholes setting and Variance Gamma (VG) setting. Using a Variance Gamma process, the distribution of the firm value process becomes asymmetric and leptokurtic. Also, the jump structure of VG processes allows random default times of the reference entities. Among structural models, the most emphasis is made on the Black-Cox model by building a relation between the survival probabilities of the Black-Cox model and the value of a binary down and out barrier option. The survival probabilities under VG setting are calculated via a Partial Integro Differential Equation (PIDE). Some applications of binary down and out barrier options, default probabilities and Credit Default Swap par spreads are also illustrated in this study.

In this work, we study the so-called combined models, generalized linear mixed models with extension to allow for overdispersion, in the context of genetics and breeding. Such flexible models accommodates cluster-induced correlation and overdispersion through two separate sets of random effects and contain as special cases the generalized linear mixed models (GLMM) on the one hand, and commonly known overdispersion models on the other. We use such models while obtaining heritability coefficients for non-Gaussian characters. Heritability is one of the many important concepts that are often quantified upon fitting a model to hierarchical data. It is often of importance in plant and animal breeding. Knowledge of this attribute is useful to quantify the magnitude of improvement in the population. For data where linear models can be used, this attribute is conveniently defined as a ratio of variance components. Matters are less simple for non-Gaussian outcomes. The focus is on time-to-event and count traits, where the Weibull-Gamma-Normal and Poisson-Gamma-Normal models are used. The resulting expressions are sufficiently simple and appealing, in particular in special cases, to be of practical value. The proposed methodologies are illustrated using data from animal and plant breeding. Furthermore, attention is given to the occurrence of negative estimates of variance components in the Poisson-Gamma-Normal model. The occurrence of negative variance components in linear mixed models (LMM) has received a certain amount of attention in the literature whereas almost no work has been done for GLMM. This phenomenon can be confusing at first sight because, by definition, variances themselves are non-negative quantities. However, this is a well understood phenomenon in the context of linear mixed modeling, where one will have to make a choice between a hierarchical and a marginal view. The variance components of the combined model for count outcomes are studied theoretically and the plant breeding study used as illustration underscores that this phenomenon can be common in applied research. We also call attention to the performance of different estimation methods, because not all available methods are capable of extending the parameter space of the variance components. Then, when there is a need for inference on such components and they are expected to be negative, the accuracy of the method is not the only characteristic to be considered.
Neste trabalho foram estudados os chamados modelos combinados, modelos lineares generalizados mistos com extensão para acomodar superdispersão, no contexto de genética e melhoramento. Esses modelos flexíveis acomodam correlação induzida por agrupamento e superdispersão por meio de dois conjuntos separados de efeitos aleatórios e contem como casos especiais os modelos lineares generalizados mistos (MLGM) e os modelos de superdispersão comumente conhecidos. Tais modelos são usados na obtenção do coeficiente de herdabilidade para caracteres não Gaussianos. Herdabilidade é um dos vários importantes conceitos que são frequentemente quantificados com o ajuste de um modelo a dados hierárquicos. Ela é usualmente importante no melhoramento vegetal e animal. Conhecer esse atributo é útil para quantificar a magnitude do ganho na população. Para dados em que modelos lineares podem ser usados, esse atributo é convenientemente definido como uma razão de componentes de variância. Os problemas são menos simples para respostas não Gaussianas. O foco aqui é em características do tipo tempo-até-evento e contagem, em que os modelosWeibull-Gama-Normal e Poisson-Gama-Normal são usados. As expressões resultantes são suficientemente simples e atrativas, em particular nos casos especiais, pelo valor prático. As metodologias propostas são ilustradas usando dados de melhoramento animal e vegetal. Além disso, a atenção é voltada à ocorrência de estimativas negativas de componentes de variância no modelo Poisson-Gama- Normal. A ocorrência de componentes de variância negativos em modelos lineares mistos (MLM) tem recebido certa atenção na literatura enquanto quase nenhum trabalho tem sido feito para MLGM. Esse fenômeno pode ser confuso a princípio porque, por definição, variâncias são quantidades não-negativas. Entretanto, este é um fenômeno bem compreendido no contexto de modelagem linear mista, em que a escolha deverá ser feita entre uma interpretação hierárquica ou marginal. Os componentes de variância do modelo combinado para respostas de contagem são estudados teoricamente e o estudo de melhoramento vegetal usado como ilustração confirma que esse fenômeno pode ser comum em pesquisas aplicadas. A atenção também é voltada ao desempenho de diferentes métodos de estimação, porque nem todos aqueles disponíveis são capazes de estender o espaço paramétrico dos componentes de variância. Então, quando há a necessidade de inferência de tais componentes e é esperado que eles sejam negativos, a acurácia do método de estimação não é a única característica a ser considerada.

En analyse multivariée de données de grande dimension, les lois de Wishart définies dans le contexte des modèles graphiques revêtent une grande importance car elles procurent parcimonie et modularité. Dans le contexte des modèles graphiques Gaussiens régis par un graphe G, les lois de Wishart peuvent être définies sur deux restrictions alternatives du cône des matrices symétriques définies positives : le cône PG des matrices symétriques définies positives x satisfaisant xij=0, pour tous sommets i et j non adjacents, et son cône dual QG. Dans cette thèse, nous proposons une construction harmonieuse de familles exponentielles de lois de Wishart sur les cônes PG et QG. Elle se focalise sur les modèles graphiques d'interactions des plus proches voisins qui présentent l'avantage d'être relativement simples tout en incluant des exemples de tous les cas particuliers intéressants: le cas univarié, un cas d'un cône symétrique, un cas d'un cône homogène non symétrique, et une infinité de cas de cônes non-homogènes. Notre méthode, simple, se fonde sur l'analyse sur les cônes convexes. Les lois de Wishart sur QAn sont définies à travers la fonction gamma sur QAn et les lois de Wishart sur PAn sont définies comme la famille de Diaconis- Ylvisaker conjuguée. Ensuite, les méthodes développées sont utilisées pour résoudre la conjecture de Letac- Massam sur l'ensemble des paramètres de la loi de Wishart sur QAn. Cette thèse étudie aussi les sousmodèles, paramétrés par un segment dans M, d'une famille exponentielle paramétrée par le domaine des moyennes M
In the framework of Gaussian graphical models governed by a graph G, Wishart distributions can be defined on two alternative restrictions of the cone of symmetric positive definite matrices: the cone PG of symmetric positive definite matrices x satisfying xij=0 for all non-adjacent vertices i and j and its dual cone QG. In this thesis, we provide a harmonious construction of Wishart exponential families in graphical models. Our simple method is based on analysis on convex cones. The focus is on nearest neighbours interactions graphical models, governed by a graph An, which have the advantage of being relatively simple while including all particular cases of interest such as the univariate case, a symmetric cone case, a nonsymmetric homogeneous cone case and an infinite number of non-homogeneous cone cases. The Wishart distributions on QAn are constructed as the exponential family generated from the gamma function on QAn. The Wishart distributions on PAn are then constructed as the Diaconis- Ylvisaker conjugate family for the exponential family of Wishart distributions on QAn. The developed methods are then used to solve the Letac-Massam Conjecture on the set of parameters of type I Wishart distributions on QAn. Finally, we introduce and study exponential families of distributions parametrized by a segment of means with an emphasis on their Fisher information. The focus in on distributions with matrix parameters. The particular cases of Gaussian and Wishart exponential families are further examined

Large passive detectors used in screening for special nuclear materials at ports of entry are characterized by poor spectral resolution, making identification of radionuclides a difficult task. Most identification routines, which fit empirical shapes and use derivatives, are impractical in these situations. Here I develop new, physics-based methods to determine the presence of spectral signatures of one or more of a set of isotopes. Gamma-ray counts are modeled as Poisson processes, where the average part is taken to be the model and the difference between the observed gamma-ray counts and the average is considered random noise. In the linear part, the unknown coefficients represent the intensites of the isotopes. Therefore, it is of great interest not to estimate each coefficient, but rather determine if the coefficient is non-zero, corresponding to the presence of the isotope. This thesis provides new selection algorithms, and, since detector data is undoubtedly finite, this unique work emphasizes selection when data is fixed and finite.

Three γ Doradus-type stars are analysed: HD139095, HD153580, and HD197541. Long-term observation campaigns have been conducted on each star, with over 300 spectra of each star being gathered for analysis using the HERCULES spectrograph at Mount John University Observatory. For each star, cross-correlation techniques were used to obtain representative line profiles for each spectrum. The analysis of these line profiles has resulted in frequency and pulsation mode identifications for these three stars. Abundance analysis has also been performed on HD139095 and HD197541, and their fundamental parameters are confirmed as being consistent with the γ Doradus class of stars. HD153580 and HD197541, previously only candidates for the class, can now be classified as bona fide γ Doradus type stars. The frequencies and modes identified in this thesis will be very useful in constraining future theoretical models, allowing us to better understand and model the interiors of γ Doradus stars.

The subject of this thesis is the analysis and modeling of the active galactic nucleus PKS 1510-089. The aim is to present a thorough background of active galactic nuclei combined with the analysis and modeling of a specific active galactic nucleus. The results will then be  linked to previous research and theories about active galactic nuclei. The data used in the analysis were retrieved from the Fermi Gamma-ray Space Telescope. A light curve analysis that extended over 12 years provided knowledge about variability and presented four interesting flaring periods. The four periods underwent a spectral analysis, and the results showed that a log parabolic curvature could best describe all four periods. The last step before the modeling was to create spectral energy distributions for all four periods to retrieve spectral points from wavelengths other than those available from Fermi. Unfortunately, there were only sufficient data for one period. That period was later used in the modeling and resulted in a well-fitted external Compton model, which was compared, with relatively good results, with previous research.

The aim of this thesis is to develop efficient and practically useful Bayesian methods for statistical inference in structural second-price auctions. The models are applied to a carefully collected coin auction dataset with bids and auction-specific characteristics from one thousand Internet auctions on eBay. Bidders are assumed to be risk-neutral and symmetric, and compete for a single object using the same game-theoretic strategy. A key contribution in the thesis is the derivation of very accurate approximations of the otherwise intractable equilibrium bid functions under different model assumptions. These easily computed and numerically stable approximations are shown to be crucial for statistical inference, where the inverse bid functions typically needs to be evaluated several million times. In the first paper, the approximate bid is a linear function of a bidder's signal and a Gaussian common value model is estimated. We find that the publicly available book value and the condition of the auctioned object are important determinants of bidders' valuations, while eBay's detailed seller information is essentially ignored by the bidders. In the second paper, the Gaussian model in the first paper is contrasted to a Gamma model that allows intrinsically non-negative common values. The Gaussian model performs slightly better than the Gamma model on the eBay data, which we attribute to an almost normal or at least symmetrical distribution of valuations. The third paper compares the model in the first paper to a directly comparable model for private values. We find many interesting empirical regularities between the models, but no strong and consistent evidence in favor of one model over the other. In the last paper, we consider auctions with both private-value and common-value bidders. The equilibrium bid function is given as the solution to an ordinary differential equation, from which we derive an approximate inverse bid as an explicit function of a given bid. The paper proposes an elaborate model where the probability of being a common value bidder is a function of covariates at the auction level. The model is estimated by a Metropolis-within-Gibbs algorithm and the results point strongly to an active influx of both private-value and common-value bidders.

At the time of the doctoral defense, the following papers were unpublished and had a status as follows: Paper 1: Epub ahead of print. Paper 2: Manuscript. Paper 3: Manuscript. Paper 4: Manuscript.

We present the modes identified for frequencies found in spectroscopic observations of the Doradus stars HD 189631, QW Puppis, and IR Draconis. A cross-correlation tech- nique was used to create mean line profiles for HD 189631. Four frequencies and modes were identified for this star: 1.6774±0.0002 d⁻¹, 1.4174±0.0002 d⁻¹, 0.0714±0.0002 d⁻¹, and 1.8228 ± 0.0002 d⁻¹ which were identified with the modes (l,m) = (1, +1), (1, +1), (2,−2), and (1, +1) respectively. A least-squares deconvolution method was implemented for line profile generation in the study of QW Puppis and IR Draconis. Three frequen- cies were identified for QW Puppis: 0.055972 ± 0.000004 d⁻¹, 0.064846 ± 0.000004, and 5.219398±0.000002 d⁻¹. These frequencies were identified with the modes (l,m) = (1,−1), (4,−1), (4, +1). Two frequencies were identified in spectra of the rapidly rotating star IR Draconis: 0.00515 ± 0.00003 d⁻¹ and 2.35538 ± 0.00004 d⁻¹; which were identified with (l,m) = (1,−1), and (1, +1) modes respectively. These mode identifications will assist in modelling the structure and interior conditions of these main sequence, non-radially pulsating stars.

Pricing of repair and maintenance (R&M) contracts is one among the most important processes carried out at Scania. Predictions of repair costs at Scania are carried out using experience-based prediction methods which do not involve statistical methods for the computation of average repair costs for contracts terminated in the recent past. This method is difficult to apply for a reference population of rigid Scania trucks. Hence, the purpose of this study is to perform suitable statistical modelling to predict repair costs of four variants of rigid Scania trucks. The study gathers repair data from multiple sources and performs feature selection using the Akaike Information Criterion (AIC) to extract the most significant features that influence repair costs corresponding to each truck variant. The study proved to show that the inclusion of operational features as a factor could further influence the pricing of contracts. The hurdle Gamma model, which is widely used to handle zero inflations in Generalized Linear Models (GLMs), is used to train the data which consists of numerous zero and non-zero values. Due to the inherent hierarchical structure within the data expressed by individual chassis, a hierarchical hurdle Gamma model is also implemented. These two statistical models are found to perform much better than the experience-based prediction method. This evaluation is done using the mean absolute error (MAE) and root mean square error (RMSE) statistics. A final model comparison is conducted using the AIC to draw conclusions based on the goodness of fit and predictive performance of the two statistical models. On assessing the models using these statistics, the hierarchical hurdle Gamma model was found to perform predictions the best

In Paper [I] we use data on Swedish public procurement auctions for internal regularcleaning service contracts to provide novel empirical evidence regarding green publicprocurement (GPP) and its effect on the potential suppliers’ decision to submit a bid andtheir probability of being qualified for supplier selection. We find only a weak effect onsupplier behavior which suggests that GPP does not live up to its political expectations.However, several environmental criteria appear to be associated with increased complexity,as indicated by the reduced probability of a bid being qualified in the postqualificationprocess. As such, GPP appears to have limited or no potential to function as an environmentalpolicy instrument. In Paper [II] the observation is made that empirical evaluations of the effect of policiestransmitted through public procurements on bid sizes are made using linear regressionsor by more involved non-linear structural models. The aspiration is typically to determinea marginal effect. Here, I compare marginal effects generated under both types ofspecifications. I study how a political initiative to make firms less environmentally damagingimplemented through public procurement influences Swedish firms’ behavior. Thecollected evidence brings about a statistically as well as economically significant effect onfirms’ bids and costs. Paper [III] embarks by noting that auction theory suggests that as the number of bidders(competition) increases, the sizes of the participants’ bids decrease. An issue in theempirical literature on auctions is which measurement(s) of competition to use. Utilizinga dataset on public procurements containing measurements on both the actual and potentialnumber of bidders I find that a workhorse model of public procurements is bestfitted to data using only actual bidders as measurement for competition. Acknowledgingthat all measurements of competition may be erroneous, I propose an instrumental variableestimator that (given my data) brings about a competition effect bounded by thosegenerated by specifications using the actual and potential number of bidders, respectively.Also, some asymptotic results are provided for non-linear least squares estimatorsobtained from a dependent variable transformation model. Paper [VI] introduces a novel method to measure bidders’ costs (valuations) in descending(ascending) auctions. Based on two bounded rationality constraints bidders’costs (valuations) are given an imperfect measurements interpretation robust to behavioraldeviations from traditional rationality assumptions. Theory provides no guidanceas to the shape of the cost (valuation) distributions while empirical evidence suggeststhem to be positively skew. Consequently, a flexible distribution is employed in an imperfectmeasurements framework. An illustration of the proposed method on Swedishpublic procurement data is provided along with a comparison to a traditional BayesianNash Equilibrium approach.

Yes
Amid much recent interest we discuss a Variance Gamma model for Rugby Union matches (applications to other sports are possible). Our model emerges as a special case of the recently introduced Gamma Difference distribution though there is a rich history of applied work using the Variance Gamma distribution – particularly in finance. Restricting to this special case adds analytical tractability and computational ease. Our three-dimensional model extends classical two-dimensional Poisson models for soccer. Analytical results are obtained for match outcomes, total score and the awarding of bonus points. Model calibration is demonstrated using historical results, bookmakers’ data and tournament simulations.

Yes
Amid much recent interest we discuss a Variance Gamma model for Rugby Union matches (applications to other sports are possible). Our model emerges as a special case of the recently introduced Gamma Difference distribution though there is a rich history of applied work using the Variance Gamma distribution – particularly in finance. Restricting to this special case adds analytical tractability and computational ease. Our three-dimensional model extends classical two-dimensional Poisson models for soccer. Analytical results are obtained for match outcomes, total score and the awarding of bonus points. Model calibration is demonstrated using historical results, bookmakers’ data and tournament simulations.

碩士
國立臺北大學
企業管理學系
92
Since 1973 Black and Scholes published the famous option pricing model, option pricing theory has become a research focus. One key factor that affects the pricing of an option is the choice of statistical distribution that governs the underlying assets return. The B-S model assumes that the change in the logarithm of the underlying asset price follows a normal distribution, which is not accurate enough in some markets. The variance gamma process is aimed at providing a model for log-return distribution that offers physical interpretation and incorporates both long-tailedness and skewness features in a log-return. Based on such a three parameter variance gamma process, Madan et al. (1998) derive a closed form for the price of an European option. The artificial neural network model has many advantages in financial derivatives and this study try to use neural network models as the evaluated model of TXO. Since the stock index option is a new financial derivatives in Taiwan, the pricing behavior of stock index options is concerned by investors as well as academic workers. This study investigates the B-S, VG and ANN models with historical, implied and GARCH(1,1) volatilities on “the Taiwan Stock Exchange Capitalization Weighted Stock Index Options.” And the empirical results indicate: 1. The implied volatility is the best estimate approach between history approach , implied approach and GARCH(1,1) approach. Any of three pricing models with implied volatility has the lowest error. 2. The VG model seems to outperform B-S model, but VG model only can moderately iron out some of the biases inherent in the B-S model and its performance in pricing is still far less than desirable. 3. The ANN model outperforms the VG model and the B-S model. 4. The ANN model with implied volatility seems to be the best model.

Tese de mestrado em Matemática Financeira, apresentada à Universidade de Lisboa, através da Faculdade de Ciências, 2013
O presente trabalho tem como objectivo debruçar-se sobre a construção de superfícies de volatilidade implícita. Toma-se, como ponto de partida, a literatura existente que nos diz que o modelo de Black-Merton-Scholes (BMS) apresenta várias limitações, sendo que a principal, para muitos, é considerar a volatilidade determinística. Neste trabalho, como forma de eliminar este problema, iremos apresentar a metodologia desenvolvida por Peter Carr e Liuren Wu (2011), de forma a construir uma superfície de volatilidade não determinística, que não seja tão difícil de obter como nos modelos de volatilidade estocástica e que seja mais rápida de estimar. Esta metodologia especifica o preço do activo subjacente e a dinâmica da volatilidade implícita, enquanto deixa a dinâmica da taxa da volatilidade instantânea variar livremente. Por sua vez, o domínio dos valores admissíveis para a superfície de volatilidade implícita inicial deriva de uma base com argumentos de não arbitragem. Com o objectivo de modelar a volatilidade implícita para os dois modelos apresentados no trabalho de Peter Carr e Liuren Wu (2011), o square-root variance model e o lognormal variance model, usamos uma variante do square-root process. Nessa construção, é usado o unscented Kalman FIlter e um algoritmo de minimização de erros, como forma de determinar os parâmetros que necessitamos para resolver a equação do modelo. Este trabalho é composto por duas vertentes, uma vertente teórica e uma vertente prática. A vertente teórica, incide sobre a metodologia apresentada por Peter Carr e Liuren Wu (2011), enquanto que a vertente prática, inclina-se sobre a construção do unscented Kalman FIlter e do algoritmo de minimização, como forma de determinar os parâmetros que necessitamos para a dinâmica da volatilidade do preço do activo subjacente.
The objective of this thesis is to construct an implied volatility surface to price options. We start from the existing literature that says Black-Merton-Scholes (BMS) has many drawbacks, being the most important one assuming deterministic volatility. In this thesis, we will present Peter Carr and Liuren Wu (2011) methodology for constructing a volatility surface that is not assumed to be deterministic, is not so complicated to determine like in stochastic volatility models and is faster to estimate. This thesis proposes a new approach, which specifies the security price and the implied volatility dynamics while leaving the instantaneous variance rate dynamics unspecified. The allowable shape for the initial implied volatility surface is then derived based on dynamic no-arbitrage arguments. This thesis presents two models for constructing volatility surfaces using a variant of the square-root process for the volatility process. This paper has two parts, one theoretical and another practical. In the theoretical part we demonstrate the paper of Peter Carr and Liuren Wu (2011) to construct volatility surfaces, while in the practical part we construct an unscented Kalman filter with a minimization algorithm to determine the parameters that we need to construct the real dynamics of the implied volatility surface of the underlying.