Dissertations / Theses on the topic 'Pricing models'

Starting from the most famous Black-Scholes model for the underlying asset price, there has been a large variety of extensions made in recent decades. One main strand is about the models which allow a jump component in the asset price. The first topic of this thesis is about the study of jump risk premium by an equilibrium approach. Different from others, this work provides a more general result by modeling the underlying asset price as the ordinary exponential of a L?vy process. For any given asset price process, the equity premium, pricing kernel and an equilibrium option pricing formula can be derived. Moreover, some empirical evidence such as the negative variance risk premium, implied volatility smirk, and negative skewness risk premium can be well explained by using the relation between the physical and risk-neutral distributions for the jump component. Another strand of the extensions of the Black-Scholes model is about the models which can incorporate stochastic volatility in the asset price. The second topic of this thesis is about the replication of exponential variance, where the key risks are the ones induced by the stochastic volatility and moreover it can be correlated with the returns of the asset, referred to as leverage effect. A time-changed L?vy process is used to incorporate jumps, stochastic volatility and leverage effect all together. The exponential variance can be robustly replicated by European portfolios, without any specification of a model for the stochastic volatility. Beyond the above asset pricing and hedging, portfolio optimization is also discussed. Based on the Merton (1969, 1971)'s reduced portfolio optimization and the delta hedging problem, a portfolio of an option, the underlying stock and a risk-free bond can be optimized in discrete time and its optimal solution can be shown to be a mixture of the Merton's result and the delta hedging strategy. The main approach is the elasticity approach, which has initially been proposed in continuous time. In addition to the above optimization problem in discrete time, the same topic but in a continuous-time regime-switching market is also presented. The use of regime-switching makes our market incomplete, and makes it difficult to use some approaches which are applicable in complete market. To overcome this challenge, two methods are provided. The first method is that we simply do not price the regime-switching risk when obtaining the risk-neutral probability. Then by the idea of elasticity, the utility maximization problem can be formulated as a stochastic control problem with only a single control variable, and explicit solutions can be obtained. The second method is to introduce a functional operator to general value functions of stochastic control problem in such a way that the optimal value function in our setting can be given by the limit of a sequence of value functions defined by iterating the operator. Hence the original problem can be deduced to an auxiliary optimization problem, which can be solved as if we were in a single-regime market, which is complete.<br>published_or_final_version<br>Statistics and Actuarial Science<br>Doctoral<br>Doctor of Philosophy

The inverse problem in finance consists of determining the unknown parameters of the pricing equation from the values quoted from the market. We formulate the inverse problem as a minimization problem for an appropriate cost function to minimize the difference between the solution of the model and the market observations. Efficient gradient based optimization requires accurate gradient estimation of the cost function. In this thesis we highlight the adjoint method for computing gradients of the cost function in the context of gradient based optimization and show its importance. We derive the continuous adjoint equations with appropriate boundary conditions for three main option pricing models: the Black-Scholes model, the Hestonâ s model and the jump diffusion model, for European type options. These adjoint equations can be used to compute the gradient of the cost function accurately for parameter estimation problems. The adjoint method allows efficient evaluation of the gradient of a cost function F(¾) with respect to parameters ¾ where F depends on ¾ indirectly, via an intermediate variable. Compared to the finite difference method and the sensitivity equation method, the adjoint equation method is very efficient in computing the gradient of the cost function. The sensitivity equations method requires solving a PDE corresponding to each parameter in the model to estimate the gradient of the cost function. The adjoint method requires solving a single adjoint equation once. Hence, for a large number of parameters in the model, the adjoint equation method is very efficient. Due to its nature, the adjoint equation has to be solved backward in time. The adjoint equation derived from the jump diffusion model is harder to solve due to its non local integral term. But algorithms that can be used to solve the Partial Integro- Differential Equation (PIDE) derived from jump diffusion model can be modified to solve the adjoint equation derived from the PIDE.<br>Ph. D.

Part I: This chapter develops a lattice method for option evaluation aiming to investigate whether the option prices reflect the shifts in the distributions of the underlying asset returns and the risk-free interest rate. More precisely we try to investigate whether the option prices reflect the switches in the correlation between the underlying and risk-free bond returns that characterise different states of the economy. For this reason we develop and test two models. In the first model we allow all the parameters to follow a regime-switching process while in the second model, in order to isolate the regime-switching correlation effect on the option prices, we allow only the correlation to follow a regime-switching process. The models developed use pentanomial lattices to represent the evolution of the regime-switching underlying assets. Our findings suggest that the option prices reflect the regime-switches and that a model which considers these switches could produce more accurate results than a single-regime model. Part II: This part develops a class of closed-form models for options on commodities evaluation under the assumptions of mean-reversion in the commodity prices and factors’ values and regime-switching in the volatilities and correlations. At first we develop novel closed-form solutions of the 1-, 2- and 3-factors models and later in the paper these three models are transformed into regime switching models. The six models (three with and three without regime-switching) are then tested and compared on real market data. Our findings suggest that the by increasing the stochastic factors and assuming regime-switching in the models their flexibility and thus their accuracy increases.

We introduce the concepts of credit risk in securitization framework. We study CDO products and their different structures and features. We analyse different pricing models used to determine the true riskiness of such products after we explain their dynamics and characteristics. We end up adopting and applying one of these models on real data to check whether the model reflect the true market value.

The ability to accurately price equity is an ineluctable requirement within businesses where decisions need to be taken daily that impact upon the future viability of that business. The Capital asset pricing model (CAPM) is the preeminent tool that has become entrenched within academia and business for exactly the purpose of costing equity capital. This study aimed to prove whether the application of the CAPM, in various forms, including the Black s CAPM, was merely a myopic inculcation of the academic and business spheres, or whether it truly reflected the empirical reality of the South African markets. The research discredited eight variations of the CAPM through a quantitative causal design, which employed t-tests and ANOVAs, tested upon a judgmental sample of the largest 160 shares on the JSE. Reaching this opprobrium would have been a Pyrrhic victory, had an alternative model not been proposed. Thus, a quartet of styles was employed in tests against both non-resource and resource shares in an attempt to generate two multi-factor models known as the Optimised Returns Score (ORS) combined models. The generated model for the non-resource shares explained 36.5% of the variation in the observed cost of equity capital, at a 95% level of significance. However, a statistically significant predictive model for resource shares was unable to be found, possibly due to the small sample size available.<br>Mini Dissertation (MBA)--University of Pretoria, 2015.<br>sn2016<br>Gordon Institute of Business Science (GIBS)<br>MBA<br>Unrestricted

Monthly returns on twenty-seven Eurobonds from July 1982 to June 1986 were examined. There were no consistent differences in returns based on the country in which a firm is located. There were consistent differences due to industry classification, with energy-related firms exhibiting higher average returns and variances. Excess returns were calculated using the capital asset pricing model and arbitrage pricing theory. The results from calculation of mean average deviation, root mean square, and R2 all indicate that the arbitrage pricing theory was a better descriptor of the Eurobond market. The excess returns were also examined using stochastic dominance. Arbitrage pricing theory never dominated the capital asset pricing model using first-order criteria, but consistently dominated using second-order criteria. The results were discussed in terms of the implications for investors and portfolio managers.

Mestrado em Mathematical Finance<br>O problema de valorização de derivados tem sido o foco da investigação em Matemática Financeira desde a sua conceção. Mais recentemente, a literatura tem-se focado por exemplo em modelos que assumem que as dinâmicas do preço do ativo subjacente são governadas por um processo de Lévy (por vezes chamado um processo com saltos). Este tipo de modelo admite a possibilidade de eventos extremos (saltos), que não são devidamente capturados por modelos clássicos do tipo Black-Scholes, alicerçados no movimento Browniano. Foi também demonstrado ao longo da última década que se as dinâmicas do preço do ativo subjacente seguem certos processos de Lévy, tais como o CGMY , o FMLS e o KoBoL, os preços das opções satisfazem uma equação diferencial parcial fracionária. Nesta dissertação, iremos mostrar que se as dinâmicas do ativo subjacente seguem o denominado Processo Estável Temperado Generalizado, que admite como caso particular os suprareferidos processos CGMY e KoBoL, então os preços das opções satisfazem igualmente uma equação diferencial parcial fracionária. Além disso, iremos implementar um método simples de diferenças finitas para resolver numericamente a equação deduzida, e valorizar opções do tipo europeu.<br>The problem of pricing financial derivatives has been the focal point of research within the field of Mathematical Finance since its conception. In recent years, one of the main areas of focus within the literature has been on models which assume that the dynamics of the price of the underlying asset are governed by a Lévy process (sometimes referred to as a jump process). This type of model admits the possibility of extreme events (jumps), which are not captured by classical Black-Scholes type models based on the Brownian motion. Over the last decades, the literature has further shown that if the dynamics of the price of the underlying is governed by certain Lévy processes, such as the CGMY , the FMLS and the KoBoL, the price processes of European-style options satisfy a variety of fractional partial differential equations (FPDEs). In this dissertation, we will show that if the underlying price dynamic follows a Generalized Tempered Stable process, which admits as particular cases the aforementioned CGMY and KoBoL processes, prices of options satisfy an FPDE of the same type. Further, we will implement a simple finite difference scheme to solve the FPDE numerically to price European-type options.<br>info:eu-repo/semantics/publishedVersion

[Truncated abstract] This thesis examines competing asset-pricing models in Australia with the goal of establishing the model which best explains cross-sectional stock returns. The research employs Australian equity data over the period 1980-2001, with the major analyses covering the more recent period 1990-2001. The study first documents that existing asset-pricing models namely the capital asset pricing model (CAPM) and domestic Fama-French three-factor model fail to meet the widely applied Merton?s zero-intercept criterion for a well-specified pricing model. This study instead documents that the US three-factor model provides the best description of Australian stock returns. The three US Fama-French factors are statistically significant for the majority of portfolios consisting of large stocks. However, no significant coefficients are found for portfolios in the smallest size quintile. This result initially suggests that the largest firms in the Australian market are globally integrated with the US market while the smallest firms are not. Therefore, the evidence at this point implies domestic segmentation in the Australian market. This is an unsatisfying outcome, considering that the goal of this research is to establish the pricing model that best describes portfolio returns. Given pervasive evidence that liquidity is strongly related to stock returns, the second part of the major analyses derives and incorporates this potentially priced factor to the specified pricing models ... This study also introduces a methodology for individual security analysis, which implements the portfolio analysis, in this part of analyses. The technique makes use of visual impressions conveyed by the histogram plots of coefficients' p-values. A statistically significant coefficient will have its p-values concentrated at below a 5% level of significance; a histogram of p-values will not have a uniform distribution ... The final stage of this study employs daily return data as an examination of what is indeed the best pricing model as well as to provide a robustness check on monthly return results. The daily result indicates that all three US Fama-French factors, namely the US market, size and book-to-market factors as well as LIQT are statistically significant, while the Australian three-factor model only exhibits one significant market factor. This study has discovered that it is in fact the US three-factor model with LIQT and not the domestic model, which qualifies for the criterion of a well-specified asset-pricing model and that it best describes Australian stock returns.

My three essays contain three studies using multi-factor asset pricing models, where all the data are based on the US market. The first study extends intertemporal CAPMs with a few macro pricing factors: inflation or the cycle of industrial production (IP). I regard this specification of such models as a multi-factor pricing model, where this multi-factor linear pricing model can alternatively be derived from a consumption-based model from a theoretical perspective. I find significant evidence that the augmented multi-factor models outperform the original ICAPM. The results show that inflation is a key additional factor in the pricing models for the 25 size/book-to-market portfolios, while the cycle of IP is another vital additional factor in pricing models for the 25 size/momentum portfolios. Moreover, I find that most pricing information contained in the momentum factor is the inclusive information of the IP cycle, where the cycle of IP is generated by using the Hodrick–Prescott filter. The second study extends another two ICAPMs and Hou, Karolyi and Kho's three-factor model with inflation. The evidence shows that inflation significantly aids the original models in pricing 25 size/book-to-market portfolios in cross-sectional tests. Hence, I provide further robust evidence that inflation is the vital factor in the factor pricing models for the 25 size/book-to-market portfolios and a few other portfolios. Inflation provides additional explanatory power beyond Fama-French’s five factors in pricing the cross-sectional variation of 25 size/book-to-market portfolios. The third study investigates the performance of multi-factor asset pricing models in explaining the cross-section variation of the large number of expanding portfolios and a set of different portfolios, where the multi-factor models refer to the Fama-French three-factor model augmented by other pricing factors. I investigate the performance of several well-regarded multi-factor models by using Hansen’s general method of momentum (GMM), which is another alternative and very robust complement/guarantee to the only regression-based procedure in the previous literature. The results continuously support the superiority of the augmented multi-factor models. In general, augmented multi-factor models outperform the original models in a sound portion of different portfolios, where the original model refers to Fama-French’s three-factor model. In conclusion, my essays shed light on a fresh type of linear asset pricing model with sound theoretical background, and my research justifies the superiority of the multi-factor pricing model over Fama-French’s three-factor model in explaining the cross-sectional variation of equity returns with robust evidence.<br>Thesis (PhD Doctorate)<br>Doctor of Philosophy (PhD)<br>Dept Account,Finance & Econ<br>Griffith Business School<br>Full Text

<p>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.</p>

Modelling defaultable contingent claims has attracted a lot of interest in recent years, motivated in particular by the Late-2000s Financial Crisis. In several papers various approaches on the subject have been made. This thesis tries to summarize these results and derive explicit formulas for the prices of financial derivatives with credit risk. It is divided into two main parts. The first one is devoted to the well-known theory of modelling the default risk while the second one presents the results concerning pricing of the defaultable models that we obtained ourselves.

We consider an important class of derivative contracts written on multiple assets which are traded on a wide range of financial markets. More specifically, we are interested in developing novel methods for pricing financial derivatives using approximation theoretic methods which are not well-known to the financial engineering community. The problem of pricing of such contracts splits into two parts. First, we need to approximate the respective density function which depends on the adapted jump-diffusion model. Second, we need to construct a sequence of approximation formulas for the price. These two parts are connected with the problem of optimal approximation of infinitely differentiable, analytic or entire functions on noncompact domains. We develop new methods of recovery of density functions using sk-splines (in particular, radial basis functions), Wiener spaces and complex exponents with frequencies from special domains. The respective lower bounds obtained show that the methods developed have almost optimal rate of convergence in the sense of n-widths. On the basis of results obtained we develop a new theory of pricing of basket options under Lévy processess. In particular, we introduce and study a class of stochastic systems to model multidimensional return process, construct a sequence of approximation formulas for the price and establish the respective rates of convergence.

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.

Includes bibliographical references (leaves 144-149).<br>This work will present an option pricing model that accommodates parameters that vary over time, whilst still retaining a closed-form expression for option prices: the Hidden Markov Option Pricing Model. This is possible due to the macro-structure of this model and provides the added advantage of ensuring efficient computation of option prices. This model turns out to be a very natural extension to the Black-Scholes model, allowing for time-varying input parameters.

Thesis (Ph. D.)--University of North Carolina at Chapel Hill, 2008.<br>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.

This thesis is concerned with the pricing of credit derivatives, in particular credit default swaps (CDSs), collateralized debt obligation (CDOs), bespoke CDOs, forward starting CDOs and options on CDOs. Two new technical devices are introduced in this thesis: one for the modelling of default probabilities of credit default swaps and the other for the pricing of portfolio credit derivatives. We are mainly interested in portfolio credit derivatives, although we also study credit default swaps (single name credit derivatives), as these instruments are the building blocks of synthetic portfolio credit derivatives. A reduced-form model is used widely in the market to infer survival probabilities from quoted CDS premiums. To price CDSs, we introduce an innovation which is based on the Principle of Maximum Entropy (PME). This approach allows us to determine the survival probability distribution that is least committal with respect to any unknown or missing information. This is in contrast to common market reduced-form approaches, where these models require an extra assumption about the functional form of the hazard rates in order to infer survival probabilities. For example, it is common practice to assume that the default of a company is triggered by the first jump of a Poisson process (which admits a hazard rate). This hazard rate is usually assumed to be piecewise constant. We compare the results of our model with the reduced-form method commonly used in the market. The second new methodology presented in this thesis, values portfolio credit deriva­tives using a discrete Markov model to create a loss distribution surface. This Markov loss model is the major contribution of this thesis. Similar to a proposal by Hull and White [51], we parameterize our loss model by the number of losses in an otherwise ho­mogeneous portfolio. Further following an approach by Liu [70], we adopt a Markov model for the loss probabilities. However, Liu's approach is a continuous time backward model, whilst ours is a discrete time forward model. Although our Markov loss model is parameterized by a small number of parameters, the model leads to successful calibration outcomes. We demonstrate the calibration process of our model to the iTraxx European investment grade series 7 (iTraxxS7) index over a range of dates. The iTraxxS7 is a stan­dardized CDO available in the market. We note that we can also apply our model to other available standardized CDOs such as the CDX index family. Our model is an aggregate approach, although many researchers would call it a top ­down approach. To be a true top-down method, our Markov loss model needs to allocate the total loss distribution of the portfolio down to the individual name level. We note that for a top-down approach, the total loss distribution of the CDO is modelled without going into the detail of individual loss distributions of names that make up the underlying port­folio of the CDO. This is in contrast with the bottom-up approach, where the total loss distribution is constructed from the individual loss distributions. To price many portfolio credit derivatives, the individual level of losses is not required. This is true for the port­folio credit derivatives considered in this thesis, except for bespoke CDOs. As bespoke CDOs have a different underlying portfolio than the standardized CDOs, a technique is required to allocate aggregate losses to the individual level. To price bespoke CDOs, we combine our Markov model with a technique proposed by Inglis and Lipton [52]. In their paper, Inglis and Lipton [52] suggest a single latent factor model be used to build up the portfolio loss distribution from the conditional survival probabilities of individual names. During the calibration process, our Markov loss model creates a loss distribution surface that can be used to price other portfolio credit derivatives. We use this loss distri­bution surface to price some exotic portfolio credit derivatives such as bespoke tranches, bespoke CDOs, forward starting CDOs, options on CDO tranches and options on CDS indexes. There is no liquid market for many exotic portfolio credit derivatives so it is dif­ficult to gauge the accuracy of our prices for these products. Although by comparing our model results with the prices obtained from base correlation techniques commonly used in the market, our model can be seen to provide reasonable prices for bespoke tranches and bespoke CDOs. We note that our approach is arbitrage free, while the base correlation methods are known to sometimes lead to arbitrage opportunities.

In this thesis the GARCH models are applied to evaluate financial options and futures. In the first application, the GARCH models in parsimonious form are studied for pricing the S&P500 options. Unlike previous studies that focus on developed formulation, the results indicate that simplified models provide effective performance and it is the simple GARCH model that yields the least valuation error. To our consideration, examining model possessing simplification is of practical importance because model estimation becomes readily accessible through available econometric software, which circumvent programming barriers in implementing alternative one’s own pricing methods. The second application consider the component GARCH models for currency option pricing. The valuation results favour the component formulations particularly in the pricing of long term contracts. Volatility modelling results indicate that the return-volatility relationship is symmetric in the long run, but over the short term asymmetry also arises in the EURUSD and GBPUSD exchange rates. The third application evaluates canola futures in Canada in relation to spot market price. Results confirm the cointegrating relationship with threshold corresponding to transaction and adjustment cost. And it is the futures market that adjusts actively to price disparities but in the meantime there is volatility spillover from futures to the spot market. Overall, our empirical assessments indicate the importance of the time varying volatility and the improvements achieved in option pricing and futures evaluation. We believe the present study’s analysis provides useful suggestions and further guidance to practitioners and investors for the pricing and trading in the equity and foreign exchange markets, also to the market agents to better evaluate price uncertainty in order to guard against adverse price changes.

American options are the most commonly traded financial derivatives in the market. Pricing these options fairly, so as to avoid arbitrage, is of paramount importance. Closed form solutions for American put options cannot be utilised in practice and so numerical techniques are employed. This thesis looks at the work done by other researchers to find an analytic solution to the American put option pricing problem and suggests a practical method, that uses Monte Carlo simulation, to approximate the American put option price. The theory behind option pricing is first discussed using a discrete model. Once the concepts of arbitrage-free pricing and hedging have been dealt with, this model is extended to a continuous-time setting. Martingale theory is introduced to put the option pricing theory in a more formal framework. The construction of a hedging portfolio is discussed in detail and it is shown how financial derivatives are priced according to a unique riskneutral probability measure. Black-Scholes model is discussed and utilised to find closed form solutions to European style options. American options are discussed in detail and it is shown that under certain conditions, American style options can be solved according to closed form solutions. Various numerical techniques are presented to approximate the true American put option price. Chief among these methods is the Richardson extrapolation on a sequence of Bermudan options method that was developed by Geske and Johnson. This model is extended to a Repeated-Richardson extrapolation technique. Finally, a Monte Carlo simulation is used to approximate Bermudan put options. These values are then extrapolated to approximate the price of an American put option. The use of extrapolation techniques was hampered by the presence of non-uniform convergence of the Bermudan put option sequence. When convergence was uniform, the approximations were accurate up to a few cents difference.

The thesis covers a wide range of topics from the credit risk modeling with the emphasis put on pricing of the claims subject to the default risk. Starting with a separate general contingent claim pricing framework the key topics are classified into three fundamental parts: firm-value models, reduced-form models, portfolio problems, with a possible finer sub-classification. Every part provides a theoretical discussion, proposal of self-developed methodologies and related applications that are designed so as to be close to the real-world problems. The text also reveals several new findings from various fields of credit risk modeling. In particular, it is shown (i) that the stock option market is a good source of credit information, (ii) how the reduced-form modeling framework can be extended to capture more complicated problems, (iii) that the double t copula together with a self-developed portfolio modeling framework outperforms the classical Gaussian copula approaches. Many other, partial findings are presented in the relevant chapters and some other results are also discussed in the Appendix.

The Black-Scholes model and corresponding option pricing formula has led to a wide and extensive industry, used by financial institutions and investors to speculate on market trends or to control their level of risk from other investments. From the formation of the Chicago Board Options Exchange in 1973, the nature of options contracts available today has grown dramatically from the single-date contracts considered by Black and Scholes (1973) to a wider and more exotic range of derivatives. These include American options, which can be exercised at any time up to maturity, as well as options based on the weighted sums of assets, such as the Asian and basket options which we consider. Moreover, the underlying models considered have also grown in number and in this work we are primarily motivated by the increasing interest in past-dependent asset pricing models, shown in recent years by market practitioners and prominent authors. These models provide a natural framework that considers past history and behaviour, as well as present information, in the determination of the future evolution of an underlying process. In our studies, we explore option pricing techniques for arithmetic Asian and basket options under a Stochastic Delay Differential Equation (SDDE) approach. We obtain explicit closed-form expressions for a number of lower and upper bounds before giving a practical, numerical analysis of our result. In addition, we also consider the properties of the approximate numerical integration methods used and state the conditions for which numerical stability and convergence can be achieved.

This thesis is motivated by the problem of modelling and analysing real share price data. Various approaches and models are considered. One approach is to consider a random walk on a discrete-time Markov chain perturbed by Gaussian noise as a model for real share price data. To implement this model a numerical algorithm is constructed to treat the NP hard Emdedding problem. A second approach to modelling share price data is to consider a random walk on the lamplighter group perturbed by Gaussian noise. This class of problems leads to interesting theoretical questions about asymptotic behaviour of random stochastic matrices. In particular, we found an asymptotic expression for the L 2 error between two independent random stochastic matrices. We apply a variety of statistical and modelling techniques to justify the models including traditional econometric transforms, regression and MLE techniques, EM algorithms, and Monte Carlo methods such as random search.

Mestrado em Actuarial Science<br>Os Modelos Lineares Generalizados (GLMs) são amplamente utilizados na precificação de seguros do ramo Não Vida. O prémio cobrado pela seguradora é calculado com base em uma tarifa. A abordagem clássica para estimar o prémio é feita assumindo a independência entre o número de sinistros e o seu custo. A partir desta independência, a frequência e a severidade dos sinistros são estimados através de GLMs separados e a tarifa é obtida combinando os dois modelos. O presente relatório fornece uma breve introdução sobre a metodologia e descreve como preparámos os dados antes da aplicação do GLM. Os modelos obtidos para os Tratamentos e Consultas de Estomatologia, uma das muitas coberturas que podem ser incluídas numa apólice de Seguro Saúde, são analisados neste relatório. O software SAS foi utilizado para construir as bases de dados e para organizar adequadamente a informação e o software R foi utilizado para o processo de modelagem. Uma vez estimados os modelos, o prémio puro foi calculado e a tarifa, para a cobertura mencionada, foi construída. Por fim, comparámos os resultados obtidos em R com as conclusões obtidas pelos meus colegas, utilizando o software implementado pela empresa. Concluímos que ambos os modelos não são significativamente diferentes, apesar de apresentarem algumas distinções estruturais.<br>Generalized Linear Models (GLMs) are being broadly used in the Non-Life Insurance Pricing. The premium charged by the insurance company is calculated based on a tariff. The most standard procedure to estimate the pure premium is by assuming that the claim counts and claim amounts are independent. From this independence, the claim frequency and severity can be forecasted by distinct GLMs and the Tariff is obtained by combining both models. The present report gives a brief introduction on the methodology and describes how we prepared the data prior to the GLM application. The models obtained for the Stomatology Treatments and Appointments, one of the many coverages that can be included in a Health Insurance policy, are analyzed in this report. The SAS software was used to construct the datasets and to properly organize the data and R was the software used for the modelling process. Once the models were estimated, the pure premium was calculated and a tariff for the mentioned coverage was constructed. Finally, we compared the results obtained by modelling the coverage in R with the output obtained by my colleagues, using the software implemented by the company. We conclude that both models are not significantly different, despite having some structural distinctions.<br>info:eu-repo/semantics/publishedVersion

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.

Includes bibliographical references<br>This thesis aims to look at option pricing under affine jump diffusion processes, with particular emphasis on using Fourier transforms. The focus of the thesis is on using Fourier transform to price European options and Barrier options under the Heston stochastic volatility model and the Bates model. Bates model combines Merton's jump diffusion model and Heston's stochastic volatility model. We look at the calibration problem and use Matlab functions to model the DAX options volatility surface. Finally, using the parameters generated, we use the two stated models to price barrier options.

In this thesis we discuss basket option valuation for jump-diffusion models. We suggest three new approximate pricing methods. The first approximation method is the weighted sum of Rogers and Shi’s lower bound and the conditional second moment adjustments. The second is the asymptotic expansion to approximate the conditional expectation of the stochastic variance associated with the basket value process. The third is the lower bound approximation which is based on the combination of the asymptotic expansion method and Rogers and Shi’s lower bound. We also derive a forward partial integro-differential equation (PIDE) for general asset price processes with stochastic volatilities and stochastic jump compensators. Numerical tests show that the suggested methods are fast and accurate in comparison with Monte Carlo and other methods in most cases.

In the commodity and energy markets, there are two kinds of risk that traders and analysts are concerned a lot about: multiple underlying risk and average price risk. Spread options, swaps and swaptions are widely used to hedge multiple underlying risks and Asian (average price) options can deal with average price risk. But when those two risks are combined together, then we need to consider Asian spread options and Asian-European spread options for hedging purposes. For an Asian or Asian-European spread call option, its payoff depends on the difference of two underlyings' average price or of one average price and one final (at expiration) price. Asian and Asian-European spread option pricing is challenging work. Even under the basic assumption that each underlying price follows a log-normal distribution, the average price does not have a distribution with a simple form. In this dissertation, for the first time, a systematic analysis of Asian spread option and Asian-European spread option pricing is proposed, several original approaches for the Black-Scholes-Merton model and a special stochastic volatility model are developed and some numerical computation tests are conducted as well.

L'obbiettivo della presente Tesi é di considerare la specificazione di mod­elli di pricing in tempo discreto (in generale, incompleti) con variabili latenti, al fine di sfruttare i vantaggi derivanti da tale contesto a tempo discreto e al fine di fornire una descrizione completa degli aspetti storici e neutrali al rischio dei prezzi dei titoli. Negli ultimi anni osserviamo un importante sviluppo di modelli di pric­ing in tempo discreto, dove la modellizzazione secondo il principio dello Stochastic Discount Factor (SDF) e la caratterizzazione della distribuzione condizionale delle variabili di stato tramite la trasformata di Laplace sem­brano fornire risultati promettenti. Più precisamente, la caratterizzazione generale di modelli di pricing in tempo discreto, usando questo tipo di approccio, e dove é assunta una speci­ficazione Compound Autoregressive (CAR ovvero affine) per le variabili di stato [vedi Darolles, Gourieroux, Jasiak (2002)], é stata proposta da Gourier­oux e Monfort (2003) e Gourieroux, Monfort e Polimenis (2002, 2003); in questi articoli viene presentata la metodologia generale di pricing e vengono specificati modelli per la Struttura a Termine e per il Rischio di Credito. Il tempo discreto é un contesto naturale per sviluppare modelli di valoriz­zazione volti a future implementazioni econometriche; infatti, i dati storici sono campionati con frequenza discreta, le transazioni finanziarie sono tipi­camente registrate a intervalli temporali discreti, la stima di parametri e i test statistici implicano dati a tempo discreto e le previsione sono fatte a orizzonti discreti. Un secondo e importante vantaggio che si ha nel lavorare in tempo dis­creto emerge quando consideriamo la classe di processi affini per applicazioni finanziarie. La classe di processi affini in tempo discreto (processi CAR) [pro­posti, come indicato sopra, da Darolles, Gourieroux, Jasiak (2002)] é molto più ampia della classe equivalente in tempo continuo proposta da Duffie, Fil­ipovic and Schachermayer (2003) : tutti i processi affini in tempo continuo campionati a istanti temporali discreti sono CAR, mentre esiste un ampio numero di processi CAR senza un processo equivalente in tempo continuo. Questa é una conseguenza del problema di embedding che caratterizza la classe affine in tempo continuo : tali processi devono essere infinitamente decomponibili, mentre tale condizione non é necessaria in tempo discreto [vedi Darolles, Gourieroux and Jasiak (2002) and Gourieroux, Monfort and Polimenis (2002)]. Nella Tesi sfrutteremo il contesto a tempo discreto anche per introdurre processi Non-Gaussiani e Non-Markoviani come le Misture di Processi Con­dizionatamente Gaussiani. Per quanto riguarda l'utilizzo della trasformata di Laplace condizionale per descrivere la distribuzione storica e neutrale al rischio delle variabili di stato, si osservi come in molte applicazione economico-finanziarie siamo por­tati in modo naturale a dover calcolare la trasformata di Laplace di tali variabili di stato. Alcuni esempi possibili sono i seguenti : (a) optimal port­folio problems (CARA utility functions, Markowitz), (b) asset pricing by thè certainty equivalence principle (CARA utility functions), (c) discrete time derivative pricing and term strueture models with exponential-affine SDFs, (d) panel duration models, (e) extreme risk [see Darolles, Gourieroux and Jasiak (2002) for details]. Vedremo, inoltre, che la trasformata di Laplace é uno strumento molto utile per caratterizzare anche la distribuzione storica e neutrale al rischio di Misture di Processi Condizionatamente Gaussiani. Per finire, la necessità di prendere in considerazione le fonti di rischio ril­evanti nell'influenzare il titolo da valorizzare, porta a considerare lo Stochas­tic Discount Factor (SDF) come strumento per caratterizzare la procedura di pricing : lo SDF é una variabile casuale (chiamata anche Pricing Kernel o State Price Deflator) che sintetizza sia l'attualizzazione temporale che la correzione per il rischio, e che porta a specificare, conseguentemente, una procedura di valorizzazione che fornisce una modellizzazione completa degli aspetti storici e neutrali al rischio. Il tempo discreto implica in generale un contesto a mercato incompleto e una molteplicità di formule di pricing; il problema della molteplicità viene ridotto imponendo una forma particolare allo SDF; il Pricing Kernel viene specificato, infatti, secondo una forma esponenziale-affine che si é dimostrata utile in molte circostanze e che troviamo sovente in letteratura [vedi Lu­cas (1978), Gerber e Shiu (1994), Stutzer (1995, 1996), Buchen e Kelly (1996), Buhlmann et al. (1997, 1998), Polimenis (2001), Gourieroux e Mon­fort (2002)]. Inoltre, uno SDF con una forma esponenziale-affine presenta proprietà tecniche interessanti : tale approccio infatti, che é basato sulla trasformata di Esscher in un contesto dinamico a tempo discreto, perme­tte di selezionare una misura martingale (di pricing) equivalente che riflette, nella formula di pricing, le diverse fonti di rischio da valorizzare. Ora, il contesto a tempo discreto, assieme ai principi di modellizzazione dello SDF esponenziale-affine e della transformata di Laplace, costituiscono gli strumenti usati nei tre capitoli fondamentali della Tesi. La Tesi analizza il ruolo che l'introduzione di variabili latenti può avere, in questa classe di modelli di pricing a tempo discreto, nello specificare metodologie di valorizzazione complete e coerenti rispetto alle indicazioni empiriche. Nei Capitoli 2 e 3 l'obbiettivo, infatti, é quello di specificare metodologie per la valorizzazione di prodotti derivati in grado di prendere in considerazione i tipici fenomeni di skewness e excess kurtosis che osserviamo nella distribuzione dei rendimenti di titoli azionari, e di riuscire quindi a repli­care le volatilità implicite di Black e Scholes (BS) e superfici di volatilità im­plicita con forme di smile a volatility skew coerenti con l'evidenza empirica1. Qui, le variabili latenti introducono cambiamenti di regime nella dinamica del titolo sottostante (cambiamenti, per esempio, tra un mercato a regime di alta e bassa volatilità) ovvero, introducono nella distribuzione storica del rendimento rischioso fenomeni come medie e varianze stocastiche. Nel Capi­tolo 4, vengono proposti modelli affini bifattoriali per la struttura a termine dei tassi di interesse (in tempo discreto) con variabili latenti; l'obbiettivo é quello di ottenere famiglie di possibili strutture a termine con forme più prossime (rispetto ai modelli unifattoriali in tempo discreto e continuo) a quelle osservate. In questo caso, le variabili latenti introducono parametri stocastici (continui e discreti) nella dinamica del fattore (tasso di interesse a breve scadenza) responsabile della forma della struttura a termine nei modelli unifattoriale. In altre parole, si vogliono definire procedure di pricing capaci di pren­dere in considerazione, in modo coerente e utile, le fonti di rischio descritte dai cambiamenti di regime e dai parametri stocastici; vogliamo specificare metodologie di valorizzazione non basate su ipotesi arbitrarie (spesso us­ate in letterature) come, per esempio, la neutralità al rischio degli investi­tori o la natura idiosincratica del rischio, e vogliamo derivare formule di pricing che hanno una forma analitica o che sono facili da implementare. L'organizzazione della Tesi é indicata nel prossimo paragrafo. 1 Questi due capitoli corrispondono a due articoli scritti con Henri Bertholon e Alain Monfort. Sintesi dei capitoli Nel CAPITOLO 1 viene inizialmente presentato il principio di modelliz­zazione dello SDF, e come é legato alla Law of One Price e al principio di Absence of Arbitrage Opportunity; successivamente, vengono descritti gli strumenti base che caratterizzano i modelli di pricing a tempo discreto sviluppati nella tesi : lo SDF esponenziale-affine, e la rappresentazione della distribuzione condizionale delle variabili di stato tramite la trasformata di Laplace considerando come esempio i processi CAR. Nel CAPITOLO 2 proponiamo una nuova procedura di valorizzazione di opzioni Europee che porta ad una generalizzazione della formula di Black e Scholes [utile, quindi, dal punto di vista delle istituzioni finanziarie]; in particolare, ci focalizziamo sulle due fonti fondamentali di cattiva specifi­cazione dell'approccio BS, ovvero l'assenza di Gaussianità e la dinamica. Gli strumenti utilizzati sono le misture in tempo discreto di processi condizion­atamente gaussiani, cioè processi {yt} tali che yt+1 é gaussiano condizion­atamente ai propri valori passati e al valore presente zt+ì di un white noise non osservabile a valori discreti. Forniamo (in un semplice caso statico) le simulazioni di volatilità implicita di BS e di superfici di volatilità implicita, e osserviamo l'abilità delle procedure di pricing che proponiamo nel repli­care smiles e volatility skews coerenti con l'evidenza empirica. Per quanto riguarda le superfici di volatilità implicita, il modello statico mostra qualche limite che é superato, con una dinamica di tipo Regime-Switching attribuita a zt+i, nel Capitolo 3. Il CAPITOLO 3 presenta una naturale evoluzione del precedente capitolo; infatti, prende in considerazione il caso in cui la variabile latente zt+\ non sia più un white noise ma, tipicamente, una Catena di Markov. Più precisa­mente, presentiamo il modello General Switching Regime per il pricing di derivati, applicato ai casi di opzioni Europee e path dependent. Studiamo inoltre le condizioni sotto le quali c'è una trasmissione di causalità (assenza di causalità istantanea, assenza di causalità, indipendenza), esistente tra la madia e la varianza stocastica, dal mondo storico al mondo neutrale al ris­chio. A questo scopo separiamo la dinamica della media e della varianza (in un caso di Hidden Markov Chain) usando due distinte variabili latenti (zit+i , Z2t+i), dove sia zu+i che Zit+\ possono prendere J possibili valori, e dove la prima variabile latente descrive la dinamica della media mentre la seconda quella della varianza. Lo scopo del CAPITOLO 4 é di introdurre parametri stocastici e cambi­amenti di regime nei modelli affini unifattoriali per la struttura a termine presentati da Gourieroux, Monfort and Polimenis (2002) [GMP (2002)], al fine di estendere la dinamica del tasso a breve termine e di ampliare, con­seguentemente, la ricchezza di curve della struttura a termine che tali modelli sono in grado di riprodurre. Vengono studiati diversi modelli alternativi e vengono presentate le simulazioni sulle possibili struttura a termine che essi sono in grado di replicare; in particolare, le strutture a termine ottenute mostrano forme con gobbe verso l'alto e verso il basso, forme con diversi gradi curvatura e con due mode. Per finire, presentiamo in problema dell' in­dividuazione di mimicking factors [un vettore Rt = (rf;i+2,..., rt,t+n) di tassi di interesse con differenti maturity] per i parametri stocastici e i cambiamenti di regime : questo é un problema interessante, dal punto di vista statistico, data l'osservabilità dei tassi di interesse. The aim of the thesis is to consider, as a new research direction, the specification of discrete time pricing models (in general incomplete) with latent variables, in order to exploit the advantages coming from the discrete time framework and in order to give a complete description of historical and risk-neutral aspects of asset prices. In the last years, we observe an important development of asset pric­ing models in discrete time, where the use of the Stochastic Discount Factor (SDF) modeling principle and the characterization of the state variables con­ditional distributions by means of the Laplace transform seem promising. More precisely, the general discrete time characterization of asset pricing models, using this kind of approach, and where a compound autoregressive (affine or CAR) specification for the state variables is assumed [see Darolles, Gourieroux, Jasiak (2002)], has been proposed by Gourieroux and Monfort (2003) and Gourieroux, Monfort and Polimenis (2002, 2003); in these pa­pers the general pricing methodology and the specifications of Affine Term Structure models, along with the Credit Risk Analysis, are presented. The discrete time is a natural framework to develop pricing models for fu­ture econometric implementations, given that all historical data are sampled discretely, financial transactions are typically recorded at discrete intervals, parameter estimation and hypothesis testing involve discrete data records, and forecasts are produced at discrete horizons. A second important advantage to work in discrete time emerges when we consider the class of affine processes for financial applications. The class of discrete time affine (CAR) processes [proposed, as indicated above, by Darolles, Gourieroux and Jasiak (2002)] is much larger than the equivalent continuous time class proposed by Duffie, Filipovic and Schachermayer (2003) : all continuous time affine processes sampled at discrete points are CAR, while there exists a large number of CAR processes without a continuous time counterpart. This is a consequence of the embedding problem that characterizes the continuous time class : these processes have to be infinitely decomposable, and this decomposition condition is not necessary in discrete time [see Darolles, Gourieroux and Jasiak (2002) and Gourieroux, Monfort and Polimenis (2002) for details]. In this Thesis, we will also exploit the discrete time framework in order to introduce non-Gaussian and non-Markovian processes like, for instance, the Mixtures of Conditionally Normal Processes. With regard to the use of the conditional Laplace transform to describe the historical and risk-neutral (pricing) distribution of the state variables, we observe that in many financial and economic applications we are naturally lead to determine the Laplace transform of the processes of interest; possible examples are the followings : (a) optimal portfolio problems (CARA utility functions, Markowitz), (b) asset pricing by the certainty equivalence princi­ple (CARA utility functions), (c) discrete time derivative pricing and term structure models with exponential-affine SDFs, (d) panel duration models, (e) extreme risk [see Darolles, Gourieroux and Jasiak (2002) for details]. In this Thesis we will see that the Laplace transform is also very convenient for the class of Mixtures of Conditionally Normal Processes. Finally, the need to take into account the relevant sources of risk that influence the asset one wants to price, lead to consider a Stochastic Discount Factor (SDF) approach to characterize the pricing procedure : the SDF is a random variable (called also Pricing Kernel or State Price Deflator) which summarizes both the time discounting and the risk correction, and which specifies, consequently, a pricing procedure that gives a complete modelisa­tion of the historical and risk-neutral (pricing) aspects. Given that discrete time implies in general an incomplete market frame­work and a multiplicity of asset pricing formulas, the multiplicity problem is reduced by imposing a special structure on the SDF; in particular, it is pos­sible to consider for the pricing kernel an exponential-affine function of the state variables which has proved useful in many circumstances and that we find frequently in the literature [see Lucas (1978), Gerber and Shiu (1994), Stutzer (1995, 1996), Buchen and Kelly (1996), Buhlmann et al. (1997, 1998), Polimenis (2001), Gourieroux and Monfort (2002)]. Moreover, a SDF with an exponential-affine form presents interesting technical properties : it is the Esscher transform approach, in a dynamic discrete time framework, which gives the possibility to select an equivalent martingale (pricing) mea­sure that reflects, in the pricing formula, the different sources of risks to be priced. Now, the discrete time framework, along with the exponential-affine SDF modeling principle and the Laplace transform approach, constitute the in­struments used in the three core chapters of the Thesis. The Thesis analyzes the role that the introduction of latent variables could play, in this class of discrete time pricing models, for the specification of com­plete and coherent, with respect to the empirical evidence, pricing methodolo­gies. In Chapters 2 and 3 the purpose is, indeed, to specify derivative pricing methodologies able to take into account time-varying stock returns skewness and excess kurtosis, that is, pricing procedure able to replicate phenomena like implied Black and Scholes volatilities and implied volatility surfaces with smile and volatility skew shapes coherent with empirical studies2. Here, the latent variables are regimes of the underlying risky asset (switches, for in­stance, between a low volatility and a high volatility regime of the market), that is, they introduce phenomena like stochastic means and variances in the historical dynamics of the stock return underlying the derivative product. In Chapter 4, the Thesis proposes discrete time two-factor affine term structure models with latent variables, able to obtain families of possible term struc­tures with shapes closer (with respect to one-factor continuous and discrete time models) to the observed ones. In this case the latent variables intro­duce discrete and continuous stochastic parameters in the dynamics of the factor (the short term interest rate) that explains the term structure of the univariate models. In other words, we want to define pricing procedure able to take into account in a coherent and useful way the sources of risk described by the switching of regimes and by the stochastic parameters; we want to spec­ify pricing procedure not characterized by arbitrary assumptions, frequently used in the literature, like, for instance, the risk-neutrality of the investors or the idiosyncratic nature of the risk. In addition, we want to provide pricing formulas which have an analytical form or which are simple to implement. The organization of the Thesis is detailed below. 2They correspond to two papers written with Henri Bertholon and Alain Monfort. Outline of the chapters In CHAPTER 1 first we present the SDF modeling principle, and its rela­tions with the Law of One Price and the Absence of Arbitrage Opportunity principle, then we consider the basic tools characterizing the discrete time pricing models developed in this Thesis : the exponential-affine SDF, the conditional Laplace transform description of the future uncertainty and the CAR processes. In CHAPTER 2 we propose a new European option pricing procedure which lead to a generalization of the Black and Scholes pricing formula [and, therefore, useful for financial institutions]. We focus on two impor­tant sources of misspecification for the Black-Scholes approach, namely the lack of normality and the dynamics. The basic tools are the mixtures of discrete time conditionally normal processes, that is to say processes {yt} such that yt+i is gaussian conditionally to its past values and the present value zt+i of a discrete value unobservable white noise process. We provide (in a static framework) simulations of implied Black-Scholes volatilities and implied volatilities surfaces, and we observe the ability of the proposed as­set pricing methodology to replicate smiles and volatility skews coherent with empirical results. With regard to implied volatility surfaces, the static model shows some limit which is overcome, with a Regime-Switching dynamics for Zt+i, in Chapter 3. CHAPTER 3 presents a natural evolution of the previous chapter, that is, it considers the case where the latent variable zt+1 is no more a white noise but, typically a Markov chain. More precisely, we present the derivative pricing General Switching Regime model applied to the cases of European and path dependent options. We also study the conditions under which there is a transmission of causality relations (absence of instantaneous casuality, absence of causality, independence), existing between the stochastic mean and variance, from the historical to the risk-neutral world. For this purpose we separate the dynamics of these two moments (in the case of a Hidden Markov Chain) using two distinct latent variables (zit+i , 22t+i)> where both Z\t+\ and Z2t+\ can take J values and where the first latent variable describe the dynamics of the mean while the second one describe the dynamics of the variance. The aim of CHAPTER 4 is to introduce stochastic parameters and switch­ing regimes in the one-factor Affine Term Structure Models proposed by Gourieroux, Monfort and Polimenis (2002) [GMP (2002) hereafter], in order to extend the dynamics of the short rate and to improve, consequently, the richness of shapes of the term structure they are able to replicate. Different models are studied and simulations of the possible term structures we are able to replicate are presented; in particular, the provided term structures show shapes with bumps both upwards and downwards, shapes with different degrees of curvature and with two modes. Finally, we present the problem to find mimicking factors [a vector Rt (rt)t+2,..., ru+n) of interest rates at different maturities] for stochastic parameters and switching regimes : this is an interesting problem, from a statistical point of view, because of the observability of the interest rates.

>Magister Scientiae - MSc<br>Warrant pricing has become very crucial in the present market scenario. See, for example, M. Hanke and K. Potzelberger, Consistent pricing of warrants and traded options, Review Financial Economics 11(1) (2002) 63-77 where the authors indicate that warrants issuance affects the stock price process of the issuing company. This change in the stock price process leads to subsequent changes in the prices of options written on the issuing company's stocks. Another notable work is W.G. Zhang, W.L. Xiao and C.X. He, Equity warrant pricing model under Fractional Brownian motion and an empirical study, Expert System with Applications 36(2) (2009) 3056-3065 where the authors construct equity warrants pricing model under Fractional Brownian motion and deduce the European options pricing formula with a simple method. We study this paper in details in this mini-thesis. We also study some of the mathematical models on warrant pricing using the Black-Scholes framework. The relationship between the price of the warrants and the price of the call accounts for the dilution effect is also studied mathematically. Finally we do some numerical simulations to derive the value of warrants.

Financial practitioners use models in order to price, hedge and measure risk. These models are reliant on assumptions and are prone to ”model risk”. Increased innovation in complex financial products has lead to increased risk exposure and has spurred research into understanding model risk and its underlying factors. This dissertation quantifies model risk inherent in Value-at-Risk (VaR) on a variety of portfolios comprised of European options written on the ALSI futures index across various maturities. The European options under consideration will be modelled using the Black-Scholes, Heston and Variance-Gamma models.

Submitted by Diego Brandão (digusmao@hotmail.com) on 2017-01-06T20:29:33Z No. of bitstreams: 1 Tese - Versao final - Diego Brandao.pdf: 1402061 bytes, checksum: 2a9d03af25fdeae9cb4300343d707aa2 (MD5)<br>Approved for entry into archive by GILSON ROCHA MIRANDA (gilson.miranda@fgv.br) on 2017-02-20T13:39:19Z (GMT) No. of bitstreams: 1 Tese - Versao final - Diego Brandao.pdf: 1402061 bytes, checksum: 2a9d03af25fdeae9cb4300343d707aa2 (MD5)<br>Made available in DSpace on 2017-03-03T12:50:29Z (GMT). No. of bitstreams: 1 Tese - Versao final - Diego Brandao.pdf: 1402061 bytes, checksum: 2a9d03af25fdeae9cb4300343d707aa2 (MD5) Previous issue date: 2016-09-23<br>The thesis consists in three articles about the estimation of asset pricing models. The first paper analyses small sample properties of Generalized Empirical Likelihood estimators for the risk aversion parameter in CRRA preferences when the economy is characterized by rare disasters. In the second article, we develop and test a methodology to assess misspeci fied asset pricing models by taking into account the smallest probability distortion necessary to assign correct prices. In the final paper, we estimate an approximate long run risks model using Brazilian data.<br>Esta tese consiste em três artigos sobre a estimação de modelos de apreçamento de ativos. No primeiro artigo, analisamos as propriedades de amostra pequena dos estimadores da classe Generalized Empirical Likelihood para o coeficiente de aversão ao risco de preferências CRRA quando a economia é suscetível a desastres. No segundo artigo, apresentamos e testamos uma metodologia de avaliação de modelos de apreçamento mal especificados que leva em conta a menor distorção de probabilidade necessária sobre a medida real para que modelo aprece corretamente ativos. No terceiro artigo, estimamos uma versão aproximada do modelo de riscos de longo prazo utilizando dados brasileiros.

In this thesis, efforts are devoted to the stochastic modeling, measurement and evaluation of credit risks, the development of mathematical and statistical tools to estimate and predict these risks, and methods for solving the significant computational problems arising in this context. The reduced-form intensity based credit risk models are studied. A new type of reduced-form intensity-based model is introduced, which can incorporate the impacts of both observable trigger events and economic environment on corporate defaults. The key idea of the model is to augment a Cox process with trigger events. In addition, this thesis focuses on the relationship between structural firm value model and reduced-form intensity based model. A continuous time structural asset value model for the asset value of two correlated firms with a two-dimensional Brownian motion is studied. With the incomplete information introduced, the information set available to the market participants includes the default time of each firm and the periodic asset value reports. The original structural model is first transformed into a reduced-form model. Then the conditional distribution of the default time as well as the asset value of each name are derived. The existence of the intensity processes of default times is proven and explicit form of intensity processes is given in this thesis. Discrete-time Markovian models in credit crisis are considered. Markovian models are proposed to capture the default correlation in a multi-sector economy. The main idea is to describe the infection (defaults) in various sectors by using an epidemic model. Green’s model, an epidemic model, is applied to characterize the infectious effect in each sector and dependence structures among various sectors are also proposed. The models are then applied to the computation of Crisis Value-at-Risk (CVaR) and Crisis Expected Shortfall (CES). The relationship between correlated defaults of different industrial sectors and business cycles as well as the impacts of business cycles on modeling and predicting correlated defaults is investigated using the Probabilistic Boolean Network (PBN). The idea is to model the credit default process by a PBN and the network structure can be inferred by using Markov chain theory and real-world data. A reduced-form model for economic and recorded default times is proposed and the probability distributions of these two default times are derived. The numerical study on the difference between these two shows that our proposed model can both capture the features and fit the empirical data. A simple and efficient method, based on the ordered default rate, is derived to compute the ordered default time distributions in both the homogeneous case and the two-group heterogeneous case under the interacting intensity default contagion model. Analytical expressions for the ordered default time distributions with recursive formulas for the coefficients are given, which makes the calculation fast and efficient in finding rates of basket CDSs.<br>published_or_final_version<br>Mathematics<br>Doctoral<br>Doctor of Philosophy

Bibliography: pages 190-209.<br>This research proposed to identify the most accurate method of pricing rights using option pricing models, including the Black Scholes model, the Cox constant elasticity of variance model and the Merton jump diffusion model, and to determine the set of input parameters that lead to the most optimal results. The empirical results indicated that on average all of the models are able to estimate the actual rights trading prices relatively well. Some models performed better than others did and these findings were consistent with the original reasonings. The market was shown to not account for the effect of dilution. The best model prices were obtained when calculating volatility over a one year historical period that included the actual rights trading period. The hypothesis regarding trading volume showed that there is a significant impact of trading volume on the estimation of accurate option prices. The filter rule of rejecting rights prices below 10 cents and 100 cents also improved the results thus showing a bias for lower priced rights to be incorrectly valued and possibly some inefficiency in this sector of the market.