Cette thèse contient deux sujets différents la simulation d'événements rares et un transport d'homotopie pour l'estimation du modèle de volatilité stochastique, dont chacun est couvert dans une partie distincte de cette thèse. Les méthodes de particules, qui généralisent les modèles de Markov cachés, sont largement utilisées dans différents domaines tels que le traitement du signal, la biologie, l'estimation d'événements rares, la finance, etc. Il existe un certain nombre d'approches basées sur les méthodes de Monte Carlo, tels que Markov Chain Monte Carlo (MCMC), Monte Carlo séquentiel (SMC). Nous appliquons des algorithmes SMC pour estimer les probabilités de défaut dans un processus d'intensité basé sur un processus stable afin de calculer un ajustement de valeur de crédit (CV A) avec le wrong way risk (WWR). Nous proposons une nouvelle approche pour estimer les événements rares, basée sur la génération de chaînes de Markov en simulant le système hamiltonien. Nous démontrons les propriétés, ce qui nous permet d'avoir une chaîne de Markov ergodique et nous montrons la performance de notre approche sur l'exemple que nous rencontrons dans la valorisation des options. Dans la deuxième partie, nous visons à estimer numériquement un modèle de volatilité stochastique, et à le considérer dans le contexte d'un problème de transport, lorsque nous aimerions trouver «un plan de transport optimal» qui représente la mesure d'image. Dans un contexte de filtrage, nous le comprenons comme le transport de particules d'une distribution antérieure à une distribution postérieure dans le pseudo-temps. Nous avons également proposé de repondérer les particules transportées, de manière à ce que nous puissions nous diriger vers la zone où se concentrent les particules de poids élevé. Nous avons montré sur l'exemple du modèle de la volatilité stochastique de Stein-Stein l'application de notre méthode et illustré le biais et la variance. / The thesis introduces simulation techniques that are based on particle methods and it consists of two parts, namely rare event simulation and a homotopy transport for stochastic volatility model estimation. Particle methods, that generalize hidden Markov models, are widely used in different fields such as signal processing, biology, rare events estimation, finance, etc. There are a number of approaches that are based on Monte Carlo methods that allow to approximate a target density such as Markov Chain Monte Carlo (MCMC), sequential Monte Carlo (SMC). We apply SMC algorithms to estimate default probabilities in a stable process based intensity process to compute a credit value adjustment (CV A) with a wrong way risk (WWR). We propose a novel approach to estimate rare events, which is based on the generation of Markov Chains by simulating the Hamiltonian system. We demonstrate the properties, that allows us to have ergodic Markov Chain and show the performance of our approach on the example that we encounter in option pricing.In the second part, we aim at numerically estimating a stochastic volatility model, and consider it in the context of a transportation problem, when we would like to find "an optimal transport map" that pushes forward the measure. In a filtering context, we understand it as the transportation of particles from a prior to a posterior distribution in pseudotime. We also proposed to reweight transported particles, so as we can direct to the area, where particles with high weights are concentrated. We showed the application of our method on the example of option pricing with Stein­Stein stochastic volatility model and illustrated the bias and variance.

Submitted by Daniel Loureiro da Silva (danloux@hotmail.com) on 2018-01-26T14:38:47Z No. of bitstreams: 1 Opção em Debêntures.pdf: 1582205 bytes, checksum: 6007183ecaa2252dd077e1fb1902f4f9 (MD5) / Approved for entry into archive by GILSON ROCHA MIRANDA (gilson.miranda@fgv.br) on 2018-01-26T18:34:06Z (GMT) No. of bitstreams: 1 Opção em Debêntures.pdf: 1582205 bytes, checksum: 6007183ecaa2252dd077e1fb1902f4f9 (MD5) / Made available in DSpace on 2018-01-30T13:02:17Z (GMT). No. of bitstreams: 1 Opção em Debêntures.pdf: 1582205 bytes, checksum: 6007183ecaa2252dd077e1fb1902f4f9 (MD5) Previous issue date: 2017-12-28 / On June 24, 2011, Law 12.431/11 was introduced, which granted a reduction of the Income Tax rate on bonds issued with the purpose of financing infrastructure projects. Since then, the so-called "Infrastructure Bonds" have increased their importance as a funding mechanism to companies in this sector. Since the publication of the Resolution of the Central Bank of Brazil (Bacen) No. 4.476, of 4 November 2016, it was allowed to include early redemption clauses at the discretion of the issuing company. Considering the peculiarities of these bonds, the methodologies already developed for pricing call options on bonds are not applicable. In this way, this work intends to develop a methodology for pricing the existence of an early redemption clause in an Infrastructure Bond covered by Law 12.431/11, considering the term structure of Brazilian interest rates and the costs of issuing bonds. / Em 24.06.2011, foi lançada a Lei n° 12.431/11, que concedeu uma redução da alíquota de Imposto de Renda (IR) em debêntures emitidas com o objetivo de financiar projetos de infraestrutura. Desde então, as chamadas “Debêntures de Infraestrutura” têm aumentado a sua importância como mecanismo de captação das empresas desse setor. A partir da publicação da Resolução do Banco Central do Brasil (Bacen) nº 4.476, de 11.04.2016, foi permitida a inclusão de cláusulas de regaste antecipado a exclusivo critério da empresa emissora. Tendo em vista as peculiaridades desses títulos, as metodologias já desenvolvidas para precificação de opções de recompra em debêntures não são aplicáveis. Desta forma, o trabalho proposto pretende desenvolver uma metodologia para precificar a existência de uma cláusula de resgate antecipado em uma debênture de infraestrutura abrangida pela Lei n° 12.431/11, considerando a estrutura a termo das taxas de juros brasileiras e os custos de emissão de debêntures.

The main goal of this dissertation was to obtain an understanding as to the true economic nature of employee share options and the problems surrounding the accounting thereof. The main conclusion of this study is that employee share options should be expensed in the income statement as and when the employee's services are performed. The reason is that employee share options are valuable financial instruments which the employer has used to compensate the employee for his services. It was also concluded that exercise date accounting and classification of outstanding employee share options as liabilities on the balance sheet is the most appropriate accounting treatment. Such accounting treatment trues up the accounting of employee share options with that of cash-settled share appreciation rights, which are economically equivalent transactions. The measurement of employee share options should be based on their fair value using an option-pricing model adapted for the specific features of employee share options. / Accounting / Thesis (M. Com. (Accounting Science))

Neste trabalho propomos um modelo de mercado através de uma discretização aleatória do movimento browniano proposta por Leão & Ohashi (2010). Com este modelo, dada uma função payoff, vamos desenvolver uma estratégia de hedging e uma metodologia para precificação de opções / In this work we propose a market model using a discretization scheme of the random Brownian motion proposed by Leão & Ohashi (2010). With this model, for any given payoff function, we develop a hedging strategy and a methodology to option pricing

我國公部門出租不動產時，其租賃契約中之終止條件及續約條件等約定，與私部門間租賃契約頗有落差。而依據不動產估價技術規則第129 條規定：「不動產之租金估計應考慮契約內容、租期長短、使用目的、稅費負擔、租金水準、變遷狀態、租約更新、變更條件及其他相關因素估計之。」表示在訂定租金時需將契約內容及可能影響因素納入考量才符合公帄原則，然公部門租賃契約中之租金訂定是否將此等特殊條件納入考量，仍有待驗證。此部分在過去文獻中較少著墨，有鑑於此，本研究欲探討租賃契約中特殊條件(如單方面停止契約之權利)對租金之影響。 本研究先以特徵價格理論為基礎，建構複迴歸模型探討影響租金之因素，發現賦予出租方可逕為終止契約之權利，對商用不動產租金具有顯著影響；再進一步以二項式評價模式分析因終止契約條件不同，在租賃契約中隱含的實質選擇權價值變化，模擬結果發現租金成長率愈高，選擇權價值愈高；無風險利率愈高，選擇權價值愈低。研究結果符合選擇權理論，除了指出目前公部門資產租賃契約宜考量具特殊條件時租金之公帄性外，亦可提供爾後租約中含特殊條件時之租金定價參考。 / Rent is affected by various factors, including macro-economic, regional, individual and other related factors such as termination or renewal options. Public sectors in Taiwan prefer using template leasing contract for standardization concerns. The template leasing contract includes some terms favorable to lessors, such as termination option for lessors. The study focuses on the term to terminate the contract for lessors, which is especially considered imposing operation risk on tenants. This study at first employed the multi-regression model to examine factors affecting the rent level. Results show that the rent is significantly affected by the termination term for lessors besides other individual, regional and macro-economic variables. We further applied the binomial option pricing model to simulate how the value of leasing contract is affected by the termination options. Results show that the more the rent tends to grow, the higher value the termination term is; and the higher the interest rate, the lower value the termination term is. Results of this study provide precious implications for rent pricing as the contracts are embedded with the termination option.

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.

Diese Dissertation widmet sich der Untersuchung singulärer stochastischer partieller Differentialgleichungen (engl. SPDEs). Wir entwickeln Erweiterungen der bisherigen Lösungstheorien, zeigen fundamentale Beziehungen zwischen verschiedenen Ansätzen und präsentieren Anwendungen in der Finanzmathematik und der mathematischen Physik. Die Theorie parakontrollierter Systeme wird für diskrete Räume formuliert und eine schwache Universalität für das parabolische Anderson Modell bewiesen. Eine fundamentale Relation zwischen Hairer's modellierten Distributionen und Paraprodukten wird bewiesen: Wir zeigen das sich der Raum modellierter Distributionen durch Paraprodukte beschreiben lässt. Dieses Resultat verallgemeinert die Fourierbeschreibung von Hölderräumen mittels Littlewood-Paley Theorie. Schließlich wird die Existenz von Lösungen der stochastischen Schrödingergleichung auf dem ganzen Raum bewiesen und eine Anwendung Hairer's Theorie zur Preisermittlung von Optionen aufgezeigt. / This thesis is concerned with the study of singular stochastic partial differential equations (SPDEs). We develop extensions to existing solution theories, present fundamental interconnections between different approaches and give applications in financial mathematics and mathematical physics. The theory of paracontrolled distribution is formulated for discrete systems, which allows us to prove a weak universality result for the parabolic Anderson model. This thesis further shows a fundamental relation between Hairer's modelled distributions and paraproducts: The space of modelled distributions can be characterized completely by using paraproducts. This can be seen a generalization of the Fourier description of Hölder spaces. Finally, we prove the existence of solutions to the stochastic Schrödinger equation on the full space and provide an application of Hairer's theory to option pricing.

[pt] Esta pesquisa investiga primeiramente, por meio de regressões, a relação entre as volatilidades implícitas das opções e os retornos futuros de 20, 40 e 60 dias das ações subjacentes no mercado acionário brasileiro. Essas regressões são então submetidas a testes de heterocedasticidade para garantir que não são regressões espúrias. Por fim, submetemos os resultados a um teste de robustez que confirma as regressões válidas e verifica a presença de autocorrelação nas séries de retornos futuros. O período analisado é de janeiro de 2011 a dezembro de 2021 em um total de onze anos completos. Apesar de apresentarem coeficientes de regressão significativos, nem todas essas regressões passam pelos testes, e sempre deve-se ter cautela ao usar uma volatilidade implícita de opção como sendo capaz de prever retornos das ações subjacentes no mercado brasileiro. / [en] This research first investigates, through regressions, the relationship between the implied volatilities of options and the future returns of 20, 40 and 60 days of the underlying stocks within the Brazilian stock market. These regressions are then subjected to heteroscedasticity tests to ensure that they are not spurious regressions. Finally, we submit the results to robustness tests to confirm the valid regressions and verify the presence of autocorrelation in the series of future returns. The period under analysis is from January 2011 to December 2021, totalling 11 years. Despite having significant regression coefficients, not all of these regressions pass the tests, and one should always exercise caution when using an option implied volatility as a predictor of underlying equity returns in the Brazilian market.

The study investigates whether the main capital market theories and pricing models provide a reasonably accurate description of the working and efficiency of capital markets, of the pricing of shares and options and the effect the risk/return relationship has on investor behaviour. The capital market theories and pricing models included in the study are Portfolio Theory, the Efficient Market Hypothesis (EMH), the Capital Asset Pricing Model (CAPM), the Arbitrage Pricing Theory (APT), Options Theory and the BlackScholes (8-S) Option Pricing Model. The main conclusion of the study is that the main capital market theories and pricing models, as reviewed in the study, do provide a reasonably accurate description of reality, but a number of anomalies and controversial issues still need to be resolved. The main recommendation of the study is that research into these theories and models should continue unabated, while the specific recommendations in a South African context are the following: ( 1) the benefits of global diversification for South African investors should continue to be investigated; (2) the level and degree of efficiency of the JSE Securities Exchange SA (JSE) should continue to be monitored, and it should be established whether alternative theories to the EMH provide complementary or better descriptions of the efficiency of the South African market; (3) both the CAPM and the APT should continue to be tested, both individually and jointly, in order to better understand the pricing mechanism of, and risk/return relationship on the JSE; (4) much South African research still needs to be conducted on the efficiency of the relatively new options market and the application of the B-S Option Pricing Model under South African conditions. / Financial Accounting / M. Com. (Accounting)

In mathematical finance, the sensitivities of option prices to various market parameters, also known as the “Greeks”, reflect the exposure to different sources of risk. Computing these is essential to predict the impact of market moves on portfolios and to hedge them adequately. This is commonly done using Monte Carlo simulations. However, obtaining accurate estimates of the Greeks can be computationally costly. Multilevel Monte Carlo offers complexity improvements over standard Monte Carlo techniques. However the idea has never been used for the computation of Greeks. In this work we answer the following questions: can multilevel Monte Carlo be useful in this setting? If so, how can we construct efficient estimators? Finally, what computational savings can we expect from these new estimators? We develop multilevel Monte Carlo estimators for the Greeks of a range of options: European options with Lipschitz payoffs (e.g. call options), European options with discontinuous payoffs (e.g. digital options), Asian options, barrier options and lookback options. Special care is taken to construct efficient estimators for non-smooth and exotic payoffs. We obtain numerical results that demonstrate the computational benefits of our algorithms. We discuss the issues of convergence of pathwise sensitivities estimators. We show rigorously that the differentiation of common discretisation schemes for Ito processes does result in satisfactory estimators of the the exact solutions’ sensitivities. We also prove that pathwise sensitivities estimators can be used under some regularity conditions to compute the Greeks of options whose underlying asset’s price is modelled as an Ito process. We present several important results on the moments of the solutions of stochastic differential equations and their discretisations as well as the principles of the so-called “extreme path analysis”. We use these to develop a rigorous analysis of the complexity of the multilevel Monte Carlo Greeks estimators constructed earlier. The resulting complexity bounds appear to be sharp and prove that our multilevel algorithms are more efficient than those derived from standard Monte Carlo.