Modelos bastante utilizados atualmente no apreçamento de derivativos de taxas de juros realizam, muitas vezes, premissas excessivamente restritivas com relação à volatilidade da série do ativo objeto. O método de Black and Scholes e o de Vasicek, por exemplo, consideram a variância da série como constante no tempo e entre as diferentes maturidades, suposição que pode não ser a mais adequada para todos os casos. Assim, entre as técnicas alternativas de modelagem da volatilidade que vêm sendo estudadas, destacam-se as regressões por kernel. Discutimos neste trabalho a modelagem não paramétrica por meio da referida técnica e posterior apreçamento das opções em um modelo HJM Gaussiano. Analisamos diferentes especificações possíveis para a estimação não paramétrica da função de volatilidade através de simulações de Monte Carlo para o apreçamento de opções sobre títulos zero cupom, e realizamos um estudo empírico utilizando a metodologia proposta para o apreçamento de opções sobre IDI no mercado brasileiro. Um dos principais resultados encontrados é o bom ajuste da metodologia proposta no apreçamento de opções sobre títulos zero cupom. / Many models which have been recently used for derivatives pricing make restrictive assumptions about the volatility of the underlying object. Black-Scholes and Vasicek models, for instance, consider the volatility of the series as constant throughout time and maturity, an assumption that might not be the most appropriate for all cases. In this context, kernel regressions are important technics which have been researched recently. We discuss in this framework nonparametric modeling using the aforementioned technic and posterior option pricing using a Gaussian HJM model. We analyze different specifications for the nonparametric estimation of the volatility function using Monte Carlo simulations for the pricing of options on zero coupon bonds and conduct an empirical study using the proposed methodology for the pricing of options on the Interbank Deposit Index (IDI) in the Brazilian market. One of our main results is the good adjustment of the proposed methodology on the pricing of options on zero coupon bonds.

Modelos bastante utilizados atualmente no apreçamento de derivativos de taxas de juros realizam, muitas vezes, premissas excessivamente restritivas com relação à volatilidade da série do ativo objeto. O método de Black and Scholes e o de Vasicek, por exemplo, consideram a variância da série como constante no tempo e entre as diferentes maturidades, suposição que pode não ser a mais adequada para todos os casos. Assim, entre as técnicas alternativas de modelagem da volatilidade que vêm sendo estudadas, destacam-se as regressões por kernel. Discutimos neste trabalho a modelagem não paramétrica por meio da referida técnica e posterior apreçamento das opções em um modelo HJM Gaussiano. Analisamos diferentes especificações possíveis para a estimação não paramétrica da função de volatilidade através de simulações de Monte Carlo para o apreçamento de opções sobre títulos zero cupom, e realizamos um estudo empírico utilizando a metodologia proposta para o apreçamento de opções sobre IDI no mercado brasileiro. Um dos principais resultados encontrados é o bom ajuste da metodologia proposta no apreçamento de opções sobre títulos zero cupom. / Many models which have been recently used for derivatives pricing make restrictive assumptions about the volatility of the underlying object. Black-Scholes and Vasicek models, for instance, consider the volatility of the series as constant throughout time and maturity, an assumption that might not be the most appropriate for all cases. In this context, kernel regressions are important technics which have been researched recently. We discuss in this framework nonparametric modeling using the aforementioned technic and posterior option pricing using a Gaussian HJM model. We analyze different specifications for the nonparametric estimation of the volatility function using Monte Carlo simulations for the pricing of options on zero coupon bonds and conduct an empirical study using the proposed methodology for the pricing of options on the Interbank Deposit Index (IDI) in the Brazilian market. One of our main results is the good adjustment of the proposed methodology on the pricing of options on zero coupon bonds.

Inflation indexed instruments are designed to help protect investors against the changes in the general level of prices. So, they are frequently preferred by investors and they have become increasingly developing part of the market. In this study, firstly, the HJM model and foreign currency analogy used to price of inflation indexed instruments are investigated. Then, the HJM model is extended with finite number of Poisson process. Finally, under the extended HJM model, a pricing derivation of inflation indexed swaps, which are the most liquid ones among inflation indexed instruments in the market, is given.

In this thesis we use techniques from white noise analysis to study solutions of semilinear stochastic differential equations in a Hilbert space H: {dX[subscript]t = (AX[subscript]t + F(t,X[subscript]t)) dt + ơ(t,X[subscript]t) δB[subscript]t, t∈ (0,T], X[subscript]0 = ξ, where A is a generator of either a C[subscript]0-semigroup or an n-times integrated semigroup, and B is a cylindrical Wiener process. We then consider applications to forward interest rate models, such as in the Heath-Jarrow-Morton framework. We also reformulate a phenomenological model of the forward rate. / Thesis (Ph.D.) -- University of Adelaide, School of Mathematical Science, 2009

In this thesis we use techniques from white noise analysis to study solutions of semilinear stochastic differential equations in a Hilbert space H: {dX[subscript]t = (AX[subscript]t + F(t,X[subscript]t)) dt + ơ(t,X[subscript]t) δB[subscript]t, t∈ (0,T], X[subscript]0 = ξ, where A is a generator of either a C[subscript]0-semigroup or an n-times integrated semigroup, and B is a cylindrical Wiener process. We then consider applications to forward interest rate models, such as in the Heath-Jarrow-Morton framework. We also reformulate a phenomenological model of the forward rate. / Thesis (Ph.D.) -- University of Adelaide, School of Mathematical Science, 2009

During the last decade several Japanese life insurance companies failed mainly due to interest losses. In fact, interest rate risk dominates mortality risk for a portfolio of business in force. When the interest rates are modelled as random variables, the yields on bonds are the sum of expected short spot rates and a risk premium for random bond prices. However, in our study, we assume a risk-neutral environment, i.e. zero risk premiums. As tools to deal with stochastic interest rates, various interest rate term structure models are considered. The Vasicek model, the Heath-Jarrow-Morton (hereafter “HJM”) approach and Cairns’ model are explained in detail. The history and nature of the very low interest rate environment in Japan is described in line with the monetary policy framework of the central bank. An unusual interest rate movement in the very low interest rate environment is identified. A modified HJM approach and Cairns’ model are chosen in our study. Cairns’ model is used to graduate the initial yield curve. The HJM approach with a specific volatility function and modified to deal with very low interest rates is used for simulating subsequent developments of the initial yield curve. After the introduction of various concepts needed to investigate a life insurance policy portfolio, we prepare for simulation by collecting information and by fitting parameters to market observations. The Yen swap curve is chosen as a base yield curve. The simulation results show how much initial capital and/or margins are needed in order to avoid the ruin of a portfolio.

我們使用一個新方法來評價美式或歐式的或有求償權，其受到本地利率和權益價值的影響。我們使用標的資產的遠期價格的樹狀圖，進而對或有求償權作定價。其中我們評價了美式與歐式的股票選擇權，以及利率期貨和利率期貨選擇權。 / We introduce a methodology for pricing American or European style contingent claims, influenced by domestic interest rates, and equity prices. Instead of using trees of short-term interest rate, bond price or forward interest rate, this tree method will use the forward prices of underlying assets to derive implied binomial spot-price tree and in turn price long term American or European options, and interest rate futures and interest rate futures options.

有鑑於附有最低保證給付投資型保險期末現金流量與選擇權如出一轍，是以應用平賭訂價理論(The Martingale Pricing Method)嵌入HJM利率模型，對隨機利率下附有最低保證給付投資型保險進行評價。並對繳費方式與利率型態兩議題所構成四種類型附有最低保證給付投資型保險作實地數據模擬與評價，以及敏感度分析。 研究結果可以歸納為四點結論。 (1) 單就附有最低保證給付投資型保險簡化版（忽略期中死亡理賠與期滿生存機率）而言： 可視為是最低保證給付折現與以之為履約價的買權組合。因此，當影響因子僅與買權有相關性時，附有最低保證給付投資型保險與理論買權的敏感度分析結果，如出一轍。連動標的期初價格與波動度變動於附有最低保證給付投資型保險影響便是實證。 (2) 延續上點論述衍生： 當影響因子同時對買權與附有最低保證給付折現具有相關性時，由於買權佔整個保險價值比重過低，是以主要影響力皆來自附有最低保證給付的變動。附有最低保證給付與固定利率折現因子變動對於保險價值影響，即反應此結果。 (3) 分別就繳費方式不同下，投保年齡與投保期限變動對於附有最低保證給付投資保險的影響而言： 躉繳型繳費方式下，由第二點結論可得，投保期限越長保費越低，是以當投保年齡越大，期中死亡率提高，且期間短的保費較高的情況下，投保年齡變動對於附有最低保證給付投資型保險影響為正向；分期繳型繳費方式下，由於條款設定不同，無法與躉繳型一概而論，反映在投保期間越長保單價值與保費皆增加，但若是比較其增加的幅度（二階條件小於零）逐漸減少，倒是與躉繳型投資保險投保期間與保費關係意思相同，只是呈現方式不同。分期繳型投資型保險保單價值與投保年齡關係，從投保期限與保費關係以及高年齡層死亡率較高，可以得知，隨著投保年齡的增加，分期繳型投資保險中因為死亡理賠的現金流量產生機會提高，而此部分期間短保單價值較低，是以投保年齡與保單價值呈現反比關係，但是保單價值平準化後的保費，源於平準因子每期存活率因投保年齡增加而減少，造成投保年齡越高，保費也越高。 (4) 就性別而言： 躉繳型附有最低保證給付投資保險，由於女性相較於男性死亡率較低，容易取得期間較長的期滿保證金，而此部分價值較低，是以女生保費較男生便宜；分期繳型附有最低保證給付投資保險，則是相反的表現，由於此部分價值較高，是以女性的保險價值高於男性，同時因女性平準因子中的存活率也比男性高，是以每期所要繳交的保費也比男性低廉。 (5) 就利率型態而言： 隨機利率下躉繳型投資型保險與固定利率下躉繳型投資保險相較，便宜許多，主要是因為利率型態為隨機，且期初利率期間結構打破水平狀態的假設，真實反應正常期初利率期間結構(Normal Interest Rate Term Structure)，是以評價出的保費較固定利率型態下的保費低廉，甚至於分期繳型附有最低保證給付投資保險，在隨機利率下，隨著投保期限增加，保費反而下降。

The thesis consists of four papers on numerical complexityanalysis of weak approximation of ordinary and partialstochastic differential equations, including illustrativenumerical examples. Here by numerical complexity we mean thecomputational work needed by a numerical method to solve aproblem with a given accuracy. This notion offers a way tounderstand the efficiency of different numerical methods. The first paper develops new expansions of the weakcomputational error for Itˆo stochastic differentialequations using Malliavin calculus. These expansions have acomputable leading order term in a posteriori form, and arebased on stochastic flows and discrete dual backward problems.Beside this, these expansions lead to efficient and accuratecomputation of error estimates and give the basis for adaptivealgorithms with either deterministic or stochastic time steps.The second paper proves convergence rates of adaptivealgorithms for Itˆo stochastic differential equations. Twoalgorithms based either on stochastic or deterministic timesteps are studied. The analysis of their numerical complexitycombines the error expansions from the first paper and anextension of the convergence results for adaptive algorithmsapproximating deterministic ordinary differential equations.Both adaptive algorithms are proven to stop with an optimalnumber of time steps up to a problem independent factor definedin the algorithm. The third paper extends the techniques to theframework of Itˆo stochastic differential equations ininfinite dimensional spaces, arising in the Heath Jarrow Mortonterm structure model for financial applications in bondmarkets. Error expansions are derived to identify differenterror contributions arising from time and maturitydiscretization, as well as the classical statistical error dueto finite sampling. The last paper studies the approximation of linear ellipticstochastic partial differential equations, describing andanalyzing two numerical methods. The first method generates iidMonte Carlo approximations of the solution by sampling thecoefficients of the equation and using a standard Galerkinfinite elements variational formulation. The second method isbased on a finite dimensional Karhunen- Lo`eve approximation ofthe stochastic coefficients, turning the original stochasticproblem into a high dimensional deterministic parametricelliptic problem. Then, adeterministic Galerkin finite elementmethod, of either h or p version, approximates the stochasticpartial differential equation. The paper concludes by comparingthe numerical complexity of the Monte Carlo method with theparametric finite element method, suggesting intuitiveconditions for an optimal selection of these methods. 2000Mathematics Subject Classification. Primary 65C05, 60H10,60H35, 65C30, 65C20; Secondary 91B28, 91B70. / QC 20100825

Submitted by Daniel Benabou (dbenabas@gmail.com) on 2018-09-24T14:05:05Z No. of bitstreams: 1 Daniel Benabou - Final.pdf: 1509370 bytes, checksum: b5f974a956000f0474b19f3aa92f13db (MD5) / Approved for entry into archive by Joana Martorini (joana.martorini@fgv.br) on 2018-09-24T15:01:39Z (GMT) No. of bitstreams: 1 Daniel Benabou - Final.pdf: 1509370 bytes, checksum: b5f974a956000f0474b19f3aa92f13db (MD5) / Approved for entry into archive by Isabele Garcia (isabele.garcia@fgv.br) on 2018-09-24T20:11:14Z (GMT) No. of bitstreams: 1 Daniel Benabou - Final.pdf: 1509370 bytes, checksum: b5f974a956000f0474b19f3aa92f13db (MD5) / Made available in DSpace on 2018-09-24T20:11:14Z (GMT). No. of bitstreams: 1 Daniel Benabou - Final.pdf: 1509370 bytes, checksum: b5f974a956000f0474b19f3aa92f13db (MD5) Previous issue date: 2018-08-24 / Com o objetivo de obter a estrutura de curvas futuras de swaps de energia, este trabalho foca na implementação numérica do modelo de Heath, Jarrow e Morton (1992) utilizando somente as informações dos contratos de swaps negociados no Sistema Elétrico Brasileiro, através do modelo discreto do HJM conhecido como Modelo de Brace, Garatek e Musiela (1997), também referido como Modelo de Mercado. A estrutura de volatilidade foi obtida de forma não-paramétrica através de curvas suaves e de vértices sintéticos obtidos por interpolação dos dados de venda de uma comercializadora tratados através do método de Análise de Componentes Principais (PCA). Os dados analisados foram contratos firmados entre o início de 2013 e o primeiro quadrimestre de 2015. / For the purpose of obtaining the structure of future swap energy curves in the Brazilian market, this paper focuses on the numerical implementation of the Heath, Jarrow and Morton model (1992) using market information regarding the swap contracts traded in the Brazilian energy system, with its multi-factor discrete form, the Brace, Garatek and Musiela (1997) model, also known as Market Model. The volatility structure is obtained with smooth curves and synthetic vertices, obtained thru swap contracts negotiated by a Brazilian energy trading company. Also, the volatility structures where analyzed with the Principal Components Analysis (PCA). The analyzed data where swap contracts stablished between the beginning of 2013 until the first quarter of 2015.