The thesis is concerned with a detailed examination of the most familiar short-rate models.Furthermore,it contains some author's own derivations of formulas for prices of interest rate derivatives and some relationships between certain discretizations of these short-rate models. These formulas are then used for calibration of ceratain chosen models to the actual market data.All the calculations are performed in R using author's own functions,which are along with the other more involved derivations placed in the appendix.

The short rate is central in the context of interest-rate markets as well as broader finance. As such, accurate modelling of this rate is of particular importance in the pricing of interest-rate options, especially during times of high volatility where increased demand is seen for simpler and lower risk investments. Recent interest has moved away from models of a pure continuous nature towards models that can account for discontinuities in the short rate. These are more representative of real world movements where the short rate is seen to jump due to current and scheduled market information. This dissertation examines this phenomenon in the context of a Vasicek short rate model and accounts for random-sized jumps at deterministic times following ideas similar to those introduced by Kim and Wright (2014). Finite difference methods are used successfully to find PDE solutions via backwards diffusion of the option value equation to its initial state. This procedure is implemented computationally and compared to Monte Carlo benchmark methods in order to assess its accuracy. In both non-jump and jump settings the method constructed was able to accurately price the call option specified and proved to be a viable means for pricing interest-rate options when stochastically-sized discontinuities are present at known times between inception and expiry. Furthermore the method showed that the stochastic discontinues in the short rate most notably affect the option price in the region around and just out of the money.

No / This article develops a novel approach for measuring market expectations and term premia in the term structure of interest rates. Key components of this approach are generic impact measures of state variables in a Gaussian dynamic term structure model. These measures are inherent in a particular state variable regardless of how other state variables are defined within the model. With the help of these measures, the approach gives rise to market expectations that predict yield changes well, and term premia with a legitimate impact on the forward curve. In my empirical analysis, I show the generic impact of the short rate on the yield curve, and present observations of the historical dynamics of market expectations and term premia. The calibrated model is also employed to study the impacts of recent unconventional monetary policies.

Title: Stochastic interest rates modeling Author: Jakub Černý Abstract: This present work studies different stochastic models of interest rates. Theoretical part of this work describes short-rate models, HJM fra- mework and LIBOR Market model. It focuses in detail on widely known short-rate models, i.e. Vašíček, Hull-White and Ho-Lee model, and on LI- BOR Market model. This part ends by valuation of interest rate options and model calibration to real data. Analytical part of the work analyses valuation of real non-standard interest rate derivative using different models. Part of this derivative valuation is comparison among models in terms of general valuation and also in terms of capturing the dynamics of interest rates. The aim of this work is to describe different stochastic models of interest rates and mainly to compare them with each other.

The scope of this study is to estimate the zero-coupon yield curve of tomorrow by using Vasicek yield curve model with the zero-coupon bond yield data of today. The raw data of this study is the yearly simple spot rates of the Turkish zero-coupon bonds with different maturities of each day from July 1, 1999 to March 17, 2004. We completed the missing data by using Nelson-Siegel yield curve model and we estimated tomorrow yield cuve with the discretized Vasicek yield curve model.

>Magister Scientiae - MSc / Pensions are essentially the only source of income for many retired workers. It is thus critical that the pension fund manager chooses the right type of plan for his/her workers.Every pension scheme follows its own set of rules when calculating the benefits of the fund’s members at retirement. Whichever plan the manager chooses for the members,he/she will have to invest their contributions in the financial market. The manager is therefore faced with the daunting task of selecting the most appropriate investment strat-egy as to maximize the returns from the financial assets. Due to the volatile nature of stock markets, some pension companies have attached minimum guarantees to pension contracts. These guarantees come at a price, but ensure that the member does not suffer a loss due to poorly performing equities.In this thesis we study four types of mathematical problems in pension fund management,of which three are essentially optimization problems. Firstly, following Blake [5], we show in a discrete time setting how to decompose a pension benefit into a combination of Euro-pean options. We also model the pension plan preferences of workers, sponsors and fund managers. We make a number of contributions additional to the paper by Blake [5]. In particular, we contribute graphic illustrations of the expected values of the pension fund assets, liabilities and the actuarial surplus processes. In more detail than in the original source, we derive the variance of the assets of a defined benefit pension plan. Secondly,we dedicate Chapter 6 to the problem of minimizing the cost of a minimum guarantee included in defined contribution (DC) pension contracts. Here we work in discrete time and consider multi-period guarantees similar to those in Hipp [25]. This entire chapter is original work. Using a standard optimization method, we propose a strategy that cal- culates an optimal sequence of guarantees that minimizes the sum of the squares of the present value of the total price of the guarantee. Graphic illustrations are included to in-dicate the minimum value and corresponding optimal sequence of guarantees. Thirdly, we derive an optimal investment strategy for a defined contribution fund with three financial assets in the presence of a minimum guarantee. We work in a continuous time setting and in particular contribute simulations of the dynamics of the short interest rate process and the assets in the financial market of Deelstra et al. [19]. We also derive an optimal investment strategy of the surplus process introduced in Deelstra et al. [19]. The results regarding the surplus are then converted to consider the actual investment portfolio per- taining to the wealth of the fund. We note that the aforementioned paper does not use optimal control theory. In order to illustrate the method of stochastic optimal control, we study a fourth problem by including a discussion of the paper by Devolder et al. [21] in Chapter 3. We enhance the work in the latter paper by including some simulations. The specific portfolio management strategies are applicable to banking as well (and is being pursued independently).

The objective of this report is to carry out a pre-study and develop a framework for how the liquidity and interest rate risk of a bank's demand deposits can be modeled. This is done by first calibrating a Vasicek short rate model and then deriving models for the bank's deposit volume and deposit rate using multiple regression. The volume model and the deposit rate model are used to determine the liquidity and interest rate risk, which is done separately. The liquidity risk is determined by a liquidity quantile which estimates the minimum deposit volume that is expected to remain in the bank over a given time period. The interest rate risk is quantified by an arbitrage-free valuation of the demand deposit which can be used to determine the sensitivity of the net present value of the demand deposit caused by a parallel shift in the market rates. Furthermore, an immunization and a replicating portfolio are constructed and the performances of these are tested when introducing the same parallel shifts in the market rates as in the valuation of the demand deposit. The conclusion of this thesis is that the framework for the liquidity risk management that is developed gave satisfactory results and could be used by the bank if the deposit volume is estimated on representative data and a more accurate model for the short rate is used. The interest rate risk framework did however not yield as reliable results and would be more challenging to implement as a more advanced model for the deposit rate is required. / Målet med denna rapport är att utveckla ett ramverk för att bestämma likviditets-och ränterisken som är relaterad till en banks inlåningsvolym. Detta görs genom att först ta fram en modell för korträntan via kalibrering av en Vasicek modell. Därefter utvecklas, genom multipelregression, modeller för att beskriva bankens inlåningsvolym och inlåningsränta. Dessa modeller används för att kvantifiera likviditets- och ränterisken för inlånings-volymen, vilka beräknas och presenteras separat. Likviditetsrisken bestäms genom att en likviditetskvantil tas fram, vilken estimerar den minimala inlånings-volymen som förväntas kvarstå hos banken över en given tidsperiod. Ränterisken kvantifieras med en arbitragefri värdering av inlåningen och resultatet används för att bestämma känsligheten för hur nuvärdet av inlåningsvolymen påverkas av ett parallellskifte. Utöver detta bestäms en immuniseringsportfölj samt en rep-likerande portfölj och resultatet av dessa utvärderas mot hur nuvärdet förändras givet att samma parallellskifte i ränteläget som tidigare introduceras. Slutsatsen av projektet är att det framtagna ramverket för att bestämma likviditetsrisken för inlåningen gav bra resultat och skulle kunna implementeras i dagsläget av banken, förutsatt att volymmodellen estimeras på representativ data samt att en bättre modell för korträntan används. Ramverket för att bestämma ränterisken gav dock inte lika tillförlitliga resultat och är mer utmanande att implementera då en mer avancerad modell för inlåningsräntan krävs.

I study, develop and implement selected interest rate models. I begin with a simple categorization of interest rate models and with an explanation why interest rate models are useful. I explain and discuss the notion of arbitrage. I use Oldrich Vasicek's seminal model (Vasicek; 1977) to develop the idea of no-arbitrage term structure modeling. I introduce both the partial di erential equation and the risk-neutral approach to zero-coupon bond pricing. I briefly comment on affine term structure models, a general equilibrium term structure model, and HJM framework. I present the Czech Treasury yield curve estimates at a daily frequency from 1999 to the present. I use the parsimonious Nelson-Siegel model (Nelson and Siegel; 1987), for which I suggest a parameter restriction that avoids abrupt changes in parameter estimates and thus allows for the economic interpretation of the model to hold. The Nelson-Siegel model is shown to fit the Czech bond price data well without being over-parameterized. Thus, the model provides an accurate and consistent picture of the Czech Treasury yield curve evolution. The estimated parameters can be used to calculate spot rates and hence par rates, forward rates or discount function for practically any maturity. To my knowledge, consistent time series of spot rates are not available for the Czech economy. I introduce two estimation techniques of the short-rate process. I begin with the maximum likelihood estimator of a square root diff usion. A square root di usion serves as the short rate process in the famous CIR model (Cox, Ingersoll and Ross; 1985b). I develop and analyze two Matlab implementations of the estimation routine and test them on a three-month PRIBOR time series. A square root diff usion is a restricted version of, so called, CKLS di ffusion (Chan, Karolyi, Longsta and Sanders; 1992). I use the CKLS short-rate process to introduce the General Method of Moments as the second estimation technique. I discuss the numerical implementation of this method. I show the importance of the estimator of the GMM weighting matrix and question the famous empirical result about the volatility speci cation of the short-rate process. Finally, I develop a novel yield curve model, which is based on principal component analysis and nonlinear stochastic di erential equations. The model, which is not a no-arbitrage model, can be used in areas, where quantification of interest rate dynamics is needed. Examples, of such areas, are interest rate risk management, or the pro tability and risk evaluation of interest rate contingent claims, or di erent investment strategies. The model is validated by Monte Carlo simulations.

Submitted by José Ignacio Valencia Díaz (jivalenciadiaz@gmail.com) on 2013-09-17T00:13:33Z No. of bitstreams: 1 Dissertacao MPFE Jose Ignacio Valencia Diaz.pdf: 1741345 bytes, checksum: b45af943bf4f6e8a2a9963c07038d9dc (MD5) / Approved for entry into archive by Suzinei Teles Garcia Garcia (suzinei.garcia@fgv.br) on 2013-09-17T12:05:59Z (GMT) No. of bitstreams: 1 Dissertacao MPFE Jose Ignacio Valencia Diaz.pdf: 1741345 bytes, checksum: b45af943bf4f6e8a2a9963c07038d9dc (MD5) / Made available in DSpace on 2013-09-17T12:54:35Z (GMT). No. of bitstreams: 1 Dissertacao MPFE Jose Ignacio Valencia Diaz.pdf: 1741345 bytes, checksum: b45af943bf4f6e8a2a9963c07038d9dc (MD5) Previous issue date: 2013-08-26 / Prediction models based on nonparametric estimation are in continuous development and have been permeating the quantitative community. Their main feature is that they do not consider as known a priori the form of the probability distributions functions (PDF), but allow the data to be used directly in order to build their own PDFs. In this work it is implemented the nonparametric pooled estimators from Sam and Jiang (2009) for drift and diffusion functions for the short rate diffusion process, by means of the use of yield series of different maturities provided by One Day Future Interbank Deposit contracts (ID1). The estimators are built from the perspective of kernel functions and they are optimized with a particular kernel format, in our case, Epanechnikov’s kernel, and with a smoothing parameter (bandwidth). Empiric experience indicates that the smoothing parameter is critical to find the probability density function that provides an optimal estimation in terms of MISE (Mean Integrated Squared Error) when testing the model with the traditional k-folds cross-validation method. Exceptions arise when the series do not have appropriate sizes, but the structural break of the diffusion process of the Brazilian interest short rate, since 2006, requires the reduction of the length of the series to the cost of reducing the predictive power of the model. This structural break represents the evolution of the Brazilian market, in an attempt to converge towards mature markets and it explains largely the unsatisfactory performance of the proposed estimator. / Modelos de predição baseados em estimações não-paramétricas continuam em desenvolvimento e têm permeado a comunidade quantitativa. Sua principal característica é que não consideram a priori distribuições de probabilidade conhecidas, mas permitem que os dados passados sirvam de base para a construção das próprias distribuições. Implementamos para o mercado brasileiro os estimadores agrupados não-paramétricos de Sam e Jiang (2009) para as funções de drift e de difusão do processo estocástico da taxa de juros instantânea, por meio do uso de séries de taxas de juros de diferentes maturidades fornecidas pelos contratos futuros de depósitos interfinanceiros de um dia (DI1). Os estimadores foram construídos sob a perspectiva da estimação por núcleos (kernels), que requer para a sua otimização um formato específico da função-núcleo. Neste trabalho, foi usado o núcleo de Epanechnikov, e um parâmetro de suavizamento (largura de banda), o qual é fundamental para encontrar a função de densidade de probabilidade ótima que forneça a estimação mais eficiente em termos do MISE (Mean Integrated Squared Error - Erro Quadrado Integrado Médio) no momento de testar o modelo com o tradicional método de validação cruzada de k-dobras. Ressalvas são feitas quando as séries não possuem os tamanhos adequados, mas a quebra estrutural do processo de difusão da taxa de juros brasileira, a partir do ano 2006, obriga à redução do tamanho das séries ao custo de reduzir o poder preditivo do modelo. A quebra estrutural representa um processo de amadurecimento do mercado brasileiro que provoca em grande medida o desempenho insatisfatório do estimador proposto.

Studier av livslängder och dödssannolikheter är avgörande för livförsäkring. Betalningar gällande livförsäkringar är helt beroende av om en individ lever eller ej, eller befinner sig i olika hälsotillstånd. För att kunna prissätta premier korrekt och avsätta reserver är det därför av stort intresse att modellera livslängden på ett så korrekt sätt som möjligt. Försäkringsbranschen använder idag historiskt beprövade och välfungerande modeller som går så långt bak i tiden som 200 år. Det finns modeller ännu längre bak i tiden, men de modeller som används idag är främst Gompertz (1826), Makeham (1860) och Lee-Carter (1992). Även om dessa modeller presterar bra är det alltid nödvändigt att undersöka om det kan finnas alternativa modeller som modellerar dödligheten bättre. I detta examensarbete tillämpas affina korträntemodeller för modellering av dödlighetsintensiteten som ligger till grund för flertalet intressanta aktuariella storheter. Då dessa modeller introducerar stokastisk dödlighet kan osäkerheten och beroendet över tid därmed beskrivas. De korträntemodeller som undersöks i arbetet och som är vanligt förekommande inom den finansiella teorin; är Ornstein-Uhlenbeck, Feller och Hull-White. Dessa modeller jämförs sedan mot varandra vad gäller modellerad dödlighetsintensitet samt förväntad återstående livslängd och ettårig dödssannolikhet. En aspekt av stokastisk dödlighetsmodellering som ej återfinns i befintlig litteratur men som undersöks i detta examensarbete är modellering av dödlighet över tid då detta är en av de mest väsentliga aspekterna inom det livförsäkringsmatematiska arbetet. Till sist i valideringssyfte utvärderas samtliga korträntemodeller genom back-testing. Den andra huvudsakliga delen av arbetet består i att generera resultat för samma storheter som ovan baserat på DUS-metoden för att på så sätt jämföra en kommersiell metod mot en mer teoretisk mindre beprövad sådan. Resultaten visar på en stor potential hos flera av korträntemodellerna kontra DUS både vad gäller modellering över åldrar och kalenderår. Däremot är inte resultaten helt felfria för enstaka kalenderår där stora spikar uppstår på grund av parametermässig felanpassning. Modelleringen av korträntemodellerna över tid var över förväntan då modellerna inte är konstruerade för att fånga avtagande trender. Detta är något som kan betraktas som en stor flexibilitet hos korträntemodellerna då de står sig väl mot Lee-Cartermodellen som används i DUS, både vad gäller ålders- och tidsmodellering av dödlighet. / Studies of life expectancy and death probabilities are crucial for life insurance. Payments for life insurance are completely dependent on whether an individual is alive or not, or is in various health conditions. In order to be able to price premiums correctly and set aside reserves, it is therefore of great importance to model life expectancy in the most accurate way possible. The insurance industry today uses historically proven well-functioning models that go as far back in time as 200 years. There are models even further back in time, but the models used today are mainly Gompertz (1826), Makeham (1860) and Lee-Carter (1992). Although these models perform well, it is always necessary to investigate whether there may be alternative models that model mortality better. In this thesis, affine short-term interest rate models are applied for modeling the force of mortality that forms the basis for most interesting actuarial variables. As these models introduce stochastic mortality, the uncertainty and dependence over time can thus be described. The three short-term interest rate models examined in this project, which are common in financial theory; are Ornstein-Uhlenbeck, Feller and Hull-White. These models are then compared against each other in terms of the modeled force of mortality as well as the expected remaining life expectancy and the one-year probability of death. One aspect of stochastic mortality modeling that is not found in the existing literature but which is examined in this thesis is the modeling of mortality over time as this is one of the most important aspects in the life insurance mathematical industry. Finally, for validation purposes, all short-term interest rate models are evaluated using back-testing. The second main part of the work consists of generating results for the same quantities as above based on the DUS method in order to compare a commercial method with more theoretical and less approved ones. The results show a great potential in several of the short-term interest rate models versus DUS both in terms of modeling over ages and calendar years. However, the results are not completely impeccable for individual calendar years where large spikes occur due to inaccurate parameter calibration. The satisfactory modeling of the short-term interest rate models over time was above the expectations as the models are not designed to capture decreasing trends. This is something that can be considered a great flexibility of the short-term interest rate models as they are more or less as accurate as the Lee-Carter model used in DUS, both in terms of age and time modeling of mortality.