In the original Vasicek model interest rates are calculated assuming that volatility remains constant over the period of analysis. In this study, we constructed a stochastic volatility model for interest rates. In our model we assumed not only that interest rate process but also the volatility process for interest rates follows the mean-reverting Vasicek model. We derived the density function for the stochastic element of the interest rate process and reduced this density function to a series form. The parameters of our model were estimated by using the method of moments. Finally, we tested the performance of our model using the data of interest rates in Turkey.

Submitted by Hagen Schoof (hagen.schoof@gmail.com) on 2011-09-19T20:54:51Z No. of bitstreams: 1 Dissertação30.pdf: 1093774 bytes, checksum: 39a1fb805474efebc7cb743bca515a10 (MD5) / Approved for entry into archive by Suzinei Teles Garcia Garcia (suzinei.garcia@fgv.br) on 2011-09-19T20:56:55Z (GMT) No. of bitstreams: 1 Dissertação30.pdf: 1093774 bytes, checksum: 39a1fb805474efebc7cb743bca515a10 (MD5) / Approved for entry into archive by Suzinei Teles Garcia Garcia (suzinei.garcia@fgv.br) on 2011-09-19T20:57:36Z (GMT) No. of bitstreams: 1 Dissertação30.pdf: 1093774 bytes, checksum: 39a1fb805474efebc7cb743bca515a10 (MD5) / Made available in DSpace on 2011-09-19T21:21:01Z (GMT). No. of bitstreams: 1 Dissertação30.pdf: 1093774 bytes, checksum: 39a1fb805474efebc7cb743bca515a10 (MD5) Previous issue date: 2011-08-17 / This work analyses the forecast of the interest rate focusing an investment strategy. First the interest rate is parameterized using the Vasicek model, then the autorregressive model is applied on the interest rate and the parameters of Vasicek’s model. The financial product used to check the performance proposed is the constant maturity swap on some maturities. The results significantly change on the sampling period and the calibration horizon without a standard behavior. / Este trabalho estuda a previsão da taxa de juros com foco em uma estratégia de investimento. Inicialmente é feita a parametrização da taxa de juros com o modelo de Vasicek para posterior aplicação do modelo autorregressivo tanto na taxa de juros quanto nos parâmetros do Vasicek. O instrumento financeiro escolhido para verificar a eficácia da metodologia proposta foi o constant matutity swap aplicado em alguns vértices. Os resultados variaram significativamente para os diferentes horizontes de calibragem e períodos de amostragem sem um padrão de desempenho.

Esse estudo avalia a adequação dos requerimentos absolutos de capital no Brasil para bancos pequenos e grandes separadamente e investiga os requerimentos de capital mínimo para risco de crédito nas diferentes abordagens de Basiléia, em especial o impacto da adoção dos modelos dos ratings internos (IRB) conforme o Edital BCB n. 37/11. Além disso, propõe e avalia a abordagem padronizada dos ratings centralizados, CRBA, para cálculo do Capital Mínimo Exigido (CME) em bancos pequenos e que é baseada na abordagem padronizada em vigor na Europa, mas voltada para dados disponíveis nas Centrais de Risco. A CRBA pertence à família dos modelos internos e busca contribuir com as recentes discussões sobre a reforma regulatória bancária na Europa e nos Estados Unidos. Para os três objetivos mencionados, as metodologias adotadas foram: 1) o Valuet-at-Risk (VaR) não paramétrico de Crédito (CVaR) de Carey (2002) e o paramétrico Creditrisk+ para estimar o capital econômico do Sistema Bancário; seguido da 2) estimação amostral e avaliação do capital regulatório para bancos pequenos e grandes nas abordagens IRB, Basileia 1, abordagem padrão simplificada (SSA); além da 3) avaliação da abordagem proposta nesse estudo, a CRBA. A performance de todas essas abordagens é avaliada frente a cenários de stress ad hoc e durante a Crise de 2008-2009. Os dados utilizados foram exposições de crédito aleatórias colhidas da Nova Central de Risco do Banco Central do Brasil (SCR). Os principais resultados desse estudo são: 1) sugerir um capital regulatório total (Patrimônio de Referência mais provisão) para bancos grandes de 17,5% baseado no CVaR paramétrico de 99,9% e, para pequenos, de 15,31% baseado no CVaR de 99%; 2) sugerir que, de todas as abordagens de Basileia II, o IRB estimado conforme o Edital BCB n. 37/2011 e para as Probabilidade de Default (PDs) calculadas por matrizes de migração do SCR, é o mais conservador; 3) sugerir que a abordagem proposta seja mais sensível ao risco de crédito do que atual brasileira, especialmente no varejo, além de oferecer um nível proteção maior contra choques aleatórios de crédito. Na Crise de 2008-2009, os bancos pequenos e grandes apresentaram respostas muito distintas a choques diversos ou quando os \"estados da economia\" se deterioravam. Os bancos pequenos não atingem o grau de diversificação necessário para minimizar perdas extremas. Por outro lado, do ponto de vista do risco sistêmico, a falência dessas entidades tem impactos muito menores que a de conglomerados bancários de porte. Finalmente, a abordagem proposta CRBA é apresentada como uma alternativa à abordagem atual no Brasil e à abordagem padronizada (SA) nos demais países, em especial na Europa. No Brasil, a CRBA cumpriria o papel de aumentar a sensibilidade a risco de crédito do CME nos bancos pequenos criando incentivos para uma gestão de risco de crédito mais cautelosa e alinhando o nível de capital dos bancos pequenos ao seu risco efetivo. Nos demais países, a CRBA é uma alternativa à abordagem padronizada, que independe da opinião das Agências de Classificação de Risco (ACRs). A CRBA traz dois benefícios: o primeiro de ampliar o escopo dos modelos internos e eliminar a dependência regulatória na opinião das ACRs, diminuindo a oportunidade de arbitragem regulatória com ratings inflacionados e corrigindo incentivos para que as ACRs sejam apenas provedoras de opiniões isentas; e o segundo, de prover os organismos supervisores com um mecanismo de controle (tracking error) sobre a qualidade de gestão de risco dos bancos pequenos por meio das Centrais de Risco. / This work analyses capital requirements adequacy in Brazil both for small and big banks individually and evaluates the minimum capital requirements for credit risk in the different Basel II approaches, especially, the impacts of IRB adoption as stated on Edital BCB n.37/11. Besides, it proposes and evaluates the Centralized Standard Ratings Based Approach (CRBA) to calculate Minimum Capital Requirements (MCR) in small banks. It is inspired in the Basel II Standard Approach (SA) disseminated in Europe, but based on information from the Credit Registers. The CRBA is an internal model approach in line with recent discussions on regulatory reform in Europe and in the US. The methodology to address these three research goals is: the non-parametric credit Value-at-Risk (VaR) or CVaR of Carey(2002) and the parametric Creditrisk+ to estimate the economic capital for the banking system; to evaluate regulatory capital in small and big banks in the IRB, Basel 1 and the Simplified Standard Approach (SSA) on the sample; and to evaluate the CRBA, proposed in this study. The performance of these approaches is confronted with ad hoc stress scenarios and within the Credit Crisis of 2008-2009. The data is comprised of credit exposures available in the Brazilian Credit Register (SCR). This work main results are: 1) to suggest a total regulatory capital (capital and provision) of 17.5% to big banks based on a parametric CVaR (99.9%) and of 15.31% to small banks based on a CVaR (99%); 2) to suggest, based on all Basel II approaches, that the IRB, as stated on Edital BCB n.31/11 and calibrated with the probabilities of default (PD) estimated with transition matrixes from the SCR, is the most conservative approach; 3) to suggest that the proposed approach is more sensitive to credit risk especially in retail and is more effective against stress chocks. Small and big banks behave differently to adverse shocks. The small banks, for instance, have problems diversifying out extreme losses when the \"states of the economy\" deteriorate. On the other hand, considering systemic risk, the bankruptcies of these institutions are much less of a problem than the ones of a big bank. Finally, the CRBA is presented as an alternative to the current approach (SSA) in Brazil and to the Standard Approach (SA) in other countries, specifically in Europe. In Brazil, the CRBA would increase the risk sensitivity of MCR on smaller banks creating incentives to more careful risk management practices and aligning their capital and risk levels. On the other countries, the CRBA is an alternative to the Standard Approach (SA) that is not dependent on Credit Rating Agencies - CRAs\' opinions and brings two additional benefits. First, it is an internal model based approach eliminating regulatory dependence on CRAs\' opinions, minimizing opportunities to regulatory arbitrage with inflated ratings and allowing CRAs to be more of a trustworthy opinion provider. Second, it provides supervisors a tracking error mechanism to evaluate risk management in small banks using Credit Registers.

In this dissertation, we investigate the spread option pricing problem with stochastic interest rate. First, we will review the basic concept and theories of stochastic calculus, give an introduction of spread options and provide some examples of spread options in different markets. We will also review the market efficiency theory, arbitrage and assumptions that are commonly used in mathematical finance. In Chapter 3, we will review existing spread pricing models and term-structure models such as Vasicek Mode, and the Heath-Jarrow-Morton framework. In Chapter 4, we will use the martingale approach to derive a partial differential equation for the price of the spread option with stochastic interest rate. In Chapter 5, we will study the spread option numerically. We will conclude this dissertation with ideas for future research.

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.

Submitted by marcelo hiroki (mhiroki@gmail.com) on 2014-09-03T02:41:10Z No. of bitstreams: 1 dissertacao_Marcelo_Hiroki_Final.pdf: 1600897 bytes, checksum: 3084ffc43e26c1311352b13d330264bf (MD5) / Approved for entry into archive by JOANA MARTORINI (joana.martorini@fgv.br) on 2014-09-03T12:07:14Z (GMT) No. of bitstreams: 1 dissertacao_Marcelo_Hiroki_Final.pdf: 1600897 bytes, checksum: 3084ffc43e26c1311352b13d330264bf (MD5) / Made available in DSpace on 2014-09-04T13:41:36Z (GMT). No. of bitstreams: 1 dissertacao_Marcelo_Hiroki_Final.pdf: 1600897 bytes, checksum: 3084ffc43e26c1311352b13d330264bf (MD5) Previous issue date: 2014-08-08 / This work aims to test the quality of forecasting of the two factor Vasicek Model coupled with the Kalman filter. Applied to an investment strategy, it runs with a Stop Loss criteria for periods in which the model does not fit the interest rates. Using BMF’s DI future contracts available for the period of 1st of March, 2007 to the 30th of May 2014, we simulated the strategy for different periods with different conditions, aiming on finding the optimal time series window that will lead to the model parameters that best fit the current term structure, also, it will be analyzed how long these parameters optimally estimate interest rate dynamic. The results were compared with the ones obtained from the vector autoregressive model of lag 1, indicating that Vasicek Model underperform compared to this model. The limitation of the two factor model on capturing the term structure dynamics does not allow the model to estimate the term structure at once denigrating the results. / Este trabalho tem o objetivo de testar a qualidade preditiva do Modelo Vasicek de dois fatores acoplado ao Filtro de Kalman. Aplicado a uma estratégia de investimento, incluímos um critério de Stop Loss nos períodos que o modelo não responde de forma satisfatória ao movimento das taxas de juros. Utilizando contratos futuros de DI disponíveis na BMFBovespa entre 01 de março de 2007 a 30 de maio de 2014, as simulações foram realizadas em diferentes momentos de mercado, verificando qual a melhor janela para obtenção dos parâmetros dos modelos, e por quanto tempo esses parâmetros estimam de maneira ótima o comportamento das taxas de juros. Os resultados foram comparados com os obtidos pelo Modelo Vetor-auto regressivo de ordem 1, e constatou-se que o Filtro de Kalman aplicado ao Modelo Vasicek de dois fatores não é o mais indicado para estudos relacionados a previsão das taxas de juros. As limitações desse modelo o restringe em conseguir estimar toda a curva de juros de uma só vez denegrindo seus resultados.

This thesis deals with the management of the fund of alternative investments with an emphasis on photovoltaic projects in the Czech Republic. The main objective is to evaluate whether, after numerous legislative changes, these projects continue to be an attractive investment alternative. The impact of legislative changes on the economy and efficiency of the projects were analyzed for fictitious projects using static and dynamic methods of investment evaluation. The analysis showed that if there was knowledge of the additional cost burden resulting from changes in legislation, then none of the evaluated projects would have been implemented. In general, changes have had the most significant impact on projects that initially appeared to be most effective. In terms of size, restrictions affected mainly smaller projects. Overall, there was a relative alignment of return. For projects with higher levels of debt, an additional cost burden could be liquidational. Then the portfolio of alternative investment fund was created, which consists of assets of two alternative projects evaluated before. This led to evaluation, whether, despite charged fees but a lower tax rate, the investment through the fund is more favorable compared to an own special purpose company. Due to the small size of the fund the hypothesis of fund being more effective mean of administration, was not confirmed.

The subject of this thesis are selected interest rate models and valuation of interest rate derivatives, especially interest rate options. Time-homogeneous one-factor short rate models, Vasicek and Cox-Ingersoll-Ross, and time-inhomogeneous short rate model, Hull{White, are treated. Heath-Jarrow-Morton framework is introduced as an alternative to short rate models, evolving the entire term structure of interest rates. The short rate models are shown to be special cases of models within the framework. The models are derived using the risk-neutral pricing methodology.

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.

This thesis focuses on modelling non-maturing deposits (NMD) and has been written in collaboration with Svenska Handelsbanken. The methodology includes regression analysis and time series analysis, with the Repo rate serving as an exogenous variable in both models. A Vasicek model is employed to generate future Repo rates, which are then used as inputs for forecasting the NMD volume. These simulated rates are then compared to forecasted Repo rates with discrete changes from an external source. The results are utilised to analyze how net interest income can vary in the case of constant volume and in the case of interest rate-dependent volume. Effective liquidity management is crucial for banks, and NMDs are an important source of funding. By using regression analysis and time series analysis, combined with the Repo rate as the exogenous variable, this thesis provides insights into the behaviour of NMD volumes, and how it is affected by the Repo rate. The models also enable the forecasting of future trends based on future Repo rates. Additionally, by using different data sets as input for future Repo rates, the behaviour of the model can be evaluated based on how well it coincides with reality. The results obtained from this analysis can also be used to compare the value and interest rate sensitivity of NMD products. In conclusion, this thesis provides an approach to modelling the NMD volumes using exogenous factors and demonstrates how this can affect the net interest income from deposit volumes. / Denna avhandling fokuserar på modellering av icke tidsbunden inlåning (non-maturing deposits, NMD) och har skrivits i samarbete med Svenska Handelsbanken. Metoden omfattar regressionsanalys och tidsserieanalys, där reporäntan fungerar som en exogen variabel i båda modellerna. En Vasicek-modell används för att generera framtida reporäntor, som sedan används som indata för att prognostisera NMD-volymen. Dessa simulerade räntor jämförs sedan med prognostiserade reporäntor med diskreta förändringar från en extern källa. Resultaten används sedan för att analysera hur räntenettot kan variera mellan fallet med konstant volym och fallet med ränteberoende volym. En effektiv likviditetshantering är avgörande för banker, och NMD:er är en viktig finansieringskälla. Genom att använda regressionsanalys och tidsserieanalys, i kombination med reporäntan som exogen variabel , ger denna avhandling värdefulla insikter i NMD-volymernas beteende och hur de påverkas av reporäntan. Modellerna gör det också möjligt att prognostisera framtida trender utifrån framtida reporäntor. Genom att använda olika datamängder som indata för de framtida reporäntorna kan modellens beteende dessutom värderas utifrån hur väl det sammanfaller med verkligheten. Resultaten från denna analys kan också användas för att jämföra NMD-produkternas värde- och räntekänslighet. Sammanfattningsvis ger denna avhandling ett tillvägagångssätt för att modellera NMD-volymerna med hjälp av exogena faktorer och visar på hur det kan påverka inlåningens räntenetto.