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1
Kang, Zhuang. "Illiquid Derivative Pricing and Equity Valuation under Interest Rate Risk." University of Cincinnati / OhioLINK, 2010. http://rave.ohiolink.edu/etdc/view?acc_num=ucin1282168157.
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Pang, Kin. "Calibration of interest rate term structure and derivative pricing models." Thesis, University of Warwick, 1997. http://wrap.warwick.ac.uk/36270/.
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We argue interest rate derivative pricing models are misspecified so that when they are fitted to historical data they do not produce prices consistently with the market. Interest rate models have to be calibrated to prices to ensure consistency. There are few published works on calibration to derivatives prices and we make this the focus of our thesis. We show how short rate models can be calibrated to derivatives prices accurately with a second time dependent parameter. We analyse the misspecification of the fitted models and their implications for other models. We examine the Duffle and Kan Affine Yield Model, a class of short rate models, that appears to allow easier calibration. We show that, in fact, a direct calibration of Duffle and Kan Affine Yield Models is exceedingly difficult. We show the non-negative subclass is equivalent to generalised Cox, Ingersoll and Ross models that facilitate an indirect calibration of nonnegative Duffle and Kan Affine Yield Models. We examine calibration of Heath, Jarrow and Morton models. We show, using some experiments, Heath, Jarrow and Morton models cannot be calibrated quickly to be of practical use unless we restrict to special subclasses. We introduce the Martingale Variance Technique for improving the accuracy of Monte Carlo simulations. We examine calibration of Gaussian Heath Jarrow and Morton models. We provide a new non-parametric calibration using the Gaussian Random Field Model of Kennedy as an intermediate step. We derive new approximate swaption pricing formulae for the calibration. We examine how to price resettable caps and floors with the market- Libor model. We derive a new relationship between resettable caplets and floorlets prices. We provide accurate approximations for the prices. We provide practical approximations to price resettable caplets and floorlets directly from quotes on standard caps and floors. We examine how to calibrate the market-Libor model.
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Twarog, Marek B. "Pricing security derivatives under the forward measure." Link to electronic thesis, 2007. http://www.wpi.edu/Pubs/ETD/Available/etd-053007-142223/.
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Pietersz, Raoul. "Pricing Models for Bermudan-Style Interest Rate Derivatives." [Rotterdam]: Erasmus Research Institute of Management (ERIM), Erasmus University Rotterdam ; Rotterdam : Erasmus University Rotterdam [Host], 2005. http://hdl.handle.net/1765/7122.
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Bouziane, Markus. "Pricing interest rate derivatives a fourier transform based approach." Berlin Heidelberg Springer, 2007. http://d-nb.info/989148165/34.
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Nohrouzian, Hossein. "An Introduction to Modern Pricing of Interest Rate Derivatives." Thesis, Mälardalens högskola, Akademin för utbildning, kultur och kommunikation, 2015. http://urn.kb.se/resolve?urn=urn:nbn:se:mdh:diva-28415.
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This thesis studies interest rates (even negative), interest rate derivatives and term structure of interest rates. We review the different types of interest rates and go through the evaluation of a derivative using risk-neutral and forward-neutral methods. Moreover, the construction of interest rate models (term-structure models), pricing of bonds and interest rate derivatives, using both equilibrium and no-arbitrage approaches are discussed, compared and contrasted. Further, we look at the HJM framework and the LMM model to evaluate and simulate forward curves and find the forward rates as the discount factors. Finally, the new framework (after financial crisis in 2008), under the collateral agreement (CSA) has been taken into consideration.
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Nyamai, Dayton. "Pricing of Interest Rate Derivatives under the Cheyette model." Thesis, Uppsala universitet, Tillämpad matematik och statistik, 2020. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-421201.
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Slinko, Irina. "Essays in option pricing and interest rate models." Doctoral thesis, Stockholm : Economic Research Institute, Stockholm School of Economics [Ekonomiska forskningsinstitutet vid Handelshögskolan i Stockholm] (EFI), 2006. http://www2.hhs.se/EFI/summary/706.htm.
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Wu, Andrew Man Kit. "Efficient lattice methods for pricing interest rate options and other derivative securities under stochastic volatility." Thesis, University of Strathclyde, 2002. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.248776.
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Mutengwa, Tafadzwa Isaac. "An analysis of the Libor and Swap market models for pricing interest-rate derivatives." Thesis, Rhodes University, 2012. http://hdl.handle.net/10962/d1005535.
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This thesis focuses on the non-arbitrage (fair) pricing of interest rate derivatives, in particular caplets and swaptions using the LIBOR market model (LMM) developed by Brace, Gatarek, and Musiela (1997) and Swap market model (SMM) developed Jamshidan (1997), respectively. Today, in most financial markets, interest rate derivatives are priced using the renowned Black-Scholes formula developed by Black and Scholes (1973). We present new pricing models for caplets and swaptions, which can be implemented in the financial market other than the Black-Scholes model. We theoretically construct these "new market models" and then test their practical aspects. We show that the dynamics of the LMM imply a pricing formula for caplets that has the same structure as the Black-Scholes pricing formula for a caplet that is used by market practitioners. For the SMM we also theoretically construct an arbitrage-free interest rate model that implies a pricing formula for swaptions that has the same structure as the Black-Scholes pricing formula for swaptions. We empirically compare the pricing performance of the LMM against the Black-Scholes for pricing caplets using Monte Carlo methods.
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Chu, Chi Chiu. "Pricing models of equity-linked insurance products and LIBOR exotic derivatives /." View abstract or full-text, 2005. http://library.ust.hk/cgi/db/thesis.pl?MATH%202005%20CHU.
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Svensson, Emma, and Viktor Tingström. "Pricing interest rate derivatives : The effects of the 2007 credit crisis." Thesis, Jönköping University, JIBS, Business Administration, 2010. http://urn.kb.se/resolve?urn=urn:nbn:se:hj:diva-13095.
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<p>The purpose of this thesis is to compare and analyze the single curve and the multiple curve frameworks used to price interest rate derivatives and to discuss the advantages of the multiple curve framework. We also describe how the overall derivative market has been affected by the 2007 credit crisis.</p>
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Bouziane, Markus [Verfasser]. "Pricing interest rate derivatives : a fourier transform based approach / Markus Bouziane." Berlin, 2008. http://d-nb.info/989148165/34.
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Frey, Roman. "Monte Carlo methods with application to the pricing of interest rate derivatives /." St. Gallen, 2008. http://www.biblio.unisg.ch/org/biblio/edoc.nsf/wwwDisplayIdentifier/03393436001/$FILE/03393436001.pdf.
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El, Menouni Zakaria. "Pricing Interest Rate Derivatives in the Multi-Curve Framework with a Stochastic Basis." Thesis, KTH, Matematisk statistik, 2015. http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-163274.
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The financial crisis of 2007/2008 has brought about a lot of changes in the interest rate market in particular, as it has forced to review and modify the former pricing procedures and methodologies. As a consequence, the Multi-Curve framework has been adopted to deal with the inconsistencies of the frameworks used so far, namely the single-curve method. We propose to study this new framework in details by focusing on a set of interest rate derivatives such as deposits, swaps and caplets, then we explore a stochastic approach to model the Libor-OIS basis spread, which has appeared since the beginning of the crisis and is now the quantity of interest to which a lot of researchers dedicate their work (F.Mercurio, M.Bianchetti and others). A discussion follows this study to set the light on the challenges and difficulties related to the modeling of basis spread.<br>Den stora finanskris som inträffade 2007/2008 har visat att nya värderingsmetoder för räntederivat är nödvändiga. Den metod baserat på multipla räntekurvor som introducerats som lösning på de problem som finanskrisen synliggjort, speciellt gällande räntespread, har givit upphov till nya utmaningar och bekymmer. I detta arbete utforskas den nya metoden baserat på multipla räntekurvor samt en stokastisk modell för räntespread. Slutsatserna och diskussionen om resultaten som presenteras tydliggör kvarvarande utmaningar vid modellering av räntespread
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Hellander, Martin. "Credit Value Adjustment: The Aspects of Pricing Counterparty Credit Risk on Interest Rate Swaps." Thesis, KTH, Matematisk statistik, 2015. http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-173225.
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In this thesis, the pricing of counterparty credit risk on an OTC plain vanilla interest rate swap is investigated. Counterparty credit risk can be defined as the risk that a counterparty in a financial contract might not be able or willing to fulfil their obligations. This risk has to be taken into account in the valuation of an OTC derivative. The market price of the counterparty credit risk is known as the Credit Value Adjustment (CVA). In a bilateral contract, such as a swap, the party’s own creditworthiness also has to be taken into account, leading to another adjustment known as the Debit Value Adjustment (DVA). Since 2013, the international accounting standards (IFRS) states that these adjustments have to be done in order to reflect the fair value of an OTC derivative. A short background and the derivation of CVA and DVA is presented, including related topics like various risk mitigation techniques, hedging of CVA, regulations etc.. Four different pricing frameworks are compared, two more sophisticated frameworks and two approximative approaches. The most complex framework includes an interest rate model in form of the LIBOR Market Model and a credit model in form of the Cox-Ingersoll- Ross model. In this framework, the impact of dependencies between credit and market risk factors (leading to wrong-way/right-way risk) and the dependence between the default time of different parties are investigated.<br>I den här uppsatsen har prissättning av motpartsrisk för en OTC ränteswap undersökts. Motpartsrisk kan definieras som risken att en motpart i ett finansiellt kontrakt inte har möjlighet eller viljan att fullfölja sin del av kontraktet. Motpartsrisken måste tas med I värderingen av ett OTC-derivat. Marknadspriset på motpartrisken är känt som Credit Value Adjustment (CVA). I ett bilateralt kontrakt, t.ex. som en swap, måste även den egna kreditvärdighet tas med i värderingen, vilket leder till en justering som är känd som Debit Value Adjustment (DVA). Sedan 2013 skall, enligt den internationella redovisningsstandarden (IFRS), dessa prisjusteringar göras vid redovisningen av värdet för ett OTC derivat. En kort bakgrund samt härledningen av CVA och DVA ar presenterade tillsammans med relaterade ämnen. Fyra olika metoder för att beräkna CVA har jämförts, två mer sofistikerade metoder och två approximativa metoder. I den mest avancerade metoden används en räntemodell i form av LIBOR Market Model samt en kreditmodell i form av en Cox-Ingersoll-Ross modell. I den här metoden undersöks även påverkan av CVA då det existerar beroenden mellan marknads
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Damberg, Petter, and Alexander Gullnäs. "Interest rate derivatives: Pricing of Euro-Bund options : An empirical study of the Black Derman & Toy model (1990)." Thesis, Örebro universitet, Handelshögskolan vid Örebro Universitet, 2012. http://urn.kb.se/resolve?urn=urn:nbn:se:oru:diva-24472.
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The market for interest rate derivatives has in recent decades grown considerably and the need for proper valuation models has increased. Interest rate derivatives are instruments that in some way are contingent on interest rates such as bonds and swaps and most financial transactions are in some way exposed to interest rate risk. Interest rate derivatives are commonly used to hedge this risk. This study focuses on the Black Derman & Toy model and its capability of pricing interest rate derivatives. The purpose was to simulate the model numerically using daily Euro-Bunds and options data to identify if the model can generate accurate prices. A second purpose was to simplify the theory of building a short rate binomial tree, since existing theory explains this step in a complex way. The study concludes that the BDT model have difficulties valuing the extrinsic value of options with longer maturities, especially out-of-the money options.
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Wang, Dan. "Interest-rate models : an extension to the usage in the energy market and pricing exotic energy derivatives." Thesis, Imperial College London, 2009. http://hdl.handle.net/10044/1/5583.
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In this thesis, we review various popular pricing models in the interest-rate market. Among these pricing models, we choose the LIBOR Market model (LMM) as the benchmark model. Based on market practice experience, we also develop a pricing model named the “Market volatility model”. By pricing vanilla interest-rate options such as interest-rate caps and swaptions, we compare the performance of our Market volatility model to that of the LMM. It is proved that the Market Volatility model produce comparable results to the LMM, while its computing efficiency largely exceeds that of the LMM. Following the recent rapid development in the commodity market, in particular the energy market, we attempt to extend the use of our proposed Market volatility model from the interest-rate market to the energy market. We prove that the Market Volatility model is capable of pricing various energy derivative under the assumption of absence of the convenience yield. In addition, we propose a new type of exotic energy derivative which has a flexible option structure. This energy derivative is named as the Flex-Asian spread options (FASO). We give examples of different option structures within the FASO framework and use the Market volatility model to generate option prices and greeks for each structure. Although the Market volatility model can be used to price various energy derivatives based on oil/gas contracts, it is not compatible with the structure of one of the most advanced derivatives in the energy market, the storage option. We modify the existing pricing model for storage options and use our own 3D-binomial tree approach to price gas storage contracts. By doing these, we improve the performance of the traditional storage model.
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Garisch, Simon Edwin. "Convertible bond pricing with stochastic volatility : a thesis submitted to the Victoria University of Wellington in fulfilment of the requirements for the degree of Masters in Finance /." ResearchArchive@Victoria e-thesis, 2009. http://hdl.handle.net/10063/1100.
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Dalmagro, Lucas Bassani. "Avaliação de derivativos de taxas de juros : uma aplicação do Modelo CIR sobre opções de IDI." reponame:Biblioteca Digital de Teses e Dissertações da UFRGS, 2015. http://hdl.handle.net/10183/127250.
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Este trabalho tem por objetivo principal aplicar o modelo de precificação de opções de taxas de juros proposto por Barbachan e Ornelas (2003), com base nos modelos de taxa de juro e avaliação de opções de Cox, Ingerssol e Ross (1985), para avaliação de opções de compra sobre o Índice de Taxa Média de Depósitos Interfinanceiros de Um Dia (IDI), negociadas na BM&FBovespa. Para estimação dos parâmetros deste modelo, foi empregado o método de Máxima Verossimilhança. Neste contexto, também fez-se uso da fórmula de precificação de opções proposta por Black (1976), adaptada para o mercado de derivativos brasileiros, conforme implementação verificada no trabalho de Gluckstern et al. (2002). Tal aplicação torna-se interessante, pois este modelo é amplamente utilizado pelo mercado brasileiro para avaliação de opções sobre o IDI. De forma a verificar a aderência dos preços teóricos gerados pelos modelos, em comparação aos preços de mercado, métricas de erro foram empregadas. De forma geral, nossos resultados mostraram que ambos os modelos apresentam erros sistemáticos de precificação, onde o modelo CIR subavalia os prêmios das opções e o modelo de Black superprecifica. No entanto, bons resultados foram encontrados ao avaliarmos opções in-the-money e out-of-money com o modelo de Black.<br>This work aims to apply the interest rate option pricing model proposed by Barbachan and Ornelas (2003), based on the interest rate model and option pricing model developed by Cox, Ingersoll and Ross (1985), to evaluate call options on the 1 day Brazilian Interfinancial Deposits Index - IDI, traded at BM&FBovespa. The Maximum Likelihood method was applied to estimate the model parameters. In this context, the option pricing formula proposed by Black (1976), adapted for the Brazilian derivative Market, was also used, according implementation verified in Gluckstern et al. (2002). This application becomes interesting because this model is widely used by the Brazilian Market to evaluate options on IDI. In order to verify the adherence of theoretical prices generated by the models, in comparison to the Market prices, error metrics were applied. In general, our results pointed out that both models presented systematic pricing errors, in which the CIR model underestimates the option prices and Black’s model overestimates. However, good results were found on the evaluation of options in-the-money and out-of-money with the Black’s Model.
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Silva, Allan Jonathan da. "A new finite difference method for pricing and hedging interest rate derivatives : comparative analysis and the case of the idi option." Laboratório Nacional de Computação Científica, 2015. https://tede.lncc.br/handle/tede/208.
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Previous issue date: 2015-06-02<br>Conselho Nacional de Desenvolvimento Científico e Tecnológico - CNPq<br>Propomos um método numérico de diferenças finitas para substituir os esquemas clássicos utilizados para solucionar EDPs em engenharia financeira. A motivação para desenvolvê-lo advém da perda de precisão na tentativa de estabilizar a solução via up-wind no termo convectivo bem como o fato de que oscilações espúrias ocorrem quando a volatilidade é baixa, o que é comumente observado nos mercados de taxas de juros. Ao contrário dos esquemas clássicos, nosso método cobre todo o espectro de volatilidade da dinâmica das taxas de juros.
Nós comparamos resultados analíticos e numéricos precificando e realizando o hedge de uma variedade de contratos financeiros de renda fixa para mostrar que o método que desenvolvemos é confiável e altamente competitivo. O método se adapta bem a derivativos exóticos de taxas de juros, incluindo um derivativo dependente da trajetória denominado Opção IDI (índice brasileiro de depósito interbancário).
O método dá ênfase à abordagem realística da capitalização discreta do índice em detrimento da capitalização contínua explorada frequentemente na literatura.<br>We propose a second order accurate numerical finite difference method to replace the classical schemes used to solving PDEs in financial engineering. The motivation for doing so stems from the accuracy loss while trying to stabilize the solution via the up-wind trick in the convective term as well as the fact that
spurious oscillation solutions occur when volatilities are low. This is actually the range that we commonly observe in the interest rate markets. Unlike the classical schemes, our method covers the whole spectrum of volatilities in the interest rate dynamics. We compare the analytical and numerical results by both pricing and
hedging a variety of fixed income financial contracts to show that the method we developed is reliable and highly competitive. The method adapts well to exotic interest rate derivative securities, including a path-dependent derivative named IDI (the Brazilian Interbank Deposit Rate Index) option. The method highlights
the use of the realistic discretely compounding interest rate scheme, in detriment of the continuously compounding case often exploited in the literature.
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Bahl, Raj Kumari. "Mortality linked derivatives and their pricing." Thesis, University of Edinburgh, 2017. http://hdl.handle.net/1842/25499.
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This thesis addresses the absence of explicit pricing formulae and the complexity of proposed models (incomplete markets framework) in the area of mortality risk management requiring the application of advanced techniques from the realm of Financial Mathematics and Actuarial Science. In fact, this is a multi-essay dissertation contributing in the direction of designing and pricing mortality-linked derivatives and offering the state of art solutions to manage longevity risk. The first essay investigates the valuation of Catastrophic Mortality Bonds and, in particular, the case of the Swiss Re Mortality Bond 2003 as a primary example of this class of assets. This bond was the first Catastrophic Mortality Bond to be launched in the market and encapsulates the behaviour of a well-defined mortality index to generate payoffs for bondholders. Pricing this type of bond is a challenging task and no closed form solution exists in the literature. In my approach, we adapt the payoff of such a bond in terms of the payoff of an Asian put option and present a new methodology to derive model-independent bounds for catastrophic mortality bonds by exploiting the theory of comonotonicity. While managing catastrophic mortality risk is an upheaval task for insurers and re-insurers, the insurance industry is facing an even bigger challenge - the challenge of coping up with increased life expectancy. The recent years have witnessed unprecedented changes in mortality rate. As a result academicians and practitioners have started treating mortality in a stochastic manner. Moreover, the assumption of independence between mortality and interest rate has now been replaced by the observation that there is indeed a correlation between the two rates. Therefore, my second essay studies valuation of Guaranteed Annuity Options (GAOs) under the most generalized modeling framework where both interest rate and mortality risk are stochastic and correlated. Pricing these types of options in the correlated environment is an arduous task and a closed form solution is non-existent. In my approach, I employ the use of doubly stochastic stopping times to incorporate the randomness about the time of death and employ a suitable change of measure to facilitate the valuation of survival benefit, there by adapting the payoff of the GAO in terms of the payoff of a basket call option. I then derive general price bounds for GAOs by employing the theory of comonotonicity and the Rogers-Shi (Rogers and Shi, 1995) approach. Moreover, I suggest some `model-robust' tight bounds based on the moment generating function (m.g.f.) and characteristic function (c.f.) under the affine set up. The strength of these bounds is their computational speed which makes them indispensable for annuity providers who rely heavily on Monte Carlo simulations to calculate the fair market value of Guaranteed Annuity Options. In fact, sans Monte Carlo, the academic literature does not offer any solution for the pricing of the GAOs. I illustrate the performance of the bounds for a variety of affine processes governing the evolution of mortality and the interest rate by comparing them with the benchmark Monte Carlo estimates. Through my work, I have been able to express the payoffs of two well known modern mortality products in terms of payoffs of financial derivatives, there by filling the gaps in the literature and offering state of art techniques for pricing of these sophisticated instruments.
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Kalavrezos, Michail. "Pricing Caps in the Heath, Jarrow and Morton Framework Using Monte Carlo Simulations in a Java Applet." Thesis, Mälardalen University, Department of Mathematics and Physics, 2007. http://urn.kb.se/resolve?urn=urn:nbn:se:mdh:diva-469.
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<p>In this paper the Heath, Jarrow and Morton (HJM) framework is applied in the programming language Java for the estimation of the future spot rate. The subcase of an exponential model for the diffusion coefficient (volatility) is used for the pricing of interest rate derivatives (caps).</p>
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Sarais, Gabriele. "Pricing inflation and interest rates derivatives with macroeconomic foundations." Thesis, Imperial College London, 2015. http://hdl.handle.net/10044/1/25266.
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I develop a model to price inflation and interest rates derivatives using continuous-time dynamics linked to monetary macroeconomic models: in this approach the reaction function of the central bank, the bond market liquidity, and expectations play an important role. The model explains the effects of non-standard monetary policies (like quantitative easing or its tapering) on derivatives pricing. A first adaptation of the discrete-time macroeconomic DSGE model is proposed, and some changes are made to use it for pricing: this is respectful of the original model, but it soon becomes clear that moving to continuous time brings significant benefits. The continuous-time model is built with no-arbitrage assumptions and economic hypotheses that are inspired by the DSGE model. Interestingly, in the proposed model the short rates dynamics follow a time-varying Hull-White model, which simplifies the calibration. This result is significant from a theoretical perspective as it links the new theory proposed to a well-established model. Further, I obtain closed forms for zero-coupon and year-on-year inflation payoffs. The calibration process is fully separable, which means that it is carried out in many simple steps that do not require intensive computation. The advantages of this approach become apparent when doing risk analysis on inflation derivatives: because the model explicitly takes into account economic variables, a trader can assess the impact of a change in central bank policy on a complex book of fixed income instruments, which is not straightforward when using standard models. The analytical tractability of the model makes it a candidate to tackle more complex problems, like inflation skew and counterparty/funding valuation adjustments (known by practitioners as XVA): both problems are interesting from a theoretical and an applied point of view, and, given their computational complexity, benefit from a tractable model. In both cases the results are promising.
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Smetaniouk, Taras. "Pricing variance derivatives using hybrid models with stochastic interest rates." College Park, Md.: University of Maryland, 2008. http://hdl.handle.net/1903/8200.
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Thesis (Ph. D.) -- University of Maryland, College Park, 2008.<br>Thesis research directed by: Applied Mathematics and Scientific Computation Program. Title from t.p. of PDF. Includes bibliographical references. Published by UMI Dissertation Services, Ann Arbor, Mich. Also available in paper.
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Wang, Shijun. "Pricing American derivatives and interest rates derivatives based on characteristic function of the underlying asset returns." Thesis, Queen Mary, University of London, 2003. http://qmro.qmul.ac.uk/xmlui/handle/123456789/1805.
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In this thesis I introduce a new methodology for pricing American options when the underlying model of the asset price allows for stochastic volatility and/or it has a multi-factor structure. Our approach is based on a decomposition of an American option price into its European options counterpart price and the early exercise premium, paid by the option holder in order to keep the right of exercising the option at any time-point before its expiration date. Based on closed form solutions of the joint characteristic function of the state variables driving the underlying model, the thesis provides analytic, integral solutions of the early exercise premium (and hence of the American option price) which enable us to build up fast and accurate numerical approximation procedures for calculating options prices. The analytic solutions that I derive express the optimal early exercise boundary in terms of prices of Arrow-Debreu type of securities reflecting the values of the options additional payoffs if they are exercised earlier, or not. Numerical results reported in the thesis show that our approach can price American options on stocks, bonds and interest rates derivatives efficiently and very fast, compared with existing methods. The efficiency gains of our method stem from the fact that it involves only one step of approximation, as the European prices embodied in the American option prices can be calculated analytically. The gains of computational speed come from the fact that our method can reduce the integral dimensions of the early exercise premium considerably.
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Rayée, Grégory. "Essays on pricing derivatives by taking into account volatility and interest rates risks." Doctoral thesis, Universite Libre de Bruxelles, 2012. http://hdl.handle.net/2013/ULB-DIPOT:oai:dipot.ulb.ac.be:2013/209649.
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Dans le Chapitre 1, nous présentons une nouvelle approche pour évaluer des options dites à barrières basée sur une méthode connue sous le nom de méthode Vanna-Volga. Cette nouvelle méthode nous permet une calibration simple et rapide sur le marché des options à barrières directement ce qui permet d'évaluer ces options avec un outil en accord avec le marché. Nous comparons également nos résultats avec ceux provenant d’autres modèles célèbres et nous étudions la sensibilité de cette méthode par rapport aux données du marché. Nous donnons une nouvelle justification théorique associée à la méthode Vanna-Volga comme étant une approximation de Taylor du premier ordre du prix de l'option autour de la volatilité dite à la monnaie.<p><p><p>Dans le Chapitre 2 de la thèse nous allons développer un modèle qui compte de la volatilité implicite du marché et de la variabilité des taux d'intérêts. Nous travaillons dans le marché particulier des taux de changes, avec un modèle à volatilité locale pour la dynamique du taux de change dans lequel les taux d'intérêts domestiques et étrangers sont également supposé stochastiques. Nous dérivons l'expression de la volatilité locale et dérivons divers résultats particulièrement utiles pour la calibration du modèle. Finalement, nous développons un nouveau modèle hybride où la volatilité du taux de change possède une composante locale et une composante stochastique et nous dérivons une méthode de calibration pour ce nouveau modèle.<p><p><p>Dans le Chapitre 3, nous allons appliquer le modèle à volatilité locale et taux d'intérêts stochastiques développé dans le précédent chapitre mais dans le cadre d'évaluation de produits dérivés associés aux assurances vie. Nous utilisons une méthode de calibration développée dans le Chapitre 2. Les produits étudiés étant exotiques, nous allons également comparer les prix obtenus dans différents modèles, à savoir le modèle à volatilité locale, à volatilité stochastique et enfin à volatilité constante pour le sous-jacent, les trois modèles étant combinés avec des taux d'intérêts stochastiques.<p><p><p>Finalement, dans le Chapitre 4 nous allons travailler avec un modèle dit de Lévy pour modéliser le sous-jacent. Nous nous intéressons à l'évaluation d'options Asiatiques arithmétiques. Comme de nombreuses options exotiques, il n'est pas possible d'obtenir un prix analytique et dans ce cas seules les méthodes numériques permettent de résoudre le problème. Dans ce Chapitre 4, nous développons une méthode basée sur la méthode de simulations de Monte Carlo et nous employons deux types de variables de contrôle permettant d'améliorer la convergence du programme. Nous développons également une méthode permettant d'obtenir une borne inférieure au prix de l'option avec une efficacité qui surpasse les autres méthodes.<p><br>Doctorat en Sciences<br>info:eu-repo/semantics/nonPublished
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Kuan, Chia-Hsuan. "The consitent pricing of interest rate options." Thesis, University of Warwick, 2002. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.250100.
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Luo, Yi. "Spread Option Pricing with Stochastic Interest Rate." BYU ScholarsArchive, 2012. https://scholarsarchive.byu.edu/etd/3269.
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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.
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Jiang, An. "American Spread Option Pricing with Stochastic Interest Rate." BYU ScholarsArchive, 2016. https://scholarsarchive.byu.edu/etd/5987.
Full textAbstract:
In financial markets, spread option is a derivative security with two underlying assets and the payoff of the spread option depends on the difference of these assets. We consider American style spread option which allows the owners to exercise it at any time before the maturity. The complexity of pricing American spread option is that the boundary of the corresponding partial differential equation which determines the option price is unknown and the model for the underlying assets is two-dimensional.In this dissertation, we incorporate the stochasticity to the interest rate and assume that it satisfies the Vasicek model or the CIR model. We derive the partial differential equations with terminal and boundary conditions which determine the American spread option with stochastic interest rate and formulate the associated free boundary problem. We convert the free boundary problem to the linear complimentarity conditions for the American spread option, so that we can go around the free boundary and compute the option price numerically. Alternatively, we approximate the option price using methods based on the Monte Carlo simulation, including the regression-based method, the Lonstaff and Schwartz method and the dual method. We make the comparisons among the option prices derived by the partial differential equation method and Monte Carlo methods to show the accuracy of the result.
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Kirriakopoulos, Konstantinos. "Optimal portfolios with constrained sensitivities in the interest rate market." Thesis, Imperial College London, 1996. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.362717.
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Strom, Christopher Solon. "Pricing and hedging in an incomplete interest rate market." Thesis, University College London (University of London), 2008. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.506807.
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This thesis explores pricing models for interest rate markets. The model used to ':describe the short rate is based on the discontinuous shot noise process. As a consequence the market is incomplete, meaning that not all securities contingent on the short rate can be replicated perfectly with a dynamically adjusted portfolio of a bond and cash. This framework is still consistent with the absence of arbitrage as evidenced by the existence of an equivalent martingale measure. This measure is not unique, however, due to the incompleteness of the market. Two approaches to pricing contingent claims are pursued. The first, risk-neutral pricing, evaluates the expected value of the pay-off at expiration under an equivalent martingale measure. A parameterized class of martingales, based on the Esscher transform, allows for the definition of a flexible set of equivalent martingale measures and results in a formula for the conditional joint Laplace transform of the short rate and its time-integral. The pricing formula for a discount bond follows trivially from these results. A method for pricing a European call option is also proposed, requiring numerical inversion of the aforementioned Laplace transform. The second approach, mean-variance hedging, addresses the incompleteness of the market. A contingent claim is priced by forming a portfolio of a bond and cash. The portfolio is dynamically updated to .mimic the pay-off of the claim at expiration. The replicating portfolio is restricted to be self-financing and predictable. This approach leads to a closed-form pricing formula for a discount bond and formulae for European call and put options, requiring the numerical Laplace inversion methods mentioned above. All this is in the context of a discrete-time model that includes as a special case a discrete-time version of the shot noise process.
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Kohler, Daniel. "Betting against uncovered interest rate parity." kostenfrei, 2008. http://www.biblio.unisg.ch/www/edis.nsf/wwwDisplayIdentifier/3513.
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Liu, Cheng. "Utility-based Futures Contract Pricing under Stochastic Interest Rate, Appreciation Rate and Dividend Yield." University of Cincinnati / OhioLINK, 2010. http://rave.ohiolink.edu/etdc/view?acc_num=ucin1283524846.
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Hatgioannides, John. "Essays on asset pricing in continuous time." Thesis, Birkbeck (University of London), 1996. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.244543.
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Xie, Yan Alice Wu Chunchi. "Immunization of interest rate risk and pricing of default risk of bond portfolios." Related Electronic Resource: Current Research at SU : database of SU dissertations, recent titles available full text, 2003. http://wwwlib.umi.com/cr/syr/main.
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Senturk, Huseyin. "An Empirical Comparison Of Interest Rate Models For Pricing Zero Coupon Bond Options." Master's thesis, METU, 2008. http://etd.lib.metu.edu.tr/upload/3/12609786/index.pdf.
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The aim of this study is to compare the performance of the four interest rate
models (Vasicek Model, Cox Ingersoll Ross Model, Ho Lee Model and Black Der-
man Toy Model) that are commonly used in pricing zero coupon bond options.
In this study, 1{5 years US Treasury Bond daily data between the dates June 1,
1976 and December 31, 2007 are used. By using the four interest rate models,
estimated option prices are compared with the real observed prices for the begin-
ing work days of each months of the years 2004 and 2005. The models are then
evaluated according to the sum of squared errors. Option prices are found by
constructing interest rate trees for the binomial models based on Ho Lee Model
and Black Derman Toy Model and by estimating the parameters for the Vasicek
and the Cox Ingersoll Ross Models.
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Chang, Po Neng, and 張博能. "Derivative Pricing Under Negative Interest Rate Environment." Thesis, 2016. http://ndltd.ncl.edu.tw/handle/13716174831640610014.
Full textAbstract:
碩士<br>國立政治大學<br>金融學系<br>104<br>Negative rate in derivatives would be discussed in our thesis. Our main contribution is to provide the empirical results for these negative pricing model by negative interest rate market data. In addition, the experiment compares the performance between traditional pricing model and these negative pricing models by positive interest rate market data. Traditional pricing model could not work effectively and consistently under negative interest rate environment. Facing the challenge of negative interest rate policy, it is quite necessary for quants to develop the new perspective of pricing financial products and view of hedging the interest rate exposure. Several studies try to use the normal distribution instead of previous convention of the log normal assumption. Recently, both shifted diffusion and free boundary model have been widely introduced in related works. Thus, these approaches bring the new concepts and inspiration for some researchers. Furthermore, the stable and correct risk metrics is also a critical issue that market participants are concerned. Three modified SABR models from different literatures would be presented and calibrated by EUR market data and USD market data in this thesis. In the long run, there are some suggestions and future studies proposed in our work for the financial product pricing and risk management in a negative interest rate capital market.
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Chen, Chi-Tsai, and 陳其財. "Exotic Interest Rate Derivative — Average Interest Rate Cap's Pricing, Hedging and Application." Thesis, 2001. http://ndltd.ncl.edu.tw/handle/64377581315706163015.
Full textAbstract:
碩士<br>國立臺灣大學<br>財務金融學研究所<br>89<br>Thesis Abstract:
Hedging interest rate risk has become one of the most common and important type of a financial manager’s risk management activities. In the last decade several instruments have been developed to help the manager to control these risks, such as swaps, forwards rate agreements, caps and collars. Caps in particular are used whenever the manager wants to have a ceiling on the borrowing costs and at the same time wants to profit form lower interest rates.
Some firms would view their objective as hedging their average cost of funds during an accounting cycle, rather than hedging individual payment. This study describe one such hedging vehicle: a cap on the average interest rate during a period.
Longstaff (1995) showed how prices for average interest rate caps can be calculated, based on the Vasicek (1977) model for interest rates. Longstaff derives analytic valuation formulas for the average rate caps, since he assumes that the final payoff depends on the short rate at the averaging points, instead of some LIBOR rate, as is common in the financial industry. In the Vasicek model, future short rates are normally distributed and hence their average is also normally distributed and option prices can be readily calculated. Three-month LIBOR rates are no longer normally distributed, since three-month bond prices are lognormally distributed in the Vasicek model.
Hence, one encounters the same difficulty in pricing average rate caps as for Asian Options on currencies. One has to calculate distributions of the sum of longnormally distributed variables, for which no analytic formulas exist. Several methodologies have been proposed for Asian Currency options (eg. Levy,1992 and Vorst,1992) .In this paper we describe the equivalents of some of these methodologies for average rate caps. Furthermore, we use the Hull-White model (1990) rather than Vasicek’s model, since the Hull-White model can be fitted to the actual term structure, which is not possible for the Vasicek model.
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Wu, Guan-shiun, and 吳冠勳. "Pricing of Interest Rate Derivatives." Thesis, 2009. http://ndltd.ncl.edu.tw/handle/26098968274018369197.
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碩士<br>國立臺北大學<br>統計學系<br>97<br>HJM model is a very general interest rate model, it only required inputs are the initial yield curve and volatility structure for pure discount bond. This paper discussed the problem of pricing a spread option on the difference of two interest rates under Heath, Jarrow and Morton (hereafter HJM) model. We know that there is no closed form of spread option. This paper will introduce a method which proposed by Borovkova,Permana and Weide (2007). By this method, we will price the yield rate spread options and the LIBOR rate spread option. Finally, we can compare with Monte Carlo simulation and confirm on accuracy of this method .
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Crotty, Michael T. "Assessing the effects of variability in interest rate derivative pricing." 2006. http://www.lib.ncsu.edu/theses/available/etd-08302006-133536/unrestricted/etd.pdf.
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Chen, Li-Shu, and 陳麗淑. "Option Pricing and Numerical Techniques for Pricing Interest Rate Derivatives." Thesis, 2001. http://ndltd.ncl.edu.tw/handle/75352731981694622130.
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碩士<br>國立交通大學<br>應用數學系<br>89<br>In the world , the securities have become very popular , with a wide variety of istrument trading in the finance and investment market . And option market becomes more and more important . Here we concentrate on models for pricing interest rate derivatives and its numerical techniques . Although Black-Scholes formula can be used to price interest rate derivatives , different instruments make different assumptions , it leads special pricing methods . In order to value interest rate derivatives accurately and consistently we need to model the whole term structure of interest rates and the associated volatilities of these rates . To be automatically consistent with the initial (observed) market data , term structure consistent models set out to model the dynamics of the entire term structure .
For most interest rate models , and for models which have some tractability but applied to pricing products which involve early exercise opportunities or complicated terminal pay-offs , we must use numerical techniques to solve them . First we construct binomial trees to represent a number of processes for short rate , and how the resulting tree can then be used to price a wide range of interest rate derivatives . Furthermore we extend it to building trinomial trees for short rate , the extra degree of freedom which this extension allows, enables us to implement short-rate models that exhibit mean reversion . A tree is constructed in such a way that approximates the stochastic differential equation for short rate and automatically returns the observed prices of pure discount bonds and possibly the volatilities of these bonds . Thus we can use these to price many interest rate derivatives .
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Lu, Yung-Chung, and 盧永忠. "The Application of the Pricing Models of Interest Rate Derivatives." Thesis, 2008. http://ndltd.ncl.edu.tw/handle/17759602224409350863.
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碩士<br>國立臺灣大學<br>財務金融組<br>96<br>This study is to analyze the application of pricing model on six interest rate derivatives which include Quanto Interest Rate Swap, Range Accrual Note, Quanto Range Accrual Note, Target Redemption Note, Constant Maturity Swap, and Quanto Constant Maturity Swap. For these interest rate derivatives, this study briefs the product term sheet attributes, interest rate model of pricing, excel input(output) format (under VBA coded), and the final output of pricing model.
The empirical results are:
1.The interest rate derivatives types are too many to understand their attributes and pricing. For the individual or financial institution investors, they need to call help from the independent financial engineering consultant or company to solve the pricing issues.
2.To develop a new type of interest rate derivatives price tool, the financial experts and the information technology geniors need to work together and share the domain know how. They also need to consider the easy use on end user side. If they can avoid too many parameters, and too man estimations input, the accuracy of the output will be more certain.
3.The computing power is the critical successful factor which we calculate the price by using the suitable interest rate model. The CPU capability, main memory, computer language, and data base, all count on the performance. Only powerful computing power can provide us to use the right model and come out the right solution.
4.In general, BGM model and Monte Carol Simulation can get more accurate output in pricing. However, BGM model sometimes generates too many intermediate calculation process. We may not get the reasonable price within limited computing time frame.
Finally, we suggest the authorization to establish an institution to help investors in price the in new interest rate derivatives. There are three advantages. First, it reduces the misunderstanding between the investors and issuers. Second, it helps the financial institutions evaluate their market risk. Finally, it activeates the financial re-intermediation market.
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Yen, Lin Ming, and 林明彥. "Pricing the Interest Rate Derivatives Under the Optimal Yield Curves." Thesis, 2003. http://ndltd.ncl.edu.tw/handle/48027914054351864483.
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碩士<br>國立彰化師範大學<br>商業教育學系<br>91<br>This study has a discussion on the process of using the Hull & White interest rate trinomial tree to price the interest rate derivatives. This study includes the choice of interest rate models, the construction of interest rate models, the analysis of B-spline function , the fitting of Taiwanese Government Bonds yield curves, and the fitting of IRS yield curves. Finally , it shows a process of pricing interest rate derivatives.
The main purpose of this study presented as following:
1. Constructing the interest rate models in this study on the basis of interest rate trinomial tree provided by Hull & White(2000).
2. Fitting the yield curves , which is the input of Hull & White interest rate model to price interest rate derivatives.
The findings of this study are as following :
1. The side of fitting yield curves: it begins with using traditional B-spline function and Powell B-spline function to fit Taiwanese Government Bonds yield curves, respectively .
2. Hull & White(2000) interest rate model is implemented by using a tree method . Furthermore, interest rate swaptions are evaluated.Using the forward induction method provided by Jamshidiam(1991) and the Arrow-Debreu security price to calibrate the interest rate tree and construct the Arrow-Debreu price tree. Next, pricing the interest rate swaptions.
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Chih-Hung, Chuang, and 莊志宏. "The Empirical Analysis of Interest Rate Model and Derivatives Pricing." Thesis, 2005. http://ndltd.ncl.edu.tw/handle/09424044841529777235.
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碩士<br>輔仁大學<br>金融研究所<br>93<br>Interest rate models have played an important role in asset pricing during modern times. People with different purposes will use them to reach their goal. This thesis begins with analyzing short-term interest rate models and then taking one step further to discuss the pricing of interest rate derivatives.
The interest rate models were discussed in this thesis including Vasicek, CIR, and CKLS models which are called equilibrium interest rate models. Furthermore, Hull-White and BDT models which are called no-arbitrage interest rate models were also explored. Using Monte Carlo simulation approach, Interest rate level effect, GARCH effect, and both effects were also incorporated in no-arbitrage models to compare with equilibrium interest rate models.
For the aspect of parameters estimated in the thesis, we can find that the data of interest rate in Taiwan’s money market has significant interest rate level effect and GARCH effect. Moreover, the interest rate level effect will be over emphasized when interest rate models only fit into interest rate level effect. To provide the relative performance of different interest rate models, we test their predictive power by estimating two series of OLS regressions (interest rate and volatility). The coefficients of determination from the regressions are used to compare models’ forecasting performance and the result indicates that no-arbitrage models incorporating level effect, GARCH effect, or both have better predictive power than equilibrium models.
In the pricing of interest rate derivatives, interest rate futures and forward rate prices are evaluated by the use of Monte Carlo simulation approach. Besides, Monte Carlo simulation approach is also used to calculate the price of discount bond and Europen option. The results are indifferent when we considerate close form solution and no close form solution in Vasicek and CIR models.
Binomial tree approach is discussed in the final part of the thesis, and we hope it would help those are interested in this field to know how it works and how to make use of it.
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Lin, Yu-Min, and 林育民. "Pricing Interest Rate Derivatives in HJM Model by Monte Carlo Method." Thesis, 2004. http://ndltd.ncl.edu.tw/handle/72556723094879159057.
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碩士<br>國立中央大學<br>財務金融研究所<br>92<br>Heath, Jarrow and Morton (hereafter HJM) model is a very general interest rate model,
their only required inputs are the initial yield curve and the volatility structure for pure
discount bond (PDB) price return. Here we provide the interest rate caps pricing model
with very general volatility structure. When we test the valuation of interest rate
derivatives in one-factor HJM model, we consider two different volatility structures as (i)
exponentially decaying (ii) humped. We use Monte Carlo simulation combined with
efficient bond return process and quasi-random sequences to price several interest rate
derivatives included PDB option, caps and swaptions. The result of this thesis is that we
can price these interest rate derivatives accurately by Monte Carlo simulation combined
with quasi-random sequences. We also show some characteristics of two-factor Gaussian
HJM model when pricing interest rate swaptions.
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Lap, Fai Tam, and 譚立暉. "Pricing Interest Rate Derivatives in Heath-Jarrow-Morton Model with Stochastic Volatility." Thesis, 2010. http://ndltd.ncl.edu.tw/handle/40752446321555108810.
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碩士<br>臺灣大學<br>財務金融學研究所<br>98<br>This article provides a flexible stochastic volatility multi-factor Heath–Jarrow–Morton term structure model, which allows forward rate correlative with its volatility, and there are N random factors affect the interest rate structure, while additional N random factors would affect volatilities (and also interest rate derivatives). This model improves the Trolle and Schwartz (2009) model, so that interest rate volatility is proportional to the short rate. This model can be converted into a finite-state variables Markov representation system, so under this model, Monte Carlo simulation can be easily used to evaluate the various interest rate derivatives, such as interest rate cap, swaption, etc.. Finally, we will analyze the impact of various parameters on the pricing result.
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Yuan, Lih-Bin, and 阮立斌. "Pricing Interest Rate Derivatives under Hump Volatility Structure With Ritchken & Chuang Model." Thesis, 2000. http://ndltd.ncl.edu.tw/handle/69386883687117009166.
Full textAbstract:
碩士<br>國立臺灣大學<br>財務金融學研究所<br>88<br>Without identifying special conditions on volatility structures, the evolution of the term structure can not be made Markovian with respect to a finite dimensional Markovian system under the generalized Heath-Jarrow-Morton (HJM) paradigm. For the flaw in implementing the non-recombining binomial lattice procedure, efficient HJM algorithm is not available for accurately pricing most types of long-term American contracts. Specifying the forward rate volatility structure as a special deterministic function of time only, Ritchken & Chuang (RC) model in the HJM paradigm develops a feasible approach to avoid the exploding-tree problem faced by HJM. In addition to incorporating full information on the term structure, the most significant advantage of RC model is that this model can price interest rate derivatives under various patterns of volatility structures, including hump volatility structure.
To find out the practical pricing ability of the RC model, the numerical analysis of this article investigates two issues. One is that the appropriate setting of volatility structures is necessary to heighten the accuracy of the pricing result. The other is the convergence speed of the RC algorithm in using different interpolation methods and the different size of the information matrix.
The result of this study shows as the following : (1) If the empirical test verifies the hump volatility structure is true, then the additional consideration of the volatility structure in the RC model will not be redundant. (2) The quadratic interpolation method is the best way to accelerate the convergence speed of the solution of the RC algorithm. The bigger size of the information matrix, the better accuracy of the numerical solution.
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Yu, Cheng-Han, and 游承翰. "Pricing Interest Rate Derivatives Products in SABR(Stochastic Alpha Beta Rho Model)-LMM Model." Thesis, 2010. http://ndltd.ncl.edu.tw/handle/83971998201356753289.
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碩士<br>國立中興大學<br>財務金融系所<br>98<br>The appearance of volatility smile has been documented since at the least the time of the USA stock market crash of 1987 .This situation has pointed out the failure of constant volatility assumption in Black and Scholes model (1973).The followed researches have also found the behavior of implied volatility is more volatile than constant or deterministic function in the options market. Therefore, joint the stochastic volatility process into traditional models in order to reflect the true dynamics of volatility in real market has been considered.
This research used SABR-LMM model proposed by Mercurio and Morini (2009) for pricing interest derivative products. This is one kind of stochastic volatility LIBOR market model consistent with SABR dynamics and develops approximations that allow for use of the SABR implied volatility formula with modified inputs.
The SABR model is proposed by Hagan et al. (2002) and is now the market standard for dealing with volatility smiles. And the LIBOR market model can generate the joint evolution of forward rates when pricing complicate term structure products.
In the empirical study, we found that SABR-LMM model is capable to calibrate the whole swaption volatility surface and capture the volatility smile accurately. Therefore, this model can reinforcement the weakness of traditional LIBOR market model for dealing with the volatility skew effect. Besides, the procedure of calibration to SABR-LMM model is more efficient than simulation due to the SABR formula. And it’s more elasticity on operation compare to LIBOR market model.
Finally, we simulated the forward rates and stochastic volatility dynamics under SABR-LMM model by using Monte Carlo method, and apply the result to CMS relative products pricing.
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Shih, Yu-Ju, and 施郁如. "Pricing and Risk Mangement of Foreign Currency Derivative on Stochastic Interest Rate and Volatility Model：The Case Study on Chunghwa Telecom Co." Thesis, 2015. http://ndltd.ncl.edu.tw/handle/q3yd3f.
Full textAbstract:
碩士<br>國立臺灣科技大學<br>財務金融研究所<br>103<br>The paper develops a stochastic interest rate and volatility model. The short rates follow CIR model and the volatility of exchange rate follow Heston model. Discuss the pricing and the risk value of foreign currency derivative on the stochastic model. With the case study, the Chunghwa Telecom Co. event occurred 4 billion Taiwan dollars unrealized loss in 2008. We will discuss in the perspective of Chunghwa Telecom Co. and try to find a better way to hedge foreign exchange rate. Because of the flexibility to adjust the conditions of contract, we provide some way to change the conditions of contract. However, we also advise to buy forward for hedge. Then, use Monte-Carlo method to evaluate the value and the risk of all the contract. Compare each value and risk of contract. The model we use is more difficult to estimate the parameters, but the results have shown that stochastic volatility model will facilitate the evaluation of foreign exchange option. The condition of contract change to reduce the down side risk will help to gain the value of contract.