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. / 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

Hydro power is the largest source for generation of electricity in the Nordic region today. This production is heavily dependent on the weather since it dictates the terms for the availability and the amount of power to be produced. Vattenfall as a company has an incentive to avoid volatile revenue streams as it facilitates economic planning and induces a positive effect on its credit rating, thus also on its bottom line. Vattenfall is a large producer of hydro power with a possibility to move the power market which adds further complexity to the problem. In this thesis the authors develop new hedging strategies which will hedge more efficiently. With efficiency is meant the same risk, or standard deviation, at a lower cost or alternatively formulated lower risk for the same cost. In order to enable comparison and make claims about efficiency, a reference solution is developed that should reflect their current hedging strategy. To achieve higher efficiency we focus on finding dynamic hedging strategies. First a prototype model is suggested to facilitate the construction of the solution methods and if it is worthwhile to pursue a further investigation. As this initial prototype model results showed that there were substantial room for efficiency improvement, a larger main model with parameters estimated from data is constructed which encapsulate the real world scenario much better. Four different solutions methods are developed and applied to this main model setup. The results are then compared to reference strategy. We find that even though the efficiency was less then first expected from the prototype model results, using these new hedging strategies could reduce costs by 1.5 % - 5%. Although the final choice of the hedging strategy might be down to the end user we suggest the strategy called BW to reduce costs and improve efficiency. The paper also discusses among other things; the solution methods and hedging strategies, the term optimality and the impact of parameters in the model.

There are many ways of modeling stochastic processes of short-term interest rates. One way is to use one-factor models which may be easy to use and easy to calibrate. Another way is to use a three-factor model in the strive for a higher degree of congruency with real world market data. Calibrating such models may however take much more effort. One of the main questions here is which models will be better fit to the data in question. Another question is if the use of a three-factor model can result in better fitting compared to one-factor models. This is investigated by using the Efficient Method of Moments to calibrate a three-factor model with a Lévy process. This model is an extension of the Chen Model. The calibration is done with Euribor 6-month interest rates and these rates are also used with the Vasicek and Cox-Ingersoll-Ross (CIR) models. These two models are calibrated by using Maximum Likelihood Estimation and they are one-factor models. Chi-square goodness-of-fit tests are also performed for all models. The findings indicate that the Vasicek and CIR models fail to describe the stochastic process of the Euribor 6-month rate. However, the result from the goodness-of-fit test of the three-factor model gives support for that model.

Coronary artery heart disease (CAD) is a common condition which can impair the quality of life and lead to cardiac infarctions. Traditional criteria during exercise tests are good but far from perfect. A lot of patients with inconclusive tests are referred to radiological examinations. By finding better evaluation criteria during the exercise test we can save a lot of money and let the patients avoid unnecessary examinations. Computers record amounts of numerical data during the exercise test. In this retrospective study 267 patients with inconclusive exercise test and performed radiological examinations were included. The purpose was to use clinical considerations as-well as mathematical statistics to be able to find new diagnostic criteria. We created a few new parameters and evaluated them together with previously used parameters. For women we found some interesting univariable results where new parameters discriminated better than the formerly used. However, the number of females with observed CAD was small (14) which made it impossible to obtain strong significance. For men we computed a multivariable model, using logistic regression, which discriminates way better than the traditional parameters for these patients. The area under the ROC curve was 0:90 (95 % CI: 0.83-0.97) which is excellent to outstanding discrimination in a group initially included due to their inconclusive results. If the model can be proved to hold for another population it could contribute a lot to the diagnostics of this common medical conditions

In this thesis a model for predicting a person’s monthly average of self-rated health the following month was developed. It was based on statistics from a form constructed by HealthWatch. The model used is a Hidden Markov Algorithm based on Hidden Markov Models where the hidden part is the future value of self-rated health. The emissions were based on five of the eleven questions that make the HealthWatch form. The questions are answered on a scale from zero to one hundred. The model predicts in which of three intervals of SRH the responder most likely will answer on average during the following month. The final model has an accuracy of 80 %.

In this thesis a corporate bond valuation model based on Dick-Nielsen, Feldhütter, and Lando (2011) and Chen, Lesmond, and Wei (2007) is examined. The aim is for the model to price corporate bond spreads and in particular capture the price effects of liquidity as well as credit risk. The valuation model is based on linear regression and is conducted on the Swedish market with data provided by Handelsbanken. Two measures of liquidity are analyzed: the bid-ask spread and the zero-trading days. The investigation shows that the bid-ask spread outperforms the zero-trading days in both significance and robustness. The valuation model with the bid-ask spread explains 59% of the cross-sectional variation and has a standard error of 56 bps in its pricing predictions of corporate spreads. A reduced version of the valuation model is also developed to address simplicity and target a larger group of users. The reduced model is shown to maintain a large proportion of the explanation power while including fewer and simpler variables. / I denna uppsats undersöks en värderingsmodell för företagsobligationer, baserad på studierna av Dick-Nielsen, Feldhütter, och Lando (2011) och Chen, Lesmond, och Wei (2007). Syftet med modellen är att kunna prissätta företagsobligationer med precision och i synnerhet hantera priseffekten av likviditet och kreditrisk. Värderingsmodellen är baserad på linjär regression och är tillämpad på den svenska marknaden. Den underliggande datan i undersökningen är tillhandahållen av Handelsbanken. Två mått av likviditet är analyserade: bid-ask-spreaden och noll-handlingsdagarna. Undersökningen visar att likviditetsmåttet för bid-ask-spreaden överträffar måttet för noll-handlingsdagarna i både signifikans och robusthet. Värderingsmodellen, med bid-ask-spreaden som likviditetsmått, förklarar 59% av variationen, mätt i justerat r-kvadrat värde. Standardfelet för modellen är 56 baspunkter. Vidare utvecklas också en reducerad version av värderingsmodellen i syfte att vara mer praktiskt användbar och tillgänglig för en större användargrupp. Undersökningen visar att den reducerade modellen bibehåller en stor del av förklaringsgraden av den ursprungliga modellen, samt att den inkluderar färre och enklare variabler.

This Master's Thesis in mathematical statistics has the two major purposes; (i) to model and measure the risk associated with a special type of reinsurance contract, the catastrophe bond, and (ii) to analyze and develop methods of portfolio optimization suitable for a portfolio of catastrophe bonds. Two pathways of modeling potential catastrophe bond losses are analyzed; one method directly modeling potential contract losses and one method modeling the underlying contract loss governing variables. The first method is simple in its structure but with the disadvantage of the inability to introduce a dependence structure between the losses of different contracts in a simple and flexible way. The second modeling method uses a stochastic number of stochastic events representation connected into a multivariate dependence structure using the theory of copulas. Results show that the choice of risk measure is of great importance when analyzing catastrophe bonds and their related risks. As an example, the measure Value at Risk often fails to capture the essence of catastrophe bond risk, which in turn means that portfolio optimization with respect to the same might lead to a systematic obscurity of risk. Two coherent risk measures were investigated, the spectral risk measure and the Expected Shortfall measure. Both measures provides good representation of the risk of a portfolio consisting of catastrophe bonds. This thesis extends and applies a well-known optimization method of Conditional Value at Risk to obtain a method of optimization of spectral risk measures. The optimization results show that expected shortfall optimization leads to portfolios being advantageous at the specific point at which it is optimized but that their characteristics may be disadvantageous at other parts of the loss distribution. Portfolios optimized for the spectral risk measure were shown to possess good characteristics across the entire loss distribution. Optimization results were compared to the popular mean-variance portfolio optimization approach. The comparison shows that the mean-variance approach handles the special distribution of catastrophe bond losses in an over-simplistic way, and that it has a severe lack of flexibility towards focusing on different aspects of risk. The spectral risk measure optimization procedure was demonstrated to be the most flexible and possibly the most appropriate way to optimize a portfolio of catastrophe bonds. / Detta examensarbete i matematisk statistik har de två huvudsyftena att; (i) modellera och mäta riskerna med en speciell typ av återförsäkringskontrakt, katastrofobligationer, och (ii) att analysera och utveckla metoder för optimering som lämpar sig för en portfölj av katastrofobligationer. Två olika inriktningar för att modellera förluster knutna till katastrofobligationer analyseras; en metod som direkt beskriver kontraktsförlusterna och en metod som beskriver de underliggande variablerna som leder till kontraktsförluster. Den första metoden är enkel till sin struktur, men med nackdelen att det inte ar möjligt att införa beroenden mellan olika kontrakts förluster på ett enkelt och flexibelt sätt. Den andra metoden använder ett betraktningssätt med ett stokastiskt antal stokastiska händelser kopplad till en beroendestrukturmed med hjälp av copulas. Resultaten visar att valet av riskmått är av stor betydelse vid analys av katastrofobligationer och dess tillhörande risker. Som ett exempel, måttet Value at Risk misslyckas att mäta risk i era fall vilket i sin tur innebär att portföljoptimering med avseende på densamma skulle kunna leda till ett systematisk döljande av risk. Två koherenta riskmått befanns vara tillfredsställande för att mäta risk relaterad till katastrofobligationer, Expected Shortfall och genom ett spektralt riskmått.   Detta examensarbete använder och tillämpar en välkänd optimeringsmetod för Conditional Value at Risk för att erhålla en metod för optimering av spektrala riskmått. Optimeringsresultaten visar att expected shortfall optimering leder till portföljer som är fördelaktiga vid den punkt där optimering skett, men att deras egenskaper kan vara långt sämre i andra delar av förlustfördelningen. Portföljer optimerade för det spektrala riskmåttet visade sig ha goda egenskaper över hela förlustfördelningen. De analyserade optimeringsmetoderna jämfördes med den populära optimeringsmetoden att minimera varians. Jämförelsen visar att varians hanterar den särskilda fördelningen av katastrofobligationsförluster på ett alltför förenklat sätt. Att minimera varians har en allvarlig brist på flexibilitet i och med att det inte är möjligt att fokusera på olika delar av fördelningen. Optimering av det spektrala riskmåttet visade sig vara det mest flexibla och kanske det bästa sättet att optimera en portfölj av katastrofobligationer

The aim of this study is to find a method that is able to locate multiple change-points in a time series with unknown properties. The methods that are investigated are the CUSUM and CUSUM of squares test, the CUSUM test with OLS residuals, the Mann-Whitney test and Quandt’s log likelihood ratio. Since all methods are detecting single change-points, the binary segmentation technique is used to find multiple change-points. The study shows that the CUSUM test with OLS residuals, Mann-Whitney test and Quandt’s log likelihood ratio work well on most samples while the CUSUM and CUSUM of squares are not able to detect the location of the change-points. Furthermore the study shows that the binary segmentation technique works well with all methods and is able to detect multiple change-points in most circumstances. The study also shows that the results can, most of the time, be improved by using a combination of the methods. / Syftet med studien är att hitta en metod som identifierar tidpunkterna för strukturella brott i en tidsserie med okända egenskaper. De metoder som undersöks är CUSUM och CUSUM av kvadrater, CUSUM test med OLS-residualer, Mann-Whitney-test samt Quandts log likelihood ratio. Eftersom alla metoder identifierar enbart en brytpunkt används binära uppdelningstekniken för att hitta multipla brytpunkter. Studien visar att CUSUM-test med OLS-residualer, Mann-Whitney-test och Quandt’s log likelihood ratio fungerar bra för de flesta stickproven medan CUSUM och CUSUM av kvadrater inte hittar tidpunkten för brytpunkterna. Vidare så visar studien att binära uppdelningstekniken fungerar bra med alla metoder och kan identifiera multipla brytpunkter i de flesta fallen. Studien visar också att resultaten för det mesta kan förbättras genom att använda en kombination av metoderna.

Det här kandidatexamensarbetet inom matematisk statistik och industriell ekonomi undersöker förklarande faktorer bakom ränteskillnad, så kallad spread, mellan amerikanska och tyska statsobligationer. De obligationer som undersöks har en löptid på fem respektive tio år. Den huvudsakliga analysen görs genom multipel linjär regression. Regressionsmodellering av statsobligationsspread är vanligt förekommande i bankvärlden och det här kandidatexamensarbetet kan utgöra en grund för vidare modellering och förbättring av befintliga modeller. Problemformuleringen och arbetsupplägget har utformats i samarbete med en svensk bank, som inte nämns vid namn på grund av sekretesskäl. Kandidatexamensarbetet består av två huvuddelar. Den första delen inom industriell ekonomi, där det undersöks vilka makroekonomiska faktorer som är relevanta. Dessa faktorer utgör även den statistiska kontexten, varför den första delens vikt inte ska underskattas. I den andra delen, inom matematisk statistik, undersöks de framtagna faktorerna genom multipel linjär regression och relaterade statistiska tester. Resultaten indikerar att skillnaden i styrränta mellan länderna är den mest signifikanta variabeln, och utgör en bra grund för att i stora drag beskriva statsobligationsspread. För de senaste fem åren påvisas en viss skillnad, då styrräntan ensam inte förklarar obligationsspread lika bra och andra faktorer istället får större vikt i modellen. / This bachelor’s thesis in Mathematical Statistics and Industrial Economics aims to determine explanatory variables of yield spread between U.S. and German government bonds. The bonds used in this thesis have maturities of five and ten years. To accomplish the task at hand, a multiple linear regression model is used. Regression models are commonly used to describe government bond spread, and this bachelor’s thesis aims to create a basis for further modeling and contribute to improvement of existing models. The problem formulation and course of action have been developed in cooperation with a Swedish bank, not named for reasons of confidentiality. Two main parts constitute this bachelor’s thesis. The Industrial Economics part investigates which macroeconomic factors are of interest in order to create the model. The economics are (in this case) the statistical context, which emphasizes the importance of this part. For the mathematical part of the thesis, a multiple linear regression and related statistical tests are performed on the chosen variables. The results of these tests indicate that the policy rate spread between the countries is the most significant variable, and in itself describes the government bond spread quite well. However, the policy rate does not seem to describe the bond spread as well regarding the last five years. This gives a hint that the importance of the variable policy spread is diminishing, while the importance of other factors is increasing.

This thesis approaches the problem of modeling the multivariate distribution of interest rates by implementing a novel tool of statistics known as functional data analysis (FDA). This is done by viewing yield curve shifts as distinct but continuous stochastic objects defined over a continuum of maturities. Based on these techniques, we provide two stochastic models with different assumptions regarding the temporal dependence of yield curve shifts and compare their performance with empirical data. The study finds that both models replicate the distributions of yield changes with medium- and long-term maturities, whereas none of the models perform satisfactory at the short segment of the yield curve. Both models, however, appear to accurately capture the cross-sectional dependence.