This thesis makes a number of contributions in the derivative pricing and risk management literature and to the growing literature that exploits information embedded in option prices. First, it develops an effective numerical scheme for importance sampling scheme of Fouque and Tullie (2002) based on a 2-dimensional lookup table of stock price and time to maturity that dramatically improves the speed of this importance sampling scheme. Second, the thesis presents an application of this importance sampling scheme in a Multi-Level Monte Carlo simulation. Such combination yields greater variance reduction compared to Multi-Level Monte Carlo or importance sampling alone. Third, it demonstrates how the Greeks can be computed using the Likelihood Ratio Method based on characteristic function, and how combining it with importance sampling leads to a significant variance reduction for the Greeks. Finally, it documents the positive relationship between the risk-neutral skewness (RNS) and future realized stock returns that is driven by the underperformance of highly negative RNS portfolio. The results provide strong evidence that the underperformance of stocks with the lowest RNS is driven by those stocks that are associated with a higher hedging demand, relative overvaluation and are also too costly or too risky to sell short. Moreover, by decomposing RNS into its systematic and idiosyncratic components, this thesis shows that the latter drives the positive relationship with future realized stock returns.

No description available.

No description available.

Segmentation of magnetic resonance images is an important part of planning radiotherapy treat-ments for patients with brain tumours but due to the number of images contained within a scan and the level of detail required, manual segmentation is a time consuming task. Convolutional neural networks have been proposed as tools for automated segmentation and shown promising results. However, the data sets used for training these deep learning models are often imbalanced and contain data that does not contribute to the performance of the model. By carefully selecting which data to train on, there is potential to both speed up the training and increase the network’s ability to detect tumours. This thesis implements the method of importance sampling for training a convolutional neural network for patch-based segmentation of three dimensional multimodal magnetic resonance images of the brain and compares it with the standard way of sampling in terms of network performance and training time. Training is done for two different patch sizes. Features of the most frequently sampled volumes are also analysed. Importance sampling is found to speed up training in terms of number of epochs and also yield models with improved performance. Analysis of the sampling trends indicate that when patches are large, small tumours are somewhat frequently trained on, however more investigation is needed to confirm what features may influence the sampling frequency of a patch. / Segmentering av magnetröntgenbilder är en viktig del i planeringen av strålbehandling av patienter med hjärntumörer. Det höga antalet bilder och den nödvändiga precisionsnivån gör dock manuellsegmentering till en tidskrävande uppgift. Faltningsnätverk har därför föreslagits som ett verktyg förautomatiserad segmentering och visat lovande resultat. Datamängderna som används för att träna dessa djupinlärningsmodeller är ofta obalanserade och innehåller data som inte bidrar till modellensprestanda. Det finns därför potential att både skynda på träningen och förbättra nätverkets förmåga att segmentera tumörer genom att noggrant välja vilken data som används för träning. Denna uppsats implementerar importance sampling för att träna ett faltningsnätverk för patch-baserad segmentering av tredimensionella multimodala magnetröntgenbilder av hjärnan. Modellensträningstid och prestanda jämförs mot ett nätverk tränat med standardmetoden. Detta görs förtvå olika storlekar på patches. Egenskaperna hos de mest valda volymerna analyseras också. Importance sampling uppvisar en snabbare träningsprocess med avseende på antal epoker och resulterar också i modeller med högre prestanda. Analys av de oftast valda volymerna indikerar att under träning med stora patches förekommer små tumörer i en något högre utsträckning. Vidareundersökningar är dock nödvändiga för att bekräfta vilka aspekter som påverkar hur ofta en volym används.

Payment service providers have unique credit portfolios with different characteristics than many other credit providers. It is therefore important to study if common credit risk estimation methods are applicable to their setting. By comparing simulation based methods for credit risk estimation it was found that combining Monte Carlo simulation with importance sampling and the asymptotic single risk factor model is the most suitable model amongst those analyzed. It allows for a combination of variance reduction, scenario analysis and correlation checks, which all are important for estimating credit risk in a payment service provider portfolio. / Betaltjänstleverantörer har unika kreditportföljer med andra egenskaper än många andra kreditgivare. Det är därför viktigt att undersöka om vanliga metoder för uppskattning av kreditrisk går att tillämpa på betaltjänstleverantörer. Genom att jämföra olika simulationsbaserade metoder för uppskattning av kreditrisk fann man att att kombinationen av Monte Carlo-simulering med Importance Sampling och en ASRF-modell är den mest lämpliga bland de analyserade metoderna. Det möjliggör en kombination av variansminskning, scenarioanalys och korrelationskontroller som alla är viktiga för att uppskatta kreditrisk i en betaltjänstleverantörsportfölj.

Mémoire numérisé par la Division de la gestion de documents et des archives de l'Université de Montréal.

Mémoire numérisé par la Division de la gestion de documents et des archives de l'Université de Montréal

Pricing different financial derivatives is an essential part of the financial industry. For some derivatives there exists a closed form solution, however the pricing of high-dimensional American-style derivatives is still today a challenging problem. This project focuses on the derivative called option and especially pricing of American-style basket options, i.e. options with both an early exercise feature and multiple underlying assets. In high-dimensional problems, which is definitely the case for American-style options, Monte Carlo methods is advantageous. Therefore, in this thesis, regression-based Monte Carlo has been used to determine early exercise strategies for the option. The well known Least Squares Monte Carlo (LSM) algorithm of Longstaff and Schwartz (2001) has been implemented and compared to Robust Regression Monte Carlo (RRM) by C.Jonen (2011). The difference between these methods is that robust regression is used instead of least square regression to calculate continuation values of American style options. Since robust regression is more stable against outliers the result using this approach is claimed by C.Jonen to give better estimations of the option price. It was hard to compare the techniques without the duality approach of Andersen and Broadie (2004) therefore this method was added. The numerical tests then indicate that the exercise strategy determined using RRM produces a higher lower bound and a tighter upper bound compared to LSM. The difference between upper and lower bound could be up to 4 times smaller using RRM. Importance sampling and Quasi Monte Carlo have also been used to reduce the variance in the estimation of the option price and to speed up the convergence rate. / Prissättning av olika finansiella derivat är en viktig del av den finansiella sektorn. För vissa derivat existerar en sluten lösning, men prissättningen av derivat med hög dimensionalitet och av amerikansk stil är fortfarande ett utmanande problem. Detta projekt fokuserar på derivatet som kallas option och särskilt prissättningen av amerikanska korg optioner, dvs optioner som både kan avslutas i förtid och som bygger på flera underliggande tillgångar. För problem med hög dimensionalitet, vilket definitivt är fallet för optioner av amerikansk stil, är Monte Carlo metoder fördelaktiga. I detta examensarbete har därför regressions baserad Monte Carlo använts för att bestämma avslutningsstrategier för optionen. Den välkända minsta kvadrat Monte Carlo (LSM) algoritmen av Longstaff och Schwartz (2001) har implementerats och jämförts med Robust Regression Monte Carlo (RRM) av C.Jonen (2011). Skillnaden mellan metoderna är att robust regression används istället för minsta kvadratmetoden för att beräkna fortsättningsvärden för optioner av amerikansk stil. Eftersom robust regression är mer stabil mot avvikande värden påstår C.Jonen att denna metod ger bättre skattingar av optionspriset. Det var svårt att jämföra teknikerna utan tillvägagångssättet med dualitet av Andersen och Broadie (2004) därför lades denna metod till. De numeriska testerna indikerar då att avslutningsstrategin som bestämts med RRM producerar en högre undre gräns och en snävare övre gräns jämfört med LSM. Skillnaden mellan övre och undre gränsen kunde vara upp till 4 gånger mindre med RRM. Importance sampling och Quasi Monte Carlo har också använts för att reducera variansen i skattningen av optionspriset och för att påskynda konvergenshastigheten.

各項信用衍生性商品中，最廣為人知的商品即為違約風險交換(credit default swap; CDS)，但由於金融市場與商品的擴張，標的資產不再侷限單一資產而是增加至數家或數百家，而多個標的資產的違約風險交換稱為一籃子違約風險交換(basket default swap; BDS)。 根據Chiang et al. ((2007), Journal of Derivatives, 8-19.)，在單因子模型中應用importance sampling (IS) 來估計違約給付金額，不僅可以確保違約事件的發生，還可以提高估計的效率，因此本文延伸此一概念，將此方法拓展至多因子模型。本文分為三種方法：一為將多個獨立因子合併為一邊際因子，並針對此邊際因子做importance sampling；二為找出其最具影響性的因子應用importance sampling；最後，我們針對portfolio C 於Glasserman ((2004), Journal of Derivatives, 24-42.) 將標的資產分為獨立兩群，我們將分段利用exponential twist及Chiang et al. (2007)所提出的單因子方法，提升違約事件發生的機率。 借由數值模擬結果，發現將多個獨立因子合併為一邊際因子的方法應用於標的資產為同質模型(homogeneous model)，會有較佳的結果；對具影響性的因子應用importance sampling的方法於各種模型之下的估計結果都頗為優秀，但其variance reduction較差且流程較不符合現實財務狀況，方法三則為特殊模型的應用，其只適用於能將標的資產獨立分群的模型，並且估計準確與否和選取exponential twist的位置有重要關係，第四節我們將同時呈現兩個不同位置的估計值與variance reduction. / Credit default swap (CDS) is the most popular in many kinds of credit derivatives, but number of obligor couldn’t be one always because of the expansions of financial market and contracts. CDS which has been contained more than one obligor is called basket default swap (BDS). According to Chiang et al. ((2007), Journal of Derivatives, 8-19.), applying importance sampling to estimate the default payment in one factor model could not only guarantee the default event occurs but also improve the efficiency of estimation. So this paper extends this concept for expanding this method to multiple factors model. There are three methods for expanding: First, merge multiple factors into a marginal factor and apply importance sampling to this marginal factor; second, apply importance sampling to the factor which has higher factor loading and third, we consider portfolio C in Glasserman ((2004), Journal of Derivatives, 24-42.) and divide total obligors into two independent groups. We would use the ways of exponential twist and the method in one factor model of Chiang et al. (2007) considered in two parts to raise the probability of default event occur. Borrow by the result of numerical simulation, method 1 has better results when obligors are homogeneous model; the results of method 2 are outstanding in each model, but its efficiency is worse and the procedure doesn’t fit with the realistic financial situation; the third method is the application of the special model, it could only apply to the model which could separate obligors independently, and the accuracy of estimates is strongly correlated to the position of exponential twist. In section 4, we would display the estimator and variance reduction in two different positions.

Importance sampling is a well known variance reduction technique for Monte Carlo simulation. For quasi-Monte Carlo integration with low discrepancy sequences it was neglected in the literature although it is easy to see that it can reduce the variation of the integrand for many important integration problems. For lattice rules importance sampling is of highest importance as it can be used to obtain a smooth periodic integrand. Thus the convergence of the integration procedure is accelerated. This can clearly speed up QMC algorithms for integration problems up to dimensions 10 to 12. (author's abstract) / Series: Research Report Series / Department of Statistics and Mathematics