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This thesis covers the topics of adjusting financial contracts for the cost of counterparty credit risk (CVA) and funding costs (FVA). A secondary topic is tackling the lack of a risk-free rate for discounting. Recent evolution in finance has shown that the value of a derivative is impacted by real world frictions. These frictions are specific to each party, and so are the valuation adjustments (xVA). Quantifying the impact on the contract value, auxiliary models for, among others, counterparty default and funding spreads are needed. Building, merging and running these auxiliary models are the responsibility of xVA desks, which have been allocated increasing resources in recent years. The main purpose of this thesis is to establish a unifying framework for calculating CVA and FVA. The central example throughout this paper is pricing of an interest rate swap (IRS) against ISS Global A/S (ISS), and quantifying counterparty credit risk and funding cost. Advancing from theoretically pricing the IRS, we calibrate a short rate Hull-White (HW) model to the swap market, develop and calibrate a Cox-Ingersoll-Ross (CIR) model for default simulation and finally establish a Piterbarg-like setup for stochastic funding. Ultimately all models are merged in a fully stochastic framework, to estimate the impact of counterparty credit risk and funding. Throughout, we treat issues such as; finding the appropriate proxy for a risk-free rate, pricing in the presence of multiple curves, calibration of models, effects of collateralization, portfolio CVA and FVA aggregation, wrong- and right-way risk and computational issues relating to xVA’s. All models are implemented in MATLAB and the intuitive understanding of every theoretical result is emphasized. Throughout the paper we support complex discussions, structures and results by visual representations. In a setup with a repo market, funding a, This thesis covers the topics of adjusting financial contracts for the cost of counterparty credit risk (CVA) and funding costs (FVA). A secondary topic is tackling the lack of a risk-free rate for discounting. Recent evolution in finance has shown that the value of a derivative is impacted by real world frictions. These frictions are specific to each party, and so are the valuation adjustments (xVA). Quantifying the impact on the contract value, auxiliary models for, among others, counterparty default and funding spreads are needed. Building, merging and running these auxiliary models are the responsibility of xVA desks, which have been allocated increasing resources in recent years. The main purpose of this thesis is to establish a unifying framework for calculating CVA and FVA. The central example throughout this paper is pricing of an interest rate swap (IRS) against ISS Global A/S (ISS), and quantifying counterparty credit risk and funding cost. Advancing from theoretically pricing the IRS, we calibrate a short rate Hull-White (HW) model to the swap market, develop and calibrate a Cox-Ingersoll-Ross (CIR) model for default simulation and finally establish a Piterbarg-like setup for stochastic funding. Ultimately all models are merged in a fully stochastic framework, to estimate the impact of counterparty credit risk and funding. Throughout, we treat issues such as; finding the appropriate proxy for a risk-free rate, pricing in the presence of multiple curves, calibration of models, effects of collateralization, portfolio CVA and FVA aggregation, wrong- and right-way risk and computational issues relating to xVA’s. All models are implemented in MATLAB and the intuitive understanding of every theoretical result is emphasized. Throughout the paper we support complex discussions, structures and results by visual representations. In a setup with a repo market, funding and collateral, we are able to

This thesis covers the topics of adjusting financial contracts for the cost of counterparty credit risk (CVA) and funding costs (FVA). A secondary topic is tackling the lack of a risk-free rate for discounting. Recent evolution in finance has shown that the value of a derivative is impacted by real world frictions. These frictions are specific to each party, and so are the valuation adjustments (xVA). Quantifying the impact on the contract value, auxiliary models for, among others, counterparty default and funding spreads are needed. Building, merging and running these auxiliary models are the responsibility of xVA desks, which have been allocated increasing resources in recent years. The main purpose of this thesis is to establish a unifying framework for calculating CVA and FVA. The central example throughout this paper is pricing of an interest rate swap (IRS) against ISS Global A/S (ISS), and quantifying counterparty credit risk and funding cost. Advancing from theoretically pricing the IRS, we calibrate a short rate Hull-White (HW) model to the swap market, develop and calibrate a Cox-Ingersoll-Ross (CIR) model for default simulation and finally establish a Piterbarg-like setup for stochastic funding. Ultimately all models are merged in a fully stochastic framework, to estimate the impact of counterparty credit risk and funding. Throughout, we treat issues such as; finding the appropriate proxy for a risk-free rate, pricing in the presence of multiple curves, calibration of models, effects of collateralization, portfolio CVA and FVA aggregation, wrong- and right-way risk and computational issues relating to xVA’s. All models are implemented in MATLAB and the intuitive understanding of every theoretical result is emphasized. Throughout the paper we support complex discussions, structures and results by visual representations. In a setup with a repo market, funding a, This thesis covers the topics of adjusting financial contracts for the cost of counterparty credit risk (CVA) and funding costs (FVA). A secondary topic is tackling the lack of a risk-free rate for discounting. Recent evolution in finance has shown that the value of a derivative is impacted by real world frictions. These frictions are specific to each party, and so are the valuation adjustments (xVA). Quantifying the impact on the contract value, auxiliary models for, among others, counterparty default and funding spreads are needed. Building, merging and running these auxiliary models are the responsibility of xVA desks, which have been allocated increasing resources in recent years. The main purpose of this thesis is to establish a unifying framework for calculating CVA and FVA. The central example throughout this paper is pricing of an interest rate swap (IRS) against ISS Global A/S (ISS), and quantifying counterparty credit risk and funding cost. Advancing from theoretically pricing the IRS, we calibrate a short rate Hull-White (HW) model to the swap market, develop and calibrate a Cox-Ingersoll-Ross (CIR) model for default simulation and finally establish a Piterbarg-like setup for stochastic funding. Ultimately all models are merged in a fully stochastic framework, to estimate the impact of counterparty credit risk and funding. Throughout, we treat issues such as; finding the appropriate proxy for a risk-free rate, pricing in the presence of multiple curves, calibration of models, effects of collateralization, portfolio CVA and FVA aggregation, wrong- and right-way risk and computational issues relating to xVA’s. All models are implemented in MATLAB and the intuitive understanding of every theoretical result is emphasized. Throughout the paper we support complex discussions, structures and results by visual representations. In a setup with a repo market, funding and collateral, we are able to

I denne opgave ses på nogle af de valg en forsikringstager kan tage i forbindelse med tegning af livsforsikring. Første del af opgaven omhandler en et-livs model hvor et antal forskellige nytteoptimeringsproblemer beskrives matematisk og løses. Der ses på tre grader af kontrol af betalingskoefficienterne som består af en præmie, en dødssum og en pensionsudbetaling. Vi løser forsikringstagerens optimeringsproblem i tilfældet hvor han vælger sine betalingskoefficienter ved tegning hvorefter de holdes konstant, vi løser tilfældet hvor kontrollerne må ændres kontinuert frem til pensionstidspunktet men holdes konstant fra da af, sidst løser vi problemet hvor forsikringstageren kontinuert må omvælge sine kontroller helt frem til kontraktens sluttidspunkt. Vi laver numeriske beregninger på de optimale forbrug og ser at forbruget er aftagende sålænge forsikrings- tagerens subjektive diskonteringfaktor er større end den tekniske (og reelle) rente. Vi sammenholder med tilsvarende resultater fra en artikel af Kraft & Steffensen (2008) og finder ligheder men også væsentlige forskelle, hvor den største forskel nok består i at artiklen tilsyneladende beregner forbruget for fastholdt dødssum. Anden del af opgaven beskriver en to-livs model, hvor manden betaler præmier og beg- ge modtager pension. Vi finder deres optimale forbrug både for statiske og dynamiske kontroller. For de statiske sammenligner vi med et-livs modellen og for det dynamiske sammenholder vi vores resultater med en artikel af Bruhn & Steffensen (2010) som om- handler optimering af dynamisk forbrug, investering og livs-forsikring for en husstand., This paper deals with the choices of individuals who wish to take out life insurance. We start out in a one-life life-death model and provide a detailed mathematical description of a number of utility optimization problems that our policyholder could be facing. We solve the problem with static premium, death sum and pension payment, i.e. they are chosen at initiation and fixed thenceforth. We also solve the problem where the controls can be chosen dynamically, one case where the controls are fixed at pension time and one where they are dynamic also through retirement. We perform numerical calculations and find that all the optimal consumption rates are decreasing when the policyholders subjective discount rate is larger than the interest rate. Our results turn out to be more or less similar to results presented in an article by Kraft & Steffensen (2008), although there are some remarkable differences in our numerical findings which partly seems to be because the article do their calculations with fixed death sum. The second part of the paper describes the utility optimization problem of a married couple where the husband pays premium and both receive pension payments resulting in 5 controls. We find static controls in a ’standard’ way and in a more special way and find that the results of the first version can be reduced to the controls from the one-life model when eliminating the two-life aspects. We also find dynamic optimal controls for the couple, which we find confirmed by an article by Bruhn & Steffensen (2010).

I denne opgave ses på nogle af de valg en forsikringstager kan tage i forbindelse med tegning af livsforsikring. Første del af opgaven omhandler en et-livs model hvor et antal forskellige nytteoptimeringsproblemer beskrives matematisk og løses. Der ses på tre grader af kontrol af betalingskoefficienterne som består af en præmie, en dødssum og en pensionsudbetaling. Vi løser forsikringstagerens optimeringsproblem i tilfældet hvor han vælger sine betalingskoefficienter ved tegning hvorefter de holdes konstant, vi løser tilfældet hvor kontrollerne må ændres kontinuert frem til pensionstidspunktet men holdes konstant fra da af, sidst løser vi problemet hvor forsikringstageren kontinuert må omvælge sine kontroller helt frem til kontraktens sluttidspunkt. Vi laver numeriske beregninger på de optimale forbrug og ser at forbruget er aftagende sålænge forsikrings- tagerens subjektive diskonteringfaktor er større end den tekniske (og reelle) rente. Vi sammenholder med tilsvarende resultater fra en artikel af Kraft & Steffensen (2008) og finder ligheder men også væsentlige forskelle, hvor den største forskel nok består i at artiklen tilsyneladende beregner forbruget for fastholdt dødssum. Anden del af opgaven beskriver en to-livs model, hvor manden betaler præmier og beg- ge modtager pension. Vi finder deres optimale forbrug både for statiske og dynamiske kontroller. For de statiske sammenligner vi med et-livs modellen og for det dynamiske sammenholder vi vores resultater med en artikel af Bruhn & Steffensen (2010) som om- handler optimering af dynamisk forbrug, investering og livs-forsikring for en husstand., This paper deals with the choices of individuals who wish to take out life insurance. We start out in a one-life life-death model and provide a detailed mathematical description of a number of utility optimization problems that our policyholder could be facing. We solve the problem with static premium, death sum and pension payment, i.e. they are chosen at initiation and fixed thenceforth. We also solve the problem where the controls can be chosen dynamically, one case where the controls are fixed at pension time and one where they are dynamic also through retirement. We perform numerical calculations and find that all the optimal consumption rates are decreasing when the policyholders subjective discount rate is larger than the interest rate. Our results turn out to be more or less similar to results presented in an article by Kraft & Steffensen (2008), although there are some remarkable differences in our numerical findings which partly seems to be because the article do their calculations with fixed death sum. The second part of the paper describes the utility optimization problem of a married couple where the husband pays premium and both receive pension payments resulting in 5 controls. We find static controls in a ’standard’ way and in a more special way and find that the results of the first version can be reduced to the controls from the one-life model when eliminating the two-life aspects. We also find dynamic optimal controls for the couple, which we find confirmed by an article by Bruhn & Steffensen (2010).

Et af de entrale spørgsmål i algebraisk topologi er udregningen af homotopigrupperne for sfærer. En af hovedopdagelserne i dette område er at disse ikke afhænger af sfærens dimension for en høj nok dimension. Disse kaldes de stabile homotopigrupper for sfærer og selvom de er blevet udregnet for relativt små dimensioner er de generelt set ikke bestemt. På den anden side er homotopigrupperne for den ortogonale gruppe og den stabile version meget velkendte. Dette speciale følger overordnet Frank Adams' artikel J(X)-IV, men er mere selvstænding. VI begynder med at definere den nulte K-gruppe for en kompakt Hausdorffrum from grothendieck-konstruktionen af isormorfiklasser af komplekse vektorbundter. Den reducerede K-gruppe for et rum med base-punkt er herefter defineret og vi viser at denne kan udvides til en multiplikativ generaliseret kohomologi-teori ved brug af bott-periodicitet. Herefter introduceres adams-operationerne, som er afbildninger på K-teori, der er definerede fra ydre prukt af moduler. Vi viser at disse er homomorfier og udregner dem for en generator for K-teorien af en sfære af dimension 2n. For det projektive bundt definerer vi thom-komplekset of thom-klassen. Multiplikation med thom-klassen giver os thom-isomorfien. Ved hjælp af thom-isomorfien og adams-operationerne definerer vi de kannibalistiske klasser og derefter e-invarianten fra adams-operationerne. Herefter defineres den komplekse J-homomorfi fra homotopigrupperne for den stabile unitære gruppe ind i de stabile homotopigrupper for sfærer. Efter at have udregnet de kannibalistiske klasser og benyttet at afbildningskeglen er et thom-kompleks udregner vi e-varianten og finder en nedre grænse for ordnen af billedet af den komplekse J-homomorfi. Ved at definere K-teori over reelle vektor bundter udregner vi også en nedre grænse for ordnen af billedet af den reelle J-homomorfi fra homotopigrupperne fra den stabile ortogonale gruppe ind i de stabi, One of the central questions in algebraic topology is the computation of the homotopy groups of spheres. One of the main discoveries in this area is that these do not depend on the degree of the sphere for the degree high enough. These are called the stable homotopy groups of spheres and though they have been computed for relatively small degrees of the homotopy groups, but they are in general not very well known. On the other hand the homotopy groups of the orthognal group and its stable version are very well known. This thesis will overall follow Frank Adams' paper J(X)-IV paper, but making it more self-contained. We begin by defining the zero'th K-group for a compact Hausdorff space as the Grothendieck contruction of isomorphism classes of complex vector bundles. The reduced K-group for a pointed space is then defined and we show that we can extend the unreduced K-theory to a multiplicative generalized cohomology theory using Bott periodicity. Next we introduce the Adams operations which are maps on K-theory defined from the exterior powers of modules. We show that these are homomorphisms and we calculate them for a generator of the K-theory of the 2n-sphere. For the projective bundle we define the Thom space as and the Thom class. Multiplication with the Thom class gives us the Thom isomorphism. Using the Thom isomorphism and the Adams operations we define the cannibalistic classes and next the e-invariant from the Adams operations. Next we define the complex J-homomorphism from the homotopy groups of the stable unitary group into the stable homotopy groups of spheres. After calculating the cannibalistic classes and using that the mapping cone is a Thom space we calculate the e-invariant and find a lower bound on the order of image of the complex J-homomorphism. Then by defining K-theory over real vector bundles we calculate the order of the image of the real J-homomorphism from the homotopy groups of the s

Et af de entrale spørgsmål i algebraisk topologi er udregningen af homotopigrupperne for sfærer. En af hovedopdagelserne i dette område er at disse ikke afhænger af sfærens dimension for en høj nok dimension. Disse kaldes de stabile homotopigrupper for sfærer og selvom de er blevet udregnet for relativt små dimensioner er de generelt set ikke bestemt. På den anden side er homotopigrupperne for den ortogonale gruppe og den stabile version meget velkendte. Dette speciale følger overordnet Frank Adams' artikel J(X)-IV, men er mere selvstænding. VI begynder med at definere den nulte K-gruppe for en kompakt Hausdorffrum from grothendieck-konstruktionen af isormorfiklasser af komplekse vektorbundter. Den reducerede K-gruppe for et rum med base-punkt er herefter defineret og vi viser at denne kan udvides til en multiplikativ generaliseret kohomologi-teori ved brug af bott-periodicitet. Herefter introduceres adams-operationerne, som er afbildninger på K-teori, der er definerede fra ydre prukt af moduler. Vi viser at disse er homomorfier og udregner dem for en generator for K-teorien af en sfære af dimension 2n. For det projektive bundt definerer vi thom-komplekset of thom-klassen. Multiplikation med thom-klassen giver os thom-isomorfien. Ved hjælp af thom-isomorfien og adams-operationerne definerer vi de kannibalistiske klasser og derefter e-invarianten fra adams-operationerne. Herefter defineres den komplekse J-homomorfi fra homotopigrupperne for den stabile unitære gruppe ind i de stabile homotopigrupper for sfærer. Efter at have udregnet de kannibalistiske klasser og benyttet at afbildningskeglen er et thom-kompleks udregner vi e-varianten og finder en nedre grænse for ordnen af billedet af den komplekse J-homomorfi. Ved at definere K-teori over reelle vektor bundter udregner vi også en nedre grænse for ordnen af billedet af den reelle J-homomorfi fra homotopigrupperne fra den stabile ortogonale gruppe ind i de stabi, One of the central questions in algebraic topology is the computation of the homotopy groups of spheres. One of the main discoveries in this area is that these do not depend on the degree of the sphere for the degree high enough. These are called the stable homotopy groups of spheres and though they have been computed for relatively small degrees of the homotopy groups, but they are in general not very well known. On the other hand the homotopy groups of the orthognal group and its stable version are very well known. This thesis will overall follow Frank Adams' paper J(X)-IV paper, but making it more self-contained. We begin by defining the zero'th K-group for a compact Hausdorff space as the Grothendieck contruction of isomorphism classes of complex vector bundles. The reduced K-group for a pointed space is then defined and we show that we can extend the unreduced K-theory to a multiplicative generalized cohomology theory using Bott periodicity. Next we introduce the Adams operations which are maps on K-theory defined from the exterior powers of modules. We show that these are homomorphisms and we calculate them for a generator of the K-theory of the 2n-sphere. For the projective bundle we define the Thom space as and the Thom class. Multiplication with the Thom class gives us the Thom isomorphism. Using the Thom isomorphism and the Adams operations we define the cannibalistic classes and next the e-invariant from the Adams operations. Next we define the complex J-homomorphism from the homotopy groups of the stable unitary group into the stable homotopy groups of spheres. After calculating the cannibalistic classes and using that the mapping cone is a Thom space we calculate the e-invariant and find a lower bound on the order of image of the complex J-homomorphism. Then by defining K-theory over real vector bundles we calculate the order of the image of the real J-homomorphism from the homotopy groups of the s

This thesis is inspired by a paper by Maryna S. Viazovska and studies the connection between modular forms and sphere packings. In particular, we prove that the E8 lattice packing is optimal in the sense that it gives a solution to the sphere packing problem in dimension 8. We start by introducing the problem and putting it in a historical context, we then proceed with a small section in which we introduce the E8 lattice and some general lattice theory. The next two sections built up a lot of theory from different areas of mathematics where we put a particularly strong emphasis on the theory of modular forms. This theory is then used in the following section which is more constructive in nature. All this theory from seemingly independent areas is then tied together in the last section and culminates in a proof, giving a solution to the sphere packing problem in dimension 8.

This thesis is inspired by a paper by Maryna S. Viazovska and studies the connection between modular forms and sphere packings. In particular, we prove that the E8 lattice packing is optimal in the sense that it gives a solution to the sphere packing problem in dimension 8. We start by introducing the problem and putting it in a historical context, we then proceed with a small section in which we introduce the E8 lattice and some general lattice theory. The next two sections built up a lot of theory from different areas of mathematics where we put a particularly strong emphasis on the theory of modular forms. This theory is then used in the following section which is more constructive in nature. All this theory from seemingly independent areas is then tied together in the last section and culminates in a proof, giving a solution to the sphere packing problem in dimension 8.

Indtil nu er der ikke udledt nogen analytisk prisformel for for amerikanske put optioner. Denne kendsgerning har ført til flere numeriske tilnærmelsesmetoder. I Black-Scholes-modellen har vi beskrevet værdifunktionen for den amerikanske put option som et optimalt stoppetidsproblem. Vi har derefter angivet løsningen af ​​det optimale stoppetidsproblem som et free-boundary problem. Dette free-boundary problem fører os til den free-boundary equation, der beskriver værdifunktionen gennem den optimale exercise boundary. Free-boundary equation er en kendt type af Volterra integralligning, som kan tilnærmes ved successive metoder. Til at løse free-boundary integralligningen har vi brugt en iterativ fixed-point metode til at tilnærme den optimale exercise boundary. Denne metode kræver et initial guess. Dette projekt handler specifikt om, hvordan forskellige initial guess påvirker den samlede beregningshastighed. Metoden vi præsenterer er vist til at kunne beregne amerikanske præmier op til en 8-cifret præcision, og vi diskuterer otte forskellige initial guess. Hvert initial guess vil enten kræve at løse et system med to ikke-lineære ligninger, løse én ikke-lineær ligning eller kræve ikke at løse nogen ligninger overhovedet. Ved at løse ikke-lineære ligninger reducerer vi det antal iterationer, der er nødvendige for en tilfredsstillende tilnærmelse. I vores implementering fandt vi, at den højeste beregningsmæssige hastighed opnås ved ikke at løse nogen ligninger og således arbejde med en ren iterativ metode. Vi fandt ud af, at et initial guess af konstant strike pris beregner den optimale boundary omkring dobbelt så hurtig som at løse én ikke-lineær ligning og tæt på en faktor 1000 hurtigere end vores mest krævende initial guess. Initial guess 'QD +', der løser én ikke-lineær ligning, er vist i litteraturen at være i stand til at overgå den rene iterative metode. Selvom vi ikke kunne eftervise dette, fandt vi et initial guess, som beregner hurtigere end QD + på alle præc, To this day no closed-form analytical price formula for American put options has been derived. This fact has lead to several numerical approximation methods. In the Black-Scholes model we have described the value function for the American put option as an optimal stopping problem. We have then stated the solution of the optimal stopping problem as a free-boundary problem. This free-boundary problem lead us to the free-boundary equation which describes the option value through the optimal exercise boundary. The free-boundary equation is a known type of Volterra integral equation, and it is approximable by successive methods. To solve the free-boundary integral equation we have used an iterative fixed-point method to approximate the optimal exercise boundary. This method requires an initial guess. The focus of this paper is specifically on how different initial guesses influence on the total computational speed. The method we present is shown to compute American premiums up to an 8 digit precision, and we discuss eight different initial guesses. Each initial guess will either require solving a system of two non-linear equations, solving one non-linear equation or require solving no equations at all. By solving non-linear equations we reduce the required number of iterations needed for a satisfactory approximation. In our implementation we found that the highest computational speed is obtained by solving no equations, and thus working with a pure iterative method. We found that an initial guess of constant strike price computed the optimal exercise boundary about twice as fast as solving one non-linear equation, and close to a factor 1000 faster than our most demanding initial guess. The initial guess ’QD+’, solving one non-linear equation, is in literature shown to be able to outperform the pure iterative method. Even though we were not able to justify this we found an initial guess which computes faster than the QD+ at every precision level, with the same level o

Indtil nu er der ikke udledt nogen analytisk prisformel for for amerikanske put optioner. Denne kendsgerning har ført til flere numeriske tilnærmelsesmetoder. I Black-Scholes-modellen har vi beskrevet værdifunktionen for den amerikanske put option som et optimalt stoppetidsproblem. Vi har derefter angivet løsningen af ​​det optimale stoppetidsproblem som et free-boundary problem. Dette free-boundary problem fører os til den free-boundary equation, der beskriver værdifunktionen gennem den optimale exercise boundary. Free-boundary equation er en kendt type af Volterra integralligning, som kan tilnærmes ved successive metoder. Til at løse free-boundary integralligningen har vi brugt en iterativ fixed-point metode til at tilnærme den optimale exercise boundary. Denne metode kræver et initial guess. Dette projekt handler specifikt om, hvordan forskellige initial guess påvirker den samlede beregningshastighed. Metoden vi præsenterer er vist til at kunne beregne amerikanske præmier op til en 8-cifret præcision, og vi diskuterer otte forskellige initial guess. Hvert initial guess vil enten kræve at løse et system med to ikke-lineære ligninger, løse én ikke-lineær ligning eller kræve ikke at løse nogen ligninger overhovedet. Ved at løse ikke-lineære ligninger reducerer vi det antal iterationer, der er nødvendige for en tilfredsstillende tilnærmelse. I vores implementering fandt vi, at den højeste beregningsmæssige hastighed opnås ved ikke at løse nogen ligninger og således arbejde med en ren iterativ metode. Vi fandt ud af, at et initial guess af konstant strike pris beregner den optimale boundary omkring dobbelt så hurtig som at løse én ikke-lineær ligning og tæt på en faktor 1000 hurtigere end vores mest krævende initial guess. Initial guess 'QD +', der løser én ikke-lineær ligning, er vist i litteraturen at være i stand til at overgå den rene iterative metode. Selvom vi ikke kunne eftervise dette, fandt vi et initial guess, som beregner hurtigere end QD + på alle præc, To this day no closed-form analytical price formula for American put options has been derived. This fact has lead to several numerical approximation methods. In the Black-Scholes model we have described the value function for the American put option as an optimal stopping problem. We have then stated the solution of the optimal stopping problem as a free-boundary problem. This free-boundary problem lead us to the free-boundary equation which describes the option value through the optimal exercise boundary. The free-boundary equation is a known type of Volterra integral equation, and it is approximable by successive methods. To solve the free-boundary integral equation we have used an iterative fixed-point method to approximate the optimal exercise boundary. This method requires an initial guess. The focus of this paper is specifically on how different initial guesses influence on the total computational speed. The method we present is shown to compute American premiums up to an 8 digit precision, and we discuss eight different initial guesses. Each initial guess will either require solving a system of two non-linear equations, solving one non-linear equation or require solving no equations at all. By solving non-linear equations we reduce the required number of iterations needed for a satisfactory approximation. In our implementation we found that the highest computational speed is obtained by solving no equations, and thus working with a pure iterative method. We found that an initial guess of constant strike price computed the optimal exercise boundary about twice as fast as solving one non-linear equation, and close to a factor 1000 faster than our most demanding initial guess. The initial guess ’QD+’, solving one non-linear equation, is in literature shown to be able to outperform the pure iterative method. Even though we were not able to justify this we found an initial guess which computes faster than the QD+ at every precision level, with the same level o

An irreducible $N\times N$-matrix $A$ over $\{0,1\}$ gives rise to a plethora of interesting mathematical objects. One can construct the \emph{one-sided topological Markov shift} $X_A$ on which the shift $\sigma_A$ acts. Alternatively, one can construct a $C^*$-algebra, the \emph{Cuntz-Krieger algebra} $\mathcal{A}$, or a groupoid $G_A$. In 2010, K. Matsumoto introduced the notion of \emph{continuous orbit equivalence} between topological Markov shifts and showed that this can be characterized in terms of $C^*$-isomorphisms between the Cuntz-Kriegers algebras which preserve a certain abelian $C^*$-subalgebra. In a recent paper, published in 2014, K. Matsumoto and H. Matui provided the final piece of a characterization of continuous orbit equivalence between Markov shifts. This includes the notions of $C^*$-algebras, groups and groupoids. In this thesis, we give a proof of this classification result. Along the way, we will see how to realize $\mathcal{A}$ as a groupoid $C^*$-algebra.

Dette speciale omhandler en computerimplementering af en spektralfølge, som er beskrevet af Totaro, der kan bruges til udregning af den rationelle kohomologi af et konfigurationsrum $F(X,n)$ af $n$-tupler af punkter i en glat kompleks projektiv kurve $X$ af genus $g$. Første kapitel introducerer nogle teoretiske forkundskaber for implementeringen. Første afsnit introducerer nogle grundlæggende algebraiske strukturer. Afsnit 2 introducerer kohomologi af topologiske rum. Afsnit 3 omhandler orienterbare mangfoldigheder og kohomologiske egenskaber ved disse. Afsnit 4 introducerer spektralfølger og afsnit 5 omhandler komplekse algebraiske varieteter. Afsnit 6 opsummerer kort resultaterne af Totaros artikel. Kapitel 2 omhandler detaljerne ved implementeringen af computerprogrammet. Afsnit 1 indeholder detaljer omkring implementeringen af de underliggende algebraer. Afsnit 2 omhandler detaljer ved elementoperationer i disse algebraer of afsnit 3 omhandler implementeringen af differentialet for spektralfølgen, samt udregningen af kohomologien. Appendiks A indeholder et par eksempler på resultater fra programmet., This thesis deals with a computer implementation of a spectral sequence by Totaro that can be used for the calculation of the rational cohomology of the configuration space $F(X,n)$ of ordered $n$-tuples of points in a smooth complex projective curve $X$ of genus $g$. The first chapter of the thesis introduces some theoretical prerequisites for the implementation. Section 1 introduces some basic algebraic objects used in the thesis. Section 2 introduces cohomology of spaces and the added algebraic structure coming from the cup product. Section 3 introduces manifolds and some cohomological properties of orientable manifolds. Section 4 briefly introduces spectral sequnces and section 5 deals with complex algebraic varieties, and finally section 6 briefly sums up the results of Totaro's paper. Chapter 2 contains the implementation details of the program. Section 1 contains details for the bases of the underlying algebras of the $E_2$-page of Totaro's spectral sequence. Section 2 contains details for operations with elements in these algebras and section 3 contains details for the differential of the spectral sequence and calculating cohomology of the algebra. Appendix A contains a few example outputs from the implementation.

Dette speciale omhandler en computerimplementering af en spektralfølge, som er beskrevet af Totaro, der kan bruges til udregning af den rationelle kohomologi af et konfigurationsrum $F(X,n)$ af $n$-tupler af punkter i en glat kompleks projektiv kurve $X$ af genus $g$. Første kapitel introducerer nogle teoretiske forkundskaber for implementeringen. Første afsnit introducerer nogle grundlæggende algebraiske strukturer. Afsnit 2 introducerer kohomologi af topologiske rum. Afsnit 3 omhandler orienterbare mangfoldigheder og kohomologiske egenskaber ved disse. Afsnit 4 introducerer spektralfølger og afsnit 5 omhandler komplekse algebraiske varieteter. Afsnit 6 opsummerer kort resultaterne af Totaros artikel. Kapitel 2 omhandler detaljerne ved implementeringen af computerprogrammet. Afsnit 1 indeholder detaljer omkring implementeringen af de underliggende algebraer. Afsnit 2 omhandler detaljer ved elementoperationer i disse algebraer of afsnit 3 omhandler implementeringen af differentialet for spektralfølgen, samt udregningen af kohomologien. Appendiks A indeholder et par eksempler på resultater fra programmet., This thesis deals with a computer implementation of a spectral sequence by Totaro that can be used for the calculation of the rational cohomology of the configuration space $F(X,n)$ of ordered $n$-tuples of points in a smooth complex projective curve $X$ of genus $g$. The first chapter of the thesis introduces some theoretical prerequisites for the implementation. Section 1 introduces some basic algebraic objects used in the thesis. Section 2 introduces cohomology of spaces and the added algebraic structure coming from the cup product. Section 3 introduces manifolds and some cohomological properties of orientable manifolds. Section 4 briefly introduces spectral sequnces and section 5 deals with complex algebraic varieties, and finally section 6 briefly sums up the results of Totaro's paper. Chapter 2 contains the implementation details of the program. Section 1 contains details for the bases of the underlying algebras of the $E_2$-page of Totaro's spectral sequence. Section 2 contains details for operations with elements in these algebras and section 3 contains details for the differential of the spectral sequence and calculating cohomology of the algebra. Appendix A contains a few example outputs from the implementation.

An irreducible $N\times N$-matrix $A$ over $\{0,1\}$ gives rise to a plethora of interesting mathematical objects. One can construct the \emph{one-sided topological Markov shift} $X_A$ on which the shift $\sigma_A$ acts. Alternatively, one can construct a $C^*$-algebra, the \emph{Cuntz-Krieger algebra} $\mathcal{A}$, or a groupoid $G_A$. In 2010, K. Matsumoto introduced the notion of \emph{continuous orbit equivalence} between topological Markov shifts and showed that this can be characterized in terms of $C^*$-isomorphisms between the Cuntz-Kriegers algebras which preserve a certain abelian $C^*$-subalgebra. In a recent paper, published in 2014, K. Matsumoto and H. Matui provided the final piece of a characterization of continuous orbit equivalence between Markov shifts. This includes the notions of $C^*$-algebras, groups and groupoids. In this thesis, we give a proof of this classification result. Along the way, we will see how to realize $\mathcal{A}$ as a groupoid $C^*$-algebra.

Dette speciale handler om parameter estimering. Vi anvender metoden introduceret i artiklen af Wood, S. N.; Statistical inference for noisy nonlinear ecological dynamic systems på en udvidet stokastisk Fitzhugh-Nagumo model, som introduceres i artiklen af Jensen, AC et al.; Markov chain Monte Carlo approach to parameter estimation in the FitzHugh-Nagumo model. Vi betragter først modellen og ser på egenskaberne for både den deterministiske model og den stokastiske model og vi studerer eksempler på de to versioner. I den næste del af specialet undersøger vi estimeringsproceduren der er beskrevet i artiklen af Wood. Til dette formål anvender vi den logaritmisk syntetiske likelihood, som vi skaber ved at bruge statistikker med helt specifikke egenskaber; vi indser, at likelihooden ligner likelihooden for normalfordelingen, hvis statistikkerne er approksimativt normalfordelt. Valget af statistikker er en iterativ proces, som grundlæggende består i at prøve sig frem. Efter at have set på likelihooden, introducerer vi Metropolis-Hasting metoden - en accepter/forkast algoritme. Vi etablerer den teoretiske ramme fra hvilken metoden er udviklet og vi undersøger den faktiske algoritme, som vi bruger til at simulere data. Tilsidst kombinerer vi modellen og metoden idet vi estimerer parametrene i modellen ved at bruge tre modifikationer af metoden samt data simuleret fra modellen. Resultaterne indikerer at det ikke er praktisk gennemførligt at estimere parametrene i Fitzhugh-Nagumo modellen. Denne konklusion opnåede vi efter at have simuleret næsten en tredjedel billion normalfordelte variable henover en periode på 512 timer (21 dage og 8 timer)., This master thesis is about parameter estimation. We use the method introduced in the article by Wood, S. N.; Statistical inference for noisy nonlinear ecological dynamic systems on an extended stochastic FitzHugh-Nagumo model, which is introduced in the paper by Jensen, A. C. et al.; Markov chain Monte Carlo approach to parameter estimation in the FitzHugh-Nagumo model. First we study the model and look at features for both the deterministic model and the stochastic model and we consider some examples of the two versions. In the next part of the thesis we study the estimation procedure described in the article by Wood. To this end we utilize the log synthetic likelihood which we create using statistics with very specific properties; We realize that the likelihood resembles the likelihood of a normal distribution if the statistics are approximately normally distributed. Choosing the statistics is an iterative process based on the fundamental trial and error approach. After studying the likelihood we introduce the Metropolis-Hasting method - an accept-reject algorithm. We establish the theoretical framework from which the method is developed and study the actual algorithm that we use to simulate data. Finally we combine the model and the method as we estimate the parameters in the model using three modifications of the method and data simulated from the model. The results indicate that using the Wood method is not a feasible way to estimate the parameters in the FitzHugh-Nagumo model. This conclusion was reached after having simulated almost a third of a trillion normally distributed variables over a period of 512 hours (21 days and 8 hours).

Specialet omhandler de mest relevante opdagelser, der er kommet af Søren Eilers' og Marius Dadarlats koncept om fuldstændig asymptotisk unitær ækvivalens i $KK$-teori. Specialet indeholder en forklarende gennemgang af artiklen hvor fuldstændig asymptotisk unitær ækvivalens er introduceret, og der bliver vist at dette fører til den samme $KK$-teori som Kasparovs. Da ækvivalensen var fastsat, tog Hyon Ho Lee det videre og fandt ud af at ækvivalensen kan være brugbar indenfor teorien om Fredholm operatorer, og definerede den essentielle codimension af et Cuntz par af projektioner til at være en klasse i $KK_h(A,B)$, og derved fik nødvendige betingelser for at løfte projektioner i corona algebraen til multiplikator algebraen. Til slut, kigger vi på M. Dadarlats definition af en pseudo-metrik som måler hvor langt Cuntz par er fra at være fuldstændig asymptotisk unitært ækvivalente, og vi viser dernæst at topologien induceret af denne pseudo-metrik på $KK(A,B)$ får $KK(A,B)$ til at blive fuldstændigt og separabelt., The thesis will be concerned with exploring the most relevant discoveries relating to Søren Eilers' and Marius Dadarlats notion of proper asymptotic unitary equivalence in $KK$-theory. The thesis contains a explanatory run-down of the paper in which the notion of proper asymptotic unitary equivalence is introduced, and it is shown that this leads to the same $KK$-theory as that of Kasparovs $KK-theory$. Having established the equivalence, Hyon Ho Lee took it further and found that the equivalence can find uses in the theory of Fredholm operators and defined the essential co-dimension of a Cuntz pair of projections to be a class of $KK_h(A,B)$, and thus obtained reasonable conditions for lifting a projection of the corona algebra to a projection in the multiplier algebra. Finally we take a look at M. Dadarlats definition of a pseudo-metric which looks at how far Cuntz pairs are from being proper asymptotic unitary equivalent, and then shows that the topo\-logy induced by that pseudo-metric on $KK(A,B)$ makes $KK(A,B)$ complete and separable.

The theory of graph C*-algebras has played an important role in the understanding of semiprojectivity of C*-algebras, most notably in the work of Spielberg. More recently, work of Katsura and the advisor in has resolved the question of when unital graph algebras are semiprojective, relating this issue to the proper infiniteness of certain projections. The core of this paper is a thorough account of this work, including the characterization of proper infiniteness of relevant projections, followed by an account of Spielberg's methods.

Dette speciale handler om parameter estimering. Vi anvender metoden introduceret i artiklen af Wood, S. N.; Statistical inference for noisy nonlinear ecological dynamic systems på en udvidet stokastisk Fitzhugh-Nagumo model, som introduceres i artiklen af Jensen, AC et al.; Markov chain Monte Carlo approach to parameter estimation in the FitzHugh-Nagumo model. Vi betragter først modellen og ser på egenskaberne for både den deterministiske model og den stokastiske model og vi studerer eksempler på de to versioner. I den næste del af specialet undersøger vi estimeringsproceduren der er beskrevet i artiklen af Wood. Til dette formål anvender vi den logaritmisk syntetiske likelihood, som vi skaber ved at bruge statistikker med helt specifikke egenskaber; vi indser, at likelihooden ligner likelihooden for normalfordelingen, hvis statistikkerne er approksimativt normalfordelt. Valget af statistikker er en iterativ proces, som grundlæggende består i at prøve sig frem. Efter at have set på likelihooden, introducerer vi Metropolis-Hasting metoden - en accepter/forkast algoritme. Vi etablerer den teoretiske ramme fra hvilken metoden er udviklet og vi undersøger den faktiske algoritme, som vi bruger til at simulere data. Tilsidst kombinerer vi modellen og metoden idet vi estimerer parametrene i modellen ved at bruge tre modifikationer af metoden samt data simuleret fra modellen. Resultaterne indikerer at det ikke er praktisk gennemførligt at estimere parametrene i Fitzhugh-Nagumo modellen. Denne konklusion opnåede vi efter at have simuleret næsten en tredjedel billion normalfordelte variable henover en periode på 512 timer (21 dage og 8 timer)., This master thesis is about parameter estimation. We use the method introduced in the article by Wood, S. N.; Statistical inference for noisy nonlinear ecological dynamic systems on an extended stochastic FitzHugh-Nagumo model, which is introduced in the paper by Jensen, A. C. et al.; Markov chain Monte Carlo approach to parameter estimation in the FitzHugh-Nagumo model. First we study the model and look at features for both the deterministic model and the stochastic model and we consider some examples of the two versions. In the next part of the thesis we study the estimation procedure described in the article by Wood. To this end we utilize the log synthetic likelihood which we create using statistics with very specific properties; We realize that the likelihood resembles the likelihood of a normal distribution if the statistics are approximately normally distributed. Choosing the statistics is an iterative process based on the fundamental trial and error approach. After studying the likelihood we introduce the Metropolis-Hasting method - an accept-reject algorithm. We establish the theoretical framework from which the method is developed and study the actual algorithm that we use to simulate data. Finally we combine the model and the method as we estimate the parameters in the model using three modifications of the method and data simulated from the model. The results indicate that using the Wood method is not a feasible way to estimate the parameters in the FitzHugh-Nagumo model. This conclusion was reached after having simulated almost a third of a trillion normally distributed variables over a period of 512 hours (21 days and 8 hours).

Specialet omhandler de mest relevante opdagelser, der er kommet af Søren Eilers' og Marius Dadarlats koncept om fuldstændig asymptotisk unitær ækvivalens i $KK$-teori. Specialet indeholder en forklarende gennemgang af artiklen hvor fuldstændig asymptotisk unitær ækvivalens er introduceret, og der bliver vist at dette fører til den samme $KK$-teori som Kasparovs. Da ækvivalensen var fastsat, tog Hyon Ho Lee det videre og fandt ud af at ækvivalensen kan være brugbar indenfor teorien om Fredholm operatorer, og definerede den essentielle codimension af et Cuntz par af projektioner til at være en klasse i $KK_h(A,B)$, og derved fik nødvendige betingelser for at løfte projektioner i corona algebraen til multiplikator algebraen. Til slut, kigger vi på M. Dadarlats definition af en pseudo-metrik som måler hvor langt Cuntz par er fra at være fuldstændig asymptotisk unitært ækvivalente, og vi viser dernæst at topologien induceret af denne pseudo-metrik på $KK(A,B)$ får $KK(A,B)$ til at blive fuldstændigt og separabelt., The thesis will be concerned with exploring the most relevant discoveries relating to Søren Eilers' and Marius Dadarlats notion of proper asymptotic unitary equivalence in $KK$-theory. The thesis contains a explanatory run-down of the paper in which the notion of proper asymptotic unitary equivalence is introduced, and it is shown that this leads to the same $KK$-theory as that of Kasparovs $KK-theory$. Having established the equivalence, Hyon Ho Lee took it further and found that the equivalence can find uses in the theory of Fredholm operators and defined the essential co-dimension of a Cuntz pair of projections to be a class of $KK_h(A,B)$, and thus obtained reasonable conditions for lifting a projection of the corona algebra to a projection in the multiplier algebra. Finally we take a look at M. Dadarlats definition of a pseudo-metric which looks at how far Cuntz pairs are from being proper asymptotic unitary equivalent, and then shows that the topo\-logy induced by that pseudo-metric on $KK(A,B)$ makes $KK(A,B)$ complete and separable.

The theory of graph C*-algebras has played an important role in the understanding of semiprojectivity of C*-algebras, most notably in the work of Spielberg. More recently, work of Katsura and the advisor in has resolved the question of when unital graph algebras are semiprojective, relating this issue to the proper infiniteness of certain projections. The core of this paper is a thorough account of this work, including the characterization of proper infiniteness of relevant projections, followed by an account of Spielberg's methods.

Systemisk risiko i finansielle markeder betyder at falitter 'smitter.' Det vil sige, at kriser opstår eller forstærkes på grund af forbindelsen mellem de finansielle institutioner. Ved finansielle kriser skabes der externaliteter på resten af samfundets økonomi, hvorfor det kan være fordelagtigt at implementere regulering af denne form for risiko. Systemisk risiko kan observeres i praksis, men er vanskelig at udregne kvantitativt. I Acharya et al.(2010), "Measuring Systemic Risk" foreslås en metode til måling af en institutions systemiske risiko, derudover foreslås også en 'systemisk skat', som kan bruges til regulering af risikoen. I denne opgave præsenteres modellen, hvorefter dans anvendelse i praksis analyseres, først på amerikanske finansielle virksomheder og dernæst på danske finansielle virksomheder. Vi konkluderer at modellen er teoretisk plausibel, men at den i praksis har betydningsfulde mangler. Derfor anbefaler vi yderligere arbejde med henblik på at konstruere en præcis metode til regulering af systemisk risiko, før en lovgivning herom kan implementeres., Systemic risk is the risk that the distress of a single institution may spread to other institutions and because of this interconnectedness strengthen, or even cause, a financial crisis. A financial crisis causes externalities on the rest of the economy and it is thus favorable to implement regulation to limit systemic risk. This risk is apparent through observation, but is difficult to measure quantitatively. A method of measuring systemic risk is proposed in Acharya et al.(2010), "Measuring Systemic Risk", in which the authors also introduce a tax-scheme that can be used to regulate this risk. This paper presents Acharya et al.'s model and analyzes its practical use on the American market and on the Danish market. Our work concludes that the theoretical model holds, but the practical method of measuring systemic risk has significant shortcomings. We recommend further work to construct an accurate method of limiting systemic risk prior to implementing government regulation.

Systemisk risiko i finansielle markeder betyder at falitter 'smitter.' Det vil sige, at kriser opstår eller forstærkes på grund af forbindelsen mellem de finansielle institutioner. Ved finansielle kriser skabes der externaliteter på resten af samfundets økonomi, hvorfor det kan være fordelagtigt at implementere regulering af denne form for risiko. Systemisk risiko kan observeres i praksis, men er vanskelig at udregne kvantitativt. I Acharya et al.(2010), "Measuring Systemic Risk" foreslås en metode til måling af en institutions systemiske risiko, derudover foreslås også en 'systemisk skat', som kan bruges til regulering af risikoen. I denne opgave præsenteres modellen, hvorefter dans anvendelse i praksis analyseres, først på amerikanske finansielle virksomheder og dernæst på danske finansielle virksomheder. Vi konkluderer at modellen er teoretisk plausibel, men at den i praksis har betydningsfulde mangler. Derfor anbefaler vi yderligere arbejde med henblik på at konstruere en præcis metode til regulering af systemisk risiko, før en lovgivning herom kan implementeres., Systemic risk is the risk that the distress of a single institution may spread to other institutions and because of this interconnectedness strengthen, or even cause, a financial crisis. A financial crisis causes externalities on the rest of the economy and it is thus favorable to implement regulation to limit systemic risk. This risk is apparent through observation, but is difficult to measure quantitatively. A method of measuring systemic risk is proposed in Acharya et al.(2010), "Measuring Systemic Risk", in which the authors also introduce a tax-scheme that can be used to regulate this risk. This paper presents Acharya et al.'s model and analyzes its practical use on the American market and on the Danish market. Our work concludes that the theoretical model holds, but the practical method of measuring systemic risk has significant shortcomings. We recommend further work to construct an accurate method of limiting systemic risk prior to implementing government regulation.