This master thesis will investigate one solution to the problem issues with nested stochastic simulation arising when the future value of a portfolio need to be calculated. The solution investigated is the Least-squares Monte-Carlo method, where regression is used to obtain a proxy function for the given portfolio value. We will further investigate how to generate an optimal regression function that minimizes the number of terms in the regression function and reduces the risk of overtting the regression.

This dissertation consists of two chapters. The first chapter establishes an algorithm for calculating capital requirements. The calculation of capital requirements for financial institutions usually entails a reevaluation of the company's assets and liabilities at some future point in time for a (large) number of stochastic forecasts of economic and firm-specific variables. The complexity of this nested valuation problem leads many companies to struggle with the implementation. The current chapter proposes and analyzes a novel approach to this computational problem based on least-squares regression and Monte Carlo simulations. Our approach is motivated by a well-known method for pricing non-European derivatives. We study convergence of the algorithm and analyze the resulting estimate for practically important risk measures. Moreover, we address the problem of how to choose the regressors, and show that an optimal choice is given by the left singular functions of the corresponding valuation operator. Our numerical examples demonstrate that the algorithm can produce accurate results at relatively low computational costs, particularly when relying on the optimal basis functions. The second chapter discusses another application of regression-based methods, in the context of pricing variable annuities. Advanced life insurance products with exercise-dependent financial guarantees present challenging problems in view of pricing and risk management. In particular, due to the complexity of the guarantees and since practical valuation frameworks include a variety of stochastic risk factors, conventional methods that are based on the discretization of the underlying (Markov) state space may not be feasible. As a practical alternative, this chapter explores the applicability of Least-Squares Monte Carlo (LSM) methods familiar from American option pricing in this context. Unlike previous literature we consider optionality beyond surrendering the contract, where we focus on popular withdrawal benefits - so-called GMWBs - within Variable Annuities. We introduce different LSM variants, particularly the regression-now and regression-later approaches, and explore their viability and potential pitfalls. We commence our numerical analysis in a basic Black-Scholes framework, where we compare the LSM results to those from a discretization approach. We then extend the model to include various relevant risk factors and compare the results to those from the basic framework.

Inom bank och försäkringsbranschen finns behov av framtidsprognoser och riskmått kopplade till finansiella instrument. För att skapa prisfördelningar, som kan användas som grund till olika riskmått, används ibland nästlad simulering. För att göra detta simuleras först en stor mängd yttre scenarion för någon tillgång, som används i ett finanisellt instrument. Vilket görs genom att priser simuleras över en tidsperiod. Detta utgör tidshorisonten varvid prisfördelningen befinner sig. Utifrån varje yttre scenario simuleras sedan ett antal inre. Som i sin tur används för att prissätta finansiella instrumentet i det yttre scenariot. En metod som används för att prisätta de yttre scenariona är Monte Carlo-metoden, vilket kräver ett stort antal inre scenarion för att prissättningen ska bli korrekt. Detta gör metoden krävande i tidsåtgång och datorkraft. Least Squares Monte Carlo-metoden är en alternativ metod som använder sig av regression och minstakvadratmetoden för att utföra prissättningen med ett mindre antal inre scenarion. En regressionsfunktion anpassas efter yttre scenarionas värden och används sedan för att omvärdera dessa, vilket minskar felen som ett mindre antal slumptal annars skulle ge. Regressionsfunktionen kan även användas för att prissätta värden utanför de som den anpassas efter, vilket gör att den kan återanvändas vid liknande beräkningar. I detta arbete undersöks hur väl Least Squares Monte Carlo-metoden beskriver prisfördelningen för korgoptioner, som är optioner med flera underliggande tillgångar. Tester utförs med olika värden för parametrarna och vikt läggs vid vilken effekt yttre scenarionas längd har, samt hur väl priserna beskrivs i prisfördelningens svansar. Resultatet är delvis svåranalyserat på grund av många extrema värden, men visade på svårigheter med prissättningen vid längre yttre scenarion. Vilket kan bero på att regressionsfunktionen som användes hade svårt att anpassa sig efter och beskriva mer spridda prisfördelningar. Metoden fungerade också sämre i den nedre delen av prisfördelningen, något som den dock delar med Standard Monte Carlo. Mer forskning behövs för att undersöka vilken effekt andra uppsättningar regressionsfunktioner skulle ha på metoden. / In the banking and insurance industry, there exists a need for forecasting and measures of risk connecting to financial instruments. To create price distributions, used to create measures of risk, nested simulations are sometimes used. This is done by simulating a large amount of outer scenarios, for some asset in a financial instrument. Which is done by simulating prices over a certain time period. This now outlines the time horizon of the price distribution. From each outer scenario, some inner scenarios are simulated. Which in turn are used to price the financial instrument in the outer scenario. A common method for pricing the outer scenarios is the Monte Carlo method, which uses a large amount of random numbers for the pricing to be accurate. This makes the method time consuming, as well as requiring large amounts of computing power. The Least Squares Monte Carlo method is an alternative method, using regression and the least squares method to perform the pricing using a smaller amount of inner scenarios. A regression function is fitted to the values of the outer scenarios and then used to revalue these, reducing the error which a smaller number of random numbers otherwise would give. The regression function can also be used to price outside of the values used for the fitting, making it reusable in similar computations. This paper examines how well the Least Squares Monte Carlo-method describes the price distribution of basket options, which are options containing several underlying assets. Tests are made for different values for the parameters used and an emphasis is laid on the effect of the time length of the outer scenarios, also, how accurate the tails of the distribution are. The results are somewhat hard to analyze,due to some extreme values, but showed difficulties for the method, when pricing longer outer scenarios. This can be due to the regression function having problems fitting to - and valuing - broader price distributions. The method also performed worse in the lower parts of the distribution, something it shares with the standard Monte Carlo method. More research is needed to ascertain the effects of other regression functions.

This thesis is focused on practical application of two methods used in non-life insurance, Nested Monte Carlo and Least squares Monte Carlo. Best estimate and 99.5% quantile was calculated using both methods and results was compared. Both methods are similar in estimates and therefore can be used for computation of capital requirement. Least squares Monte Carlo seem more favourable, because it significantly reduces computation time.

[pt] Este trabalho tem por objetivo o estudo e a aplicação do Método de Mínimos Quadrados de Monte Carlo com diferentes bases polinomiais - Potência, Laguerre, Legendre e Hermite A - na precificação de Opções Asiáticas Americanas (Amerasian) tanto em sua modalidade de compra quanto em sua modalidade de venda. Os resultados encontrados ratificam a possibilidade de utilização alternativa de diversas bases polinomiais. Além disso, verifica-se a convergência em cada um dos experimentos, sem perder de vista a possibilidade de que haja, para cada tipo de Amerasian precificada, uma base polinomial específica que, marginalmente, mostra-se mais precisa. / [en] This work aims at studying and applying the Least-Squares Monte Carlo Method by using different polynomial basis - Power, Laguerre, Legendre and Hermite A - in pricing American Asian Options, either call or put. The results found ratify the possibility of an alternated use of several polynomial bases. Besides, each of the experiments is checked for convergence, taking into account that there may be an optimal polynomial basis for each kind of Amerasian option which is marginally more accurate regarding its pricing.

本文採用BGM模型評價兩個配息型態不同的利率連結商品。利用BGM模型，我們可以直接透過蒐集市場資料，即可描述LIBOR利率的期間結構。同時，對模型內遠期利率波動度與相關係數進行校準(Calibration)，使評價更為正確。 而本文評價的第一個商品為「三年期美元每日計息雙指標利率連動債」，第二個商品則是「10年期歐元逆浮動連動債」。使用BGM模型，並透過最小平方蒙地卡羅模擬，考慮提前買回條款及計算各期的配息，分別求得兩個商品的合理價格並計算避險參數。此外，從發行商與投資人的角度，分別給予避險與投資建議。 關鍵字：利率連動債、每日計息、逆浮動、BGM模型、LIBOR Market Model、Least-Squares Monte Carlo

Public-Private-Partnership (P3) is gaining momentum as the delivery method for the development of public infrastructure. These projects, however, are exposed to economic risks. If the private parties are not comfortable with the level of the risks, they would not participate in the project and, as a result, the infrastructure will most likely not be realized. As an incentive for participation in the P3 project, private parties are sometimes offered guarantees against unfavorable economic risks. Therefore, the valuation of these guarantees is essential for deciding whether or not to participate in the project. While previous works focused on the valuation of guarantees, the incorporation of credit risk in the value of the P3 projects and the guarantees has been neglected. The effect of credit risk can be taken into account by using the rigorous Credit Value Adjustment method (CVA). CVA is a computationally demanding method that the valuation methods currently in the literature are not capable of handling. This research offers a novel approach for the valuation of guarantees and P3 projects which is computationally superior to the existing methods. Because of this computational efficiency, CVA can be implemented to account for credit risk. For the development of this method, a continuous stochastic differential equation (SDE) is derived from the forecasted curve of an economic risk. Using the SDE, the partial differential equation (PDE) governing the value of the guarantees will be derived. Then, the PDE will be solved using Finite Difference Method (FDM). A new feature for this method is that it obtains exercise strategies for the Australian guarantees. The present work extends the literature by providing a valuation method for the cases that multiple risks affect P3 projects. It also presents an approach for the valuation of the Asian style guarantee, a contract which reimburses the private party based on the average of risk factor. Finally, a hypothetical case study illustrates the implementation of the FDM-based valuation method and CVA to obtain the value of the P3 project and the guarantees adjusted for the counterparty credit risk.

Public-Private-Partnership (P3) is gaining momentum as the delivery method for the development of public infrastructure. These projects, however, are exposed to economic risks. If the private parties are not comfortable with the level of the risks, they would not participate in the project and, as a result, the infrastructure will most likely not be realized. As an incentive for participation in the P3 project, private parties are sometimes offered guarantees against unfavorable economic risks. Therefore, the valuation of these guarantees is essential for deciding whether or not to participate in the project. While previous works focused on the valuation of guarantees, the incorporation of credit risk in the value of the P3 projects and the guarantees has been neglected. The effect of credit risk can be taken into account by using the rigorous Credit Value Adjustment method (CVA). CVA is a computationally demanding method that the valuation methods currently in the literature are not capable of handling. This research offers a novel approach for the valuation of guarantees and P3 projects which is computationally superior to the existing methods. Because of this computational efficiency, CVA can be implemented to account for credit risk. For the development of this method, a continuous stochastic differential equation (SDE) is derived from the forecasted curve of an economic risk. Using the SDE, the partial differential equation (PDE) governing the value of the guarantees will be derived. Then, the PDE will be solved using Finite Difference Method (FDM). A new feature for this method is that it obtains exercise strategies for the Australian guarantees. The present work extends the literature by providing a valuation method for the cases that multiple risks affect P3 projects. It also presents an approach for the valuation of the Asian style guarantee, a contract which reimburses the private party based on the average of risk factor. Finally, a hypothetical case study illustrates the implementation of the FDM-based valuation method and CVA to obtain the value of the P3 project and the guarantees adjusted for the counterparty credit risk.

國內結構債市場業已蓬勃發展，市場模型亦相當適合結構債評價。本文在市場模型下，因市場模型不具馬可夫性質，運用最小平方蒙地卡羅法針對三連結標的為LIBOR的結構債進行評價。 / The market of the structured notes has been blossoming. The lognormal forward LIBOR model is more suitable for the valuation of structured notes than do the traditional interest rate models. In this article, we perform three case studies of the valuation of the structured notes linked to LIBOR in lognormal forward LIOBR model. It is easier to implement the lognormal forward LIBOR model by Monte Carlo simulation due to the non-Markovian property. Therefore, the least-squares Monte Carlo approach is used to deal with the callable feature of the structured notes in our case studies.

[pt] No modelo de Pesquisa e Desenvolvimento (P&D) para o setor farmacêutico, considerando a proteção da patente, Schwartz fez uma contribuição muito importante para a precificação do valor de uma empresa. Alem de considerar a possibilidade dos eventos catastróficos, o autor aplicou uma ferramenta simples, mas ao mesmo tempo bastante poderosa, até então não utilizada pela literatura para tal fim (denominada Mínimos Quadrados de Monte Carlo). Na última etapa do seu modelo (período pós patente), o autor aproxima o processo por um valor constante (obtido por meio de uma estimativa), acreditando que o mercado absolverá imediatamente a tecnologia e, com isto, a simplificação não alterará significativamente o resultado final.Contudo, com o avanço da globalização e o desenvolvimento dos meios de comunicação, o mercado tem-se tornado cada vez mais dinâmico e competitivo. Uma vez que as táticas clássicas já se tornaram acessíveis a todos (e. g. produção em escala ou corte dos custos), elas não são mais suficientes para manter a lucratividade de uma empresa. Diversos estudiosos e instituições renomadas (como por exemplo, BNDES) observaram que, para sobreviverem a isso, as empresas precisam suplantar a concorrência, através de um conjunto de capacitações denominadas bens intangíveis, o que na maioria das ocasiões é uma operação bastante complexa. Devido a essas evidencias a absorção de uma nova tecnologia não necessariamente é uma atividade imediata (contrariando as suposições de Schwartz). Com o intuito de uma melhor descrição da realidade, esta dissertação propõe um aperfeiçoamento matemático (não mais aproximando a etapa pós-patente) e algorítmico do modelo de Schwartz. O modelo foi criado a partir de um conjunto de conceitos da Microeconomia, Opções Reais e das Métricas de Avaliação do BNDES, previamente desenvolvido com o auxílio da equipe do BNDES. Os resultados obtidos na simulação do exemplo teórico, a qual foi analisada uma empresa de TI que pretende desenvolver uma nova tecnologia, não apresentaram nenhuma incoerência, indicando, desta forma, nenhum erro sob a ótica matemática ou algorítmica e confirmando, ao mesmo tempo, a importância dos ativos intangíveis, que, por sua vez, propiciaram maiores ganhos, antes desprezados. / [en] In the model of Research and Development (R & D) for the pharmaceutical industry, considering patent protection, Schwartz made an important contribution to the valuation of a company. Besides considering the possibility of catastrophic events, the author applied a simple tool, but at the same time very powerful, hitherto not used in the literature for this purpose (called Least Squares Monte Carlo). In the last stage of its type (post patent), the author approaches the process with a constant value (obtained via an estimate), believing that the market immediately absolve the technology and, thus, the simplification does not significantly change the outcome final. Canted, with the advance of globalization and the media, the market has become increasingly dynamic and competitive. Since the classic tactics have become accessible to all (egg scale production or cut costs), they are no longer sufficient to maintain the profitability of a company. Several renowned scholars and institutions (such as BNDES) observed that, to survive this, companies must overcome competition through a set of skills called intangibles, which in most cases is a complex operation. Because of these evidences the absorption of new technology is not necessarily an immediate activity (contrary to the assumptions of Schwartz). Aiming at a better description of reality, this study proposes a mathematical processing (no longer approaching the post-patent) and the algorithmic model of Schwartz. The model was created from a set of concepts of microeconomics, Real Options and Metrics Assessment of BNDES, previously developed with the assistance of staff from BNDES. The simulation results of the theoretical example, which was considered an IT company that aims to develop a new technology, did not show any inconsistency, indicating thus no error in the mathematical or algorithmic perspective and confirming at the same time, importance of intangible assets, which, in turn, lead to the greatest earnings before slighted.