Uma série de observações advindas da medida da distância de supernovas tipo IA, idade das estrelas mais antigas, anisotropias da radiação cósmica de fundo, entre outras, evidenciam que o universo está passando por uma fase de expansão acelerada. Essa expansão está sendo causada por uma componente misteriosa denominada energia escura, que representa cerca de $70\\%$ do conteúdo total do universo, e cuja natureza é desconhecida. Para descrever a energia escura vários modelos têm sido propostos, entre eles, podemos destacar a energia do vácuo (constante cosmológica) e um campo escalar dinâmico (quintessência). Também são considerados modelos em que a energia escura interage com outro componente misterioso, a matéria escura. Existem vários testes observacionais para vincular os parâmetros desses modelos. Nesta dissertação, exploraremos um método baseado na idade de galáxias em altos redshifts e na idade do universo, conhecido em inglês como lookback time. / A number of observations arising from the measurement of distance of type IA Supernovae, age of oldest stars, anisotropy of cosmic microwave background, among others, show that the universe is undergoing a phase of accelerated expansion. This expansion is being caused by a mysterious component called dark energy, which represents about $70\\%$ of the total content of the universe, and whose nature is unknown. To describe the various dark energy models have been proposed, among them we highlight the vacuum energy (cosmological constant), and a dynamic scalar field (quintessence). Are also considered models in which dark energy interacts with another mysterious component, the dark matter. There are several observational tests to constraint the parameters of these models. In this dissertation, we explore a method based on age of galaxies at high redshift and the age of the universe, known as lookback time.

Uma série de observações advindas da medida da distância de supernovas tipo IA, idade das estrelas mais antigas, anisotropias da radiação cósmica de fundo, entre outras, evidenciam que o universo está passando por uma fase de expansão acelerada. Essa expansão está sendo causada por uma componente misteriosa denominada energia escura, que representa cerca de $70\\%$ do conteúdo total do universo, e cuja natureza é desconhecida. Para descrever a energia escura vários modelos têm sido propostos, entre eles, podemos destacar a energia do vácuo (constante cosmológica) e um campo escalar dinâmico (quintessência). Também são considerados modelos em que a energia escura interage com outro componente misterioso, a matéria escura. Existem vários testes observacionais para vincular os parâmetros desses modelos. Nesta dissertação, exploraremos um método baseado na idade de galáxias em altos redshifts e na idade do universo, conhecido em inglês como lookback time. / A number of observations arising from the measurement of distance of type IA Supernovae, age of oldest stars, anisotropy of cosmic microwave background, among others, show that the universe is undergoing a phase of accelerated expansion. This expansion is being caused by a mysterious component called dark energy, which represents about $70\\%$ of the total content of the universe, and whose nature is unknown. To describe the various dark energy models have been proposed, among them we highlight the vacuum energy (cosmological constant), and a dynamic scalar field (quintessence). Are also considered models in which dark energy interacts with another mysterious component, the dark matter. There are several observational tests to constraint the parameters of these models. In this dissertation, we explore a method based on age of galaxies at high redshift and the age of the universe, known as lookback time.

CETTE THÈSE EST CONSACRÉE À LA QUANTIFICATION AVEC DES APPLICATIONS À LA FINANCE. LE CHAP.1 RAPPELLE LES BASES DE LA QUANTIFICATION ET LES MÉTHODES DE RECHERCHE DE QUANTIFIEURS OPTIMAUX. AU CHAP.2 ON ÉTUDIE LE COMPORTEMENT ASYMPTOTIQUE, DANS L^S, DE L'ERREUR DE QUANTIFICATION ASSOCIÉE À UNE TRANSFORMATION LINÉAIRE D'UNE SUITE DE QUANTIFIEURS OPTIMALE DANS L^R. ON MONTRE QU'UNE TELLE TRANSFORMATION PERMET DE RENDRE LA SUITE TRANSFORMÉE L^S TAUX OPTIMALE POUR TOUT S, POUR UNE LARGE FAMILLE DE PROBABILITÉS. LE CHAP.3 ÉTUDIE LE COMPORTEMENT ASYMPTOTIQUE DE LA SUITE DU RAYON MAXIMAL ASSOCIÉE À UNE SUITE DE QUANTIFIEURS L^R OPTIMALE. ON MONTRE QUE DÈS QUE SUPP(P) EST NON BORNÉ CETTE SUITE TEND VERS L'INFINI. ON DONNE, POUR UNE GRANDE FAMILLE DE PROBABILITÉS, LA VITESSE DE CONVERGENCE VERS L'INFINI. LE CHAP.4 EST CONSACRÉ AU PRICING D'OPTIONS DE TYPE LOOKBACK ET À BARRIÈRRE. ON ÉCRIT CES PRIX SOUS UNE FORME QUI NOUS PERMET DE LES ESTIMER PAR MONTE CARLO, PAR UNE MÉTHODE HYBRIDE MONTE CARLO-QUANTIFICATION ET PAR PUR QUANTIFICATION.

Lookback option is a well-known path-dependent option where its payoff depends on the historical extremum prices. The thesis focuses on the binomial pricing of the American floating strike lookback put options with payoff at time $t$ (if exercise) characterized by \[ \max_{k=0, \ldots, t} S_k - S_t, \] where $S_t$ denotes the price of the underlying stock at time $t$. Build upon the idea of \hyperlink{RBCV}{Reiner Babbs Cheuk and Vorst} (RBCV, 1992) who proposed a transformed binomial lattice model for efficient pricing of this class of option, this thesis extends and enhances their binomial recursive algorithm by exploiting the additional combinatorial properties of the lattice structure. The proposed algorithm is not only computational efficient but it also significantly reduces the memory constraint. As a result, the proposed algorithm is more than 1000 times faster than the original RBCV algorithm and it can compute a binomial lattice with one million time steps in less than two seconds. This algorithm enables us to extrapolate the limiting (American) option value up to 4 or 5 decimal accuracy in real time.

Lookback option is a well-known path-dependent option where its payoff depends on the historical extremum prices. The thesis focuses on the binomial pricing of the American floating strike lookback put options with payoff at time $t$ (if exercise) characterized by \[ \max_{k=0, \ldots, t} S_k - S_t, \] where $S_t$ denotes the price of the underlying stock at time $t$. Build upon the idea of \hyperlink{RBCV}{Reiner Babbs Cheuk and Vorst} (RBCV, 1992) who proposed a transformed binomial lattice model for efficient pricing of this class of option, this thesis extends and enhances their binomial recursive algorithm by exploiting the additional combinatorial properties of the lattice structure. The proposed algorithm is not only computational efficient but it also significantly reduces the memory constraint. As a result, the proposed algorithm is more than 1000 times faster than the original RBCV algorithm and it can compute a binomial lattice with one million time steps in less than two seconds. This algorithm enables us to extrapolate the limiting (American) option value up to 4 or 5 decimal accuracy in real time.

La constatation que les prix des actifs boursiers sautent brusquement a conduit à étudier des modèles de marchés avec sauts. Cette thèse va dans cette direction. On y considère des marchés dirigés par des martingales normales qui ont la propriété de représentation chaotique: les martingales vérifiant une équation de structure déterministe, la martingale d'Azéma, etc. On trouve des stratégies de couverture pour les options européennes, asiatiques et Lookback soit par la formule d'Itô, soit par la formule de Clark-Ocone selon la plus appropriée. L'application du calcul de Malliavin au calcul des Greeks est traitée pour les options asiatiques dans le cas d'un marché dirigé par un processus de Poisson. On traite aussi de couverture dans un modèle à volatilité stochastique avec sauts où le prix de l'actif risqué est dirigé par un processus somme d'un mouvement brownien et d'un processus de Poisson 2-dimensionnels. Le marché est incomplet et il existe une infinité de mesures martingales équivalentes. On minimise l'entropie pour choisir telle mesure. Sous celle-ci on calcule la stratégie minimisant la variance.

Optioner säljs och köps idag flitigt av många olika anledningar. En av dessa kan vara spekulation kring framtida händelser för aktiepriser där optioner har fördelar jämfört med aktier i form av en hävstångseffekt. En annan anledning för optionshandel är för att hedga (säkra) risker vilket ställer krav på att innehavet av optionen ska kompensera den negativa effekt som riskerna bidrar till. Med andra ord, om det finns en risk för ett negativt framtida scenario som man inte vill riskera att utsätta sig för så kan optioner vara rätt verktyg att använda sig av.   Risker finns idag överallt, i olika former, vilket har bidragit till att efterfrågan av optioner har ökat enormt de senaste årtiondena. Dock kan risker vara både komplexa och varierande vilket har lett till att mer komplexa optioner har utvecklats för att mätta den efterfrågan som utvecklats på marknaden. Dessa, mer komplexa optioner, kallas exotiska optioner och de skiljer sig från de vanliga europeiska och amerikanska köp- och säljoptionerna. Däribland hittar vi bland annat lookback-optioner i form av bland annat köpoptioner på maximum och kvantiloptioner vilka är två av de huvudsakliga optionerna som diskuteras i uppsatsen.   Det har länge varit känt hur man prissätter europeiska köp- och säljoptioner via Black-Scholes-Mertons modell men desto fler komplexa optioner som tillkommer på marknaden desto mer komplicerade prissättningsmodeller utvecklas. Till skillnad från europeiska köp- och säljoptioner vars utdelning beror på aktiepriset på lösendagen så är lookback-optioner beroende av aktieprisets rörelse under hela kontraktstiden. Detta medför att prissättningen av dessa beror av fler parametrar än i Black-Scholes-Mertons modell, bland annat ockupationstiden för den stokastiska process som beskriver aktiepriset, vilket bidrar till andra prissättningsmodeller.   Uppsatsen har som syfte att redogöra för modellen som används vid prissättningen av kvantiloptioner samt presentera hur deras egenskaper förhåller sig till andra typer av lookback-optioners egenskaper. Det presenteras i rapporten att kvantiloptioner liknar vissa typer av lookback-optioner, mer bestämt köpoptioner på maximum, och att kvantiloptioners egenskaper faktiskt konvergerar mot köpoptioner på maximums egenskaper då kvantilen närmar sig 1. Utifrån detta resonemang så kan det finnas fördelar i att använda kvantiloptioner snarare än köpoptioner på maximum vilket investerare bör ta i hänsyn när, och om, kvantiloptioner introduceras på marknaden. / Options are today used by investors for multiple reasons. One of these are speculation about future market movements, here ownership of options is advantageous over usual ownership of shares in the underlying stock in terms of a leverage effect. Furthermore, investors use options to hedge different kinds of risks that they are exposed to, this demands that the option compensates the possible negative effect that the risk brings to the table. In other words, if there is a risk of a future negative scenario which the investor is risk averse to, then owning specific options which neutralize this risk could be the perfect tool to use.   Risks are today seen all over the market in different shapes which have created a great demand for options over the last decades. However, since risks can be both complex and range over multiple business areas, investors have demanded more complex options which can neutralize the risk exposures. These, more complex options, are called exotic options, and they differ from the regular American and European options in the way they behave with respect to the underlying stock. Amongst these exotic options, we can find different kind of lookback options as well as quantile options which are two of the main options that are discussed in this thesis.   It has been known for a while how to price European call and put options by the Black-Scholes-Merton model. However, with more complex options also comes more complex pricing models and unlike the European options’ payoff which depend on the underlying stock price at time of maturity, the lookback option’s and quantile option’s payoff depend on the stock price movement over the total life span of the option contract. Hence, the pricing of these options depends on more variables than the classic Black-Scholes-Merton model include. One of these variables is the occupation time of the stochastic process which describes the stock price movement, this leads to a more complex and extensive pricing model than the general Black-Scholes-Merton’s model.   The objective of this thesis is to derive the pricing model that is used for quantile options and prove that the properties of quantile options are advantageous when compared to some specific lookback options, viz. call options on maximum.  It is concluded in the thesis that quantile options in fact converges to the call option on maximum for quantiles approaching 1. However, quantile options come with some different properties which potentially makes them a good substitute for the call option on maximum. This is a relevant factor for investors to consider when, and if, quantile options are introduced to the market.

本論文主要結合了「匯率連動選擇權」與「多點重設型選擇權」、「履約價回顧型選擇權」，除了評價與分析四款匯率連動多點重設型賣權以及匯率連動履約價回顧型賣權，並且探討重設點之選擇對於賣權價格之影響，使其理論與模型更為一般化，發行券商或銀行欲發行此類商品時，更能夠依據模型做更進一步之風險控管，藉以降低避險損失。 / For the most part, this article combines the quanto option with the multiple-reset put and lookback put. In addition to price and analyze the four specific types of quanto multiple-reset puts and quanto lookback puts, this article also provides a more comprehensive study on how the frequency of resetting in exercise price affects the quanto puts’ price and risk. When issuers issue this kind of financial product, based on the model in this article, they will be able to better control the risk further and secure the investment return by diminishing the loss of hedging.

In mathematical finance, the sensitivities of option prices to various market parameters, also known as the “Greeks”, reflect the exposure to different sources of risk. Computing these is essential to predict the impact of market moves on portfolios and to hedge them adequately. This is commonly done using Monte Carlo simulations. However, obtaining accurate estimates of the Greeks can be computationally costly. Multilevel Monte Carlo offers complexity improvements over standard Monte Carlo techniques. However the idea has never been used for the computation of Greeks. In this work we answer the following questions: can multilevel Monte Carlo be useful in this setting? If so, how can we construct efficient estimators? Finally, what computational savings can we expect from these new estimators? We develop multilevel Monte Carlo estimators for the Greeks of a range of options: European options with Lipschitz payoffs (e.g. call options), European options with discontinuous payoffs (e.g. digital options), Asian options, barrier options and lookback options. Special care is taken to construct efficient estimators for non-smooth and exotic payoffs. We obtain numerical results that demonstrate the computational benefits of our algorithms. We discuss the issues of convergence of pathwise sensitivities estimators. We show rigorously that the differentiation of common discretisation schemes for Ito processes does result in satisfactory estimators of the the exact solutions’ sensitivities. We also prove that pathwise sensitivities estimators can be used under some regularity conditions to compute the Greeks of options whose underlying asset’s price is modelled as an Ito process. We present several important results on the moments of the solutions of stochastic differential equations and their discretisations as well as the principles of the so-called “extreme path analysis”. We use these to develop a rigorous analysis of the complexity of the multilevel Monte Carlo Greeks estimators constructed earlier. The resulting complexity bounds appear to be sharp and prove that our multilevel algorithms are more efficient than those derived from standard Monte Carlo.

本文針對路徑相依選擇權(path dependent option)商品，提供一個一般化且有效率評價方法。由於路徑相依選擇權的種類很多，而大部分的美式路徑相依選擇權都沒有封閉解(closed-form)，或是封閉解的數學計算過於複雜，而造成評價的困難。此時，透過數值方法可以對路徑相依選擇權定出理論價值。但是選定一個有效率的數值方法是主要的困難，理論上，樹狀模型及蒙地卡羅的數值方法都可以評價路徑相依選擇權，而蒙地卡羅法在評價美式選擇權時較困難，相對而言，使用樹狀模型可以評價美式的選擇權的一個不錯的方法。 自從CRR(Cox, Ross and Rubinstein, 1979)發展二項樹模型(Binomial Tree model)來評價選擇權後，二項樹模型一直被廣泛的應用，此方法基本的概念假設股價的變動為間斷(Discrete)的，且股價呈現上漲或下跌兩種情形，這樣可以容易地來評價歐式及美式的選擇權。之後Boyle(1988)更發展三元樹模型(Trinomial Tree)，股價比CRR更多了持平的情形，這樣比CRR多考慮了一種股價行為的模式，實證得知三元樹在穩定性及收斂度上比二項樹表現較佳。 上述二項樹模型及三元樹模型受到節點重合(recombined)的特性，而路徑相依選擇權同一個節點若由不同歷史路徑所產生時，其報酬(payoff)是不同的，報酬可能因為歷史路徑的不同產生很多的情形，所以當路徑相依選擇權的條件越複雜時，要評價一個路徑相依選擇權有其困難性。 本文分別以二項樹模型及三元樹模型來評價路徑相依的選擇權，而為了解決節點可能存在之前的路徑問題，放鬆條件使得節點不再結合一起(non-recombined)，如此所有的節點將可以被紀錄，不會有不能評價路徑相依選擇權的問題，但在此情況下會產生另一個問題，節點數隨著切割期數的上昇呈指數成長，使得電腦計算較無效率。 針對路徑選擇權本文提出一個有效率的路徑相依選擇權方法，稱為動態樹狀法(Dynamic Tree Model, DTM)，此評價方法建構在風險中立定價(risk-neutral)的理論基礎上。在每一期時間點檢查是否有相同的路徑資訊和標的物現價，若發生路徑資訊和標的物的現價相同且有重複的節點時，可以預期的，這些節點未來長出的子股價樹也會相同，因此不必重複節點，浪費電腦記憶體空間及運算時間，而將此節點予以合併，以達到減少節點個數目的。若遇到不同的路徑資訊或不同標的物現價的節點時，則予以產生。 動態樹狀法將真正需要的節點加以產生，其目的能降低節點數目，改善計算效率，而將此方法廣泛地應用在其他不同的路徑相依選擇權上。而根據不同路徑相依選擇權，我們必須將有用的路徑資訊存在節點上。本文將提出一般化的模型，使用policy設計樣式，以二項樹及三元樹為例，並選擇不同的路徑相依選擇權產品－障礙選擇權、回顧選擇權、亞式選擇權為例，求其理論價值，而實務上通常是間斷(discrete)觀察，我們將討論間斷觀測的情形，比較其觀察點、效率、精確度、節點數目、允許誤差之探討，並提出建議，也能夠廣泛應用在其它路徑相依選擇權上。