Probability theory plays a crucial role in the study of the geometry of Banach spaces. In the literature, notions from probability theory have been formulated and studied in the measure free setting of vector lattices. However, there is little evidence of these vector lattice techniques being used in the study of geometry of Banach spaces. In this thesis, we fill this niche. Using the l-tensor product of Chaney-Shaefer, we are able to extend the available vector lattice techniques and apply them to the Lebesgue-Bochner spaces. As a consequence, we obtain new characterizations of the Radon Nikod´ym property and the UMD property.

Student Number : 8804388Y - MSc Dissertation - School of Computational and Applied Mathematics - Faculty of Science / This dissertation considers the pricing and hedging of contingent claims in a general semimartingale market. Initially the focus is on a complete market, where it is possible to price uniquely and hedge perfectly. In this context the two fundamental theorems of asset pricing are explored. The market is then extended to incorporate risk that cannot be hedged fully, thereby making it incomplete. Using quadratic cost criteria, optimal hedging approaches are investigated, leading to the derivations of the minimal martingale measure and the variance-optimal martingale measure. These quadratic approaches are then applied to the problem of minimizing the basis risk that arises when an option on a non-traded asset is hedged with a correlated asset. Closed-form solutions based on the Black-Scholes equation are derived and numerical results are compared with those resulting from a utility maximization approach, with encouraging results.

Neste trabalho, estudamos o passeio aleatório do elefante introduzido em (SCHUTZ; TRIMPER, 2004). Um processo estocástico não Markoviano com memória de alcance ilimitada que apresenta transição de fase. Nosso objetivo é demonstrar a convergência quase certa do passeio aleatório do elefante nos casos subcrítico e crítico. Além destes resultado, também apresentamos a demonstração do Teorema Central do Limite para ambos os regimes. Para o caso supercrítico, vamos demonstrar a convergência do passeio aleatório do elefante para uma variável aleatória não normal com base nos artigos (BAUR; BERTOIN, 2016), (BERCU, 2018) e (COLETTI; GAVA; SCHUTZ, 2017b). / In this work we study the elephant random walk introduced in (SCHUTZ; TRIMPER, 2004), a discrete time, non-Markovian stochastic process with unlimited range memory that presents phase transition. Our objective is to proof the almost sure convergence for the subcritical and critical regimes of the model. We also present a demonstration of the Central Limit Theorem for both regimes. For the supercritical regime we proof the convergence of the elephant random walk to a non-normal random variable based on the articles (BAUR; BERTOIN, 2016), (BERCU, 2018) and (COLETTI; GAVA; SCHUTZ, 2017b).

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In this dissertation, I consider a range of topics related to the role played by information in modern asset pricing theory. The primary research focus is twofold. First, I synthesize existing research in insider trading and seek to stimulate an expansion of the literature at the intersection of work in the insider trading and financial economics areas. Second, I present the case for using Peter Bossaerts's (2004) Efficiently Learning Markets (ELM) methodology to empirically test asset pricing models. The first chapter traces the development of domestic and international insider trading regulations and explores the legal issues surrounding the proprietary nature of information in financial markets. I argue that, practically, the reinvigoration of the insider trading debate is unfortunate because, in spite of seemingly unending efforts to settle the debate, we are no closer to answering whether insider trading is even harmful, much less worthy of legal action. In doing so, I challenge the conventional wisdom of framing insider trading research as a quest for resolution to the debate. By adopting an agnostic perspective on the desirability of insider trading regulations, I am able to clearly identify nine issues in this area that are fruitful topics for future research. The second chapter studies prices and returns for movie-specific Arrow-Debreu securities traded on the Iowa Electronic Markets. The payoffs to these securities are based on the movies' initial 4-week U.S. box office receipts. We employ a unique data set for which we have traders' pre-opening forecasts to provide the first direct test of Bossaerts's (2004) ELM hypothesis. We supplement the forecasts with estimated convergence rates to examine whether the prior forecast errors affect market price convergence. Our results support the ELM hypothesis. While significant deviations between initial forecasts and actual box-office outcomes exist, prices nonetheless evolve in accordance with efficient updating. Further, convergence rates appear independent of both the average initial forecast error and the level of disagreement in forecasts. Lastly, the third chapter revisits the theoretical justifications for Bossaerts's (2004) ELM, with the goal of providing clear, intuitive proofs of the key results underlying the methodology. The seemingly biggest hurdle to garnering more widespread adoption of the ELM methodology is the confusion that surrounds the use of weighted modified returns when testing for rational asset pricing restrictions. I attack this hurdle by offering a transparent justification for this approach. I then establish how and why Bossaerts's results extend from the case of digital options to the more practically relevant class of all limited-liability securities, including equities. I conclude by showing that the ELM restrictions naturally lend themselves to estimation and testing of asset pricing models, using weighted modified returns, in a Generalized Method of Moments (GMM) framework.

無匯率風險之跨通貨股權交換是新型態的衍生性金融商品,投資人可用此從事跨國投資,而不承擔匯率風險.本文立基於Dravid,Richardson and Sun(1993), Amin and Bodurhta(1995), 及Lin(1997), 首次推導間斷時間,可計算之無匯率風險下跨通貨單向及雙向股權交換之評價模型.在現金流量法之下,股價指數及匯率的隨機過程設定為lognormal process,利率則跟隨HJM模型.文中發現影響無匯率風險之跨通貨股權交換價格的主要變數為股價指數的波動度,外國股價指數與匯率間的相關性,以及兩國利率差距,與匯率的波動度無關.最後,以三個例子說明此新型態金融工具之運作情形. 1. Preliminary....................................................................................................................................................................1 1.1 Background...............................................................................................................................................................1 1.2 Brief Literature Review on the Pricing of Equity Swaps...........................................................................................3 1.3 The Aim of This Study.............................................................................................................................................4 1.4 Chapter Outline and Results......................................................................................................................................5 2.Introduction to Equity Swaps..........................................................................................................................................7 2.1 Basic Equity Swaps...................................................................................................................................................7 2.2 Variants of Baisc Equity Swaps...............................................................................................................................11 2.3 International Investment Environments....................................................................................................................12 2.4 Cross-Currency Equity Swaps.................................................................................................................................14 3. Literature Review...........................................................................................................................................................19 3.1 Marshall,Sorensen, and Tucker(1992)......................................................................................................................19 3.2 Rich(1995)................................................................................................................................................................20 3.3 Jarrow and Turnbull(1996).......................................................................................................................................22 3.4 Lin(1997)..................................................................................................................................................................25 3.5 Chance and Rich(1998).............................................................................................................................................27 4. Valuation of Cross-Currency Two-Way Equity Swaps Without Currency Risks Under Constant Interest Rates.........35 5. Valuation of Cross-Currency Two-Way Equity Swaps Without Currency Risks Under Stochastic Interest Rates.......43 6. Valuation of Cross-Currency One-Way Equity Swaps Without Currency Risks..........................................................55 7. Case Study......................................................................................................................................................................61 7.1 Case One...................................................................................................................................................................61 7.2 Case Two..................................................................................................................................................................63 7.3 Case Three.................................................................................................................................................................64 8. Conclusions...................................................................................................................................................................67 Appendixes........................................................................................................................................................................69 References .........................................................................................................................................................................92 / Based on Dravid,Richardson, and Sun(1993), Amin and Bodurtha(1995), and Lin(1997), this thesis first derives the computable discrete-time pricing formulas for the cross-currency one-way and two-way equity swaps without currency risks, which are exotic financial derivatives used for cross-border investments without the exchange rate exposure. Under the cash flow approach, equity indexes and the exchange rate are modeled by the lognormal processes, and the interest rate processes follow the HJM model. The swap price is shown to depend on the volatilities of equity indexes, the interaction between the foreign equity index and the exchange rate, as well as the interest rate differential of two countries. It does ont depend on the volatility of the exchange rate. Finally, three cases illustrate the usage of these two exotic financial instruments.

<p>The main objective of this paper is to prove that liquidity costs do exist in option pricingtheory. To achieve this goal, a martingale approach to option pricing theory is usedand, from a model by Jarrow and Protter [JP], a sound theoretical model is derived toshow that liquidity risk exists. This model, derived and tested in this extended theory,allows for liquidity costs to arise. The expression liquidity cost is used in this paper tomeasure liquidity risk relative to the option price.</p>

<p>Aggressive fiscal and monetary policies by governments of countries and central banks in developed markets could somehow push inflation to some very high level in the long run. Due to the decreasing of pension fund benefits and increasing inflation rate, pension companies are selling inflation-linked products to hedge against inflation risk. Such companies are seriously considering the possible effects of inflation volatility on their investment, and some of them tend to include inflationary allowances in the pension payment plan. In this dissertation we study the management of pension funds of the defined contribution type in the presence of inflation-recession. We study how the fund manager maximizes his fund&rsquo / s wealth when the salaries and stocks are affected by inflation. In this regard, we consider the case of a pension company which invests in a stock, inflation-linked bonds and a money market account, while basing its investment on the contribution of the plan member. We use a benchmarking approach and martingale methods to compute an optimal strategy which maximizes the fund wealth.</p>

The main objective of this paper is to prove that liquidity costs do exist in option pricingtheory. To achieve this goal, a martingale approach to option pricing theory is usedand, from a model by Jarrow and Protter [JP], a sound theoretical model is derived toshow that liquidity risk exists. This model, derived and tested in this extended theory,allows for liquidity costs to arise. The expression liquidity cost is used in this paper tomeasure liquidity risk relative to the option price.

The thesis studies the martingale properties, probabilistic methods and efficient unbiased Monte Carlo simulation methods for various time-homogeneous diffusion models commonly used in mathematical finance. Some of the popular stochastic volatility models such as the Heston model, the Hull-White model and the 3/2 model are special cases. The thesis consists of the following three parts: Part I: Martingale properties in time-homogeneous diffusion models: Part I of the thesis studies martingale properties of stock prices in stochastic volatility models driven by time-homogeneous diffusions. We find necessary and sufficient conditions for the martingale properties. The conditions are based on the local integrability of certain deterministic test functions. Part II: Analytical pricing methods in time-homogeneous diffusion models: Part II of the thesis studies probabilistic methods for determining the Laplace transform of the first hitting time of an integral functional of a time-homogeneous diffusion, and pricing an arithmetic Asian option when the stock price is modeled by a time-homogeneous diffusion. We also consider the pricing of discrete variance swaps and discrete gamma swaps in stochastic volatility models based on time-homogeneous diffusions. Part III: Nearly Unbiased Monte Carlo Simulation: Part III of the thesis studies the unbiased Monte Carlo simulation of option prices when the characteristic function of the stock price is known but its density function is unknown or complicated.