Texas’ Mass Emission Cap and Trade program is a mandatory Nitrous Oxide (NOx) abatement program for medium and large stationary sources located in the Houston-Galveston ozone non-attainment area. Effected companies are required to upgrade equipment to meet the current best achievable NOx control technology (BACT) standards or to purchase emission credits in sufficient quantity to cover the difference in emissions between existing equipment and equipment meeting the BACT standard. With over 260 participating companies, the market for emission credits is ever changing, making it difficult to evaluate whether the lowest cost decision is to upgrade equipment or to purchase NOx emission credits. Because equipment upgrades are capital investments, a well informed, rational decision can have a significant impact on the corporate balance sheet. The objective of this research is to aid the decision maker by predicting credit prices based on a Geometric Brownian Motion model based on historical NOx emission credit transactions. The predicted credit price is useful in evaluating the likelihood of the equipment upgrade option being a favorable or unfavorable decision. For the examined cases, modeled results indicate that equipment upgrade is the more cost effective option. / text

The paper examines the impact of fees on the return of a fund investment using different simulation and fee structure models. The results show that fees have a significant expected impact, particularly for well-performing funds. Two simulation models were used, the Geometric Brownian Motion (GBM) model and Merton Jump Diffusion (MJD) model. Two fee structures were also analysed for each simulation, a High-water mark fee structure and a Hurdle fee structure. Comparing the GBM and MJD models, the two tend to generate very similar fee statistics even though the MJD model's day-to-day returns fit better with empirical data. When comparing the HWM and Hurdle fee models, larger differences are observed. While overall average fee statistics are similar, the performance fee statistics are significantly higher in the Hurdle fee structure for assets achieving higher returns, e.g. at least an 8% annual return. However, the HWM fee structure tends to generate higher performance fees for assets with low returns. Regression models are also developed for each combination of the simulation model and fee structure. The regression models reflect the above conclusions and can for investors serve as simple key indicators to estimate expected fund fee payments. The GBM regression results are likely more useful than the MJD regression results, as the parameters of the former are easier to calculate based on historical return data. / Uppsatsen undersöker effekten av avgifter på avkastningen av en fondinvestering med hjälp av olika simuleringar och avgiftsmodeller. Resultaten visar att avgifter förväntas ha en betydande påverkan, särskilt för fonder som genererar hög avkastning. Två simuleringar användes, Geometric Brownian Motion (GBM) och Merton Jump Diffusion (MJD). Två avgiftsstrukturer analyserades också för varje simulering, en High-water mark avgiftsstruktur och en Hurdle avgiftsstruktur. Jämförelse mellan GBM och MJD-modellerna visar att de två tenderar att generera mycket liknande avgiftsstatistik trots att MJD-modellens dagliga avkastning passar bättre med empiriska data. Vid jämförelse av HWM- och Hurdle avgiftsmodellerna observeras större skillnader. Medan den övergripande genomsnittliga avgiftsstatistiken är liknande för avgiftsmodellerna, är resultatbaserade avgifterna betydligt högre i Hurdle avgiftsstrukturen för tillgångar som uppnår högre avkastning, t.ex. minst 8% årlig avkastning. Däremot tenderar HWM-avgiftsstrukturen att generera högre resultatbaserade avgifter för tillgångar med låg avkastning. Regressionsmodeller utvecklades också för varje kombination av simulering och avgiftsstruktur. Regressionmodellerna återspeglar ovanstående slutsatser och kan för investerare fungera som enkla nyckeltal för att uppskatta förväntad kostnad av fondavgifter. GBM-regressionsresultaten är sannolikt mer användbara än MJD-regressionsresultaten, eftersom parametrarna för den förra är lättare att beräkna baserat på historisk avkastningsdata.

The Black-Scholes formula was the widely-used model for option pricing, this formula can be use to calculate the price of option by using current underlying asset prices, strike price, expiration time, volatility and interest rates. The European call option price from the model is a convex and increasing with respect to the initial underlying asset price. Assume underlying asset prices follow a generalized geometric Brownian motion, it is true that option prices increasing with respect to the constant interest rate and volatility, so that the volatility can be a very important factor in pricing option, if the volatility process £m(t) is constant (with £m(t) =£m for any t ) satisfying £m_1 ≤ £m(t) ≤ £m_2 for some constants £m_1 and £m_2 such that 0 ≤ £m_1 ≤ £m_2. Let C_i(t, S_t) be the price of the call at time t corresponding to the constant volatility £m_i (i = 1,2), we will derive that the price of call option at time 0 in the model with varying volatility belongs to the interval [C_1(0, S_0),C_2(0, S_0)].

This Master thesis studies Asian perpetuities, which is a term standing for European type of options with an average asset as the underlying asset and the execution time of the option in infinity. Assuming Geometric Brownian motion model of price of an asset, the goal of this thesis is to study behavior of the average of the asset price. Three different types of averaging are considered: arithmetic, geometric and harmonic average. The average values of the log-normals maintain the known distribution only for the geometric average. As it is shown in the thesis; however, when the average is examined on infinite time horizon, the arithmetic and harmonic averages maintain the inverse gamma distribution or gamma distribution, respectively. This result enables the computation of the price of Asian perpetuity which is also examined in the thesis. 1

Yes / We introduce a continuous time model for stock prices in a general factor representation with the noise driven by a geometric Brownian motion process. We derive the theoretical hitting probability distribution for the long-until-barrier strategies and the conditions for statistical arbitrage. We optimize our statistical arbitrage strategies with respect to the expected discounted returns and the Sharpe ratio. Bootstrapping results show that the theoretical hitting probability distribution is a realistic representation of the empirical hitting probabilities. We test the empirical performance of the long-until-barrier strategies using US equities and demonstrate that our trading rules can generate statistical arbitrage profits.

This paper explores different covariance matrix estimators in application to geometric Brownian motion. Particular interest is given to shrinkage estimation methods. In collaboration with Söderberg & Partners risk management team, the goal is to find an estimation that performs well in low-data scenarios and is robust against erroneous model assumptions, particularly the Gaussian assumption of the stock price distribution. Estimations are compared by two criteria: Frobenius norm distance between the estimate and the true covariance matrix, and the condition number of the estimate. By considering four estimates — the sample covariance matrix, Ledoit-Wolf, Tyler M-estimator, and a novel Tyler-Ledoit-Wolf (TLW) estimator — this paper concludes that the TLW estimator performs best when considering the two criteria.

The financial market is a stochastic and complex system that is challenging to model. It is crucial for investors to be able to model the probability of possible outcomes of financial investments and financing decisions in order to produce fruitful and productive investments. This study investigates how Monte Carlo simulations of random walks can be used to model the probability of future stock returns and how the simulations can be improved in order to provide better accuracy. The implemented method uses a mathematical model called Geometric Brownian Motion (GBM) in order to simulate stock prices. Ten Swedish large-cap stocks were used as a data set for the simulations, which in turn were conducted in time periods of 1 month, 3 months, 6 months, 9 months and 12 months. The two main parameters which determine the outcome of the simulations are the mean return of a stock and the standard deviation of historical returns. When these parameters were calculated without weights the method proved to be of no statistical significance. The method improved and thereby proved to be statistically significant for predictions for a 1 month time period when the parameters instead were weighted. By varying the assumptions regarding price distribution with respect to the size of the current time period and using other weights, the method could possibly prove to be more accurate than what this study suggests. Monte Carlo simulations seem to have the potential to become a powerful tool that can expand our abilities to predict and model stock prices. / Den finansiella marknaden är ett stokastiskt och komplext system som är svårt att modellera. Det är angeläget för investerare att kunna modellera sannolikheten för möjliga utfall av finansiella investeringar och beslut för att kunna producera fruktfulla och produktiva investeringar. Den här studien undersöker hur Monte Carlo-simuleringar av så kallade random walks kan användas för att modellera sannolikheten för framtida aktieavkastningar, och hur simuleringarna kan förbättras för att ge bättre precision. Den implementerade metoden använder den matematiska modellen Geometric Brownian Motion (GBM) för att simulera aktiepriser. Tio svenska large-cap aktier valdes ut som data för simuleringarna, som sedan gjordes för tidsperioderna 1 månad, 3 månader, 6 månader, 9 månader och 12 månader. Huvudparametrarna som bestämmer utfallet av simuleringarna är medelvärdet av avkastningarna för en aktie samt standardavvikelsen av de historiska avkastningarna. När dessa parametrar beräknades utan viktning gav metoden ingen statistisk signifikans. Metoden förbättrades och gav då statistisk signifikans på en 1 månadsperiod när parametrarna istället var viktade. Metoden skulle kunna visa sig ha högre precision än vad den här studien föreslår. Det är möjligt att till exempel variera antagandena angående prisernas fördelning med avseende på storleken av den nuvarande tidsperioden, och genom att använda andra vikter. Monte Carlo-simuleringar har därför potentialen att utvecklas till ett kraftfullt verktyg som kan öka vår förmåga att modellera och förutse aktiekurser.

In the literature, many methods are proposed to value American options. However, due to computational difficulty, there are only approximate solution or numerical method to evaluate American options. It is not easy for general investors either to understand nor to apply. In this thesis, we build up an option pricing and virtual asset model system, which provides a friendly environment for general public to calculate early exercise boundary of an American option. This system modularize the well-handled pricing models to provide the investors an easy way to value American options without learning difficult financial theories. The system consists two parts: the first one is an option pricing system, the other one is an asset model simulation system. The option pricing system provides various option pricing methods to the users; the virtual asset model system generates virtual asset prices for different underlying models.

In this thesis, we expand the option pricing and virtual asset model system by Cheng (2005) and include new simulations and maximum likelihood estimation of the parameter of the stochastic differential equations. For easy manipulation of general users, the interface of original option pricing system is modified. In addition, in order to let the system more completely, some stochastic models and methods of pricing and estimation are added. This system can be divided into three major parts. One is an option pricing system; The second is an asset model simulation system; The last is estimation system of the parameter of the model. Finally, the analysis for the data of network are carried out. The differences of the prices between estimator of this system and real market are compared.

Stochastic hybrid dynamic systems that incorporate both continuous and discrete dynamics have been an area of great interest over the recent years. In view of applications, stochastic hybrid dynamic systems have been employed to diverse fields of studies, such as communication networks, air traffic management, and insurance risk models. The aim of the present study is to investigate properties of some classes of stochastic hybrid dynamic systems. The class of stochastic hybrid dynamic systems investigated has random jumps driven by a non-homogeneous Poisson process and deterministic jumps triggered by hitting the boundary. Its real-valued continuous dynamic between jumps is described by stochastic differential equations of the It\^o-Doob type. Existing results of piecewise deterministic models are extended to obtain the infinitesimal generator of the stochastic hybrid dynamic systems through a martingale approach. Based on results of the infinitesimal generator, some stochastic stability results are derived. The infinitesimal generator and stochastic stability results can be used to compute the higher moments of the solution process and find a bound of the solution. Next, the study focuses on a class of multidimensional stochastic hybrid dynamic systems. The continuous dynamic of the systems under investigation is described by a linear non-homogeneous systems of It\^o-Doob type of stochastic differential equations with switching coefficients. The switching takes place at random jump times which are governed by a non-homogeneous Poisson process. Closed form solutions of the stochastic hybrid dynamic systems are obtained. Two important special cases for the above systems are the geometric Brownian motion process with jumps and the Ornstein-Uhlenbeck process with jumps. Based on the closed form solutions, the probability distributions of the solution processes for these two special cases are derived. The derivation employs the use of the modal matrix and transformations. In addition, the parameter estimation problem for the one-dimensional cases of the geometric Brownian motion and Ornstein-Uhlenbeck processes with jumps are investigated. Through some existing and modified methods, the estimation procedure is presented by first estimating the parameters of the discrete dynamic and subsequently examining the continuous dynamic piecewisely. Finally, some simulated stochastic hybrid dynamic processes are presented to illustrate the aforementioned parameter-estimation methods. One simulated insurance example is given to demonstrate the use of the estimation and simulation techniques to obtain some desired quantities.