This thesis analyzes perpetual American options and in particular aimsto identify what kind of behavior the implied volatility of perpetual Americanput options display. Using two different approaches, we derive the valuefunction for the American put option. Moreover, by assuming the volatilityfunction of the underlying model to be decreasing we find that this inducesan implied volatility skew. Furthermore, this claim is illustrated with numericalcalculations.

A trajectory is the movement of an object in space as time goes by,aka ‘spatiotemporal’. Aside from all the properties an ordinary trajectorypossesses, the primary characteristic of a distributed trajectoryis the uncertainty of its positioning. Each movement will generate a(probability) distribution so that the location for each record is notdeterministic. This thesis will present the dynamics of distributedtrajectories under Mobile Phone Network (MPN) of certain areas inSweden, by demonstrating a series of visualisations and how they areapplied in the real world with estimation and simulation of certainspatiotemporal Markov models. Our main objective is to develop andtest estimation and simulation algorithms that can scale to arbitrarilylarge MPN datasets.

Three problems in distributed computing and non-parametric computationalstatistics are explored to develop scalable distributed algorithms. The problemsare rigorous global optimization, trend detection in time series, and nonparametricdensity estimation with tree-based histograms. Scalable algorithmsare implemented for each problem using the fault-tolerant distributed frameworkof Apache Spark.

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In this thesis, we analyze the SEK/USD exchange rate by using the multivari-ate time series models. First we sort out the general workflow. Then we selectthe variables by the Granger causality test and design different combinationsof variables for models. After building VAR models and TVP-VAR modelsfor different variables,the analysis focuses on comparison of the impulse re-sponse function and the variance decomposition. To evaluate the performanceof the models, we split the real data into training data and validation data, andcompare the prediction accuracy using the validation data.

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