Option valuation is one of the more applied areas of mathematics. Options are financial derivatives whose value depends on the value of an underlying asset. They are frequently used in hedging to minimize the risk when trading in the underlying stock and therefore require accurate pricing. Though quite some research has been conducted on standard option types such as European options and American options, more exotic options such as conditional Asian options or Asian tail options are less known and therefore enjoyed far less attention from the research community. In this bachelor thesis we will focus on computing the correct price for the Asian option and various types of Asian options. A short introduction on these types of options will be given, after which mathematical models for pricing these types of options will be derived. Several numerical methods such as the Monte Carlo method and the finite difference method (applied to the Black-Scholes partial differential equation) will be used to approximate the true value of these types of options.

In financial mathematics, stochastic processes are regularly used to describe observed financial indicators such as stocks, options, futures or interest rates. Identifying the underlying dynamics of observed financial time series is crucial in risk management, as it greatly affects pricing and hedging strategies. The large number of available stochastic processes make selecting the most suitable stochastic process a non-trivial problem. Additionally, realisations of stochastic processes are elements of the path space which is infinite-dimensional and non-locally compact. Given these observations, we find that model selection methods from classical statistics, such as distribution metrics, are inadequate. In this thesis a signature-based model recognition method is proposed. The goal of this model is to select the most suitable stochastic process to describe an observed financial time series. Signatures are transformations from a path to an infinite-length sequence of properties of that path, which makes signatures a highly interesting approach to construct input features which can be used by a machine learning model. In our evaluation, we use the aforementioned methodology to distinguishing between various classification settings of Arithmetic Brownian Motion, Geometric Brownian Motion and the exponential jump diffusion process, both in a binary and multi-class classification setting. This evaluation shows that the proposed method can adequately distinguish between the same model with different parameters, models with and without jumps and models with different jump sizes and jump intensities.