Numerical methods in financial mathematics with applications to option pricing and anomaly detection

In the wake of the 2008 global financial crisis, financial institutions confront intricate mathematical challenges in managing diverse risks. While many proposals to tackle these challenges can be found in the academic literature, some prove impractical due to high computational costs. Furthermore, risk priorities vary among financial entities. Pension funds and insurers may emphasize mortality risk, tied to unexpected deviations in policyholders' lifespans. Conversely, banks, dealing with stocks and interest rates, prioritize market risk, particularly in the face of sudden asset price fluctuations known as "price jumps." Addressing modern concerns, many financial institutions also invest in Anti-Money Laundering infrastructures, deploying anomaly detection algorithms to discern legitimate from illicit transactions. These algorithms, falling under the data mining category, seek anomalous observations deviating from normal transaction patterns. The thesis responds to these challenges, striving for practical and accurate alternatives. Regarding mortality risk, it explores novel solutions for joint and survivor annuity pricing based on urn models, considering correlations in lifetimes. The proposed method also handles incomplete data, a common occurrence in survival studies. With respect to market risk, the thesis analyzes a novel jump-diffusion asset price model that embeds contagion effects: jumps on the asset price trigger further jumps, simulating the arrival of news in the market. Finally, the thesis proposes a new anomaly detection algorithm, combining ideas from different methods, that is parameter-free. An explanation method is also introduced in order to understand why certain transactions are flagged as anomalies.