Stochastic Modelling in Finance

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News coming from financial markets has a profound impact on people's everyday lives. Rising interest rates lead to larger mortgage payments. Fluctuations of crude oil prices are propagated into gas price tags. The credit rating of a government affects the yield rate of bonds issued to cover the budget deficit. The importance and complexity of financial markets explain the rapid development of financial mathematics in the last three decades. Mathematical modelling is a main research approach to describing and studying complex real-world systems such as financial markets where millions of transactions are made every day. Stochastic processes are a perfect instrument for the simulation of such dynamically changing uncertain systems with many coupled degrees of freedom. That is why stochastic modelling is at the core of financial mathematics. The focus of this research is the development and implementation of methods to help quantify and control financial risk. Modern day financial markets offer and trade increasingly more complex products. A path-dependent derivative is a financial contract whose value depends on the time history of values of other, more basic, underlying financial variables. Such options have become increasingly popular over the last few years because of the greater precision with which they allow investors to choose or avoid exposure to well-defined sources of risk. The numerical part of the proposed project is devoted to high-performance computational methods for pricing and hedging path-dependent financial instruments. To take into account realistic market effects such as price jumps, stochastic volatility, liquidity issues, and credit risk, many sophisticated financial models have been proposed. However, models used by practitioners for controlling risk are often too simplistic. By developing more realistic but still tractable stochastic models along with pricing and hedging algorithms and software, we can facilitate the implementation of recent achievements of financial mathematics in industry. Such models provide a flexible framework to account for many real market effects. On the other hand, we are able to construct closed-form solutions or efficient numerical algorithms for pricing various financial instruments.