Stochastic volatility models with applications in finance

Financial institutions, like banks and insurance companies, are responsible for carefully managing the huge amount of assets collected from different sources (e.g. investors and the government). Although different financial institutions have various investing strategies, they are all facing the risk coming from many unpredictable future events, like the financial crisis. The financial derivatives, serve as insurance for financial institutions, and provide the practical tools to hedge many main kinds of risks. Therefore, the financial derivatives should not be free (just like you should pay the health insurance before having it). To price the financial derivative is not easy, and thus the stochastic calculus as the ideal tool comes in. The double Heston model is one of those well–known practical pricing models which computes the price of many derivatives including European options. This thesis implements the double Heston model in an optimized way which reduces the pricing time by about 15%. The sensitivity analysis of the pricing model is also very important since it provides us the window to mathematically observe the risk. Another contribution of this thesis is: a novel application of the automatic algorithmic differentiation algorithm in order to achieve a more stable, more accurate, and faster sensitivity analysis for the double Heston model (compared to the classical finite difference methods). Lastly, our study presents a novel hybrid calibration method, which mixes the Differentiation Evolution algorithm and the Levenberg–Marquardt algorithm. Our new hybrid model significantly accelerates the calibration process.