In this thesis, I present three research topics on quantitative finance. The first research is about the calibration of the stochastic local volatility model, where we have developed two robust and efficient methods. The first method is to calculate the joint density by solving the Kolmogorov forward partial differential equation (PDE). The second method is a direct fitting method. It is shown that they both methods will recover the input market European option prices. The second research topic is about the optimal quoting policy of an option market maker, where I provide the optimal bid and ask quotes for maximizing the expected terminal wealth function at maturity and minimizing the variance risk simultaneously by using the standard tools in optimal stochastic control. In the...[ Read more ]

In this thesis, I present three research topics on quantitative finance. The first research is about the calibration of the stochastic local volatility model, where we have developed two robust and efficient methods. The first method is to calculate the joint density by solving the Kolmogorov forward partial differential equation (PDE). The second method is a direct fitting method. It is shown that they both methods will recover the input market European option prices. The second research topic is about the optimal quoting policy of an option market maker, where I provide the optimal bid and ask quotes for maximizing the expected terminal wealth function at maturity and minimizing the variance risk simultaneously by using the standard tools in optimal stochastic control. In the numerical experiment, Monte Carlo simulations are used to demonstrate that our strategy better manage the inventory risk, when the underlying asset price process is assumed to follow a Heston model. The third topic is to specify an optimal delta hedging ratio for the option driven by historical data without a model for the underlying asset price. The data-driven approach is designed to achieve the minimal variance in delta hedging, which also means that it can better explain the option price movement. Compared with previous research on this topic, we apply more machine learning methodologies and investigate more features. As a result, I obtain better hedging effect, and the added new features are proved to be important.

Keywords: stochastic local volatility model, calibration, option market making, stochastic control, optimal delta hedging, machine learning