Levy processes and (quasi-)Monte Carlo methods in finance

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One of the assumptions in the famous Black-Scholes model used in financial derivative pricing is that the log returns of an asset follow the normal or Gaussian distribution. Although it is widely used in practice, this model has serious drawbacks, such as heavy tail, volatility smile / surface, etc. The stochastic volatility models under the BS framework can improve the results to some extent. A quite different approach is to replace the driving processes for the underlying assets by more realistic ones. Levy processes provide a large variety of distributions to serve such purposes. Empirical studies showed that a special class of Levy process based on the generalized hyperbolic distributions can fit the real financial data much better than the Black-Scholes model. A lot of important work on derivative pricing under the generalized hyperbolic distributions has been done in recent years. However, the use of new distributions results in a lacking of closed formulas and more complex problems that need to be explored. On numerical aspects, efficient numerical methods such as Monte Carlo/quasi-Monte Carlo methods are desperately needed.   There are two main objectives of my  research. The first one is to construct new quasi-random sequences, also known as the low-discrepancy sequences. The second one is to study problems related to financial derivative pricing and risk management under more realistic models and their numerical computation methods.   This research should be of high interest for both academics and industry.