Some Limit Theorems In Mathematical Finance

The highest order of convergence for traditional binomial models is O(1/n), and convergence is not smooth. We extend Chang-Palmer (2007) method for single parameter by properly choosing the two introduced parameters such that the option price of our model smoothly converges to the corresponding price of European option or digital option at the rate of O(1/n), this enlarges the space of the two parameters defined, and makes our tree converge faster to be of order o(1/n). We propose another family of binomial trees with even numbers of steps under Joshi’s (2010) inspiration for odd numbers of steps, and prove that our trees could be made converge smoothly of any finite positive order by properly choosing the coefficients in the expansion of the moving-up probability.To study the hedging error between the discrete-time and ideal continuous rebal-ancing strategies is also a hot topic. We consider the L2convergence for the general Levy-Ito processes under some technical conditions. Moreover, under some additional conditions, we use the idea of Tankov and voltchkova (2009) for equal distant time lag rebalancing, and prove the stable convergence of the total hedging error about Levy-Ito processes in nonequal distant rebalancing.In addition, from the results of Hayashi and Mykland (2005) for continuous diffusion processes, we prove stable convergence of the relative and total hedging error for discrete data-driven strategy about more general Levy-Ito processes. Note that the limit is not a martingale, but it can be made to converge in law to a martingale by setting a threshold for the renormalized error process.For nonparametric tests in finance, in the past the study of convergence of the threshold estimator of the integrated volatility (IV) on Levy processes was based on finite activity of jumps. We use the method of Mancini (2011) for allowable infinite active jumps to study the speed of convergence of the threshold version of Bipower variation, and analyze the speed of convergence in different cases. We find that this method also can be applied to the study of speed of convergence of the threshold version of integrated quarticity (IQ), and we discuss the speed of convergence in different cases.