Monte Carlo Simulation of Boundary Crossing Probabilities with Applications to Finance and Statistics

Sercan Gür

Publication: ThesisDoctoral thesis

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Abstract

This dissertation is cumulative and encompasses three self-contained research articles. These essays share one common theme: the probability that a given stochastic process crosses a certain boundary function, namely the boundary crossing probability, and the related financial and statistical applications.

In the first paper, we propose a new Monte Carlo method to price a type of barrier option called the Parisian option by simulating the first and last hitting time of the barrier. This research work aims at filling the gap in the literature on pricing of Parisian options with general curved boundaries while providing accurate results compared to the other Monte Carlo techniques available in the literature. Some numerical examples are presented for illustration.

The second paper proposes a Monte Carlo method for analyzing the sensitivity of boundary crossing probabilities of the Brownian motion to small changes of the boundary. Only for few boundaries the sensitivities can be computed in closed form. We propose an efficient Monte Carlo procedure for general boundaries and provide upper bounds for the bias and the simulation error.

The third paper focuses on the inverse first-passage-times. The inverse first-passage-time problem deals with finding the boundary given the distribution of hitting times. Instead of a known distribution, we are given a sample of first hitting times and we propose and analyze estimators of the boundary. Firstly, we consider the empirical estimator and prove that it is strongly consistent and derive (an upper bound of) its asymptotic convergence rate. Secondly, we provide a Bayes estimator based on an approximate likelihood function. Monte Carlo
experiments suggest that the empirical estimator is simple, computationally manageable and outperforms the alternative procedure considered in this paper.
Original languageEnglish
Awarding Institution
  • WU Vienna
Publication statusPublished - 1 Apr 2019

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