Modeling and pricing financial assets under long memory processes
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An important research area in financial mathematics is the study of long memory phenomenon in financial data. Long memory had been known long before suitable stochastic models were developed. Fractional Brownian motion (FBM) can be used to characterize this phenomenon. This thesis examines the use of FBM and its long memory parameter H, from the view point of estimation method, approximation, and numerical performance.How to estimate the long memory parameter H is important in financial pricing. This thesis starts by reviewing the performance of some existing preliminary methods for estimating H. It is then applied to some Malaysia financial data. Although these methods are easy to use, their performance are in doubts, in particular these methods can only get an estimator of H, without providing the dynamic, long-memory behaviour of financial price process.This thesis is therefore concerned with the estimation of the dynamic, long-memory behaviour of financial processes. We propose estimation methods based on models of two stochastic differential equations (SDEs) perturbed by FBM, that play important role in option pricing and interest rate modelling. These models are the geometric fractional Brownian motion (GFBM) and the fractional Ornstein-Uhlenbeck (FOU) model, respectively. These methods are able to obtain H and other parameters involved in the models. The efficiency of these methods are investigated through simulation study. We applied the new methods to some financial problems.We also extend this study to filtering the SDE driven by FBM in multidimensional case. We propose a novel approximation scheme to this problem. The convergence property is also established. The performance of this method is evaluated through solving some numerical examples. Results demonstrate that methods developed in this thesis are applicable and have advantages when compared with other existing approaches.
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