Traditional financial modeling is based on semimartingale processes with stationary and independent
increments. However, empirical investigations on financial data does not always
support these assumptions. This contradiction showed that there is a need for new stochastic
models. Fractional Brownian motion (fBm) was proposed as one of these models by Benoit
Mandelbrot. FBm is the only continuous Gaussian process with dependent increments. Correlation
between increments of a fBm changes according to its self-similarity parameter H. This
property of fBm helps to capture the correlation dynamics of the data and consequently obtain
better forecast results. But for values of H different than 1/2, fBm is not a semimartingale and
classical Ito formula does not exist in that case. This gives rise to need for using the white noise
theory to construct integrals with respect to fBm and obtain fractional Ito formulas. In this
thesis, the representation of fBm and its fundamental properties are examined. Construction of
Wick-Ito-Skorohod (WIS) and fractional WIS integrals are investigated. An Ito type formula
and Girsanov type theorems are stated. The financial applications of fBm are mentioned and
the Black& / Scholes price of a European call option on an asset which is assumed to follow a
geometric fBm is derived. The statistical aspects of fBm are investigated. Estimators for the
self-similarity parameter H and simulation methods of fBm are summarized. Using the R/S methodology of Hurst, the estimations of the parameter H are obtained and these values are used to evaluate the fractional Black& / Scholes prices of a European call option with different
maturities. Afterwards, these values are compared to Black& / Scholes price of the same option
to demonstrate the effect of long-range dependence on the option prices. Also, estimations
of H at different time scales are obtained to investigate the multiscaling in financial data. An
outlook of the future work is given.
| Identifer | oai:union.ndltd.org:METU/oai:etd.lib.metu.edu.tr:http://etd.lib.metu.edu.tr/upload/12613736/index.pdf |
| Date | 01 October 2011 |
| Creators | Inkaya, Alper |
| Contributors | Hayfavi, Azize |
| Publisher | METU |
| Source Sets | Middle East Technical Univ. |
| Language | English |
| Detected Language | English |
| Type | M.S. Thesis |
| Format | text/pdf |
| Rights | To liberate the content for public access |
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