Option pricing with fractional brownian motion

Download
index.pdf
2011
İnkaya, Alper
Show full item record
175
views
73
downloads
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.
Brownian motion processes
http://etd.lib.metu.edu.tr/upload/12613736/index.pdf
https://hdl.handle.net/11511/21107
Graduate School of Applied Mathematics, Thesis

Suggestions

An approach for decentralized process modeling
Turetken, Oktay; Demirörs, Onur (2007-05-20)
This paper describes a method for organizations to perform process modeling in a decentralized and concurrent manner. The approach is based on the idea that modeling organizations' processes can be performed by individuals actually performing the processes. Instead of having a central and devoted group of people to analyze, understand, model and improve processes, real performers are held responsible to model and improve their own processes concurrently. The paper also summarizes the lessons learned by appl...
Investigation of fractional black scholes option pricing approaches and their implementations
Hergüner, Ecem; Uğur, Ömür; Department of Financial Mathematics (2015)
One of the fundamental research areas in the financial mathematics is option pricing. With the emergence of Black-Scholes model, the partial differential equations (PDE) for option pricing have started to be used widely. PDEs are adopted for both finding numerical and analytical solutions and developing new models for option pricing. One of the significant PDE is fractional Black-Scholes PDE. Essentially, a PDE can become non-local with fractionalization and this non-localization enables to expand the time ...
Process Maintenance through Component-Process Replacement
Manzer, Ayesha; Doğru, Ali Hikmet (2006-03-01)
An enterprise is represented by its process model that is constructed by the integration of smaller processes corresponding to value-added contributors. Replacing the sub-processes can modify the super-process. Component processes are represented in task systems in order to discover how process attributes will be preserved after integration. This approach is especially versatile if virtual enterprises are formed over the Internet through integrating the published processes of core competencies. The virtuall...
Analytical modeling of a time-threshold based multi-guard bandwidth allocation scheme for cellular networks
Candan, Idil; Salamah, Muhammed (2009-05-28)
In this paper, the analytical modeling of a time-threshold based multi-guard bandwidth allocation scheme is presented using a two-dimensional markov chain. The main idea of the scheme is based on monitoring the elapsed real time of handoff calls and according to a time threshold (t(e)), a handoff call is either slightly-prioritized or fully-prioritized. A slightly-prioritized handoff call has higher priority than a new call and lower priority than a fully-prioritized handoff call. Also in this paper, the nu...
Feature based design of rotational parts based on step
Fidan, Tahir; Kılıç, Engin; Department of Mechanical Engineering (2004)
The implicit and low-level part definition data, provided by geometric modeling cannot be used by downstream applications. Therefore, feature based modeling concept has been introduced to integrate CAD and downstream applications. However, due to the lack of implicit and explicit standard representations for features and unmanageable number of possible predefined features without standardization, feature based modeling approach has proved to be inadequate. STEP AP224 provides a standard for both implicit an...
Citation Formats
A. İnkaya, “Option pricing with fractional brownian motion,” M.S. - Master of Science, Middle East Technical University, 2011.
METU IIS - Integrated Information Service open@metu.edu.tr