Stochastic Clock and Financial Mathematics

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Yansori, Sina | 2011

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Type of Document: M.Sc. Thesis
Language: Farsi
Document No: 41408 (02)
University: Sharif University of Technology
Department: Mathematical Sciences
Advisor(s): Zohuri Zangeneh, Bijan
Abstract:
Brownian motion played a central role throughout the twentieth century in probability theory. The same statement is even truer in finance, with the introduction in 1900 by the French mathematician Louis Bachelier of an arithmetic Brownian motion (or a version of it) to represent stock price dynamics. This process was pragmatically transformed by Samuelson in 1965 into a geometric Brownian motion ensuring the positivity of stock prices. More recently, the elegant martingale property under an equivalent probability measure derived from the no-arbitrage assumption combined with Monroe's theorem on the representation of semi martingales has led to write asset prices as time-changed Brownian motion. Independently, Clark [1973] had the original idea of writing cotton future prices as subordinated processes, with Brownian motion as the driving process. Over the last few years, time changes have been used to account for different speeds in market activity in relation to news arrival as the stochastic clock goes faster during periods of intense trading. In this project, We try to uncover new classes of processes in asset price modeling
Keywords:
Brownian motion ; Martingales ; Stochastic Volatility Model ; Time-Changed Stochastic Process ; Levy Process ; Financial Modeling

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