Thesis advisor: Nancy Rallis / The theory of Brownian motion is an integral part of statistics and probability, and it also has some of the most diverse applications found in any topic in mathematics. With extensions into fields as vast and different as economics, physics, and management science, Brownian motion has become one of the most studied mathematical phenomena of the late twentieth and early twenty-first centuries. Today, Brownian motion is mostly understood as a type of mathematical process called a stochastic process. The word "stochastic" actually stems from the Greek word for "I guess," implying that stochastic processes tend to produce uncertain results, and Brownian motion is no exception to this, though with the right models, probabilities can be assigned to certain outcomes and we can begin to understand these complicated processes. This work reaches to attain this goal with regard to Brownian motion, and in addition it explores several applications found in the aforementioned fields and beyond. / Thesis (BA) — Boston College, 2007. / Submitted to: Boston College. College of Arts and Sciences. / Discipline: Mathematics. / Discipline: College Honors Program.

Obsahem této práce je popis problematiky opcí. Stěžejním cílem této práce je získat reálný odhad hodnoty opce. K ocenění opcí bude použit základní Black-Scholesův model, který bude rozvinut o vliv dividend. Oceňování je založené na mnohokrát opakované předpovědi budoucí hodnoty podkladové akcie. Tato metoda se pokouší napodobit skutečnou situaci pomocí numerické simulace Monte Carlo.

Doctor of Philosophy(PhD) / This thesis builds on the real estate lease model of Grenadier which consists of the Black Scholes PDE and an upper reflecting boundary condition. Extending the method of images of Buchen, a new technique was developed to solve this class of problems. Problems that previously required difficult integration can now be solved with algebra and simple integrals. In addition, the compound option in this framework is solved using this new technique. To the best of our knowledge the solution of the compound problem has not been published. An interesting symmetry between this class of problems and the lookback option was also discovered and described in this thesis. The extension of the method of images to include problems with the reflecting boundary condition in the context of real estate leases was presented at the Financial Integrity Research Network Doctoral Tutorials at the University of Technology, Sydney, in 2006. The presentation was awarded the ``FIRN Best Paper Award''. This paper has been submitted to the Journal of Financial Mathematics for publication. The solution to the compound problem in the context of the upward-only market review option is the subject of the next paper.

<p>The famous Black-Scholes (BS) model used in the option pricing theory</p><p>contains two parameters - a volatility and an interest rate. Both</p><p>parameters should be determined before the price evaluation procedure</p><p>starts. Usually one use the historical data to guess the value of these</p><p>parameters. For short lifetime options the interest rate can be estimated</p><p>in proper way, but the volatility estimation is, as well in this case,</p><p>more demanding. It turns out that the volatility should be considered</p><p>as a function of the asset prices and time to make the valuation self</p><p>consistent. One of the approaches to this problem is the method of</p><p>uncertain volatility and the static hedging. In this case the envelopes</p><p>for the maximal and minimal estimated option price will be introduced.</p><p>The envelopes will be described by the Black - Scholes - Barenblatt</p><p>(BSB) equations. The existence of the upper and lower bounds for the</p><p>option price makes it possible to develop the worse and the best cases</p><p>scenario for the given portfolio. These estimations will be financially</p><p>relevant if the upper and lower envelopes lie relatively narrow to each</p><p>other. One of the ideas to converge envelopes to an unknown solution</p><p>is the possibility to introduce an optimal static hedged portfolio.</p>

The famous Black-Scholes (BS) model used in the option pricing theory contains two parameters - a volatility and an interest rate. Both parameters should be determined before the price evaluation procedure starts. Usually one use the historical data to guess the value of these parameters. For short lifetime options the interest rate can be estimated in proper way, but the volatility estimation is, as well in this case, more demanding. It turns out that the volatility should be considered as a function of the asset prices and time to make the valuation self consistent. One of the approaches to this problem is the method of uncertain volatility and the static hedging. In this case the envelopes for the maximal and minimal estimated option price will be introduced. The envelopes will be described by the Black - Scholes - Barenblatt (BSB) equations. The existence of the upper and lower bounds for the option price makes it possible to develop the worse and the best cases scenario for the given portfolio. These estimations will be financially relevant if the upper and lower envelopes lie relatively narrow to each other. One of the ideas to converge envelopes to an unknown solution is the possibility to introduce an optimal static hedged portfolio.

This thesis studies binomial and trinomial lattice approximations in Black-Scholes type option pricing models. Also, it covers the basics of these models, derivations of model parameters by several methods under different kinds of distributions. Furthermore, the convergence of the binomial model to normal distribution, Geometric Brownian Motion and Black-Scholes model is discussed. Finally, the connections and interrelations between discrete random variables under the Lattice approach and continuous random variables under models which follow Geometric Brownian Motion are discussed, compared and contrasted.

No description available.

Black-Scholes Model is a widely used mathematical model for stock price behaviors, of which the return is assumed to be normally distributed. But this 'normally distributed' assumption is doubted and proved to be not true by realistric data. The main goal of this thesis is to explore polynomial-normal distribution, and use this distribution in the stock return, as a non-normal extension of the Black-Scholes Model. We will develop the properties of polynomial-normal distribtuion in the thesis, and also give the European call and put option price formulas under this model, and show how to use this model to estimate real stock returns.

Doctor of Philosophy(PhD) / This thesis builds on the real estate lease model of Grenadier which consists of the Black Scholes PDE and an upper reflecting boundary condition. Extending the method of images of Buchen, a new technique was developed to solve this class of problems. Problems that previously required difficult integration can now be solved with algebra and simple integrals. In addition, the compound option in this framework is solved using this new technique. To the best of our knowledge the solution of the compound problem has not been published. An interesting symmetry between this class of problems and the lookback option was also discovered and described in this thesis. The extension of the method of images to include problems with the reflecting boundary condition in the context of real estate leases was presented at the Financial Integrity Research Network Doctoral Tutorials at the University of Technology, Sydney, in 2006. The presentation was awarded the ``FIRN Best Paper Award''. This paper has been submitted to the Journal of Financial Mathematics for publication. The solution to the compound problem in the context of the upward-only market review option is the subject of the next paper.

Univ., Diss.--Lausanne, 2002. / Zsfassung in engl. Sprache.