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# Theses and Dissertations - Department of Mathematics

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### Browsing Theses and Dissertations - Department of Mathematics by Subject "Finance"

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Item A non-geometric Brownian motion model estimated by Markov chain approximation(University of Alabama Libraries, 2011) Chang, Hung Yu; Wang, Pu; University of Alabama TuscaloosaShow more The pricing of most contingent claims is continuously monitored the movement of the underlying assets that follow geometric Brownian motion. However, for exotic options, the pricing of the underlying assets is difficult to be obtained analytically. In reality, numerical methods are employed to monitor discretized path-dependent options since complexity of exotic options increases the difficulty of obtaining the closed-form solutions. In this dissertation, we propose a Markov chain method to discretely monitor the underlying asset pricing of an European knock-out call option with time-varying barriers. Markov chain method provides some advantages in computation since the discretized time step can be partitioned to match with the number of the underlying non-dividend paying asset prices. Compared to Monte Carlo simulation, Markov chain method can not only efficiently handle the case where the initial asset price is close to a barrier level but also effectively improve the accuracy of obtaining the price of a barrier option. We study an European knock-out call option with either constant or time-varying barriers. Under risk-neural measure, the movement of the underlying stock price is said to follow a non-geometric Brownian motion. Furthermore, we are interested to estimate the parameter p value that generates optimal payoff of a knock-out option with time-varying barriers. However, implied volatility is an essential factor that affects the movement of the underlying asset price and determines whether the barrier option is knocked out or not during the lifetime of the option.Show more Item An optimal approximation for the payoffs of variance swaps in static replication(University of Alabama Libraries, 2014) Chen, Qiang; Wang, James L.; Wu, Zhijian; University of Alabama TuscaloosaShow more In this dissertation, we create a portfolio of simple vanilla put and call options as an optimal approximation of nonlinear payoffs by using static replication (1995, 1998) [1, 2] under certain measure which is called E(a,b,N,f). More specifically, we focus on the static replication of variance swaps payoffs because of their popularity in current financial market [3]. The analysis is motivated by the following reasons. Due to the limited availability of strike prices with traded vanilla options, static replication is only an approximation [1]. Bradie and Jain (2008) [4] used Black-Scholes and Heston stochastic volatility model to find the optimal approximation. Liu (2010) [5] created three approximation methods. In order to improve the approximation, we use a new measure for the static replication to construct the replicating portfolio with lower cost compared with the current methods.Show more