The purpose of this paper is threefold. First, the paper builds on the work done previously done in the area of option implied probability distribution functions (PDFs) by extending the methods described by Breeden and Litzenberger (1978) to individual equity options. Second, it describes a closed-form, onto mapping from a two-dimensional volatility surface to the risk-neutral PDF. Lastly the paper performs an event study on the implied risk-neutral PDFs of companies which are the target of corporate takeover. While there was not sufficient data to determine any statistical relationship, there is observational evidence that option market implied PDFs may be predictive of future takeovers.

Radial basis function (RBF) approximation, is a new extremely powerful tool that is promising for high-dimensional problems, such as those arising from pricing of basket options using the Black-Scholes partial differential equation. The main problem for RBF methods have been ill-conditioning as the RBF shape parameter becomes small, corresponding to flat RBFs. This thesis employs a recently developed method called the RBF-QR method to reduce computational cost by improving the conditioning, thereby allowing for the use of a wider range of shape parameter values. Numerical experiments for the one-dimensional case are presented  and a MATLAB implementation is provided. In our thesis, the RBF-QR method performs better  than the RBF-Direct method for small shape parameters. Using Chebyshev points, instead of a standard uniform distribution, can increase the accuracy through clustering of the nodes towards the boundary. The least squares formulation for RBF methods is preferable to the collocation approach because it can result in smaller errors  for the same number of basis functions.

In this dissertation we first formulate the Black-Scholes equation with a tensor (or matrix) diffusion coefficient into a conservative form and present a convergence analysis for the two-dimensional Black-Scholes equation arising in the Hull-White model for pricing European options with stochastic volatility. We formulate a non-conforming Petrov-Galerkin finite element method with each basis function of the trial space being determined by a set of two-point boundary value problems defined on element edges. We show that the bilinear form of the finite element method is coercive and continuous and establish an upper bound of order O(h) on the discretization error of method, where h denotes the mesh parameter of the discretization. We then present a finite volume method for the resulting equation, based on a fitting technique proposed for a one-dimensional Black-Scholes equation. We show that the method is monotone by proving that the system matrix of the discretized equation is an M-matrix. Numerical experiments, performed to demonstrate the usefulness of the method, will be presentd.

Option is an asymmetric contract between two parties with future payoff derived from the price of underlying asset. Methods of pricing di erent types of options under more or less general assumptions have been extensively studied since the Nobel price winning works of Black and Scholes [1] and Merton [12] were published in 1973. A new way of pricing options with the use of Mellin transforms have been recently introduced by Panini and Srivastav [15] in 2004. This thesis offers a brief introduction to option pricing with Mellin transforms and a revision of some of the recent research in this field.

Asian option is an option which payoff depends on the average underlying price over some some specific time period. Although there is no closed form solution of asian option, appropriate change of variable and Num¡¦eraire would reduce some terms of equation satisfies the Asian call price function. This thesis presents asian option¡¦s properties and process of reduction terms.

A num¡¦eraire is the unit of account in which other assets are denominated. One usually takes the num¡¦eraire to be the currency of a country. In some applications one must change the num¡¦eraire due to the finance considerations. And sometimes it is convenient to change the num¡¦eraire because of modeling considerations. A model can be complicated or simple, depending on the choice of thenum¡¦eraire for the method. When change the num¡¦eraire, denominating the asset in some other unit of account, it is no longer a martingale under ˜P . When we change the num¡¦eraire, we need to also change the risk-neutral measure in order to maintain risk neutrality. The details and some applications of this idea developed in this thesis.

The Black-Scholes formula was the widely-used model for option pricing, this formula can be use to calculate the price of option by using current underlying asset prices, strike price, expiration time, volatility and interest rates. The European call option price from the model is a convex and increasing with respect to the initial underlying asset price. Assume underlying asset prices follow a generalized geometric Brownian motion, it is true that option prices increasing with respect to the constant interest rate and volatility, so that the volatility can be a very important factor in pricing option, if the volatility process £m(t) is constant (with £m(t) =£m for any t ) satisfying £m_1 ≤ £m(t) ≤ £m_2 for some constants £m_1 and £m_2 such that 0 ≤ £m_1 ≤ £m_2. Let C_i(t, S_t) be the price of the call at time t corresponding to the constant volatility £m_i (i = 1,2), we will derive that the price of call option at time 0 in the model with varying volatility belongs to the interval [C_1(0, S_0),C_2(0, S_0)].

This article emphasizes on fitting GARCH option pricing model with Taiwan stock market. Duan¡¦s(1995) NGARCH option pricing model is adopted. Duan solved the European option by simulation, this article follow the method and extents to pricing American option. In general, simulation approach is not convenient to solve American options as well as European options. However, the least-squares method proposed by Longstaff and Schwartz is a simple and powerful tool, so this article tests the method. The NGARCH model has parameters, and base on loglikelihood function, we fit the model with empirical observations to obtain parameters. Then we can simulate the stock prices, once stock prices are simulated, the option value can be priced. Since the article simulates the option, there should be the antithetic approaches instead of simulation. In practice, the Black-Schoels model is the benchmark for pricing European option, so this article compares the simulated European options with Black-Scholes. For American option, this article compares the simulated American options which are priced by least-squares method with trinomial tree (finite difference method).

This article examines the hedging positions derived from the Black-Scholes(B-S) model and the GARCH(1,1) models, respectively, when the log returns of underlying asset exhibits GARCH(1,1) process. The result shows that Black-Scholes and GARCH options deltas, one of the hedging parameters, are similar for near-the-money options, and Black-Scholes options delta is higher then GARCH delta in absolute terms when the options are deep out-of-money, and Black-Scholes options delta is lower then GARCH delta in absolute terms when the options are deep in-the-money. Simulation study of hedging procedure of GARCH(1,1) and B-S models are performed, which also support the above findings.

由於忽略員工認股選擇權的稀釋性會造成偏誤的企業評價，本研究利用以認購權證為基礎的改良評價模型，並配合會計研究的剩餘淨利模型，欲探討Warrant-Pricing Model與 Black-Scholes-Model之差異。由於現行國際會計準則與美國會計準則都已明確規定員工認股選擇權需依公平價值認列為費用，我國會計公報未來必定朝此方向修改，為因應使用公平價值法對員工認股權評價，本文對於財報附註揭露之表達提出建議，以提供會計人員與審計人員進行財務報表編製與查核工作時為參考。 / Because employee stock option (ESO) has some special conditions which make them different from all the options transferring in markets, we can not use the general option pricing model, such as: Black-Scholes-Model, to price ESO. By using Warrant-Pricing Model and the residual income model, this research introduces us the differences between Warrant-Pricing Model and Black-Scholes-Model. Moreover this research leads to the conclusion that Warrant-Pricing Model can price ESO more properly, and it is helpful in evaluating company equity and pricing stock. This research also provide some advice to auditors and accountants on financial statement disclosures.