This paper aims to use Monte Carlo methods to price American call options on equities using the variance reduction technique of control variates and to price American put options using the binomial model. We use this information to form option positions. This project was done a part of the masters capstone course Math 573: Computational Methods of Financial Mathematics.

<p>In this work we consider a problem of the</p><p>computation of the components of the hedging portfolio structure. In</p><p>literature often one can find valuations and estimations of the</p><p>fair price of American options. But the formulas for hedging portfolio</p><p>are interesting as well and are known for very particular cases</p><p>only. In our work we study different cases of American Put and Russian</p><p>options on finite and infinite horizon.</p>

<p>In this thesis a new method for the option pricing will be introduced with the help of the Mellin transforms. Firstly, the Mellin transform techniques for options on a single underlying stock is presented.After that basket options will be considered. Finally, an improvement of existing numerical results applied to Mellin transforms for 1-basket and 2-basket American Put Option will be discussed concisely. Our approach does not require either variable transformations or solving diffusion equations.</p> / thesis

In this thesis a new method for the option pricing will be introduced with the help of the Mellin transforms. Firstly, the Mellin transform techniques for options on a single underlying stock is presented.After that basket options will be considered. Finally, an improvement of existing numerical results applied to Mellin transforms for 1-basket and 2-basket American Put Option will be discussed concisely. Our approach does not require either variable transformations or solving diffusion equations. / thesis

In this work we consider a problem of the computation of the components of the hedging portfolio structure. In literature often one can find valuations and estimations of the fair price of American options. But the formulas for hedging portfolio are interesting as well and are known for very particular cases only. In our work we study different cases of American Put and Russian options on finite and infinite horizon.

This dissertation considers an American put option written on a single underlying which does not pay dividends, for which no closed form solution exists. As a conse- quence, numerical techniques have been developed to estimate the value of the Amer- ican put option. These include analytical approximations, tree or lattice methods, ¯nite di®erence methods, Monte Carlo simulation and integral representations. We ¯rst present the mathematical descriptions underlying these numerical techniques. We then provide an examination of a selection of algorithms from each technique, including implementation details, possible enhancements and a description of the convergence behaviour. Finally, we compare the estimates and the execution times of each of the algorithms considered.

美式賣權已經存在很長的時間，由於沒有公式解，目前只能利用數值分析方法(numerical analysis approach)和解析近似法(analytic approximations) 來評價它。這類的評價方法在文獻中相當多，但對這些方法的完整的比較卻相當貧乏。本文整理了27種評價方法和186種在文獻中常被引用的美式賣權契約，這些契約包含了各種不同狀態(有股利、沒有股利、價內、價平、價外、短到期日、長到期日)，後續的研究者可以用這些美式賣權契約來驗證他們的方法。本文實作其中14種方法並應用於上述的186種美式賣權契約上。這14種方法包含了樹狀法、有限差分法、蒙地卡羅法與解析近似法。從這些數值的結果中，本文根據精確度與計算效率整理出各種方法的優缺點與適用的時機。 　由本文之數值分析，我們得到下列幾點結論：1.Binomial Black and Scholes with Richardson extrapolation of Broadie and Detemple (1996)與Extrapolated Flexible Binomial Model of Tian (1999)這二種方法在這14種方法中，在速度與精確度的考量下是最好的方法；2.在精確度要求在root mean squared relative error大約1%的情形下，解析近似法是最快的方法；3.Least-Squares Simulation method of Longstaff and Schwartz (2001)在評價美式賣權方面並不是一個有效的方法。 / American put option has existed for a long time. They cannot be valued by closed-form formula and require the use of numerical analysis methods and analytic approximations. There exists a great deal of methods for pricing American put option in related literatures. But a complete comparison of these methods is lacking. From literatures, we survey 27 methods and 186 commonly cited option contracts, including options on stock with dividend, non-dividend, in-the-money, at-money and out-of-money, short maturity and long maturity. In addition, we implement 14 methods, including lattice approaches, finite difference methods, Monte Carlo simulations and analytic approximations, and apply these methods to value the 186 option contracts above. From the numerical results, we summarize the advantages and disadvantages of each method in terms of speed and accuracy: 1.The binomial Black and Scholes with Richardson extrapolation of Broadie and Detemple (1996) and the extrapolated Flexible Binomial Model of Tian (1999) are both efficient improvements over the binomial method. 2.With root mean squared relative error about 1%, the analytic approximations are faster than the numerical analysis methods. 3.The Least-Squares Simulation method of Longstaff and Schwartz (2001) is not an effective method for pricing American put options.

Many problems in financial world are being modeled by means of differential equation. These problems are time dependent, highly nonlinear, stochastic and heavily depend on the previous history of time. A variety of financial products exists in the market, such as forwards, futures, swaps and options. Our main focus in this thesis is to use the numerical analysis tools to solve some option pricing problems. Depending upon the inter-relationship of the financial derivatives, the dimension of the associated problem increases drastically and hence conventional methods (for example, the finite difference methods or finite element methods) for solving them do not provide satisfactory results. To resolve this issue, we use a special class of numerical methods, namely, the mesh free methods. These methods are often better suited to cope with changes in the geometry of the domain of interest than classical discretization techniques. In this thesis, we apply these methods to solve problems that price standard and non-standard options. We then extend the proposed approach to solve Heston' volatility model. The methods in each of these cases are analyzed for stability and thorough comparative numerical results are provided.

Many problems in financial world are being modeled by means of differential equation. These problems are time dependent, highly nonlinear, stochastic and heavily depend on the previous history of time. A variety of financial products exists in the market, such as forwards, futures, swaps and options. Our main focus in this thesis is to use the numerical analysis tools to solve some option pricing problems. Depending upon the inter-relationship of the financial derivatives, the dimension of the associated problem increases drastically and hence conventional methods (for example, the finite difference methods or finite element methods) for solving them do not provide satisfactory results. To resolve this issue, we use a special class of numerical methods, namely, the mesh free methods. These methods are often better suited to cope with changes in the geometry of the domain of interest than classical discretization techniques. In this thesis, we apply these methods to solve problems that price standard and non-standard options. We then extend the proposed approach to solve Heston' volatility model. The methods in each of these cases are analyzed for stability and thorough comparative numerical results are provided.

Philosophiae Doctor - PhD / Many problems in financial world are being modeled by means of differential equation. These problems are time dependent, highly nonlinear, stochastic and heavily depend on the previous history of time. A variety of financial products exists in the market, such as forwards, futures, swaps and options. Our main focus in this thesis is to use the numerical analysis tools to solve some option pricing problems. Depending upon the inter-relationship of the financial derivatives, the dimension of the associated problem increases drastically and hence conventional methods (for example, the finite difference methods or finite element methods) for solving them do not provide satisfactory results. To resolve this issue, we use a special class of numerical methods, namely, the mesh free methods. These methods are often better suited to cope with changes in the geometry of the domain of interest than classical discretization techniques. In this thesis, we apply these methods to solve problems that price standard and non-standard options. We then extend the proposed approach to solve Heston' volatility model. The methods in each of these cases are analyzed for stability and thorough comparative numerical results are provided. / South Africa