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# Dissertation代寫案例-美式期權American Options American Options Are Finanical Derivatives

Chapter 1 - Introduction引言

American options are financial derivatives, an instrument whose value is derived from an underlying asset, usually a stock. Black and Scholes (1973) described an option as: "a security giving the right to buy or sell an asset, subject to certain conditions, within a specified period of time".

The main question of this dissertation is how American options can be valued. The option value is only known with certainty when the option is exercised, either at maturity or not. When the owner decides to exercise the option or it is the option maturity time, it is possible to determine the price of the option as the strike will be exchanged by the asset in the case that the conditions are favourable for the owner of the option. When the one buys the option, she does not know what will be the future price of the underlying asset, and assuming it follows a random process it is hard to put a price on such contract without knowing what will be the price change. This non linear feature of the option makes calculating the price to pay for such contracts a challenging process and has been the focus of a large number of financial studies and publications.
本文的主要問題是美式期權的定價問題。期權價值只有在期權到期或未到期行使時才能確定。當期權所有人決定行使期權或期權到期時，如果條件對期權所有人有利，則可以確定期權的價格，因為資產將交換行使權。當一方購買期權時，她不知道標的資產的未來價格是多少，并且假設它遵循一個隨機過程，在不知道價格變化會是什么的情況下，很難在這樣的合同上定價。期權的這種非線性特征使得計算此類合同的支付價格成為一個具有挑戰性的過程，并已成為大量金融研究和出版物的焦點。

This dissertation deals with the most popular methods for pricing American options and their implementation in MatLab?, including a graphic user interface.
本文主要研究美式期權定價的幾種常用方法及其在MatLab中的實現？，包括圖形用戶界面。

The methods studied include the Black and Scholes (1973) European option pricing as the starting point, followed by the Barone Adesi and Whaley (1987) analytical approximation. Then the binomial and trinomial lattice methods presented in Cox, Ross and Rubinstein (1979) are considered also as the Finite difference approximations models AAA. The most sophisticated method is the Least Squares Monte Carlo simulation presented in Longstaff and Schwartz (2001).

Chapter 2 provides a survey of some of the most relevant publications in American Option Pricing, with focus on analytical approximations, lattice and finite difference methods, more precisely, binomial and trinomial trees, explicit, implicit and Crank Nicolson Scheme, and also on Monte Carlo Simulation.

Chapter 3 provides a description of the methods used, their advantages, disadvantages and limitations. Here the required equations will be derived and the solution for the pricing of American options will be provided.

On Chapter 5 results and their comparison are shown for the different methods used, with the required figures to support the numerical answers.

In the final chapter the dissertation is concluded and a summary of the findings is provided, also as with further work on this subject.

Chapter 2 - Literature Survey文學觀點

Black and Scholes (1973) and Merton (1973) developed the first analytical closed form solution for the pricing of European type options and certain types of American options, such as American call options on non dividend paying stocks. "The option pricing model developed by Black and Scholes and extended by Merton gives rise to partial differential equations governing the value of an option" Schwartz (1976).

The Black and Scholes (1973) model valued European options on non dividend paying stocks, and with a number of quite restrictive assumptions, constant and known interest rates, the markets are frictionless with no transaction costs and penalties for short selling. The Black and Scholes (1973) model also assumes that the underlying stocks follow a random walk. Due to all this assumptions the pricing model Black and Scholes (1973) proposed was of easy use, and there is only the need to input the required values on the proposed pricing equation. The model they have proposed does not take into consideration early exercise of the option so it is inaccurate for pricing American Options.

One of the most popular analytical approximation models that starts from the Black and Scholes (1973) model and adjusts it to consider the scenario of early exercise strategies is the work by Baron Adesi and Whaley (1987) which was based on the paper by MacMillan (1986).

When closed form solutions, like the Black and Scholes (1973) valuation model cannot be derived, numerical methods must be developed. These are computational methods where the values for the underlying assets are modelled up to maturity and the price of the options is derived from them. In the case of American options this is a complex process, as the modelled price changes may have to be adjusted to include dividend payments and the derivation of the option price must also include the possibility of early exercise.

Cox, Ross and Rubinstein (1979) developed a simple discrete time lattice model to deal with the complexity of option valuation, as they considered the methods of Black and Scholes (1973) "quite advanced and have tended to obscure the underlying economics" Cos, Ross and Rubinstein (1979). The use of lattice models such as the one by Cox, Ross and Rubinstein (1979) is the simplicity of its application.

The most significant drawback of the Cox, Ross and Rubinstein (1979) model, is to increase its accuracy the number of time intervals must increase, in order to approach a continuous time model, which will significantly increase the computational time, needed for processing the entire tree in order to derive the option value.

Geske and Shastri (1985) give a good description of the finite difference method: "The finite difference technique analyze the partial differential equation (...) by using discrete estimates of the changes in the options value for small changes in time or the underlying stock price to form equations as approximations to the continuous partial derivatives." Usually the approximations is done using forward, backward or central difference theorem, which respectively result in the explicit, implicit and Crank Nicolson schemes, the procedure used in this study will be shown further in the paper.

In this case as with most of the methods for pricing options, the most significant drawback is the duality between accuracy and processing time. In order to increase accuracy the time and stock change steps must be smaller, increasing their number and the number of computations to make, this issue also affects the stability and convergence of the methods.

Another approach used for solving the option pricing problem, especially for path dependent American options is the use of simulation. This means that the option price is derived from a simulated underlying asset price, usually using a Monte Carlo simulation method. Boyle (1977) and Schwartz (1977) pioneered the use of Monte Carlo simulation which is nowadays used to price complex options contracts. The Monte Carlo simulation method is very powerful in terms of its flexibility to generate the returns of the underlying asset of the options, by changing the random variables used to generate the process a new returns distribution may be easily obtained, Boyle (1977).

Chapter 3 - Pricing American Options Methods美式期權定價方法

3.1 Asset Prices Models

(3.1.1)

3.2 Analytical Approximation by Barone Adesi and Whaley (1987)

Barone Adesi and Whaley (1987) developed a method to approximate analytically and easily the price of American options. They considered that the American and European option pricing equation is represented by the partial differential equation (3.2.1) developed by Black and Scholes (1987) and Merton (1987),

Barone Adesi and Whaley (1987) assumed that if this is true, then the early exercise premium of the American option, which is the price difference between the American and the European call option prices (3.2.2), can be represented by the same partial differential equation (3.2.3).

(3.2.2)

(3.2.4)

Where (3.2.5), (3.2.6) and (3.2.7). Equation (3.2.4) "is a second order ordinary differential equation with two linearly independent solutions of the form . They can be found by substituting (3.2.8) into" equation (3.2.4) Barone Adesi and Whaley (1987),

(3.2.9)

With a general solution of the form, (3.2.10)

From (3.2.9) we have the value for so the only value missing is . This can be calculated interactively considering another boundary condition of American call options. We know that in early exercise the payoff will never be higher than S - X, so from a critical underlying asset value the option payoff curve must be tangent to the S - X curve, which means that below the critical asset value the pricing equation is represented by (3.2.11), Barone Adesi and Whaley (1987).

The algorithm presented by Barone Adesi and Whaley (1987) for the above pricing problem is presented further in the paper in the section dedicated to the implementation of the American option pricing models.

3.3 Lattice Methods

Cox, Ross and Rubinstein (1979) proposed a model where the underlying asset would go up or down from one time step to the next by a certain proportional amount and with a certain probability until maturity. Due to the up and down characteristic of the asset price model these type of models are characterised by a binomial tree or, in the cases of the existence of a third possible movement, they are characterised by a trinomial tree, therefore named as Binomial or Trinomial models

The price of the option would be recursively derived from maturity, due to the boundary condition as has been referenced before that the price of the option is only known with certainty at maturity.

3.3.1 Binomial Tree Model

The model starts being built for a American option of a non dividend paying stock and after that the scenario of dividend payments and optimal early exercise strategy is considered.

The tree formed using these specifications from Cox, Ross and Rubinstein (1979), can have the following graphical representation

The option is price is calculated from the asset price binomial tree. The maturity boundary condition for an American option, is that the payoff is equal to , we already have S at each maturity node from the asset price model, so we can calculate backwards the price of the option as the expectation of the future payoff of the option.
期權價格由資產價格二叉樹計算得出。美式期權的到期邊界條件是，收益等于，在資產價格模型的每個到期節點上都有S，因此我們可以向后計算期權的價格，作為對期權未來收益的預期。

At each node we calculate the expectation of the future payoffs, where the price of the option will be a compound of expectations. These can be represented by the multi period case for a call as in Cox, Ross and Rubinstein (1979),

The option prices are calculated as the expectation of the option's future payoffs using their respective weighted risk neutral probabilities of an up movement and a down movement and then discounted at the risk free rate .
期權價格計算為期權未來收益的預期，使用其各自的加權風險中性概率（上升和下降），然后以無風險利率貼現。

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