### Chapter 1 – Introduction

American options are fiscal derived functions, an instrument whose value is derived from an implicit in plus, normally a stock. Black and Scholes ( 1973 ) described an option as: “ a security giving the right to purchase or sell an plus, capable to certain conditions, within a specified period of clip ” .

The chief inquiry of this thesis is how American options can be valued. The option value is merely known with certainty when the option is exercised, either at adulthood or non. When the proprietor decides to exert the option or it is the option adulthood clip, it is possible to find the monetary value of the option as the work stoppage will be exchanged by the plus in the instance that the conditions are favorable for the proprietor of the option. When the one buys the option, she does non cognize what will be the future monetary value of the implicit in plus, and presuming it follows a random procedure it is difficult to set a monetary value on such contract without cognizing what will be the monetary value alteration. This non additive characteristic of the option makes ciphering the monetary value to pay for such contracts a challenging procedure and has been the focal point of a big figure of fiscal surveies and publications.

This thesis deals with the most popular methods for pricing American options and their execution in MatLab^{®}, including a in writing user interface.

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 estimate. Then the binomial and trinomial lattice methods presented in Cox, Ross and Rubinstein ( 1979 ) are considered besides as the Finite difference estimates theoretical accounts AAA. The most sophisticated method is the Least Squares Monte Carlo simulation presented in Longstaff and Schwartz ( 2001 ) .

The analysis of the different option pricing methods in this thesis follow most of the premises made by Black and Scholes ( 1973 ) , the short term involvement rate and the dividend are assumed to be known and changeless, the underlying stock follows a log normal distributed geometric Brownian gesture, the markets are frictionless and eventually it exists the possibility of organizing a riskless portfolio, dwelling of the option and implicit in stock.

The thesis is organised as follows: a brief literature study is provided in the following Chapter. The analytical estimate method and the numerical methods used are described on Chapter 3 and their execution in Matlab environment is given in chapter 4. Numeric consequences are given in Chapter 5. The decision and future developments are presented in Chapter 6.

Chapter 2 provides a study of some of the most relevant publications in American Option Pricing, with focal point on analytical estimates, lattice and finite difference methods, more exactly, binomial and trinomial trees, explicit, implicit and Crank Nicolson Scheme, and besides on Monte Carlo Simulation.

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

Chapter 4 focal point on the algorithms used and their execution on the MatLab environment, besides as the processs for the development of the GUI for easier user interface.

On Chapter 5 consequences and their comparing are shown for the different methods used, with the needed figures to back up the numerical replies.

In the concluding chapter the thesis is concluded and a sum-up of the findings is provided, besides as with farther work on this topic.

### Chapter 2 – Literature Survey

Black and Scholes ( 1973 ) and Merton ( 1973 ) developed the first analytical closed signifier 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 theoretical account developed by Black and Scholes and extended by Merton gives rise to partial differential equations regulating the value of an option ” Schwartz ( 1976 ) .

Black and Scholes ( 1973 ) develop their theoretical account on the footing of the no arbitrage theory, “ If options are right priced in the market, it should non be possible to do certain net incomes by making portfolios of long and short places in options and their implicit in stocks ” Black and Scholes ( 1973 ) .

The Black and Scholes ( 1973 ) theoretical account valued European options on non dividend paying stocks, and with a figure of rather restrictive premises, changeless and known involvement rates, the markets are frictionless with no dealing costs and punishments for short merchandising. The Black and Scholes ( 1973 ) theoretical account besides assumes that the underlying stocks follow a random walk. Due to all this assumptions the pricing theoretical account Black and Scholes ( 1973 ) proposed was of easy usage, and there is merely the demand to input the needed values on the proposed pricing equation. The theoretical account they have proposed does non take into consideration early exercising of the option so it is inaccurate for pricing American Options.

One of the most popular analytical estimate theoretical accounts that starts from the Black and Scholes ( 1973 ) theoretical account and adjusts it to see the scenario of early exercising schemes is the work by Baron Adesi and Whaley ( 1987 ) which was based on the paper by MacMillan ( 1986 ) .

Baron Adesi and Whaley ( 1987 ) see that the Black and Scholes ( 1973 ) partial differential equation must use to the early exercising premium as this is merely the difference between the American and the European option monetary values, which are besides priced by the same partial differential equation. After some transmutation they end with an easy solvable through an synergistic procedure 2nd order differential equation.

When closed signifier solutions, like the Black and Scholes ( 1973 ) rating theoretical account can non be derived, numerical methods must be developed. These are computational methods where the values for the implicit in assets are modelled up to adulthood and the monetary value of the options is derived from them. In the instance of American options this is a complex procedure, as the modelled monetary value alterations may hold to be adjusted to include dividend payments and the derivation of the option monetary value must besides include the possibility of early exercising.

Cox, Ross and Rubinstein ( 1979 ) developed a simple distinct clip lattice theoretical account to cover with the complexness of option rating, as they considered the methods of Black and Scholes ( 1973 ) “ rather advanced and have tended to befog the implicit in economic sciences ” Cos, Ross and Rubinstein ( 1979 ) . The usage of lattice theoretical accounts such as the one by Cox, Ross and Rubinstein ( 1979 ) is the simpleness of its application.

The most important drawback of the Cox, Ross and Rubinstein ( 1979 ) theoretical account, is to increase its truth the figure of clip intervals must increase, in order to near a uninterrupted clip theoretical account, which will significantly increase the computational clip, needed for treating the full tree in order to deduce the option value.

Others such as Hull and White ( 1988 ) , ( 1993 ) and Trigeorgis ( 1991 ) have extended the theoretical account of Cox, Ross and Rubinstein ( 1979 ) .

Hull and White ( 1988 ) present a survey of the usage of lattice theoretical accounts for underlying assets with known dividends alternatively of known divided outputs. They besides consider the usage of a control random variable to monetary value a option numerically, by a the lattice theoretical account, utilizing the monetary value of a similar option calculated analytically. While Trigeorgis ( 1991 ) proposes “ a log transformed fluctuation of binomial option pricing designed to get the better of jobs of consistence, stableness and efficiency encountered in the Cox, Ross and Rubinstein ( 1979 ) ” concentrating on the pricing of alien options. Hull and White ( 1993 ) besides present an application of binomial and trinomial processs for alien way dependant options, where they developed a theoretical account faster than Monte Carlo simulation and faster than other numerical methods.

Normally the analytical processs are applicable to simple final payments of the American Options, but in the instances where this is non possible numerical solutions must be developed. Geske and Shastri ( 1985 ) give a elaborate comparing of the lattice methods to the different numerical methods, finite difference methods and other simulation methods.

The theoretical account proposed by Brennan and Schwartz ( 1978 ) for valuing options was the first attack that used the finite difference method. This attack was used due to the fact that most of the times an analytical solution for the option pricing job does non be. The finite difference method uses the heat equation derived from the Black and Sholes PDE to obtain an estimate of the option monetary value. Courtadon ( 1998 ) goes farther to cut down the estimate mistake of the Brennan and Schwartz ( 1978 ) theoretical account but lone applies his findings merely to simple option wage offs.

Geske and Shastri ( 1985 ) give a good description of the finite difference method: “ The finite difference technique analyze the partial differential equation ( … ) by utilizing distinct estimations of the alterations in the options value for little alterations in clip or the implicit in stock monetary value to organize equations as estimates to the uninterrupted partial derived functions. ” Normally the estimates is done utilizing frontward, backward or cardinal difference theorem, which severally result in the explicit, implicit and Crank Nicolson strategies, the process used in this survey will be shown further in the paper.

In this instance as with most of the methods for pricing options, the most important drawback is the dichotomy between truth and processing clip. In order to increase accuracy the clip and stock alteration stairss must be smaller, increasing their figure and the figure of calculations to do, this issue besides affects the stableness and convergence of the methods.

Another attack used for work outing the option pricing job, particularly for path dependent American options is the usage of simulation. This means that the option monetary value is derived from a fake implicit in plus monetary value, normally utilizing a Monte Carlo simulation method. Boyle ( 1977 ) and Schwartz ( 1977 ) pioneered the usage of Monte Carlo simulation which is presents used to monetary value complex options contracts. The Monte Carlo simulation method is really powerful in footings of its flexibleness to bring forth the returns of the implicit in plus of the options, by altering the random variables used to bring forth the procedure a new returns distribution may be easy obtained, Boyle ( 1977 ) .

Boyle ( 1977 ) introduces the Monte Carlo technique for pricing European option where there is a dividend payment, but Schwartz ( 1977 ) was the true innovator, pricing American options, with the implicit in plus paying distinct dividends, and besides deducing an optimum scheme for early exercising of the option, which is the important point for pricing American type options. Schwartz ( 1997 ) focused on a peculiar type of contract, warrants, so in equity his first theoretical account is non precisely on an American type option.

Tilley ( 1993 ) was one of the first to to the full concentrate on the pricing of American option utilizing a Monte Carlo simulation method as he mentioned that simulation methods were reserved for alien options or other complex debt merchandises. His findings are merely applied to American options on non dividend paying stocks, but he develops an of import portion of the theoretical account which is the optimum early exercising option.

Carriere ( 1996 ) presents a development of the Monte Carlo simulation method presented by Tilley ( 1993 ) . The paper by Carriere ( 1996 ) presents a theoretical account where the optima early exercising scheme is based on conditional outlooks of Markov procedures by transporting a nonparametric arrested development on the fake implicit in plus return waies.

Brodie and Glasserman ( 1997 ) extended the old surveies by sing an upper and lower meeting bounds of the option monetary value. These estimated bounds are calculated utilizing a high and a low prejudice, which “ Uniting the two calculators yields a assurance interval for the true monetary value. ” Brodie and Glasserman ( 1997 )

One of the most of import documents, and likely one of the most used 1s, is the paper by Longstaff & A ; Schwartz ( 2001 ) . Their Least Squares Monte Carlo ( LSM ) rating theoretical account is really simple and consecutive forward which combined with the truth of the method made it celebrated. Their greatest progress can be described as: “ The key to this attack is the usage of least squares to gauge the conditional expected final payment to the option holder from continuance ” Longstaff & A ; Schwartz ( 2001 ) . They applied their theoretical account to a series of alien way dependent American options with great success.

### Chapter 3 – Pricing American Options Methods

### 3.1 Asset Prices Models

The Black and Scholes ( 1973 ) and Merton ( 1973 ) pricing methods which are the footing for most of this paper assume that the stock returns follow a Geometric Brownian gestures, with the stock monetary values log usually distributed.

The stock returns can be represented by the undermentioned stochastic differential equation,

( 3.1.1 )

Where St is the plus monetary value at clip T, is the assets expected return, is the assets instantaneous volatility and Wt is a Wiener procedure.

### 3.2 Analytic Estimate by Barone Adesi and Whaley ( 1987 )

Barone Adesi and Whaley ( 1987 ) developed a method to come close analytically and easy the monetary value 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 ) ,

( 3.2.1 )

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

( 3.2.2 )

( 3.2.3 )

The above equation after some transmutation, shown on Barone Adesi and Whaley ( 1987 ) paper, and using an estimate of a term be givening to zero, yields the undermentioned quadratic equation,

( 3.2.4 )

Where ( 3.2.5 ) , ( 3.2.6 ) and ( 3.2.7 ) . Equation ( 3.2.4 ) “ is a 2nd order ordinary differential equation with two linearly independent solutions of the signifier. They can be found by replacing ( 3.2.8 ) into ” equation ( 3.2.4 ) Barone Adesi and Whaley ( 1987 ) ,

( 3.2.9 )

With a general solution of the signifier, ( 3.2.10 )

When the American option boundary conditions are applied to the above solution and sing, so must be equal to 0 as when the plus monetary value tends to zero so does the option monetary value, ensuing in the undermentioned American call option pricing equation, Barone Adesi and Whaley ( 1987 ) ,

( 3.2.11 )

From ( 3.2.9 ) we have the value for so the lone value losing is. This can be calculated interactively sing another boundary status of American call options. We know that in early exercising the final payment will ne’er be higher than S – Ten, so from a critical underlying plus value the option final payment curve must be tangent to the S – Ten curve, which means that below the critical plus 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 job is presented farther in the paper in the subdivision dedicated to the execution of the American option pricing theoretical accounts.

### 3.3 Lattice Methods

Cox, Ross and Rubinstein ( 1979 ) proposed a theoretical account where the implicit in plus would travel up or down from one clip measure to the following by a certain relative sum and with a certain chance until adulthood. Due to the up and down feature of the plus monetary value theoretical account these type of theoretical accounts are characterised by a binomial tree or, in the instances of the being of a 3rd possible motion, they are characterised by a trinomial tree, hence named as Binomial or Trinomial theoretical accounts

The monetary value of the option would be recursively derived from adulthood, due to the boundary status as has been referenced before that the monetary value of the option is merely known with certainty at adulthood.

This means that the monetary value of the option is calculated at adulthood and recursively at each node up to the initial value, by dismissing backwards at the hazard free rate and several chances. Due to the feature of American options, the theoretical account has to look into if it is optimum to exert the option at each node or if it has the advantage to go on to the following 1, for illustration on the instance of dividend payments.

In the instance that it is optimum to exert the option at a certain node, its monetary value will be equal to the intrinsic value at that same node. Every node will be checked for the optimality of exerting the option or non, until we have reached the initial point where we want to monetary value the option.

### 3.3.1 Binomial Tree Model

The theoretical account starts being built for a American option of a non dividend paying stock and after that the scenario of dividend payments and optimum early exercising scheme is considered.

As referenced before the stock goes up and down by a certain sum signifier one period to the following, if u is the up motion and d the down motion, so they can be calculated as, ( 3.3.1.1 ) and ( 3.3.1.2 ) as in Cox, Ross and Rubinstein ( 1979 ) . In no arbitrage conditions it is possible to cipher the chance of the up and down motions, with the up being defined as, ( 3.3.1.3 ) where from the definition of chance and the down motion as ( 3.3.1.4 ) .

The tree formed utilizing these specifications from Cox, Ross and Rubinstein ( 1979 ) , can hold the undermentioned graphical representation

The option is monetary value is calculated from the plus monetary value binomial tree. The adulthood boundary status for an American option, is that the final payment is equal to, we already have S at each adulthood node from the plus monetary value theoretical account, so we can cipher backwards the monetary value of the option as the outlook of the future final payment of the option.

At each node we calculate the outlook of the hereafter final payment, where the monetary value of the option will be a compound of outlooks. These can be represented by the multi period instance for a call as in Cox, Ross and Rubinstein ( 1979 ) ,

The option monetary values are calculated as the outlook of the option ‘s future final payments utilizing their respective weighted hazard impersonal chances of an up motion and a down motion and so discounted at the hazard free rate r. The Binomial value is found for each node, get downing at the concluding clip measure, and working backwards to the