The Least-Squares Method for American Option Pricing

Transcription

1 U.U.D.M. Proect Report 29:6 The Least-Squares Method for Aerican Option Pricing Xueun Huang and Xuewen Huang Exaensarbete i ateatik, 3 hp + 5 hp Handledare och exainator: Macie Kliek Septeber 29 Departent of Matheatics Uppsala University

2

3 Abstract This article presents how to use the least-squares (LS) regression ethod to price the Aerican options on basis of the algorith in a paper by Cleent, Laberton & Protter[]. The key to LS is the approxiation of the conditional expectation functions which deterine the optial exercise strategy. In this paper, through the detailed description of the algorith and presentation of convergence, it shows how to estiate the conditional expectation by using the LS to value the Aerican options. Moreover, we also copare the siulation results with the historical data and experient with alternative polynoials to analyze testing results. Key words: Aerican Option, Least-Squares-Method, Monte-Carlo Procedure, Convergence, Optial Stopping Tie, Dynaic Prograing Principle. 2

4 Acknowledgeents First of all, we give the greatest thanks to our supervisor, Professor. Macie Kliek. Thank you for offering us this challenging and interesting topic to help us coplete our aster thesis in financial atheatics progra and for your patience and suggestions in the revision of this paper. Secondly, we would like to give sincere thanks to all teachers in Departent of Matheatics. Particularly, we need to thank Erik Ekströ and Prof. Johan Tysk since we learn so uch relevant knowledge fro financial atheatics II and III, which provides the prerequisite for our thesis. Last but not least, we owe the honest thanks to our parents. They support us not only in the studying, but also in our lives. Their endless love and care help us overcoe the difficulties and always ake us feel confident. Thank you!! We love you forever! 3

5 Contents Introduction 5 2 An introductory exaple 6 3 Theoretical fraework of valuation 3. The Snell Envelope The valuation of algorith The least-squares regression ethod Convergence 7 4. Notation Convergence results Nuerical siulation 9 5. Siulating fro Geoetric Brownian Motions Assessing the least-squares regression Using different nubers of paths and regressors Altering polynoial failies Open probles Multiple assets Coputational coplexities Conclusion and future work 25 8 Appendixes 26 Appendix A: Proofs..26 Appendix B: Tables..29 Appendix C: Figures.35 Appendix D: Codes Reference 4 4

6 . Introduction The global financial crisis of 28 has been leading to a world-wide econoic revolution. At the beginning, it originates fro finance but quickly and broadly spreads to other fields. It is the econoic crisis that causes a disastrous ipact to the global econoy in a short tie: banks close down, copanies go bankrupt, enterprises cobination and reconstruction, uneployent rate rapidly increases, the stagnation in trades of iports and exports, stock-option arket crashes... Econoists analyze and predict that the econoic crisis aybe gradually recover until the iddle of 2, however, it will be a long and tough process. The causes of financial crisis have coplexities and diversities, such as arket instability, higher interest rates in ortgage and real estate and those are inevitable products of long-ter ibalance in developent between financial arkets and financial institutions. On the other hand, stocks, bonds, derivatives, especially options, as coponents of financial arkets, play an iportant role in econoy. Recently, ore and ore scholars have been interested in ethods and techniques to price different types of options, such as finite difference ethod, Monte-Carlo ethod and the siulation technique. In this paper, we will discuss how to price Aerican options with the least-squares ethod. In finance, an option is a contract between a buyer and a seller that gives the buyer the right but not the obligation to buy or to sell a particular asset (the underlying asset) at a later day at an agreed price. In return for granting the option, the seller collects a payent (the preiu) fro the buyer. A call option gives the buyer the right to buy the underlying asset; a put option gives the buyer of the option the right to sell the underlying asset. If the buyer chooses to exercise this right, the seller is obliged to sell or buy the asset at the agreed price. The buyer ay choose not to exercise the right and let it expire. The underlying asset can be a piece of property, shares of stock or soe other securities. For exaple, buying a call option provides the right to buy a specified quantity of a security at an agreed aount, known as the strike price at soe tie on or before expiration, while buying a put option provides the right to sell. Upon the option holder's choice to exercise the option, the party who sold, or wrote the option, ust fulfill the ters of the contract. Generally, the price of an option depends on various factors, such as the strike price, the dividends, interest rates and the aturity tie. Unlike European option, the Aerican option can be exercised at any tie before or at aturity in paper by Bensoussan[2], which leads to highly coplicated coputations and akes it ipossible to find analytical solutions. In order to solve this proble, people use the ost faous nuerical ethod the Binoial Model suggested by Cox, Ross & Rubinstein[3]. However, the coputational cost in binoial odel exponentially increases with the nuber of 5

7 assets, which akes the speed of calculation obviously slow down. Thus, siulation technique sees to be a better choice. One of the first to give solutions to pricing Aerican options using siulation technique was Tilley[4]. Until 996, Carriere[5] proposed an alternative ethod to price the Aerican option in ters of optial stopping ties. A recent paper by Longstaff & Schartz [6] presents a siple and effective ethod to approxiate the value of Aerican options. The ain idea is to estiate the conditional expectation by the orthogonal proection on space generated by a finite nuber of basis functions in the siulation by using least-squares ethod so as to copare the payoff fro iediate exercise with the expected payoff fro continuation at each exercise date alone each siulated path and finally ake the optial exercise decision. Another recent paper by Cleent, Laberton &Protter gives a detailed analysis of convergence theores and proves the alost sure convergence of the coplete algorith under general conditions. In addition, they also deterine the rate of convergence of Monte-Carlo procedure. The purpose of this paper is to apply the LS regression ethod to price the Aerican options based on the exaple in [6] but with the algorith in []. The reark 2. in [] explains the difference between these two approaches. We will give specific description of the algorith with clear presentation of the atheatical results. As a suppleentation, we will also explain how the Snell Envelope in discrete tie provides a powerful way of representing the value of an Aerican option in [2] and Karatzas[7]. Theoretically, the ost technical part of our work is the analysis of algorith and convergence. In nuerical experient, we pay ore attention to copare the siulation results with historical data fro the Montreal Stock Exchange Market and copare the estiations obtained fro those alternative specifications of the cross-sectional regression odel. Furtherore, we exaine the results of changing the nuber of siulated paths and the regressors. As we know, both the nuber of paths and the nuber of regressors should go to infinity to approxiate the conditional expectation well. The structure of the paper is as follows. Section 2 presents an introductory exaple of the siulation approach. Section 3 provides theoretical fraework for description of the algorith. Section 4 shows the convergence results. Section 5 illustrates the nuerical tests and analysis of epirical results. Section 6 gives the open probles for several ipleentation issues. Section 7 concludes. 2. An Introductory Exaple In this part, we quote the sae exaple in [6] but based on the algorith in [] to value an Aerican put option using the LS regression ethod. In [6], the regression involves only in the oney paths considering the efficiency of the algorith and 6

8 siplicity of nuerical coputations. However, here we stick to the original algorith (involving all paths: in the oney, at the oney, out of the oney) to price an Aerican option. The key to optially exercise an Aerican option is specifying the expected value of continuation. According to the algorith, we approxiate the conditional expectation by the orthogonal proection onto space generated by a set of basis functions and provide an efficient estiate of conditional expectation functions using the LS ethod. Through the coparison between the iediate exercise value and the payoff fro continuation, we obtain the optial stopping rule for this Aerican option. The saple paths are generated under the risk-neutral easure and are shown in the following atrix. The strike price is.. The riskless rate is 6% and we discount it back to tie t, naely: exp (-.6). Stock price paths Path t= t= t=2 t= We need to axiize the value of Aerican put option at each exercise date along each siulated path. First of all, assue that we exercise the option at the final date (t=3), in this case, the Aerican option is exactly the sae to the European option. We have the following cash-flow atrix. Cash-flow atrix at tie 3 Path t= t=2 t= Now, we assue that the put is at tie 2. It requires that the option holder should decide whether to exercise the option iediately or hold it until expiration (t=3). Here, we consider all the paths and obtain a regression atrix at tie 2. 7

9 Regression at tie 2 Path Y X.* * * * * * * * X are the stock prices at tie 2 and Y denote the corresponding discounted cash flows received at tie 3 if the put is not exercised at tie 2. We use the LS regression ethod to estiate the conditional expectation function by regressing Y on three 3 3 siple basis functions, X and X : E[Y X ] X +.37 X. We insert X into the approxiation of the conditional expectation function and copare the values with the iediate exercise values at tie 2 and give the following atrix below. Optial early exercise decision at tie 2 Path Exercise Continuation This coparison iplies that for the paths 4, 5, 6, 7, it is optial to exercise the option at tie 2 which results in the atrix as follows. Cash-flow atrix at tie 2 Path t= t=2 t=

10 We redo this recursive process again to exaine whether the option should be exercised at tie. Siilarly, we obtain regression atrix at tie. Regression at tie Path Y X.* * *.9476* * * * * * X denote the stock prices at tie and Y are the corresponding discounted cash-flows. The approxiation of the conditional expectation function at tie is E[Y X ] X +.72 X 3. Substituting the X into this regression and coparing with the iediate exercise values, we have the atrix below. Optial early exercise decision at tie Path Exercise Continuation The coparison above shows that for the paths 4, 6, 7, 8, it is optial to exercise the option at tie. The relevant option cash-flow atrix and stopping rule are as follows. Option cash-flow atrix Path t= t=2 t=

11 Stopping rule Path t= t=2 t= Fro the stopping rule atrix above, it is clear to see that the optial strategy should be chosen at the exercise dates where there is a one in the atrix. Having given the cash-flows generated by the Aerican put option at each exercise date along each siulated path, we only need to discount each cash-flow back to tie zero and ake the average of all paths, then the option can be valued approxiately. We obtain the sae value of.44 for the Aerican put option as described in [6]. Through this siple but very typical exaple, we illustrate how to price an Aerican option using the LS ethod according to the algorith in []. Under the shade of failure with nuerical ethods, the LS, without doubt, provides a ore effective and powerful approach and the ore iportant thing is the LS is quite easily to be ipleented in siulation techniques to value the Aerican-style options. 3. Theoretical fraework of valuation In this section, we give detailed description of the general LS algorith for pricing the Aerican options. Furtherore, we also illustrate how to use the least-squares ethod to approxiate the conditional expectation functions as entioned in the introductory exaple. 3. The Snell Envelope In this short part, we briefly explain how the Snell Envelope in discrete tie provides a way of pricing the Aerican options and introduce the optional stopping tie for the pricing of Aerican options. To ake this algorith easier to understand, first of all, we introduce several iportant concepts: Definition 3..: A probability space [8] ( Ω, F, P) is a easure space, where the set Ω is called the saple space of all possible outcoes, the σ -algebra F, as the subsets of Ω, are called events, the easure P is the probability easure defined on

12 the eleents of F. Definition 3..2: A filtration on (, F) Ω is a faily { } F = F of σ -algebras F F such that s < Fs F. Now we consider an underlying probability space: ( Ω, F, P), equipped with a discrete tie filtration ( F ) =,,..., L, with a finite tie horizon L. A general Aerican option ay be exercised at any tie before or at the terinal date L. To odel this, we need to consider an adapted payoff process Z = ( Z ) =,..., L, which is a sequence of real-valued square integrable rando variables: E 2 2 Ζ = Ζ( ω) dρ ( ω) <, ω Ω (3..2.) Ω Definition 3..3: An n-diensional stochastic process { } a artingale with respect to a filtration { } F if Μ on ( Ω, F, P) is called (I) (II) Μ is F - easurable for all ; E Μ < for all ; (III) E t F Μ =Μ for all t Notice: if (III) satisfies: E t F Μ Μ, we call the process superartingale. E Μ F Μ, we call the process subartingale. t We hope to price an Aerican option at any tie, so we also construct a price processu = ( U ) =,..., L. At the final exercise date, clearly, we haveu L = Z L. However, what is the price of the process at tie L -?? Obviously, the option holder now can decide either to exercise the option to earn ZL or wait until tie L to earn Z L. In latter, the option is equivalent to a European contingent clai held over the tie period L - to L, so the writer of the option needs to invest Ee Z F r( L ( L )) ( L L ) in a replicating portfolio in order to generate Z L at tie L. Hence, we can give that the

13 r price of the option at tie L - should be { ( L ( L U )) L ax ZL, E( e UL FL ) } =. We iterate this arguent and deduce that the price for this option at tie - is r( ( )) { } U = ax Z, E( e U F ). If we can discount both the pay-off and the price processes by defining U = r e U r and Z = e Z, then the arguent above can be rewritten as the for: U = ax { Z, E( U F ) }. Particularly, if Z E( U F ), then we have the artingale property[9]: U = E( U F ). Now, we turn to the properties of processes that are entioned above. The optial stopping proble for the pay-off process Z consists of axiizing Ε( Z ) over all stopping ties. Definition 3..4: A rando variable taking values in is a stopping tie if, for = F. any, { } Definition 3..5: Let Z = ( Z ) =,..., L be an adapted sequence of real-valued integrable rando variables. The Snell Envelope of Z is the sequence U = ( U ) =,..., L defined by U = ess sup E( Z F ), L. Τ L, where Τ, L denotes the set of all stopping ties with values in {, +,..., L }. Theore 3..6: The Snell Envelope U of Z satisfies the following properties: () UL = ZL =, L (2) U ax { Z, E( U + F )} (3) U is the sallest superartingale which doinates Z. Proof. See the appendix A. 2

14 Definition 3..7: A stopping tie is called optial if E( Z F ) = sup E( Z F ). ' ' Τ, L Since the stopping ties play an iportant role in pricing the Aerican options, with the help of the Snell Envelope, the optial stopping ties can be characterized. Theore 3..8: A stopping tie is optial if and only if it satisfies: () U = Z (2) The stopped process U is a artingale. (Where a b denotes the iniu of the nubers a and b) Proof. See the appendix A. 3.2 The valuation of algorith Consistent with no-arbitrage paradig, we assue that the existence of an equivalent artingale easure Q[] for the financial arkets and assue that the Aerican option can only be exercised at L discrete ties. Fro the theore 3..8, we can derive U = E( Z F ) since U k is a artingale, = =. Therefore, we have U = E( Z F ), with where in{ k U k Z k } = in{ k Uk = Zk}. Substituting U + = E( Z F ) + + into the second property of the theore 3..8, we obtain that { } { } U = ax Z, E( E( Z F ) F ) = ax Z, E( Z F ). Thus, the dynaic prograing principle of the theore 3..6 can be rewritten in ters of the optial stopping ties as follows: L = L = +, { } { } ( ) + L Z E Z F Z< E ( Z F + + ) (3.2.), condition holds In the forula above, { } = condition, which deterines the value of, otherwise the stopping tie. Also, this forula highlights the fact that if we can know how to approxiate the conditional expectations of payoffs, the value of Aerican option can be priced. The first equation in 3.2. tells the holder to exercise the option at the final exercise date if it is in the oney, otherwise, the option will be expired. When exercising the option prior to the terinal expiration date, the second equation 3

15 requires the option holder to deterine whether to exercise iediately or to hold the option for next exercise date. Therefore, the optial stopping proble changes to coparing the iediate exercise value with the conditional expectation fro continuation. Fro the introductory exaple in section 2, we know that the value of iediate exercise is easily to obtain. However, the ain proble in (3.2.) is that the conditional expectation is unknown, so for the next step, we focus on the estiation of conditional expectations. The otivation for approxiating the conditional expectation can be given in ters of proection theory of Hilbert spaces. Now, we liit our attentions to the square-integrable payoff functions in (3..2.). This space is often denoted by L 2 ( Ω, F, P) and it is an exaple of a Hilbert space: a coplete space equipped with a for of inner-product: f, h = f( xhxdx ) ( ). In addition, we assue the underlying odel to be a Markov chain[], which eans the description of the present state fully captures all the inforation that could influence the future evolution of the process. Therefore, we have E ( Z ) ( ) F = E Z X + +. ( X ) = is an F -adapted Markov chain and presents a process of stock Here,,..., L price and we denote Z = f(, X ) as an adapted pay-off process for the option. Since Now it is tie to approxiate the conditional expectation with respect to X. 2 L is a Hilbert space, it has a countable orthogonal basis: φ ( X ) = ( φ ( X), φ ( X),..., φ ( X)). As an eleent of the 2 2 L space of square-integrable functions, the conditional expectation can be represented as a linear cobination of the eleents of the basis[2]. Hence, (3.2.) can be written as follows: L = L = + +, L Z ( Z ) Z Ρ ( Z ) <Ρ + + P ( Z ) = α i φ ( X ), here i denotes the usual scalar product in + and (3.2.2) α are coefficients of linear cobination of eleents of basis: obtain an approxiation for the value function: α φ ( X ). So we U Z E Z = ax(, ( )) (3.2.3) 4

16 Fro (3.2.2), it is not difficult to see that the proection onto vector space becoes a key for approxiating the conditional expectation. The coefficients α are unknown and ust be estiated, so we use the Monte-Carlo procedure[3] to approxiate the siulation procedure. Just like the introductory exaple, we consider N siulated n n n paths of the Markov chain ( X ) =,..., and denote Z = f(, X ) as the relevant n N pay-off process. Furtherore, we estiate the coefficients nn,, α with α and the nn,, coefficients α can be calculated as the solutions to the following expression: N n n n n n n n in ( αφ( X ) 2 2( )... ( ) ( n,, N)) n + α φ X + α φ X Z { } α + n = This is the Least-Squares proble and will be discussed in part (3.2.4) Hence, we can estiate recursively the stopping tie with N siulated paths by: nn,, L = L nn,, nn,, = n N,,, { ( n + Z X ) n N } { Z ( X n L α φ + i < α iφ ) } (3.2.5) We substitute the coefficients fro (3.2.4) into α iφ ( X ) and use these values as N, n the linear approxiation to P ( Z ) in (3.2.2) based on the N siulated paths. Now, + the option can be valued by discounting each cash flow back to tie zero and average over all the siulated paths. Finally, we obtain the following approxiation for option value: N N, n U = ax( Ζ, Ζ nn,, ) (3.2.6) N n= 3.3 The Least-Squares Regression Method In section 3.2, we use the least-squares regression ethod to copute the value functions in the approxiation of the conditional expectation. Now, we present the specific ipleentation of least-squares regression ethod. The least-squares ethod assues that the best-fit curve of a given type is the curve that has the inial su of the deviations squared fro a given set of data. Suppose that the data set: ( x, y),( x2, y2),...,( xn, yn) and a fitting curve f ( x ) which has the deviation fro each data point: d = y f( x ), d = y f( x ),..., d = y f( x ) n n n

17 The least-squares ethod produces the best fitting curve with the property: 2 the Miniu Least-Square Error = d [ ( )] 2 i = yi f xi n n i= i= Now we look back to our case. We assue that the nuber of regressors is and the nuber of siulated paths is N. Thus, we need to calculate the coefficients α in n, (3.2.4) by using the LS ethod to deterine the value of P ( Z ) in (3.2.2). Let + Z Η: E Z X, X,..., X = Pro ( Z) 2 N Μ. X denote the stock prices and Z(Y in the introductory exaple) are the corresponding discounted cash-flows. Suppose that φ, φ2,..., φ are non-zero vectors in Hilbert Space and M is a subspace generated by vectorsφ, φ 2,..., φ. According to the orthogonal proection, we have ProΜ ( Z) = αφ + α2φ2+,..., + αφ, Z = ( Z, Z2,..., Z N ) T T Hence, α = ( α, α2,..., α ) is a solution of Φ α = B. Here, Φ= φ φ, B = φ b. In our case, we are assuing three linear independent basis vectors in 2 L space: 3 3 φ = (,...,), φ2 = ( X,..., X N), φ3 = ( X,..., X N) N N N So the atrices Naely, T φ φ = (,...,) Z = φ = ( X,..., X ), b= 2 N 3 3 φ3 = ( X,..., XN ) Z N < φφ, >... < φφ, > α < Z, φ> = < φ, φ >... < φ, φ> α < Z, φ > Φ α B (3.3.) Here, <>, denotes the inner-product. Generally, any Euclidean space n with n inner-product is < ( x... xn),( z... zn) >= xizi = xz xnzn. i= 6

18 If φ φ2,,..., φ are linear independent, there is a unique solution for the syste of equations Φ α = B, so we haveα =Φ B. Φ is the inverse atrix of Φ. Now we can calculate the coefficients of the approxiation of the conditional expectation functions in section 2 as follows: N N N N 3 Xi Xi Zi i= i= i= i= α N N N N 2 4 α 2 = Xi Xi Xi XZ i i i= i= i= i= α 3 N N N N Xi Xi Xi XiZi i= i= i= i= (3.3.2) Fro calculation above, it is siple to see that the coefficients of the approxiation of the conditional expectation functions depend on the choices of different basis functions and the results will aybe influence the optial early exercise decision, the stopping rules and even the values of the Aerican options. In addition, the nuber of regressors and the nuber of siulated paths will also affect the approxiating results of conditional expectation. In the next section, we will show the convergence results: both the nuber of regressors and the nuber of paths should tend to infinity; we can obtain an ideal approxiation of the conditional expectation. 4. Convergence Fro (3.2.6), it sees reasonable that the price of estiation will converge to the true price of the Aerican option if the approxiation of the conditional expectation converges to the true expectation function. Therefore, in this section, we will show the convergence results of the conditional expectation approxiation: for any fixed nuber of basis functions, N, U converges to U as N tends to infinity, and that U converges to U as goes to infinity. 4. Notation Before presenting the convergence results, we give soe necessary notations to ake the theores and proofs easier and ore clearly to understand. Fro (3.2.4), the estiator α N, has the explicit expression: 7

19 α = ( Χ ) ( ) (4..) N N, N, n n Z nn,, φ X + N n= for L, where Χ N, is an atrix, with coefficients given by (3.3.). For any vectors α = ( α,..., α ), α = ( α,..., α ), z = ( z,..., z ) and, N, N, N L L L L x x x L L = (,..., ), a vector pay-off function G = ( G,..., G L ) can be defined: GL( α, z, x) = zl G( α, z, x) = z + G (,, ) { z ( x ) α z x α iφ + } { z< α iφ ( x ) } (4..2) Copared with (3.2.2) and (3.2.5), we have: G ( α, Z, X) = Z G Z X Z N, n n n ( α,, ) = nn,, (4..3) Thus, (4..) can be rewritten as: α = ( Χ ) ( α,, ) φ ( ) (4..4) N N, N, N, n n n G+ Z X X N n= 4.2 Convergence result Before stating the convergence results, we have to propose two assuptions such that the optial early exercise decision deterined fro the approxiation is correct. Assuption : The siulated paths are independent. Assuption 2: For L, P( α i φ ( X ) = Z ) = The first assuption is ipleented during the Monte Carlo procedure. The second assuption will be considered in the proofs of the convergence theores. Also, this assuption will ake sure that the optial stopping tie can be correctly n identified when the pay-off process Z nn,, converges to Z. Theore 4.2. Assue that the easurable real valued basis functions φ ( X ) is total in L 2. For L, we have li EZ ( F) = EZ ( F) in L

20 We proceed by induction on. For = L, theore 4.2. is obviously true. If it is true for +, we only need to prove that the theore holds for ( L ). Fro (4..3), (4..2) and (3.2.), we have Hence, we obtain Z = Z { Z + Z ( )} { Z α iφ X + < α iφ ( X ) } Z = Z + Z + { Z E( Z F )} { Z E( Z F ) < + + } EZ ( Z F) = ( Z EZ ( F))( ) { Z α φ ( X) } { Z E( Z F ) + i + } + EZ ( Z F) + + { Z < α iφ ( X ) } The second ter on the right hand side converges to zero by assuption, so we ust 2 prove that the first ter converges to zero in L. Proof. See the reference []. Theore Assue that Assuption 2 is satisfied. Then U N, converges alost surely to U as N tends to infinity. N n Theore requires us to prove that Ζ nn,, in (3.2.6) converges alost N surely to E( Z ) in (3.2.3) as N tends to infinity for L. Now we define the function σ ( α ) = EG [ ( α, Z, X)]. Using (4..3), we only need to prove that n= Proof. See the appendix A. N N, n n li ( α,, ) σ ( α ), N N n= G Z X = L (4.2.2.) 5. Nuerical siulation Earlier we have given a specific exaple to illustrate how the least-square regression ethod could be applied to the valuation of Aerican put options. In this part, we will show how to price the Aerican options using nuerical siulation based on the LS ethod and test the results against the actual prices. Also, we copare the LS with the 9

21 option calculator in Montreal Stock Exchange to obtain the difference in early exercise value. We will assue that the only stochastic factors are the prices of the underlying stocks. Moreover, we will assue that the risk neutral dynaics of these stock prices can be specified as Geoetric Brownian Motions (GBM). 5. Siulating fro a GBM We are interested in pricing an Aerican put option, where the risk-neutral stock price process satisfies the following stochastic differential equation (SDE): ds() t = rs() t dt + σ S() t dw () t (5..) where W is a standard Brownian process and r and σ are constants. The stock does not pay dividends. Given the initial stock price S, the well-known solution of (5..) can be written as follows: 2 St () = S exp ( r σ ) t+ σwt () 2 (5..2) Fro the properties of the Brownian process, siulated values of St () can be obtained fro the forula: 2 St () = S exp ( r σ ) t+ σ tzt () 2, (5..3) where Zt () N(,) satisfies the standard noral distribution. So a sequence of values at discrete ties < t t2... tn = L can be obtained fro 2 St ( ) St ( )exp + = ( r σ )( t+ t) + σ ( t+ t) Zt ( + ), (5..4) 2 Fro (5..4), it is seen that the stock price can be siulated fro a GBM at each single point. With those stock prices, the coparison between the payoff fro iediate exercise and the expected payoff fro keeping the option alive at each exercise date is easily to be done, which leads to the optial decision to price the Aerican options eventually. 5.2 Assessing the LS regression Here, we are interested in exaining the results of using different nubers of paths and regressors than those used in Longstaff & Schwartz (2) and the results of changing various polynoials. As convergence theore entioned above, both the nuber of paths and the nuber of regressors should tend to infinity so as to estiate the conditional expectation arbitrarily well. Furtherore, alternative specifications of 2

22 the cross-sectional regression odel should be exained. According to the Montreal Stock Exchange arket, we choose First Quantu Minerals (FM) as the underlying asset. We pay attention to options with the date June 2, 29 to expiration, and correspondingly we set 3 days and 6 days as possible exercise points. The options all have a strike price of 48. To be precise, we consider two different initial stock prices S =47.22 and The volatility of return is 7% and in all cases an annual interest rate of 5% is assued Using different nubers of paths and regressors In Longstaff & Schwartz (2, section 3) Aerican style options are priced using siulated paths and the first three weighted Laguerre polynoials in the regressions. However, other cobinations in paths and regressors can be considered and prices approxiations with various properties could probably be obtained. To be precise, the LS regression ethod potentially has two kinds of biases. Firstly, there is an approxiation bias as the conditional expectation function is estiated. This results in a low bias whichh can vanish as the nuber of regressors increase. Secondly, there is a bias fro using the sae paths approxiate the conditional expectation function and to calculate the option price. This leads to a high bias which possibly disappears as the nuber of siulated paths increase. In our nuerical tests, we increase the nubers of regressors,, fro 2 to 5 and ake use of siulated paths, N, fro 5 to 3. The benchark with whichh we copare is an options calculator fro Montreal Stock Exchange. Here, we siply introduce this calculator and show how it works for pricing options. We illustrate a chart of options calculator fro hoepage of Montreal Stock Exchange arket and set the sae input values used in our nuerical siulations. 2

23 The options calculator is provided as an educational tool intended to illustrate and to assist users in learning how options work. Values generated by the options calculator only provide a reference. The theoretical calculations ade by the calculator are based on option pricing odels. The odels used can be approxiations and the results they produce can differ depending on the odel which is used. In addition, this options calculator can evaluate the preiu of index and equity options, for either Aerican or European exercise style, with or without dividend. The input values in the chart above include the underlying price, strike price, annual volatility, and so on. The expiration onth or the days to expiration include all calendars days (including holidays); for dividend paying stock options and Aerican options, it uses the nuerical binoial odel (Cox, Ross and Rubinstein, 979) with the axiu nuber of steps 6. In Table 2 we report the results for the options with two initial stock prices 47.2 and 38.5, correspondingly the exercise dates 3 and 6. The first obvious thing to note fro the table is that without considering the nuber of paths, the bias decreases with the nuber of regressors growing. It eans that the convergence can be guaranteed when the nuber of regressors is increased. Particularly, when the LS with a low nuber of regressors, =2 or 3, results in a larger bias; and with ore regressors, =5, for 6 exercise days, it produces upward biased prices. This corresponds to the approxiation bias entioned above which potentially vanishes as the nuber of regressors is increased. Fro Table 2 to Table 5, we can also see that when the nuber of paths used increases, for a reasonable nuber of regressor =3, the effect of which could potentially be the high bias tends to disappear. Meanwhile, the standard error of the estiate decreases fro increasing the nuber of paths used. The siilar conclusion can also be obtained fro other tables but we need to choose a reasonable nuber of regressors =3 or = Altering polynoial failies Now we divert our attentions to the results in nuerical experients with alternative polynoial failies. Three different polynoials used in this paper can be found in Table in Appendix. To be siple, the first polynoial that we choose consists of k cobinations of onoials: Qk ( x) { x } k = =. The second one that we use is the weighted Laguerre polynoial: L ( x ). The eleents of the Laguerre faily have the k property of being utually orthogonal on the interval [, ) with respect to the weighting function wx ( ) = e x. The third one that we use is the shifted Legendre polynoial: Sk ( x ), which also shares the orthogonal property on the interval (,). Another advantage with the Shifted Legendre polynoials is that no coputationally 22

24 intensive weights have to be calculated since the weighting function wx ( ) =. Instead of using the exact forula for those polynoials, we use the recursive forula for calculating the eleents, since it is coputationally uch easier. The first three ters have been given in Table. We present the results with different polynoial failies fro Table 2 to Table 3. First of all, it is obviously to note fro Table 6 to Table 9 that the nuber of regressors and the nubers of paths both increase to achieve arbitrarily close approxiation with the Laguerre polynoials, as it is the case with the siple onoials fro Table 2 to Table 5. Additionally, it also shows that the ain benefit fro increasing the nuber of paths is again to lower the standard errors. Hence, fro Table 6 to Table 9, it guarantees that using the standard error as a precision both the nuber of regress,, and the nuber of paths, N, should tend to infinity to obtain convergence of the estiated price. Tables,,2,3 illustrate the siilar conclusions with the Shifted Legendre polynoials. However, it is not difficult to observe that the standard errors becoe saller when the Laguerre and Shifted Legendre polynoials are used as regressors, especially for soe larger siulated paths. Particularly, for sall nubers of regressor, =2, the bias is uch saller with the Shifted Legendre polynoials copared with two others and with the sae nubers of paths, N=2 or 3, the estiated prices are ore close to the actual option prices. Therefore, the Legendre polynoials see to be a better choice than either of the other polynoials. 6. Open probles In this section, we discuss several iportant nuerical and ipleentation issues associated with the ulti-diensional probles and coputational coplexities. 6. Multiple assets The nuerical siulations we have ipleented in section 5 benchark the perforance of the Least-squares ethod for -diensional probles which can also be solved by other techniques, such as FD and BM. However, for the ultiple assets, those techniques becoe infeasible in practice. A siple exaple of ultiple stochastic factors is the case in which there are two or ore underlying assets. Options on ultiple underlying assets typically can be divided into three types [5]: Rainbow options, Basket options and Quanto options. The ost coon options with ultiple assets have payoffs as a function of the axiu, iniu or the average of the asset prices: 23

25 + The payoff of axiu option= ( K ax( S, S2,..., S n )) (6..) + The payoff of iniu option= ( K in( S, S2,..., S n )) (6..2) n + The arithetic average option= ( K α S ) (6..3) = n The geoetric average option= ( K S α + ) (6..4) = where S, =,..., n are the prices of the underlying assets, α is the ratio of the th asset takes in the total underlying assets, and K is the strike price. These types of payoff functions have been used widely in the literature. Furtherore, Boyle and Tse present how to trade these types of options [6] and Lars Stentoft gives detailed description about how to price options on ultiple assets with nuerical ethods [3]. 6.2 Coputational Coplexities Extensive nuerical tests indicate that the results fro the LS algorith are rearkably relevant to the coputational coplexities. In section 5, we used three different polynoial failies as basis functions to estiate the conditional expectation functions. Specially, the weighted Laguerre polynoials include an exponential ter, which will result in the coputational underflows if applying the polynoials directly. Thus, for the nuerical convenience, we siplify the weighted Laguerre polynoials to the for listed in Table in Appendix B to obtain the reasonable option values. Another solution to this proble is that to renoralize the Aerican put exaple by dividing all cash flows and prices by the strike price and estiating the conditional expectation function in the renoralized space. Note that the option value is unaffected since we discount back the un-noralized value of the cash flows along each path to obtain its value. However, altering the cross-sectional inforation will have an obvious effect on the running tie. Adding regressors will lead to an increase in the coputational tie. Moreover, increasing the nuber of path to siulate will also increase tie to calculate the estiated option prices. More than 6 siulated paths will lead to out-of-eory in our nuerical siulations with MATLAB coands. Here we present a easure of coputational tie according to different polynoials for the different cobinations between regressors and paths. It is calculated per second on a Genuine Intel T24,.83GHz, with.99gb of RAM. The results are reported in Table 4, 5 and 6. Corresponding to the tables, we plot the Figure 2, 3, and 4. A key advantage of siulation techniques is that we can do the siulations using 24

26 parallel coputing architecture. With the parallel algorith, we could generate paths each on 3 CPUs instead of generating 3 paths based on a single CPU. Without doubt, it will largely increase the coputational speed and decrease the CPU tie; eanwhile, the parallel algorith will obviously save the coputer s eory occupation. Moreover, in any large-scale application fields, the coputational speed and eory-consuing in coputers are far ore iportant copared with the cost of hardware. Fro the perspective of the LS algorith, the constraint on parallel coputation is that the regression needs to use the cross-sectional inforation in the siulations and the drawback aybe involves little loss of coputational efficiency. 7. Conclusion and Future work In this article, we have presented a detailed analysis of the Least Square (LS) ethod proposed in Longstaff & Schwartz (2). The approach is intuitive, accurate and easy to apply. Firstly, the detailed descriptions of the LS algorith and the convergence theore are perfored. Next, we illustrate the LS algorith using a nuber of nuerical experients with different polynoial failies and copare this ethod with an existing options calculator. Through the coparison of bias and standard errors, it suggests that the LS algorith is able to approxiate closely the actual option values. Meanwhile we also show that the Shifted Legendre polynoial is a better choice as the specification of the cross-sectional regressors. Finally, we discuss coputational coplexities. We conclude that considering the coputational tie to calculate an option price, the preferred choice, for fewer regressors =2 or 3, is to use the siple onoials. Several extensions of our fraework can be carried out, such as to approxiate the option values for at the oney and out of the oney and to apply the LS algorith to the ultiple assets. Furtherore, a trade-off between the coputational tie and the precision of the estiated option price can be easured. 25

27 Appendix A: Proofs A: Proof of Theore 3..6 Proof: (): When = L, the equality UL = ZL is obtained fro definition (3..5). (2): Step : Now we consider L. It is obvious that U Z. Using definition (3..5), we also have: E( U F ) = E( ess sup E( Z F ) F ) = ess sup E( E( Z F ) F ) Τ +, L Τ +, L Since the inclusion: Τ+, L Τ, L, E( EZ ( F+ ) F) = EZ ( F) U. Hence, U ax { Z, E( U + F )}. Step 2: If Τ, L, we choose ( + ) Τ +, L (where a b denotes the axiu of the nubers a and b) so Hence, U ax { Z, E( U + F )}. EZ ( F) = Z + EZ ( F) { = } { > } = Z + EZ ( F) { = } { > } ( + ) = Z + E( U F ) { = } { > } + ax( Z, E( U F )) + (3): The definition of superartingale in (3..3) follows iediately fro the theore (3..6): U E( U + F) as does the fact that U doinates Z. To prove the sallest doination property, we choose W = ( W ) =,..., as another superartingale L which doinates the process Z. We ust prove that W doinates U. Clearly, WL ZL = UL. Since W is a superartingale, WL E( WL FL ) E( UL FL ). W doinates Z, WL ZL and so WL ax( ZL, E( UL FL )) = UL. So the general result now follows by a recursive arguent. 26

28 A2: Proof of Theore 3..8 The proof is based on one proposition and one corollary: Proposition Define inf { U Z } = =. Then is a stopping tie and the stopped sequence U is a artingale. Proof: We have { } { } { } { } = k = U > Z... Uk > Zk Uk = Zk Fk According to the definition (3..4), is a stopping tie. To prove U is a artingale, we use the artingale transfor to write U = U + ( U U ) { k } k k k = Therefore, U( + ) U = { }( U U ) { }( U E( U F)) + = Now, we take expectation with respect to F. Since { } { } C F + =, we obtain EU ( U F) = E(( U EU ( F)) F) = ( + ) + + According to the definition (3..3), U is a artingale. Corollary 2 The stopping tie satisfies: U = E( Z F ) = sup E( Z F ) Τ, L Proof: Fro proposition : is a artingale and the definition of, we have: U U = U = E( U F ) = E( U F ) = E( Z F ) L Since stopped sequence U is a superartingale, we have U E( U F ) = E( U F ) E( Z F ) U = sup E( Z F ) L Τ, L Proof: If U is a artingale and satisfies U = Z, we can obtain U = E( U F ) = E( Z F ) Fro Corollary 2, we know U = sup E( Z F ), so EZ ( F ) = sup E( Z F ) ' ' Τ, L ' Τ, L ' 27

29 By the definition (3..7), stopping tie is optial. Conversely, if is optial, fro Corollary 2, we have: U = E( Z F ) E( U F ) U (the inequality hold since U is a superartingale which doinates Z) Hence, EZ ( F) = EU ( F) Z = U Using the superartingale property again: U E( U F) E( U F) = U By the property of conditional expectation: E( U F) = E( E( U F ) F), so we have E( U F) = E( E( U F) F) U = E( U F), thus, U is a artingale. A3: Proof of Theore The proof is based on two leas: Lea For L, we have: L G( a, Z, X) G( b, Z, X) ( Zi )( ) { Z b iφ( X ) a b φ( X )} Proof. See the reference [][3]. L i= i= i i i i i i Lea 2 Assue that Assuption 2 is satisfied for L, then α N, converges alost surely to Proof. See the reference []. α N n Proof: Observe that we can obtain convergence if Ζ nn,, converges to N n= E( Z ). Fro Assuption, the siulated paths are independent, so we have that σ ( α ) = EG [ ( α, Z, X )] converges to n n E( Z ). Therefore, it suffices to show that N N, n n n n li ( G( α, Z, X ) G( α, Z, X )) = N N n= We note W G Z X G Z X N N, n n n n N = ( ( α,, ) ( α,, )) N n=. By lea, we have: 28

30 W G Z X G Z X N N, n n n n N ( ( α,, ) ( α,, )) N n= N L L n Z, { Z n i } ( X n ) N ( X n ) N α φ α α φ n= = = By lea 2, α converges alost surely to N, α, we have for each ε > : li sup li sup N L L n WN Z N N N n= = = L L n = E Z = = { Z n ( n α i φ X ) ε} { Z n ( n α iφ X ) ε} where the last equality follows by Assuption, Letting ε go to zero, we obtain the convergence since by Assuption 2, for L, P( α i φ ( X ) = Z ) = Appendix B: Tables Table : Polynoial failies Polynoial failies First eber Second eber Third eber Definition Monoials Q ( x ) = Q ( x) = x Q ( x) = x 2 2 Qk ( x) = x k Laguerre Shifted Legendre L ( x ) = L( x) = x 2 x k x 4x+ 2 e d k x L2 ( x) = Lk ( x) = ( x e ) k 2 k! dx k k ( ) d k ( ) [( (2 ) ) ] 2 k! dx S ( x ) = S ( x) = 4x 2 2 S 2 ( x ) = 24 x 24 x+ 2 4 S x = x k k k Notes: This table presents the different orthogonal polynoial failies used in the paper. The first three ebers of the respective failies are shown together with the explicit definitions of the polynoials [4]. Table 2: Price estiates for the LS algorith using various nubers of Monoials in the cross-sectional regression,, and given nubers of paths in siulations N=5. S n LS Cal His. Bias S.E. =2 S= S2=

31 =3 S= S2= =4 S= S2= =5 S= S2= Notes: This table shows price estiates for an Aerican put option with one and two onths to expiration, 3 and 6 exercise dates, and a strike price of 48. The initial stock price is 47.2 and 38.5 and the volatility is 7%. The annual interest rate is 5%. Price estiates are calculated for the different cobinations of the nuber of regressors and siulated paths N=5. The reported value LS are calculated based on the LS regression ethod. We also report the standard errors in the colun headed S.E. The bias is the difference between LS and the benchark value fro an options calculator in Montreal Stock Exchange with 6 steps, the value is and.822. Table 3: Price estiates for the LS algorith using various nubers of Monoials in the cross-sectional regression,, and given nubers of paths in siulations N=. S n LS Cal His. Bias S.E. =2 S= S2= =3 S= S2= =4 S= S2= =5 S= S2= Notes: See Table with N=. Table 4: Price estiates for the LS algorith using various nubers of Monoials in the cross-sectional regression,, and given nubers of paths in siulations N=2. S n LS Cal His. Bias S.E. =2 S= S2= =3 S= S2= =4 S= S2= =5 S=

32 S2= Notes: See Table 2 with N=2. Table 5: Price estiates for the LS algorith using various nubers of Monoials in the cross-sectional regression,, and given nubers of paths in siulations N=3. S n LS Cal His. Bias S.E. =2 S= S2= =3 S= S2= =4 S= S2= =5 S= S2= Notes: See Table 2 with N=3. Table 6: Price estiates for the LS algorith using various nubers of Laguerre Polynoials in the cross-sectional regression,, and given nubers of paths in siulations N=5. S n LS Cal His. Bias S.E. =2 S= S2= =3 S= S2= =4 S= S2= =5 S= S2= Notes: See Table 2. Table 7: Price estiates for the LS algorith using various nubers of Laguerre Polynoials in the cross-sectional regression,, and given nubers of paths in siulations N=. S n LS Cal His. Bias S.E. =2 S= S2=

33 =3 S= S2= =4 S= S2= =5 S= S2= Notes: See Table 2 with N=. Table 8: Price estiates for the LS algorith using various nubers of Laguerre Polynoials in the cross-sectional regression,, and given nubers of paths in siulations N=2. S n LS Cal His. Bias S.E. =2 S= S2= =3 S= S2= =4 S= S2= =5 S= S2= Notes: See Table 2 with N=2. Table 9: Price estiates for the LS algorith using various nubers of Laguerre Polynoials in the cross-sectional regression,, and given nubers of paths in siulations N=3. S n LS Cal His. Bias S.E. =2 S= S2= =3 S= S2= =4 S= S2= =5 S= S2= Notes: See Table 2 with N=3. Table : Price estiates for the LS algorith using various nubers of Shifted Legendre Polynoials in the cross-sectional regression,, and given nubers of 32

34 paths in siulations N=5. S n LS Cal His. Bias S.E. =2 S= S2= =3 S= S2= =4 S= S2= =5 S= S2= Notes: See Table 2. Table : Price estiates for the LS algorith using various nubers of Shifted Legendre Polynoials in the cross-sectional regression,, and given nubers of paths in siulations N=. S n LS Cal His. Bias S.E. =2 S= S2= =3 S= S2= =4 S= S2= =5 S= S2= Notes: See Table 2 with N=. Table 2: Price estiates for the LS algorith using various nubers of Shifted Legendre Polynoials in the cross-sectional regression,, and given nubers of paths in siulations N=2. S n LS Cal His. Bias S.E. =2 S= S2= =3 S= S2= =4 S= S2= =5 S= S2= Notes: See Table 2 with N=2. 33