U.U.D.M. Project Report 2009:16

Examensarbete i matematik, 30 hp + 15 hpHandledare och examinator: Maciej KlimekSeptember 2009

The Least-Squares Method for American Option Pricing

Xuejun Huang and Xuewen Huang

Department of MathematicsUppsala University

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Abstract

This article presents how to use the least-squares (LS) regression method to price the American options on basis of the algorithm in a paper by Clement, Lamberton & Protter[1]. The key to LS is the approximation of the conditional expectation functions which determine the optimal exercise strategy. In this paper, through the detailed description of the algorithm and presentation of convergence, it shows how to estimate the conditional expectation by using the LS to value the American options. Moreover, we also compare the simulation results with the historical data and experiment with alternative polynomials to analyze testing results. Key words: American Option, Least-Squares-Method, Monte-Carlo Procedure, Convergence, Optimal Stopping Time, Dynamic Programming Principle.

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Acknowledgements

First of all, we give the greatest thanks to our supervisor, Professor. Maciej Klimek. Thank you for offering us this challenging and interesting topic to help us complete our master thesis in financial mathematics program and for your patience and suggestions in the revision of this paper. Secondly, we would like to give sincere thanks to all teachers in Department of Mathematics. Particularly, we need to thank Erik Ekström and Prof. Johan Tysk since we learn so much relevant knowledge from financial mathematics 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 overcome the difficulties and always make us feel confident. Thank you!! We love you forever!

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Contents 1 Introduction 5

2 An introductory example 6

3 Theoretical framework of valuation 10 3.1 The Snell Envelope .....................................................................................10 3.2 The valuation of algorithm .........................................................................13 3.3 The least-squares regression method ..........................................................15

4 Convergence 17 4.1 Notation ......................................................................................................17 4.2 Convergence results ....................................................................................18

5 Numerical simulation 19 5.1 Simulating from Geometric Brownian Motions..........................................20 5.2 Assessing the least-squares regression .......................................................20

5.2.1 Using different numbers of paths and regressors…………………21 5.2.2 Altering polynomial families .........................................................22

6 Open problems 23 6.1 Multiple assets ........................................................................................... 23 6.2 Computational complexities .......................................................................24

7 Conclusion and future work 25

8 Appendixes 26 Appendix A: Proofs …………………………………………………………..26 Appendix B: Tables …………………………………………………………..29 Appendix C: Figures ………………………………………………………….35 Appendix D: Codes …………………………………………………………...37

9 Reference 40

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1. Introduction The global financial crisis of 2008 has been leading to a world-wide economic revolution. At the beginning, it originates from finance but quickly and broadly spreads to other fields. It is the economic crisis that causes a disastrous impact to the global economy in a short time: banks close down, companies go bankrupt, enterprises’ combination and reconstruction, unemployment rate rapidly increases, the stagnation in trades of imports and exports, stock-option market crashes... Economists analyze and predict that the economic crisis maybe gradually recover until the middle of 2010, however, it will be a long and tough process. The causes of financial crisis have complexities and diversities, such as market instability, higher interest rates in mortgage and real estate and those are inevitable products of long-term imbalance in development between financial markets and financial institutions. On the other hand, stocks, bonds, derivatives, especially options, as components of financial markets, play an important role in economy. Recently, more and more scholars have been interested in methods and techniques to price different types of options, such as finite difference method, Monte-Carlo method and the simulation technique. In this paper, we will discuss how to price American options with the least-squares method.

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 payment (the premium) from 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 may choose not to exercise the right and let it expire. The underlying asset can be a piece of property, shares of stock or some other securities. For example, buying a call option provides the right to buy a specified quantity of a security at an agreed amount, known as the strike price at some time 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, must fulfill the terms of the contract.

Generally, the price of an option depends on various factors, such as the strike

price, the dividends, interest rates and the maturity time. Unlike European option, the American option can be exercised at any time before or at maturity in paper by Bensoussan[2], which leads to highly complicated computations and makes it impossible to find analytical solutions.

In order to solve this problem, people use the most famous numerical

method—the Binomial Model suggested by Cox, Ross & Rubinstein[3]. However, the computational cost in binomial model exponentially increases with the number of

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assets, which makes the speed of calculation obviously slow down. Thus, simulation technique seems to be a better choice.

One of the first to give solutions to pricing American options using simulation

technique was Tilley[4]. Until 1996, Carriere[5] proposed an alternative method to price the American option in terms of optimal stopping times. A recent paper by Longstaff & Schartz [6] presents a simple and effective method to approximate the value of American options. The main idea is to estimate the conditional expectation by the orthogonal projection on space generated by a finite number of basis functions in the simulation by using least-squares method so as to compare the payoff from immediate exercise with the expected payoff from continuation at each exercise date alone each simulated path and finally make the optimal exercise decision. Another recent paper by Clement, Lamberton &Protter gives a detailed analysis of convergence theorems and proves the almost sure convergence of the complete algorithm under general conditions. In addition, they also determine the rate of convergence of Monte-Carlo procedure.

The purpose of this paper is to apply the LS regression method to price the

American options based on the example in [6] but with the algorithm in [1]. The remark 2.1 in [1] explains the difference between these two approaches. We will give specific description of the algorithm with clear presentation of the mathematical results. As a supplementation, we will also explain how the Snell Envelope in discrete time provides a powerful way of representing the value of an American option in [2] and Karatzas[7]. Theoretically, the most technical part of our work is the analysis of algorithm and convergence. In numerical experiment, we pay more attention to compare the simulation results with historical data from the Montreal Stock Exchange Market and compare the estimations obtained from those alternative specifications of the cross-sectional regression model. Furthermore, we examine the results of changing the number of simulated paths and the regressors. As we know, both the number of paths and the number of regressors should go to infinity to approximate the conditional expectation well.

The structure of the paper is as follows. Section 2 presents an introductory

example of the simulation approach. Section 3 provides theoretical framework for description of the algorithm. Section 4 shows the convergence results. Section 5 illustrates the numerical tests and analysis of empirical results. Section 6 gives the open problems for several implementation issues. Section 7 concludes.

2. An Introductory Example In this part, we quote the same example in [6] but based on the algorithm in [1] to value an American put option using the LS regression method. In [6], the regression involves only in the money paths considering the efficiency of the algorithm and

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simplicity of numerical computations. However, here we stick to the original algorithm (involving all paths: in the money, at the money, out of the money) to price an American option. The key to optimally exercise an American option is specifying the expected value of continuation. According to the algorithm, we approximate the conditional expectation by the orthogonal projection onto space generated by a set of basis functions and provide an efficient estimate of conditional expectation functions using the LS method. Through the comparison between the immediate exercise value and the payoff from continuation, we obtain the optimal stopping rule for this American option. The sample paths are generated under the risk-neutral measure and are shown in the following matrix. The strike price is 1.10. The riskless rate is 6% and we discount it back to time t, namely: exp (-0.06).

Stock price paths Path t=0 t=1 t=2 t=3

1 1.00 1.09 1.08 1.34 2 1.00 1.16 1.26 1.54 3 1.00 1.22 1.07 1.03 4 1.00 0.93 0.97 0.92 5 1.00 1.11 1.56 1.52 6 1.00 0.76 0.77 0.90 7 1.00 0.92 0.84 1.01 8 1.00 0.88 1.22 1.34

We need to maximize the value of American put option at each exercise date along each simulated path. First of all, assume that we exercise the option at the final date (t=3), in this case, the American option is exactly the same to the European option. We have the following cash-flow matrix.

Cash-flow matrix at time 3 Path t=1 t=2 t=3

1 -- -- 0.00 2 -- -- 0.00 3 -- -- 0.07 4 -- -- 0.18 5 -- -- 0.00 6 -- -- 0.20 7 -- -- 0.09 8 -- -- 0.00

Now, we assume that the put is at time 2. It requires that the option holder should decide whether to exercise the option immediately or hold it until expiration (t=3). Here, we consider all the paths and obtain a regression matrix at time 2.

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Regression at time 2 Path Y X

1 0.00*0.94176 1.08 2 0.00*0.94176 1.26 3 0.07*0.94176 1.07 4 0.18*0.94176 0.97 5 0.00*0.94176 1.56 6 0.20*0.94176 0.77 7 0.09*0.94176 0.84 8 0.00*0.94176 1.22

X are the stock prices at time 2 and Y denote the corresponding discounted cash flows received at time 3 if the put is not exercised at time 2. We use the LS regression method to estimate the conditional expectation function by regressing Y on three simple basis functions 1, X and 3X : E[Y| X ]≈0.6685-0.7081 X +0.1137 3X . We insert X into the approximation of the conditional expectation function and compare the values with the immediate exercise values at time 2 and give the following matrix below.

Optimal early exercise decision at time 2 Path Exercise Continuation

1 0.02 0.0480 2 0.00 0.0050 3 0.03 0.0511 4 0.13 0.0863 5 0.00 -0.0027 6 0.33 0.1759 7 0.26 0.1418 8 0.00 0.0122

This comparison implies that for the paths 4, 5, 6, 7, it is optimal to exercise the option at time 2 which results in the matrix as follows.

Cash-flow matrix at time 2 Path t=1 t=2 t=3

1 -- 0.00 0.00 2 -- 0.00 0.00 3 -- 0.00 0.07 4 -- 0.13 0.00 5 -- 0.00 0.00 6 -- 0.33 0.00 7 -- 0.26 0.00 8 -- 0.00 0.00

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We redo this recursive process again to examine whether the option should be exercised at time 1. Similarly, we obtain regression matrix at time 1.

Regression at time 1 Path Y X

1 0.00*0.94176 1.09 2 0.00*0.94176 1.16 3 0.07*0.94176*0.94176 1.22 4 0.13*0.94176 0.93 5 0.00*0.94176 1.11 6 0.33*0.94176 0.76 7 0.26*0.94176 0.92 8 0.00*0.94176 0.88

X denote the stock prices at time 1 and Y are the corresponding discounted cash-flows. The approximation of the conditional expectation function at time 1 is E[Y| X ] ≈2.044-2.715 X +0.720 X 3. Substituting the X into this regression and comparing with the immediate exercise values, we have the matrix below.

Optimal early exercise decision at time 1 Path Exercise Continuation

1 0.01 0.0171 2 0.00 0.0184 3 0.00 0.0391 4 0.17 0.0982 5 0.00 0.0150 6 0.34 0.2967 7 0.18 0.1069 8 0.22 0.1455

The comparison above shows that for the paths 4, 6, 7, 8, it is optimal to exercise the option at time 1. The relevant option cash-flow matrix and stopping rule are as follows.

Option cash-flow matrix Path t=1 t=2 t=3

1 0.00 0.00 0.00 2 0.00 0.00 0.00 3 0.00 0.00 0.07 4 0.17 0.00 0.00 5 0.00 0.00 0.00 6 0.34 0.00 0.00 7 0.18 0.00 0.00

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8 0.22 0.00 0.00

Stopping rule Path t=1 t=2 t=3

1 0 0 0 2 0 0 0 3 0 0 1 4 1 0 0 5 0 1 0 6 1 0 0 7 1 0 0 8 1 0 0

From the stopping rule matrix above, it is clear to see that the optimal strategy should be chosen at the exercise dates where there is a one in the matrix. Having given the cash-flows generated by the American put option at each exercise date along each simulated path, we only need to discount each cash-flow back to time zero and make the average of all paths, then the option can be valued approximately. We obtain the same value of 0.1144 for the American put option as described in [6].

Through this simple but very typical example, we illustrate how to price an American option using the LS method according to the algorithm in [1]. Under the shade of failure with numerical methods, the LS, without doubt, provides a more effective and powerful approach and the more important thing is the LS is quite easily to be implemented in simulation techniques to value the American-style options.

3. Theoretical framework of valuation In this section, we give detailed description of the general LS algorithm for pricing the American options. Furthermore, we also illustrate how to use the least-squares method to approximate the conditional expectation functions as mentioned in the introductory example.

3.1 The Snell Envelope In this short part, we briefly explain how the Snell Envelope in discrete time

provides a way of pricing the American options and introduce the ”optional stopping time” for the pricing of American options. To make this algorithm easier to understand, first of all, we introduce several important concepts:

Definition 3.1.1: A probability space [8] ( , , )F PΩ is a measure space, where the set

Ω is called the sample space of all possible outcomes, the σ -algebra F , as the subsets of Ω , are called events, the measure P is the probability measure defined on

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the elements of F .

Definition 3.1.2: A filtration on ( , )FΩ is a family { }0

F j jF

≥= of σ -algebras

jF F⊂ such that 0 s js j F F≤ < ⇒ ⊂ .

Now we consider an underlying probability space: ( , , )F PΩ , equipped with a

discrete time filtration 0,1,...,( )j j LF = , with a finite time horizon L∈ . A general

American option may be exercised at any time before or at the terminal date L . To

model this, we need to consider an adapted payoff process 0,...,( )j j LZ Z == , which is a

sequence of real-valued square integrable random variables:

22 ( ) ( )j jE dω ω

Ω⎡ ⎤Ζ = Ζ Ρ < ∞⎣ ⎦ ∫ , ω∈Ω (3.1.2.1)

Definition 3.1.3: An n-dimensional stochastic process { }0j j≥

Μ on ( , , )F PΩ is called

a martingale with respect to a filtration { }0j j

F≥

if

(I) jΜ is jF - measurable for all j ;

(II) jE ⎡ ⎤Μ < ∞⎣ ⎦ for all j ;

(III) t j jE F⎡ ⎤Μ =Μ⎣ ⎦ for all t j≥

Notice: if (III) satisfies: t j jE F⎡ ⎤Μ ≤Μ⎣ ⎦ , we call the process supermartingale.

t j jE F⎡ ⎤Μ ≥Μ⎣ ⎦ , we call the process submartingale.

We hope to price an American option at any time j, so we also construct a price

process 0,...,( )j j LU U == . At the final exercise date, clearly, we have L LU Z= . However,

what is the price of the process at time L -1?? Obviously, the option holder now can

decide either to exercise the option to earn 1LZ − or wait until time L to earn LZ . In

latter, the option is equivalent to a European contingent claim held over the time

period L -1 to L , so the writer of the option needs to invest ( ( 1))1( )r L L

L LE e Z F− − −− in a

replicating portfolio in order to generate LZ at time L . Hence, we can give that the

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price of the option at time L -1 should be { }( ( 1))1 1 1max , ( )r L L

L L L LU Z E e U F− − −− − −= .

We iterate this argument and deduce that the price for this option at time j -1 is

{ }( ( 1))1 1 1max , ( )r j j

j j j jU Z E e U F− − −− − −= . If we can discount both the pay-off and the

price processes by defining rjj jU e U−=

∼ and rj

j jZ e Z−=∼

, then the argument above

can be rewritten as the form: { }11 1max , ( )jj j jU Z E U F−− −=∼ ∼ ∼

. Particularly, if

1 1( )j j jZ E U F− −≤∼ ∼

, then we have the martingale property[9]: 1 1( )j j jU E U F− −=∼ ∼

.

Now, we turn to the properties of processes that are mentioned above. The

optimal stopping problem for the pay-off process Z consists of maximizing ( )ZτΕ

over all stopping timesτ .

Definition 3.1.4: A random variable τ taking values in is a stopping time if, for

any j∈ , { } jj Fτ = ∈ .

Definition 3.1.5: Let 0,...,( )j j LZ Z == be an adapted sequence of real-valued integrable

random variables. The Snell Envelope of Z is the sequence 0,...,( )j j LU U == defined

by

,

sup ( )j L

j jU ess E Z Fττ∈Τ

= − , 0 j L≤ ≤ .

where ,j LΤ denotes the set of all stopping times with values in { j , j +1 ,..., L }.

Theorem 3.1.6: The Snell Envelope U of Z satisfies the following properties:

(1) L LU Z=

(2) { }1max , ( )jj j jU Z E U F+= , 0 1j L≤ ≤ −

(3) U is the smallest supermartingale which dominates Z. Proof. See the appendix A.

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Definition 3.1.7: A stopping timeτ is called optimal if 0,

0 ' 0'

( ) sup ( )L

E Z F E Z Fτ ττ ∈Τ

= .

Since the stopping times play an important role in pricing the American options, with the help of the Snell Envelope, the optimal stopping times can be characterized. Theorem 3.1.8: A stopping timeτ is optimal if and only if it satisfies:

(1) U Zτ τ=

(2) The stopped process jUτ ∧ is a martingale.

(Where a b∧ denotes the minimum of the numbers a and b) Proof. See the appendix A.

3.2 The valuation of algorithm Consistent with no-arbitrage paradigm, we assume that the existence of an equivalent martingale measure Q[10] for the financial markets and assume that the American option can only be exercised at L discrete times.

From the theorem 3.1.8, we can derive 00 0( )U E Z Fτ= since

0 kUτ ∧ is a martingale,

where { }0 min 0 k kk U Zτ = ≥ = . Therefore, we have ( )jj jU E Z Fτ= , with

{ }minj k kk j U Zτ = ≥ = . Substituting 11 1( )

jj jU E Z Fτ ++ += into the second property

of the theorem 3.1.8, we obtain that

{ } { }1 11max , ( ( ) ) max , ( )j j

jj j j j jU Z E E Z F F Z E Z Fτ τ+ ++= = .

Thus, the dynamic programming principle of the theorem 3.1.6 can be rewritten in

terms of the optimal stopping times jτ as follows:

{ } { }1( ) ( )1 1

1 1 ,0 1L

j jZ E Z F Z E Z Fj j j jj j

Lj j L

τ τ

ττ τ +

≥ <+ +

=⎧⎪⎨ = + ≤ ≤ −⎪⎩

(3.2.1)

In the formula above, { }condition holds

otherwise

1, 1

0, condition

⎧= ⎨⎩

, which determines the value of

the stopping time. Also, this formula highlights the fact that if we can know how to approximate the conditional expectations of payoffs, the value of American option can be priced. The first equation in 3.2.1 tells the holder to exercise the option at the final exercise date if it is in the money, otherwise, the option will be expired. When exercising the option prior to the terminal expiration date, the second equation

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requires the option holder to determine whether to exercise immediately or to hold the option for next exercise date. Therefore, the optimal stopping problem changes to comparing the immediate exercise value with the conditional expectation from continuation. From the introductory example in section 2, we know that the value of immediate exercise is easily to obtain. However, the main problem in (3.2.1) is that the conditional expectation is unknown, so for the next step, we focus on the estimation of conditional expectations.

The motivation for approximating the conditional expectation can be given in

terms of projection theory of Hilbert spaces. Now, we limit our attentions to the square-integrable payoff functions in (3.1.2.1). This space is often denoted by

2 ( , , )L F PΩ and it is an example of a Hilbert space: a complete space equipped with a

form of inner-product: , ( ) ( )f h f x h x dx⟨ ⟩ = ∫ . In addition, we assume the underlying

model to be a Markov chain[11], which means the description of the present state fully captures all the information that could influence the future evolution of the

process. Therefore, we have1 1

( ) ( )j jj jE Z F E Z Xτ τ+ +

= .

Here, 0,...,( )j j LX = is an jF -adapted Markov chain and presents a process of stock

price and we denote ( , )j jZ f j X= as an adapted pay-off process for the option.

Now it is time to approximate the conditional expectation with respect to jX .

Since 2L is a Hilbert space, it has a countable orthogonal basis:

1 2( ) ( ( ), ( ),..., ( )).mmX X X Xφ φ φ φ= As an element of the 2L space of square-integrable

functions, the conditional expectation can be represented as a linear combination of the elements of the basis[12]. Hence, (3.2.1) can be written as follows:

1( ) ( )

1 1

1 1 ,0 1

mLm mj j

m mZ Z Z Zm mj j j jj j

L

j j Lτ τ

τ

τ τ +⎧ ⎫ ⎧ ⎫⎪ ⎪ ⎪ ⎪⎨ ⎬ ⎨ ⎬⎪ ⎪ ⎪ ⎪⎩ ⎭ ⎩ ⎭

≥Ρ <Ρ+ +

⎧ =⎪

= + ≤ ≤ −⎨⎪⎩

(3.2.2)

1( ) ( )m

j

m m mj j jP Z X

τα φ

+= i , here” i” denotes the usual scalar product in m and m

jα

are coefficients of linear combination of elements of basis:1

( )m

m mj jXα φ∑ . So we

obtain an approximation for the value function:

1

0 0max( , ( ))mmU Z E Z

τ= (3.2.3)

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From (3.2.2), it is not difficult to see that the projection onto vector space becomes a

key for approximating the conditional expectation. The coefficients mjα are unknown

and must be estimated, so we use the Monte-Carlo procedure[13] to approximate the simulation procedure. Just like the introductory example, we consider N simulated

paths of the Markov chain 1,...,( )nj n NX = and denote ( , )n n

j jZ f j X= as the relevant

pay-off process. Furthermore, we estimate the coefficients mjα with , ,m n N

jα and the

coefficients , ,m n Njα can be calculated as the solutions to the following expression:

{ }, ,1

21 1 2 2

1

m in ( ( ) ( )... ( ) ( ))n m Nn m jj

Nn n n n n n n

j j m m jn

X X X Zτα

α φ α φ α φ+∈ =

+ + −∑ (3.2.4)

This is the Least-Squares problem and will be discussed in part 3.3. Hence, we can estimate recursively the stopping time with N simulated paths by:

{ } { }

, ,

, , , ,1, ,( ) ( )

1 1 ,0 1

m n NLm n N m n Nj jm N m Nn m n n m nZ X Z Xj j j jj j

L

j j Lα φ α φ

τ

τ τ +≥ <

⎧ =⎪⎨ = + ≤ ≤ −⎪⎩ i i

(3.2.5)

We substitute the coefficients from (3.2.4) into , ( )m N m nj jXα φi and use these values as

the linear approximation to 1

( )mj

mjP Z

τ + in (3.2.2) based on the N simulated paths. Now,

the option can be valued by discounting each cash flow back to time zero and average over all the simulated paths. Finally, we obtain the following approximation for option value:

, ,1

,0 0

1

1max( , )n m N

Nm N n

n

UN τ

=

= Ζ Ζ∑ (3.2.6)

3.3 The Least-Squares Regression Method In section 3.2, we use the least-squares regression method to compute the value functions in the approximation of the conditional expectation. Now, we present the specific implementation of least-squares regression method.

The least-squares method assumes that the best-fit curve of a given type is the curve that has the minimal sum of the deviations squared from a given set of data.

Suppose that the data set: 1 1 2 2( , ), ( , ),..., ( , )n nx y x y x y and a fitting curve ( )f x which

has the deviation from each data point:

1 1 1 2 2 2( ), ( ),..., ( )n n nd y f x d y f x d y f x= − = − = − .

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The least-squares method produces the best fitting curve with the property:

the Minimum Least-Square Error [ ]22

1 1

( )n n

i i ii i

d y f x= =

= = −∑ ∑

Now we look back to our case. We assume that the number of regressors is m and the

number of simulated paths is N. Thus, we need to calculate the coefficients ,n mjα in

(3.2.4) by using the LS method to determine the value of 1

( )mj

mjP Z

τ + in (3.2.2). Let

1 2: , ,..., ( )NZ E Z X X X Pro ZΜ∈Η ⎡ ⎤ =⎣ ⎦ . X denote the stock prices and Z(Y in the

introductory example) are the corresponding discounted cash-flows. Suppose that

1 2, ,...., mφ φ φ are non-zero vectors in Hilbert Space and M is a subspace generated by

vectors 1 2, ,...., mφ φ φ . According to the orthogonal projection, we have

1 1 2 2( ) ,..., m mPro Z α φ α φ α φΜ = + + + , 1 2( , ,..., )NZ Z Z Z=

Hence, 1 2( , ,..., )mα α α α= is a solution of .BαΦ = Here, , .T TB bφ φ φΦ = = In

our case, we are assuming three linear independent basis vectors in 2L space:

3 31 2 1 3 1(1,...,1) , ( ,..., ) , ( ,..., )N N

N N N

X X X Xφ φ φ= = =

So the matrices 1 1

2 13 3

3 1

(1,...,1)( ,..., ) ,

( ,..., )

TN

NN

ZX X b

ZX X

φφ φ

φ

⎛ ⎞= ⎛ ⎞⎜ ⎟ ⎜ ⎟= = =⎜ ⎟ ⎜ ⎟

⎜ ⎟⎜ ⎟= ⎝ ⎠⎝ ⎠

Namely,

1 1 1 1 1

1

, . . . , ,. . . . .. . . . .. . . . ., . . . , ,

m

m m m m m

B

Z

Zα

φ φ φ φ α φ

φ φ φ φ α φΦ

< > < > < >⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥=⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥< > < > < >⎣ ⎦ ⎣ ⎦ ⎣ ⎦

(3.3.1)

Here, ,< > denotes the inner-product. Generally, any Euclidean space n with

inner-product is 1 1 1 11

( ... ), ( ... ) ...n

n n i i n ni

x x z z x z x z x z=

< >= = + +∑ .

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If 1 2, ,...., mφ φ φ are linear independent, there is a unique solution for the system of

equations BαΦ = , so we have 1Bα −= Φ . 1−Φ is the inverse matrix of Φ . Now we

can calculate the coefficients of the approximation of the conditional expectation functions in section 2 as follows:

13

1 1 1 11

2 42

1 1 1 13

3 4 6 3

1 1 1 1

1 1

N N N N

i i ii i i i

N N N N

i i i i ii i i iN N N N

i i i i ii i i i

X X Z

X X X X Z

X X X X Z

ααα

−

= = = =

= = = =

= = = =

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥ = ⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎣ ⎦ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

∑ ∑ ∑ ∑

∑ ∑ ∑ ∑

∑ ∑ ∑ ∑

(3.3.2)

From calculation above, it is simple to see that the coefficients of the approximation of the conditional expectation functions depend on the choices of different basis functions and the results will maybe influence the optimal early exercise decision, the stopping rules and even the values of the American options. In addition, the number of regressors and the number of simulated paths will also affect the approximating results of conditional expectation. In the next section, we will show the convergence results: both the number of regressors and the number of paths should tend to infinity; we can obtain an ideal approximation of the conditional expectation.

4. Convergence From (3.2.6), it seems reasonable that the price of estimation will converge to the true price of the American option if the approximation of the conditional expectation converges to the true expectation function. Therefore, in this section, we will show the convergence results of the conditional expectation approximation: for any fixed

number of basis functions m, ,0m NU converges to 0

mU as N tends to infinity, and that

0mU converges to 0U as m goes to infinity.

4.1 Notation Before presenting the convergence results, we give some necessary notations to make the theorems and proofs easier and more clearly to understand.

From (3.2.4), the estimator ,m Njα has the explicit expression:

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18  

, ,1

, , 1

1

1( ) ( )n m Nj

Nm N m N n m nj j j

n

Z XN τ

α φ+

−

=

= Χ ∑ (4.1.1)

for 1 1j L≤ ≤ − , where ,m NjΧ is an m m× matrix, with coefficients given by (3.3.1).

For any vectors , , ,1 1 1 1 1( ,..., ), ( ,..., ), ( ,..., )m m m m N m N m N L

L L Lz z zα α α α α α− −= = = ∈ and

1( ,..., ) LLx x x= ∈ , a vector pay-off function 1( ,..., )LG G G= can be defined:

{ } { }1( ) ( )

( , , )

( , , ) 1 ( , , )1

mL L

m mj j jm m m mz x z xj j j j j j

G z x z

G z x z G z xα φ α φ

α

α α+≥ <

⎧ =⎪⎨ = +⎪⎩ i i

(4.1.2)

Compared with (3.2.2) and (3.2.5), we have:

, ,,

( , , )

( , , )

mj

n m Nj

mj

m N n n nj

G Z X Z

G Z X Zτ

τ

α

α

=

= (4.1.3)

Thus, (4.1.1) can be rewritten as:

, , 1 ,1

1

1( ) ( , , ) ( )N

m N m N m N n n m nj j j j

n

G Z X XN

α α φ−+

=

= Χ ∑ (4.1.4)

4.2 Convergence result Before stating the convergence results, we have to propose two assumptions such that the optimal early exercise decision determined from the approximation is correct. Assumption 1: The simulated paths are independent.

Assumption 2: For 1 j L≤ ≤ , ( ( ) ) 0j j jP X Zα φ = =i

The first assumption is implemented during the Monte Carlo procedure. The

second assumption will be considered in the proofs of the convergence theorems. Also, this assumption will make sure that the optimal stopping time can be correctly

identified when the pay-off process , ,n m Nj

nZτ

converges toj

Zτ .

Theorem 4.2.1 Assume that the measurable real valued basis functions ( )m jXφ is

total in 2L . For 1 j L≤ ≤ , we have lim ( ) ( )m jjj jm

E Z F E Z Fττ→+∞= in 2L .

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We proceed by induction on j . For j L= , theorem 4.2.1 is obviously true. If it is

true for 1 j + , we only need to prove that the theorem holds for j ( 1)j L≤ − . From

(4.1.3), (4.1.2) and (3.2.1), we have

{ } { }

{ } { }

1

1

( ) ( )

( ) ( )1 1

1 1

1 1

m mj j

j j

j Z Z

j

m m m mX Xj j j j j j

Z E Z F Z E Z Fj j j jj j

Z Z Z

Z Z Z

τ τ

τ τ

α φ α φ

τ τ

+

+

≥ <

≥ <+ +

= +⎧⎪⎨ = +⎪⎩

i i

Hence, we obtain

{ } { }

{ }

( )1( )

( )

( ) ( ( ))( )1

( )11

1 1

1

jZ E Zj j F

Z

m mZ Xj j j

m m Xj j j

F Fj j

Fj

E Z Z Z E Zm jj jj

E Z Zm jj

τα φ

α φ

τ ττ

ττ

≥ +

−≥

<

− = −+

−++

+

i

i

The second term on the right hand side converges to zero by assumption, so we just prove that the first term converges to zero in 2L . Proof. See the reference [1].

Theorem 4.2.2 Assume that Assumption 2 is satisfied. Then ,0m NU converges almost

surely to 0mU as N tends to infinity.

Theorem 4.2.2 requires us to prove that , ,

1

1n m Nj

Nn

nN τ=

Ζ∑ in (3.2.6) converges almost

surely to ( )mj

E Zτ

in (3.2.3) as N tends to infinity for 1 j L≤ ≤ . Now we define the

function ( ) [ ( , , )]m mj jE G Z Xσ α α= . Using (4.1.3), we only need to prove that

,

1

1lim ( , , ) ( ), 1N

m N n n mj jN n

G Z X j LN

α σ α→∞

=

= ≤ ≤∑ (4.2.2.1)

Proof. See the appendix A.

5. Numerical simulation Earlier we have given a specific example to illustrate how the least-square regression method could be applied to the valuation of American put options. In this part, we will show how to price the American options using numerical simulation based on the LS method and test the results against the actual prices. Also, we compare the LS with the

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option calculator in Montreal Stock Exchange to obtain the difference in early exercise value. We will assume that the only stochastic factors are the prices of the underlying stocks. Moreover, we will assume that the risk neutral dynamics of these stock prices can be specified as Geometric Brownian Motions (GBM).

5.1 Simulating from a GBM We are interested in pricing an American put option, where the risk-neutral stock price process satisfies the following stochastic differential equation (SDE):

( ) ( ) ( ) ( )dS t rS t dt S t dW tσ= + (5.1.1)

where W is a standard Brownian process and r and σ are constants. The stock does

not pay dividends. Given the initial stock price 0S , the well-known solution of (5.1.1)

can be written as follows:

20

1( ) exp ( ) ( )2

S t S r t W tσ σ⎧ ⎫= − +⎨ ⎬⎩ ⎭

(5.1.2)

From the properties of the Brownian process, simulated values of ( )S t can be

obtained from the formula:

20

1( ) exp ( ) ( )2

S t S r t tZ tσ σ⎧ ⎫= − +⎨ ⎬⎩ ⎭

, (5.1.3)

where ( ) (0,1)Z t N∼ satisfies the standard normal distribution. So a sequence of

values at discrete times 1 20 ... Nt t t L< ≤ ≤ ≤ = can be obtained from

21 1 1 1

1( ) ( )exp ( )( ) ( ) ( ) ,2j j j j j j jS t S t r t t t t Z tσ σ+ + + +

⎧ ⎫= − − + −⎨ ⎬⎩ ⎭

(5.1.4)

From (5.1.4), it is seen that the stock price can be simulated from a GBM at each single point. With those stock prices, the comparison between the payoff from immediate exercise and the expected payoff from keeping the option alive at each exercise date is easily to be done, which leads to the optimal decision to price the American options eventually.

5.2 Assessing the LS regression Here, we are interested in examining the results of using different numbers of paths and regressors than those used in Longstaff & Schwartz (2001) and the results of changing various polynomials. As convergence theorem mentioned above, both the number of paths and the number of regressors should tend to infinity so as to estimate the conditional expectation arbitrarily well. Furthermore, alternative specifications of

 

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21 

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show from ed in

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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 made by the calculator are based on option pricing models. The models used can be approximations and the results they produce can differ depending on the model which is used. In addition, this options calculator can evaluate the premium of index and equity options, for either American 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 month or the days to expiration include all calendars days (including holidays); for dividend paying stock options and American options, it uses the numerical binomial model (Cox, Ross and Rubinstein, 1979) with the maximum number of steps 60.

In Table 2 we report the results for the options with two initial stock prices 47.12 and 38.5, correspondingly the exercise dates 30 and 60. The first obvious thing to note from the table is that without considering the number of paths, the bias decreases with the number of regressors growing. It means that the convergence can be guaranteed when the number of regressors is increased. Particularly, when the LS with a low number of regressors, m=2 or 3, results in a larger bias; and with more regressors, m=5, for 60 exercise days, it produces upward biased prices. This corresponds to the approximation bias mentioned above which potentially vanishes as the number of regressors is increased. From Table 2 to Table 5, we can also see that when the number of paths used increases, for a reasonable number of regressor m=3, the effect of which could potentially be the high bias tends to disappear. Meanwhile, the standard error of the estimate decreases from increasing the number of paths used. The similar conclusion can also be obtained from other tables but we need to choose a reasonable number of regressors m=3 or m=4.

5.2.2 Altering polynomial families Now we divert our attentions to the results in numerical experiments with alternative polynomial families. Three different polynomials used in this paper can be found in Table 1 in Appendix. To be simple, the first polynomial that we choose consists of

combinations of monomials: { }0

( ) kk k

Q x x∞

== . The second one that we use is the

weighted Laguerre polynomial: ( )kL x . The elements of the Laguerre family have the

property of being mutually orthogonal on the interval [0, )∞ with respect to the

weighting function ( ) xw x e−= . The third one that we use is the shifted Legendre

polynomial: ( )kS x , which also shares the orthogonal property on the interval (0,1) .

Another advantage with the Shifted Legendre polynomials is that no computationally

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intensive weights have to be calculated since the weighting function ( ) 1w x = . Instead

of using the exact formula for those polynomials, we use the recursive formula for calculating the elements, since it is computationally much easier. The first three terms have been given in Table 1. We present the results with different polynomial families from Table 2 to Table 13. First of all, it is obviously to note from Table 6 to Table 9 that the number of regressors and the numbers of paths both increase to achieve arbitrarily close approximation with the Laguerre polynomials, as it is the case with the simple monomials from Table 2 to Table 5. Additionally, it also shows that the main benefit from increasing the number of paths is again to lower the standard errors. Hence, from Table 6 to Table 9, it guarantees that using the standard error as a precision both the number of regress, m, and the number of paths, N, should tend to infinity to obtain convergence of the estimated price. Tables 10,11,12,13 illustrate the similar conclusions with the Shifted Legendre polynomials. However, it is not difficult to observe that the standard errors become smaller when the Laguerre and Shifted Legendre polynomials are used as regressors, especially for some larger simulated paths. Particularly, for small numbers of regressor, m=2, the bias is much smaller with the Shifted Legendre polynomials compared with two others and with the same numbers of paths, N=20000 or 30000, the estimated prices are more close to the actual option prices. Therefore, the Legendre polynomials seem to be a better choice than either of the other polynomials.

6. Open problems In this section, we discuss several important numerical and implementation issues associated with the multi-dimensional problems and computational complexities.

6.1 Multiple assets The numerical simulations we have implemented in section 5 benchmark the performance of the Least-squares method for 1-dimensional problems which can also be solved by other techniques, such as FD and BM. However, for the multiple assets, those techniques become infeasible in practice.

A simple example of multiple stochastic factors is the case in which there are two or more underlying assets. Options on multiple underlying assets typically can be divided into three types [15]: Rainbow options, Basket options and Quanto options. The most common options with multiple assets have payoffs as a function of the maximum, minimum or the average of the asset prices:

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The payoff of maximum option= 1 2( max( , ,..., ))nK S S S +− (6.1.1)

The payoff of minimum option= 1 2( min( , ,..., ))nK S S S +− (6.1.2)

The arithmetic average option=1

( )n

j jj

K Sα +

=

−∑ (6.1.3)

The geometric average option=1

( )jn

jj

K Sα +

=

−∏ (6.1.4)

where , 1,...,jS j n= are the prices of the underlying assets, jα is the ratio of the j

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. Furthermore, Boyle and Tse present how to trade these types of options [16] and Lars Stentoft gives detailed description about how to price options on multiple assets with numerical methods [13].

6.2 Computational Complexities Extensive numerical tests indicate that the results from the LS algorithm are remarkably relevant to the computational complexities. In section 5, we used three different polynomial families as basis functions to estimate the conditional expectation functions. Specially, the weighted Laguerre polynomials include an exponential term, which will result in the computational underflows if applying the polynomials directly. Thus, for the numerical convenience, we simplify the weighted Laguerre polynomials to the form listed in Table 1 in Appendix B to obtain the reasonable option values. Another solution to this problem is that to renormalize the American put example by dividing all cash flows and prices by the strike price and estimating the conditional expectation function in the renormalized space. Note that the option value is unaffected since we discount back the un-normalized value of the cash flows along each path to obtain its value. However, altering the cross-sectional information will have an obvious effect on the running time. Adding regressors will lead to an increase in the computational time. Moreover, increasing the number of path to simulate will also increase time to calculate the estimated option prices. More than 60000 simulated paths will lead to “out-of-memory” in our numerical simulations with MATLAB commands. Here we present a measure of computational time according to different polynomials for the different combinations between regressors and paths. It is calculated per second on a Genuine Intel T2400, 1.83GHz, with 0.99GB of RAM. The results are reported in Table 14, 15 and 16. Corresponding to the tables, we plot the Figure 2, 3, and 4.

A key advantage of simulation techniques is that we can do the simulations using

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parallel computing architecture. With the parallel algorithm, we could generate 1000 paths each on 30 CPUs instead of generating 30000 paths based on a single CPU. Without doubt, it will largely increase the computational speed and decrease the CPU time; meanwhile, the parallel algorithm will obviously save the computer’s memory occupation. Moreover, in many large-scale application fields, the computational speed and memory-consuming in computers are far more important compared with the cost of hardware. From the perspective of the LS algorithm, the constraint on parallel computation is that the regression needs to use the cross-sectional information in the simulations and the drawback maybe involves little loss of computational efficiency.

7. Conclusion and Future work In this article, we have presented a detailed analysis of the Least Square (LS) method proposed in Longstaff & Schwartz (2001). The approach is intuitive, accurate and easy to apply. Firstly, the detailed descriptions of the LS algorithm and the convergence theorem are performed. Next, we illustrate the LS algorithm using a number of numerical experiments with different polynomial families and compare this method with an existing options calculator. Through the comparison of bias and standard errors, it suggests that the LS algorithm is able to approximate closely the actual option values. Meanwhile we also show that the Shifted Legendre polynomial is a better choice as the specification of the cross-sectional regressors. Finally, we discuss computational complexities. We conclude that considering the computational time to calculate an option price, the preferred choice, for fewer regressors m=2 or 3, is to use the simple monomials.

Several extensions of our framework can be carried out, such as to approximate the option values for at the money and out of the money and to apply the LS algorithm to the multiple assets. Furthermore, a trade-off between the computational time and the precision of the estimated option price can be measured.

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Appendix A: Proofs

A1: Proof of Theorem 3.1.6

Proof: (1): When j L= , the equality L LU Z= is obtained from definition (3.1.5).

(2): Step 1: Now we consider 0 1j L≤ ≤ − . It is obvious that j jU Z≥ .

Using definition (3.1.5), we also have:

1, 1,

1 1 1( ) ( sup ( ) ) sup ( ( ) )j L j L

j j j j j jE U F E ess E Z F F ess E E Z F Fτ ττ τ+ +

+ + +∈Τ ∈Τ

= − = −

Since the inclusion: 1, ,j L j L+Τ ⊂ Τ , 1( ( ) ) ( )j j j jE E Z F F E Z F Uτ τ+ = ≤ .

Hence, { }1max , ( )jj j jU Z E U F+≥ .

Step 2: If ,j Lτ ∈Τ , we choose 1,( 1) j Ljτ +∨ + ∈Τ (where a b∨ denotes the

maximum of the numbers a and b) so

{ } { }

{ } { }

{ } { }

( 1)

1

1

( ) 1 ( 1 )

1 1 ( )

1 1 ( )

max( , ( ))

j j jj j

j j jj j

j j jj j

j j j

E Z F Z E Z F

Z E Z F

Z E U F

Z E U F

τ ττ τ

ττ τ

τ τ

= >

∨ += >

+= >

+

= +

= +

= +

≤

Hence, { }1max , ( )jj j jU Z E U F+≤ .

(3): The definition of supermartingale in (3.1.3) follows immediately from the

theorem (3.1.6): 1( )j j jU E U F+≥ as does the fact that U dominates Z. To prove the

smallest domination property, we choose 0,...,( )j j LW W == as another supermartingale

which dominates the process Z. We must prove that W dominates U. Clearly,

L L LW Z U≥ = . Since W is a supermartingale, 1 1 1( ) ( )L L L L LW E W F E U F− − −≥ ≥ . W

dominates Z, 1 1L LW Z− −≥ and so 1 1 1 1max( , ( ))L L L L LW Z E U F U− − − −≥ = . So the general

result now follows by a recursive argument.

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A2: Proof of Theorem 3.1.8 The proof is based on one proposition and one corollary:

Proposition 1 Define { }0 inf 0 j jj U Zτ = ≥ = . Then 0τ is a stopping time and the

stopped sequence 0jU τ∧ is a martingale.

Proof: We have { } { } { } { }0 0 0 1 1... k k k k kk U Z U Z U Z Fτ − −= = > > = ∈∩ ∩ ∩

According to the definition (3.1.4), 0τ is a stopping time.

To prove 0jU τ∧ is a martingale, we use the martingale transform to write

{ }0 00 11

1 ( )j

j k kkk

U U U Uτ τ∧ −≤=

= + −∑

Therefore, { } { }0 0 0 0( 1) 1 1 11 11 ( ) 1 ( ( ))j j j j j j jj jU U U U U E U Fτ τ τ τ+ ∧ ∧ + + ++ ≤ + ≤− = − = − . Now, we

take expectation with respect to jF . Since { } { }0 01 Cjj j Fτ τ+ ≥ = ≤ ∈ , we obtain

0 0( 1) 1 1( ) (( ( )) ) 0j j j j j j jE U U F E U E U F Fτ τ+ ∧ ∧ + +− = − =

According to the definition (3.1.3), 0jU τ∧ is a martingale.

Corollary 2 The stopping time 0τ satisfies:

00,

0 0 0( ) sup ( )L

U E Z F E Z Fτ ττ∈Τ

= =

Proof: From proposition 1:0jU τ∧ is a martingale and the definition of 0τ , we have:

0 0 0 00 0 0 0 0( ) ( ) ( )LU U E U F E U F E Z Fτ τ τ τ∧ ∧= = = =

Since stopped sequence Uτ is a supermartingale, we have

0,

0 0 0 0 0 0( ) ( ) ( ) sup ( )L

LU E U F E U F E Z F U E Z Fτ τ τ ττ

∧∈Τ

≥ = ≥ ⇒ =

Proof: If jUτ ∧ is a martingale and τ satisfies U Zτ τ= , we can obtain

0 0 0( ) ( )U E U F E Z Fτ τ= =

From Corollary 2, we know 0,

0 ' 0'sup ( )

L

U E Z Fττ ∈Τ

= , so 0( )E Z Fτ0,

' 0'sup ( )

L

E Z Fττ ∈Τ

=

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By the definition (3.1.7), stopping time τ is optimal. Conversely, if τ is optimal, from Corollary 2, we have:

0 0 0 0( ) ( )U E Z F E U F Uτ τ= ≤ ≤

(the inequality hold since U is a supermartingale which dominates Z)

Hence, 0 0( ) ( )E Z F E U F Z Uτ τ τ τ= ⇒ =

Using the supermartingale property again: 0 0 0 0( ) ( )jU E U F E U F Uτ τ∧≥ ≥ =

By the property of conditional expectation: 0 0( ) ( ( ) )jE U F E E U F Fτ τ= , so we have

0 0( ) ( ( ) ) ( )j j j jE U F E E U F F U E U Fτ τ τ τ∧ ∧= ⇒ = , thus, jU τ∧ is a martingale.

A3: Proof of Theorem 4.2.2 The proof is based on two lemmas:

Lemma 1 For 1 1j L≤ ≤ − , we have:

{ }1

( ) ( )( , , ) ( , , ) ( )( 1 )

L L

j j ii j i j

Z b X a b Xi i i i i iG a Z X G b Z X Z

φ φ

−

= =− ≤ −

− ≤ ∑ ∑ i

Proof. See the reference [1][13].

Lemma 2 Assume that Assumption 2 is satisfied for 1 1j L≤ ≤ − , then ,m Njα

converges almost surely to mjα

Proof. See the reference [1].

Proof: Observe that we can obtain convergence if , ,11

1n m N

Nn

nN τ=

Ζ∑ converges to 0

( )E Zτ .

From Assumption 1, the simulated paths are independent, so we have that

1 1( ) [ ( , , )]m m n nE G Z Xσ α α= converges to 0

( )E Zτ . Therefore, it suffices to show that

,

1

1lim ( ( , , ) ( , , )) 0N

m N n n m n nj jN n

G Z X G Z XN

α α→∞

=

− =∑

We note ,

1

1 ( ( , , ) ( , , ))N

m N n n m n nN j j

n

W G Z X G Z XN

α α=

= −∑ . By lemma 1, we have:

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29  

{ }

,

11

1 1 1,( ) ( )

1 ( ( , , ) ( , , ))

1 1j j j j

Nm N n n m n n

N j jnN L L

nj

n j jm Nn m n m nZ X Xj j

W G Z X G Z XN

ZN α φ α α φ

α α=

−

= = =− ≤ −

≤ −

≤

∑

∑∑ ∑ i

By lemma 2, ,m Njα converges almost surely to m

jα , we have for each 0 :ε >

{ }

{ }

1

N N 1 1 1

1

1 1

( )

( )

1lim sup lim sup 1

= 1

N L Ln

N jn j j

L Lnj

j j

n m nZ Xj j j

n m nZ Xj j j

W ZN

E Z

α φ ε

α φ ε

−

→∞ →∞ = = =

−

= =

− ≤

− ≤

≤

⎡ ⎤⎢ ⎥⎣ ⎦

∑∑ ∑

∑ ∑

i

i

where the last equality follows by Assumption 1, Letting ε go to zero, we obtain the

convergence since by Assumption 2, for 1 j L≤ ≤ , ( ( ) ) 0j j jP X Zα φ = =i

Appendix B: Tables

Table 1: Polynomial families

Polynomial families

First member

Second member

Third member Definition

Monomials 0 ( ) 1Q x = 1( )Q x x= 22 ( )Q x x= ( ) k

kQ x x=

Laguerre 0 ( ) 1L x = 1( ) 1L x x= −

2

24 2( )2

x xL x − += ( ) ( )

!

x kk x

k k

e dL x x ek dx

−=

Shifted Legendre 0 ( ) 1S x = 1( ) 4 2S x x= − 2

2 ( ) 24 24 4S x x x= − + 2( 1)( ) [(1 (2 1) ) ]2 !

k kk

k k k

dS x xk dx

−= − −

Notes: This table presents the different orthogonal polynomial families used in the paper. The first three members of the respective families are shown together with the explicit definitions of the polynomials [14]. Table 2: Price estimates for the LS algorithm using various numbers of Monomials in the cross-sectional regression, m, and given numbers of paths in simulations N=5000.   S  n  LS  Cal  His.  Bias  S.E. m=2  S1=47.12  30  4.002  4.227  4.150  ‐0.225  ‐0.148   S2=38.5  60  10.692  10.822  10.500  ‐0.130  0.192 

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m=3  S1=47.12  30  4.059  4.227  4.150  ‐0.168  ‐0.091   S2=38.5  60  10.603  10.822  10.500  ‐0.219  0.103 m=4  S1=47.12  30  4.105  4.227  4.150  ‐0.122  ‐0.045   S2=38.5  60  10.732  10.822  10.500  ‐0.090  0.232 m=5  S1=47.12  30  4.141  4.227  4.150  ‐0.086  ‐0.009   S2=38.5  60  10.858  10.822  10.500    0.036  0.358  

Notes: This table shows price estimates for an American put option with one and two months to expiration, 30 and 60 exercise dates, and a strike price of 48. The initial stock price is 47.12 and 38.5 and the volatility is 71%. The annual interest rate is 5%. Price estimates are calculated for the different combinations of the number of regressors and simulated paths N=5000. The reported value LS are calculated based on the LS regression method. We also report the standard errors in the column headed S.E. The bias is the difference between LS and the benchmark value from an options calculator in Montreal Stock Exchange with 60 steps, the value is 4.227 and 10.822. Table 3: Price estimates for the LS algorithm using various numbers of Monomials in the cross-sectional regression, m, and given numbers of paths in simulations N=10000.   S  n  LS  Cal  His.  Bias  S.E. m=2  S1=47.12  30  3.938  4.227  4.150  ‐0.289  ‐0.212   S2=38.5  60  10.714  10.822  10.500  ‐0.108  0.214 m=3  S1=47.12  30  4.074  4.227  4.150  ‐0.153  ‐0.076   S2=38.5  60  10.727  10.822  10.500  ‐0.095  0.227 m=4  S1=47.12  30  4.109  4.227  4.150  ‐0.118  ‐0.041   S2=38.5  60  10.723  10.822  10.500  ‐0.099  0.223 m=5  S1=47.12  30  4.145  4.227  4.150  ‐0.082  ‐0.005   S2=38.5  60  10.754  10.822  10.500  ‐0.068  0.254  

Notes: See Table with N=10000.  

 

Table 4: Price estimates for the LS algorithm using various numbers of Monomials in the cross-sectional regression, m, and given numbers of paths in simulations N=20000.   S  n  LS  Cal  His.  Bias  S.E. m=2  S1=47.12  30  3.961  4.227  4.150  ‐0.266  ‐0.189   S2=38.5  60  10.831  10.822  10.500  0.009  0.331 m=3  S1=47.12  30  4.102  4.227  4.150  ‐0.125  ‐0.048   S2=38.5  60  10.680  10.822  10.500  ‐0.142  0.180 m=4  S1=47.12  30  4.057  4.227  4.150  ‐0.170  ‐0.093   S2=38.5  60  10.838  10.822  10.500  0.016  0.338 m=5  S1=47.12  30  4.122  4.227  4.150  ‐0.105  ‐0.028 

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  S2=38.5  60  10.757  10.822  10.500  ‐0.065  0.257  

Notes: See Table 2 with N=20000.  Table 5: Price estimates for the LS algorithm using various numbers of Monomials in the cross-sectional regression, m, and given numbers of paths in simulations N=30000.   S  n  LS  Cal  His.  Bias  S.E. m=2  S1=47.12  30  3.981  4.227  4.150  ‐0.246  ‐0.169   S2=38.5  60  10.881  10.822  10.500  0.059  0.381 m=3  S1=47.12  30  4.116  4.227  4.150  ‐0.111  ‐0.034   S2=38.5  60  10.594  10.822  10.500  ‐0.228  0.094 m=4  S1=47.12  30  4.072  4.227  4.150  ‐0.155  ‐0.078   S2=38.5  60  10.881  10.822  10.500  0.059  0.381 m=5  S1=47.12  30  4.131  4.227  4.150  ‐0.096  ‐0.019   S2=38.5  60  10.746  10.822  10.500  ‐0.076  0.246  

Notes: See Table 2 with N=30000.  Table 6: Price estimates for the LS algorithm using various numbers of Laguerre Polynomials in the cross-sectional regression, m, and given numbers of paths in simulations N=5000.   S  n  LS  Cal  His.  Bias  S.E. m=2  S1=47.12  30  3.873  4.227  4.150  ‐0.354  ‐0.277   S2=38.5  60  10.755  10.822  10.500  ‐0.067  0.255 m=3  S1=47.12  30  4.094  4.227  4.150  ‐0.133  ‐0.056   S2=38.5  60  10.719  10.822  10.500  ‐0.103  0.219 m=4  S1=47.12  30  4.012  4.227  4.150  ‐0.215  ‐0.138   S2=38.5  60  10.981  10.822  10.500  0.159  0.481 m=5  S1=47.12  30  4.123  4.227  4.150  ‐0.104  ‐0.027   S2=38.5  60  10.779  10.822  10.500  ‐0.043  0.279  Notes: See Table 2.  

 

Table 7: Price estimates for the LS algorithm using various numbers of Laguerre Polynomials in the cross-sectional regression, m, and given numbers of paths in simulations N=10000.   S  n  LS  Cal  His.  Bias  S.E. m=2  S1=47.12  30  3.970  4.227  4.150  ‐0.257  ‐0.180   S2=38.5  60  10.693  10.822  10.500  ‐0.129  0.193 

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m=3  S1=47.12  30  4.107  4.227  4.150  ‐0.120  ‐0.043   S2=38.5  60  10.872  10.822  10.500  0.050  0.372 m=4  S1=47.12  30  4.022  4.227  4.150  ‐0.205  ‐0.128   S2=38.5  60  10.787  10.822  10.500  ‐0.035  0.287 m=5  S1=47.12  30  4.057  4.227  4.150  ‐0.170  ‐0.093   S2=38.5  60  10.754  10.822  10.500  ‐0.068  0.254  

Notes: See Table 2 with N=10000.  Table 8: Price estimates for the LS algorithm using various numbers of Laguerre Polynomials in the cross-sectional regression, m, and given numbers of paths in simulations N=20000.   S  n  LS  Cal  His.  Bias  S.E. m=2  S1=47.12  30  3.975  4.227  4.150  ‐0.252  ‐0.175   S2=38.5  60  10.722  10.822  10.500  ‐0.100  0.222 m=3  S1=47.12  30  4.130  4.227  4.150  ‐0.097  ‐0.020   S2=38.5  60  10.791  10.822  10.500  ‐0.031  0.291 m=4  S1=47.12  30  4.130  4.227  4.150  ‐0.097  ‐0.020   S2=38.5  60  10.672  10.822  10.500  ‐0.150  0.172 m=5  S1=47.12  30  4.135  4.227  4.150  ‐0.092  ‐0.015   S2=38.5  60  10.669  10.822  10.500  ‐0.153  0.169  

Notes: See Table 2 with N=20000.  

 

Table 9: Price estimates for the LS algorithm using various numbers of Laguerre Polynomials in the cross-sectional regression, m, and given numbers of paths in simulations N=30000.   S  n  LS  Cal  His.  Bias  S.E. m=2  S1=47.12  30  3.975  4.227  4.150  ‐0.252  ‐0.175   S2=38.5  60  10.766  10.822  10.500  ‐0.056  0.266 m=3  S1=47.12  30  4.124  4.227  4.150  ‐0.103  ‐0.026   S2=38.5  60  10.736  10.822  10.500  ‐0.086  0.236 m=4  S1=47.12  30  4.150  4.227  4.150  ‐0.077  0.000   S2=38.5  60  10.749  10.822  10.500  ‐0.073  0.249 m=5  S1=47.12  30  4.160  4.227  4.150  ‐0.067  0.010   S2=38.5  60  10.807  10.822  10.500  ‐0.015  0.307  

Notes: See Table 2 with N=30000.  

 

Table 10: Price estimates for the LS algorithm using various numbers of Shifted Legendre Polynomials in the cross-sectional regression, m, and given numbers of

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paths in simulations N=5000.   S  n  LS  Cal  His.  Bias  S.E. m=2  S1=47.12  30  4.025  4.227  4.150  ‐0.202  ‐0.125   S2=38.5  60  10.800  10.822  10.500  ‐0.022  0.300 m=3  S1=47.12  30  4.098  4.227  4.150  ‐0.129  ‐0.052   S2=38.5  60  10.879  10.822  10.500  0.057  0.379 m=4  S1=47.12  30  4.000  4.227  4.150  ‐0.227  ‐0.150   S2=38.5  60  10.870  10.822  10.500  0.048  0.370 m=5  S1=47.12  30  4.030  4.227  4.150  ‐0.197  ‐0.120   S2=38.5  60  10.806  10.822  10.500  ‐0.016  0.306  

Notes: See Table 2.  Table 11: Price estimates for the LS algorithm using various numbers of Shifted Legendre Polynomials in the cross-sectional regression, m, and given numbers of paths in simulations N=10000.   S  n  LS  Cal  His.  Bias  S.E. m=2  S1=47.12  30  4.079  4.227  4.150  ‐0.148  ‐0.071   S2=38.5  60  10.760  10.822  10.500  ‐0.062  0.260 m=3  S1=47.12  30  4.100  4.227  4.150  ‐0.127  ‐0.050   S2=38.5  60  10.632  10.822  10.500  ‐0.190  0.132 m=4  S1=47.12  30  4.022  4.227  4.150  ‐0.205  ‐0.128   S2=38.5  60  10.866  10.822  10.500  0.044  0.366 m=5  S1=47.12  30  4.180  4.227  4.150  ‐0.047  0.030   S2=38.5  60  10.765  10.822  10.500  ‐0.057  0.265  

Notes: See Table 2 with N=10000.  

 

Table 12: Price estimates for the LS algorithm using various numbers of Shifted Legendre Polynomials in the cross-sectional regression, m, and given numbers of paths in simulations N=20000.   S  n  LS  Cal  His.  Bias  S.E. m=2  S1=47.12  30  4.146  4.227  4.150  ‐0.081  ‐0.004   S2=38.5  60  10.706  10.822  10.500  ‐0.116  0.206 m=3  S1=47.12  30  4.165  4.227  4.150  ‐0.062  0.015   S2=38.5  60  10.799  10.822  10.500  ‐0.023  0.299 m=4  S1=47.12  30  4.079  4.227  4.150  ‐0.148  ‐0.071   S2=38.5  60  10.762  10.822  10.500  ‐0.060  0.262 m=5  S1=47.12  30  4.120  4.227  4.150  ‐0.107  ‐0.030   S2=38.5  60  10.783  10.822  10.500  ‐0.039  0.283  

Notes: See Table 2 with N=20000. 

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Table 13: Price estimates for the LS algorithm using various numbers of Shifted Legendre Polynomials in the cross-sectional regression, m, and given numbers of paths in simulations N=30000.   S  n  LS  Cal  His.  Bias  S.E. m=2  S1=47.12  30  4.118  4.227  4.150  ‐0.109  ‐0.032   S2=38.5  60  10.702  10.822  10.500  ‐0.120  0.202 m=3  S1=47.12  30  4.157  4.227  4.150  ‐0.070  0.007   S2=38.5  60  10.787  10.822  10.500  ‐0.035  0.287 m=4  S1=47.12  30  4.181  4.227  4.150  ‐0.046  0.031   S2=38.5  60  10.708  10.822  10.500  ‐0.114  0.208 m=5  S1=47.12  30  4.160  4.227  4.150  ‐0.067  0.010   S2=38.5  60  10.732  10.822  10.500  ‐0.090  0.232  

Notes: See Table 2 with N=30000.  

 

Table 14: Computational time with Monomials for different combinations of the number of regressors, m, and number of paths, N.   m=2  m=3  m=4  m=5 N=5000  t=5.31  t=5.47  t=5.48  t=5.53 N=10000  t=16.23  t=16.39  t=17.09  t=17.66 N=20000    t=52.22  t=51.97  t=52.51  t=53.67 N=30000  t=107.25  t=108.01 t=108.86 t=108.63 

 

Table 15: Computational time with Laguerre Polynomials for different combinations of the number of regressors, m, and number of paths, N.   m=2  m=3  m=4  m=5 N=5000  t=5.40  t=5.52  t=5.84  t=5.53 N=10000  t=16.17  t=16.09  t=16.25  t=16.39 N=20000    t=52.34  t=52.16  t=52.41  t=52.48 N=30000  t=108.47  t=108.52 t=108.72 t=109.08 Table 16: Computational time with Legendre Polynomials for different combinations of the number of regressors, m, and number of paths, N.   m=2  m=3  m=4  m=5 N=5000  t=5.42  t=5.50  t=5.52  t=5.59 N=10000  t=16.27  t=16.03  t=16.34  t=16.39 N=20000    t=55.42  t=52.11  t=52.36  t=52.61 N=30000  t=108.13  t=108.13 t=108.34 t=108.98 

Notes: Adding regressors will not lead to a significant increase in the computational

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time for different polynomial families but with the same path. The number of simulated paths results in a significant increase in the computational time. This effect will be particularly important for options where a large proportion of the observations are applied in regressions.  

Appendix C: Figures

Figure 1: This figure shows the first three members of the polynomial families used plotted on the interval (0,1); the relevant formulas can be found in Table 1. The solid line corresponds to m=1, the dashed line to m=2, and the dotted line to m=3.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

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Figure 2: This figure reports that the computational time (y-axis) can be measured for different combinations of the number of regressors and the number of simulated paths (x-axis) using ordinary monomials.

Figure 3: This figure reports that the computational time (y-axis) can be measured for different combinations of the number of regressors and the number of simulated paths (x-axis) using Laguerre Polynomials.

Figure 4: This figure reports that the computational time (y-axis) can be measured for different combinations of the number of regressors and the number of simulated paths

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(x-axis) using Legendre Polynomials.

Appendix D: Codes

1 // Codes for numerical simulations with the LS method// 2 close all 3 %%basic inputs 4 %numbers of simulated paths 5 N=5000; 6 %number of exercise points 7 n= 60; 8 %Volatility 9 sigma=0.71; 10 %Initial stock price 11 S0=38.5; 12 %Strike price 13 K=48; 14 %Interest rate 15 r= 0.05; 16 %Time to maturity 17 T=1/6; 18 dt=T/n; 19 %Standard Brownian increments 20 dW= sqrt(dt)*randn(n-1,N); 21 %Discount factor 22 disc= exp(-r*dt); 23 %%simulate asset processes 24 S=S0*exp(cumsum((r-1/2*sigma^2)*dt+sigma*dW)); 25 S=[ones(1,N)*S0; S]'; 26 %Initialize the payoff matrix 27 P= zeros(N,n); 28 P(:,n)= max(0,-S(:,n)+K); 29 for NN=n-1:-1:2 30 y= max(0,-S(:,NN)+K); 31 yex=[]; 32 X=[];

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33 Y=[]; 34 for i=1:N 35 X=[X;S(i,NN)]; 36 if y(i)>0 37 yex=[yex;y(i)]; 38 Y=[Y;(disc.^[1:n-NN])*P(i,NN+1:n)']; 39 else yex=[yex;0]; 40 Y=[Y;0]; 41 end 42 end 43 %Construct a matrix with basis function 1, X, X^2 44 A=[ones(size(yex)) X X.*X]; 45 %Least-Square Regression: 46 b=inv(A'*A)*A'*Y; 47 %continuation value 48 yco=A*b; 49 %stopping rules 50 for i=1:N 51 if y(i)>0 52 if (yex(i)>yco(i)) 53 P(i,:)=0; 54 P(i,NN)=yex(i); 55 end 56 57 end 58 end 59 end 60 price=sum(P*disc.^[0:n-1]')/N Now we give the detailed explanations of the numerical simulations with the LS algorithm used in MATLAB codes above. 1) We give the basic inputs such as initial stock price, time to maturity, numbers of

paths to simulate, numbers of exercise point, etc. Since the stock prices can be specified as GBM from (5.1.3) and (5.1.4) under the risk neutral dynamics, we initialize a N n× matrix for the stock prices S.

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2) We initialize a N n× payoff matrix P and construct a 1N × vector y for the

payoff function of American put option. Moreover, we create three matrices: yex for immediate exercise values, X for stock prices at exercise values and Y for discounted cash flows.

3) Considering all simulated paths ( 0 y > and 0 y ≤ ), we initialize the matrices yex,

X and Y using option payoff.

4) Start iterating for each time step along each simulated paths backwards. 4.1 Create a matrix of regression independent variables using basis function 1, X and X^2 and set X=stock price at the node. 4.2 Perform the least squares regression to calculate parameters for coefficients of basis functions. 4.3 Compute back continuation values using the parameter values obtained. 4.4 Compare continuation values with the immediate exercise values. If the immediate payoff is larger, then exercise the option and set the exercise value in the payoff matrix. Otherwise, set it to discounted value of continuation at next time step. 4.5 Repeat 4.1 until the simulations are completed.

5) The option can be valued by discounting each cash flow back to time zero, and averaging over all paths.

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Reference [1] Emmanuelle Clement, Damien Lamberton and Philip Protter, 2002, ”An Analysis of a Least Squares Regression Method for American Option Pricing”, Finance and Stochastic, 449-471. [2] Bensoussan.A, 1984, ”On the Theory of Option Pricing,” Acta Applicandae Mathematicae, 2, 139-158. [3] Cox, J.C., Ross, S.A. and Rubenstein, M, 1979, ”Option Pricing: A Simplified Approach”, J. Fin. Econ. 7, 229-263. [4] Tilley, J.A. 1993, ” Valuing American Options in a Path Simulation Model,” Transactions of the Society of Actuaries, 45, 83-104. [5] Carriere, J. 1996, ”Valuation of Early-Exercice Price of options Using Simulations and Non-parametric Regression,” Insurance: Mathematics and Economics, 19, 19-30. [6] Longstaff Francis.A and Eduardo S.Schwartz, 2001, ”Valuing American Options by Simulation: A Simple Least-Squares Approach,” Review of Financial Studies 14, 113-148. [7] Ioannis Karatzas, 1988, ”On the Pricing of American Options”, Applied Mathematics and Optimization, 17, 37-60. [8] D.L.Cohn, 1980, ”Measure Theory”, Birkhauser, Chapter 1. [9] Bernt Oksendal, 2003, ”Stochastic Differential Equation: An Introduction with applications”, 6, 43-64 [10] Tomas Björk, 2003, ”Arbitrage Theory in Continuous Time”, Oxford University, 2nd edition, 146-152 [11] David Stirzaker, 2005, ”Stochastic Processes & Models”, St John’s College, Oxford, 107-147. [12] Royden, H.L. 1988, ”Real Analysis”, Prentice Hall, Inc., New Jersey. [13] Lars Stentoft, 2004,”Least Squares Monte-Carlo and GARCH Methods for American Options: Theory and Applications”, University of Aarhus, Denmark, 6-70. [14] Judd, K.L. 1998,”Numerical Methods in Economics”, MIT, Cambridge, Massachusetts.

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[15] Shangli, Jiang, 2003, ”Mathematical Modeling and Methods of Option Pricing”, Beijing High educational press, 202-222. [16] Boyle, P. P. and Tse, Y. K., 1990, “An Algorithm from Computing Values of Options on the Maximum and Minimum of several assets”, Journal of Financial and Quantitative Analysis, 25, 215-227. [17] Heath. D., R. Jarrow, and A. Morton, 1992, “Bond Pricing and the Term Structure of Interest Rate”, Econometrics, 77-106.

Note: This master thesis is contributed by Xuejun Huang and Xuewen Huang, and

the division of work between us follows: Xuejun Huang: Introduction (Part 1). Introductory example explaining the algorithm, detailed description of the algorithm, presentation of the Snell Envelope, convergence and the proofs of some theorems (Part 2, 3, 4, Appendix A). Numerical simulation with relevant Tables and Figures (Part 5, Appendix B and C). Open problem and conclusion (Part 6, 7). Xuewen Huang: Introduction (Part 1). Numerical simulation with Codes and Tables (Part 5, Appendix B, and D). Conclusion and future work (Part 7)