## Chapter 1 Introduction

In Taiwan’…nancial market, in order to adpat to ”Globalization” and satisfy the demand of investors owning the knowledge and the training of modern investment, there are many …nancial innovations which are created and issued by securities companies, especially warrants, which are more popular than other derivatives. In our thesis, we are forcus on the warrants whose strike prices are related to the arithmetic moving average of the underlying stock price. The most prominent examples are moving-average-reset and moving- average-lookback warrants. A moving-average-reset warrant is struck at a series of decreasing contract-speci…ed prices over a monitoring window based on the moving average. With the moving-speci…ed-lookback condition the warrant becomes more complicated, which is struck at the minmum moving average of the underlying stock price over a monitoring window. There issues a great portion of these combounded warrants in Taiwan.

Moving average is often considered as a technical measure for short-term trends in stock prices. Hence, it is straightforward to associate the mov- ing average with the strike price. The advantage is, …rst, it is too violently changing as only considering the stock price as the reset date approaches and, second, to provide an better way to determine the strike price of the options.

Recently, there has been a little research on the pricing of moving-average-
reset and moving-average-lookback options. In this thesis we will forcus on
the moving-average-lookback option (henceforth MAL), as the slightly sim-
pler moving-average-reset option can be handled similarly. Pricing moving-
average-style options is di¤ucult. Firs, let M_{t}^{5}be the 5-day movingg average
at the tth trading day, where t ¸ 4 and St denotes the stock price at day t.

1

2 CHAPTER 1. INTRODUCTION Then

M_{t}^{5}=

P_{t}
i=t¡ 4Si

5

The moving average M_{t+1}^{5} is related to not M_{t}^{5} but also M_{t}^{5}_{¡ 1}; M_{t}^{5}_{¡ 2}; :::;

M_{t}^{5}_{¡ 4}: In pricingg arithmetic average MAL, we should solve the problem of the
nonnormality of the sum of lognormal distribution, and the non-Morkovian
property mentioned above. These two issues combine to increase the di¢ culty
of pricing.

There are three major way to value derivatives. The …rst is to derive closed-form solutions of derivatives by partial-di¤erential-equation (PDE) or martingale method. We can value the option from PDE by …nite d¤erence techniques by transforming the problem from a path-dependent one to a Markovian problem. However, the PDE of arithmetic MAL has never been derived and the strike based on all past moving-average term is very di¢ cult to value by …nite di¤erence approach.The second, we can price American- style derivatives on the tree algorithm, especially the CRR model (Cox, Ross, and Rubinstein (1979)). The last is Least-Squares Monte-Carlo (henceforth LSM) simulation approach we will forcus on, especailly used most e¤ectively and powerful to not only price strongly path-denpendent and multifactor derivatives, but also to solve the problem of early exercise of American-style deratives on Monte-Carlo simulation before.

It is easy to solve the problem of path-dependence about the moving- average term by Monte-Carlo simulation. There exists, nevertheless, a bot- tleneck of pricing American-style derivatives on Monte-Carlo simulation ap- proach all the time. The famous technology is introduced by Boyle, Broadie

& Glasserman (1997). Their approach is more closely related to the tree method, and should produces an upward bias and a downward bias estimates and average both to obtain the unbias value. Besides the method needs to simulate several paths from each point to obtain an unbias estimator of the American option price, which resulting in the curse of dimesionality that the lattice methods also su¤ers from.

A new and simpler simulation based method to price American options has recently been proposed by Longsta¤ & Schwartz (2001). The idea is to estimate the conditional expectation of the payo¤ from continuing to keep the option alive at each possible exercise point from a cross-sectional regression using the simulated paths. Based on LSM methods, we can solve the problem of determining the optimal early exercise strategy of American-style options.

We also consider the developed CRR model (Kao (2002)) as a benchmark

3 on pricing arithmetic MALs. It will be found that the LSM approach is very close to the price calculated by the CRR model. With the LSM algorithm, a detail analysis of American-style AMALs will be presented to understand the properties of this derivatives. Besides we will …nd even if any changeable and complicated derivatives, such as MALs discussed in this thesis, it is not hard to price with the LSM method.

The remainder of this thesis is organized as follows. Chapter 2 reviews ba- sic concepts and pricing technologies on simulation and tree model. Chapter 3 describes the underlying theoretical framework. Chapter 4 covers the pric- ing of arithmetic MALs and numerical analysis of di¤erent arithmetic MALs.

Chapter 5 describes how to choose the number of regressors and alternative family of basis funtions. Chapter 6 summarizes results and concludes.

4 CHAPTER 1. INTRODUCTION

## Chapter 2

## Preliminaries on Options Pricing

In this chapter, we review fundamental concepts and pricing techniques used in later chapters. The …rst, the Monte-Carlo simulation technique must be introduced. The second, we will review the tree model.

### 2.1 Simulation and option pricing

There exists a major problem with numerical methods is that they are not easily extended to more than one of stochastic factors. In the Tree Model, the number of nodes grows exponentially as the number of stochastic factors increases. In the Finite Di¤erence, it can only calculate less than three stochastic factors generally. So it is possible method to solve the problem of multidimensional by using simulation. Furthermore, time should be divided into a number of segments in the simulation method. We get the next period price by a random walk, and the number of nodes remains constant through time. Besides there needs a large number M of simulated paths by Law of Large Number for convergence. At last the estimator is gotten by the average of the prices over the paths.

If we want to get the stochastic variables, such as stock prices, interest rate, volatility, or dependence on multiple stock prices, which can be included in the simulation. How many paths to use and how many steps to partition time to expiration into should be decided on. In general the more simulated paths , the more precise the estimator of stochastic factor is. In the same way,

5

6 CHAPTER 2. PRELIMINARIES ON OPTIONS PRICING increasing the number of steps would con…rm that the estimator converges to the exact true price.

We will show how to price options using simulation. The American option is assumed to be exercisable at a …nite nubmer of equally spaced points in time. We can specify the risk discrete exercise feature by using of Geometric Brownian Motion (GBM). Finally, we describe how to price options using the simulation.

### 2.1.1 Simulating from a Geometric Brownian Motion

To simulate a GBM by the stochastic Di¤erential Equation (SDE) dS(t) = rS(t)dt + ¾S(t)dW (t) (2:1)

where W is a standard Wiener process and r and ¾ are assumed constant, we use the well known solution to (1). Given a starting level of S(0) this is

S(t) = S(0) expf(r ¡ 1

2¾^{2})t + ¾ W (t)g (2:2)

From the propeties of the Wiener process simulated value of S(t) at a a single point in time can be obtained from the formula

S(t) = S(0) expf(r ¡ 1

2¾^{2})t + ¾p

tZg (2:3)

where Z~N(0; 1). A sequence of values at discrete date 0 · t1 · t2 ·
::· t^{N} = T is obtained by setting

S(ti+1) = S(ti) expf(r ¡ 1

2¾^{2})(ti+1¡ t^{i}) + ¾^{q}(ti+1¡ t^{i})Z(ti+1)g (2:4)
where Z(ti+1)~IIN (0; 1).

### 2.1.2 Pricing European options using simulation

The price of European put option is the expectation under the risk neutral measure of the present value of its payo¤ given as

p´ p(S(0); T) = E[e^{¡ rT} max(X ¡ S(T); 0)]

2.1. SIMULATION AND OPTION PRICING 7 And we can get an estimate of the price by the formula

PN= 1 N

XN

j=1

e^{¡ rT}max(X ¡ S^{j}(T ); 0)

where N is the number of simulated paths and Sj(T ) is the value of the underlying stock at expiration of the option for path numbe j.

### 2.1.3 Pricing American options using simulation

The key of pricing American options with simulation is determining the op- timal exercise strategy. We write the price of American put options as

P ´ P (S(0); T) = max

0<¿· T E[e^{¡ rT}max(X¡ S(¿); 0)]

where the maximization is over stopping times ¿ · T adapted to …ltration generated by the relevant stock price process S(t). The problem is that at any possible exercise time, the holder of an American option should compare the payo¤ from immediate exercise to the expected payo¤ from continuation.

The optimal decision is to exercise if the exercise value is positive and larger than the expected payo¤ from continuation.Using next period values of the underlying asset to determine the expected value along each path of contin- uing to keep the option alive would lead to biased price estimates. The main reason of making the estimator biased is to consider the expected payo¤ from continuation as perfect forsight (see Broadie & Glasserman(1997)). Hence, we can not simply estimate the price P by

PN= 1 N

XN

j=1

max¿ [e^{¡ r¿}max(X¡ Sj(¿); 0)]

Note that we want to prevent this bias, but the best way is to simulate several paths from each possible exercise point thus resulting in multidimen- sionality. However, Longsta¤ & Schwartz (2001) provide a very powful idea to estimate the conditional expectation of the payo¤ from continuing to keep the option alive, using the cross-sectional information in the simulation.

The main motivation of the LSM approach can be given in terms of Hilbert Spaces, the space of square-integrable functions with the norm

hf(x); g(x)i =

Z

f (x)g(x)dx

8 CHAPTER 2. PRELIMINARIES ON OPTIONS PRICING The theory of Hilber spaces tells us that any function G(xn) belonging to this space can be represented as a countable linear combination of bases for this vector space. We can write

G(x_{n}) = ^{X}^{1}

k=0

a_{k}Á_{k}(x_{n}) (2:5)

where fÁ^{k}(x)g^{1}k=1 form a basis (See Royden(1988)). In pratice we use a

…nite linear combination to approximate G(xn) which we denote GK(xn),
where K is the number of basis functions used. The simplest approximation
way is using least squares regression. when the coe¢ cients fa^{k}g^{K}_{k=0} in (15)
are estimated, we have to simulate N paths s.t. N ¸ K + 1 ,i.e. there will
exist data points (yj; xj), j = 1; :::; N; by solving the minimization problem

min

fakg^{K}k=0

XN

j=1

(a0Á0(xj) + a1Á1(xj) + ::: + aKÁK(xj)¡ yj)^{2}
With the parameter estimates fa^{k}g^{K}k=0 we estimate GK(x) with

G^K (x) =

XK

k=0

a^k Ák(x) (2:6)

In general, G^{^}K (x)! G^{K}(x) as N ! 1. Letting G(x) = E[yjx], where
y is the payo¤ from continuing to keep the option alive, x represents the
current state, and fÁ^{k}(x)g^{K}_{k=0} is a set of independent variables, the condi-
tional expectation function G(x) can be arbitrarily approximated as N and
K both tend to in…nitely. And the approximated G(x) is used to determine
the optimal exercise strategy.

### 2.2 Tree Models and Auxiliary State Vari- able

In this Section, we review two usefule pricing techniques. The …rst, the CRR model, is mainly used to solve American-style options. The second, auxil- iary state varialbes approach, is a general method to price path-dependent derivatives on the tree.

2.2. TREE MODELS AND AUXILIARY STATE VARIABLE 9

### 2.2.1 The CRR model

The CRR model is one simplest but very powerful of tree models introduced in Cox, Ross, and Rubinstein (1979).

Time to expiration is also divided into a number of segments, denote each
unit of time as 4t = ^{T}_{n}, where T is time to expiration and n is the number
of partitions. From Eq. (2.2), we can obtain the expected value of the stock
price change after a small time 4t is S^{0}e^{r}^{4t} and the variance of the stock
price change after 4t time is ¾^{2}4t. Now consider the discrete-time version
of Eq. (2.1) and change the normal di¤usion to a discrete random variable,
4W . It follows that

4S^{t}= rSt¡ 4t4t + ¾ S^{t}¡ 4t4W:

Assume 4W follows the Bernoulli distribution such that St+4t=

( Stu; with probability p, Std; with probability 1¡ p,

where u and d are the proportional change of Stin the up and the down state. We let 4W satisfy the mean and variance function mentioned above.

This yields the following conditions,

e^{r}^{4t} ¼ pu + (1 ¡ p)d

¾^{2}4t ¼ pu^{2}+ (1¡ p)d^{2}¡ [pu + (1 ¡ p)d]^{2}
ud = 1

We obtain a possible solution :

p = e^{r}^{4t}¡ d
u¡ d
u = e^{¾}^{4t}
d = e^{¡ ¾ 4t}

The stock price on node N (i; j) reachable from the root with j up and i¡ j down moves is

S(i; j) = S0u^{j}d^{i}^{¡ j}

10 CHAPTER 2. PRELIMINARIES ON OPTIONS PRICING and the value of derivatives C on node N(i; j) can be obtained by the backward induction formula :

C(i; j) = e^{¡ r4t}[pC (i + 1; j + 1) + (1¡ p)C(i + 1; j)] (2:5)

for i = 0; 1; :::; n and j = 0; 1; :::; i; where e^{¡ r4t}[pC (i + 1; j + 1) + (1¡
p)C (i + 1; j)] is called as the expected payo¤ from continuation on the CRR
model. When pricing American-style options, we change Eq. (2.5) into
C (i; j) = max(e^{¡ r4t}[pC(i +1; j +1)+(1¡ p)C(i+1; j)]; S(i; j)¡ X) (2:6)

where X is the strike price of call option, and S(i; j) ¡ X is the value of immediate exercise at node N(i; j). The option value emerges in C(0; 0):

### 2.2.2 Auxiliary State Variable

This section draws on Dai (1999), which provides a general method for pric- ing path-dependent derivatives on tree Model. Auxiliary state varialbes are memory space to record the past information needed in dealing with the path dependency. Let C(i; j; k) denote the option value on node N(i; j). In addition to i and j, which provide the information of time and the current stock price, we need an additional k to record the information arising from path dependency.

To apply backward induction, we have to allocate enough auxiliary state variables for all the possible situations at each node.The size of auxiliary state variables depends on the number of possible situations determined by path dependency. This technique is not suitable for cases which need very large sizes of auxiliary state variables such as Asian options. However, the auxil- iary state varialbes approach is useful in pricing longer time path-dependent derivatives, such as ”weekly”. When we allow approximation, the alleged shortcoming of this approach no longer exists.

## Chapter 3

## The LSM Valuation Algorithm

We will describe the general LSM algorithm in theory later. The valuation algorithm of LSM can be applied on the general derivative pricing paradigms, such as Black and Scholes (1973), Merton(1973), Cox, Ingersoll, and Ross (1985), Heath, and so on.We also present several convergence results for the algorithm.

### 3.1 The LSM valuation framework

Assuming an underlying complete probability space (¤ ; F,P) and …nite time
[0; T ], where the state space ¤ is the set of all possible realizations of the
stochastic economy between time 0 and T and has typical element w repre-
senting a sample path, F is the sigma …eld of distinguishable events at tine
T , and P is a probability measure de…ned on the elements of F . We de…ne
F =fF^{t}; t2 [0; T]g to be the augmented …ltration generated by the relevant
price processes for the securities in the economy, and assume that FT = F .
Consistent with the no-arbitrage paradigm, we assume the existence of an
equivalent martingale measure Q for this economy.

We restrict our attention to payo¤s that are elements of the space of
square-integrable (or …nite-variance) functions L^{2}(¤ ; F; Q). The value of an
American option equals the maximum is taken over all stopping times with
respect to the …ltration F . We present the path of cash ‡ows generated by
the option , denoted as C(w; s; t; T) , conditional on the option not being
exercised at or before time t and on the optionholder following the optimal
stopping strategy for all s, t < s · T.

11

12 CHAPTER 3. THE LSM VALUATION ALGORITHM The objective of the LSM algorithm is to provide a pathwise approxima- tion to the optimal stopping rule that maximizes the value of the American option. In practice, many American options are continuously exercisable;

the LSM algorithm can be used to approximate the value of these options by taking the exercising times to be su¢ ciently large.

At the …nal expiration date (T ) of the option, the option is exercised if it is in the money, or expire if out of money. At exercise time tM <

T , however, the holder of an American option must determine whether to exercise immediately or to keep alive.

At time ti, the payo¤ from immediate exercise is known to the investor, but the cash ‡ow from continuation are unknown. No-arbitrage valuation theory, however, implies the expected payo¤ from continuation assuming that it cannot be exercised until after tM, is given by taking the expectation of the remaining discounted cash ‡ows C(w; s; t; T) with respect to the risk- neutral pricing measure Q. Speci…cally, at time tm, the value of continuation G(w; tm) can be represented as

G(w; tm) = EQ[exp(¡ ^{Z} ^{t}^{i}

tm

r(w; s)ds)C(w; ti; tm; T )jFtm]

where r(w; t) is the riskfree rate, and the expectation is conditional on the information set Ftm at time tm. With this representation, the problem of optimal exercise reduces to comparing the immediate exercise value with this conditional expectation, and then exercise as soon as the immediate exercise value is positive and greater than and equal to the conditional expectation.

### 3.2 The LSM algorithm

The LSM approach uses least squares to approximate the conditional ex-
pectation function at tM¡ 1; tM¡ 2; :::; t1. We work backwards to generate the
cash ‡ows C(w; s; t; T ) recursively. At a special time tM¡ 1 we can repre-
sent the unknown G(w; tM¡ 1) as a linear combination of a countable set of
F_{t}_{m}_{¡ 1}-measurable basis functions.

When the conditional expectation is an element of the L^{2}space of square-
integerable functions. Since L^{2}is a Hilbert space, it has a countable ortho-
normal basis and the conditional expectation can be represented as a linear
function of the elements of the basis.

3.2. THE LSM ALGORITHM 13
As an example, assume that x(ti) is the value of the asset underlying the
option and that X follows a Markov process^{1}. We choose the set of laguerre
polynomials as the basis functions (as Longsta¤ and Schwartz (2001)).

L0(X) = w(X)

L1(X) = w(X)(1¡ X)

L_{2}(X) = w(X))(1¡ 2X + X^{2}=2)
L_{n}(X)) = w(X)e^{X}

n!

d^{n}

dX^{n}(X^{n}e^{¡ X})

where w(X) = exp(¡ ^{X}_{2}): With this speci…cation, G(w; t_{M}_{¡ 1}) can be rep-
resented as

G(w; tM¡ 1) =

X1

k=0

akLj(X) where the ak coe¢ cients are constants.

To implement the LSM approach, we approximate G(w; tM¡ 1) using M <

1 basis functions mentioned above, and denote this approximation GM(w; t_{M}_{¡ 1}).

GM(w; tM¡ 1) is estimated by regressing the discounted values of C(w; s; tM¡ 1; T ) on the basis functions across paths where the option is in the money. We use only in-the-money paths in the estimation since the exercise decision is only related with the in-the-money option. And we need a …nite number of basis to obtain an accurate approximation to the conditional expectation function.

Since the basis functions are independently and identically distributed across
paths, the existence of moments of Theorem 3.5 of White (1984) shows that
the …tted value of this regression G^{^}K (w; tM¡ 1) converges in mean square
and in probability to G(w; tM¡ 1) as the number N of paths goes to in…nity.

Furthermore, Theorem 1.2.1 of Amemiya (1985) implies thatG^{^}K (w; tM¡ 1) is
the best linear unbiased estimator of G_{K}(w; t_{M}_{¡ 1}) based on a mean-squared
metric.

Once the conditional expectation function at time tM¡ 1 is estimated, we can determine whether early exercise at time tM¡ 1is optimal for in-the-money

1For Markovian problems, only current values of the state varialbes are necessary. For non-Markovain problems, both current ans past realizations of the state varialbes can be included in the basis functions and the regressions.

14 CHAPTER 3. THE LSM VALUATION ALGORITHM
path w by comparing the immediate exercise value with G (w; t^{^} M¡ 1), and
repeating for each in-the-money path. Once exercise decision is indenti…ed,
the option payo¤ C(w; s; tM¡ 1; T ) can be approximated based on cash ‡ows
along path w after the determination of optimal exercise strategy at time
tM¡ 1: The recursion process is rolling back and repeating until the exercise
decisions at each exercise time along each path have been determined. The
American option is then valued by strating at time 0, moving forward along
each path until the …rst stopping time occurs, discounting the payo¤ from
exercise back to time 0, and then averaging the payo¤ over all paths w.

When there are two state variables X and Y , the set of basis functions should include terms in X and in Y , as well as cross-products term, X Y . Contrary to other methods with higher-dimensional problems, the number of basis functions does not grows exponentially but grow a slower rate with convergence result.

### 3.3 The LSM algorithm to pricing Amercian option in mathematics

### 3.3.1 The presentation of pricing American call op- tions

The following is the detail of implementation of LSM algorithm.

1. Simulation of stock paths:

Simulate a large number of paths (N) of asset prices using an exact formula like (4), and choose the number of steps (M ) su¢ ciently large to approximate continuous exercise. Following (a) let Sj(ti) denote the asset price along path j at time ticorresponding to step i, where j = 1; :::; N and i = 1; :::; M . 2. Calculation of the payo¤ matrix:

Let P (for payo¤ ) be a N £ M matrix, with typical element f^{j;i}. At time
tM = T (the expiration date of the option) the payo¤ along each path is the
maximum between zero and the value of exercising the option. Hence, we
can de…ne the elements of the last column as

3.3. THE LSM ALGORITHM TO PRICING AMERCIAN OPTION IN MATHEMATICS15

fM;j = max(Sj(T )¡ X; 0); 1 · j · N

OLS is used to estimate the conditional expectation of the payo¤ that the option is kept alive (see (14)) by working backwards at each time ti; 0 < i <

M. First, the rule for choosing the paths is where the option is in-the-money
denoted asN. For a put option we de…ne^{~} N=^{~} fj : S^{j}(ti)¡ X(t) > 0; 1 · j ·
Ng. For any j 2N , the payo¤ from continuation is the payo¤ along the path^{~}
until expiration of the option discounted back using the risk-free interest rate

yj(ti) =

XM

k=i+1

e^{¡ r(t}^{k}^{¡ t}^{i}^{)}fk;j

This is the dependent variables. We need to transform these dependent
variables to independent ones, such as Xj(ti) = h(xj(ti)); where h(xj(tj)) is
a transformation of the state variables. If the underlying asset is only one
stochastic factor, it is su¢ ce to explain variations in the dependent variable
as xj(ti) = Sj(tj): Following (c) we approximate the conditional expectation
G(x_{j}(t_{i}))´ E[yj(t_{i})jxj(t_{i})] as

G (x) = X^ j(ti)¯ (ti)

where ¯ (ti) is a vector of coe¢ cients. This is the linear regression model yi(ti) = Xj(ti)¯ (ti) + ui(ti), the parameters can be estimated by

¯ (t^ i) = (X(ti)^{0}X(ti))^{¡ 1}X(ti)^{0}y(ti)

The …tted valuesy (t^{^} _{i}) = X(t_{i})¯ (t^{^} _{i}), which corresponds to the estimated
conditional expectation of the payo¤ when the option is kept alive, are used
to determine if it is optimal to exercise the option at time ti.If the …tted
value is larger than the value of immediate exercise X ¡ S^{j}(ti), fi;j are set
equal to the value of immediate exercise X ¡ Sj(ti) and in all other values
fn;ji < n· N, are set to equal to zero. That is,

fi;j =

( K ¡ Sj(ti) and fj;n= 0; i < n· N , X ¡ Sj(ti) >y^{^}_{j} (ti)

0 , otherwise ; j 2N .^{~}

16 CHAPTER 3. THE LSM VALUATION ALGORITHM 3. Calculating the value of the option:

When ti = 0 the value of the option is calculated from the payo¤ matrix by discounting the payo¤s to period zero using the risk-free rate and averaging across the simulated paths. Since there is at most one nonzero element along each path in P this can be written as

PN= _{N}^{1} ^{P}^{N}_{j =1}^{P}^{M}_{i=0}[e^{¡ rt}^{i}max(fi;j;0)]

The neutral price process follows a GBM and the option has discrete exercise features.

### 3.4 Convergence results

How well the LSM algorithm performs is using a realistic number of paths and basis functions, it is useful to examine the theoretical convergence of the algorithm to the true value G(X) of the American option.

First, we present the bias of the LSM algorithm when the American option is continuously exercisable.

Proposition 1 For any …nite K, M, and vector µ 2 R^{K}^{£ (M¡ 1)}representing
the coe¢ cients for the K basis functions at each of M ¡ 1 early exercise dates,
let N denote the number of simulated paths, G(X) denote the true value of
the American-style option and LSM (w; M; K) denote the discounted cash

‡ ow resulting from following the LSM rule of exercising when the immediate
exercise value is positive and greater than or equal to G^{^}K (wi; tm) as de…ned
by µ. Then the following inequality holds almost surely,

G(X )¸ lim

N!1

1 N

XN

i=1

LSM(wi; K; M)

The LSM algorithm is considered as a stopping rule for an American-style option. The value of an American-style option is based on the stopping rule that maximizes the value of the option.

The result is particularly useful since it provides an objective criterion for convergence, that is any result simulated by the LSM algorithm has a upper bound. As a criterion example, we can increase K until the value implied by the LSM algorithm no longer increases. It is very useful and important property in the LSM algorithm.

3.4. CONVERGENCE RESULTS 17 The following convergence result for the LSM algorithm is di¢ cult since we need to consider limits as the number of exercisable dates M; the number of basis functions K, the number of paths N go to in…nitely. Consider the following proposition.

Proposition 2 Assume that the value of an American option depends on a single state variable X with support on (0; 1) which follows a Markov processes. Assume further that the option can only be exercised at times t1and t2, and that the conditional expectation function G(w;t1) is absolutely continuous and

Z _{1}

0 e^{¡ X}F^{2}(w; t1)dX <1

Z _{1}

0 e^{¡ X}F_{K}^{2}(w; t1)dX <1
Then for any ² > 0, there exists an K < 1 such that

Nlim!1Pr[jG(X)¡ 1 N

XN

i=1

LSM(wi; K; M)j > ²] = 0

Intuitively this result means that when K is large enough and N ! 1, the LSM algorithm results in a value for American option within ² of the true value, where ² is selected arbitrarily.An important implication of this result is that the number of basis functions result in a desired of accuracy need not go to in…nity.

18 CHAPTER 3. THE LSM VALUATION ALGORITHM

## Chapter 4 Pricing

## Moving-Average-Lookback options

There exists more than two variables, such as stock price , moving-average term, and strike price, etc... in valuing Moving-Average-Lookback options (MVALs), but it is easy to value with the LSM approach.In order to con…rm that the value is almost approximate to the true value, the value of the CRR model is taken as a benchmark compared with the LSM approach. (see Kao (2002)). In Taiwan, the issued warrants are almost Arithmetic Moving- Average options, so we will forcus on and price Arithmatic Moving-Average- Lookback options (AMVALs). To the end, the empirical results about the di¤erent contracts of American-style AMALs will be presented for you.

### 4.1 De…ning the AMVALs

Let 0 = t0< t1< t2< ::: < tns · T; where ns is the number of trading days
before reset dates Ts, and ti be the time points when the moving average
is calculated. The ti are expected to correspond to trading dates as closing
prices. Assuming that the time interval between monitoring times are equal
and 4t = Ts=n_{s}; i.e., t_{i} = i4t. and n = T=4t. De…ne Si ´ St_{i}:, the stock
price at time i4t: The Arithmetic moving average at time t^{i} equals

ma(i) ´

Pi

j =i¡ a+1Si

a ; a¡ 1 · i · n^{s}:
19

20CHAPTER 4. PRICING MOVING-AVERAGE-LOOKBACK OPTIONS
The minimum a-day Arithmetic moving average as of the reset date tn_{s} =
Ts is de…ned as

ma(k) ´ min

a¡ 1· t· k
P_{t}

i=t¡ a+1Si

a ; a¡ 1 · t · k; k = t_{k}
4t

Note that time to expiration is divided into a number of periods, i.e., it is evaluated at discrete times. The payo¤ function of the MAL at expiration date ti is

f_{i} =

( max(S_{i}¡ X; 0);n^{s}< i· n

max(S_{i}¡ Xi; 0); a¡ 1 < i · ns (4:1)
X_{i} =

( max(min(m_{a}; UB); LB); n_{s}< i· n

max(min(ma(i); U B); LB); a¡ 1 < i · n^{s} (4:2)

where Xs is the strike price of the option determined at reset date Ts: U B is the upper bound of the strike price is set to S0; the initial stock price. The change of U B may happen at times t between ta¡ 1 and Ts. LB, the lower bound of the strike price, is determined by the contract and …xed. Eq. (4.2) means that the strike price of the option, X, is struck at the minimum a-day moving average but range between LB and UB.

### 4.2 Pricing American-Style AMALs

We will described the details how to prcing the American-Style AMALs (AA- MALs) by using the LSM approach. There are two scenarios de…ned for distinguishing the option is early exercised after reset date and before. We denote the former as scenario 1 and the latter as scenario 2. Note that we assume the strike price can be reset every day before Ts on scenario 2. The improved CRR model for AAMALs will be introduced simply.

### 4.2.1 The LSM methods

Now, we describe how to price AAMALs with LSM. As in section 3.3, the

…rst step is to generate the stock price matix from t1to T: Simulate a large number of paths (M ) of stock prices using the formula like (2.4). Ts is denoted as the reset date. The time must be classi…ed into two parts, i.e., before the reset date and after the reset date, we set the former ns steps and

4.2. PRICING AMERICAN-STYLE AMALS 21
the latter n^{0} steps. Note that n is equal to nsplus n^{0}.

And we save the stock price matix S(i; j), where i refers to stop at time ti

and j represent the j-th paths. The strike vector M (i; j) is determined by the rule of max(min(ma(i; j); UB); LB) before Ts.

After reset date, the strike vector is determined at the reset date, denoted as Mns: Giving the expiration conditions, Sn;j ¡ Mns;j; we use a constant, the …rst two Laguerre polynomials evaluated at the stock price, the …rst two Laguerre polynomials evaluated at the strike price, and the cross products of these Laguerre polynomials up to third-order terms. Thus we use a total of eight basis functions in the regressions. Thus, least squares regression is done on the following model after reset date Ts:

y_{i;j} = ¯_{0}+ ¯_{1}LS + ¯_{2}LM_{n}_{s} + ¯_{3}LSM_{n}_{s}_{;j}
+¯_{4}LS(1¡ Si;j) + ¯_{5}LM_{n}^{0}(1¡ Mns;j)
+¯6LSMns(Si;j¡ 1

2S_{i;j}^{2} Mns;j)
+¯7LSMns(Mns;j¡ 1

2M_{n}^{2}_{s}_{;j}Si;j) (4:2)
LS = exp(¡ Si;j

2 )
LM_{n}_{s} = exp(¡ Mns;j

2 ) LSMns = exp(¡ Si;;jMns;j

2 )

where yi;j is that the stock price vector of the j-th path after the time tiis
never early exercised at or before the time t_{i} based on the optimal exercise
strategy of the LSM rule: To avoid any form of numerical over‡ow, and to get
as precise results as possible, both payo¤ yi;j and the stock price Si;j and the
strike Mns;j are normalized by dividing the initial stock price S0. Regressing
with Eq. (4.2), we obtain this conditional expected payo¤ function at time
i4t.

^

Ei (C_{i;j}jSi;j; M_{n}_{s}_{;j}) = y^{^}_{i;j}=¯^{^}_{0}+¯^{^}_{1}LS+ ¯^{^}_{2}LM_{n}_{s}+¯^{^}_{3} LSM_{n}0

+¯^{^}_{4}LS(1¡ Si;j)+ ¯^{^}_{5}LM_{n}_{s}(1¡ Mns;j)
+¯^{^}6LSM_{n}^{0}(Si;j¡ 1

2Si;j^{2} Mn^{0};j)

22CHAPTER 4. PRICING MOVING-AVERAGE-LOOKBACK OPTIONS

+¯^{^}_{7} LSMns(Mns;j¡ 1

2M_{n}^{2}_{s}_{;j}Si;j) (4:3)
where ¯^{^}_{i}denotes the OLS estimator of ¯i:Compare the exercise value,
Si;j ¡ Mns;j , and the expected value of continuation, E(Ci;jjSi;j; Mns;j); to
determine the option value with respect to each path at time ti ¸ T^{s} as
follows :

fi;j =

( Si;j ¡ M^{n}s;j; if Si;j¡ M^{n}s;j > Ei(Ci;jjS^{i;j}; Mns;j)

0 , otherwise (4:4)

After repeating the procedure in a backward fashion for i = n ¡ 1 to n^{s};
we can get the value of scenario 1 by discounting the value in C(ns; j) for all
j , averaging over all paths, and then discounting the value at time 0 with
e^{¡ rT}^{s}:

The value of scenario 2 is stated two steps di¤erent from scenario 1. The

…rt step is to replace Mi;j , 0 < i < ns into Mns;j in Equation (4.2), (4.3) and (4.4). The second step is repeating the …rst step until the time 0, discounting the value of all cash ‡ows to time 0 and averaging all discounted payo¤ over all paths, which is the value of scenario 2.

### 4.2.2 The CRR Model

We proceed to price the American-style AMAL on the CRR model in this section. Recall that ns is the number of trading days before the reset date.

Let L denote the number of periods between two adjacent monitoring time points (which will coincide with daily closing times). By making 4t a day, we make L the number of trading points per day. The number of trading points before the reset date, N, is equal to nsL. We will build the binomial tree up to the reset date.

In order to speed up the algorithm and becasue moving averages involes only daily closing prices, we simplify the N-period tree based on ideas from Ritchken and Trevor (1999). Although there are N periods before the reset date, we only care about nodes on monitoring days, i.e., at times 0, 4t, 24t, ..., n4t. We therefore merge every L levels of the binomial tree into one, creating an (L + 1)-ary tree with n periods in the process. There are more details introduced in Kao’s Master thesis (2002).

4.3. NUMERICAL RESULTS 23

### 4.3 Numerical Results

In this section, we do some empirical works and studies on the AAMALs with the LSM method. There are …ve cases for di¤erent contracts of AAMALs analysis, and we will take the value of scenario 1 on the CRR model on Kao (2002) as a benchmark.

### 4.3.1 Case 1 : Stock Price v.s. Volatility

We use the LSM method to price 5-day the scenario 1 and scenario 2 AA- MALs and European-Style AMALs, and use the CRR method to price 5-day the scenario 1 AAMALs. Assume UB = 45, r = 3%; q = 0; T = 1; and Ts = 1=12. We will vary St and ¾ in the experiment and …x L = 3 on the CRR model. The results are tabulated in Table 4.1.

We obtain the following observation. First, the prices of senario 1 cal- culated by the LSM method is not di¤ erenet from those by the CRR model within 0.08. Therefore, we can have the con…dence in the LSM algorithm.

Second, the option value increases with ¾. Third, if the stock price is less than the UB at time 0; the special appearance is that the value of AAMALs calculated by the LSM method is undervalued to one by the CRR model;

on the contraty, it is overestimated, which shows that there exists ”slight”

negative and positive bias in the LSM algorithm with the benchmark of the CRR model. If the CRR model is very close to the true value, the value calculated by LSM must be undervalued to one by the CRR model based on proposition 1. We think that maybe the more simulated paths in more complexed contract are needed to see the consistent result as proposition 1.

Next we check the relation between scenario 1 and scenario 2. Because
the reset date Ts is one month, it is too short to see their di¤erence. The
di¤erence between scenario 2 and scenario 1 only moves the highest up to
0.006. It means that when Ts is less than two months and the dividend
yield is very low, the possibility of early exercise before T_{s}for the AAMAL is
low. It is so interesting that the di¤erence between scenario 1 and European-
style is very close to each other, which means an American-style AMALs
call options will also not be exercised early with no dividend payment or low
dividend rate.

24CHAPTER 4. PRICING MOVING-AVERAGE-LOOKBACK OPTIONS

### 4.3.2 Case 2 : Stock Price v.s. LB

Assume UB = 50, r = 2%; q = 4%; ¾ = 50%; T = 1; and Ts= 1=12. Suppose
there are 22 trading days in a month, so ns = 22, and we set n^{0} = 50: We
will vary St and ¾ in the experiment and …x L = 3 on the CRR model. We
will vary St and LB in the experiment. The results are tabulated in Table
4.2.

We make the following observations. Fisrt, not surprisingly, the option value decreases with LB, and increases with ¾ . Second, compared with the tree algorithm, there also exists biases in the LSM method, but all of their di¤erences are less than the highest 0.16, although a large proporsition of the di¤erences are positive. Next, the di¤ erences between scenario 1 and scenario 2 are also close to 0, due to the reset date is too short, and the option-holder would not exercise early before Ts. Besides, the European MALs value are similar to the value of scenario 1, too.

S0 ¾ Scenario1

CRR LSM Scenario2 European 40 0.3 5.0496 4.9931 4.9944 4.9511

(0.0002) (0.0002) (0.0005) 0.4 6.5839 6.5626 6.5651 6.5038

(0.0022) (0.0022) (0.0017) 0.5 8.1042 8.0247 8.0262 7.9423

(0.0002) (0.0002) (0.0004) 45 0.3 6.7462 6.7828 6.7844 6.72409 (0.0004) (0.0004) (0.0008) 0.4 8.5769 8.6568 8.6612 8.6044

(0.0030) (0.0030) (0.0028) 0.5 10.3431 10.4185 10.4219 10.3414 (0.0059) (0.0062) (0.0044) 50 0.3 9.3899 9.4242 9.4292 9.3691

(0.0048) (0.0045) (0.0013) 0.4 11.2034 11.2612 11.2642 11.1978 (0.0009) (0.0010) (0.0007) 0.5 13.0503 13.1141 13.1182 13.0427 (0.0044) (0.0043) (0.0027)

Table 4.1 : The parameters are UB = 45, LB = 40.5, r = 3%, ¾ = 0, T

= 1, T = 1 (n^{0}=50), Ts = 1/12 (ns = 22), a = 5 and L = 3 for the CRR

4.3. NUMERICAL RESULTS 25 model. The numbers in parentheses are standard error of the price.In each simulation a total of 50,000 antithetic paths.

S_{0} LB Scenario1

CRR LSM Scenario2 European 45 42.5 8.7468 8.6955 8.6975 8.4149

(0.0005) (0.0006) (0.0017) 55 12.7241 12.8498 12.9212 12.6029 (0.0050) (0.0039) (0.0047) 65 19.4002 19.5083 19.5400 18.9262 (0.0065) (0.0070) (0.0068) 45 45 8.1833 8.1394 8.1465 7.9350

(0.0013) (0.0014) (0.0019) 55 12.6691 12.8097 12.8606 12.5923 (0.0035) (0.0034) (0.0041) 65 19.3984 19.5575 19.6455 19.0297 (0.0071) (0.0076) (0.0089)

Table 4.2 : he parameters are UB = 50, r = 2%, q = 4%, ¾ = 50%, a =
5, T = 1 (n^{0}=50), Ts = 1/12 (ns = 22), and L = 3 for the CRR model. The
numbers in parentheses are standard error of the price.In each simulation a
total of 50,000 antithetic paths.

### 4.3.3 Case 3 : Dividend Rate v.s. Reset Date

In order to examine the option values of the AGMAL between scenario 1 and scenario 2 by the LSM method and the CRR method, the most important factors q and Ts are varied. Assume UB = 50, r = 2%; ¾ = 30%; LB = 45;

T = 1; and a = 3. Suppose there are 22 trading days in a month, so ns = 22
for Ts = 1=12, ns = 44 for Ts = 2=12, and ns = 66 for Ts = 3=12 and we
set n^{0} = 50 for all, and …x L = 3 on the CRR model. The pricing results
by Monte Carlo simulation are based on 50,000 paths : 25,000 plus 25,000
antithetic, and the scenario 1 , scenario 2 and European-style by LSM are
based on the same sample paths.

Table 4.3 shows that the prices are senstive to Ts and q, respectivelly.

First, the option value incearses with Ts but decreases with q; and there is little di¤erence when Ts is at most two months whatever the value of q is: And even if Ts is three months long, the di¤erence is still insigni…cant.

26CHAPTER 4. PRICING MOVING-AVERAGE-LOOKBACK OPTIONS Second, compared with the tree algorithm, as q increases at the same reset date Ts, the variations between the LSM and the CRR on sceanrio 1 raise more. The second result shows that after the company of the underlying stock pays the dividends, the original strike price UB must also be reset to the same proportinal change as the proportional change of the stock price;

otherwise, as dividend rate increases, the stock price will drop o¤ much to
reach the LB quickly before the reset date T_{s}, especially relatively long reset
date, the probability of early exercise before Tsis large. Besides these results
also show that the probability of early exercise for AAMAL before Ts with a
relatively long reset period is larger than relatively short one.

T_{s} 1/12 2/12 3/12

q CRR LSM LSM2 CRR LSM LSM2 CRR LSM LSM2

2% 6.778 6.843 6.860 7.069 7.124 7.247 7.216 7.278 7.541 (0.001) (0.001) (0.001) (0.001) (0.001) (0.000) 4% 6.322 6.380 6.400 6.607 6.656 6.838 6.750 6.808 7.157

(0.002) (0.001) (0.001) (0.001) (0.000) (0.000) 6% 5.923 5.984 6.073 6.203 6.256 6.480 6.343 6.406 6.849

(0.000) (0.000) (0.002) (0.002) (0.000) (0.000) Table 4.3 : CRR and LSM are calculated on Scenario 1, and LSM2 is

calculated on Scenario 2. The parameters are S_{0} = UB = 50, LB = 45, r =
2%, ¾ = 30%, T = 1 (n^{0}=50), Ts= 1/12 (ns = 22), a = 3 and L = 3 for the
CRR model. The numbers in parentheses are standard error of the price.In
each simulation a total of 50,000 antithetic paths.

### 4.3.4 Case 4 : Reset Date v.s. Di¤ erent Reset Condi- tion

In Taiwan, there are many various Moving-Average Options contract issued.

We want to know the ralation of these di¤erent contracts, so vary two im- portant factors di¤erenct reset conditions and the length of reset date. Two Moving-Average Reset Options are added. The …rst is that the AAMALs, denoted as RS9. The second is that it would be reset to 98%, 96%, 94%, 92%, and 90% of the initial strike price if the 3-day average price of the stock price would fall to 98%, 96%, 94%, 92%, and 90%, denoted as RS5. The last is that it would be reset to 95% and 90% of the initial strike price if the

4.3. NUMERICAL RESULTS 27 3-day average price of the stock price would fall to 95% and 90%, denoted as RS2.

Assume St= 50; UB = 50, r = 2%; q = 0; ¾ = 30%; T = 1; and a = 3.

Suppose there are 22 trading days in a month, so ns = 22 for Ts = 1=12,
ns = 44 for Ts = 2=12, and ns = 66 for Ts = 3=12 and we set n^{0} = 50 for
all, and …x L = 3 on the CRR model. The pricing results by Monte Carlo
simulation are based on 50,000 paths : 25,000 plus 25,000 antithetic, and the
scenario 1 , scenario 2 and European-style by LSM are based on the same
sample paths. The pricing results appear in Table 4.4.

We make the following observations. First, the higher reset frequency, the more valuable the option value is, and we can …nd that the RS9 values are the highest in the three di¤erent reset conditions. Second, as reset frequency increases whatever the reset date Ts is, the premiums between scenario 2 and scenario 1 are not drawn out a conclusion. These results show that as the reset frequency become higher, the probability of Moving-Average options being in-the-money is larger, i.e., the AMAL call option would become more valuable.

### 4.3.5 Case 5 : Moving-Average Number v.s. Volatility

In Taiwan, the Securities often issue di¤erent moving-average number con- tracts. We want to know how the di¤erent moving-average number would a¤ect the option value. So the important factors a and ¾ are varied. Assume St = 45; UB = 45, r = 3%; q = 0; ¾ = 30%; T = 1; and Ts = 1=12. The pricing results by Monte Carlo simulation are based on 50,000 paths : 25,000 plus 25,000 antithetic, and the scenario 1 , scenario 2 and European-style by LSM are based on the same sample paths. The pricing results are on Table 4.5.

We make the following observations. The option value decreases with a, due to the strike price is reset smooth as a moves up. This result shows that as the monitoring interval becomes longer, the AMAL call option would become less valuable. Due to use a total of 100,000 antithetic paths more than 50,000, the estimates on the value are almost undervalued to those on the CRR model.

28CHAPTER 4. PRICING MOVING-AVERAGE-LOOKBACK OPTIONS

Ts Contract Scenario 1

CRR LSM Scenario 2 European

1 RS9 7.4514 7.4410 7.4384 7.3802

(0.0021) (0.0022) (0.0037)

RS5 6.9787 6.9726 6.8802

(0.0014) (0.0012) (0.0019)

RS2 6.9198 6.9172 6.8307

(0.0029) (0.0029) (0.0026)

3 RS9 8.0087 7.9994 7.9237 7.8282

(0.0032) (0.0020) (0.0031)

RS5 7.1073 7.0064 6.9040

(0.0011) (0.0007) (0.0012)

RS2 7.0525 6.9518 6.8512

(0.0027) (0.0021) (0.0032)

Table 4.4:The parameters are S0 = UB = 50, LB = 45, r = 2%, q = 0,

¾ = 30%, T = 1, a = 3, T = 1 (n^{0}=50), ns= 22 for Ts = 1/12 case and
ns = 66 for Ts = 3/12 case , and L = 3 for the CRR model. The numbers
in parentheses are standard error of the price.In each simulation a total of
50,000 antithetic paths.

a ¾ Scenario 1

CRR LSM Scenario 2 European 3 0:3 6.8329 6.9044 6.9071 6.8625

(0.0011) (0.0010) (0.0037) 0:4 8.6679 8.7378 8.7428 8.6802

(0.0022) (0.0012) (0.0019) 0:5 10.4341 10.5083 10.5129 10.4462 (0.0056) (0.0054) (0.0026) 6 0:3 6.7014 6.8197 6.8227 6.7212

(0.0006) (0.0006) (0.0031) 0:4 8.5285 8.6328 8.6356 8.5218

(0.0035) (0.0036) (0.0012) 0:5 10.2955 10.4100 10.4142 10.3062 (0.0039) (0.0028) (0.0032)

Table 4.5:The parameters are S0= UB = 45, LB = 40.5, r = 3%, q = 0,
T = 1, a = 3, T = 1 (n^{0}=50), Ts = 1/12 (ns= 22), and L = 3 for the CRR

4.3. NUMERICAL RESULTS 29 model. The numbers in parentheses are standard error of the price.In each simulation a total of 150,000 antithetic paths.

30CHAPTER 4. PRICING MOVING-AVERAGE-LOOKBACK OPTIONS

## Chapter 5

## What to choose the robustness of LSM?

In the previous sections we we showed that it is possible to value AAMAL us-
ing LSM method and that the price estimates are not di¤erent from the CRR
method. We now examine alternative speci…cations of the cross-sectional re-
gressions models. What we are looking for is the best way to approximate
g_{K}(x) in Eq. (2.6). To this end it is natural to work with members of di¤erent
families of polynomials, fÁkg^{1}k=0:

### 5.1 Altering the number of regressors

In LSM method, it is argued that increasing the number of regressors should be able to obtain an accurate approximation, and it is suggested that the number be increased until the option value implied by the LSM algorithm no longer increases. In order to examine the practical use of this suggestion we formulate the cross-sectional regressions of AAMALs as

y(t_{i}) = ¯_{0}+^{X}^{K}

k=1[¯_{k}w(S_{t}_{i})LS_{k}_{¡ 1}+ ®_{k}w(M(t_{i}))LM_{k}_{¡ 1}]
+^{X}^{K}_{k=1}°kw(S(ti)M(ti))LSMk¡ 1+ u(ti) (5:1)

where w(S(ti)) = exp(¡ ^{¡ S(t}_{2}^{i}^{)}); w(M (ti)) = exp(¡ ^{¡ M(t}_{2} ^{i}^{)}); and w(S(ti)M(ti))
31

32 CHAPTER 5. WHAT TO CHOOSE THE ROBUSTNESS OF LSM?

= exp(¡ ^{¡ S(t}^{i}^{)M(t}2 ^{i}^{)}), LSk = ^{e}^{Sti}_{k!} ^{d}

k(s^{k}_{ti}eSti)

dS_{ti}^{k} ; LMk = ^{e}^{Mti}_{k!} ^{d}

k(M_{ti}^{k}eMti)

dM_{ti}^{k} ; LSMk =

Pk

i=0 e^{Sti Mti}
k!

@^{k}(s^{k}_{ti}eSti Mti)

@S^{i}_{ti}@M_{ti}^{k}^{¡ i} and consider increasing K from one to three.

Penal A of Table 5.6 shows the result for four speci…cations in the scenario 1 contitional on both of St= 45 and 50 associated with ¾ = 0:3 and 0:4 from Talbe 5.4. Changing the number K from one to two, the price estimate increases by signi…cant amounts for all of the four speci…cations. And the value of scenario 1 is, obviously, signi…cantly less than one of European-style when K = 1, which means that losing the correlation between the stock price and the strike price on the least square regreesion model would result in a very great amount of bias. Increasin K to three does not have the same large e¤ect, although all the estimates increase. Thus, depending on how the suggestion in LS is interpreted we should choose K = 2 or 3.

### 5.2 Using alternative polynomial families

Even though the di¤erent elements of the family fL^{k}g^{1}k=0have the property of
being mutually orthogonal with respect to the weighting function exp(¡ ^{S}_{2}),
it is not clear why using them. If there exists more than two stochastic
factors, the number of regressors would increases with individual terms and
the cross product terms. In this section, we try to work the simplest family
of ordinary monomials Although they are not orthogonal, they produce very
close approximations. Furthermore, they are much simple compared to the
Laguerre polynomial. Thus, we formulate the cross-sectional regressions as

C(t_{i}) = ¯_{0}+^{X}^{K}

k=1¯_{k}S^{k}(t_{i}) +^{X}^{K}

k=1®_{k}M^{k}(t_{i})

+^{X}^{K}_{k=2}^{¡ 1}^{X}^{k}_{i=1}^{¡ 1}°kS^{i}(ti)M^{k}^{¡ i}(ti) + u(ti) (5:2)
and again we increase K from one to three.

Panel B of Table 5.6 shows the e¤ect of increasing K on the family of
monomials. There exsits an interesting result that compared with the family
of Laguerre polynomials, the option values with K = 1 are insigni…cantly
di¤erent from one with K = 2: When K = 1, we never add the interset term
into the regression model, i.e., St_{i}Mt_{i} but the values appear more similar to
those with K = 2 than the result of the family of Laguerre polynomials, but
the values are also less than the European-style value. The reason, we think,

5.2. USING ALTERNATIVE POLYNOMIAL FAMILIES 33 is that there exists some correlation between the stock price and the strike price and the orthogonal work of both stochastic factors are not done on the family of monomials, so although we regress the stock price and the strike price without interset term, the model could still contribute their correlation e¤ect from individual stock price and strike price. Therefore, comparing the changes in the price when increasing K from two to three, the penal B also shows that monomials may converge faster, as the price estimates with K = 2 are signi…cant di¤erent from the CRR model.Thus, the rule of suggestion of increasing one more K is less important when replacing the Laguerre polynomials by ordinay monomials.

S0 ¾ CRR K = 1 K = 2 K = 3 European

45 0.3 6.7462 6.0481 6.7844 6.7435 6.7409 (0.0013) (0.0020) (0.0006) (0.0004) 0.5 10.3431 8.5768 10.4219 10.5622 10.3414

(0.0028) (0.0042) (0.0029) (0.0044) 50 0.3 9.3899 9.0532 9.4292 9.4839 9.3691

(0.0011) (0.0025) (0.0007) (0.0013) 0.5 13.0503 11.4582 13.1182 13.2399 13.0427

(0.0034) (0.0043) (0.0042) (0.0047) Panel A : Di¤erent numbers of weighted Laguerre polynomials

S0 ¾ CRR K = 1 K = 2 K = 3 European

45 0.3 6.7462 6.7181 6.7629 6.7456 6.7409 (0.0009) (0.0012) (0.0012) (0.0004) 0.5 10.3431 10.3378 10.4447 10.3433 10.3414

(0.0056) (0.0051) (0.0054) (0.0044) 50 0.3 9.3899 9.2560 9.3814 9.3890 9.3691

(0.0007) (0.0013) (0.0011) (0.0013) 0.5 13.0503 12.8223 13.2527 13.0490 13.0427

(0.0044) (0.0043) (0.0042) (0.0047) Penal B : Di¤ erent numbers of Monomials

Table 5.6 : The parameters of scenario 1are UB = 45, LB = 40.5, r =
3%, q = 0, T = 1 (n^{0}=50), Ts = 1/12 (ns= 22), and a = 3 , and L = 3 on
the CRR model, The numbers in parentheses are standard error of the price.

In each simulation a total of 100,000 antithetic paths.

34 CHAPTER 5. WHAT TO CHOOSE THE ROBUSTNESS OF LSM?

## Chapter 6 Conclusion

This thesis presents a simple new technology for approximating the value of the American-style AMALs on the LSM approach. It is not hard to …nd that the LSM algorithm not only help solve the pricing of complex deriv- atives, especially path-dependnce problem, and also disencumber the main problem of early exercise of American-style derivatives on the simulation all the time. We …nd, oh, My God, even if one does not learn any knowledge of …nancial engenering and other technologies of …nancial computation, he can only price any derivatives which is too complicated and hard to value with the LSM algorithm. This approach is intuitive, accurate, easy to apply and computationally e¢ cient. We illustrate this technique using a number analyses of complicated derivatives AAMALs to let us know the properties of this …nancial commodity popularly issued in Taiwan.

The family of basis functions of the cross-sectional regressions, i.e. the Laguerre polynomials, is compared with the simple family of ordinary Monon- ials, each of which leads to a trade-o¤ betwenn the time used to calculate a price and the precision of that price. Comparing the method-speci…c trade- o¤ reveals that the preferred basis functions uses K = 2 or 3 simple ordinary polynomials instead of K = 3 Laguerre polynomials.

At last, we make the comment that with the ability to value American options, the applicability of simulation techniques becomes much broader and more promising, particularly in multiple factors, such as more than three factors, the LSM method is much easier to extend than and superior to the tree models.

35

36 CHAPTER 6. CONCLUSION

## Bibliography

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