國立臺灣大學財務金融學系(所)碩士論文

國立臺灣大學財務金融學系(所)碩士論文 Department and Graduate Institute of Finance

College of Management

指導老師﹕呂育道 教授 Advisor：Dr. Yuh-Dauh Lyuu

控制變異數法在美式選擇權之應用

Variance Reduction Methods for Monte Carlo Valuation of American Options

研究生﹕陳芳婷

Advisee：Fung-Ting Chen

中華民國96年7月 July, 2007

碩士論文

控 制 變 異 數 法 在 美 式 選 擇 權 之 應 用 陳 芳 婷 撰

96

國立臺灣大學

財務金融學系

(所

)

摘要 摘要 摘要 摘要

自從 Longstaff and Swartz (2001)提出的最小平方估計法 (least-squares Monte Carlo)，解決了蒙地卡羅模擬法難以用於美式選擇權之訂價的一大缺點。

於是，蒙地卡羅模擬法簡單、易懂，且易於應用至多資產商品的特性，使得蒙 地卡羅模擬廣泛地被用於選擇權的評價問題上。然而，蒙地卡羅模樣通常需要 大量的模擬路徑，才能得到較好的估計；這使得評價變得極為耗時。

本研究即是探討兩種降低變異的方法，希望能藉此提昇蒙地卡羅的模擬效 率。這兩種降低變異的方法分別是由 Rasmussen (2005)以及 Duan and Simonato (2001)所提出來的。本研究將之分別應用到美式賣權及極大值買權的評價，結 果發現由 Rasmussen (2005)所提出來的方法，皆能有效地降低模擬的變異程 度。

Abstract

For many complex options, analytical solutions are not available. In these cases

a Monte Carlo simulation is computationally inefficient, the variance reduction

method can be used to improve the efficiency of a Monte Carlo simulation.

In this thesis we apply the two variance reduction methods proposed by

Rasmussen (2005) and Duan and Simonato (1998) in American option pricing. We

find that the variance reduction method proposed by Rasmussen can provide

significant improvement of efficiency than Duan and Simonato even the

combination of these two methods does not perform better than only using the

variance reduction methods proposed by Rasmussen. We also apply this variance

reduction method proposed by Rasmussen in the valuation of two-, three- or five

max-call options and we find that they can provide significant improvement both on

efficiency and accuracy for pricing.

Content

1. Introduction .………1

2. The American Option Valuation Problem..………....6

3. Monte Carlo Valuation with Variance Reduction Method.………...8

3.1. Control Variates Method ……….8

3.2. Empirical Martingale Simulation ………..12

4. The LSM Approach………...15

4.1. The Choice of Basis Functions………..…………....16

4.2. Accuracy, Stability and Convergence………..………...17

4.3. The Control Variates Improved LSM………..……..………....18

5. Numerical Results……….21

5.1. American Put………..21

5.2. American Rainbow Options………...28

6. Conclusion………...33

Bibliography………...34

1. Introduction

Closed-form pricing formulas have been derived for many European options

under a variety of financial models, the most notable being the Black-Sholes

formula under the geometric Brownian motion model. American-type options are

options with flexible early exercise features. Examples are American equity and

fixed-income options and convertible bonds. These contracts arise in virtually all

major financial markets. However, when the option is American-type, the possibility

of early exercise should be considered for the determination of the optimal early

exercise policy. This often leads to highly complicated calculations.

There is a long and rich history of numerical methods for pricing American-style

contingent claims. Among the earliest approaches are the binomial lattice of Cox et

al. (1979) and the explicit finite difference scheme of Brennan and Schwartz (1977).

These methods work particularly well for American options on a single underlying

asset. However, many American-style options have been introduced that depend on

multiple underlying assets or state variables. Multidimensional generalizations of the

Cox et al. binomial method were proposed in Boyle (1988), Boyle et al. (1989), and

others. A related approach involves extensions of the finite difference method to

higher dimensions were proposed by Mitchell and Griffiths (2001). Adapting

for options on two or three state variables, but because their computational effort

grows exponentially with the number of state variables, these methods are

impractical for higher dimensional problems. This is the so-called curse of

dimensionality. In contrast, simulation methods do not suffer from this difficulty.

This thesis considers adapting the Monte Carlo approach for pricing American

options.

The conditional expectations involved in the iterations of dynamic programming

cause the main difficulty for the development of Monte Carlo techniques. Boyle

(1977) first proposed Monte Carlo simulation for the pricing of European claims.

However, it was not until much later that the possibility of using Monte Carlo

simulation for pricing American-style options was suggested by Bossaerts (1989)

and Tilley (1993). Tilley ranked simulated stock price from the maximum to

minimum and divided them into several groups. He computed the holding value by

averaging the discounted payoff within each group and used these holding values to

find an exercise boundary. In this method, stock price is the only factor determining

whether to exercise or not. Barraquand and Martineau (1995) developed a method

which is closely related to that of Tilley (1993) but easier to extend. The idea is to

partition the state space of simulated paths into a number of cells in such a way that

the payoff from the option is approximately equal across the paths in the particular

cell. The probabilities of moving to different cells in the next period conditional on

the current cell can then be calculated from the simulated paths. With these

probabilities in place, the expected value of keeping the option alive until the next

period can be calculated, and a strategy for exercise determined. Broadie and

Glasserman (1997) proposed a convergent algorithm based on simulated trees. Their

method generated both lower and upper bounds so that valid confidence intervals on

the true Bermudan price can be determined. The simulation tree method removes the

exponential dependence of the computation time (CPU time) on the problem

dimension; however, the CPU time is still exponential in the number of exercise

opportunities. A new and somewhat simpler simulation based method to price

American options has recently been proposed by Longstaff and Schwartz (2001).

The idea is to estimate the conditional expectation of the payoff from keeping the

option alive at each possible exercise point from a simple least squares

cross-sectional regression using the simulated paths. They show how to price

different types of path-dependent options using this least-squares Monte Carlo (LSM)

approach.

In these papers, authors introduced numerical methods based on Monte Carlo

techniques. The starting point of these methods is to replace the time interval of

exercise dates by a fixed finite subset. This amounts to approximating the American

option by a Bermudan option, i.e., options with discretely exercisable features not

continuous ones. The solution of the discrete optimal stopping problem reduces to an

effective implementation of the dynamic programming principle.

Although there are many works about least squares regression methods, most

have paid only little attention to the issue of variance reduction. Only a few articles

go beyond applying antithetic variates. In this thesis we consider the application of

control variates to the valuation of American- or Bermudan-type options. In Broadie

and Glasserman (1997), the European option’s payoff at expiry is used as a control

variate. They found that the control variates work quite well for out-of-the-money

options, but are less effective for deep-in-the-money options. They also found that

payoff processes of European options and American options are less correlated when

American options are deep-in-the-money than when American options are

out-of-the-money. The reason is that out-of-the-money American options might have

lower probability to be exercised than deep-in-the-money American options. As a

result, European options are highly correlated with out-of-the-money American

options but less correlated with deep-in-the-money American options. In other

words, traditional European option is not a very good control variate for

deep-in-the-money options.

Another idea is based on a simple observation that simulated sample paths for

the underlying asset price almost always fail to possess the martingale property even

though the theoretical model uses the assumption of martingale. The failure to

ensure the martingale property has particularly serious consequences in the later

time interval when there is more time division. It often requires a very large number

of simulation repetitions to dampen these simulation errors. Duan and Simonato

(1998) proposed a correction to the standard procedure by ensuring that the

simulated sample paths all satisfy the martingale property in each time interval. This

correction is referred to as empirical martingale simulation (EMS).

In this thesis we apply the two variance reduction methods proposed by

Rasmussen (2005) and Duan and Simonato (1998) in American option pricing. We

also compare these two methods and combine them. We find that this new variance

reduction method, which combines the above two methods, cannot provide

significant improvement of efficiency and accuracy for pricing. Then we also apply

variance reduction methods in the valuation of max-call options and we find that

they can provide significant improvement on efficiency and accuracy for pricing.

2. The American Option Valuation Problem

The problem of valuing an American option consists of finding an optimal

exercise strategy and then valuing the expected discounted payoff from this strategy

under the equivalent martingale measure.¹ We let

V

_r denote the time t solution to

this problem, that is,

 

 

 Ε 

=

∈

F

t

T

sup |

) ,

( τ

τ

τ

β

β

X

V

_Q

T t t

t

where

{ } X

t _0≤t≤T is the payoff process adapted to the filtration and T( T

t

, ) denotes

the set of stopping time

τ

satisfying

T

τ

t

≤ ≤

We can easily define a lower bound on the American option price at time t denoted

by

L

_t, since for any given exercise strategy or stopping time

τ

we have

t Q t

t

t

X V

L

β β

β

_τ

τ

 ≤



 

 Ε 

= | F

_t

For an upper bound we refer the reader to Theorem 1 of Anderson and Broadie

1 In what follows we assume that the financial market is defined for the finite horizon [0, T] on a complete filtered probability space(Ω,F,{F_t}_0≤t≤T,Ρ). Here the state Ω is the set of all realizations of the financial market,

F

is the sigma algebra of events at time T, and Ρis a probability measure defined on F. The filtration

{ }

Ft 0≤_t≤_T is assumed to be generated by the price processes of the financial market and augmented with the null sets of

F

, and assuming

F =

T

F

. We furthermore assume that using the numeraire process

{ } β

t 0≤_t≤_T there exists a measure

Q

equivalent to Ρ under which all asset prices relative to the numeraire are martingales.

(2004) and leave out the details here.

In the following sections we give numerical results based on the single-asset

American put option, using the same combinations of underlying asset prices, time

to expiry and volatilities as in Table 1 of Lonstaff and Schwartz (2001). The payoff

process of the single-asset put option for strike K is given by

) 0 , (

max _t

t

K S

X

= −

In this case, holders can exercise on the following set of equidistant points only,

, ) / (

e d T

t

_e = for e=0,1, 2,K,d and is hence a Bermudan rather than an American

option.

3. Monte Carlo Valuation with Variance Reduction Method

3.1. Control Variates for Monte Carlo Simulation (Rasmussen (2005))

Given a stopping time

τ

∈T (

t

,

T

), we want to determine the following

conditional expectation given the information at time t :





 



=  | F

_t

τ τ Q

t t

β Ε X β

L

(1)

Using the underlying model to generate N independent paths of the variables

determining the payoff process

{ } X

t _0≤t≤T and the numeraire process

{ } β

t 0≤_t≤_T, the

crude Monte Carlo estimate is

∑

=

=

N

i i

i

t (N) t

τ τ

β

X N

β

L

1

1

where

X

_τⁱ

β

_τⁱ is the discounted payoff from th

i

path using the exercise strategy

given by the stopping time

τ .

To reduce the variance of the Monte Carlo estimate of the American option, we

can replace the path estimate

X

_τⁱ

β

_τⁱ with the following path estimate

(

t^Q

[ ]

ⁱ

)

i i t

i

i τ i

τ θ Υ

E

Υ

β

X

β

Z

τ

τ + −

= (2)

for some appropriately chosen F -measurable random variable _t

θ

_t, where

Y

ⁱis the

th

i

observation of a random variable for which we can easily compute the time t

conditional expectation. The Monte Carlo estimate using control variates is then

given by

[ ] (

t^{Q i}

)

(N) t t t (N) t

N

i i

i

t (N) t

Y Ε Υ β θ

L β Z N β L

τ τ

− +

=

= ∑

=

1

1

where

∑

∑

=

=

=

=

N

i i (N)

t N

i i

i τ t

(N)

t

Y

, Υ N

β

X N

β

L

τ 1

1

1 1

By the standard ordinary least-squares theory, the optimal choice of

θ

_t is

[ ] [ ] ^Υ

,Υ β X

Q t

τ τ Q t

t

Var

*

= − Cov

θ

(3)

which results in the following minimum variance

[ ]

( )

( )CV

2 t

Var 1 Var 1

N

Q t Q τ

t t t τ τ

τ

L X

ρ X β ,Υ

N β

β

   

= −

   

   

where

[ ]

[ ] [ ]

Cov

Var Var

Q

t τ τ

τ

t Q Q

τ t τ τ t

X β ,Υ ρ X ,Υ

β X β Υ

 

 ≡

  ⋅

Hence the most effective control variates Y are obtained by having the largest

possible correlation, either positive of negative, with the discounted payoff from the

Bermudan option.

A good control variate should have the following two properties: it should be

highly correlated with the payoff of the option in question and its conditional

expectation should be easy to compute. When looking for a control variate for the

Bermudan option, the corresponding European option would be our general choice.

Let

W

_T be the value of a self-financing portfolio at expiry of the Bermudan option.

Using the discounted value of this portfolio, the European option control variate

Y

_T

is then defined by

T T T

Y W

= β

By construction of the equivalent martingale measure Q, the process

{ } Y

t _0≤t≤T

defined by

[

_T |Ft

]

Q

t Ε

Y

Y

=

is a martingale.

The optimal European option control variates is given by

T T T

Y X

= β

(4)

for which we have

t t

T Q T t

C Ε X

Y β ^ _ ⁼ β

 



=  | F

_t (5)

the discounted European option price. This control variate clearly satisfies the

second properties mentioned; however, its correlation with the payoff of the

Bermudan option is not high enough, especially for in-the-money options. Therefore,

we replace the control variate (4) with the control variates





 



= 

=

_τ

τ τ

τ

β β

_T

^{| F}

Q

W

T

W Ε

Y

(6)

Thus, rather than sampling the discounted payoff process at expiry of the option,

Rasmussen (2005) suggests sampling the discounted value process at the time of

exercise of the Bermudan option. The conditional expectation of control variates (4)

is

[ ]

t

Q

Y Y

Ε _τ |F_t =

which is identical to the expectation of the control variate (4).

It is intuitively true that European option and American option payoffs are less

correlated when the options are in-the-money. From an option holder’s viewpoint,

the estimate accuracy is of greatest importance for in-the-money options because

this is where the critical exercise decisions have to be made. As a result, Rasmussen

He has shown that the application of the control variates to the valuation of

American or Bermudan options can be very effective if we sampled these control

variates at the exercise time rather than at expiry.

3.2. Empirical Martingale Simulation (Duan and Simonato (1998))

The theoretical works for contingent claim pricing mostly rely on the absence

of arbitrage opportunities. The martingale connection to the arbitrage-free price

system was first observed by Cox and Ross (1976) and later formalized by Harrison

and Kreps (199). For the ease of exposition, we consider a price system consisting of

two securities, one risky and the other risk-free. The risky security, a common stock,

does not pay dividends and its price, denoted by

S

(t), has the following dynamics

under the risk-neutral probability measure Q:

( )





 



 − +

=

^S

^o

∫

^t

^{r s ds} ∫

^t

^{s dW s}

t s

0 0

2( ) ( ) ( )

5 . 0 exp

)

(

σ σ

where r is the continuously compounded return on the risk-free security,

σ

(s) is

the instantaneous standard deviation of the asset return and

W

(s) is a standard

Brownian motion under probability measure Q. It is easy to verify that the

discounted asset price is indeed a Q-martingale in that, for any

τ

≥ t ≥0,

[ ^{e S}

^{( )|}

^F ] ^{e S}

⁽

^t

⁾

Ε^{Q −rτ}

τ

_t = ^−rt

where

^Ε

^Q

[] ^⋅

denotes the expectation operator under the risk-neutral measure Q and

F the information filtration up to time t. t

In a typical Monte Carlo simulation, this martingale property almost always

fails in the simulated sample. Valuation often requires a very large number of

simulation repetitions to dampen simulation errors. Duan and Simonato (1998)

proposed a simple transformation for the original simulation asset process and

adjusted them to satisfy the martingale property. Their transformation steps are listed

below:

(1) Define the discrete times by

t , t , t ,

_{0 1 2}

K , t

_{m, where}

t

0 is the current time.

(2) Simulate asset prices at times

t , t , t ,

_{0 1 2}

K , t

_m for each simulated path

N

i

=1,2,K, and define the ith simulated asset price at time

t

_j by

).

( _j

i

t S

(3) Compute the ith simulated asset returns at time

t

_j defined as

) (

) ) (

(

1

−

=

j i

j i

j i

t S

t t S

R

, for all

j

=1,2,K,

m

;

i

=1,2,K,

N

(4). Set

S

ⁱ

( t

₀

) = S

ⁱ

( t

₀

) = S

₀

)

for all

i

=1,2,K,

N

In this equation,

S

ⁱ(

t

_j) )

is

the adjusted EMS asset price at the ith sample path and at time

t

_j and

S

₀

is the initial asset price at time

t

₀.

(5). Do a recursive procedure for all

j

=1,2,K,

m

;

i

=1,2,K,

N

) (

) ) (

(

) 1 (

) (

) ( ) ( ) (

0 0

1 0

1

j j i

j i

N

i

j rt i

j

j i j i j i

t Z

t S Z t S

t N Z

e t Z

t R t S t Z

j

=

=

⋅

=

∑

=

−

−

)

)

where r is the risk-free interest rate.

Although they had proposed a correction to the standard Monte Carlo

simulation procedure and imposed the martingale property on the collection of

simulated sample paths, in their paper they only applied this modification within

some specific parameters (e.g., maturities less than 1 year) and European-type

options. This thesis will apply the modification to American options with longer

maturities and different volatility parameters and combine this simulation

adjustment to the control variates method proposed by Rasmussen (2005) for

American options.

4. The LSM Approach with Control Variates

The main assumption of the LSM approach is that the time t conditional

expectation in (1) can be expressed as a linear combination of a countable set of

F -measurable basis functions as follows: t

∑

^∞

=

=

1 j

j t j t t

t

a F

L

β

(7)

The implement this assumption in the LSM approach, (7) can be approximated with

a finite sum at a given level. We let

L

^M_t

β

_t denote this approximation when M

basis functions are used. We then have

∑

=

=

M

j j t j t t

M

t

a F

L

β

1 (8) Using cross-sectional observations of the Monte Carlo generated state variable (such

as stock price), the coefficients

a

_t^j,

j

=1,2,K,

M

, are determined by least-squares

regression, where the current basis functions

F

_t^j,

j

=1,2,K,

M

, are independent

variables and the discounted payoff process

X

_t

β

_t is the dependent variable. As

argued by Longstaff and Schwartz (2001), we only need the approximation where

the option is in-the-money at time t and include these observations in the regression.

With a total of N Monte Carlo generated paths, we let

a

_t^{j( N)},

j

=1,2,K,

M

, denote

the time t coefficients determined from the regression using only the in-the-money

paths. Hence the approximation used in the implementation denoted by

L

^M(N)

β

is

given by

∑

=

=

M

j

j t N j t t

M

t

a F

L

1 ) (

β

(9)

Given the time t value of the state variable summarized by the basis

functions

F

_t^j, j=1, 2, …, M, we exercise the option whenever the time t exercise

value exceeds the corresponding conditional expectation. In other words, we will

exercise when the following inequality is satisfied

t M t

t

t

L

X β β ^≥ 4.1. The Choice of Basis Functions

For the sake of simplicity and to maintain some financial intuition, we use the

following basis functions corresponding to

M

=3

.

All functions depend on the

current time t and the current stock price

S

_t, i.e.,

F

_t^j =

f

_j(

t

,

S

_t), where

) , ( )

, (

) , ( ) , (

) , (

) , (

3 2 1 0

s t C s s t f

s t C s t f

s s t f

K s t f

⋅

=

=

=

=

Above, K is the strike price of the Bermudan put option, s is the current asset price

and

C

( s

t

, ) is the time t European put option price expiring at time T and with its

other parameters identical to the Bermudan option.

4.2. Accuracy, Stability and Convergence of the LSM Approach

The accuracy of the LSM exercise strategy is solely determined by the accuracy

of the following two approximations:

t N M t

t M t

t M t

t t

L L

L L

β β

β β

) (

≅

≅

With an infinite computational budget, we should require that

L

^M_t ^(N) →

L

_t as

∞

→

N

and

M

→∞. However, in Clement et al (2002), the convergence of

t M

t

L

L

→ as

M

→∞ is established when an orthogonal set of basis functions is

used. They also considered the convergence of

L

^M_t ^(N) →

L

_t as

N

→∞. Finally,

they established a central limit result for the rate of convergence of the LSM

algorithm. These results are very important especially from a theoretical point of

view. However, from a practical point of view, we are more concerned with the

performance given a finite sample of N paths and a finite number of basis functions

M. Hence we focus in the following section on improving the above approximation

for a given number of paths and basis functions. The object is to replace (8) with a

more accurate approximation.

4.3. Improvement of LSM with Control Variates

In traditional LSM approach, we only choose the discounted payoff

X

_t

β

_t of

in-the-money paths to do regression for evaluating the conditional expectation value

 

 



=  | F

_t

τ τ Q

t t

β Ε X β L

as in (1). In this section, we reuse and generalize the concept of control variates as

the Monte Carlo variation in section 3.1. We replace the discounted payoff

X

_t

β

_t

with the random variable

Z

_t

β

_t :

(

τ

[ ]

τ

)

τ τ τ

τ

θ Υ E Υ

β X β

Z

_Q

t

t

−

+

=

(10)

where

Y

_τ is the control variates sampled from European option’s payoffs at

exercise points as in (6), and

θ

_t is an appropriately chosen F -measurable random _t

variable. Rather than a point estimate as in (3), we now use a functional estimate of

∗

θ

t . By the definition of

[ ^X

t

^β

t

^Y

_τ

]

Q

t ,

Cov and V

^ar

t^Q

[ ] ^Y

_τ we get

[ ] [ ] ( [ ] )

( )

( )

²

( )

²

Q 2 2 Q

Q Q

Q

E

t t

t t t t t

t t

t t

t

t

Y Y

L Y LY

Y E Y E

X Y E X Y

E

−

⋅



 



−

−

=

−

⋅

 

− 





⋅



 





−

∗ =

β β

β θ β

τ τ

τ τ τ τ

τ τ

where

L

_t

β

_t is defined in (1),

Y

_t is defined in (5), and

( ) Y

² t and

( LY )

t

β

t are

defined by

( ) [ ]

( )

 

 



  ⋅





 

≡ 

≡

τ t Q t t t

t

τ Q t t

β Y E X β

LY

, Y E

Y

^{2 2}

respectively. Generalizing the assumptions of LSM as (7), we assume that the time t

conditional expectations

Y

_t,

( ) Y

² t and

( LY )

t

β

t can also be expressed as a

countable sum of the same set of F -measurable basis functions, i.e., _t

( )

( )

_j

t j

j t t

t

j t j

j t t

j t j

j t t

F LY d

F c Y

F b Y

⋅

=

⋅

=

⋅

=

∑

∑

∑

∞

=

∞

=

∞

=

1 1 2

1

, ,

β

Again we can approximate the conditional expectations by truncating the sums at the

same level M. Let

Y

_t^M,

( ) Y

² t^M and

( )

t M

LY

t

β

denote the approximations as in (8).

Now we have

( )

( )

_j

t M

j j t t

M t

j t M

j j t M

t

j t M

j j t M

t

F LY d

F c Y

F b Y

⋅

=

⋅

=

⋅

=

∑

∑

∑

=

=

=

1 1 2

1

, ,

β

We can approximate

θ

_t^∗ with

θ

_t^M given by

( )

( ) Y

²

( Y M )

²

L Y LY

t M t

M t t M t t M t

M

t −

⋅



 



−

=

β β

θ

Using the concept of control variates in (10), we can approximate the time t

conditional expectation of the discounted payoff from following the strategy

τ

by

CV , M

L

t given by the following expression:

(

t

)

M t M

t M

t M

Y Υ β θ

L β

L

t t

t

= + −

CV ,

5. Numerical Results

5.1. American Put

In this chapter, we use some numerical examples² to compare the efficiency

and accuracy of three different variance reduction methods. First, we use normalized

antithetic paths, i.e., we use traditional moment matching simulation and antithetic

method, and denote this method as “AN+MM” in the following. Second, we

replicate the simulated paths in the first method and use discounted European

options sampling at exercise as control variates and denote this as “CV-at-exercise.”

Third, we also replicate the simulated paths in the first method but only adjusting the

underlying process to satisfy martingale property and denote this as “EMS” method.

In order to show the efficiency and accuracy of the AN+MM, CV-at-exercise

and EMS, we use binomial-based put prices as the benchmark and compare their

standard error (S.E.) and mean absolute percentage error (MAPE) to see the

efficiency and accuracy of these three methods (AN+MM, CV-at-exercise and EMS).

Also we will use ratios of S.E and MAPE between AN+MM and CV-at-exercise or

AN+MM and EMS to see whether CV-at-exercise or EMS can improve the

efficiency or accuracy of the LSM approach.

From the results of CV-at-exercise method in the second category in Table 1,

we find the standard errors in CV-at-exercise are only one tenth of the AN+NN

method in the third column of the second category. These improvements are more

significant when options are at-the-money (

S

=40) or out-of-the-money (

S

=42,44)

and with shorter maturity (

t

=1). This can be confirmed by observing the smaller

ratio of S.E of CV divided by S.E of AN+MM listed in the third column of the

second category. But from the next two columns, improvement in terms of MAPE is

not very significant in CV-at-exercise method except when option is

deep-out-of-the-money (

S

=44). Also from the third column of third category in

Table 1, we find that standard errors in the EMS method are not less than the

AN+MM method but the MAPE in this method are smaller than AN+MM method

when options are in-the-money (

S

=36,38).

In Table 1, the improvements of standard errors using CV-at-exercise are quite

significant. Now we try to combine CV-at-exercise and EMS and evaluate if the

resulting method can improve the efficiency or accuracy in pricing Bermudan

options compared with the AN+MM method. The results are shown in Table 2.

As before, we use the parameters of Longstaff and Schwartz (2001) in the

following numerical examples whose results are listed in Table 2. In our second test,

we combine the CV-at-exercise method and the EMS method, denoted by CV+EMS.

We also compare the result of CV+EMS with CV-at-exercise and list the results in

Table 2. In order to more clearly compare the results, we find the ratios of their S.E

and MAPE and list the results in Table 3. From the results in Table 3, we find only

when

σ

=0.2 and options are deep-in-the-money (

S

=36), can CV+EMS obtain

smaller S.E. than CV-at-exercise in all our maturity parameters. In other parameters,

the improvement in terms of S.E. in CV+EMS method is not significant. Also, the

improvement in terms of MAPE in CV+EMS is not very significant in most

parameters of our test except when

σ

=0.5and options are deep-in-the-money

(

S

=36).

Table 1. This is Monte Carlo valuation of the Bermudan options. We use the binomial tree-based

exercise strategy as the benchmark and compare three different variance reduction methods. First, we only use antithetic pair paths and do moment matching method (AN+MM) in the simulation. Second, we replicate the AN+MM and use European options sampling at exercise as control variates (CV-at-exercise). Last, we also replicate the AN+MM and use empirical martingale simulation (EMS) to adjust underlying asset processes. All simulations are based on 2000 pairs of antithetic paths. All options have strike K=40, and the interest rate equals r=0.06. The current stock price S, the volatility σ, and the time to expiry T are given.

AN+MM CV-at-exercise EMS

S t σ binomial

model price S.E.

(%)

MAPE

(%) price S.E.

(%)

CV/

AN+MM

MAPE (%)

CV/

AN+MM price S.E.

(%)

EMS/

AN+MM

MAPE (%)

EMS/

AN+MM 0.2 4.4845 4.4885 2.6433 0.0891 4.4726 0.2777 0.1051 0.2645 2.9693 4.4874 2.6627 1.0074 0.0642 0.7213 1

0.4 7.0997 7.1392 4.1982 0.5568 7.0852 0.2724 0.0649 0.2043 0.3670 7.1143 4.9029 1.1678 0.2059 0.3699 0.2 4.8512 4.8498 3.3725 0.0287 4.8474 0.5526 0.1639 0.0790 2.7514 4.8500 3.3443 0.9916 0.025 0.8713 36

2

0.4 8.5310 8.5397 4.6569 0.1019 8.5328 0.6612 0.1420 0.0207 0.2032 8.5383 6.5843 1.4139 0.0857 0.8415 0.2 3.2529 3.2672 2.2367 0.4402 3.2420 0.2285 0.1022 0.3355 0.7622 3.2586 2.3066 1.0313 0.1747 0.3969 1

0.4 6.1805 6.1843 4.2744 0.0610 6.1782 0.2400 0.0561 0.0372 0.6090 6.1816 6.0043 1.4047 0.0185 0.3034 0.2 3.7546 3.7524 2.7642 0.0591 3.7564 0.4651 0.1683 0.0481 0.8138 3.7526 2.9799 1.0780 0.0542 0.9171 38

2

0.4 7.6990 7.6943 4.7164 0.0605 7.6929 0.5717 0.1212 0.0792 1.3098 7.6983 7.0241 1.4893 0.0092 0.1515 0.2 2.3130 2.3369 2.6482 1.0347 2.2966 0.2157 0.0815 0.7081 0.6844 2.3388 2.8427 1.0735 1.1139 1.0766 1

0.4 6.3028 5.3505 4.3863 15.1086 5.2802 0.2161 0.0493 16.2243 1.0738 5.3539 6.1141 1.3939 15.0553 0.9965 0.2 2.8800 2.8947 2.7106 0.5089 2.8532 0.4146 0.1529 0.9298 1.8271 2.8951 3.1117 1.1480 0.5249 1.0315 40

2

0.4 6.9036 6.9458 4.6295 0.6117 6.8654 0.5427 0.1172 0.5527 0.9035 6.9502 7.3925 1.5968 0.6755 1.1042 0.2 1.6239 1.6400 2.8891 0.9903 1.6163 0.1686 0.0583 0.4685 0.4731 1.6409 3.3076 1.1449 1.0492 1.0594 1

0.4 4.6137 4.6182 4.8191 0.0982 4.6128 0.2047 0.0425 0.0192 0.1958 4.6252 6.9900 1.4505 0.2486 2.5310 0.2 2.2249 2.2274 3.0713 0.1132 2.2277 0.4061 0.1322 0.1242 1.0971 2.2291 3.5747 1.1639 0.1908 1.6856 42

2

0.4 6.2670 6.2690 4.8283 0.0312 6.2680 0.4679 0.0969 0.0157 0.5040 6.2682 8.1596 1.6899 0.0199 0.6403 0.2 1.1212 1.1300 2.6878 0.7858 1.1198 0.1496 0.0557 0.1210 0.1539 1.1309 2.9862 1.1110 0.8653 1.1011 1

0.4 3.9616 3.9797 5.6508 0.4576 3.9502 0.1908 0.0338 0.2877 0.6288 3.9657 7.6675 1.3569 0.1034 0.2260 0.2 1.6907 1.7082 3.1561 1.0339 1.6885 0.3569 0.1131 0.1295 0.1253 1.7088 3.6018 1.1412 1.0719 1.0367 44

2

0.4 5.6781 5.6620 4.8480 0.2835 5.6764 0.3953 0.0815 0.0307 0.1081 5.6682 7.9300 1.6357 0.1749 0.6167

Table 2. This is Monte Carlo valuation of the Bermudan options. We use the binomial tree-based exercise

strategy as the benchmark and compare two variance reduction methods. First we use CV-at-exercise method. Second, we combine CV-at-exercise and EMS method and denote this as EMS+CV method. All these simulations are based on 2000 pairs of antithetic paths. All options have strike K=40, and the interest rate equal to r=0.06. The current stock price S is given the same value as in Table 1. In order to see the effect of the volatility σ and the time to expiry T , we give additional values to these two parameters besides in Table 1.

T=0.25 T=0.5

σ S

binomial

model CV S.E

(%) MAPE CV+

EMS S.E

(%) MAPE binomia

l model CV S.E

(%) MAPE CV+

EMS S.E

(%) MAPE 4.0000 4.0000 0.0000 0.0000 4.0000 0.0000 0.0000 4.0000 4.0007 0.0052 0.0002 4.0007 0.0052 0.0002 2.0000 2.0000 0.0105 0.0000 2.0000 0.0105 0.0000 2.0000 1.9948 0.0190 0.0026 1.9948 0.0190 0.0026 0.2245 0.2215 0.0549 0.0134 0.2215 0.0546 0.0134 0.2587 0.2541 0.0934 0.0178 0.2541 0.0949 0.0178 0.0019 0.0018 0.0053 0.0618 0.0018 0.0051 0.0666 0.0082 0.0080 0.0141 0.0288 0.0080 0.0141 0.0284 0.05

36 38 40 42

44 0.0000 0.0000 0.0000 * 0.0000 0.0000 * 0.0001 0.0001 0.0011 0.3318 0.0001 0.0011 0.3318 4.0000 4.0023 0.0171 0.0006 4.0023 0.0171 0.0006 4.0000 4.0018 0.0348 0.0004 4.0018 0.0348 0.0004 2.0000 2.0003 0.7655 0.0001 2.0003 0.7629 0.0002 2.0123 2.0148 0.4546 0.0012 2.0148 0.4477 0.0012 0.5861 0.5819 0.0494 0.0071 0.5819 0.0495 0.0071 0.7372 0.7304 0.1057 0.0092 0.7304 0.1064 0.0092 0.1051 0.1051 0.0177 0.0003 0.1051 0.0197 0.0002 0.2173 0.2178 0.0570 0.0021 0.2177 0.0576 0.0021 0.1

36 38 40 42

44 0.0104 0.0102 0.0140 0.0219 0.0102 0.0136 0.0208 0.0493 0.0489 0.0273 0.0085 0.0489 0.0262 0.0087 4.0480 4.0472 0.3861 0.0002 4.0472 0.3669 0.0002 4.2105 4.2122 0.3066 0.0004 4.2122 0.2947 0.0004 2.4718 2.4701 0.0814 0.0007 2.4701 0.0785 0.0007 2.8242 2.8212 0.1404 0.0011 2.8212 0.1389 0.0011 1.3527 1.3472 0.0507 0.0041 1.3472 0.0513 0.0041 1.7921 1.7825 0.0964 0.0053 1.7825 0.0959 0.0053 0.6700 0.6685 0.0357 0.0022 0.6685 0.0353 0.0022 1.0928 1.0921 0.0801 0.0007 1.0921 0.0805 0.0007 0.2

36 38 40 42

44 0.2937 0.2934 0.0273 0.0010 0.2934 0.0289 0.0010 0.6362 0.6361 0.0543 0.0001 0.6361 0.0603 0.0001 4.5098 4.5090 0.1621 0.0002 4.5090 0.1495 0.0002 5.0238 5.0219 0.1863 0.0004 5.0219 0.1737 0.0004 3.1775 3.1732 0.0622 0.0013 3.1732 0.0614 0.0013 3.8403 3.8365 0.1376 0.0010 3.8365 0.1339 0.0010 2.1314 2.1249 0.0468 0.0031 2.1249 0.0477 0.0031 2.8760 2.8646 0.1010 0.0039 2.8647 0.1002 0.0039 1.3800 1.3787 0.0357 0.0009 1.3787 0.0372 0.0009 2.1361 2.1308 0.0836 0.0025 2.1308 0.0831 0.0025 0.3

36 38 40 42

44 0.8569 0.8564 0.0355 0.0006 0.8563 0.0343 0.0006 1.5595 1.5579 0.0743 0.0010 1.5579 0.0813 0.0010 5.1170 5.1138 0.1156 0.0006 5.1138 0.1038 0.0006 5.9823 5.9777 0.1664 0.0008 5.9777 0.1570 0.0008 3.9069 3.9033 0.0546 0.0009 3.9033 0.0528 0.0009 4.9086 4.9074 0.1267 0.0003 4.9074 0.1256 0.0003 2.9128 2.9056 0.0440 0.0025 2.9056 0.0446 0.0025 3.9653 3.9526 0.0984 0.0032 3.9527 0.1008 0.0032 2.1422 2.1377 0.0360 0.0021 2.1377 0.0357 0.0021 3.2317 3.2285 0.0924 0.0010 3.2286 0.0957 0.0010 0.4

36 38 40 42

44 1.5469 1.5445 0.0417 0.0016 1.5445 0.0379 0.0015 2.5772 2.5753 0.0859 0.0007 2.5753 0.0878 0.0007 5.8042 5.8021 0.0719 0.0004 5.8021 0.0664 0.0004 6.9490 6.9351 0.1388 0.0020 6.9351 0.1354 0.0020 4.6715 4.6698 0.0482 0.0004 4.6697 0.0474 0.0004 5.9766 5.9741 0.1251 0.0004 5.9741 0.1291 0.0004 3.6939 3.6861 0.0443 0.0021 3.6861 0.0452 0.0021 5.0539 5.0400 0.0948 0.0028 5.0400 0.0983 0.0028 0.5

36 38 40

*When T=0.25 and σ=0.05, the option price of binomial model is nearly zero. In this case, MAPE is infinite and has no meaning..

T=1 T=2

σ S

binomial

model CV S.E

(%) MAPE CV+

EMS S.E

(%) MAPE

binomial

model CV S.E (%) MAPE CV+

EMS S.E (%) MAPE 4.0000 4.0045 0.0170 0.0011 4.0045 0.0170 0.0011 4.0000 4.0014 0.0376 0.0003 4.0014 0.0356 0.0003 2.0000 2.0082 0.0374 0.0041 2.0081 0.0354 0.0041 2.0000 2.0142 0.0583 0.0071 2.0142 0.0587 0.0071 0.2825 0.2728 0.1600 0.0342 0.2729 0.1654 0.0341 0.2936 0.2786 0.2646 0.0509 0.2786 0.2645 0.0509 0.0176 0.0177 0.0369 0.0066 0.0177 0.0378 0.0070 0.0243 0.0243 0.1118 0.0015 0.0242 0.1106 0.0035 0.05

36 38 40 42

44 0.0007 0.0006 0.0080 0.0906 0.0006 0.0079 0.0912 0.0019 0.0019 0.0247 0.0185 0.0018 0.0245 0.0267 4.0000 4.0076 0.1311 0.0019 4.0076 0.1286 0.0019 4.0000 4.0234 0.7864 0.0058 4.0234 0.7863 0.0058 2.0554 2.0424 0.2309 0.0063 2.0424 0.2310 0.0063 2.1077 2.0950 0.4239 0.0060 2.0950 0.4237 0.0060 0.8894 0.8775 0.1849 0.0134 0.8775 0.1894 0.0134 1.0188 1.0012 0.3712 0.0173 1.0012 0.3814 0.0173 0.3558 0.3549 0.1522 0.0026 0.3549 0.1555 0.0026 0.4874 0.4847 0.3222 0.0055 0.4847 0.3363 0.0056 0.1

36 38 40 42

44 0.1276 0.1279 0.0766 0.0025 0.1279 0.0780 0.0026 0.2267 0.2273 0.2072 0.0027 0.2273 0.2087 0.0026 4.4845 4.4722 0.3610 0.0027 4.4722 0.3475 0.0027 4.8512 4.8475 0.5288 0.0008 4.8476 0.5208 0.0008 3.2529 3.2421 0.2553 0.0033 3.2420 0.2533 0.0033 3.7546 3.7562 0.4683 0.0004 3.7562 0.4766 0.0004 2.3130 2.2967 0.2157 0.0071 2.2967 0.2235 0.0071 2.8800 2.8535 0.4498 0.0092 2.8536 0.4612 0.0092 1.6239 1.6163 0.1686 0.0047 1.6163 0.1750 0.0047 2.2249 2.2272 0.3702 0.0010 2.2271 0.3828 0.0010 0.2

36 38 40 42

44 1.1212 1.1192 0.1396 0.0018 1.1192 0.1450 0.0018 1.6907 1.6887 0.3168 0.0012 1.6887 0.3287 0.0012 5.7476 5.7399 0.2565 0.0013 5.7399 0.2506 0.0013 6.6160 6.5993 0.5140 0.0025 6.5993 0.5262 0.0025 4.7055 4.7062 0.2296 0.0001 4.7062 0.2361 0.0001 5.1707 5.7116 0.4534 0.1046 5.7116 0.4815 0.1046 3.8016 3.7820 0.2002 0.0052 3.7820 0.2024 0.0052 4.8797 4.8461 0.4179 0.0069 4.8461 0.4395 0.0069 3.1002 3.0987 0.1858 0.0005 3.0987 0.1879 0.0005 4.2214 4.2240 0.4182 0.0006 4.2241 0.4457 0.0006 0.3

36 38 40 42

44 2.4738 2.4689 0.1682 0.0020 2.4688 0.1736 0.0020 3.6232 3.6126 0.3780 0.0029 3.6127 0.4028 0.0029 7.0997 7.0854 0.2724 0.0020 7.0855 0.2725 0.0020 8.5310 8.5338 0.5882 0.0003 8.5339 0.6032 0.0003 6.1805 6.1784 0.2400 0.0003 6.1784 0.2427 0.0003 7.6990 7.6943 0.5167 0.0006 7.6944 0.5507 0.0006 6.3028 5.2804 0.2161 0.1622 5.2805 0.2223 0.1622 6.9036 6.8665 0.4681 0.0054 6.8665 0.5161 0.0054 4.6137 4.6129 0.2047 0.0002 4.6129 0.2139 0.0002 6.2670 6.2682 0.4657 0.0002 6.2683 0.5115 0.0002 0.4

36 38 40 42

44 3.9616 3.9503 0.1908 0.0028 3.9503 0.1998 0.0028 5.6781 5.6767 0.4536 0.0002 5.6768 0.5056 0.0002 8.5205 8.5181 0.2791 0.0003 8.5182 0.2760 0.0003 10.4470 10.4480 0.6471 0.0001 10.4475 0.7124 0.0000 7.6491 7.6438 0.2661 0.0007 7.6438 0.2748 0.0007 9.6697 9.6552 0.5931 0.0015 9.6547 0.6680 0.0015 6.7993 6.7754 0.2316 0.0035 6.7755 0.2476 0.0035 8.9128 8.8707 0.5587 0.0047 8.8703 0.6297 0.0048 6.1322 6.1311 0.2179 0.0002 6.1312 0.2321 0.0002 8.3091 8.3051 0.5202 0.0005 8.3047 0.5774 0.0005 0.5

36 38 40 42

44 5.4964 5.4900 0.1974 0.0012 5.4901 0.2185 0.0012 7.7441 7.7442 0.4541 0.0000 7.7438 0.5161 0.0000

Table 3. This is Monte Carlo valuation of the Bermudan options. We select S.E. and MAPE of CV-at-exercise

and CV+EMS from Table 2 and define new ratios called S.E-ratio and MAPE-ratio. S.E.-ratios are derived by S.E of EMS+CV divided by S.E. of CV-at-exercise and MAPE-ratios are derived by MAPE of EMS+CV divided by MAPE of CV-at-exercise.

S.E.-ratio ( EMS+CV/CV ) MAPE-ratio ( EMS+CV/CV )

σ t S=36 S=38 S=40 S=42 S=44 σ t S=36 S=38 S=40 S=42 S=44

0.25 0.9045 1.0000 0.9941 0.9608 * 0.25 0.9557 0.9874 1.0003 1.0780 * 0.5 1.0001 1.0000 1.0151 0.9985 1.0889 0.5 1.0009 0.9998 0.9999 0.9878 1.0000

1 1.0002 0.9448 1.0341 1.0247 0.9776 1 1.0002 0.9978 0.9989 1.0562 1.0069 0.05

2 0.9456 1.0060 0.9997 0.9892 0.9940 0.05

2 1.0075 0.9997 0.9999 2.3503 1.4443 0.25 1.0000 0.9965 1.0011 1.1127 0.9728 0.25 1.0005 1.1457 1.0003 0.6734 0.9488 0.5 0.9998 0.9848 1.0066 1.0122 0.9609 0.5 1.0010 0.9844 1.0007 0.9635 1.0295 1 0.9812 1.0003 1.0243 1.0216 1.0191 1 1.0045 1.0009 1.0000 0.9927 1.0362 0.1

2 0.9999 0.9993 1.0275 1.0437 1.0070 0.1

2 1.0008 1.0008 1.0002 1.0075 0.9884 0.25 0.9503 0.9638 1.0125 0.9894 1.0577 0.25 0.9909 1.0031 0.9999 0.9977 1.0138 0.5 0.9610 0.9890 0.9942 1.0061 1.1122 0.5 1.0026 1.0013 0.9992 0.9900 1.2660 1 0.9624 0.9922 1.0362 1.0381 1.0390 1 1.0030 1.0046 1.0000 1.0042 0.9995 0.2

2 0.9849 1.0177 1.0254 1.0342 1.0374 0.2

2 0.9936 1.0080 0.9987 0.9823 0.9968 0.25 0.9222 0.9866 1.0191 1.0412 0.9658 0.25 0.9360 0.9955 0.9989 0.9997 1.0535 0.5 0.9322 0.9731 0.9925 0.9950 1.0939 0.5 1.0088 0.9957 0.9993 0.9940 1.0211 1 0.9770 1.0281 1.0109 1.0111 1.0324 1 0.9948 1.0017 0.9994 1.0079 1.0070 0.3

2 1.0238 1.0622 1.0518 1.0658 1.0656 0.3

2 0.9966 1.0001 1.0008 1.0115 0.9947 0.25 0.8982 0.9671 1.0138 0.9906 0.9078 0.25 1.0011 1.0018 1.0018 1.0031 0.9831 0.5 0.9436 0.9917 1.0248 1.0361 1.0220 0.5 0.9947 0.9924 0.9977 0.9949 1.0070 1 1.0003 1.0114 1.0287 1.0454 1.0471 1 0.9949 0.9913 1.0000 0.9779 0.9992 0.4

2 1.0256 1.0657 1.1026 1.0984 1.1147 0.4

2 1.0544 0.9782 0.9984 1.0939 0.9175 0.25 0.9237 0.9830 1.0217 1.0183 1.0287 0.25 0.9962 1.0021 1.0007 1.0045 0.9800 0.5 0.9752 1.0314 1.0364 1.0614 1.0172 0.5 0.9991 0.9879 0.9994 1.0004 1.0003 1 0.9891 1.0326 1.0691 1.0654 1.4106 1 0.9741 0.9884 0.9980 0.9208 0.9844 0.5

2 1.1010 1.1263 1.1270 1.1099 1.1367 0.5

2 0.4883 1.0348 1.0100 1.0902 4.1065

*When T=0.25 and σ=0.05, the option price of binomial model is nearly zero. In this case, MAPE is infinite and has no meaning..

5.2. American Rainbow Options

To investigate the application of control variates to a more complex example

than the single-asset put option investigated in the previous numerical examples, we

now investigate the n-asset Bermudan rainbow option for the case of n=2,3, and 5

assets. We use the same parameters as the example in Andersen and Broadie (2004)

without any use of variance reduction and compare the results.

Andersen and Broadie use a set of 13 basis functions involving the highest and

second highest asset prices, as well as polynomials of these, together with the value

of European max-call option on the two largest assets and polynomials of this.

Inspired by their choice and the concept of Rasmussen (2005), we use another set of

basis functions.

For the case of two assets, the European max-call option price can easily be

computed according to Stulz (1982). Although the pricing formula of multi-asset

max-option has been derived by Johnson (1987) the computation of multivariate

normal cumulated probability requires numerical integration. To circumvent this

shortcoming we therefore choose to use combinations of two-asset max-call options

as basis functions in our regression and as control variates. We list our basis

functions as following. For the case of two assets, the six basis functions are