Pricing Asian Options Using Path Bundling

transcript

(2000)

Edwin H. Neave School of Business, Queen's University

George L. Ye

Department of Finance & Management Science, Saint Mary's University

Abstract This paper presents a new methodology to price European and American Asian options on a

recombining binomial process. The method uses state descriptions for bundles of paths rather than

for individual paths, thus reducing the amount of computation needed. Depending on the size of

the problem and the availability of computational resources, our methods give either an exact

solution or an approximate solution with upper and lower bounds on its error. In comparison to

other approaches, our methods are both more nearly accurate and very fast.

Key words: options pricing, Asian options.

JEL Classification: G1

This paper can be downloaded from the Social Science Research Network Electronic Paper Collection:

http://papers.ssrn.com/abstract=284888

Pricing Asian options using path bundling

Edwin H. Neave and George L Ye

Edwin H. Neave: School of Business Queen's University Kingston, Ontario Canada K7L 3N6 Tel. 613-533-2348 Fax:613-533-2321 Email: neave@qsilver.queensu.ca

George L. Ye: Department of Finance & Management Science Saint Mary's University Halifax, Nova Scotia Canada B3H 3C3 Tel. 902-420-5733 Fax: 902-496-8101 Email: gye@stmarys.ca

Abstract

This paper presents a new methodology to price European and American

Asian options on a recombining binomial process. The method uses state

descriptions for bundles of paths rather than for individual paths, thus reducing the

amount of computation needed. Depending on the size of the problem and the

availability of computational resources, our methods give either an exact solution

or an approximate solution with upper and lower bounds on its error. In

comparison to other approaches, our methods are both more nearly accurate and

very fast.

Key words: options pricing, Asian options.

JEL Classification: G1

1. INTRODUCTION

This paper prices fixed strike arithmetic average Asian calls. Asian options have

been actively traded in the over-the-counter markets since at least the mid-1980s. These

options are mostly used in currency exchange, interest rate, and commodity trading

markets (see Ritchken, Sankarasubramanian, and Vijh 1993, Jarrow and Turnbull 1996),

but also in equity markets (Bouaziz, et. al. 1994). Indeed, according to a survey of U.S.

non-financial firms conducted by CIBC World Markets and the Wharton School (Bodnar,

Marston, and Hayt, 1998), Asian options are the most commonly used exotic options.

In comparison to standard instruments, Asian options offer two important

advantages. First, the average feature means that Asian options can effectively reduce the

effects of extreme price movements at or near maturity. Thus, Asian options can be used

to avoid price biases and price manipulation - a concern when the underlying asset is

either infrequently traded, or illiquid, as is the case with commodities such as oil and

some metals (Heenk, Kemna, and Vorst 1992, Fung 1992, Kemna and Vorst 1990).

Second, as an established hedging instrument, Asian options are cheaper than the standard

European options (Chance and Rich 1996, Vorst 1996)1.

In spite of the rising popularity of Asian options, their valuation has presented a

challenge to both academics and practitioners. Even in the simplest possible settings of

the underlying asset price, such as the Black-Scholes model in continuous time or the

binomial tree model in discrete time, valuation presents the difficulty of determining the

distribution of process arithmetic averages. Conventionally, asset prices are assumed

either to have lognormal distributions in continuous time, or a multiplicative binomial in

1The prices of Asian options are usually lower than the prices of their standard counterparts. However, as pointed out in Turnbull and Wakeman (1991) back-starting Asians can have higher values.

discrete time.2 Because the sum of lognormal variates is not itself lognormal, and has no

explicit distribution function, it is impossible to derive an analytical solution using the

Black-Scholes (1973) approach in continuous time.

The literature identifies five approaches to valuing European Asian options. The

first uses an analytic solution to derived an approximate payoff distribution. Levy (1992)

uses a Wilkinson approximation to the lognormal, based on its first two moments.

Turnbull and Wakeman (1991) use an approximation based on the lognormal’s first four

moments. Levy and Turnbull (1992), observe that Levy’s solution is a good

approximation only when the asset process’ volatility is less than 20%; Turnbull and

Wakeman’s solution when the volatility lies between 20% and 30%. Bessel processes are

used in Geman and Yor (1993) and the reciprocal Gamma distribution is proposed in

Milevsky and Posner (1998). Although these approaches are usually very fast in

computations, their accuracy is suspect for larger volatility. Moreover, they can only be

used to value European Asian options.

The second approach is to derive an approximate analytic solution of the PDE.

Among others, Rogers and Shi (1995) derive a lower bound for the price of backward-

starting, fixed strike Asian call options. Bouaziz, Briys, and Crouhy (1994) derive a

closed-form solution for the valuation of European forward-starting, floating strike, Asian

call options. Chacko and Das (1997) find a closed-form solution for binary Asian options

on interest rates.

The third approach is to solve the PDE numerically using the finite-difference

method. Dewynne and Wilmott (1995a, b) solve a two dimensional PDE for floating

strike Asian options. Alziiary, Decamps, and Koehl (1997) concentrate on the pricing of a

backward-starting fixed strike Asian option by solving a one dimensional PDE. They

argue that the PDE method is typically faster than Monte Carlo simulation methods, but

gives similar accuracy. However, Barraquand and Pudet (1996) argue that standard finite-

2 Many authors regard the binomial process as an approximation to the continuous time process. For a

difference methods are inaccurate. Zvan, Forsyth and Vetzal (1998) propose a

modification by using a so-called high-order flux limiter.

The fourth and probably most popular approach is Monte Carlo simulation. The

work in this area includes Kemna and Vorst (1990), Haykov (1993), Boyle (1977), among

others. Although Monte Carlo simulation is widely used in options pricing it is relatively

slow in computations since a huge number of sample paths are required, especially when

high accuracy is demanded.

Even though the literature has viewed it as incurring important computational

difficulties (see e.g., Turnbull and Wakeman 1991), the fifth approach uses discrete time

and a recombining tree. The difficulties arise because the number of paths to be

enumerated grows exponentially in the time steps of the binomial tree: for instance, in a

100-time step binomial tree, there could be as many as 2100 ~ 1030 separate paths to

tabulate, a computationally infeasible task. The difficulties are overcome by using a

bundling technique that eliminates the need to evaluate all paths separately. Neave (1997)

and Neave and Stein (1998) use this approach to value European options in third or fourth

degree polynomial time rather than exponential time. Their state description uses groups

of price paths and reduces enumeration without significantly sacrificing solution accuracy.

The present paper extends this approach.

American Asian options have recently begun to trade actively, but they are even

more difficult to price than their European counterparts. Hansen and Jorgensen (1998)

derive an analytical approximation formula for floating strike options. Since the values of

American Asian options satisfy the PDE with free boundary conditions, Hansen and

Jorgensen use a finite-difference method. Others using similar approaches include

Dewynne and Wilmott (1995a, b), Zvan, Forsyth and Vetzal (1998).

contrary view, see Bertsimas, Kogan and Lo (2000).

Ritchken, Sankarasubramanian, and Vijh (1993) suggest modeling the price

process as a binomial tree. Hull and White (1993) use a linear interpolation procedure in a

recombining binomial tree to compute upper bounds on the values of American Asian

options. Chalasani, et. al. (1998) extend Hull and White's approach by choosing the

average values of the arithmetic averages over the paths with the same geometric averages

as the interpolation points, and provide an upper bound and a lower bound on American

Asian option values.

The geometric averages of lognormal variates are still lognormal, and therefore, as

first proposed by Curran (1992, 1994), the literature often calculates the expected payoffs

by conditioning on geometric averages. Kemna and Vorst (1990) use geometric averages

as a control variate in the Monte Carlo simulation; Chalasani et. al. (1998) use them as the

linear interpolation points; Neave (1997), and Neave and Stein (1998), use geometric

averages as a conditioning variable to define the bundles with which they work.

Despite all the foregoing, the search for satisfactory solutions is not over. As

observed by Turnbull and Wakeman (1991), there is a practical need for a model that can

quickly and accurately price options.3 This paper offers a method of pricing both

European and American options that is both highly accurate (in some cases exact) and

very fast computationally.

We price both European and American options. To price European options, we

bound the exact solution above and below, then reduce the difference between the bounds.

The procedure involves working with sets of paths conditioned on a common time T asset

price and a common geometric mean. These sets may have more than a single arithmetic

average, but in general have many fewer distinct arithmetic averages than there are paths.

Moreover, the arithmetic averages within a set typically differ from each other by much

less than averages taken from different sets. Finally, it is easy to obtain the maximum and

minimum arithmetic averages in each such set. We offer two variants of the method. The

3 Quick, accurate determination of hedge ratios is also important.

first, our loose bound approach, is very fast and at least as accurate as others in the

literature (e.g., Hull and White 1993). The second, our tight bound approach, permits

reducing approximation error to any desired magnitude, even zero, by incurring additional

computation costs.4 We examine the details of the accuracy-computation time tradeoff

below.

We price American Asian options by using our European valuation methods

recursively. Again, our algorithms are faster and more accurate than existing methods.

Moreover they permit obtaining exact valuations if sufficient computational resources are

expended, again as detailed below.

As regards accuracy, we distinguish model from computational error. We assume

the underlying asset price process is represented exactly by the binomial5 and focus on

any error created by our approximate solution techniques. Our work shows that the

computations are very fast, the pricing is more nearly accurate than appears in the

literature, and with enough computation can be made exact if desired.

The rest of the paper is organized as follows. Section 2 introduces the basic model

and the path bundling technique. Section 3 introduces the Gaussian generating functions

and explains how they are used in organizing the calculations. Section 4 values European

Asian calls, Section 5 American Asian calls, and Section 6 concludes.

2. BASIC MODEL AND PATH BUNDLING

Asian options have many variants. We consider those whose payoffs are

determined by the arithmetic average of the underlying prices at a set of discrete time

4 The error-magnitude/computation-cost tradeoff can be controlled by the computer operator. 5 Even though our computations can be interpreted as studying the convergence of a discrete to a continuous solution, this is not our primary purpose. Our purpose is to understand and minimize the valuation error in a discrete model, under the assumption that the discrete process is an accurate representation of the problem. Thus we are not primarily concerned with such issues as how quickly the binomial solution converges to a continuous time solution.

points, possibly extending from prior the option’s creation to its maturity. In all of this

paper’s computations, the arithmetic averages are determined by sampling at every time

point over the life of the option.

2.1 Basic Model

We model the asset price process as a recombining binomial. Let St be the asset

price at time t, t=0, 1, ... , T. For simplicity, we assume S0=1. Then, following the CRR

model, we can write

= −−

p-1 ;p ;

tt t=1,2,…,T, (2.1)

teu ∆= σ , (2.2)

σ is the volatility, ∆t is the time interval, and p is the martingale probability with

−−=

uuuRp . (2.3)

We assure the existence of p by assuming that neither the prices of the underlying assets

nor the averages of asset prices admit arbitrage opportunities. The symbol R is the one

period risk free discount rate, assumed to be a constant throughout this paper.

Let { t(n): n = 0, 1, …, N; 0 = t(0) < t(1) < … < t(N) = T } be the set of time

points defining the prices used for calculating the arithmetic average. Denote the average

itself by a random variable AN :

nntN S

1 . (2.4)

If we choose a binomial tree so that T =kN, where k is a positive integer, the prices for

averaging are sampled every k steps. For simplicity, we choose k=1 in this paper, i.e., the

length of a time step is equal to the length of the process time interval. Thus, we select T

= N, t(n) = n∆t; n = 0, 1, … , N, and

011 . (2.4′)

Since the number of distinct averages attained by a process increases as the sampling

interval decreases, our choice of T = N presents us with the greatest computational

challenge in the present setting. Our methods can readily be amended to situations where

T < N, but for brevity we do not present such models here.

Let prob{path} denote the martingale probability of the occurrence of a path.

Then, using the risk neutral pricing technique, the value of the arithmetic average call at

time zero, Ca, is

∑ +−=all paths

pathNa K)(Aprob{path},K)(T;SC 0 (2.5)

where K is the fixed strike price. To evaluate Asian calls using Equation (2.5), we need to

calculate the expectation, under the martingale, of the call payoffs. To overcome

computational difficulties, we group the payoffs according to a path-bundling technique.

Our technique is designed to keep approximation error to a minimum.

2.2 Path Bundling

We define bundles as sets of paths with a common ending price. We also define

sub-bundles – bundle subsets whose paths also have a common geometric average. We

later show these definitions arise naturally from properties of the process’ Gaussian

generating function.

Let Jt , t=0, 1, ... , T, be a random variable that represents the total number of up

moves of the underlying asset price up to time t. Then the total number of down moves is

t-Jt , and using (2.1), we can write the asset price at any time t ≤ T as

tuS −= 2 , (2.6)

2.2.1 Bundles

In a T-step binomial tree, let B(T, j) denote the sets of paths of length T and having

j up-moves, i.e., j = JT. We call B( T, j ), j = 0, 1, … , T a bundle of paths. In any given

bundle, all paths attain the same time T asset price: ST = u 2j-T , j = 0, 1, … , T. Obviously,

there are T+1 bundles in the tree, and all paths in B(T, j) have the identical probability of

occurrence, pj(1-p)T-j, j = 0, 1, …, T.

Let Ca(T,j) denote the expected payoffs of an arithmetic average call in B( T, j ).

−− −

jTjTa jTCpp

),()1()( . (2.7)

2.2.2 Sub-bundles

From equation (2.6), the geometric average of a path can be expressed as

Define the absolute geometric average index v of a path by:

. (2.8)

Clearly, since

2 +−

T uuG , (2.9)

we can use the index v to represent a geometric average.

Example 1: For a path { u0, u-1 u0 u1}, we have J0=0, J1=0, J2 =1, J3= j = 2. Therefore,

the path is in B(3, 2), v=3, and

.0)1332()

3 uuuG == +×−

Proposition 1. Given a bundle B(T, j), then:

a) the set of geometric average indexes is6

{ v-, v-+1,…, v+},

6 Although v+, v- and some other notations in the following context specify a given sub-bundle, the subscripts t, j, v are suppressed to eliminate notational clutter.

v- = j(j+1)/2 (2.10)

v+ = v- +j(T-j). (2.11)

b) the number of geometric averages in B(T,j) is equal to j(T-j)+1.

A proof of this proposition is given in Ye (1999) • .

Example 2: For the bundle B(3, 2) in Example 1, from equation (2.10)-(2.11), we have

v- = 3 and v+=5. Therefore, we have three geometric averages in B(3, 2): GT=u0, u1/2, and

u1, corresponding to v=3, 4, and 5, respectively.

Let B( T, j, v) denote the set of paths in B(T, j) that have the same geometric

average index v. We call the subsets of paths B(T, j, v), j = 0, 1, … ,T; v = v-, v-+1, … , v+

sub-bundles. By definition, all paths in a sub-bundle B(T, j ,v) have the same terminal

asset price ST=u2 j-T , and an identical geometric average, indexed by v.

Let vjTg ,, denote the number of paths in a sub-bundle B(T, j, v) and let Ca(T, j, v)

be the expected payoff of an Asian call in B(T, j, v). Then the price of a European Asian

call is

),,()1()( ,,0

vjTCgppRTC a

jTjTa ∑∑

−− −= (2.12)

The next section shows that vjTg ,, is an easily calculated Gaussian binomial coefficient.

Hence, we can readily calculate the expected payoff of an Asian call in each sub-bundle

B(T, j, v), i.e. Ca(T, j, v).

2.3 Three Types of Sub-bundles

Our method distinguishes among three types of sub-bundles: those containing only

paths whose arithmetic averages exceed the option strike price (positive sub-bundles),

those containing only paths whose averages never exceed the strike price (zero sub-

bundles), and those containing paths for which some averages lie above, others below the

strike price (divided sub-bundles). All types of sub-bundles need to be identified, but the

contributions of all paths in the positive and zero sub-bundles require only minimal

further calculation. Divided sub-bundles can readily be valued approximately, but require

further investigation if the original approximation error is to be reduced. However there

are relatively few divided sub-bundles and therefore, as we show below, investigating

them does not necessarily complicate the valuation process significantly.

The contribution of a positive sub-bundle to the option’s value can be determined by

calculating the mean of the sub-bundle averages. The contribution of a zero sub-bundle is

zero. The contribution of a divided sub-bundle can be bounded above and below by

determining the maximum and minimum averages in the sub-bundle. These distinctions

form the essence of our valuation methods and reduce calculations to polynomial

magnitudes in T. The rest of this section describes sub-bundles’ properties more formally.

For a sub-bundle B(T, j, v), let m1, m2, …, mN be its N distinct arithmetic averages,

m1 < m2 < … < mN ,

and let xn be the number of paths in B(T,j,v) with an arithmetic average of mn. Let xn be

the frequency of mn ; n = 1, 2, … , N. Then the expected sub-bundle payoff is:

+−=N

vjTa Kmx

)(1),,( . (2.13)

Type 1: Positive Sub-bundles. B(T, j, v) is called a positive sub-bundle if all distinct

arithmetic averages in B(T, j, v) are greater than, or equal to the strike price K, i.e.

K ≤ m1 .

If B(T, j, v) is a positive sub-bundle, then the expected value of the payoffs in the sub-

bundle is:

)(1),,(1,,

vjTa Kmx

KM vjT −= ,, (2.14)

vjTvjT mx

1 , (2.15)

i.e., MT,j,v is the sub-bundle mean of the arithmetic averages. Equation (2.14) shows that

to compute the expected payoff in a positive sub-bundle, we need only compute the sub-

bundle mean of the arithmetic averages.

Type 2: Zero Sub-bundles. A sub-bundle B(T,j,v) is called a zero sub-bundle if none of

the distinct arithmetic averages in B(T,j,v) exceeds the strike price K, i.e.,

mN ≤ K .

Obviously, the expected payoff in a zero sub-bundle is

Ca(T,j,v) = 0. (2.16)

Type 3: Divided Sub-bundles. B(T, j, v) is called a divided sub-bundle if some of the

distinct arithmetic averages are greater than K while the others are less than K, i.e.,

m1 < K < mN .

Thus, if B(T, j, v) is a divided sub-bundle, there is a number n': 1 < n' < N, such that

vjTa Kmx

)(1),,( . (2.17)

Sub-bundle types are identified by exploiting properties of the Gaussian

generating function discussed in the next section. The following example illustrates the

importance of each type.

Example 3: Consider an Asian call with parameters u=1.15, p=0.51, T=12,K=1. For

these data σ = 0.40. In total, this problem has 4096 paths that can be grouped into 299

sub-bundles. There are 155 positive sub-bundles, 141 zero sub-bundles, and only 3

divided sub-bundles.

3. GAUSSIAN GENERATING FUNCTION AND SUB-BUNDLE MEANS

The Gaussian generating function discussed in this section is used to study

problem structure. The generating function and its coefficients can be used to describe the

number of sub-bundles in a bundle, the number of paths in a sub-bundle, the sub-bundle

geometric average, the maximal and minimal arithmetic averages in each sub-bundle, the

sub-bundle mean of arithmetic averages, recursion relations between sub-bundles, and

recursion relations between sub-bundle arithmetic averages.

3.1 Gaussian Generating Function, Gaussian Binomial Coefficients

We call f(x) the generating function of the sequence { a0, a1, a2, . . . an , ... } if

f(x)=a0 +a1 x + a2 x2 + . . . + an xn + .... (3.1)

The Gaussian Generating Function is defined as

nr<0 1

≤−

−∏r

n,r qq(q)=g (3.2)

gn,0( q )=1.

Berman and Fryer (1972) show that gn, r (q) can be expressed in the form

gn, r (q) =A0 + A1 q + A2 q2 + ... + Ar(n-r) qr(n-r) , (3.3)

where Ai, i=0,1,...,r(n-r), are positive integers called Gaussian binomial coefficients. The

coefficients are denoted as

= ( A0, A1, A2, ... Ar(n-r) ). (3.4)

Proposition 2 establishes that the Gaussian binomial coefficients describe the

number of paths in a sub-bundle.

Proposition 2 . Let j,T be positive integers, j < T. Let vjTg ,, be the number of paths in the

sub-bundle B(T, j, v). By equation (3.4),

T = ( A0, A1, A2, ... Aj(T-j) ) . (3.5)

vjTg ,, = *vA (3.6)

where v* = v-j(T-j).

For the proof of Proposition 2, see Ye(1999).•

Example 4: For the bundle B(3, 2) as in example 2 with T = 3 and j = 2, the Gaussian

generating function is:

)q)(q()q)(q()q(g , 11

23 −−−−=

= q2+q+1.

That is, the Gaussian Binomial coefficients are

( )1,1,123 =

Thus, since v*=v-2, the numbers of paths in the three sub-bundles B(3,2,3), B(3,2,4) and

B(3,2,5) are g3,2,3 = g3,2,4 = g3,2,5 =1, respectively.

3.2 Convolution of Gaussian Binomial Coefficients

The contribution of a positive sub-bundle to the option payoff can be found by

using the mean value of the arithmetic averages in the sub-bundle; it is not necessary to

use the distribution of distinct arithmetic averages. In order to determine the averages’

mean value, it is necessary to describe the frequencies with which sub-bundle paths attain

given asset prices at any point during the sampling period. An array of frequencies can be

determined, in essence, by finding the frequency with which paths attain a given asset

price at a given point in time, then generalizing that information. This section describes

the theoretical underpinnings to calculating this frequency distribution. An example of a

frequency distribution is given in Table 1.

Let a ai in= ={ } 0 and b bi i

m= ={ } 0 be two sets of Gaussian binomial coefficients. Let

c a bi kk i

−∑0

i= 0, 1, ... , n+m. (3.7)

i0 = max { i - m, 0 } , (3.8)

i1 = min { i, n } , (3.9)

c ci in m= =

+{ } 0 . (3.10)

Then we call c the convolution of a and b, denoted by

.bac ⊗= (3.11)

The following proposition relates the convolutions of Gaussian binomial coefficients to

the number of paths in a sub-bundle that pass through a particular middle node. These

quantities define intermediate steps in generating the data of Table 1.

Proposition 3. Let stvjTn ,

,, be the number of paths in a sub-bundle B(T, j, v) which pass

through a node defined by t-s up moves and s down moves. Let

,})({ 0,

−−

== sjtT

sti (3.12)

I=s(t-s)+(j-s)(T-t-j+s). (3.13)

)(,)(*

,,, jcn st

stjTsvst

vjT +−−−= . (3.14)

Thus, the number of paths in a sub-bundle B(T, j, v) which pass through a node is equal to

a coefficient of the convolution of two Gaussian generating functions.

For a proof of Proposition 3, see Ye(1999).•

3.3 Frequency Distributions

Proposition 3 shows theoretically how we can generate frequency distributions of

the prices attained by paths in a bundle. However, in practice it is simplest to calculate the

frequency distributions recursively. As already mentioned the frequency distributions are

used to calculate the sub-bundle means of arithmetic averages, the principal statistic for

evaluating the contributions of positive and zero sub-bundles. The same frequency

distributions are also used to establish recursion relations between subbundles, and

between sub-bundle arithmetic averages.

Definition. For a sub-bundle B(T, j, v) and any integer i, 0 ≤ i ≤ T, the total number of

paths in the sub-bundle passing through the nodes at which the asset price is ui-T+j is

called the sub-bundle frequency of index i, denoted by f T j vi, , , i=0,1,...,T.

It is clear from the definition that

∑=−

ivjT nf

,,,,, . i=j-T, j-T+1, …, T. (3.15)

The frequency distribution of distinct averages in B(T, j, v) is defined as a T+1

dimensional vector:

( )TvjTvjTvjT ffF ,,

0,,,, ,...,= . (3.16)

The frequency distributions are symmetric and we exploit this property to write the

frequency distribution in a bundle B(T, j) as a NT,j × (T+1) matrix

( )]2/)([,...,,,, jTjvvvvjTjT FF

−+= −−= (3.17)

NT, j=[j(T-j)/2]+1. (3.18)

As well, we can write the frequency distribution in a T-period problem as a NT × (T+1)

matrix

( )]2/[,...,1,0, TjjTT FF

=≡ (3.19)

+−=]2/[

).1]2/)(([T

jT jTjN (3.20)

In calculation, we fix T and generate a frequency distribution matrix FT by forward

induction. This matrix is used as a database. From Equation (3.20), the size of FT is

proportional to T3. Since the matrix can be used repeatedly once it has been generated, it

is thought a part of ‘hardware’, and its computing time is not an important concern. As

already mentioned, the frequency distribution matrix for the 6-period binomial tree is

given in Table 1.

Table 1 about here

3.4 Sub-bundle Means of Arithmetic Averages

Given the fT, j, v we can compute the sub-bundle mean of arithmetic averages by

vjT ufgT

,, )1(. (3.21)

Proof. Let ] [~,, •vjTE be the martingale expectation operator conditioning on sub-bundle

B(T, j, v), and let AT refer to the set of arithmetic averages. By definition,

∑∑

tsstvjT

TvjTvjT

(3.22) .)1(1

Changing the order of the summation in Equation (3.22) and using Equation (3.16) gives

Equation (3.21).•

4. VALUATION OF EUROPEAN ASIAN CALLS

Given FT we can compute the expected contribution for any given positive sub-

bundle exactly, and we also know that any zero sub-bundles make no contribution. Thus

the only source of approximation error is associated with divided sub-bundles.

We next present our two approximation methods. Each of these approaches

calculates upper and lower bounds to the contribution of a divided sub-bundle. The

combination of upper and lower bound is called a set of bounds, and the difference

between the upper and lower bound is called a gap. The gap measures the maximum

approximation error created, on a sub-bundle basis, by the approximation method.

Choosing between the loose bound and the tight bound approach involves

tradeoffs. Each of the two approaches uses the maximum and minimum arithmetic

averages7 in a sub-bundle, but in different ways. The tight bound approach can make the

gaps progressively tighter, but the loose bound approach is computationally cheaper. Our

loose upper bound is relatively less accurate than its tight counterpart; our loose lower

bound is about as good its tighter counterpart. In valuing European calls it is

computationally cheaper to employ a loose lower bound irrespective of whether we use a

tight or a loose upper bound. On the other hand, we must use tight upper and lower

bounds when valuing American calls because our backward recursion calculations employ

data for which approximation errors are introduced when the data are obtained by forward

induction.

7 The maximum and minimum averages are easy to determine with the aid of the generating function.

4.1 Loose Bound Approach

This section shows how to derive loose bounds. The loose bound approach uses

the first and second moments of the distribution of arithmetic averages in a sub-bundle,

while the tight bound approach uses the entire distribution.

4.1.1 Loose lower bounds

Consider a divided sub-bundle B(T, j, v). Rewrite Equation (2.13),

+−=N

vjTa Kmx

)(1),,(

m1< K< mN ,

Then it is easy to verify that

−≥ ∑ Kmx

1),,( (4.1)

( )+−≥ KMvjTC vjTa ,,),,( . (4.2)

Thus, inequality (4.2) provides a lower bound of the expected payoff in a divided sub-

bundle B(T, j, v), denoted by

( )+−≡ KMKvjTLLB vjT ,,);,,( . (4.3)

4.1.2 Loose upper bound

Let m be a distinct arithmetic average in a divided sub-bundle B(T,j,v). Then

011 . (4.4)

Define the squared average of m as

11 . (4.5)

Define f(m;K) by

)()2();( 1211

2 mmbmmmmaKmf −++−= , for m1<m<mN , (4.6)

)2/(},min{ 211

21 mmmmmKKma NNN +−−−= , (4.7)

b=(m N –K-a)/(mN –m1 ) . (4.8)

Proposition 4. Given any strike price K, then )0,max();( KmKmf −≥ , for any m in a

divided sub-bundle.

A proof is provided in Ye(1999).•

For a divided sub-bundle B(T, j, v), we have

)];([),,( ,, KmfEvjTC vjTa ≤ . (4.9)

Denote the right side of (4.9) by LUB(T,j,v;K). By definition,

KvjTLUB1,,

);(1);,,( . (4.10)

In order to compute LUB(T,j,v;K), let

][ 2,,,, mEV vjTvjT ≡ , (4.11)

where VT,j,v is called the sub-bundle mean of squared arithmetic averages. By definition,

ttvjTvjT SE

2,,,, )(

11 (4.12)

)()2();,,( 1,,21,,1,, mMbmMmVaKvjTLUB vjTvjTvjT −++−= . (4.13)

Using the same ideas as those used to derive MT,j,v , V T,j,v can be calculated using the

frequency distributions based on the following formula:

TjiivjT

vjTvjT uf

,,,, )1(

1 . (4.14)

4.2 Tight Bound Approach

While the loose bound approach is based on the first two moments of the sub-

bundle distribution of distinct averages, the tight bound approach is based on the entire

distribution. Thus the tight bound approach provides more nearly accurate valuation, but

at increased computation cost. This section implements the tight bound approach by

developing a forward induction procedure to generate the distinct arithmetic averages in

sub-bundle.

4.2.1 Distributions of arithmetic averages in sub-bundles

Let at be the arithmetic average of prices along a path up to time t, with a0 =1. Let

ht ≡ (t+1) at , (4.15)

i.e., ht is the path sum of prices. Let v* ≡ v-v-, and H(t ,s, v*) be the set of all ht 's in a sub-

bundle B(t, s, v), say,

H(t ,s, v*)={ ht1, ht2, …, htn}. (4.16)

where t=0,1,…,T; s=0,1,…,t; v* =0,1,…,s(t-s).

Let H+(t, s, v*) be the set whose elements are equal to one plus elements in H(t, s, v*)

multiplied by u, and H-(t, s, v*) be the set whose elements are equal to one plus elements

in H(t, s, v*) divided by u. That is,

H+(t, s, v*)={ 1+uht1, 1+uht2 ,…,1+uhtn } (4.17)

H-(t, s, v*)={ 1+u-1ht1, 1+u-1ht2, …,1+u-1htn }. (4.18)

Then, for t > 1, it is easy to verify that

H(t,s, v*) = H+(t-1,s-1,s+ v*-t ) ∪ H-(t-1,s, v*). (4.19)

Note that H(0, 0, 0*) = {1}. Then the above equations imply a recursive mechanism on

sub-bundles. According to this mechanism, we develop a forward induction procedure to

generate the distinct arithmetic averages and their frequencies in sub-bundles.

To illustrate the importance of finding exact distributions, Figure 1 plots the

cumulative probability distribution of distinct arithmetic averages in B(20, 10, 105). For

comparison, we also plot the cumulative probability function of the lognormal distribution

with the same mean and variance as those of the exact distribution. In B(20, 10, 105),

there are 5448 paths, but only 52 distinct arithmetic averages. It is known that the

cumulative distribution function is not a discrete approximation to the lognormal, and

indeed on the basis of Figure 1 the exact distribution seems difficult to fit into any

familiar class.

Figure 1 about here

4.2.2 Exact valuation of European Asian calls

Having determined the exact distribution of distinct averages for each divided sub-

bundle, it is straightforward to compute expected payoffs using Equation (2.14). The

exact distributions permit calculating the exact values of European Asian calls. However,

when a large number of time steps is used, the number of distinct arithmetic averages in a

sub-bundle can become very large, and calculating their distribution can take considerable

computing time. Thus an alternative approach is desirable.

4.2.3 Tight bounds on values

To speed up the computations, we restrict the number of the distinct arithmetic

averages that we tabulate. We require the number to be no greater than a preset parameter

L. If the number of the distinct arithmetic averages in a sub-bundle is greater than L, we

approximate low-frequency occurrences of distinct averages with averages by increasing

the frequencies with which more commonly observed averages occur. This method

introduces relatively small approximation errors. Moreover, if we combine low-frequency

averages with larger (smaller) observed averages, the approximate expected payoff in that

sub-bundle will be an upper (lower) bound on its expected payoff. Note finally that if L is

sufficiently large the method will give exact valuations.8

Using the approximate distribution, it is straightforward to calculate tight upper

and lower bounds on the expected payoff in the sub-bundle. The difference between the

tight upper and lower bound is a decreasing function of L, and by construction the

valuation will be exact if L is sufficiently large. Here the tight upper and lower bounds are

used to value European options.

As an example, we compute the tight upper bounds and lower bounds in

B(20,10,105) with changing strike prices and L = 100. The respective curves are plotted in

Figure 2. For comparison purposes, the exact values and the loose bounds are also plotted

in the same figure. Since the tight and loose bounds are not calculated in the same way,

there is no reason to suppose that one set would dominate the other for all possible

exercise prices and all possible values of L. Nevertheless, Figures 2(a) and 2(b) suggest

that the tight bounds usually outperform the previously calculated loose bounds,

especially the loose upper bound. Moreover, if L is sufficiently large, then by construction

the tight bounds cannot underperform the loose bounds.

Figure 2 about here

4.3 Computation

8 As a practical matter L may only have to be a few hundred to give exact valuations.

The value of an Asian call is a weighted sum of the expected payoffs for the

positive and the divided sub-bundles. We showed that we could easily calculate the exact

values of the expected payoffs for positive sub-bundles, and the values are known to be

zero for the zero sub-bundles. For divided sub-bundles, we can calculate the two sets of

bounds discussed above. The calculations are programmed in C++, and implemented on a

Pentium 200 computer.

For purposes of illustration, we consider a ‘plain vanilla’ European Asian call

whose parameters are given in Table 2. We calculate the loose bounds of the call prices

for varied numbers of time steps up to 100, and the tight bounds up to 48 with L=100. The

results are reported in Table 3.9

Table 2 and Table 3 about here

The left half of Table 3 shows gaps of 0.0008 or less. The gap grows from the

interactoon of two opposing effects created by an increasing number of time steps. First,

the difference between the upper and lower bound within each sub-bundle becomes

smaller as T increases. However, the number of the divided sub-bundles increases, and

thus more approximation errors are introduced. When the second effect dominates the

first, the overall gap grows. If we use the average of the upper bound and lower bound as

an approximate value of a call, then a gap of 0.0008 implies a pricing error of about

0.0004. The table also reports the computing times needed to obtain bounds as T varies.

The computations are very fast. For instance, it only takes 10 seconds to obtain loose

bounds when 60 time steps are used and about 3 minutes when 100 time steps are used.

9 In our calculations the number of observations defining the arithmetic average increases with T. We prefer this method of comparison, rather than one depending on a fixed number of observation points and a declining length of time interval, because it gives a more stringent test of the computing times required by our method. In other words, our estimates of computational difficulty with increasing T are never less than the time needed to value an option whose averages depend only on a fixed number of observations.

The right half of Table 3 shows the results for tight bounds. It shows that the

accuracy attained by using the tight bound approach is typically in fifth significant digit,

while that attained by using the loose bounds is typically in third or fourth significant

digit. More importantly, by adjusting the parameter L, the tight bound approach can attain

any desired degree of accuracy, given sufficient computer resources. For example, in the

cases where the number of time steps is not greater than 18, because the largest number of

the distinct arithmetic averages in the sub-bundles is less than L = 100, we obtain the

exact solutions.

The tight bound computations take longer than their loose bound counterparts. For

instance, when 48 time steps are used, it takes more than 3 minutes of computing time for

the tight bounds, but only 3 seconds for the loose bounds. Depending on the demand for

accuracy, and the availability of computing resources, we can choose between those two

approaches.

We see from the table that as the number of time steps increases, both the lower

and upper bounds increase monotonically. This phenomenon has been explained in Hull

and White (1993, hereafter HW). Since in our tabulations an increasing number of time

steps implies an equal increase in the number of sampling points determining the average,

the call value is influenced by both effects. Table 3 suggests that the lower bounds

converge quite quickly to a constant as the number of time steps increases, but the upper

bounds keep increasing. Further computational experience is needed to determine which

bound tends to be closer to the exact value of the call.

To compare our methods with those of HW we compute both tight and loose

upper bounds as well as loose lower bounds for European Asian calls with various strike

prices and maturities. The parameters for the calls are as the following: the initial stock

price is $50, the risk-free interest rate is 10% per year, and the stock price volatility is

30% per year. The 40-time-steps binomial tree is used, and L is 100. The strike prices are

equal to 40, 45, 50, 55, and 60, and the years to maturity are equal to 0.5, 1, 1.5, and 2.

The results are reported in Table 4.

Table 4 about here

First, Table 4 shows that our tight upper bounds outperform the HW upper bounds

on almost all occasions. Second, we compare our loose upper bounds with HW upper

bounds. It seems for short maturity, say, 6 months to one year, our loose upper bounds are

tighter than the HW's. For longer maturities, say, 1.5 to 2 years, HW's upper bounds are

tighter than our loose upper bounds. Nevertheless, the differences between those two

bounds are in the fourth or fifth significant digit. Third, while HW's method can only

calculate upper bounds, we also provide a lower bound and hence a controlled

approximation error. Moreover, the gaps between our upper bounds and lower bounds are

small enough to allow us to control approximation error relatively tightly. Since HW do

not report computing times, we cannot compare the speed of their method with ours.

4.4 Accuracy vs. Volatility

As mentioned in our introduction, the methods of Levy (1992) and Turnbull and

Wakeman (1991) can obtain satisfactory accuracy only when the volatility is small, say

less than 0.40. To study the validity of our approach for high volatility, we calculate upper

bounds and lower bounds when volatility is equal to 40% and 80%. The call's parameters

are those presented in Table 2, and L=100. The numbers of time steps in the tree are

increased from 6 up to 48. The results are reported in Table 5. When the number of time

steps is not greater than 18, we obtain exact solutions for all values of volatility studied.

When more than 18 time steps are used in calculation, an increase in volatility results an

increase in the gap between the upper bounds and the lower bounds. For example, when

48 time steps are used, the gaps increase from 0.0001 to 0.0009 when the volatility

increases from 40% to 80%. However, even in this latter case the errors are still confined

to the fourth significant digit.

Table 5 about here

The importance of volatility to call pricing is well known. To demonstrate the

sensitivity in the present case, Figure 3 presents a graph of loose lower and upper bounds

when T ranges from 6 up to 100, and volatilities range from 40% to 42.5% in increments

of 0.5%. Figure 3 shows that when a 0.5% increase of volatility occurs, the lower bound

shifts up to a level higher than the upper bound with the original volatility. For example,

the lower bound curve with volatility 0.405 is of higher magnitude than that of the upper

bound curve with volatility 0.400. This implies that if the estimation error of volatility is

greater than 0.5%, there is little point in expending effort to reduce the gaps below the

magnitudes reported here.

Figure 3 about here

For instance, assuming the true value of the volatility is 0.4000, when 100 time

steps are used, we obtain an upper bound equal to 0.138405. If the estimated value of

volatility is 0.405, i.e. 0.005 higher than the true value, then the lower bound calculated is

0.138818. This shows that the pricing error introduced by incorrect volatility estimation

exceeds the approximation error of our method. Since errors in volatility estimation are

commonly much greater than 0.005, an approximate solution more accurate than those we

develop does not make sense in practice, whatever its theoretical appeal.

5. VALUATION OF AMERICAN ASIAN CALLS

Having obtained the exact distributions of arithmetic averages, it is

straightforward to price American Asian calls by backward induction on sub-bundles.

Since the previously discussed method of calculating bounds is recursive, it can readily be

adapted to value American calls as shown next.

5.1 Exact Valuation

The problem can be formulated as follows. Let C(t, s, v*, h) be the value of an

American Asian call when the path is in a sub-bundle B(t, s, v*) and the path sum up to t

is equal to h. Using the notation in section 4.2.1, we have h ∈ H(t, s, v*).

At time T, by definition,

C(T, s, v*, h) = (h/(t+1) - K)+. (5.1)

For any t<T, then at t+1, the path will be in B(t+1, s+1, v*) when an up move occurs with

probability p, or in B(t+1 ,s, v* +s ) when a down move occurs with probability q.

If the distributions of arithmetic averages are exact, there exist path sums h1 ∈

H(t+1 ,s+1, v*) and h2 ∈ H(t+1 ,s, v* +s) such that

h1=h+uSt=h+u2s-t+1, (5.2)

h2=h+u-1 St=h+u2s-t-1. (5.3)

Thus, we have the exact value:

C(t, s, v*, h)=max{(h/(t+1)-K)+,(pC(t+1, s+1, v*, h1)+qC(t+1, s, v*+s, h2))/R}. (5.4)

Then, by carrying out the backward induction on sub-bundles, we can get the exact

American Asian call price.

5.2 Approximate Valuation

If the distributions of arithmetic averages are approximate, there exist integers i and j,

with hi ∈ H(t+1 ,s+1, v*) and hj ∈ H(t+1 ,s, v* +s) , such that

hi ≥ h+uSt > hi+1, (5.5a)

hj ≥ h+u-1 St > hj+1. (5.5b)

h1 = hi , (5.6a)

h2 = hj . (5.6b)

Then, using Equation (5.4), we can obtain an upper bound of the American call price.

Similarly, we can obtain a lower bound of the American call price. Note that the gap can

be reduced as desired, even to zero, by increasing L.

Table 6 reports computing accuracy and speed. The input parameters are those in

Table 2. The time steps are from 6 up to 30, and L=100 and 200. If more time steps are

used, a greater L is needed to obtain the same level of accuracy. For time steps not greater

than 18, with L = 100, we get exact values with fourth or fifth significant digit accuracy.

For time steps equal to 24 or 30, with L=200, we get fourth significant digit accuracy. The

computing speed is fast. For example, when 24 time steps are used and L = 100, we can

value an American call in one minute with accuracy to four significant digits.

Table 6 about here

We also compare the accuracy of our method with those of HW and of Chalasani,

et al. (1998). Table 7 shows that our upper bound calculated with L = 100 is similar to

HW with h = 0.005; and our upper bounds calculated with L ≥ 150 are tighter than HW

with h = 0.003. Next, we get both the upper bounds and lower bounds, while HW only get

upper bounds. When L = 300 is used, we get the exact value.

Table 7 about here

Since HW does not indicate computing times, we cannot offer exact comparisons.

Nevertheless, our method is very fast and can calculate the exact value in less than one

minute. In comparison to results presented in Chalasani et al., our upper bound with L ≥

150 and lower bound with L ≥ 200 are tighter. In other words, we obtain more accurate

solution in 15 seconds computing times using a PC. Another advantage, unique to our

method, is that we can control approximation error, getting any desired degree of accuracy

by adjusting L.

6. CONCLUSIONS

This paper provides upper and lower bounds on the values of both European and

American arithmetic average fixed strike calls. For European Asian calls, our loose

bounds are better than those in the literature and as quick or quicker to calculate. Our tight

bounds, calculated using forward induction on sub-bundles, take longer to calculate but

can obtain as close an approximation to the true value as desired. Our values for

American Asian calls are more nearly accurate than those of competing methods and are

also obtained quickly.

Acknowledgements

We wish to thank Phelim Boyle, Frank Milne, Ieuan Morgan, and Wayne Yu for

helpful comments and constructive suggestions on an earlier draft of the paper. We

would also like to thank the participants at the Computational Finance’99 conference,

New York, the APFA’99 conference, Melbourne, the 6th MFS conference, Toronto, and

the NFA’99 conference, Calgary, for helpful comments.

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TABLE 1. Frequency distribution matrix for 6-time-steps binomial lattice

i V 0 1 2 3 4 5 6

j=0 0 1 1 1 1 1 1 1

j=1 1 2 3

3 4 5 6 7

1 0 0 0 0

2 2 2 1 1

2 3 6 5 6

1 1 4 5 7

1 1 2 3 6

0 0 0 0 1

0 0 0 0 0

6 7 8 9

1 0 0 0 0

2 2 2 2 0

2 3 6 7 6

2 2 6 10 12

0 0 0 2 3

0 0 0 0 0

TABLE 2 Process parameters

Year to Maturity 1.5 Spot Asset Price 1 Strike Price 1 Volatility (p.a.) 40% Interest Rate (p.a.) 10% Number of Time steps 100

TABLE 3. European Asian call option prices and computing times

Loose Bound Tight Bound

T Lower Bound

Upper Bound

Lower Bound

Upper Bound

6 0.136519 0.136519 0.000000 0 0.136519 0.136519 0.000000 0

12 0.137026 0.137048 0.000022 0 0.137036 0.137036 0.000000 0

18 0.137213 0.137265 0.000052 0 0.137231 0.137231 0.000000 1

24 0.137322 0.137422 0.000100 0 0.137339 0.137355 0.000016 4

30 0.137391 0.137543 0.000152 1 0.137399 0.137442 0.000043 14

36 0.137439 0.137645 0.000206 1 0.137441 0.137502 0.000061 31

42 0.137474 0.137735 0.000261 2 0.137476 0.137560 0.000084 65

48 0.137501 0.137816 0.000315 3 0.137502 0.137605 0.000103 118

54 0.137522 0.137894 0.000372 6

60 0.137538 0.137965 0.000427 10

66 0.137552 0.138035 0.000473 18

72 0.137564 0.138104 0.000540 28

78 0.137574 0.138170 0.000596 44

84 0.137582 0.138235 0.000653 68

90 0.137590 0.138300 0.000710 95

100 0.137600 0.138405 0.000805 170

Notes: 1) Initial stock price $1.00, strike price $1.00, risk-free interest rate 0.10 per annum, stock price volatility 0.40 annually, time to maturity 1.5 years. CT is computation time in seconds. In calculations of the tight bounds, L = 100. 2) Valuation bounds are calculated using a C++ program in a Pentium 200 computer. The gaps are the differences between the upper and lower bounds.

TABLE 4. Comparisons of accuracy: European Asian calls

Years to Maturity Strike Price 40 45 50 55 60

H&W upper bound 10.755 6.363 3.012 1.108 0.317

loose upper bound 10.75480 6.36145 3.00839 1.10557 0.31647

tight upper bound 10.75455 6.36078 3.00762 1.10456 0.31552 0.5

lower bound 10.75439 6.36044 3.00733 1.10413 0.31499

H&W upper bound 11.545 7.616 4.522 2.420 1.176

tight upper bound 11.54415 7.61398 4.51984 2.41750 1.17471

1.0 lower bound 11.54351 7.61315 4.51900 2.41632 1.17309

H&W upper bound 12.285 8.670 5.743 3.585 2.124

tight upper bound 12.28435 8.66875 5.74141 3.58436 2.12340 1.5

lower bound 12.28320 8.66737 5.73989 3.58221 2.12063

H&W upper bound 12.953 9.582 6.792 4.633 3.057

tight upper bound 12.95358 9.58156 6.79128 4.63251 3.05756

lower bound 12.95186 9.57961 6.78905 4.62958 3.05338 Notes: Initial stock price $50, risk-free interest rate 10% per annum, stock price volatility 0.30 per year. The 40-time-steps binomial tree model is used in computations. In calculations of the tight bounds, the greatest number of distinct arithmetic average intervals, L, is equal to 100. H&W upper bounds are quoted from Hull and White(1993).

TABLE 5 Accuracy and volatility for tight bounds Volatility 0.4 Volatility 0.8

Tight Lower Bound

Tight Upper Bound Gap Tight

Lower Bound Tight

Upper Bound Gap

6 0.136519 0.136519 0.000000 0.231945 0.231945 0.000000

12 0.137036 0.137036 0.000000 0.232966 0.232966 0.000000

18 0.137231 0.137231 0.000000 0.233474 0.233474 0.000000

24 0.137339 0.137355 0.000016 0.233682 0.233829 0.000147

30 0.137399 0.137442 0.000043 0.233796 0.234085 0.000289

36 0.137441 0.137502 0.000061 0.233822 0.234327 0.000510

42 0.137476 0.137560 0.000084 0.233901 0.234570 0.000669

48 0.137502 0.137605 0.000103 0.233963 0.234853 0.000890

Notes: Initial stock price $1, strike price $1, risk-free interest rate 10% per year, time to maturity 1.5 years, and L = 100.

TABLE 6. American Asian call option prices and computing times

L=100 L=200 T Lower

bound Upper bound Gap C.T.

(sec.) Lower bound

Upper bound Gap C.T.

(sec.) 6 0.141268 0.141268 0. 0

12 0.146479 0.146479 0. 1

18 0.148574 0.148633 0.000059 5

24 0.149445 0.150294 0.000849 60 0.149670 0.149981 0.000311 165

30 0.150001 0.151745 0.001744 160 0.150395 0.151168 0.000773 980

Notes: Initial stock price $1.00, strike price $1.00, risk-free interest rate 0.10 per annum, stock price volatility 0.40 annually, time to maturity 1.5 years. C.T. is the computing times in Pentium 200.

TABLE 7. Comparisons of accuracy: American Asian call

Tight Bounds

L 100 150 200 250 300

Upper bound 4.81552 4.81343 4.81273 4.81258 4.81248

Lower bound 4.80988 4.81161 4.81225 4.81240 4.81248

Gap 0.00564 0.00182 0.00048 0.00018 0.00000

C.T.(sec.) 9 12 15 20 47

Chalasani, et al. Bounds

Upper bound 4.814

Lower bound 4.812

Gap 0.002

H&W Upper Bounds

H 0.1 0.05 0.01 0.005 0.003

Upper bound 5.197 4.971 4.823 4.815 4.814

Notes: The initial stock price is $50, the strike price is $50, the risk-free interest rate is 0.10 per annum, σ = 0.30, T = 20. H&W upper bounds are quoted from Hull and White(1993). Chalasani, et al. bounds are quoted from Chalasani, et al. (1998). C.T. is the computing times in Pentium 200.

FIGURE 1 Cumulative probabilities of the exact distinct arithmetic averages in B(20,10,105) and the lognormal variate. The cumulated probabilities of the exact distinct arithmetic average (DAA) are calculated by using our forward induction procedure; the lognormal distribution is one with the mean and the variance the same as those of the exact distribution.

lognormal

FIGURE 2.Upper bounds and lower bounds in B(20,10,105). 'Exact' is the sub-bundle expected payoff as a function of the strike price, 'U-loose', 'L-loose' are loose bounds, 'U-100' , 'L-100' are upper bound and lower bound calculated by our tight bound approach with L=100.

(a) upper bounds

(b) lower bounds

0.0071.

Strike

U-loose

Strike

L-loose

FIGURE 3. Asian option prices vs volatility and time steps: loose bounds. The loose lower bounds and upper bounds are calculated using our C++ program in a Pentium 200 computer. The inputs are shown in Table 2.

6 12 18 24 30 36 42 48 54 60 66 72 78 84 90

0.400L

0.400U

0.405L

0.405U

0.410L

0.410U

0.415L

0.415U0.420L

0.420U0.425L

0.425U

time step

tility

12345678

S1 S2 S3 S4 S5 S6 S7 S8 S9S1

Vertical Axis: Values of Arithmetic Averages, B(8, 0). Annual Volatility = 1.39 Axis Labelled 0 to 8: Numbers of paths in B(8, 0, v); v = -16, 14, … , 16. Axis Labelled S1 to S17: Defines sub-bundles B(8, 0, 16) to B(8, 0, -16) in that order. The strike price is represented by a plane parallel to and above the plane shown.

Pricing Asian Options Using Path Bundling

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