price options, and of order T3 to value geometric average strike price options. ... options. However, in practice both American and European arithmeti...

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SOLUTIONS STRIKE

FOR AVERAGE PRICE

EUROPEAN

ON A RECOMBINING EDWIN

RANDOM H.

SPOT

AND

OPTIONS WALK

NEAVE

h3STRACT

This paper finds exact valuations for three kinds of European average options on a discrete time, recombining multiplicative binomial stock price process. For the types of average option described below, the methods of this paper can reduce the evaluations dramatically: instead of the customary 2T enumerations it is only necessary to record payoffs of order T2 to value geometric average spot price options, and of order T3 to value geometric average strike price options. The enumerations for arithmetic average spot price options offer similar computational savings, but the number of distinct paths requiring to be recorded is less easy to describe analytically. Nevertheless, the ratio of these distinct paths to total paths is a strictly decreasing proportion of T. The paper uses backward recursion formulae analogous to the ChapmanKolmogorov equations to find option payoff distributions, which are then valued under a martingale. The recursions are structured so that families of options can be valued without needing to recalculate the payoff distributions. 1. INTRODUCTION Reiner (1991), extending and generalizing the work of Goldman, Sosin, and Shepp (1979) and of Goldman, Sosin and Gatto (1979), finds analytic solutions for a wide variety of continuous time path dependent options. However, in practice both American and European arithmetic average options remain difficult to value. For, given the conventional choice of a geometric diffusion for the continuous time stock price process valuation requires calculating a sum of lognormal variates which is not Note: An earlier version of this paper was presented to the Annual Meeting of the Canadian Economics Association, Charlottetown, 1992. I am indebted to Phelim Boyle, Peter Carr, Kent Daniel and Stuart Turnbull for constructive suggestions. However, I remain solely responsible for any remaining difficulties.

154

Edwin

H. Neave

itself lognormal. Moreover most such options calculate their averages from discrete observations rather than using the continuous averaging assumed by Reiner. To overcome these difficulties the payoff distribution is usually approximated; cf. Turnbull and Wakeman (1991), Levy (1991), Levy and Turnbull (1992). This paper uses a discrete time analogue of the geometric diffusion process to obtain exact solutions for the arithmetic average spot price option and two other European average options(r). Path dependent options on a binomial process are usually thought to require 2T enumerations. However, in the cases considered below (for values of T depending on the particular option) this paper’s methods can dramatically reduce the necessary calculations and as a result it becomes computationally feasible to compare the present discrete methods with other approximate methods. Such comparisons are likely to prove useful in practice. For existing approximation methods exhibit relatively large errors for the high volatilies characteristic of, for example, oil or aluminum contracts; cf. Levy (1991). The paper’s discrete time approach both constitutes a natural choice for averages defined at discrete points in time (cf. Turnbull and Wakeman (1991)) and offers other advantages. First, the methods extend readily to valuing American options, albeit at increases in computational cost. For example, they can be used to find the exact discrete form of the payoffs for the “buy at the low” American options approximated in Barone-Adesi and Chesney (1991, 1992). Second, the paper’s methods can be extended to study the relative efficacy of the bundling methods recently developed by Ho (1992) and by Tilley (1992). 2. APPROACH

This paper finds option payoff distributions recursively, and evaluates them under a martingale. For the multiplicative process assumed, the solutions require enumerations of order T2 for geometric average spot price options, order T3 for geometric average strike price options(2). The paper also finds exact values for arithmetic average spot price options, @)The terminology is intended to be consistent with current usage. The paper substitutes “average spot price” for “Asian” option and uses “average strike price” to highlight the differences between types of average options; cf. Reiner (1991), Rubinstein (1991), Turnbull and Wakeman (1991). @)The same computational savings can be obtained for arithmetic averages on an additive process, as shown in Section 7.

Exact solutions for average spot and average strike price etc.

155

and with similar computational savings, but in this last case the number of computations is less easy to characterize analytically. 2.1. MODEL Let {St}, the stock price process, be a recombining, binomial random walk:

multiplicative

t E { 1,2, . . . , T}, where U is a random variable assuming the values u > 1 with probability

p

and u-l

with probability

1 -p = q .

The realized stock price cannot become negativec3), and is finite for finite values of T and u. For fixed T, the price can be kept below an upper bound M* by choosing u such that 1 < u < (M*)liT. The European options to be valued are next defined formally. the exercise date be T, and let G be the geometric average price

Let

1/V+1)

P.21 Then a geometric payoff of

P.31

GE

average

[1

.

spot price

(Asian)

fist t=o

call has a time T

C(G, K) = max{G - K, 0) ,

where K is the strike price, and a geometric (Asian) put has a time T payoff defined by

average

(3)The continuous time limit of the price process is lognormal; Rubinstein (1985) or Huang and Litzenberger (1988).

spot

price

see Cox and

156

Edwin

P.41

H. Neave

P(G, I<) = max{K

The geometric

average

strike

- G,O} .

price call has a time T payoff of

C(ST, G) = max{ST - G, 0) ,

P.51 and the geometric

WI

average

strike

price put has a time T payoff of

I’(&-, G) = max{G - ST, 0) . Finally, letting H be the arithmetic

average,

P.71 the arithmetic

average

spot price

call has a time T payoff(4) of

P.81

C( H, I() = max{ H - K, 0} ,

and the arithmetic

average

P-91

P(H, K) = max{K

spot price

put has a time T payoff of

- H, 0) .

The rest of the paper is organized as follows. Section 3 records both the assumptions necessary to use martingale based valuation methods and a useful convention for describing the paths of realized prices. Section 4 shows how to value the geometric average spot price call by recursively generating its payoff distribution and then valuing the payoffs under the martingale. The put can be valued using the same c4)Reiner (1991) finds solutions for time weighted averages. The methods of this paper can be adapted to the time weighted case, but at the cost of additional computational complexity.

Exact solutions

for average spot and average strike price etc.

157

distribution. Similarly, Section 5 shows how to value the geometric average strike price call, and Section 6 the arithmetic average spot price call. Section 7 concludes with sketches of valuing arithmetic average options on additive random walks, and of valuing options with early exercise privileges. Appendices illustrate the calculations for each type of option.

3. PRELIMINARIES 3.1.

THE

MARTINGALE

MEASURE

Assume the following conditions i) markets are perfectly competitive, with atomistic transactors, no transactions costs, and homogeneously distributed information; ii) markets have reached an equilibrium (i.e., there are no remaining, profitable arbitrage opportunities); and iii) markets are dynamically complete. Conditions i) to iii) are both necessary and sufficient for the existence of an unique martingale under which securities can be priced using expected value calculations. The martingale is determined from the normalized stock price pro cess defined by letting Rt reflect the t-period accumulation of $1 at the risk free interest factor R E 1 + r, where r is the single-period risk free interest ratec5). Given the normalized

PI

stock prices Sz’ = St/Rt

,

and recalling that St evolves according to [2.1], there is a (conditional) probability & such that

WI

s; = (p*us; + q*u-‘S;)/R

(‘)The paper’s results extend to a stocastically varying risk free rate at the expense of recording the sequences of path probabilities. In this case the rates {rt} are assumed to become known no later than time t, and it is necessary to take expectations over the possible time t realized values of the random interest rate as well as over the stock price outcomes.

158

Edwin

H. Neave

where q* z 1 - pt. Then {St} is a martingale measure p*, where from [2.4], and [2.2]:

WI

under the probability

p’ = (R - u-‘)/(U - u-l) .

3.2. PATHS OF REALIZED PRICES In the rest of this paper it will be convenient to describe the paths of realized prices using patterns of the prices’ exponents. For these purposes, divide the paths of realized prices by the common factor So to obtain, for T = 1 and T = 2:

u”ul

WI

7L”zL-l

and

P.51

The paths in [3.4] for T = 1 can now be described concisely by the row vectors (0,l) and (0, -l), and similar notation can be used for T = 2,3,4, . . . The path probabilities are described by the appropriate binomial distributions; for T = 1 they are respectively p and q; for T = 2p2, pq, pq and q2; and similarly for values of T > 2. The foregoing conventions can now be used to find payoff distributions recursively. To illustrate, the paths in [3.4] for T = 1 can be interpreted as a continuation of those paths with length T = 2 which begin with an initial upward move. More precisely, the exponents of the paths in [3.4] are related to those of the last two prices along the first two paths in [3.5] by:

WI

Exact solutions

for

average

159

spot and average strike price etc.

Similarly, when the paths in [3.4] are interpreted as a continuation paths beginning with a downward move they are related by:

of

f3.71 Thus by combining [3.6], [3.7], and appropriately dimensioned arrays of zeroes, the paths described in [3.5] can be related linearly to the decrip tions in [3.4]. Moreover, the calculations can be continued recursively for T = 3,4, . . . . Variants of the foregoing methods can be used to characterize the sums of path exponents or other data relevant to the terms of a given options. Since the data needed to value each type of option differ according to its terms, the precise nature of the recursive calculations differs correspondingly. However, all the recursions presented below apply to the concepts illustrated by [3.6] and [3.7]. 4. GEOMETRIC AVERAGE SPOT PRICE OPTIONS

To value options with payoffs given by [2.3] or [2.4], it is necessary to find the density function of stock prices’ geometric averages, which can in turn conveniently be obtained from the density function for the sums of path exponents. Denote the latter random variable by V. Examining the price paths in [3.4] shows that when T = 1, V assumes the values in Cl E {1,-l}

,

and when T = 2 it follows from [3.5] that V assumes the values in cz E {3,1,

-1, -3)

.

More generally, it can be verified that V assumes the b(T + 1,2) + 1 values in: CT E {b(T

+ 1,2),

b(T + 1,2) - 2,. . . , -b(T

where b(z, y) E

X! y!(x -y)!

.

+ 172))

,

Edwin

160

H. Neave

Next, let BT be a vector whose components BT,( denote the probabilities of the outcomes in CT; i.e.,

BT,( =

Prob {V = b(T+

1,2) - 2i) ;

i E {O,l,. . ., b(T+ 1,2)} .

Even though the values in CT are easy to express analytically, the vectors do not have similar simple expressions. Nevertheless, the BT can be found recursively using the simplified approach illustrated in Section 3. To see the value of proceeding recursively, consider the alternative of obtaining the BT using generating functions:

BT

PROPOSITION 1. If the process [2.1] continues for T periods, the number of paths with sum of exponents V is equal to the number of ways the indices of the coeficients of the generating function

P.11

hT(s)

= fi[at

+ a-t

. s]

t=1

sum to V. Proof.

See Appendix

I.

If [4.1] is used directly to calculate the values of BT, computations of order 2T are involved. Even so, [4.1] suggests a simpler approach, because it implies:

w21

b-+1(s)= (UT+1+ a-T-lsh(s) ,

from which it follows immediately

P.31

that:

&-+I = P(&-, O(T+I))+ q(O(z-+I), BT) ,

where O(T) denotes a column vector of T zeroes. Thus [4.3] can be used to calculate the vectors BT recursively, and with far fewer enumerations than are required by the generating function approach. To begin the process, recall from the discussion following [3.4] and [3.5] that

BI = (P,9)’ ,

Exact

solutions

for

average

spot

and

avemge

strike

price

161

etc.

and B2 = (P2m,pq,q2)'

,

where ’ denotes transpose. The example of Appendix III uses

Bg.

Given BT, the options defined in [2.3] and [2.4] can be valued under the martingale. Thus the time zero value of the geometric average spot price call defined by [2.3] is

WI

co(G, I<) = c [max[(G - K),O]/RT] GEIYT

where rewriting

. b; ,

[2.2], G = SoUV/(T+‘)

and rT

14.51

E

{GIV

E C,}

.

In [4.4] b;, the martingale probability that the path geometric average is G, is given by the appropriate component of B& where Bg is the analogue to BT under the martingale. An example is worked in Appendix III, both by the paper’s recursive methods and by backward induction. The objective probability

that the call will have positive value,

Prob {G > K} = Prob

{V/(T

+

1) > K} ,

can also be calculated readily from BT. Finally, the value of the put whose payoffs are given by [2.4] can be calculated using the same methods. 5. GEOMETRIC

AVERAGE

STRIKE

PRICE OPTIONS

The main problem in valuing geometric average strike price options; i.e., options with payoffs given in [2.5] and [2.6], is to generate the joint probabilities of G and e, where e is a random variable indicating the exponents of realized prices at time T. As before, it is convenient to find G from V, and we therefore wish to obtain the joint distribution of V and e. It is a more complex distribution than that of Section 4, but can still be obtained recursively.

Edwin

162

H. Neave

Denote the matrix of joint probabilities for the outcomes V and e of a T period walk beginning at the origin by CT. Begin by considering Cr, written as in Table 1. (The relations of Cr to the process [2.1] can be established by recalling [3.4]). Table

l-

Cl

and

headings

V\e

1

1

P

-1

-1

9

Recalling [3.6] and [3.7], it is easy to check that the matrix of joint probabilities for a T = 1 walk with origin +1 is the same as Cr, but refers to V E (3,l) and e E (2,O). The matrix of joint probabilities for a T = 1 walk with origin -1 is also the same, but refers to V E (-1, -3) and e E (0, -2). Then, regarding the foregoing outcomes as continuations of a T = 2 walk gives the joint distribution of the possible outcomes, written as Cz in Table 2. Table

2 - C2 and

headings

V\e

2

0

3

P2

1

-2

P9

-1

P9

-3

q2

The matrices CT can be obtained recursively for T > 2. Given the number of distinct outcomes to the binomial process, CT has T + 1 columns, and (from Section 4), b(T + 1,2) + 1 rows. It is straightforward to show inductively that for all T:

WI

CT+1

=Pc$+&$,

where

L5.21

c$s

CT

oW+1,2)+l)

O(T+i,T+l)

OT+l

Exact solutions for average spat and average strike price etc.

163

and

b31

O(T+l)

c$ E

O(T+l,T+l)

[email protected](T+1,2)+1)

CT

The matrix C$ indicates the joint probabilities of paths with length T + 1 conditional on the first move being upward; C$ the joint prob abilities conditional on the first move being downward. The notation O(S,T) indicates an S by T matrix of zeroes. From 15.11 and the [email protected] ciated calculations it is easily seen that the symmetric structure of CT is preserved under the recursion, as is the binomial distribution of its column totals. The call value at time zero is obtained by calculating the expected present values of the call payoffs under the martingale, using martingale probabilities taken from [5.1]. That is,

[5.4]

%(S, G) = c GENT

c

max[(Ue -Us),

O] . [So/RT] . ~2,~

&ET

where rT is the set of geometric averages defined in [4.5], and I& equals c;,, , an element of C$. Finally, CG is the analogue under the martingale of CT. An example using CY~is worked in Appendix IV. The matrix CT contains (T + 1) . [b(T + 1,2) + l] cells, of which 2b(T+2,3) have the value zero. Thus for the process [2.1] the sell at the average put and buy at the average call valuation problems are cubic in T. The simplified expression for the number of cells with positive values is [T3 + 5T + 6]/6. Finally, the objective probability of a positive payoff, Prob {V/(T + 1) < e}, can also be calculated from CT. 6. ARITHMETIC AVERAGESSPOT PRICE OPTIONS

Valuing the arithmetic average spot price option on the continuous analogue of [2.1] is difficult because the lognormal distribution is not stable, with the result that the arithmetic average of lognormally distributed stock prices is not itself lognormal. In contrast, solving the discrete time valuation problem using the now familiar recursive methods is relatively straightforward. From [3.5], it is apparent that to calculate the unweighted arithmetic average of, say, the second path, it is only necessary to record

Edwin

164

H. Neave

that the price Seu” occurs twice and that the price Seu’ occurs once. Similarly, for higher values of T any two paths for which each realized price occurs the same number of times (but not necessarily in the same temporal order) have the same arithmetic average. Moreover, all paths with the same arithmetic average have the same sum of path exponents, a fact useful for ordering the distinct [email protected]). Thus to calculate the arithmetic average of the stock prices given by [2.1] it is only necessary to tabulate paths which are distinct in terms of different stock price realizations, ignoring the order in which the realizations occur. For the data presented in Table 5 below, this number is much smaller than 2T, and as will be shown its ratio to the number of total paths is a strictly decreasing function of T(‘). The distributions of arithmetic averages can be generated using procedures similar to those of Sections 4 and 5. To record the necessary information, let & be a matrix recording path sums V and the frequencies of realized stock prices S. (For unweighted averages, the temporal order of the realized prices is irrelevant and is not recorded). It is now convenient to record the path probabilities separately rather than in the body of the matrix as in Section 5. Using [3.5], the matrix 02 can be written (headings are included for clarity) as shown in Table 3. ‘Pable

v\s

3 -

DCI and

headines

-2

-1

0

1

2

Path Prob

3

1

1

1

P2

1

2

1

-1 -3

1

Pg

1

2

Pq

1

1

q2

The matrices DT can be generated recursively using ideas similar to those in [3.6] and [3.7]. It is convenient to employ the following steps, which illustrate generating 0s from D2. (‘)The condition is only necessary; paths can have different arithmetic averages even if the sum of their exponents is the same. (7)Some arithmetic mean options’ averaging periods are shorter than the option’s maturity, in which case the valuations require still fewer calculations.

Exact solutions

for average spot and average strike price etc.

165

- First, record the distribution of paths conditional on the first move being upward, followed by the distribution of paths conditional on the first move being downward. Use a matrix similar in form to Dz, say 0:. - Second, increase the entries in the zero column of 0; by T/N, where N is the sum of the components in the relevant row of 0;. This step reflects the fact that at time T + 1 the paths will be one step longer. - Third, reorder the rows of D.$’ so that the path sums are again monotonically decreasing, thus making it easier to find and remove redundant paths in the next step. (This step is only necessary for values of T > 3). - Fourth, for any given path sum in 03, add the components and the probabilities of any rows whose ratios of positive components are identical. Retain one row containing the foregoing sums and delete the remaining (now redundant) rows used to form the sums. Applying these four steps results in D3, as shown in Table 4. Note that the row headed V = 0 represents two paths.

Table

4 - D3 and

v\s

-3

headings -2

6

-1

0

1

2

3

Prob

1

P3

4

P%

2

Pg2

0

P29 + Pq=

-2

w2

-4

Pf12

-6

1

The matrices DT, T > 3, can be generated sequentially actly the same methods.

q3

using ex-

While analytic expressions for the number of separate paths which must be recorded can be determined as sums of binomial coefficients, the procedure is tedious and offers little or no advantage over recursive

166

Edwin

H. Neave

computation. However, Proposition 2 gives an upper bound on the proportion of paths which must be enumerated. PROPOSITION 2. distinct paths (as defined Proof.

See Appendix

For any value of T > 5, let pi be the ratio by DT) to total paths. Then PT > p~+l.

of

II.

The data of Table 5 record the number of distinct paths required for arithmetic average calculations when T < 18. Note also that for problems with T > 18, Proposition 2 ensures that PT < 0.0737.

lkble

T

5 - Distinct

paths

for arithmetic

average

Distinct Paths

calculations

PT

1

2

1.0000

2

4

1.0000

3

7

0.8750

4

13

0.8125

5

22

0.6875

6

40

0.6250

7

66

0.5156

8

118

0.4609

9

192

0.3750

10

338

0.3301

11

546

0.2666

12

948

0.2314

13

1526

0.1863

14

2618

0.1598

15

4208

0.1284

16

7146

0.1090

17

11482

0.0876

18

19332

0.0737

The data of the table, especially when considered in relation to the results of Sections 4 and 5, suggest that the number of distinct rows may

Exact solutions

for average spot and average strike price etc.

167

grow only polynomially, but a proof of this conjecture is not presently available. Given DT, the options defined in [2.8] and [2.9] can be valued using methods similar to those of Section 3. For example, the value of the arithmetic average spot price call is

WI

%(H,K) = C [max[(H- W, OII~I .G HEAT

where H is defined in [2.7] and AT is the set of distinct paths defined in DT. In [6.2] dk, the martingale probability that the path arithmetic average is H, is given by the appropriate component of D$, where D$ is the analogue to DT under the martingale. An example is worked in Appendix V. The value of the put is calculated methods outlined above.

similarly,

using [2.8] and the

7. CONCLUSIONS AND EXTENSIONS

This paper showed how to value geometric average spot price, geometric average strike price, and arithmetic average spot price options on a recombining multiplicative random walk. The following extensions to other classes of options are immediate. 7.1. ADDITIVE RANDOM WALKS Suppose the process [2.1] is replaced by the recombining random walk:

V-11

additive

s, = Is&l + u ;

t E {1,2 ,...) T}, where U assumes the value u > 0 with probability p and -U with probability q. (The values of Se, u and T should be chosen so that the realized stock prices remain positive as well as finite). Then the valuation procedures in Sections 4 and 5 extend immediately to arithmetic average options. In this case BT and CT record to the net sum of changes in the stock price.

Edwin H. Neave

168 7.2.

AMERICAN OPTIONS

Whether the process is [2.1] or [7.1], this paper’s recursively generated data describe the temporal evolution of payoff distributions and hence can be used to value early exercise privileges along the lines of the continuous time work in Barone-Adesi and Chesney (1991). In the present discrete time case, the payoff distributions are first generated by forward recursion. The valuation calculations are then begun at time T, using the last of the payoff distributions. The valuation calculations are repeated, working recursively backward one stage at a time until reaching time 0. At each stage t, the current payoffs from exercising the option are compared with the expected present value of future exercise calculated under the martingale, the larger value recorded, and the backward calculations continued. The paper’s economical path characterizations can still be used, but since the payoffs must now be examined explicitly at each stage, the number of computations increases T-fold. 7.3. SUMMARY Similar methods could be employed for other path dependent op tions. For example, amending the methods to allow for an averaging period which differs from the number of periods to exercise is straightforward. In general, if the payoff distributions became more complex than those considered here, the method’s advantage over explicitly enumerating all paths is lessened correspondingly. The degree of computational complexity in valuing a given path dependent option is indicated by the dimension of the appropriate matrix of joint probabilities; i.e., by the degree of path dependence. APPENDIX I. PROOF OF PROPOSITION I Let U,, a binomial random variable, represent (the exponents of) the possible outcomes at time t. Reasoning similar to that illustrated by [3.6] and [3.7] can be used to show that:

Proof.

V = TU1 + (T - 1)U2 +. . . + UT .

The since V is clearly a sum of binomial random variables it has a generating function of the form [4.1]; cf. the discussion in Feller (1957, ch. 11).

Exact solutions for average spot and avemge strike price etc.

169

Interpretation: The coefficients at and a-t are interpreted respectively as p and q, and their products give the path probabilities. For example, if T = 3 the product aia2a-as describes a path whose sum is zero and whose probability is p2q. The power of s, (and the number of negative signs in any product of coefficients) indicates the number of downward moves in the path to which it refers.

APPENDIX

II. PROOF

OF PROPOSITION

2

Since the distinct paths are generated recursively as described above, when T increases to T + 1 the numerator of pi at most doubles (some of the new paths may not be distinct from existing paths) while the denominator doubles exactly. Thus pi cannot be increasing in T, whatever the value of T. To show that PT is strictly decreasing in T 2 5, it suffices to show that in calculating &+I from DT there are always some paths which can be combined. This result is established by generalizing two special cases, one for even and one for odd values of T. First consider T = 5. Consider the path -1 - 2 - 101, which starts with a downward move. (The path description omits the first zero indicating that the path begins at the origin. The chosen path is feasible because each component differs from its predecessor by exactly 11I). This path has a corresponding path beginning with an upward move, namely 10 - 1 - 2 - 1. Hence these two paths can be combined in Ds. This example extends to all subsequent odd periods by repeating the initial sequence -1 - 2; i.e. for T = 7, begin with the path -1 - 2 - 1 - 2101. Now consider T = 6 and the price path -101210. A corresponding path beginning with an upward move is 10 - 1012. This example can be used for all even periods greater than T = 6 by repeating the price sequence -10; e.g., for T = 8, consider -10 - 101210. Proof.

Edwin

170

III.

APPENDIX

Recursive Assume

Path

GEOMETRIC

H. Neave AVERAGE

SPOT

PRICE

CALL

calculation strike

price

is 1.00

tL=

1.1000

p* =

0.528571

R=

1.0100

q’ =

0.471428

Sums

ProbVec

Path B5

Average

Option

EV of

Payoff

Payoff

15

1

0.0413

1.2691

0.2691

0.0111

13

1

0.0368

1.2294

0.2294

0.0084

11

1

0.0368

1.1909

0.1909

0.0070

9

2

0.0696

1.1537

0.1537

0.0107

7

2

0.0696

1.1176

0.1176

0.0082

5

3

0.1024

1.0827

0.0827

0.0085

3

3

0.0949

1.0488

0.0488

0.0046

1

3

0.0949

1.0160

0.0160

0.0015

-1

3

0.0914

0.9842

0.0000

0.0000

-3

3

0.0914

0.9535

0.0000

0.0000

-5

3

0.0847

0.9236

0.0000

0.0000

-7

2

0.0554

0.8948

0.0000

0.0000 0.0000

-9

2

0.0554

0.8668

0.0000

-11

1

0.0261

0.8397

0.0000

0.0000

-13

1

0.0261

0.8134

0.0000

0.0000

-15

1

0.0233

0.7880

0.0000

0.0000

Option

value

at T = 5

0.060076

Option

value

at T = 0

0.057160

Exact solutions

Backward

for average spot and average strike price etc.

induction

calculation

0

1

stage

2

3

171

4

5 Option Values

0.057160

0.099823

0.157265

0.209843

0.247871

0.269058 0.229374

0.101649

0.171657

0.190930

0.135328

0.153689 0.153689 0.117613

0.066043

0.082664 0.048808

0.037537

0.067816

0.117613

0.100135

0.082664

0.004385

0.033017

0.048808 0.016011

0.008379

0.016011 0 0

0 0

0.010537

0.020135

0.038474

0.066043

0.082664 0.048808

0.008379

0.016011 0

0

0 0 0 0

0

0

0 0 0

0

0 0 0 0 0

Value

of option

at

T =0

0.057160

172

Edwin

APPENDIX

IV.

GEOMETRIC

1.1000 1.0100

U=

R=

H.

Neave

AVERAGE

p* =

CALL

calculation

C5: frequencies End values Path sums 15 13 11 9 7 5 3 1 -1 -3 -5 -7 -9 -11 -13 -15

C5:

PRICE

0.526571 0.471423 1.01

q* = R=

Recursive

STRIKE

5

3

1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0

1

-1

-3

-5

0 0 0 0 0 0 1

0 0 0 0 0 0 0

0 0 0 0 0 0 0

1 2 2 2 1 1 0 0 0

0 0 0 1 1 1 1 1 0

0 0 0 0 0 0 0 0 1

probabilities 5 15 13 11 9 7 5 3 1 -1 -3 -5 -7 -9 -11 -13 -15

0.041258 0 0 0 0 0 0 0 0 0 0 0 0 0 0

3

1

-1

-3

-5

0

0 0 0

0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0.036798 0.036798 0.036798 0.036798 0.036798 0 0 0 0 0 0 0 0 0 0

0.032820 0.032820 0.065640 0.065640 0.065640 0.032820 0.032820 0 0 0 0 0 0

0.029272 0.029272 0.058544 0.058544 0.058544 0.029272 0.029272 0 0 0

0.026107 0.026107 0.026107 0.026107 0.026107 0

0.023285

Exact solutions

Calculation

for average

of option

spot and average strike price etc.

173

values

5

3

-1

1

-5

-3

15

0.0141

0.0000

0.0000

0.0000

0.0000

0.0000

13

0.0000

0.0037

0.0000

0.0000

0.0000

0.0000

11

0.0000

0.0052

0.0000

0.0000

0.0000

o.oooa

9

0.0000

0.0065

0.0000

0.0000

o.ooQo

0.0000

7

0.0000

0.0079

0.0000

0.0000

0.0000

0.0000

5

0.0000

0.0091

0.0011

0.0000

0.0000

0.0000

3

0.0000

0.0000

0.0034

0.0000

0.0000

0.0000

1

0.0000

0.0000

0.0055

0.0000

0.0000

0.0000

-1

0.0000

0.0000

0.0038

0.0000

0.0000

0.0000

-3

0.0000

0.0000

0.0048

0.0000

0.0000

0.0000

-5

0.0000

0.0000

0.0000

0.0000

0.0000

0.0000

-7

0.0000

0.0000

0.0000

0.0004

0.0000

0.0000

-9

0.0000

0.0000

0.0000

0.0012

0.0000

0.0000

-11

0.0000

0.0000

0.0000

0.0000

0.0000

0.0000

-13

0.0000

0.0000

0.0000

0.0000

0.0000

0.0000

-15

0.0000

0.0000

0.0000

0.0000

Option

value

0.0000 at

T = 0

0.0000 0.063534

Edwin

174

Backward

induction

H. Neave

calculation

Paths

V/V 0

1 1

0

+ 1)

2

3

2

3

4 4

5 3

2.5000 2.1667 1.8333

0

1

2

3

2

3

0

1

2

3

2

1

1.5000

0

1

1

2

1 1

1 1

2 0

3 1

1.5000

0

2 2

0

1 1

0 2

-1 3 1

0.8333

0 0

2 2

1

1.1667 0.8333 0.5000

0

1 1

0

1

0

1

2

0

1

0

1

0.5000

0

1 1

0 0

1

0

1 -1

0 0

-1 1

0.1667 0.1667

0

1.1667

0

1

0

-1

0

-1

-0.1667

0

1

0

-1

-2

-1

-0.5000

0

1

0

-1

-2

-3

-0.8333

0

0

-1 -1

1 1

2 2

3 1

0.8333 0.5000

0

-1

0

1

0.1667

0

-1 -1

0 0

1

0

1 -1

0 0

-1 1

-0.1667 -0.1667

0

-1

-1

0

-1

-0.5000

0

-1

-2

-1

-0.8333

0

0

0

-1 -1

-1 -1

-2

-1

-2 0

-3 1

-1.1667 -0.5000

0

-1

-2

-1

0

-1

-0.8333

0

-1

-2

-1

-2

-1

-1.1667

0

-1

-2

-1

-2

-3

-1.5000

0

-1

-3

-2

-1

-1.5000

0 0

-1 -1

-2 -2 -2

-3 -3

-2 -4

-3 -3

-1.8333 -2.1667

0

-1

-2

-3

-4

-5

-2.5000

0

0

0 0 0

Exact solutions

Backward

induction

stage.3

for average

spot and avemge strike price etc.

175

calculation 0

0.063534

1 0.078056

2 0.104482

3 0.152557

4

5 0.341451

0.226129

0.101625 0.073303

0.140069

0.092793

0.177310

0.009072

0.017335

0.119765

0.213386

0 0.052796

0 0 0.050082

0.075182

0.017335 0.026790

0.051191

0.043954

0.083988

0 0.023002

0 0

0

0 0.048599

0.067674

0.10 1035

0.153857

0.248335 0.051191

0.043954

0.083988

0.060581

0.115759

0 0.031704

0 0

0 0

0.028243

0.043633

0.076688

0.146537

0.007497

0.014326

0.022140

0.042306

0 0 0.011587

0 0 Option

value

at T = 0

0.063534

0 0

176

Edwin

V.

APPENDIX

ARITHMETIC

Assume

strike

tL=

1.1000

p* =

0.5286

R=

1.0100

q* =

0.4714

Types

and

price

of distinct 2

3

4

5

1

1

1

1

1

1

PRICE

CALL

1

1

1

2

1

# Paths

Sum

Probs

Prob

1

15

50

0.041270

1

13

41

0.036804

22

1

11

41

0.036804

2

4

42

2

9

32 41

0.069625

1

3

2

1

7

32

0.032821

2

2

11

1

7

41

0.036804

4

6

2

2

5

32 32

0.065643

111

1

5

41

0.036804

3

1

3

32

0.032821

2

4

4

2

3

23 32

0.062091

3

9

6

2

3

1

23 32 32

0.094913

6

9

3

3

-1

23 23 32

0.091361

2

4

4

2

-3

23 32

0.062091

3

3

1

-3

23

0.029270

2

6

4

2

-5

23 23

0.058540

1

1

2

1

-5

14

0.026102

1

1

2

2

1

-7

14

0.026102

2

3

1

1

-7

23

0.029270

2

4

4

2

2

-9

14 23

0.055372

2

2

1

1

1

-11

14

0.026102

1

-13

14

0.026102

1

-15

05

0.023278

12111 1

SPOT

D5

1

3

1

paths:

1

12

1

AVERAGE

0

11

2

Neave

is 1.00

frequencies

-5-4-3-2-l

H.

1

1

1

1

1 1

2

9

16

38

60

38

16

9

2

1

192

Exact solutions

for

average

spot and average

strike price etc.

177

Average Price

Option Payoff

Pr from D5

1.2859

0.2859

0.0413

0.0118

1.2394

0.2394

0.0368

0.0088

1.1970

0.1970

0.0368

0.0073

1.1585

0.1585

0.0696

0.0110

1.1200

0.1200

0.0328

0.0039

1.1235

0.1235

0.0368

0.0045

1.0850

0.0850

0.0656

0.0056

1.0917

0.0917

0.0368

0.0034

1.0500

0.0500

0.0328

0.0016

1.0532

0.0532

0.0621

0.0033

1.0182

0.0182

0.0949

0.0017

0.9864

0.0000

0.0914

0.0000

0.9574

0.0000

0.0621

0.0000

0.9545

0.0000

0.0293

0.0000

0.9256

0.0000

0.0585

0.0000

0.9311

0.0000

0.0261

0.0000

0.8993

0.0000

0.0261

0.0000

0.8967

0.0000

0.0293

0.0000

0.8704

0.0000

0.0554

0.0000

0.8441

0.0000

0.0261

0.0000

0.8202

0.0000

0.0261

0.0000

0.7985

0.0000

0.0233

0.0000

Option

value

at

T = 0

EV of Payoff

0.059945

Edwin

178

Backward 1

0

0.059944

0.104144

H. Neave

induction 2

0.163888

calculation 3

0.219436

4

5

0.261361

0.2859 0.2394 0.1970 0.1585 0.1585 0.1200 0.0850 0.0532 0.1235 0.0850 0.0500 0.0182 0.0182 0.0000 0.0000 0.0000 0.0917 0.0532 0.0182 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

0.177080 0.138961

0.105075

0.069307

0.070765

0.039360

0.104308

0.034654 0.004980

0.009515 0

0.011653

0.022266

0.042544

0.72804 0.009515

0

0 0

0

0

0 0

0

0 0

Option

value

0.059944

at T = 0

1.2859 1.2394 1.1970 1.1585 1.1585 1.1200 1.0850 1.0532 1.1235 1.0850 1.0500 1.0182 1.0182 0.9864 0.9574 0.9311 1.0917 1.0532 1.0182 0.9864 0.9864 0.9545 0.9256 0.8993 0.9574 0.9256 0.8967

0.8704 0.8704 0.8441 0.8202 0.7985

BIBLIOGRAPHY (1)

, Options to trade currencies as the of Alberta, Faculty of Business, Institute for financial research,l991, working paper 5-91. G. BARONE-ADESI, M. CHESNEY , American path dependent options, University of Alberta, Faculty of Business, Institute for financial research, working paper, 1992. A. CONZE, R. VISWANATHAN, European path dependent options, Finance, forthcoming, 1991. J.C. COX, M. RUBINSTEIN , Options markets, Englewood Cliffs, NJ.: PrenticeHall, 1985. G.

BARONE-ADESI,

‘Most Favorable’

(2)

(3) (4)

M.

CHESNEY

rate, University

Exact solutions

for average spot and average strike price etc.

179

(5) W. FELLER , An introduction to probability theory and its applications, 2nd edition, Wiley, New York, 1957. (6) M.B. GOLDMAN, H.B SOSIN, M. GATTO , Path dependent options: buy at the low, sell at the high, Journal of finance 34, 1979, 1111-1127. (7) M.B. GOLDMAN, H.B. SOSIN, L.A. SHEPP , On contingent claims that insure es-post optimal stock market timing, Journal of finance 34, 1979, 401-412. (8) T.S.Y. HO, Managing illiquid bonds and the linear path space, Journal of fixed income, June 1992, 80-94. (9) H. HUA, A solution guide to ‘Foundations for Financial Economics’, Mass. MIT Sloan School of Management, Cambridge, 1989. (10) CHI-FU HUANG, R.H. LITZENBERGER, Foundations for Financial Economics, New York: North-Holland, 1988. (11) A.G.Z. KEMNA, A.C.F. VORST , A pricing method for options based on average asset values, Journal of banking and finance 14, 1990, 113-129. (12) E. LEVY , A note on pricing European average options, Nomnra bank international working paper, 1991. (13) E. LEVY, S.M. TURNBULL , Average intelligence, Risk magazine, 1992. (14) I.G. MORGAN, E.H. NEAVE, A discrete time model for pricing treasury bills, forward and futures contracts, Actuarial approach to financial risks 3, first AFIR International Colloquium, 169-189, Paris, 1990. (15) E. S. REINERI, Valuation of path dependent options, Department of chemical engineering, University of California, Berkeley, 1991. (16) P. RITCHKEN, L. SANKARASUBRAMANIAN, A.M. VIJH, Averaging options for capping total costs, Financial management, autumn (1991), 35-41. (17) M. RUBINSTEIN, Exotic flavors for the options gourmet, Futures and options world, march (1991), 56-57. (18) J.A. TILLEY , Valuing American options in a path simulation model, Morgan Stanley & Co., Inc. (Forthcoming, transactions of the Society of Actuaries) 1992. (19) S.M. TURNBULL, L.M. WAICEMAN, A quick algorithm for pricing European average options, Journal of financial and quantitative analysis 26, 1991, 377-389.

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