binomial valuation of lookback options - ummdirectory.umm.ac.id/data elmu/jurnal/j-a/journal...

E-mail address: [email protected] (S. Babbs).

1After the original version of this paper, the author became aware that Reiner (1991a, b) had alsosuggested that working in units of the underlying security price could reduce the problem of valuinglookback options to one involving a single stochastic variable, which might then be approximatedby a binomial scheme.

Reiner's papers do not encompass the case where the current gap between the underlying securityprice and its extremum does not correspond to a whole number of space steps. It also allows thepossibility of that gap being zero at both the beginning and the end of a time step. It thus resemblesa scheme devised by the present author in the late 1980s, and which appeared to give inferiorconvergence properties to the scheme reported here.

Journal of Economic Dynamics & Control24 (2000) 1499}1525

Binomial valuation of lookback options

Simon Babbs!,"

!Bank One, NA, 1 Bank One Plaza, d0036, Chicago, IL 60670, USA"University of Warwick, UK

Abstract

We present a straightforward and computationally e$cient binomial approximationscheme for the valuation of lookback options. This enables us to value Americanlookback options. Previous research on lookback options has assumed that the contractsare based on the extrema of the continuously observed price of the underlying security; inpractice, however, contracts are often based on the extrema of prices sampled at a "niteset of "xed dates. We adapt our binomial scheme to investigate the impact on the value ofthe options. ( 2000 Elsevier Science B.V. All rights reserved.

1. Introduction1

Goldman et al. (1979a) provided closed-form valuation formulae for a Euro-pean-style option to buy (sell) a non-dividend paying security at the lowest(highest) price achieved by that security over the life of the option. When theirpaper was published, such options were not traded. Subsequently, such

0165-1889/00/$ - see front matter ( 2000 Elsevier Science B.V. All rights reserved.PII: S 0 1 6 5 - 1 8 8 9 ( 9 9 ) 0 0 0 8 5 - 8

2Conze and Viswanathan (1991) rederived Goldman, Sosin and Gatto's results, together withvaluations of various related European-style contingent claims. They also present upper bounds forthe value of the early exercise right in American lookback options. However, the authors' claim thatthese bounds are &tight'must be interpreted with caution, since their own illustrative results (Table 1,p. 1902) show that their upper bound for the value of a new 6-month American lookback put canexceed 160% of the value of its European counterpart.

&lookback' or &hindsight' options have gained a modest but enduring placeamong the range of derivative securities available over-the-counter from a num-ber of "nancial institutions. This provides perhaps the most notable instance todate of a theoretical valuation model prompting the commercial introduction ofnew types of contingent claim.

From a commercial point of view, one might have expected American-stylelookback options to have appeared alongside their European-style counter-parts. Yet American lookback options are almost unheard-of. This contrastsmarkedly with the case of ordinary puts and calls, where American options, bothover-the-counter and exchange-traded, are commonplace. We believe that thissituation is due, in no small measure, to the unavailability of an e$cientvaluation technique for American lookback options.2 In this paper, we remedythat situation by presenting a straightforward and computationally e$cientbinomial valuation scheme. Our proof that the resulting option values convergeto the correct levels as the time step length is reduced to zero has the attractionsof being direct and of using only basic mathematics; the approach it adoptsappears to transfer readily to other contexts, and may therefore be a usefulcontribution.

Previous research on lookback options has assumed that the contracts arebased on the extrema of the continuously observed price of the underly-ing security; in practice, however, contracts are often based on the extrema ofprices sampled at a "nite set of "xed dates, typically at daily intervals. We adaptour binomial scheme to investigate the impact on the value of the options. Itemerges that even a switch from continuous to daily sampling has a signi"cantimpact on option values, even when options are far from expiry; our resultstherefore provide opportunities for improved pricing and, hence also, riskmanagement.

The remainder of the paper is organized as follows. Section 2 rehearses theassumptions of the model and the resulting abstract general valuation formulafor contingent claims. We generalize slightly the assumptions of Goldman et al.(1979a) by allowing the underlying security to pay a constant continuousproportional dividend. This extension accommodates, inter alia, the foreigninterest rate when the security is one unit of a foreign currency. Our expositionmakes use of the modern &martingale' approach to contingent claimsanalysis stemming from Harrison and Kreps (1979). Section 3 formalises the

1500 S. Babbs / Journal of Economic Dynamics & Control 24 (2000) 1499}1525

3Until Section 8, we implicitly follow the existing literature in assuming that the contracts arebased on the extrema of the continuously observed security price process.

4 I.e. maximum (minimum) in the case of a put (call) option.

5Numerous technical details are omitted in the interest of brevity. The concerned reader familiarwith the &martingale' approach to contingent claims analysis should be able to rectify the omissionswithout di$culty.

payo!s3 of lookback options and derives closed-form valuation formulae forEuropean-style exercise.

Section 4 discusses why the valuation techniques for American-style ordinaryputs and calls do not carry over readily to lookback options. In Section 5, weovercome these problems by means of a change of numeraire and a binomiallattice scheme in a new variable. Section 6 provides illustrative results to assessthe accuracy of our binomial scheme as applied to European lookback options,against the closed-form formulae obtained in Section 3, and to illustrate thevalue of the early exercise right in American lookback options. Combining thesewith further results tends to refute a conjecture by Garman [1987] that the &earlyexercise premium' of an American lookback option (i.e. the excess of the value ofthe American option over that of an otherwise identical option) is well approxi-mated by the early exercise premium of an American ordinary option with strikeequal to the extreme4 security price so far achieved. Section 7 considers theconvergence properties of our binomial scheme as the length of each discretetime step tends to zero.

In Section 8 we modify the binomial scheme to investigate the impact ofbasing lookback contracts on the extrema of a "nite sample of prices of theunderlying security, and present illustrative results. Section 9 concludes thepaper.

2. Assumptions and abstract valuations results5

We assume a frictionless competitive market is open continuously in asecurity whose price follows as Ito process:

dS/S"k dt#p dZ (1)

where Z( ) is a standard Brownian motion; p'0 is the constant price volatility;and the instantaneous expected return k is an essentially arbitrary stochasticprocess adapted to the Brownian "ltration. The security pays a continuousproportional dividend at a constant yield y. Interpretations of the securityinclude that of a stock, and of a unit of a &foreign' currency. In the latter case,S corresponds to the exchange rate, expressed in the form of the &domestic'currency price of one unit of the &foreign' currency, and y corresponds to the

S. Babbs / Journal of Economic Dynamics & Control 24 (2000) 1499}1525 1501

6 It su$ces for this paper for us to consider claims with a single payo! only.

7Harrison and Kreps' analysis required that pay-o!s be square integrable under the originalprobability measure, but also goes through under the alternative requirement that pay-o!s besquare integrable under the EMM. Standard results in stochastic calculus can be used to verify thatthis latter requirement is satis"ed for all contingent claims considered in this paper.

&foreign' interest rate (Garman and Kohlhagen, 1983). Goldman et al. (1979a)assume y"0.

We also assume that riskless borrowing and lending are available at the sameconstant interest rate r50.

It is now well-known, using seminal results of Harrison and Kreps (1979),that, given the above assumptions, there exists a unique pricing operator forclaims contingent on the security price process. Moreover, this pricing operatorcan be represented in terms of the expected value of the discounted payo! 6 fromthe contingent claim, where the expectation is taken under an &equivalentmartingale measure' (EMM), a reassignment of probabilities, under which thesecurity price process (1) can be rewritten as:

dS/S"(r!y) dt#p dZH (2)

where ZH( ) is a standard Brownian motion under the EMM. In particular, ifa contingent claim which gives a payo! <(t@) at some * possibly stochasticstopping* time, t@, is traded at an earlier time t, than the value of the claim att is given by:

<(t)"EHt[e~(t{~t)r <(t@)], (3)

where EHt[ ] denotes the conditional expectation operator at t under the

EMM.7

3. Lookback pay-o4s and valuation of European options

A European (American) lookback call option contracted at time zero a!ordsits holder the right to purchase one unit of the security, on an exercise date equalto (at or before) some speci"ed "xed expiry date ¹, at a price equal to the lowestvalue of S attained between time zero and the exercise date. Hence, as Goldmanet al. (1979a) show, the payo! upon exercise at time t@ is

<#!--

(t@)"S(t@)!S*/&

(t@) (4)

where

S*/&

(t),infMS(u): 04u4tN. (5)


8We commence with puts rather than calls because, by doing so, the ensuing analysis will linkmost neatly into the requisite mathematics in Harrison (1985).

The pay-o! to a lookback put option is analogously given as

<165

(t@)"S461

(t@)!S(t@) (6)

where

S461

(t),supMS(u): 04u4tN. (7)

To value the European put8 we note that t@,¹ and that

S461

(¹)"expMX461

(¹)N

where

X461

(t),supMln S(u): 04u4tN.

Applying Ito's lemma to (2) yields that, under the EMM, ln S( ) followsa straightforward Brownian motion with drift:

d ln S"mH dt#p dZH

where

mH"r!y!12

p2.

We can then obtain the probability distribution function of X461

(¹), condi-tional on X(t) and X

461(t), from e.g. Harrison (1985, Section 1.9, pp. 14}15), as

F(x)"GU(b

1(x))!e2(x~X(t))mH@p2U(!b

2(x)) if x5X

461(t),

0, otherwise,

where U( ) denotes the standard Normal distribution function, and

b1(x)"

x!X(t)!(¹!t)mH

p J¹!t,

b2(x)"

x!X(t)#(¹!t)mH

p J¹!t.

Hence, by substituting into (3),

<165

(t)"e~(T~t)rPR ex dF(x)dx!e~(T~t)yS(t)


9The term &strike bonus option' arises from the potential increase in the price at which the optionallows its holder to sell the security, if and when the security price attains a new supremum.

which, upon evaluating the integral, reveals that <165

(t) can be expressed as thesum of two components:

<165

(t)"P(t)#B165

(t) (8)

where P(t) is the value of an ordinary European put option with strike equal toS461

(t), i.e.

P(t)"!S(t) e~(T~t)yU(z1!pJ¹!t)#S

461(t) e~(T~t)rU(z

1)

with

z1"b

1(X

461(t))

and B165

(t) is the value of what Garman (1987) calls the &strike bonus option',9given for the general case yOr by

B165

(t)"S(t)

a Ge~(T~t)yU(pJ¹!t!z1)!e~(T~t)rA

S461

(t)

S(t) BaU(!z

2)H(9a)

where a"2(r!y)p2 and z2"b

2(X

461(t)) while, for y"r

B165

(t)"S(t)e~(T~t)ypJ¹!t M/(z2)!z

2U(!z

2)N (9b)

where /( ) is the standard Normal density function.To value the otherwise identical call option, we exploit the relation

X*/&

(t),infMln S(u): 04u4tN

"!supM!ln S(u): 04u4tN

and apply the same machinery, obtaining

<#!--

(t)"C(t)#B#!--

(t) (10)

where C(t) is the value of an ordinary European call option with strike equal toS*/&

(t), i.e.

C(t)"S(t) e~(T~t)yU(pJ¹!t!z3)!S

*/&(t) e~(T~t)rU(!z

3)


10The necessary formulae could also be obtained, via L'Hopital's rule, from those for theyOr case.

with z3"b

1(X

*/&(t)) and where, for yOr

B#!--

(t)"S(t)

a Ge~(T~t)rAS*/&

(t)

S(t) BaU(z

4)!e~(T~t)yU(z

3!pJ¹!t)H (11a)

with z4"b

2(X

*/&(t)) while, for y"r

B#!--

(t)"S(t) e~(T~t)ypJ¹!t M/(z4)#z

4U(z

4)N. (11b)

Remark 1. The valuation formulae of Goldman et al. (1979a, b) correspond to thecase y"0(r. The formulae for the general yOr case were independently derivedby Babbs (1986) and Garman (1987). Babbs (1986) also covers the y"r case, forwhich the formulae given for yOr would involve division by zero.10 The observa-tion that the valuation formulae could be so arranged that certain of the termscorrespond to the values of ordinary European options is due to Garman (1987).

Remark 2. Goldman et al. (1979a, b) derived their results using the approach ofCox and Ross (1976a, b) which entails establishing the possibility of a perfecthedge before moving to the arti"cial &risk neutral' probabilities we nowadaysterm the EMM. This forced them to undertake a lengthy technical discussion toestablish that the sensitivity of the option value to S vanishes when S achievesa new extremum. As Goldman et al. (1979b, p. 405) indicated, it is a testimony tothe power of Harrison and Kreps' (1979) results that they enable us to obtainvaluations without any such discussion. See also footnote 11.

4. Di7culties posed by American-style exercise

As in the more familiar case of ordinary puts and calls, the valuation ofAmerican lookback options is complicated by the need to determine simulta-neously the value of the option and the optimal exercise rule. Experience withordinary options suggests a number of approaches, based either on a partialdi!erential equation satis"ed by the option value, or upon a binomial approxi-mation scheme. As we will now show, these approaches do not readily transferto the lookback case.

It is trivial to verify that the value processes of European lookback options, asgiven by (8)}(11b), satisfy

!r<#L<Lt

#(r!y) SL<LS

#

1

2p2S2

L2<LS2

"0 (12)


11Brie#y, the argument runs as follows. The discounted option value process must be a martin-gale under the EMM. Since the information "ltration is Brownian, the martingale is continuous, andhence predictable (see, e.g. Elliott, 1982, Corollary 6.34, p. 57). It must therefore have zero drift (see,e.g. Elliott, 1982, Lemma 11.39, p. 121). By the relevant extension of Ito's lemma (see, e.g. Harrison,1985, Proposition 6.3, p. 67), the drift consists of an absolutely continuous term corresponding to theLHS of (12), plus a term due to the sensitivity of the option value to S when S achieves a newextremum. But such times constitute, pathwise, a set of Lebesgue measure zero (see, e.g. Harrison,1985, Proposition 1.6, p. 15); the two terms can therefore sum to zero only if each separatelyvanishes.

Note the contrast between the ease with which martingale methods enable this result to beobtained, and the lengths to which Goldman et al. (1979a, pp. 1115, 1116, 1124, 1125) had to go usingthe methods available to them. It is unclear how to extend Goldman et al.'s line of argument to theAmerican case.

subject to the respective boundary conditions (4) and (6) for t@,¹. It can beshown that the American options also satisfy (12)11 subject to the additionalrespective early exercise boundary conditions:

<#!--

(t)5S(t)!S*/&

(t), 04t4¹,

<165

(t)5S461

(t)!S(t), 04t4¹.

By analogy with techniques developed for ordinary options, we might then hopeeither to construct a "nite di!erence scheme for these partial di!erential equa-tions (cf. e.g. Brennan and Schwartz, 1977) or obtain an approximate solution bylargely analytical methods (cf. e.g. Barone-Adesi and Whaley, 1987). Unfortu-nately, it is hard to see how the "nite di!erence or analytic approximationapproaches could accommodate the presence of the path-dependent term S

461(t)

(or S*/&

(t) as appropriate) as an argument of < and its derivatives, or in theboundary conditions. (In the case of ordinary options, by contrast, this term isreplaced by the "xed exercise price.)

The same di$culty besets the application to lookback options of the binomialapproximation developed for ordinary options by Cox et al. (1979) (CRR).Because of the Markovian nature of the security price under the EMM, thevalue of an ordinary option depends only on time and the current price of thebinomial branching process over a recombinant lattice, therefore, CRR limitthe number of calculations involved in valuing an ordinary option, to beinggoverned by the number of nodes in the lattice* i.e. (n#1)(n#2)/2 where n isthe number of discrete time steps into which the outstanding life of the option isdivided* rather than by the total number of price paths* i.e. 2n; this a!ordsgreat computational e$ciency. Unfortunately, in the case of lookback options,the need to keep track of current extreme values and not just current spot valuesmeans that not all the necessary information would be carried by individualnodes of the lattice. If the lattice structure for S is retained, therefore, the need tokeep track of the additional, path-dependent, extreme price information will


12Extensive combinatoric analysis would be required to see how far short of O(2n) the growthcould be contained. The &reduced lattice' in Rubinstein (1990) suggests that O(n3) is attainable.(Contrast the O(n2) in this paper.)

cause the number of calculations to grow substantially faster than the number ofnodes * leading to a signi"cant loss of computational e$ciency.12

In the next section, we introduce a change of numeraire, following which wepresent a binomial scheme in a new variable, that achieves for lookback optionslevels of computational e$ciency similar to those which CRR attained forordinary options.

5. A simple binomial scheme for lookbacks

So far, we have been implicitly content to use as numeraire the units ofaccount in which the security price is expressed. We now adopt units of thesecurity as our numeraire. This is most intuitively grasped when the security isa &foreign' currency* our change of numeraire then consists simply of workingin &foreign' rather than &domestic' currency terms. A mere change of numeraire iseconomically neutral * in particular (8)}(11b) still hold and require onlydivision by S(t) to express them in our new units. However, the shift simpli"esthe analysis considerably since the payo!s of lookback options each becomefunctions of single variables rather than of two: thus (4) becomes

<f#!--

(t@)"1!expMM(t@)N (13)

where the superscript reminds us we are working in units of the security, and

M(¹),lninfMS(u): 04u4tN

S(t)

while (5) becomes

<f165

(t@)"expMN(t@)N!1 (14)

where

N(t),lnsupMS(u): 04u4tN

S(t).

It is natural to re-express all price processes in terms of our new numeraire.We therefore work not with S, but with

R( ),1/S( ).


13The key to (17) is that, under the new EMM, the forward price of &domestic' currency, expressedin units of &foreign' currency (i.e. expM(y!r)(¹!t)NR(t)) must be martingale.

14See Harrison (1985) for the requisite mathematics.

R represents the price of cash in terms of units of the security; when S is anexchange rate, R expresses the same exchange rate in the reciprocal form of the&foreign' currency price of one unit of &domestic' currency. With this change, andsimple manipulations,

M(t),ln R(t)!supMln R(u): 04u4tN40 (15)

and

N(t),ln R(t)!infMln R(u): 04u4tN50. (16)

A change of numeraire entails a change of EMM. Under the new EMM,13

d ln R"(y!r!12p2) dt#p dZI (17)

where ZI ( ) is, once more, a standard Brownian motion.As in Section 3, we focus on lookback put options. A precisely analogous

treatment can be applied to lookback calls.Under the EMM, the process for N di!ers from that of R solely by the

presence of a re#ecting barrier at zero:14

dN"(y!r!12

p2) dt#p dZI #d¸ (18)

where

¸(t),supGmaxG!lnR(u)

R(0), 0H: 04u4tH.

Note that this process for N has constant parameters, and that the put pay-o!(14) depends solely on N. To value the lookback put, therefore, we may use N asour sole state variable. We now proceed to construct a binomial approximationscheme based on N.

Consider the evaluation at time t of a lookback put expiring at ¹. We dividethe period from t to ¹ into n discrete time steps of length h,(¹!t)/n.

The CRR approximation to (17) would consist of a binomial lattice in which,

for each time step, ln R( ) rises or falls by an amount pJh with respective &riskneutral' probabilities p and 1!p where

p"expM(y!r)hN!expM!pJhN

expMpJhN!expM!pJhN. (19)

Since (18) di!ers from (17) only by the presence of a re#ecting barrier at zero, weare prompted to form our binomial approximation scheme for N by modifying


Fig. 1. Lattice scheme with a re#ecting barrier.

15Where N(t) is an integer multiple of pJh, notably for new options at t"0, certain nodes of thelattice coincide. This could be exploited to achieve even greater computational e$ciency. In thegeneral case, however, the loss of accuracy caused by approximating N(t) by adjacent integermultiples would appear to outweigh the reduction in the number of nodes that would result fora given number of time steps.

the CRR scheme to incorporate such a barrier. Speci"cally, we require

N(t#( j#1)h)"GN(t#jh)#pJh with probability p,

DN(t#jh)!pJhD with probability 1!p.(20)

This generates a recombinant lattice, with a re#ecting barrier at zero, asillustrated in Fig. 1. The nodes shown with un"lled discs result from an oddnumber of re#ections at the barrier. We label the nodes occurring at the end ofthe jth time step as (0, j ), (1, j ), 2, ( j, j ) with node (0, j) farthest from the barrierat zero and moving monotonically closer (see Fig. 1). The overall number ofnodes for an n-time step lattice is precisely the same as that in the CRR latticefor R.15

Although there is no one-to-one correspondence between nodes in our latticefor N and the CRR lattice for R, each movement in N nevertheless mirrorsa movement in R. Hence the probabilities p and 1!p are the correct &riskneutral' probabilities in the lattice for N. (This is easily con"rmed directly.) Thus


16We use node references as function arguments in the obvious way, and drop put subscripts andf superscripts.

the valuation of the lookback put takes place by backwards recursion over thelattice as follows. We deal with the case of European exercise "rst, and thenindicate the modi"cations for American exercise.

The transformed payo! equation (14) gives us16

<(i, n)"expMN(i, n)N!1 ∀i (21)

the terminal values of N being related to the node index i by

N(i, n)"GN(t)#(n!2i)pJh, 04i4k,

(n!i#1!f )pJh, odd (i!k)'0,

(n!i#f )pJh, even (i!k)'0,

(22)

where

k ( f )"integer (fractional) part of N(t)/pJh. (23)

For time steps j"n, n!1, 2, 1, we apply the backwards recursion:

<(i, j!1)"e~yhMp<(i, j)#(1!p)<(id, j)N i"0, 2, j!1 (24)

where (id, j) is the node such that

N(id, j)"DN(i, j!1)!pJhD

i.e. where

id"G

i#2 if k4i(j!1,

i#1 otherwise.

The value of the option (in units of our numeraire) is given by <(0, 0). For anAmerican lookback put, we replace the RHS of (24) by the immediate exercisevalue: expMN(i, j!1)N!1 at any node at which that value is greater. (N(i, j!1)is given by (22), with j!1 replacing n.)

6. Numerical examples

Table 1 shows results of using our binomial scheme, with various numbers oftime steps, to value new 6-month European and American call and put options.The row for in"nite time steps contains the analytic solution, where available.

Note that all the results for 150 time steps are within two pennies of the mostaccurate results quoted. In proportion to the initial security price, this is the


Table 1Lookback option values under continuous sampling

Calls Puts

Time steps European or American! European American

10 12.88 8.97 10.2750 13.13 9.23 10.19

100 13.16 9.26 10.18150 13.17 9.27 10.18200 13.18 9.28 10.18500 13.19 9.29 10.17

6000 13.19 9.29 10.17R 13.19 9.29 na

!We assume a non-dividend paying stock wih initial price 100 and volatility 20%. The interest rateis 10%. (For European options, this permits comparison with Table 1 on p. 1902 of Conze andViswanathan (1991)). Conze and Viswanathan (1991) have shown that an American lookback call ona non-paying stock is worth the same as its European counterpart. Our binomial scheme alsoproduces identical values.

17 I.e. maximum (minimum) in the case of a put (call) option.

18A lookback option is said to be at-the-money if the current security price represents theextremum observed so far.

same degree of accuracy as that reported for the CRR scheme for valuingordinary options, as reported by Cox et al. (1979, pp. 258, 259). Further results(not reported here) indicate that the relative accuracy of our scheme holds upwell for away-from-the-money options or those of shorter maturity, but falls o!slightly for higher levels of volatility. Nevertheless, bearing in mind that thenumber of nodes in our binomial scheme grows with the number of time steps inprecisely the same way as in the CRR scheme, we feel fully justi"ed in claimingessentially the same high level of computational e$ciency.

Garman (1987) has conjectured that the value of an American lookbackoption is well approximated by the sum of the values of an American ordinaryoption with strike equal to the extreme17 security price so far achieved, and ofwhat Garman terms the &strike bonus option' * a European option whosepay-o! is the increment between the current and "nal extreme security prices, orbetween the "nal spot price and "nal extreme price, whichever is the smaller.This conjecture can be more simply expressed as the surmise that the earlyexercise premium on an American lookback option is well approximated by theearly exercise premium on an American ordinary option with strike equal to theextreme security price so far achieved. From the last two columns of Table 1,we deduce that the early exercise premium on the six-month at-the-money18


lookback put is 10.17!9.29"0.88; the early exercise premium on the corres-ponding ordinary put is 0.52. We feel that the quality of approximation in thisparticular case is rather poor. To be fair, however, Garman's conjecture wasadvanced in the context of a paper on lookback options on foreign exchange.More typical values of interest rates and volatility in a foreign exchange contextmight be, say, r"10%, y"8%, p"11%; for these parameter values the earlyexercise premia on the six-month at-the-money lookback and ordinary puts are0.20 and 0.12, respectively. As in the previous case, the early exercise premiumon ordinary options is only roughly 60% of the corresponding "gure forlookback options. While &well approximated' allows a certain latitude, our viewis that Garman's conjecture must be rejected.

7. Convergence

The burgeoning literature on the convergence of discrete-time valuationmethods for contingent claims to their continuous-time counterparts (see, e.g.Cox et al., 1979; Cutland et al., 1991; He, 1990; Hull and White, 1990; Nelsonand Ramaswamy, 1990, does not cover path-dependent payo!s such as those onlookbacks. Our reformulation of the lookback valuation problem, in Section 5,removed the path dependency at the expense of basing the analysis on processessubject to a re#ecting barrier at zero. The only kind of barriers considered in theextant convergence literature are where the di!usion coe$cient vanishes atthe barrier (see especially Nelson and Ramaswamy, 1990). The question of theconvergence of our binomial scheme for lookbacks, therefore, is novel. Ourmethod of resolving it appears to be novel also.

An exhaustive treatment of convergence issues is beyond the scope of thepresent paper. We will con"ne ourselves to what we regard as the key issue: asthe number of time steps increases to in"nity, does the resulting value ofa European lookback put, as computed under our binomial scheme, converge toits continuous-time counterpart as given by the analytical results of Section 3?The demonstration we will present carriers over, mutatis mutandis, to lookbackcalls, and extends to American lookbacks by arguments such as those inCarverhill and Webber (1990).

The mathematical machinery used to establish convergence has often beenquite elaborate. Thus, for example, Cutland et al. (1991) use non-standardanalysis; He (1990) uses martingale convergence theory and requires stringentpiecewise smoothness and growth conditions on the pay-o! of the contingentclaim; Babbs (1990) makes extensive use of moment generating functions anda novel style of dominated convergence. By contrast, the method of resolvingconvergence questions in this paper uses only basic mathematics. It cancertainly be applied to the binomial scheme of Cox et al. (1979) for ordinaryoptions, and we conjecture that it is of widespread applicability.


19We show in the Appendix that (28) holds even at the re#ecting barrier, where

N(id, j)ON(i, j!1)!pJh.

We present here an intuitive version of our argument; and indicate thetechnical details requiring additional attention. That attention is relegated tothe Appendix.

Placing hats over values of the option, according to our binomial scheme, toemphasise the distinction from the true value at the corresponding values ofN and of time, we may de"ne the error at node (i, j) by

g(i, j),<K (i, j)!<(N(i, j), t#jh). (25)

Substituting (25) into the fundamental recursion (24), i.e.

<(i, j!1)"e~yhMp<(i, j)#(1!p)<(id, j)N,

we obtain

g(i, j!1)"e~yhMp<(N(i, j), t#jh)#(1!p)<(N(id, j), t#jh)N

!<(N(i, j!1), t#( j!1)h)#e~yhMpg (i, j)#(1!p)g(id, j)N(26)

Taking Taylor's expansions of the < terms on the RHS of (26) around(N(i, j!1), t#( j!1)h), we "nd that

<(N(i, j), t#jh)"<#hL<Lt

#pJhL<LN

#

1

2p2h

L2<LN2

#O(h3@2) (27)

where all functions on the RHS of (27) are evaluated at (N(i, j!1), t#( j!1)h;likewise,19

<(N(id, j), t#jh)"<#h

L<Lt

!pJhL<LN

#

1

2p2h

L2<LN2

#O(h3@2). (28)

Substituting (27) and (28) into (26), and applying a "rst-order Taylor's expansionto the "rst instance of e~yh, and to p as given by (19),

g(i, j!1)"hG!y<#L<Lt

#Ay!r!1

2p2B

L<LN

#

1

2p2h

L2<LN2H#O(h3@2)

#e~yhMpg(i, j)#(1!p)g(id, j)N. (29)

We show in the Appendix that < satis"es the PDE:

!y<#L<Lt

#Ay!r!1

2p2B

L<LN

#

1

2p2

L2<LN2

"0. (30)


20This problem* among others* is probably why Cox and Rubinstein (1985, pp. 208}209) referto an argument based on Taylor's expansions as o!ering only &intuitive con"rmation' that the CRRbinomial technique for valuing ordinary options converges to the Black}Scholes PDE.

21The closest approximations to continuous availability are provided by a modest number ofcurrencies and securities traded on exchanges. Certain foreign exchange rates are monitored byReuters on a more or less continuous basis, as currency trading activity moves between successive"nancial centres around the world. Even these data, however, are interrupted by weekends and bankholidays. Various futures and other exchanges monitor the prices of all transactions throughouttrading hours, and even publish daily high and low prices.

Hence,

Dg(i, j!1)D4O(h3@2)#maxMDg(l, j)D: l"0,2, j!1N. (31)

The interpretation of (31) is that the backwards valuation recursion adds anerror of at most O(h3@2) at each step; since the number of steps is inverselyproportional to h, the accumulated error at the base node (0, 0) is thereforeO(h1@2) and so tends to zero.

The validity of the "nal step in the above argument depends on the variouserrors being uniformly of the stated orders (i.e. in the sense of being uniformlybounded upon division by the speci"ed powers of h). This in turn depends on thederivatives of < being su$ciently well-behaved over the region of the binomialapproximation. Unfortunately, the time derivative of < is unbounded as weapproach the expiry date ¹, and the space derivative is unbounded in N.20 Inthe Appendix, we detail a solution to this problem, based on a purely technicalmodi"cation of the binomial scheme so that it operates not over the timeinterval [t, ¹ ] but over [t, ¹!d] for arbitrarily small d, and so that the optionvalue is replaced by its &discounted intrinsic value' for very large N.

Remark. The PDE (30) could, of course, be used as the basis of an alternativenumerical technique for valuing lookbacks. We prefer our binomial scheme,however, for two reasons. Firstly, binomial methods maximize intuitive clarity.Secondly, we can adapt our binomial scheme (see the next Section) to cope withbasing the security price extremum upon "nite sampling; by contrast, thevalidity of the PDE is con"ned to the continuous sampling case.

8. Finite sampling

So far, this paper has maintained the assumption, made by the existingliterature, that lookback contracts are based on the extrema of the entirecontinuous sample path of the underlying security price. In practice, continuousand undisputable records of securities prices are generally unavailable,21 andlookback contracts are often based on the extrema of a "nite sample of prices,


22Once more we focus on puts; analogous arguments apply to calls.

taken at speci"ed intervals, typically daily. Obviously, limiting the set of pricesof which the extremum is taken reduces the payo!s * and thus earlier values* of the options, but the magnitude of the reduction has not previously beeninvestigated.

As we shall see below, a European lookback option under "nite sampling canbe regarded as a specialised variant of an option on the maximum of severalrisky assets, one for each outstanding sampling date. Where the number ofsampling dates is large, however, this insight is not of great computationalbene"t. In fortunate contrast, our binomial approximation scheme can beadapted quite readily; moreover, it can be applied to American lookbacks also.

Consider a European lookback put22 option to sell one unit of the securityat time ¹ at the maximum of the security prices observed at times¹

1(¹

2(2(¹

Q. The payo! is

<(¹)"maxMS(¹1)!S(¹), 2, S(¹

Q)!S(¹), 0N. (32)

Making the change of variables introduced in Section 5 enables us to transform(32) into

<f(¹)"maxGR(¹)

R(¹1)!1, 2,

R(¹)

R(¹Q)!1, 0H. (33)

Thinking of the ratios R(¹)/R(¹1), 2, R(¹)/R(¹

Q) as &asset prices', the form

of (33) is precisely that of the pay-o! on a call option on the maximum of Q riskyassets, with unit strike price. This is not just an idle thought, since, under theEMM, the logarithms of these &asset prices' are jointly Normally distributedwith conditional (i.e. given information available at t) means, for q"1, 2, Q:

EtCln

R(¹)

R(¹q)D"G

lnR(t)

R(¹q)#Ay!r!

1

2p2B(¹!t), ¹

q4t,

Ay!r!1

2p2B (¹!¹

q), ¹

q't,

(34a)

and (co-)variances, for q1, q2"1, 2, Q:

covt Cln

R(¹)

R(¹q1

), ln

R(¹)

R(¹q2

)D"(¹!maxMt, ¹q1

, ¹q2

N)p2. (34b)

We may therefore apply the analytical valuation results for options on themaximum of several risky assets, due to Johnson (1987) and Babbs and Salkin


Fig. 2. Modi"ed lattice scheme for "nite sampling.

(1987). Unfortunately, those results involve multidimensional Normal integrals,and thus become computationally cumbersome in high dimensions, i.e. when thenumber of outstanding sampling dates is large. Nevertheless, for small numbersof sampling dates, the above insight will provide us with a check on the accuracyof our modi"ed binomial approximation scheme, to which we now turn.

Fresh consideration of the pay-o! of the lookback put reveals that we maystill write (allowing for exercise of American options before ¹):

<f(t@)"expMN(t@)N!1

as long as we make the modi"ed de"nition:

N(t),ln R(t)!minMln R(¹q): ¹

q4tN. (35)

From (35), we can see that, between sampling dates, the increments of N areidentical to those of ln R( ). At sampling dates, if N has become negative, it isreset to zero. As in the continuous sampling case, N is the sole state variablerequired.

The modi"cations required to our binomial approximation scheme are intuit-ively obvious: rather than re#ecting N upwards whenever it would otherwisebecome negative, we allocate sampling dates to time points (i.e. the ends of timesteps) and simply reset to zero any negative values of N occurring at those times.


23Fig. 2 relates to the &normal' case where k50. The formal description is complicated by theneed to cover also the &inverted' case where k(0 which can arise when the valuation date t liesbetween sampling dates.

24Clearly, this entire allocation procedure involves an element of approximation, but whoseimpact will tend to zero as the time step length is reduced.

25The "rst case of this node count is obvious. The second case #ows from the fact that all nodesare reset to zero at jH. The third case adds the additional nodes which arise at j, arising from nodesnot reset to zero at jH. The fact that we have forced both jH and j@ to be even provides importantsimpli"cations.

The resulting kind of lattice is illustrated in Fig. 2, and is described formally asfollows.23

We allocate the earliest (outstanding) sampling date to the earliest subsequenttime point that falls an even number of time steps before the expiry of the option.Each later sampling date is allocated to the time point at one or other end of thetime step in which it falls; again, the time point is chosen so as to lie an evennumber of time steps before the expiry of the option. The emphasis on evennumbers of time steps avoids the proliferation of nodes that would otherwiseoccur if some values of N were reset to zero, while positive values of N at nodes

in#uenced by earlier resettings were at odd multiples of pJh ; at the same time,we preserve exactly the timing of the "nal sampling date in the especiallysensitive case where it coincides with the expiry of the option.24

Let jH be the number of the time step to whose end we allocate the "rst samplingdate (if any) to cause some values of N to be reset to zero. More formally, de"neA,M j'minMk, 0N: n!j even, and M¹

1, 2, ¹

QNW(t, t#jh]OHN

then

jH"GminM j3AN, AOH,

R, A"H.

We say that &sampling occurs at' l5jH if n!l is even and M¹1, 2, ¹

QNW

[t#(l!1)h, t#lh]OH.Clearly, the number, I

j, of nodes at the end of the jth time step, and the

corresponding values of N, depend on whether the time step is before or after thejHth. In the latter case, the number and values of the nodes depend on the lengthof time since the most recent sampling date and upon whether all or only someof the nodal values of N were reset to zero at jH. Thus25

Ij"G

j#1, if j(jH,

2j!j@#2!jH

2, if j5jH and k#jH(0,

2j!j @#2!jH

2#integer part of

2j!j@#2#k

2, if j5jH and k#jH50


26We allow here for the case that the expiry date ¹ is not a sampling date, creating a possibility ofa lookback option expiring out-of-the-money.

where

j@,maxMl: jH4l4j and &sampling occurs at' lN, j"jH, 2, n.

The values of N at the end of the jth time step, indexed (0, j ), 2, (Ij!1, j),

are given for j(jH, by

N(i, j)"N(t)#( j!2i)pJh, i"0, 2, Ij!1

while, for j5jH, if k#jH(0,

N(i, j)"( j!jH!2i)pJh, i"0, 2, Ij!1

and, if k#jH50,

N(i, j)"GN(t)#( j!2i)pJh, i"0, 2, I

jH!2,

( j!jH!i#IjH!1)pJh, odd (i!I

jH#2)'0,

N(t)#( j!i!IjH#2)pJh, even (i!I

jH#2)'0.

The evolution of N over the lattice is described, in terms of risk neutralprobabilities, by the relation that, if N(t#( j!1)h)"N(i, j!1), then

N(t#jh)"GN(i

u, j) with probability p,

N(id, j) with probability 1!p,

where for i5Ij!1 (i.e. for the lowest nodes just prior to sampling),

iu"i

d"I

j!1 whereas, for i(I

j!1, i

u"i and

id"G

i#2, j'jH5!k and IjH!24i(I

j!2,

i#1, otherwise.

We evaluate the option over this lattice by the terminal condition26

<(i, n)"maxMeN(i, n)!1, 0N, i"0, 2, In!1

and the backwards recursion over j"n, n!1, 2, 1:

<(i, j!1)"e~yhMp<(iu, j)#(1!p)<(i

d, j), i"0, 2, I

j!1.

The option value (in units of our numeraire) is given by <(0, 0). The resultspresented in Table 2 and Fig. 3 illustrate the impact of "nite sampling upon thevalues of lookback options.

The most striking feature to emerge from Table 2 is that alterations insampling frequency have a signi"cant impact on option value. Even a switch


Table 2Lookback option values under "nite sampling!

European puts American puts

Binomial time steps Numericalintegration

Binomial time steps

Sampling 200 2000 6000 200 2000 6000

Start/end 3.39 3.40 3.40 3.40 3.92 3.92 3.92Quarterly 4.55 4.57 4.57 4.57 5.18 5.19 5.19Monthly 6.15 6.18 6.19 6.19 6.91 6.94 6.94Weekly 7.57 7.65 7.66 na 8.42 8.49 8.50Daily 8.25 8.62 8.64 na 9.13 9.49 9.51Continuous 9.28 9.29 9.29 9.29" 10.18 10.17 10.17

!This table employs the same stock price, interest rate and volatility assumptions as Table 1, andvalues the same European and American put options, save that, except in the "nal line of Table 2, thecontract is based on the maximum stock price observed at various regular intervals* correspond-ing roughly to the frequencies indicated* rather than continuously. The fourth column of values isbased upon the insight that a lookback put option based on "nite sampling at Q dates can be viewedas an option on the maximum of Q assets, and uses numerical integration to compute the expectedpay-o! (under the EMM of Sections 2 and 3). The integration scheme is an adaptation of that ofHaselgrove (1961) and required at most 3500 computations of the integrand to converge to twodecimal places."Closed-form formula as in Table 1.

27This explanation does, however, beg the question of why, under daily sampling, a binomialvaluation using 200 time steps does not produce a value closer to the result under continuoussampling. Intuitively expressed, the answer is that, whereas under continuous sampling, N isre#ected upwards as soon as it reaches zero, under "nite sampling N is merely reset to zero at thenearest convenient time point; for the parameter values in the table, the cumulative e!ect of theselatter more &feeble' re#ections is appreciable. Moreover, we decided that &convenient' time pointsoccur only at even numbers of time steps from expiry; thus, for example, with only 200 time steps inall, only 100 provide &convenient' time points * coming closer to sampling every two days ratherthan each day, let alone continuously.

from continuous to daily sampling reduces option values by about 0.65.Throughout the table, changes in sampling frequency a!ects European andAmerican option values more or less equally. The results obtained by numericalintegration con"rm the validity of the adapted binomial scheme. For "nite butfrequent sampling, larger numbers of time steps (approximately 2000 for weeklyor daily sampling) are required to obtain the degree of accuracy obtained with150 steps under continuous sampling; this should not surprise us since the "nerdetail of a "nite sampling scheme is inevitably lost unless the number of timesteps is several times the number of sampling dates.27


Fig. 3. Lookback option values: continuous versus daily sampling.

Fig. 3 illustrates the varying impact of "nite sampling as the maturity of anoption shortens by plotting against time the values of an at-the-money Euro-pean put under continuous and daily sampling, using the same stock price,interest rate and volatility data as in the tables. The striking feature of the graphis the way the di!erence between the value of the option under continuous anddaily sampling holds up until just before expiration: from 0.66 at six months, thedi!erence declines only very gradually until a few days before expiry; with oneday to expiry the di!erence has only fallen to 0.42.

9. Summary and concluding remarks

The paper by Goldman et al. (1979a) on European lookback optionsprompted the commercial introduction of the contingent claims they analysed.The virtual absence of American lookback options may be due in part tothe unavailability of a straightforward and computationally e$cient valua-tion technique. We have now provided just such a technique in the form ofa binomial approximation scheme. Moreover, the method by which we estab-lished convergence was novel, direct, and straightforward; it may well be ofwider use.

Existing research on lookback contracts has assumed that they are based oncontinuous sampling of the price of the underlying security. We have adaptedour binomial scheme to cover contracts under which sampling occurs at a "xed"nite set of dates; the impact on option values was signi"cant. We note that thenature of the sampling regime may also have a signi"cant impact on other typesof path-dependent contingent claims, such as the related claims analysed by


Conze and Viswanathan (1991), and* especially when the underlying securityprice is close to the barrier/dropout level* on &barrier' and &dropout' options.

This paper has been restricted to binomial approximation, largely for ease ofexposition. A number of authors (see e.g. Kamrad and Ritchken, 1991) havereported gains in computational e$ciency from replacing binomial approxima-tion schemes for conventional options by multinomial ones. We would expectsimilar gains to be achievable for lookback options.

We hope that our paper will stimulate attempts to develop convenientapproximation techniques for other types of contingent claims, e.g. &averageprice' and &average strike' options, whose payo! functions contain path-dependent terms.

10. Disclaimer

Views expressed in this paper are those of the author, and not necessarilythose of any organisation to which he is or has been a$liated.

Acknowledgements

The author wishes to thank Rodney Coleman, Chris Rogers and MichaelSelby for interesting discussions on random walks with barriers, and JohnBarratt, Gerald Moore and Paul Wilmott for bringing Haselgrove (1961) to hisattention.

Appendix A

The treatment of convergence in Section 7 deferred a number of matters forfurther attention in this Appendix; we deal with them in turn. We "rst establishthat (28) remains valid where (i, j!1) is a node adjacent to the re#ecting barrier.

For such a node, as noted in footnote 19, N(id, j)ON(i, j!1)!pJh; instead,

we have N(i, j!1)"jpJh where j equals either f or 1!f [as de"ned in (23)],

and N(id, j)"(1!j)pJh. The equivalent expansion to (28) is

<(N(id, j), t#jh)"<#h

L<Lt

#(1!2j)pJhL<LN

#

1

2(1!2j)2p2h

L2<LN2

#O(h3@2) (A.1)

where all functions on the RHS are evaluated at (jpJh, t#( j!1)h).


Now, if we make a Taylor's expansion of L</LN about (jpJh, t#( j!1)h)and exploit the fact that (L</LN)(0, u)"0, ∀u we obtain

0"L<LN

(0, t#( j!1)h)"L<LN

(jpJh, t#( j!1)h)

!jpJhL2<LN2

(jpJh, t#( j!1)h)#O(h). (A.2)

Substituting (A.2) into (A.1), we can re-arrange to obtain (28) asrequired.

Secondly, we establish that the option value < (as measured in units of thesecurity price) satis"es the PDE (30).

As noted in Section 5, following our change of numeraire, N serves as the solestate variable for <, and the process for N is given, under the EMM, by (18).Hence, applying the relevant extension of Ito's lemma (see, e.g. Harrison, 1985,Proposition 6.3, p. 67) to the discounted option value e~yt< as a function ofN and time, we obtain

d(e~yt<)"e~ytG!y<#L<Lt

#Ay!r!1

2p2B

L<LN

#

1

2p2

L2<LN2H dt

#e~ytL<LN K

N/0d¸#e~ytp

L<LN

dZI . (A.3)

Now, the discounted option value must be a martingale under theEMM. By arguments identical to those set out in footnote 11, therefore, the"rst and second terms on the RHS of (A.3) must separately vanish.The coe$cient of dt in the former of those terms is precisely the LHS of(30).

The third, and perhaps most important task of this Appendix is to modify ourbinomial scheme so that the error terms arising at each time step, as expressed in(27)}(31), are uniformly of the orders stated, while ensuring that the errorspropagated from the values of g( ) at the boundaries of the binomial scheme cansimultaneously be made arbitrarily small. As indicated in Section 7, this willcomplete our proof of convergence.

Consideration of (8)}(9a) reveals that, from the expiry date, ¹, where the timederivative of < does not exist,< is in"nitely di!erentiable, and thus the Taylor'sexpansions re#ected in (27)}(31) hold uniformly on compact sets. We will seek toexploit this.

Close to the expiry date, ¹, or for large values of N at any time, theuncertainty as to the eventual pay-o! of the option shrinks towards zero.Formally, we de"ne H as the di!erence between the option value < and the


28The same convergence properties emerge in the y"r case, for which (9b) applies in place of (9a).

option's &discounted forward intrinsic value':

H(N, u),<(N, u)!e~(T~u)yMeN~(T~u)(y~r)!1N. (A.4)

If we re-express (8)}(9a)28 in units of the security price, and in terms of N, andsubstitute in (A.4), elementary manipulations yield

H(N, u)"A1#1

aBe~(T~u)yU(pJ¹!u!z1)!eN~(T~u)rU(!z

1)

!

eaN~(T~u)r

aU(!z

2)

P0 as NPR, uniformly on [t, ¹].

Moreover, we can deduce from (A.4) that H( ) is a continuous function whichvanishes on u"¹. Over any bounded range of values of N, therefore, H( ) tendsuniformly to zero as uP¹.

We are now ready to present our modi"ed binomial scheme. For any e'0,choose NH, d'0 such that:

DH( ) )D(e2

outside [0, NH]][t, ¹!d].

Run the binomial scheme over the interval [t, ¹!d] instead [t, ¹] as pre-viously, and, at the "nal time step ( j"n) and for any node (i, j) at whichN(i, j)'NH, set the option value equal to the &discounted forward intrinsicvalue' i.e. set

<K (i, j)"expMN(i, j)!(¹!t!jh)rN!expM!(¹!t!jh)yN.

Then, assuming without loss of generality that we con"ne attention toh4 some h

0, we know that all our Taylor's expansions are taking place

within the compact set [0,NH#pJh0]][t, ¹!d] and thus hold uni-

formly. Moreover, on any boundary of our binomial scheme * by which wemean at j"n or for N(i, j)'NH, the error term g(i, j) is bounded by e/2.We deduce that (31) holds uniformly in h, and by recursion across reducing


values of j gives us:

Dg(i, j)D"O(h1@2)#e/2(e as hP0

thus establishing our desired convergence result.

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