American options with lookback payoff - Department of Mathematics

AMERICAN OPTIONS WITH LOOKBACK PAYOFF

MIN DAI∗ AND YUE KUEN KWOK†

Abstract. We examine the early exercise policies and pricing behaviors of one-asset Americanoptions with lookback payoff structures. The classes of option models considered include floatingstrike lookback options, Russian options, fixed strike lookback options and pricing model of dynamicprotection fund. For each class of the American lookback options, we analyze the optimal stoppingregion, in particular the asymptotic behavior at times close to expiration and at infinite time toexpiration. The inter-relations between the price functions of these American lookback options areexplored. The mathematical technique of analyzing the exercise boundary curves of lookback optionsat infinitesimally small asset value is also applied to the American two-asset minimum put optionmodel.

Key words. Lookback options, American feature, free boundary problems, two-asset minimumput option

AMS subject classifications. Mathematics Subject Classifications (1991): 90A09, 91B28,93E20

1. Introduction. In this paper, we consider the theoretical analysis of the op-timal exercise policies of an American option with lookback payoff. An Americanlookback option involves the combination of two exotic features: early exercise fea-ture and lookback feature. Like other American option models, the analysis of anAmerican lookback option requires the solution of a free boundary value problem.The solution procedure involves the determination of the free exercise boundary thatseparates the stopping region and continuation region. The analysis is further com-plicated by the presence of the path dependent lookback state variable. For floatingstrike lookback options, the analysis is easier since the dimensionality of the pricingmodel can be reduced through homogeneity of the price function. This is achieved bytaking the asset price as the numeraire. However, for American fixed strike lookbackoptions, the exercise boundary is a two-dimensional curve in the state space describedby the asset price and the lookback state variable.

Several earlier papers on American lookback options concentrate on the analysisof the Russian option [7, 17, 18], which is essentially a perpetual zero-strike fixed strikelookback call option. There have been only a few papers which analyze the optimalexercise behaviors of finite time American lookback options. Yu et al. [22] developfinite difference algorithms to compute the exercise boundaries of both Americanfixed strike and floating strike lookback options. In a sequel of two papers [15, 16],Lai and Lim propose the Bernoulli walk approach to compute the price functionsand optimal exercise boundaries of American fixed strike and floating strike lookbackoptions. They also obtain analytic price formulas for American lookback optionsusing a decomposition, which expresses the price as the sum of the correspondingEuropean value and an early exercise premium. Dai et al. [4] analyze the exercisepolicies of American floating strike lookback options with quanto payoff. These quantooptions involve an underlying foreign currency asset but the payoffs are denominatedin domestic currency.

∗Department of Mathematics, National Singapore University, Singapore 117543([email protected]).

†Correspondence author. Department of Mathematics, Hong Kong University of Science andTechnology, Clear Water Bay, Hong Kong, China ([email protected]).

1

2 MIN DAI AND YUE KUEN KWOK

We would like to provide a more comprehensive and thorough analysis of theexercise behaviors of the commonly traded American lookback options. Our analysisframework relies more on the partial differential equation approach, as opposed tothe usual stochastic approach in most earlier works (say, [14, 15]). For the sake ofcompleteness, we attempt to provide a comprehensive list of analytic properties of theexercise boundaries and stopping regions of the lookback option models. The classesof American lookback option models considered in this paper include the floatingstrike and fixed strike lookback call and put options, Russian options and pricingmodel of dynamic protection fund. We analyze the exercise boundary of each class oflookback options, in particular the asymptotic behavior at times close to expirationand at infinite time to expiration. The inter-relations between the price functions ofthese American lookback options are explored. We observe that our mathematicaltechnique developed for analyzing the exercise boundary at infinitesimally small assetvalue for lookback options can be extended to American two-asset minimum putoption model. For all types of American lookback options considered in this paper, weperformed numerical calculations to compute the corresponding exercise boundaries.These plots of exercise boundaries serve as the verification to all results derived fromthe theoretical studies of the optimal exercise policies.

2. Floating strike lookback options. In this section, we explore some analyticproperties of the price functions and optimal exercise policies of the American floatingstrike lookback options. The usual assumptions of the Black-Scholes option pricingframework are adopted in this paper. Let S denote the price of the underlying assetof the lookback option, whose stochastic dynamics under the risk neutral measure isgoverned by

dS

S= (r − q)dt + σ dZ,(2.1)

where t is the calendar time, r is the riskless interest rate, σ and q are the volatilityand dividend yield of S, respectively, and Z is the standard Wiener process. We writeτ as the time to expiry, 0 ≤ τ < ∞. Let m and M denote the realized minimumvalue and realized maximum value, respectively, of the asset price over the lookbackmonitoring period (continuous monitoring is assumed) up to the current time. Thepayoff functions of the American floating strike lookback call and lookback put aretaken to be

(αS − m)+ and (M − αS)+

respectively, where α is a positive parameter value, 0 < α < ∞, and x+ = max(x, 0).When α = 1, we recover the usual lookback payoffs. While lookback options areless attractive to investors due to their high option premium, the parameter α allowsflexible adjustment of the resulting option premium. For example, we may take αto be less (greater) than one in the floating strike call (put) payoff so as to achieveoption premium reduction. Furthermore, the addition of the parameter α in thepricing model facilitates our asymptotic analysis of the exercise boundary curves atthe limit of infinitesimally small asset value.

2.1. American floating strike lookback call. Let Cf`(S, m, τ ) denote theprice function of an American floating strike lookback call with payoff (αS − m)+.The linear complementarity formulation that governs Cf`(S, m, τ ) is given by (see [12]

AMERICAN OPTIONS WITH LOOKBACK PAYOFF 3

and [21])

∂Cf`

∂τ−LCf` ≥ 0, Cf` ≥ αS − m,

(2.2) (∂Cf`

∂τ− LCf`

)[Cf` − (αS − m)] = 0, S > m > 0, τ > 0,

with auxiliary conditions:

∂Cf`

∂m

∣∣∣∣S=m

= 0(2.3)

Cf`(S, m, 0) = (αS − m)+.

The operator L is defined by

L =σ2

2S2 ∂2

∂S2+ (r − q)S

∂

∂S− r.

Note that the payoff upon early exercise is guaranteed to be positive so that we canreplace the payoff function (αS −m)+ by αS −m. However, we cannot do so for theterminal payoff at τ = 0. The dimension of the above formulation can be reduced byone if we define the following transformation of variables:

η =m

Sand Cf`(η, τ ) =

Cf`(S, m, τ )S

.(2.4)

This is equivalent to take S as the numeraire. The new linear complementarity for-mulation for Cf`(η, τ ) is given by

∂Cf`

∂τ− LCf` ≥ 0, Cf` ≥ α − η,

(2.5) (∂Cf`

∂τ− LCf`

)[Cf` − (α − η)] = 0, 0 < η < 1, τ > 0,


∂Cf`

∂η

∣∣∣∣η=1

= 0

(2.6)Cf`(η, 0) = (α − η)+,

where the operator L is given by

L =σ2

2η2 ∂2

∂η2+ (q − r)η

∂

∂η− q.

RemarkThe normal reflection condition in Eq. (2.6) plays a crucial role in distinguishing theoptimal exercise policies of American lookback options from usual American options.The auxiliary condition is derived from the observation that the lookback option valueis insensitive to the running extremum value when the current asset value equals the


extremum value. This is because the probability that the current extremum valueremains to be the realized extremum value at maturity is essentially zero when thecurrent asset value and running extremum value are equal (see [10]). In a more recentwork, Peskir [17] presents a proof on the normal reflection condition for the finite timeRussian option. A similar proof can be mimicked for an American lookback optionwith more general lookback payoff.

The holder optimally exercises the lookback call whenever S reaches sufficientlyhigh level. In terms of η, the holder chooses to exercise when η ≤ η∗, where thethreshold η∗ has dependence on τ . The domain of the pricing model can be dividedinto two regions: the stopping region S = {(η, τ ) : 0 < η ≤ η∗(τ ), 0 < τ < ∞} insidewhich it is optimal to exercise the option and the continuation region SC = {(η, τ ) :η∗(τ ) < η ≤ 1, 0 ≤ τ < ∞} inside which it is optimal to continue to hold the option.Upon exercise, we have Cf` = α − η so that the stopping region is defined by

S = {(η, τ ) : 0 < η ≤ 1, 0 ≤ τ < ∞ and Cf`(η, τ ) = α − η}.

The analysis of the optimal exercise policies amounts to the analysis of the analyticproperties of η∗(τ ) that separates the continuation and stopping regions. Some of theanalytic properties of η∗(τ ) are summarized in Proposition 2.1.

Proposition 2.1The exercise boundary η∗(τ ; α) of the American floating strike lookback call optionobserves the following properties:

(i) Suppose (η, τ ) ∈ SC , then (λ1η, λ2τ ) ∈ SC for all λ1 ≥ 1, λ2 ≥ 1.(ii) The line η = 1 always lies inside SC for finite value of α.(iii) The behavior of η∗(τ ; α) near expiry, τ → 0+, is given by

η∗(0+; α) = min(1, α,

q

rα)

.

When q > 0, η∗(0+; α) is guaranteed to be positive so that there exists at least aline segment: τ = 0, where 0 < η < η∗(0+; α), in the stopping region. Property (ii)reveals that the line η = 1 lies in the continuation region. Hence, we can conclude thatboth the continuation and stopping regions exist in the η-τ plane. Further, by virtueof (i), the free boundary η∗(τ ; α) that separates the stopping and continuation regionscan be deduced to be monotonically decreasing with respect to τ . In conclusion, forq > 0, there exists the monotonic free boundary η∗(τ ; α) such that Cf` = α − ηfor η ≤ η∗(τ ; α), τ > 0. The details of the proof of Proposition 2.1 is presented inAppendix A. Further asymptotic properties of η∗(τ ; α) with respect to τ → ∞ andα → ∞ are stated in Proposition 2.2

Proposition 2.2When q > 0, the asymptotic behaviors at τ → ∞ and α → ∞ of the exercise boundaryη∗(τ ; α) of the American floating strike lookback call option are summarized as follows.

(i) Write η∗∞(α) as limτ→∞ η∗(τ ; α); η∗

∞(α) is given by the solution of the rootinside the interval (0, 1) of the following algebraic equation

(η∗∞)λ+−λ− =

λ+

λ−

(1 − λ−)η∗∞ + λ−α

(1 − λ+)η∗∞ + λ+α


where

λ± =r − q

σ2+

12±

√(r − q

σ2+

12

)2

+2q

σ2.

(ii) limα→∞

η∗(τ ; α) = 1 for all τ .

The proof of Proposition 2.2 is presented in Appendix B. From the monotonicdecreasing property of η∗(τ ; α) with respect to τ and the finiteness property of η∗

∞(α)for q > 0, we infer that η∗(τ ; α) > 0 exists for all τ when q > 0. When α → ∞, thecontinuation region vanishes.

When the underlying asset is non-dividend paying, q = 0, we have η∗(0+; α) = 0.Furthermore, since η∗(τ ; α) is monotonically decreasing with respect to τ , we deducethat η∗(τ ; α) = 0 for τ > 0. That is, the stopping region does not exist when q = 0.Interpreted in financial sense, it is never optimal to exercise the American floatingstrike lookback call at any asset price level if the underlying asset is non-dividendpaying. Such result agrees intuitively with a similar result of the usual American call.

Figure 1 shows the plot of η∗(τ ; α) against τ at varying values of α. The parametervalues used in the calculations are: r = 0.04, q = 0.02 and σ = 0.3. The monotonicityproperties of η∗(τ ; α) with respect to τ and α and the asymptotic behaviors at τ → 0+

and τ → ∞ as shown in the plots do agree with the results stated in Propositions 2.1and 2.2. Our calculations give the following asymptotic values for η∗(τ ; α):

η∗(0+; 0.5) = 0.25, η∗(∞; 0.5) = 0.1023,η∗(0+; 1) = 0.5, η∗(∞; 1) = 0.1988,η∗(0+; 2) = 1, η∗(∞; 2) = 0.3617,η∗(0+; 10) = 1, η∗(∞; 10) = 0.7947.

2.2. American floating strike lookback put. Let Pf`(S, M, τ ) denote theprice function of an American floating strike lookback put with payoff (M − αS)+.The Russian option is the perpetual version of the American floating strike lookbackput with α = 0. In a similar manner, we use S as the numeraire and define

ξ =M

Sand Pf`(ξ, τ ) =

Pf`(S, M, τ )S

.(2.7)

The linear complementarity formulation for Pf`(ξ, τ ) is given by

∂Pf`

∂τ− LPf` ≥ 0, Pf` ≥ ξ − α,

(2.8) (∂Pf`

∂τ− LPf`

)[Pf` − (ξ − α)] = 0, 1 < ξ < ∞, τ > 0,


∂Pf`

∂ξ

∣∣∣∣ξ=1

= 0

(2.9)Pf`(ξ, 0) = (ξ − α)+.

Similarly, we have the free boundary ξ∗(τ ) that separates the stopping region{(ξ, τ ) : ξ ≥ ξ∗(τ ), 0 ≤ τ < ∞} and the continuation region {(ξ, τ ) : 1 ≤ ξ <


ξ∗(τ ), 0 ≤ τ < ∞}. The analytic properties of ξ∗(τ ) are summarized in Proposition2.3.

Proposition 2.3The free boundary ξ∗(τ ; α) observes the following properties:

(i) ξ∗(τ ; α) is monotonically increasing with respect to τ and α.(ii) The behavior of ξ∗(τ ; α) near expiry, τ → 0+, is given by

ξ∗(0+; α) = max(1, α,

q

rα)

.

(iii) Write ξ∗∞(α) as limτ→∞ ξ∗(τ ; α); ξ∗∞(α) is given by the solution of the rootinside the interval (1,∞) of the following algebraic equation:

(ξ∗∞)λ+−λ− =λ+

λ−

(1 − λ−)ξ∗∞ + λ−α

(1 − λ+)ξ∗∞ + λ+α.

In particular, when q = 0, we have

ξ∗∞(α) = ∞.

As a remark, it is well known that it is never optimal to exercise a Russian optionwhen the underlying asset is non-dividend paying [18]. The above result shows thatsuch optimal exercise policy holds even for non-zero value of α (Russian option is thespecial case of α = 0).

The ideas behind the proof of Proposition 2.3 are similar to those used in provingPropositions 2.1 and 2.2. In Figure 2, we show the plot of ξ∗(τ ; α) against τ withdifferent values of α. The parameter values used in the calculations are: r = 0.02, q =0.04 and σ = 0.3. We obtained the following asymptotic values for ξ∗(τ ; α):

ξ∗(0+; 0) = 1, ξ∗(∞; 0) = 3.4939,ξ∗(0+; 0.5) = 1, ξ∗(∞; 0.5) = 4.8536,ξ∗(0+; 1) = 2, ξ∗(∞; 1) = 6.6068,ξ∗(0+; 2) = 4, ξ∗(∞; 2) = 10.7613.

The monotonic behaviors of ξ∗(τ ; α) as exhibited by the plots in Figure 2 are consistentwith the results stated in Proposition 2.3.

3. Fixed strike lookback options. We now consider the pricing behaviors andoptimal exercise policies of American fixed strike lookback options, where the payoffinvolves the strike price K and either realized maximum value M or realized minimumvalue m. The payoff functions of the American fixed strike lookback call and lookbackput are given by

(M − K)+ and (K − m)+,

respectively. We also consider American option model with lookback payoff of theform

max(M, K),

which is related to the pricing model of dynamic protection fund with early with-drawal right [7, 9]. According to the guarantee clause, the fund holder acquires moreunits of the fund from the fund sponsor whenever the fund value falls below the guar-anteed protection floor. The early withdrawal right embedded in the protection fund


resembles the early exercise right of an American option. When we set K = 0 in thepayoff max(M, K), the option model becomes the finite-time Russian option.

It is tempting to seek possible fixed-floating symmetry relations between Americanlookback call and put options, similar to those obtained by Detemple [6] for usualAmerican options. While it is possible to obtain symmetry relations between thegrant-date price functions of European lookback options (with no dependence on therunning extremum value), such relations do not hold for the in-progress counterparts.We do not expect to have nice fixed-floating symmetry relations between the pricefunctions of in-progress American lookback options.

3.1. American fixed strike lookback call. Let Cfix(S, M, τ ; K) denote theprice function of an American fixed strike lookback call with payoff (M − K)+. Thelinear complementarity formulation that governs Cfix(S, M, τ ; K) is given by

∂Cfix

∂τ− LCfix ≥ 0, Cfix ≥ (M − K),

(3.1) (∂Cfix

∂τ−LCfix

)[Cfix − (M − K)] = 0, 0 < S < M, τ > 0,


∂Cfix

∂M

∣∣∣∣S=M

= 0,

(3.2)Cfix(S, M, 0) = (M − K)+.

Let S(K) denote the stopping region of the American fixed strike lookback call withstrike price K. Inside S(K), the price function equals the exercise payoff, that is,

S(K) = {(S, M, τ ) ∈ {0 < S ≤ M} × (0,∞) : Cfix(S, M, τ ) = (M − K)+}.

Propositions 3.1–3.2 summarize the characterization of the optimal exercise policy ofthe American fixed strike lookback call and the analytic properties of the stoppingregion.

Proposition 3.1The stopping region S(K) and the price function Cfix(S, M, τ ; K) of the Americanfixed strike lookback call observe the following properties:

(i) Cfix(S, M, τ ; K2) − Cfix(S, M, τ ; K1) ≤ K1 − K2 if K1 > K2,(ii) S(K1) ⊂ S(K2) if K1 > K2,(iii) Suppose (S, M, τ ) ∈ S(K) and 0 < λ1 ≤ 1, λ2 ≥ 1, 0 < λ3 ≤ 1, we have

(λ1S, λ2M, λ3τ ) ∈ S(K).

The proof of Proposition 3.1 is presented in Appendix C. In Figure 3, we plot theexercise boundary that separates the stopping region and continuation region in theS-M plane, and use M∗(S, τ ; K) to denote the exercise boundary. Such representationreveals the dependence of the critical realized maximum value M∗ on S, τ and K. Byvirtue of (iii) in Proposition 3.1, we deduce that the stopping region lies to the upperleft side of the exercise boundary in the S-M plane. Hence, we may rewrite S(K) inthe following alternative form:

S(K) = {(S, M, τ ) ∈ {0 < S ≤ M} × (0,∞) : M > M∗(S, τ )}.


Further properties on M∗(S, τ ; K) are summarized in Proposition 3.2.

Proposition 3.2Let M∗(S, τ ; K) denote the exercise boundary of the American fixed strike lookbackcall in the S-M plane, then M∗(S, τ ; K) observes the following properties:

(i) limτ→0+

M∗(S, τ ; K) = K for all S,

(ii) M∗(S, τ ; K) is a monotonically increasing with respect to S and τ ,(iii) lim

S→0+M∗(S, τ ; K) = K for all τ ,

(iv) When K = 0, M∗(S, τ ; 0) is a linear function of S. Furthermore,M∗(S, τ ; 0)

Sis a monotonically increasing function of τ and

limS→∞

M∗(S, τ ; K)S

=M∗(S, τ ; 0)

Sfor K > 0.(3.3)

Part (i) gives the zeroth order asymptotic expansion of M∗(S, τ ; K) as τ → 0+

(see [15] for a higher order asymptotic expansion of M∗(S, τ ; K) as τ → 0+). Onecan prove Part (i) by following a similar approach as that of (iii) in Proposition 2.1.Part (ii) is a corollary of part (iii) in Proposition 3.1. The proof of parts (iii) and (iv)in Proposition 3.2 is presented in Appendix D.

In Figure 3, we show the plot of the exercise boundaries of the American fixedstrike lookback call option with varying values of maturity τ in the S-M plane. Theparameter values used in the calculations are: K = 1, r = 0.02, q = 0.04 and σ = 0.3.The exercise boundary corresponding to the zero-strike lookback call is a straight line,the slope of which depends on τ . By virtue of Eq. (3.3), the exercise boundaries forthe non-zero strike lookback call options tend to those of their zero-strike counterpartsas S → ∞. Note that M∗(S, τ ; 0)/S = ξ∗(τ ; 0), where ξ∗(τ ; α) denotes the exerciseboundary in the pricing model for Pf`(ξ, τ ) [see Eqs. (2.8, 2.9)]. Our calculationsgive the following numerical values for ξ∗(τ ; 0):

ξ∗(∞; 0) = 3.4939ξ∗(2; 0) = 2.0300

ξ∗(0.5; 0) = 1.5450.

The finite-time Russian option is seen to be identical to the zero-strike Americanfixed strike lookback call. Let VRus(S, M, τ ) denote the price function of the finite-time Russian option so that

VRus(S, M, τ ) = Cfix(S, M, τ ; 0).(3.4)

Since K does not appear in the price function VRus(S, M, τ ), the asset value S canbe used as a numeraire. We may write

VRus(ξ, τ ) =VRus(S, M, τ )

Swhere ξ =

M

S.(3.5)

This explains why M∗(S, τ ; 0)/S becomes independent of S. More detailed theoreticalanalysis of the price function VRus(S, M, τ ) can be found in Peskir’s paper [16].


The exercise boundaries plotted in Figure 3 do agree with our financial intuitionabout the optimal early exercise policies of the American fixed strike lookback calloptions. Either S → 0+ or τ → 0+, the chance of achieving a higher realized maximumvalue M becomes vanishingly small, so it becomes optimal to exercise even when Mreaches the level K. When the asset price is very high, M∗(S, τ ; K) becomes almostinsensitive to the strike price K since the value K has only small effect on the exercisepayoff. Hence, when S → ∞, the asymptotic behavior of M∗(S, τ ; K) as stated inEq. (3.3) is observed.

3.2. American fixed strike lookback put. Consider an American fixed strikelookback put with payoff (K − m)+, the linear complementarity formulation thatgoverns its price function Pfix(S, m, τ ) is given by

∂Pfix

∂τ− LPfix ≥ 0, Pfix ≥ (K − m),

(3.6) (∂Pfix

∂τ−LPfix

)[Pfix − (K − m)] = 0, 0 < m < S, τ > 0,


∂Pfix

∂m

∣∣∣∣S=m

= 0(3.7)

Pfix(S, m, 0) = (K − m)+ .

In a similar manner, we let m∗(S, τ ; K) denote the exercise boundary that sep-arates the stopping region and continuation region in the S-m plane. The analyticproperties of m∗(S, τ ; K) are summarized in Proposition 3.3.

Proposition 3.3The exercise boundary m∗(S, τ ; K) of the American fixed strike lookback put satisfiesthe following properties:

(i) limτ→0+

m∗(S, τ ; K) = K for all S,

(ii) m∗(S, τ ; K) is monotonically increasing with respect to S,(iii) lim

S→∞m∗(0, τ ; K) = K for all τ ,

(iv) limS→0+

m∗(S, τ ; K)S

= 1 for all τ .

Parts (i) - (iii) in Proposition 3.3 can be proven by using similar argumentsas those used in proving parts (i) - (iii) in Proposition 3.2. The proof of (iv) inProposition 3.3 is interesting and challenging. It relies on the asymptotic result onη∗(τ ; α) as stated in (ii) in Proposition 2.2 (see Appendix E for details).

Figure 4 shows the plot of the exercise boundaries m∗(S, τ ; K) of the Americanfixed strike lookback put with varying values of maturity τ in the S-m plane. Theparameter values used in the calculations are: K = 1, r = 0.04, q = 0.02 and σ = 0.3.According to (iii) and (iv) in Proposition 3.3, the exercise boundaries are seen to tendasymptotically to m = K as S → ∞ and m = S as S → 0+.

3.3. American lookback option with payoff max(M, K). Let VM (S, M, τ )denote the price function of the American option with lookback payoff max(M, K).


First, we argue from financial intuition that VM (S, M, τ ) should be insensitive to thecurrent realized maximum value of asset price M when M < K, that is,

∂VM

∂M= 0 for M < K.(3.8)

The option payoff is given by K if the future realized maximum value of asset priceis less than or equal to K; otherwise, the payoff equals the future realized maximumvalue. In either case, the current realized maximum value M does not enter into thepayoff function. Hence, VM (S, M, τ ) does have dependence on M when M < K. Onthe other hand, when M ≥ K, the future realized maximum value is always greaterthan or equal to K, so the payoff is simply given by M . This is the same payoff asthat of the finite-time Russian option. Hence, we have

VM (S, M, τ ) = VRus(S, M, τ ) for M ≥ K.(3.9)

By virtue of the continuity property of the price function VM (S, M, τ ) with respectto M , we then have

VM (S, M, τ ) ={

VRus(S, M, τ ) for M ≥ KVRus(S, K, τ ) for M < K

.(3.10)

For M ≥ K, VM and VRus should share the same optimal exercise policy. At M =K, the exercise boundary of the finite-time Russian option is given by S = K/ξ∗(τ ; 0).Hence, for M < K, the American option with payoff max(M, K) will be exercisedoptimally when S ≤ K/ξ∗(τ ; 0) and unexercised if otherwise.

In Figure 5, we plot the stopping region and continuation region in the S-M planeof the American option with payoff max(M, K). The set of parameter values usedin the calculations are: K = 1, r = 0.02, q = 0.04 and σ = 0.3. When M ≥ K,the stopping region and continuation region for fixed value of τ are separated by theoblique line: M = Sξ∗(τ ; 0). On the other hand, when M < K, the exercise boundarybecomes the vertical line: S = K/ξ∗(τ ; 0).

3.4. A related two-asset American option model. As a slight departurefrom the option models with lookback payoff structures, we consider the optimalexercise policies of a two-asset American option with a put payoff on the minimum oftwo asset values. There have been several comprehensive papers that analyze the earlyexercise policies of two-asset American options [2, 5, 9, 13, 14, 19, 20]. We would liketo demonstrate that the mathematical technique of analyzing the exercise boundariesof the American fixed strike lookback put option at S → 0+ can be adopted to resolvethe mystery on the asymptotic behaviors of the exercise boundaries of the two-assetAmerican minimum put option at infinitesimally small asset values.

Let S1 and S2 denote the prices of the two underlying assets, whose dynamicsunder the risk neutral measure are governed by

dSi

Si= (r − qi)dt + σi dZi i = 1, 2,(3.11)

where dZ1 dZ2 = ρ dt, ρ is the correlation coefficient between the two Wiener processesdZ1 and dZ2. The exercise payoff is given by (K−min(S1, S2))+, where K is the strikeprice. Let Pmin(S1, S2, τ ; K) denote the price function of this two-asset Americanminimum put option. Let S2(K) denote the continuation region in the S1-S2 plane,


with dependence on K. The linear complementarity formulation for Pmin(S1, S2, τ ; K)is given by

∂Pmin

∂τ− L2Pmin ≥ 0, Pmin ≥ (K − min(S1, S2))+,

(3.12) [∂Pmin

∂τ−L2Pmin

][Pmin − (K − min(S1, S2))+] = 0,

0 < S1 < ∞, 0 < S2 < ∞, τ > 0.

The operator L2 is defined by

L2 =σ2

1

2S2

1

∂2

∂S21

+ ρσ1σ2S1S2∂2

∂S1∂S2+

σ22

2S2

2

∂2

∂S22(3.13)

+ (r − q1)S1∂

∂S1+ (r − q2)S2

∂

∂S2− r.

In Figure 6, we show the plot of the exercise boundaries of the two-asset Americanminimum put option in the S1-S2 plane. The following set of parameter values are usedin the calculations: K = 1, r = 0.02, q1 = 0, q2 = 0.03, σ1 = σ2 = 0.3 and ρ = 0.5.The whole line S1 = S2 always lie in the continuation region. The continuationregion is bounded by the two branches of the exercise boundaries. In the regionS1 > S2, we let S∗

2 (S1, τ ) denote the exercise boundary at time to expiry τ . Weobserve that the curve S∗

2 (S1, τ ) tends to the line S1 = S2 as S1 → 0+ and tends tosome asymptotic limit as S1 → ∞. Similar phenomena occur in the region S2 > S1,where the exercise boundary at time to expiry τ is represented by S∗

1 (S2, τ ). For theabove set of parameter values chosen for the option model, we obtain

limS1→∞

S∗2 (S1, 0.1) = 0.6277, lim

S1→∞S∗

2(S1, 1) = 0.4855, limS1→∞

S∗2 (S1,∞) = 0.2268,

limS2→∞

S∗1 (S2, 0.1) = 0.8118, lim

S2→∞S∗

1(S2, 1) = 0.6100, limS2→∞

S∗1 (S2,∞) = 0.3077.

Some of the analytic properties of the exercise boundaries S∗1 (S2, τ ) and S∗

2 (S1, τ ) aresummarized in Proposition 3.4.

Proposition 3.4Let S∗

1 (S2, τ ) and S∗2 (S1, τ ) denote the exercise boundaries at time to expiry τ in the

two respective regions, S2 > S1 and S1 > S2, in the S1-S2 plane of the two-assetAmerican minimum put option. The exercise boundaries and the continuation regionobserve the following properties:

(i) Let S∗1,P (τ ) and S∗

2,P (τ ) denote the exercise boundary of the one-asset Amer-ican put option with the underlying asset S1 and S2, respectively. We have

limS2→∞

S∗1 (S2, τ ) = S∗

1,P (τ ) and limS1→∞

S∗2 (S1, τ ) = S∗

2,P (τ ).

(ii) Both S∗1 (S2, τ ) and S∗

2 (S1, τ ) are monotonically decreasing with respect totime to expiry and monotonically increasing with respect to the asset pricelevel.

(iii) The whole line S1 = S2 is contained completely inside the continuation region.(iv) At infinitesimally small asset values, we have


limS1→0+

S∗2 (S1, τ )

S1= 1 and lim

S2→0+

S∗1 (S2, τ )

S2= 1 for all τ.(3.14)

All exercise boundaries tend asymptotically to the line S1 = S2 as S1 and S2

both tend to zero.

The intuition behind the asymptotic properties stated in part (i) of Proposi-tion 3.4 is quite obvious. When S1 → ∞, Pmin(S1, S2, τ ; K) −→ P (S2, τ ; K), whereP (S2, τ ; K) denotes the price function of the one-asset American put option with un-derlying asset S2. We would expect that both option models follow the same optimalexercise strategy, thus leading to the asymptotic properties stated in (i). The proofof these asymptotic properties can be pursued by following similar arguments usedin the proof of Proposition 4.8 in Villeneuve’s paper [20]. Also, the monotonicityproperties of S∗

1 (S2, τ ) and S∗2 (S1, τ ) have been discussed in other papers (say [2] and

[20]). Property (iii) states that when S1 = S2, it is never optimal to exercise thetwo-asset American minimum put option. This optimal exercise policy is similar tothat of the two-asset American maximum call option. The proof of (iii) can followa similar argument presented by Detemple et al . [5] on the American maximum calloption. The proof of the asymptotic behavior of the exercise boundaries at S1 → 0and S2 → 0 requires specifically the technique developed in the proof of property (iii)in Proposition 3.3. The proof of part (iv) of Proposition 3.4 is presented in AppendixF.

4. Conclusion. This paper demonstrates the richness of the optimal exercise be-haviors adopted by holders of the American options with payoff structures involvinglookback state variables. The analysis of the optimal exercise policies of an Ameri-can lookback option is complicated by the presence of an additional lookback statevariable. For fixed strike lookback options, we characterize the exercise behaviors byanalyzing the analytic properties of the stopping region and continuation region inthe two-dimensional state space (asset price and lookback state variable). For floatingstrike lookback options, the dimension of the pricing model can be reduced by oneif the asset price is used as the numeraire. We reveal the close relationship betweenthe price functions of the finite-time Russian option and the dynamic protection fundwith withdrawal right. For the American put option on the minimum value of twoassets, the exercise region consists of two branches of exercise surfaces. Compared toearlier works, our analyses provide more comprehensive understanding of the optimalexercise policies of commonly traded American lookback options. In particular, weprovide more precise description of the asymptotic behaviors of the exercise bound-aries. All the optimal exercise policies of American lookback options derived from ourtheoretical studies have been verified by plots of the exercise boundaries obtained vianumerical calculations.

REFERENCES

[1] H. BREZIS and A. FRIEDMAN, Estimates on the support of solutions of parabolic variationalinequalities, Illinois Journal of Mathematics, 20(1) (1976), pp. 82–97.

[2] M. BROADIE and J. DETEMPLE, The valuation of American options on multiple assets,Mathematical Finance, 7(3) (1997), pp. 241–286.


[3] M. DAI, A closed-form solution for perpetual American floating strike lookback options, Journalof Computational Finance, 4(2) (2000), pp. 63–68.

[4] M. DAI, H.Y. WONG and Y.K. KWOK, Quanto lookback options, Mathematical Finance,14(3) (2004), pp. 445-467.

[5] J. DETEMPLE, S. FENG and W. TIAN, The valuation of American call options on theminimum of two dividend-paying assets, Annals of Applied Probability, 13(3) (2003), pp.953-983.

[6] J.B. DETEMPLE, American options: Symmetry properties, edited by J. CVITANIC, E.JOUINI and M. MUSIELA, Cambridge University Press, Cambridge (2001) pp. 67-104.

[7] J.D. DUFFIE and J.M. HARRISON, Arbitrage pricing of Russian options and perpetuallookback options , Annals of Applied Probabilities, 3(3) (1993), pp. 641–651.

[8] H.U. GERBER and G. PAFUMI, Pricing dynamic investment fund protection, North Amer-ican Actuarial Journal, 4(2) (2000), pp. 28–41.

[9] H. GERBER and E. SHIU, Martingale approach to pricing perpetual American options ontwo stocks, Mathematical Finance, 3 (1996), pp. 87–106.

[10] M.B. GOLDMAN, H.B. SOSIN AND M.A. GATTO, Path dependent options: buy at the low,sell at the high, Journal of Finance, 34(5) (1979) pp. 1111-1127.

[11] J. IMAI, and P. BOYLE, Dynamic fund protection, North American Actuarial Journal, 5 (3)(2001) pp. 31–51.

[12] P. JAILLET, D. LAMBERTON AND B. LAPEYRE, Variational inequalities and the pricingof American options , Acta Applicandae Mathematicae, 21 (1990), pp. 263-289.

[13] L.S. JIANG, Analysis of pricing American options on the maximum (minimum) of two riskyassets, Interfaces and Free Boundaries, 4 (2002), pp. 27–46.

[14] J. KAMPEN, On American derivatives and related obstacle problems, International Journalof Theoretical and Applied Finance, 6(2003), pp. 565–591.

[15] T.L. LAI and T.W. LIM, Exercise regions and effective valuation of American lookback op-tions, Mathematical Finance, 14 (2004), pp. 249–269.

[16] T.L. LAI and T.W. LIM, Efficient valuation of American floating-strike lookback optionsusing a decomposition technique, working paper of Stanford University (2004).

[17] G. PESKIR, The Russian option: Finite horizon, Finance and Stochastics, 9(2) (2005), pp.251.

[18] L.A. SHEPP and A.N. SHIRYAEV, The Russian option: Reduced regret, Annals of AppliedProbability 3 (1993), pp. 631–640.

[19] K. TAN and K. VETZAL, Early exercise regions for exotic options , Journal of Derivatives,3, (1995), pp. 42-56.

[20] S. VILLENEUVE, Exercise regions of American options on several assets, Finance andStochastics, 3 (1999), pp. 295–322.

[21] P. WILMOTT, J. DEWYNNE and J. HOWISON, Option pricing: Mathematical models andcomputation, Oxford Financial Press, Oxford (1993).

[22] H. YU, Y.K. KWOK and L. WU, Early exercise policies of American floating and fixed strikelookback options, Nonlinear Analysis, 47 (2001), pp. 4591–4602.


APPENDIX A — Proof of Proposition 2.1(i) First, we show that if (η, τ ) ∈ SC , then (η, λ2τ ) ∈ SC for λ2 ≥ 1. By applying

the comparison principle, one can show that∂Cf`

∂τ> 0. This is consistent

with the financial intuition that the price function of any American option isan increasing function of τ . Suppose (η, τ ) lies in the continuation region, then

Cf`(η, τ ) > α−η. By virtue of∂Cf`

∂τ> 0, we deduce that Cf`(η, λ2τ ) > α−η

for λ2 ≥ 1. Hence, (η, λ2τ ) also lies in the continuation region.Next, we show that if (η, τ ) ∈ SC , then (λ1η, τ ) ∈ SC for λ1 ≥ 1. If suffices

to show that

(A.1)∂

∂η[Cf`(η, τ )− (α − η)] ≥ 0.

We write U (η, τ ) = Cf`(η, τ ) − (α − η), then the linear complementarityformulation for U (η, τ ) is given by

∂U

∂τ− LU ≥ rη − qα, U ≥ 0,

(∂U

∂τ− LU

)U = 0, 0 < η < 1, τ > 0,


∂U

∂η

∣∣∣∣η=1

= 1 and U (η, 0) = (η − α)+.

Both the initial condition (η − α)+ and the non-homogeneous term rη − qα

are increasing functions of η, and∂U

∂η

∣∣∣∣η=1

> 0. By virtue of the comparison

principle, we deduce that∂U

∂η≥ 0.

(ii) We prove by contradiction. Suppose there exists τ0 > 0 such that (1, τ0) ∈ S,by applying Eq. (A.1), we can show that (η, τ0) ∈ S for η < 1. We then have

Cf`(η, τ0) = α − η, η < 1.

This implies

∂Cf`

∂η= −1 at (1, τ0),

which contradicts the Neumann boundary condition stated in Eq. (2.6).(iii) A necessary condition for (η, τ ) lying inside S is given by

(∂

∂τ− L

)(α − η) = αq − rη ≥ 0,

that is, η ≤ q

rα. Hence, we should have η∗(0+) ≤ q

rα. Since the exercise

payoff must be non-negative, so another necessary condition is given by η ≤ α.


Lastly, the feasible region for η is {η : η ≤ 1}. Combining all three necessaryconditions, we should have

η∗(0+) ≤ min(1, α,

q

rα)

.

Suppose η∗(0+) < min(1, α,

q

rα), then for η ∈

(η∗(0+), min

(1, α,

q

rα))

, wehave

∂Cf`

∂τ

∣∣∣∣τ=0

= LCf`

∣∣∣∣τ=0

= L(α − η) = rη − αq < 0.

This contradicts with∂Cf`

∂τ≥ 0 for all τ . Hence, we obtain

η∗(0+) = min(1, α,

q

rα)

.

APPENDIX B — Proof of Proposition 2.2(i) Write η∗

∞(α) = limτ→∞

η∗(τ ; α) and C∞f` (η) = lim

τ→∞Cf`(η, τ ), then C∞

f` (η) satis-fies the following differential equation:

L C∞f` = 0, η∗

∞ < η < 1,

subject to the auxiliary conditions:

C∞f`(η

∗∞) = α − η∗

∞,∂Cf`

∂η(η∗

∞) = −1,∂Cf`

∂η(1) = 0.

The general solution to C∞f` (η) is given by

C∞f`(η) = A1η

λ+ + A2ηλ− , η∗

∞ < η < 1.

Applying the auxiliary conditions, we obtain

A1 =(1 − λ−)η∗

∞ + λ−α

(λ− − λ+)(η∗∞)λ+

and A2 =(1 − λ+)η∗

∞ + λ+α

(λ+ − λ−)(η∗∞)λ−

,

and η∗∞ satisfies the non-linear algebraic equation

(B.1) (η∗∞)λ+−λ− =

λ+

λ−

(1 − λ−)η∗∞ + λ−α

(1 − λ+)η∗∞ + λ+α

.

The above algebraic equation has two roots, one lies in (0, 1) and the otherlies in (1,∞) (the proof of these properties can be found in [3]). Here, η∗

∞corresponds to the root in (0, 1). Hence, the results in part (i) are established.

(ii) When α → ∞, the non-linear algebraic equation (B.1) reduces to

(η∗∞)λ+−λ− = 1

so that the solution for η∗∞ becomes 1. Also, η∗(0+) = 1 when α becomes

sufficiently large. Since η∗(τ ) is monotonically decreasing with respect to τ ,and η∗(0+) = η∗(∞) = 1 as α → ∞, we can deduce that

limα→∞

η∗(τ ; α) = 1 for all τ.


APPENDIX C — Proof of Proposition 3.1(i) Define the function V (S, M, τ ; K) = Cfix(S, M, τ ; K) + K. Similar to Eqs.

(3.1–3.2), the linear complementarity formulation for V (S, M, τ ; K) is givenby

∂V

∂τ−LV ≥ rK, V ≥ max(M, K)

[∂V

∂τ−LV − rK

][V − max(M, K)] = 0,


∂V

∂M

∣∣∣∣S=M

= 0 and V (S, M, 0; K) = max(M, K).

By virtue of the comparison principle, we have

V (S, M, τ ; K1) ≥ V (S, M, τ ; K2) if K1 > K2,

and hence the result.(ii) From (i), for K1 > K2, we have

(C.1) Cfix(S, M, τ ; K1) − (M − K1) ≥ Cfix(S, M, τ ; K2) − (M − K2).

Suppose (S, M, τ ) ∈ SC(K2), where SC(K2) denotes the continuation region.In the continuation region, the option value is strictly greater than the exercisepayoff so that

Cfix(S, M, τ ; K2) > M − K2.

Combining with Inequality (C.1), we can deduce

Cfix(S, M, τ ; K1) > M − K1,

so that (S, M, τ ) ∈ SC(K1). Hence, we establish SC(K2) ⊂ SC(K1); and soS(K1) ⊂ S(K2).

(iii) Since Cfix(S, M, τ ) is monotonically increasing with respect to both S and τ ,and the exercise payoff is independent of S and τ , we deduce that if (S, M, τ ) ∈S(K), then

(λ1S, M, λ3τ ) ∈ S(K) for all 0 < λ1 ≤ 1 and 0 < λ3 ≤ 1.

Next, we would like to show that (S, M, τ ) ∈ S(K) would imply (S, λ2M, τ ) ∈S(K), for all λ2 ≥ 1. Suppose (S, M, τ ) ∈ S(K), then (S/λ2, M, τ ) ∈ S(K)for λ2 ≥ 1. Furthermore, by virtue of the linear homogeneity property of theprice function and the price function and the result in (i), we obtain

Cfix(S, λ2M, τ ; K) = λ2Cfix

(S

λ2, M, τ ;

K

λ2

)

≤ λ2

[Cfix

(S

λ2, M, τ ; K

)+(

1 − 1λ2

)K

]

= λ2

[M − K +

(1 − 1

λ2

)K

]= λ2M − K.


On the other hand, the option value Cfix(S, λ2M, τ ; K) cannot fall below theexercise payoff λ2M − K. Combining the results, we then have

Cfix(S, λ2M, τ ; K) = λ2M − K,

that is, (S, λ2M, τ ) ∈ S(K). Hence, we obtain the desired result.

APPENDIX D — Proof of Proposition 3.2(iii) It is clear that M∗(0+, τ ; K) ≥ K. From the monotonic increasing property

of M∗(S, τ ; K) with respect to S, suppose we can show that the line M = M0

lies in the stopping region in the S-M plane for any M0 > K, then one candeduce that M∗(S, τ ; K) → K as S → 0+. This is because the minimumvalue of M∗(S, τ ; K) is achieved when S is approaching zero from above, andthis minimum value is K. We write Ufix(S, τ ) = Cfix(S, M0, τ )− (M0 − K).The linear complementarity formulation of Ufix(S, τ ) is given by

(∂

∂τ−L

)Ufix ≥ −r(M0 − K), Ufix ≥ 0,

[(∂

∂τ−L

)Ufix

]Ufix = 0

with initial condition: Ufix(S, 0) = 0. Since the right-hand term −r(M0−K)is always negative and the initial value has compact support, we apply thetheorem by Brezis and Friedman [1] that the solution Ufix(S, τ ) has compactsupport too. The stopping region is non-empty, that is, there exists (S, τ )such that Cfix(S, M0, τ ) = M0 − K for any M0 > K. Hence, the line M =M0 ∈ S(K) for any M0 > K.

(iv) When K = 0, the American fixed strike lookback call is the same as theAmerican floating strike lookback put [with α = 0 in Eq. (2.8)]. The mono-tonically increasing property of ξ∗(τ ) = M∗(S, τ ; 0)/S follows directly fromProposition 2.3(i).

For K > 0, by virtue of the linear homogeneity property of M∗(S, τ ; K),we obtain

limS→∞

M∗(S, τ ; K)S

= limS→∞

M∗( SK

, τ ; 1)

SK

= limK→0

M∗( SK

, τ ; 1)

SK

= limK→0

M∗(S, τ ; K)S

=M∗(S, τ ; 0)

S.

APPENDIX E — Proof of Proposition 3.3(iv) First, we consider the proof with q > 0, whose arguments rely on the existence

of η∗(τ ; α). Since η∗(τ ; α) does not exist when q = 0, we will deal with thespecial case of zero dividend separately later. For α ≥ 1, we observe that

(K − m)+ ≤ (K − αS)+ + αS − m

so that

(E.1) Pfix(S, m, τ ; K) ≤ αP

(S, τ ;

K

α

)+ Cf`(S, m, τ ; α),


where P

(S, τ ;

K

α

)denotes the price function of the American vanilla put

option with strike priceK

α. Let S∗

P

(τ ;

K

α

)be the critical asset price of the

American vanilla put with payoff(

K

α− S

)+

. Consider the point (S, m) in

the S-m plane which lies inside the region

Rα ={

(S, m) : m ≤ Sη∗(τ ; α) and S ≤ S∗P

(τ ;

K

α

)},

(S, m) lies in the corresponding stopping region of both the American floatingstrike call and American vanilla put. We then have

(E.2) P

(S, τ ;

K

α

)=

K

α− S and Cf`(S, m, τ ; α) = αS − m.

Now, we argue that (S, m) also lies in the stopping region of the Americanfixed strike put. To establish the claim, it suffices to show that

(E.3) Pfix(S, m, τ ; K) = K − m.

Combining the results in Eqs. (E.1) and (E.2), we obtain Pfix(S, m, τ ; K) ≤K − m. Since the option value of the American fixed strike put cannot fallbelow its exercise payoff, the result in Eq. (E.3) is then established.

Lastly, we take the limit α → ∞ and observe that

limα→∞

η∗(τ ; α) = 1 and limα→∞

S∗P

(τ ;

K

α

)= 0

for all τ . As α → ∞, Rα shrinks to an infinitesimally small triangular wedgewith the oblique side: S = m. Hence, we can deduce that as S → 0+ andfor all values of τ , all the exercise boundaries m∗(S, τ ; K) tend to the obliqueasymptotic line: S = m.

Lastly, we consider the case where q = 0. We add the parameter q inthe price function Pfix(S, m, τ ; K, q) and exercise boundary m∗(S, τ ; q), andwrite the corresponding stopping region as S(q) with dependence on q. Fromthe pricing property

Pfix(S, m, τ ; K, 0) ≤ Pfix(S, m, τ ; K, q),

we deduce that

S(q) ⊂ S(0), q > 0.

Hence, we have m∗(S, τ ; 0) ≥ m∗(S, τ ; q) so that

m∗(S, τ ; q)S

≤ m∗(S, τ ; 0)S

≤ 1, q > 0.

Since we have establishedm∗(S, τ ; q)

S→ 1 as S → 0, so lim

S→0+

m∗(S, τ ; 0)S

=

1.


APPENDIX F — Proof of Proposition 3.4(iii) We only show the proof of

limS1→0+

S∗2 (S1, τ )

S1= 1.

The proof of the other limiting property in Eq. (3.14) can be pursued in asimilar manner. Following a similar approach in Appendix E, we employ thefollowing inequality

(F.1) (K − min(S1, S2))+ ≤ (K − αS2)+ + (αS2 − min(S1, S2))+,

and examine the stopping region Sα of the American two-asset option withpayoff (αS2 − min(S1, S2))+. Also, we let S∗

2,P be the critical asset price of

the American put with payoff(

K

α− S2

)+

. By applying inequality (F.1) and

following a similar argument presented in Appendix E, one can show that thestopping region of the two-asset American minimum put option is containedinside

Rα ={

(S1, S2) : (S1, S2) ∈ Sα and S2 ≤ S∗2,P

(τ ;

K

α

)}.

The asymptotic behavior of S∗2 (S1, τ ) at infinitesimally small value of S1 is

established once we can show that the boundaries of Rα are bounded by theline S1 = S2 as α → ∞.

Let Vα denote the price function of the American two-asset option withpayoff (αS2 − min(S1, S2))+, α ≥ 1. We let x = S1/S2 and define Wα =Vα/S2. The exercise boundary of the American option model Wα(x, τ ) hastwo branches, and let them be denoted by x∗

h(τ ) and x∗` (τ ). The continuation

region is represented by {(x, τ ) : x∗` (τ ) < x < x∗

h(τ ), 0 ≤ τ < ∞}. The linearcomplementarity formulation of Wα(x, τ ) is given by

∂Wα

∂τ− 1

2(σ2

1 − 2ρσ1σ2 + σ22)x

2∂2Wα

∂x2− (q2 − q1)x

∂Wα

∂x+ q2Wα = 0,

x∗` (τ ) < x < x∗

h(τ ), τ > 0,


Wα(x∗` , τ ) = α − x∗

` ,∂Wα

∂x(x∗

` , τ ) = −1,

Wα(x∗h, τ ) = α − 1,

∂Wα

∂x(x∗

h, τ ) = 0,

Wα(x, 0) ={

α − x if x ≤ 1α − 1 if x > 1 .

For q2 > 0, one can show that x∗`(τ ) and x∗

h(τ ) are monotonic functions ofτ . Also, x∗

` (0+) = x∗

h(0+) = 1 when α >q1

q2. Similar to Property (ii) in

Proposition 2.2, we would like to establish the following asymptotic results

(F.2) limα→∞

x∗` (τ ; α) = 1 and lim

α→∞x∗

h(τ ; α) = 1


so that the boundary of Rα will be bounded by S1 = S2 as α → ∞. Byvirtue of the monotonicity properties of x∗

` (τ ) and x∗h(τ ) with respect to τ ,

the asymptotic properties in (F.2) are valid if we can show

(F.3) limα→∞

x∗` (∞; α) = 1 and lim

α→∞x∗

h(∞; α) = 1.

When q2 = 0, x∗`(τ ) does not exist but lim

α→∞x∗

h(τ ; α) = 1 remains valid. Thearguments in the proof presented below have to be modified slightly for thisdegenerate case.

The proof of Eq. (F.3) requires the solution of W∞α (x), the perpetual limit

of Wα(x, τ ). The governing equation for W∞α (x) is given by

12(σ2

1 − 2ρσ1σ2 + σ22)x

2d2W∞α

dx2+ (q2 − q1)x

dW∞α

dx− q2W

∞α = 0,

x∗`(∞) < x < x∗

h(∞),


W∞α (x∗

` (∞)) = α − x∗` (∞),

dW∞α

dx(x∗

` (∞)) = −1,

W∞α (x∗

h(∞)) = α − 1,dW∞

α

dx(x∗

h(∞)) = 0.

By following a similar approach in Appendix B, we can show that

limα→∞

x∗h(∞; α)

x∗` (∞; α)

= 1,

and hence the relations in Eq. (F.3) are established.


0 5 10 15 20 25 30 35 40 45 500

0.25

0.5

0.75

1

τ

η*(τ

;α)

Continuation Region

α=10

α=2

α=1

α=0.5

Stopping Region

FIG 1. The critical threshold η∗(τ ; α) of the American floating strike lookbackcall option is plotted against τ for different values of α. The parameter values of thepricing model are: r = 0.04, q = 0.02 and σ = 0.3.

0 5 10 15 20 25 30 35 40 45 501

2

3

4

5

6

7

8

9

10

11

τ

ξ*(τ

;α)

Continuation Region

Stopping Region

α=0

α=0.5

α=1

α=2

FIG 2. The critical threshold ξ∗(τ ; α) of the American floating strike lookbackput option is plotted against τ for different values of α. The parameter values of thepricing model are: r = 0.02, q = 0.04 and σ = 0.3.


0 0.5 1 1.5 2 2.5 30

0.5

1

1.5

2

2.5

3

3.5

4

4.5

S

M M=Sξ*(2;0) M=Sξ*(0.5;0)

Infeasible Region

M=S

Stopping Region

Continuation Region

τ=2 τ=0.5τ=∞

M=Sξ*(∞;0)

FIG 3. The exercise boundaries (solid curves) of the American fixed strike look-back call option with varying values of maturity τ are plotted in the S-M plane. Ata given τ , the stopping region is lying to the left and above of the correspondingexercise boundary. The dotted lines are asymptotic lines of the exercise boundaries,corresponding to the exercise boundaries of the zero-strike counterparts. The stop-ping region of the Russian option lies to the left of the dotted line: M = Sξ∗(∞; 0).The parameter values used in the calculations are: K = 1, r = 0.02, q = 0.04 andσ = 0.3.

0 1 2 3 4 5 6 7 8 90

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

S

m

Continuation Region

Infeasible Region

m=S

Stopping Region

τ=0.5τ=2

τ=∞

FIG 4. The exercise boundaries of the American fixed strike lookback put optionwith varying values of maturity τ are plotted in the S-m plane. All exercise boundariestend to the oblique asymptotic line: m = S as S → 0+, and the horizontal asymptoticline: m = K as S → ∞. The parameter values used in the calculations are: K =1, r = 0.04, q = 0.02 and σ = 0.3.


0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.5

1

1.5

2

2.5

3

S

M

Infeasible Region

Continuation Region

Sto

ppin

g R

egio

n

M=Sξ*(∞;0) M=Sξ*(2;0) M=Sξ*(0.5;0)

τ=0.5 τ=2 τ=∞

M=S

FIG 5. The exercise boundaries of the American option with payoff functionmax(M, K) with varying values of maturity τ are plotted in the S-M plane. Theparameter values used in the calculations are: K = 1, r = 0.02, q = 0.04 and σ = 0.3.

0 0.2 0.4 0.6 0.8 1 1.20

0.2

0.4

0.6

0.8

1

1.2

S1

S2

S1* (S

2,∞) S

1* (S

2,1) S

1* (S

2,0.1)

S2* (S

1,0.1)

S2* (S

1,1)

S2* (S

1,∞)

Stopping Region

Stopping Region Continuation Region

S1=S

2

FIG 6. The exercise boundaries of the two-asset American minimum put optionwith varying values of maturity τ are plotted in the S1-S2 plane. The continuationregion is bounded between the two branches of the exercise boundaries. The parametervalues used in the calculations are: K = 1, r = 0.02, q1 = 0, q2 = 0.03, σ1 = σ2 = 0.3and ρ = 0.5.

American options with lookback payoff - Department of Mathematics

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