Kimura American Installment

8/6/2019 Kimura American Installment

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SIAM J. APPL. MATH. c 2009 Society for Industrial and Applied MathematicsVol. 70, No. 3, pp. 803824

AMERICAN CONTINUOUS-INSTALLMENT OPTIONS:

VALUATION AND PREMIUM DECOMPOSITION

TOSHIKAZU KIMURA

Abstract. Installment options are weakly path-dependent contingent claims in which the pre-mium is paid discretely or continuously in installments, instead of paying a lump sum at the timeof purchase. This paper deals with valuing American continuous-installment options written ondividend-paying assets. The setup is a standard BlackScholesMerton framework where the price ofthe underlying asset evolves according to a geometric Brownian motion. The valuation of installmentoptions can be formulated as an optimal stopping problem, due to the flexibility of continuing orstopping to pay installments as well as the chance of early exercise. Analyzing cash flow generatedby the optimal stop, we can characterize asymptotic behaviors of the stopping and early exerciseboundaries close to expiry. Combining the PDE and Laplace transform approaches, we obtain theLaplace transform of the initial premium in an explicit form, which is decomposed into the valueof the associated European vanilla option with the same payoff plus the premiums of early exerciseand halfway cancellation. We also obtain a pair of nonlinear equations for the Laplace transformsof the b oundaries. An Abelian theorem of Laplace transforms enables us to obtain a concise resultfor the perpetual case. We show that numerical inversion of these Laplace transforms works well forcomputing both the option value and the boundaries.

Key words. continuous installments, American-style options, free boundary problem, Laplacetransforms, premium decomposition, numerical inversion

AMS subject classifications. 91B28, 91B70, 60G40

DOI. 10.1137/080740969

1. Introduction. Installment options or pay-as-you-go options are contingentclaims in which a small amount of up-front premium is paid at the time of purchase,and then a sequence of installments are paid up to a fixed maturity. Installmentoptions are path-dependent in a weak sense such that their historical paths affectthe value but the option payoff does not contain the paths explicitly. The holder

has to pay installments to keep the contract alive, although she/he has the rightto terminate the contract by stopping the payments at any time, in which case thecontract is canceled and the payoff vanishes. If the option is not worth the NPV (netpresent value) of the remaining payments, she/he does not have to continue to payfurther installments. Hence, an optimal stopping problem arises for the installmentoption even in European style. For American-style installment options, the holderalso has the right to exercise the option at any time until maturity, and hence theoption holder is faced with three decisionscancel, exercise, or do nothingduringthe trading interval.

Installment options may appeal to an investor who is willing to pay a little extrafor the opportunity of terminating the contract and reducing losses caused by her/hisvoid investment position. Actually, installment options have been traded actively in afew markets, e.g., installment warrants on Australian stocks listed on the Australian

Stock Exchange (ASX) [3, 4], a 10-year warrant with nine annual payments offeredReceived by the editors November 16, 2008; accepted for publication (in revised form) May 5,

2009; published electronically July 22, 2009. This research was supported in part by the Grant-in-Aid for Scientific Research (A) (grant 20241037) of the Japan Society for the Promotion of Science(JSPS) in 20082012.

http://www.siam.org/journals/siap/70-3/74096.htmlGraduate School of Economics and Business Administration, Hokkaido University, Nishi 7, Kita

9, Kita-ku, Sapporo 060-0809, Japan ([email protected]).

803


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804 TOSHIKAZU KIMURA

by Deutsche Bank [10], and so on. Also, many life insurance contracts and capitalinvestment projects can be thought of as installment options [12]. However, therehave been relatively few studies on installment options.

An installment option with payments at prespecified dates is usually referred

to as a discrete-installment option, whereas its continuous-time limit in which pre-mium is paid at a certain rate per unit time is referred to as a continuous-installmentoption. For discrete-installment options, Ben-Ameur, Breton, and Francois [4] devel-oped a dynamic programming algorithm for computing the American option valueapproximated by a piecewise-linear interpolation, which is applied to valuing ASXinstallment warrants with dilution effects. For European-style discrete-installmentoptions, Davis, Schachermayer, and Tompkins [10, 11] applied the concepts of com-pound options and NPV to obtain no-arbitrage bounds of the initial call premiumin the BlackScholesMerton framework, and then to examine dynamic and statichedging strategies. Based on a compounding structure, Griebsch, Kuhn, and Wystup[14] derived a closed-form pricing formula in terms of multidimensional cumulativenormal distribution functions.

As for European continuous-installment options, Alobaidi, Mallier, and Deakin [2]

analyzed asymptotic properties of the optimal stopping boundary close to maturity.Kimura [19] applied the Laplace transform approach to the European case for obtain-ing transforms of the initial premium, its Greeks, and the optimal stopping boundary,and then he analyzed them quantitatively by a numerical transform inversion. ForAmerican continuous-installment options, Ciurlia and Roko [9] derived an integralrepresentation [6, 15, 17] of the initial premium and applied the multipiece exponen-tial function (MEF) method [16] to this representation. The MEF method, however,generates discontinuous optimal stopping and early exercise boundaries, which is aserious obstacle to decision-making of the option holder. Caperdoni and Ciurlia [5]analyzed the perpetual American case. The purpose of this paper is to value Amer-ican continuous-installment options by using the Laplace transform approach, whichgenerates smooth numerical solutions for the boundaries and provides a much simplersolution for the perpetual case.

This paper is organized as follows. The principal focus is on the call case toavoid tedious repetition. We summarize the corresponding results for the put casein Appendices A through C, except for computational results. In section 2, basedon an optimal stopping problem, we analyze asymptotic properties of the stoppingand early exercise boundaries at expiry. In section 3, applying the Laplace transformapproach to a PDE for the initial premium, we obtain an explicit Laplace transform ofthe premium, which is decomposed into the value of the associated European vanillaoption with the same payoff plus the premiums of early exercise and halfway cancel-lation. This premium decomposition enables us to derive concise Laplace transformsfor some Greeks. With the aid of the Abelian theorem of Laplace transforms, weprovide the perpetual result in section 4. In section 5, to see the power of the Laplacetransform approach and to clarify detailed properties of the initial premium as well

as the stopping and early exercise boundaries, we show some computational resultsboth for call and put cases. Finally, in section 6, we give further remarks as well asdirections for future research.

2. Free boundary formulation. Suppose an economy with finite time period[0, T], a complete probability space (, F,P), and a filtration F (Ft)t[0,T]. ABrownian motion process W (Wt)t[0,T] is defined on (, F) and takes values in R.The filtration F is the natural filtration generated by W and FT = F.


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AMERICAN CONTINUOUS-INSTALLMENT OPTIONS 805

Let (St)t[0,T] be the price process of the underlying asset. For S0 given, assumethat (St)t[0,T] is a geometric Brownian motion process,

dSt = (r )Stdt + StdWt, t [0, T],

where the coefficients (r,,) are constant. Here r represents the risk-free rate ofinterest, the continuous dividend rate, and the volatility of asset returns. Theasset price process (St)t[0,T] is represented under the equivalent martingale measureP, which indicates that the asset has mean rate of return r, and the process W is aP-Brownian motion.

Consider an American-style continuous-installment call option with the maturitydate T and strike price K. The payoff at maturity is given by (ST K)+, where(x)+ = x 0. Let q > 0 be the continuous installment rate, which means thatthe holder continuously pays an amount q dt in time dt, while the asset itself paysa continuous dividend in the amount of St dt to the holder at the same time. LetC C(t, St; q) denote the value of the continuous-installment call option at timet [0, T]. In the absence of arbitrage opportunities, the value C(t, St; q) is a solutionof an optimal stopping problem

C(t, St; q) = ess supe,s

E

1{esT}e

r(Tt)(ST K)+(2.1)

+ 1{e


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0 0.2 0.4 0.6 0.8 1

t

60

80

100

120

140

exercise region

continuation region

stopping region

Bt

At

Fig. 1. Stopping, exercise, and continuation regions of an American continuous-installmentcall option (T = 1, K = 100, r = 0.05, = 0.04, = 0.2, q = 5).

where Lt,S stands for the differential operator

Lt,Sv = vt

+ 122S2

2v

S2+ (r )Sv

S rv for v C1,2(C).

The boundary conditions of this PDE are given by

(2.3)

lim

SAtC(t, S; q) = 0, lim

SBtC(t, S; q) = Bt K,

limSAt

C

S= 0, lim

SBt

C

S= 1.

The first two (value matching) conditions imply that the initial premium is continuousacross the boundaries, while the last two (smooth pasting) conditions imply that theslope is continuous. The terminal condition is clearly given by

(2.4) C(T, S; q) = (S K)+.It is sometimes convenient to work with the equations where the current time t is

replaced by the remaining time to expiry T t. For notational convenience, wewrite S ST = St, and we refer to (S)[0,T] as the backward running process of(St)t[0,T]. For > 0, let C(, S; q) = C(T , S; q), A = AT, and B = BT.On the basis of the PDE (2.2), Ciurlia and Roko [9] proved that the value functionof the continuous-installment call option has an integral representation, which can berewritten in the time-reversed form as

C(, S; q) =

c(, S) q

0

er(u)

d(S,

Au, u)

du(2.5)

+ 0Se(u)d+(S, Bu, u) (rK q)er(u)d(S, Bu, u)du,

where ( ) is the standard normal cumulative distribution function defined by

(x) =

x

(y)dy with (x) =12

e12

x2 , x R,


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d(x, y, ) =log(x/y) + (r 122)

,

and

c(, S) is the time-reversed value of the associated European vanilla call option;

i.e., it is given by the so-called BlackScholes formula,(2.6) c(, S) = Sed+(S,K,) Kerd(S,K,).

Theorem 2.1. The terminal values of the stopping and early exercise boundaries

at expiry are given by

AT = K and BT = max

rK q

, K

.

Proof. From the value matching conditions in (2.3) and (2.5), the time-reversed

boundaries A and B are implicitly defined byB K = c(, B) q

0

er(u)d( B, Au, u)du(2.7)+0

Be(u)d+( B, Bu, u) (rK q)er(u)d( B, Bu, u)du

and

0 = c(, A) q 0

er(u)

d( A, Au, u)du(2.8)+

0

Ae(u)d+( A, Bu, u) (rK q)er(u)

d(

A, Bu, u)

du.

Using (2.6) and rearranging the terms in (2.7) and (2.8), we have

AK

=NA()

DA()and

BK

=NB()

DB(),

where

NA() = er

d( A, K , )+ r

0

er(u)

d( A, Bu, u)du+

q

K

0

er(u)

d( A, Au, u) d( A, Bu, u) ,DA() = e

d+( A, K , ) +

0

e(u)d+( A, Bu, u)du,NB() = e

r

d( B, K , ) 1 + r 0

er(u)

d( B, Bu, u)du+

q

K

0

er(u)

d( B, Au, u) d( B, Bu, u) ,DB() = e

d+( B, K , ) 1 + 0

e(u)

d+( B, Bu, u)du.


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sum (integral) of the time-reversed value c(, S; q) for (infinitely many) different val-ues of the maturity T R+ [0, +), and hence for R+, which makes LCTs bewell defined. Following this interpretation, we assume hereafter that is a positivereal number.

Lemma 3.1. Let c

(, S) = LC[c(, S)] be the LCT of the vanilla call value forthe backward running process. Then,

c(, S) =

1(S), S < K,

2(S) +S

+ K

+ r, S K,

where for i = 1, 2,

(3.1) i(S) =K

1 2

+

1 r

+ r3i

S

K

i,

and the parameters 1 1() > 1 and 2 2() < 0 are two real roots of thequadratic equation

(3.2) 1222 +

r 1

22

( + r) = 0.Proof. The vanilla call value c(t, S) satisfies the same PDE as (2.2) with the

boundary conditions

(3.3) limS0

c(t, S) = 0 and limS

dc

dS<

and the terminal condition c(T, S) = (SK)+. Hence, the call value for the backwardrunning process c(, S) c(T , ST) = c(t, St) for = T t can be obtained bysolving the PDE

(3.4)

c + 122S2 2

cS2 + (r )ScS rc = 0, S > 0,with the conditions of the same form as (3.3) and the initial condition c(0, S) =(S K)+. Taking the LCTs of (3.4) and the boundary conditions, we see thatc(, S) satisfies the ordinary differential equation (ODE)

(3.5) 122S2

dc

dS2+ (r )Sdc

dS ( + r)c + (S K)+ = 0, S > 0,

with the boundary conditions

(3.6) limS0

c(, S) = 0 and limS

dc

dS< .

It is straightforward to solve (3.5) with (3.6) as well as the continuity conditions of

c(, S) and its first derivative at S = K. Assuming a general solution of the form

(3.7) c(, S) =

2i=1

ai

S

K

i, S < K,

2i=1

bi

S

K

i+

S

+ K

+ r, S K,


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From the smooth pasting conditions in (3.12) and as well as the continuity conditionsof C(, S; q) and its first derivative at S = K, we can determine the four constantsai and bi (i = 1, 2), which are given by,

(3.13)

ai = (1)i K

i

ig(A, B)(A)3i ,

bi = (1)i K1 2

+

1 r

+ r3i

,

where

(3.14) g(A, B) =

B

+2i=1 ibi BK i

(A)1(B)2 (A)2(B)1 .

From the value matching conditions in (3.12), we see that A and B are given bysolving a pair of nonlinear equations

(3.15)

(A)1+2g(A, B) =q

+ r

121 2 ,

1

2(A)1(B)2 1

1(A)2(B)1

g(A, B) +

2i=1

bi

B

K

i=

B

+ rK q

+ r.

Rearranging the terms in (3.13)(3.15) and using Lemma 3.1, we obtain the desiredresults (3.8)(3.10) after somewhat cumbersome calculations.

Remark3. The solvability of the nonlinear equations in (3.10) is an open problemdue to its complexity. However, we will see in section 5 that the roots A and B can

be stably computed by the standard Newton method where the initial values, say A0and B0 , are chosen from the terminal values given in Theorem 2.1, i.e.,

A0 = K and B0 = max

rK q

, K

.

We see from Theorem 3.2 that the total premium in the continuation regionC = (At, Bt) has the decomposition

(3.16) C(t, S; q) + Kt = c(t, S) + c(t, S; q) for S (At, Bt),

where

(3.17) Kt = LC1 q + r = q 0 erudu = qr 1 er(Tt)

is the NPV of the future payment stream at time t [0, T], and hence the left-hand side of (3.16) represents the total amount of premiums to be paid during thetrading interval [t, T]. Due to the additional rights to exercise early or to terminatethe contract, the total payments charged for an installment option is greater thanthat for a standard vanilla option if the holder keeps his position until maturity. The


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quantity c(t, S; q) represents the sum of the premiums of early exercise and halfwaycancellation, which is obtained by inverting its LCT as

(3.18) c(t, S; q) = c(T , S; q) = c(, S; q) = LC1[c (, S; q)].

Using the relation (x) = 1 (x) (x R), we can rewrite the integral representa-tion (2.5) as

(3.19) C(t, S; q) = c(t, S) Kt + ec(t, S; q) + hc(t, S; q),

where, for = T t,

ec(t, S; q)

=

0

Se(u)

d+(S, Bu, u) rKer(u)d(S, Bu, u)du

denotes the premium of early exercise, and

hc(t, S; q) = q

0

er(u)

d(S, Au, u)+ d(S, Bu, u) du

denotes the premium of halfway cancellation. From the premium decomposition(3.16), we have

c(t, S; q) = ec(t, S; q) + hc(t, S; q).

It should be noted that the sum c(t, S) + ec(t, S; q) does not coincide with thevalue of the associated American vanilla option, because the early exercise bound-ary (Bt)t[0,T] is implicitly affected by the stopping boundary (At)t[0,T] except for

q > 0.By virtue of the premium decomposition (3.16), we obtain some Greeks ofC(t, S; q) in hybrid forms, as follows.

Corollary 3.3. For S (At, Bt), we have

C =C

S= c + LC1[c ],

C =2C

S2= c + LC1[c ],

C = C

= c + qer + LC1[c ],

where, for d

d(S,K,),

c = e

d+

,

c =e

S

d+

,

c = Se

2

d+

+ Se

d+ rKerd,


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and

c =1

S

q

+ r

121 2

S

A

2

S

A

1 11(S)

,

c =1

S2

q + r

121 2

(21)

SA

2 (11)

SA

1 1(11)1(S)

,

c =

q

+ r

121 2

1

2

S

A

2 1

1

S

A

1 1(S)

.

Proof. The results for C and C can be obtained from the premium decompo-sition (3.16), the BlackScholes formula (2.6) for c(t, S), and the relations

c = LC

cS

=

cS

and c = LC

2cS2

=

2cS2

,

while the result for C follows from (3.16), (2.6), and

c = LC

c

= {c (, S; q) c(0, S; q)} = c (, S; q),

because

c(0, S; q) = c(T, S; q) = C(T, S; q) + KT c(T, S)= (S K)+ (S K)+ = 0,

which completes the proof.Remark 4. We see from (3.8) and (3.11) that there exists a parity relation among

the Greeks for c (, S; q) such that

(3.20) 122S2c + (r )Sc + c = rc ,

and hence that one of the Greeks can be computed by the other two Greeks and c .

4. Perpetual continuous-installment call option. Consider the case withinfinite maturity, i.e., T = . Let C(S; q) denote the value of the perpetualcontinuous-installment call option for the initial asset price S. Applying the Abeliantheorem of LTs to the LCT C(, S; q) in Theorem 3.2, we can obtain the perpetualresult as follows.

Theorem 4.1. For S (A, B),

C(S; q) = 1

1

(A)2S

1 + 12

(A)1S

2

(A)1 (B)

21 (A)

2 (B)

11

q

r

(4.1)

=q

r

12

1 2

1

1

S

A

1

+1

2

S

A

2

q

r,(4.2)

where 1 > 1 and 2 < 0 are two real roots of the quadratic equation

(4.3) 122()2 + (r 122) r = 0,


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and the constants A and B are the optimal threshold levels given by

(4.4)

A =q

r

12

1 2

21 11

,

B = A = qr

12

1 22 1

in terms of > 1, which is the unique solution of the equation

(4.5) 2(1 1)

1 1(2 1)

2 = (1 2)

1 rKq

.

Proof. By virtue of the final-value theorem of LTs, the asymptotic value C(S; q)can be obtained by

C(S; q) = limT

C(t, S; q) = lim

C(, S; q) = lim0+

C(, S; q).

From (3.13) and (3.14), we see that for i = 1, 2lim

0+ai = (1)i K

i

i

(A)3i

(A)

1 (B)

21 (A)2 (B)11 ,

lim0+

bi = 0,

where A = lim0+ A(), B = lim0+ B

(), and i = lim0+ i() (i = 1, 2)are given by the roots of (4.3), and hence that (4.1) holds for S (A, B). Letting 0+ in (3.15) and using the change of variables

:=BA

> 1,

we have

(4.6)

B

2 1 =q

r

12

1 2,

B

2 1

11

1 +1

2

2

= B K+ q

r.

We obtain B and A in (4.4) as well as the alternative expression in (4.2) fromthe first equation in (4.6) and the relation A = B/. Equation (4.5) for can bederived by substituting B in (4.4) into the second equation in (4.6). If we set

() 2(1 1)

1 1(2 1)

2 (1 2)

1 rKq

,

then it is easy to see thatlim1

() = (1 2)rK

q> 0,

lim

() = ,

() = 12

(1 1)

11 (2 1)

21

< 0 for > 1,


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and hence that the root of the equation () = 0 uniquely exists in the interval(1, ).

Define the Greeks of C(S; q) as

C =dC

dS , C =

d2CdS2 , and

C = dC

d .

Clearly, C = 0. For C and C , from (4.2) we immediately have the following.Corollary 4.2. For S (A, B),

C =1

S

q

r

12

1 2

S

A

2

S

A

1

,

C =1

S2q

r

12

1 2

(2 1)

S

A

2

(1 1)

S

A

1

.

Corollary 4.3. Let C = (A, B) denote the continuation region of theperpetual American continuous-installment call option. Then, as the installment rate

q increases, C shrinks monotonically and vanishes in the limit, i.e., limq C = .Proof. From the equation () = 0, we have

d

dq=

(1 2)rKq21

2

(1 1)11 (2 1)21

< 0,lim

q(1) = 2(

1 1) 1(2 1) (1 2) = 0,

which proves the assertion.Remark 5. Caperdoni and Ciurlia [5] also analyzed the perpetual continuous-

installment call and put options. For the call case, they assumed a general form ofC(S; q) as

C(S; q) = a1S1 + a2S

2

q

r

, S

(A, B),

for constants ai (i = 1, 2) and obtained a set of four nonlinear equations for a1, a2,A, and B, which have to be solved numerically. However, there is no proof forthe existence and uniqueness of these roots. No doubt, Theorem 4.1 provides a muchmore useful representation of the perpetual solution.

If we fix the installment rate as q = rK, then from (4.5) we obtain

(4.7) =

1(

2 1)

2(1 1)

11

2

.

It is worth noting that this solution has a form similar to the early exercise boundaryof the Russian option [24]; see also Proposition 4 of Kimura [18]. Another explicitsolution of (4.5) for can be found when = 0, which is

(4.8) =

1 rKq

22r.

From > 1, we have q > r K , for which the value of the perpetual continuous-installment call option does not coincide with that of the perpetual Europeancontinuous-installment call option with the same contractual features; see Remark 4of Kimura [19].


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Table 2

Initial premiums of American continuous-installment put options (t = 0, K = 100, r = 0.05, = 0.04, = 0.2).

q S T = 0.5 T = 1 T = 5 T = 10 T =

1 95 7.505 8.872 12.577 13.742 14.456100 4.853 6.400 10.456 11.719 12.492105 2.914 4.448 8.630 9.958 10.778

5 95 6.224 6.653 7.045 7.052 7.052100 3.403 3.921 4.378 4.385 4.385105 1.492 1.976 2.422 2.429 2.429

10 95 5.369 5.430 5.444 5.444 5.444100 2.250 2.337 2.356 2.356 2.356105 0.507 0.579 0.595 0.595 0.595

60 80 100 120 140

S

10

20

30

40

A me ri can E uro pean

(a) call case

60 80 100 120 140

S

10

20

30

40

AmericanEuropean

(b) put case

Fig. 2. American vs. European continuous-installment option values (t = 0, T = 1, K = 100,q = 5, r = 0.05, = 0.04, = 0.2).

of the simultaneous equations in (3.10) ((B.3)) or the root () of the equation in(4.5) ((C.4)), we simply use the Newton method. For finite-lived cases, the 6-pointextrapolation is used in the GaverStehfest method. We see from these tables that thevalues of initial premium become insensitive to the length T of the trading interval asq grows, which implies that the perpetual value gives an accurate approximation forfinite-lived cases, especially when the position is in-the-money. On the contrary, forsmall q, the option values are sensitive to T when the position is out-of-the-money.

To see the early exercise premium of the American continuous-installment option,we compare the values of American and European continuous-installment call (put)options in Figure 2(a) (2(b)), where the intrinsic value (S K)+ ((K S)+) isdrawn in a dashed line. We compute the American option values by the GaverStehfest method combined with the 4-point extrapolation and the Newton method,whereas the European values are computed by the algorithm developed in Kimura [19].From these figures, we observe that (i) numerical inversion mostly works well evenaround the smooth-pasting points; (ii) there is almost no difference between the twocurves when the position is out-of-the-money; (iii) the difference between two curves

looks almost constant at least when the position is in-the-money. Obviously, thisdifference represents the early exercise premium of the American option. The finding(iii) can be stated more precisely by using the asymptotic value of the Europeancontinuous-installment option obtained in Kimura [19, Remark 1]. For the call case,the asymptotic result is written as

c(t, S; q) Se(Tt) Ker(Tt) qr

1 er(Tt)

for large S K,


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figures, the curves for q = 5 in Figure 3 and those for = 0.03 in Figure 4 are drawn indashed lines. We can certify in these figures that the terminal values at maturity areconsistent with the theoretical results in Theorem 2.1. From Figure 3, we see that thearea of continuation, region C, becomes narrower as q increases. This property wasalso shown in numerical experiments of Ciurlia and Roko [9], in which they proved forthe call case that C vanishes if q , by using the integral representation (2.5); cf.Corollary 4.3 in this paper. We observe from Figure 4(a) that the region C for the callcase also becomes narrower as increases, and that the stopping boundary (At)t[0,T]is less sensitive to the value of than the early exercise boundary (Bt)t[0,T]. However,for the put case in Figure 4(b), we see that the region C slightly grows as increases,although both boundaries (Ft)t[0,T] and (Gt)t[0,T] are almost insensitive.

6. Concluding remarks. In this paper, for American continuous-installmentcall/put options with a given installment rate q, we analyzed their initial premiums,stopping and early exercise boundaries, and some hedging parameters by combiningthe PDE approach with LCTs. From a financial point of view, the question of howto determine an appropriate installment rate q is also an important problem. For

discrete-installment options with premium payments q0 at time t0 = t and q1 at timest1, . . . , tn (t, T), a uniform installment plan with q0 = q1 can be considered asa reasonable candidate for determining q1. For continuous-installment options, theuniform installment plan implies the null initial premium, i.e., C(t, S; q) = 0 for thecall case, and the uniform installment rate, say q, can be obtained by solving aminimization problem

q = inf{q > 0 | C(t, S; q) = 0} .

As pointed out by Griebsch, Kuhn, and Wystup [14], the minimization is requiredto obtain a unique rate, which is due to the fact that C(t, S; q) can never becomenegative, as it is always possible to stop paying installments immediately. Of course,we could consider more sophisticated installment plans, which can be formulated asan optimization problem. This is a direction of future research.

Like finite-difference methods and lattice models, we believe that the PDE/ LCTapproach has the potential power to deal with further generalizations to, e.g., (i)options with a general payoff function at expiry, (ii) exotic options whose initialpremium satisfies the inhomogeneous PDE in (2.2), and (iii) price processes with

jumps [23], stochastic volatility [20], and so on. These generalizations also may beconsidered as directions for future research.

Appendix A. Free boundary formulation for puts. In much the same wayas for the call case, we can also consider the put case with the maturity date T,strike price K, and installment rate q. Let P P(t, S; q) denote the value of thecontinuous-installment put option at time t [0, T]. Then, the put value P(t, S; q)(t [0, T]) is a solution of an optimal stopping problem similar to (2.1),

P(t, S; q) = esssupe,s

E 1{esT}er(Tt)(K ST)+(A.1)+ 1{e


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0 0.2 0.4 0.6 0.8 1

t

60

80

100

120

140

stopping region

continuation region

exercise region

Gt

Ft

Fig. 5. Stopping, continuation, and exercise regions of an American continuous-installmentput option (T = 1, K = 100, r = 0.05, = 0.04, = 0.2, q = 5).

respectively. To distinguish the boundaries of these regions from those for the callcase, we denote the exercise boundary by (Ft)t[0,T] and the stopping boundary by(Gt)t[0,T]. Since the put value P(t, S; q) is nonincreasing in S, unlike the call case,we see that (Ft)t[0,T] becomes a lower boundary and (Gt)t[0,T] an upper boundary;see Figure 5.

The put value P(t, S; q) satisfies the same PDE as (2.2),

(A.2) Lt,S P(t, S; q) = q, Ft < S < Gt,together with the boundary conditions

(A.3)

limSFt

P(t, S; q) = K Ft, limSGt

P(t, S; q) = 0,

limSFt

P

S = 1, limSGtP

S = 0,

and the terminal condition

(A.4) P(T, S; q) = (K S)+.Corresponding to (2.5), we have the integral representation

P(, S; q) = p(, S) q 0

er(u)d(S, Gu, u)du(A.5)

+

0

(rK+ q)er(u)

d(S, Fu, u) Se(u)

d+(S,

Fu, u)du,

where p(t, S) is the value of the associated European vanilla put option, defined by

(A.6) p(, S) = Kerd(S,K,) Sed+(S,K,).In much the same way as in Theorem 2.1, we can prove that

FT = min

rK + q

, K

and GT = K,


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from which we obtain FT = K for = 0, and FT min

r

, 1

K for > 0; cf. Corol-lary 2.2 and (2.9).

Appendix B. Valuation with LCTs for puts. For American vanilla call andput options, it is well known that there exist certain symmetric relations betweenthe values and the early exercise boundaries of American options; see McDonaldand Schroder [21]. However, such symmetric relations do not hold for continuous-installment options with constant installment rate, which can be easily verified bychecking the integral representations (2.5) and (A.5). This means that results parallelto Theorem 3.2 and Corollary 3.3 should be given separately for the continuous-installment put option. We simply provide those results without proofs.

Lemma B.1. Let p(, S) = LC[p(, S)] be the LCT of the vanilla put value forthe backward running process. Then

p(, S) =

1(S) S

+ +

K

+ r, S < K,

2(S), S K.Theorem B.2. The LCT P(, S; q) for the American continuous-installment

put option is given by

(B.1) P(, S; q) =

K S, S [0, F],

p(, S) + p(, S; q) q

+ r, S (F, G),

0, S [G, ),where p (, S; q) is defined by

(B.2) p (, S; q) =q

+ r

121 2

1

2

S

G

2 1

1

S

G

1 2(S),

and F and G are given by solving a pair of nonlinear equations

(B.3)

q

+ r

121 2

1

2

F

G

2 1

1

F

G

1

= 1(F) + 2(F) +

F +K+ q

+ r,

q

+ r

121 2

F

G

2

F

G

1

= 11(F) + 22(F) +

F.

We see from Theorem B.2 that the total premium for the put option also has thedecomposition

(B.4) P(t, S; q) + Kt = p(t, S) + p(t, S; q),

where p(t, S; q) represents the sum of the premiums of early exercise ep(t, S; q) andof halfway cancellation hp(t, S; q). From the integral representation (A.5), they are


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given by

ep(t, S; q)

=

0 rKer(u)d(S, Fu, u) Se(u)d+(S, Fu, u)duand

hp(t, S; q) = q

0

er(u)

d(S, Fu, u)+ d(S, Gu, u)du.

Corollary B.3. For S (Ft, Gt), we have

P =P

S= p + LC1[p ],

P =2P

S2= p + LC1[p ],

P = P

= p + qe

r

+ LC1

[

p ],

where for d d(S,K,),p = e

d+ 1 ,

p = c =e

S

d+

,

p = Se

2

d+ Sed++ rKerd,

and

p =

1

S q + r 121 2 SG2

SG1

22(S) ,p =

1

S2

q

+ r

121 2

(21)

S

G

2 (11)

S

G

1 2(21)2(S)

,

p =

q

+ r

121 2

1

2

S

G

2 1

1

S

G

1 2(S)

.

Appendix C. Perpetual continuous-installment put option. For the putcase, we can also have parallel results. Actually, the continuation region C =(F, G) for the put option also shrinks as q . For later convenience, wewrite down the results on the value P(S; q) and its Greeks without their proofs.

Theorem C.1. For S (F, G),

P(S; q) = 1

1

(G)2S

1 + 12

(G)1 S

2

(G)

1 (F)

21 (G)2 (F)11

q

r(C.1)

=q

r

12

1 2

1

1

S

G

1

+1

2

S

G

2

q

r,(C.2)


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where i (i = 1, 2) are two real roots of the quadratic equation (4.3), and the constantsF and G are the optimal threshold levels given by

(C.3) F =

q

r

12

1

2 2

1 ,G =

F

=q

r

12

1 2

21 11

in terms of (0, 1), which is a unique solution of the equation

(C.4) 2(1 1)

1 1(2 1)

2 = (1 2)

1 +rK

q

.

Corollary C.2. For S (F, G),

P =dP

dS=

1

S

q

r

12

1

2

S

G2

S

G1

,

P =d2P

dS2=

1

S2q

r

12

1 2

(2 1)

S

G

2

(1 1)

S

G

1

.

For = 0, (C.4) for also can be solved explicitly, obtaining

(C.5) =

1 +

rK

q

22r

.

Unlike the call case, however, the solution is defined for arbitrary q > 0.

Acknowledgments. I would like to thank Kazuaki Kikuchi of Sumitomo MitsuiBanking Corporation for his contribution to these computational experiments. Special

thanks are also due to the anonymous referees for their comments.

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Kimura American Installment