# 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 (kimura@econ.hokudai.ac.jp).

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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806 TOSHIKAZU KIMURA

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

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