Optimal Exercise forDerivative SecuritiesJérôme DetempleSchool of Management, Boston University, Boston, Massachusetts 02215;email: detemple@bu.edu

Annu. Rev. Financ. Econ. 2014. 6:459–87

First published online as a Review in Advance onOctober 22, 2014

The Annual Review of Financial Economics isonline at financial.annualreviews.org

This article’s doi:10.1146/annurev-financial-110613-034241

Copyright © 2014 by Annual Reviews.All rights reserved

JEL code: G13

Keywords

derivatives, options, American-style, valuation, exercise region,exercise boundaries

Abstract

This article reviews the literature on American-style derivatives. Thepresentation stresses some of the major developments in the field.The focus is on the determination of optimal exercise policies and thestructure of derivatives’ prices. Illustrative examples highlight thecomplexity of the optimal exercise decision.

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1. INTRODUCTION

American-style derivatives, that is, derivatives with early exercise provisions, are available onnumerous exchanges. Typical examples are options on indices (e.g., OEX contract on S&P 100Index), on exchange-traded products such as ETFs (exchange-traded funds) or ETNs (exchange-traded notes) (e.g., BOND contract on Pimco Total Return ETF), on futures (e.g., LO contract onLight Sweet Crude Oil futures), and on bonds (e.g., OZF contract on 5 Yr T-Note). Otherexamples include flexible options with early exercise provisions. Products in this category can becustomized as needed and can be written on a variety of underlying assets including stocks.Exercise decisions are timing decisions. The latter arise in a multitude of contexts, such as capitalbudgeting decisions of firms, durables consumption decisions of households, and employmentdecisions of individuals and corporations. In all cases, determining the best timing decision iscritical. Incorrect choices result in losses of value and can have significant negative ramifications.

The valuation of American-style derivatives is a complex exercise. Unlike its European coun-terpart, the value of anAmerican claimdepends on the exercise decision. Proper valuation requiresthe identification of the optimal exercise policy. Conversely, the best exercise policy dependson the value that can be achieved. The intrinsic link between these two aspects explains thecomplexity of the problem. It also motivates the academic interest in the subject.

The literature on American derivatives is vast. A complete survey is beyond the scope of thisreview. This reviewpresents a selection ofmethods that have been proposed to identify the optimalexercise policy and the associated value of a claim. The article focuses mostly on significantadvances and notable results. This is complemented by short descriptions of a few additionalresearch topics and citations to the relevant literature.

Section 2 starts with a description of the decision problem. Section 3 continues with a briefpresentation of fundamental pricing methods for American derivatives. The emphasis is onmethods based on partial differential equations (PDEs) and on risk-neutral valuation. Section 4focuses on the structure of prices and optimal exercise decisions. Section 5 contains applicationsto parametric models and specific contractual forms. These examples illustrate the usefulness ofsome of the general results and show the potential complexity of exercise decisions associatedwithseemingly simple contractual forms. Section 6 reviews a less typical derivative contract, high-lighting complications that can arise. Section 7 discusses an extension to amodelwith jumps in theunderlying price. Section 8 provides conclusions and suggested directions for future research.

2. MODEL

Consider an American derivative written on an asset with price S. The financial market includesa riskless asset (money market account) paying interest at the nonnegative rate r(Y) per unit time.The underlying asset price S, along with a relevant state variable Y, evolves according to

dSt ¼ StððmðSt,YtÞ� dðSt,YtÞÞdt þ sðSt,YtÞdWtÞdYt¼

�myðSt,YtÞdt þ syðSt,YtÞ

�rðSt,YtÞdWt þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� rðSt,YtÞ2

qdWy

t

�,

8><>: ð1Þ

where (W, Wy) are independent Brownian motions. The coefficient m(St, Yt) is the expectedasset return, d(St, Yt) is the dividend yield, and s(St, Yt) is the return volatility. The coefficientsmy(St, Yt) and sy(St, Yt) are the expected change and the volatility of the change in thestate variable, and r(St, Yt) is the correlation between the asset return and the change in thestate variable. All coefficients are functions of the pair (S, Y). Coefficients are assumed to besuch that the price process S is nonnegative (S 2 Rþ), and 0 is an absorbing barrier (Sv ¼ 0 for

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v� t if St¼ 0). The state variable process Y takes values in a domainDy ⊆R. Let D ¼Rþ3Dy bethe domain for the pair (S, Y). Also, d(St, Yt) is nonnegative, and s(St, Yt) is positive. The marketprice of W-risk is u(S, Y) [ s(S, Y)�1 (m(S, Y) � r(Y)).

The state variable affecting the evolution of the underlying asset return is assumed to bemarketed. Thus, there exists a traded or synthetic asset that hedges the risk Wy specific to Y.Let dHt=Ht ¼ mhðSt,YtÞdt þ shðSt,YtÞdWy

t be the return on that asset, and assume that themarket price ofWy-risk, uy(S, Y)[ sh(S, Y)�1(mh(S, Y)�r(Y)), is well defined. The market underconsideration is complete.1

Equation 1 is general enough to accommodate reasonable specifications for the evolution ofstock market indices, currencies, and commodity prices. It also allows for stochastic fluctuationsin the return volatility, the dividend yield, and the short rate.

The American derivative is a contingent claim with exercise payoff g(S), where g(×) :Rþ → Rþis a continuous function, almost everywhere differentiable. The claim is American style,meaning that it can be exercised at any time prior to the maturity date T of the contract. Whetherto exercise is the choice of the holder of the claim. The objective is to price the derivative securityand find the optimal exercise policy.

3. PRICING METHODOLOGIES

This section reviews a selection of pricing approaches for American claims. The focus is on PDEs(Section 3.1) and risk-neutral valuation (Section 3.2). References to other approaches are alsoprovided (Section 3.3).

3.1. Partial Differential Equations

The earliest approach to the valuation of American derivatives treats the valuation problem asa free boundary problem. The value of the contract is shown to satisfy a PDE.

Theorem 1: Let V(S, Y, t) be the value of the American derivative with payoff g(S)and suppose that V(S, Y, t) 2 C2,1(D3 ½0,T�). The price of the claim solves thefundamental valuation equation

0 ¼ �rðYÞV þ ∂V∂t

þ ∂V∂S

S�rðYÞ � dðS,YÞ�þ ∂V

∂Ymy�ðS,YÞþ 1

2∂2V∂S2

S2sðS,YÞ2

þ ∂2V∂S∂Y

SsðS,YÞrðS,YÞsyðS,YÞ þ 12∂2V∂Y2 s

yðS,YÞ2, ð2Þ

subject to the boundary conditions

ðiÞ VðS,Y,TÞ ¼ gðSÞ for ðS,YÞ 2D

ðiiÞ Vð0,Y, tÞ ¼ gð0Þ for ðY, tÞ 2Dy � ½0,T�ðiiiÞ VðS,Y, tÞ ¼ gðSÞ for ðS,Y, tÞ ¼ �

BðY, tÞ,Y, t�ðivÞ ∂VðS,Y, tÞ

∂S¼ g0ðSÞ for ðS,Y, tÞ ¼ �

BðY, tÞ,Y, t�,

8>>>>>>>><>>>>>>>>:

ð3Þ

where

1The underlying probability space is (V, F , P). The information filtration is the filtration generated by the pair (W, Wy).

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my�ðS,YÞ[myðS,YÞ � syðS,YÞðrðS,YÞuðS,YÞ þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� rðS,YÞ2

quyðS,YÞÞ, ð4Þ

and B(Y, t) is the optimal exercise boundary. The system comprising Equations 2and 3 is the free boundary characterization of the pricing function.

The fundamental valuation formula takes the form of a PDE, as described by Equation 2. ThePDE governs the behavior of the option price prior to exercise. It is subject to the boundaryconditions in Equation 3. Condition i is the terminal condition. It stipulates that the priceat the maturity date be equal to the exercise payoff. Condition ii reflects the assumption thatS ¼ 0 is an absorbing barrier for the underlying price. Condition iii is known as the value-matching condition. It mandates that the derivative’s price be equal to the exercise payoff at thepoint (S, Y, t)¼ (B(Y, t),Y, t), where exercise takes place. Condition iv is the smooth-pasting orhigh-contact condition. It ensures that immediate exercise is optimal at (S,Y, t)¼ (B(Y, t),Y, t).

All the conditions in Equations 2 and 3 reflect no-arbitrage considerations. The PDE(Equation 2) ensures that an instantaneously riskless portfolio, constructed by investing in thestock and the option, earns the same return as an investment in the money market account.Boundary conditions i, ii, and iii enforce the absence of arbitrages at the terminal date, theabsorption date, and the exercise date, respectively. The smooth-pasting condition (iv) is alsomotivated by no-arbitrage considerations. If the derivative of the price does not equal thederivative of the payoff function at an exercise point, i.e., limx→S∂Vðx,Y, tÞ=∂S�g0ðSÞ at (S,Y, t)¼(B(Y, t),Y, t), there exists a riskless portfolio composed of the underlying asset and the claimreturning more than the riskless rate.2

The valuationPDE (Equation 2) has a couple of notable features. First, the expected return of theunderlying asset price, as shown in the third term, has been adjusted by substituting the interest rate rin place of the driftm. Second, the expected return of the derivative security is set equal to the interestrate, as shown by the first term. Both features are consequences of the no-arbitrage restriction

Et

�dVV

�� rðYtÞ ¼ bs

tðmt � rtÞ þ bht ðmh

t � rtÞ,where

bst ¼

CovðdV=V, dS=SÞVarðdS=SÞ ¼ VSStst þ VYs

yt rt

Vst

bht ¼

CovðdV=V, dH=HÞVarðdH=HÞ ¼

VYsyt

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� r2t

qVsh

t.

In these expressions, Et[×] is the conditional expectation at time t, VS and VY are partial derivatives,and arguments of functions are omitted to simplify notation. This condition sets the risk premium ofthe derivative equal to a weighted sum of the risk premia of the underlying asset and of the assethedging state variable risk. Theweights reflect the exposures of the derivative to the sources of risk andare givenby thebeta coefficientsbs

t ,bht . In effect, the derivative’s return satisfies the standard restriction

2The generalized Ito rule for continuous functions that are not everywhere differentiable shows that the derivative’s price hasa local time component. This local time component can be extracted by constructing an instantaneously riskless portfolio.

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found in the multi-beta capital asset pricing model (Merton 1973a). Using Ito’s lemma to calculatethe expected return of the derivative and collecting terms gives the valuation PDE (Equation 2).

Theoptimal exerciseboundaryB(Y, t) appearing in conditions iii and iv is not known ex ante andneeds to be determined, along with the derivative’s price. For this reason, the system comprisingEquations2and3 is calleda freeboundaryproblem.Also, theboundary is a two-dimensional surfaceparametrized by the state variable Y and time t. This follows from the asset price specification(Equation 1) incorporating stochastic fluctuations in the interest rate, dividend rate, and returnvolatility. In this setting, the exercise trigger depends on the level of the state variable.

The earliest attempt to formulate the valuation of American warrants as a free boundaryproblem goes back toMcKean (1965). Following Samuelson (1965), the warrant is discounted atan arbitrary rate. The warrant price is then shown to satisfy an equation similar to Equation 2, butwith modified coefficients, along with the relevant boundary conditions in Equation 3. Black &Scholes (1973) and Merton (1973b) introduce valuation arguments based on the absence ofarbitrage opportunities. For special cases of the underlying price model (Equation 1), they derivethe relevant valuation equations and boundary conditions for European options.Merton (1973b)also considers the valuation of American options when the underlying asset pays dividends. Hisarticle motivates the value-matching and smooth-pasting conditions. It also gives an explicitsolution for the price of a perpetual put.

A vast literature uses the free boundary approach to price or derive properties of Americanclaims. The approach also serves as the basis for several numerical schemes. Van Moerbeke(1976); Kuske & Keller (1998); Evans, Kuske & Keller (2002); Ekstrom (2004); and Chen andcoauthors (2008, 2013) establish properties of the exercise boundary. Schwartz (1977),Brennan & Schwartz (1977), and Courtadon (1982) develop finite difference algorithms tocompute prices. Ramaswamy & Sundaresan (1985) and Brenner, Courtadon & Subrahmanyam(1985) apply finite difference methods to American options on futures contracts. Approximationsof the fundamental pricing equation are proposed by MacMillan (1986) and Barone-Adesi &Whaley (1987). Methods based on transformations of the valuation equation are developed byWu & Kwok (1997) and Nielsen, Skavhaug & Tveito (2002). A moving boundary approach ispresented by Muthuraman (2008). Goodman & Ostrov (2002); Chen & Chadam (2006); andMitchell, Goodman & Muthuraman (2014) develop boundary evolution methods.

3.2. Risk-Neutral Valuation

An alternative to the free boundary formulation relies on the risk-neutral valuation principle andthe central concept of a risk-neutral measure Q.

Thekey idea is already implicit inEquation 2. As noted above, the valuation equation prices thederivative so as to equate its expected return to the rate of interest. It also substitutes the risk-freerate for the underlying asset’s expected return. These features suggest a connection betweenno-arbitrage pricing and risk-neutral pricing.

Theorem 2: Let t� be the optimal exercise date. For t� t�, the price of the Americanderivative with payoff g(S) is given by

VðSt,Yt, tÞ ¼ supt2Sð½t,T�Þ

E�t

bt,tgðStÞ

, ð5Þ

whereS([t,T]) is the set of stopping times of theBrownian filtration, bt,t [ e�R t

trvdv is the

discount factor at the locally risk-free rate, and E�t [×] is the expectation under the risk-

neutral measure Q. The evolution of (S,Y) under the risk-neutral measure is given by

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8<:

dSt ¼ St�rðYtÞ � dðSt,YtÞ

�dt þ sðSt,YtÞdW�

t

dYt¼ my�ðSt,YtÞdt þ syðSt,YtÞ�rðSt,YtÞdW�

t þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� rðSt,YtÞ2

qdWy�

t

�, ð6Þ

wheremy�(St,Yt) is defined in Equation 4, andW� andWy� areQ-Brownianmotions.The optimal exercise time t� is the solution of the stopping time problem at the initialdate, t� ¼ arg supt2S([0,T])E�[b0,tg(St)]. At t�, immediate exercise is optimal:VðSt� ,Yt� , t�Þ ¼ gðSt� Þ.

Equation 5 is intuitive. It states that the price of the American derivative is the highest presentvalue that can be achieved by exercising at or before maturity. Present value is calculated bydiscounting the realized payoff at the instantaneous risk-free rate and taking the expectation underthe risk-neutral measure. Maximization of the present value is over the set of feasible exercisedates, i.e., the set of stopping times of the Brownian filtration.3

Risk-neutral valuation involves the replacement of the drift component m(S, Y) of the un-derlying asset return by the risk-free rate r(Y). It also adjusts the drift of the state variable by therisk premium my�(S, Y) � my(S, Y). To preserve the trajectories of (S, Y), the driving processesbecome dW�

t ¼ dWt þ uðSt,YtÞdt and dWy�t ¼ dWy

t þ uyðSt,YtÞdt. The processes W� and Wy�

are Brownian motions under Q. Under the risk-neutral measure, the expected return of theunderlying asset equals the risk-free rate. The expected change of the state variable becomesmy�(St, Yt). Volatilities and correlations are not affected.

The risk-neutral valuation procedure was originally introduced by Cox & Ross (1976), whofocus on European options. Foundations for the approach can be found in Harrison & Kreps(1979) and Harrison & Pliska (1981).

Performing the following transformation can provide additional perspective. MultiplyingEquation 5 by the discount factor shows that

btVðSt,Yt, tÞ ¼ supt2Sð½t,T�Þ

E�t ½btgðStÞ�[Zt ð7Þ

for t � t�. Unlike V(St, Yt, t), the process Zt [ supt2Sð½t,T�ÞE�t

btgðStÞ

can be defined for all

t 2 [0, T], not just t � t�. This process is called the Snell envelope of the discounted payoff bg(S).The Snell envelope corresponds to the value function of the collection of optimal stopping timeproblems fsupt2Sð½t,T�ÞE�

tbtgðStÞ

: t 2 ½0,T�g indexed by time. The stopping time problem with

starting time t has value function Zt. Its solution is a stopping time t(t) in S([t, T]). As Equation 7shows, the value of the American claim prior to exercise is the adjusted value of the Snell enve-lope, VðSt,Yt, tÞ ¼ b�1

t Zt. At the optimal exercise date, VðSt� ,Yt� , t�Þ ¼ b�1

t� Zt� ¼ gðSt� Þ andt(t�) ¼ t�. For t < t�, it is suboptimal to exercise, implying VðSt,Yt, tÞ ¼ b�1

t Zt > gðStÞ andt(t)> t. It follows that the optimal exercise time of the claim is the first time at which it is optimalto stop. That is, t� ¼ t(0) ¼ inf {t 2 [0, T] : Zt ¼ btg(St)} ¼ inf {t 2 [0, T] : t(t) ¼ t}.4

Stopping time problems such as Equation 7 have been extensively studied. Standard referencesfocusing on Markovian models include Fakeev (1971), Bismut & Skalli (1977), and Shiryaev

3The Brownian filtration is the filtration generated by the BrownianmotionsW andWy. This is the set of trajectories generatedby (W,Wy).The randomtime t is a stopping time of the filtration if {t� t}2F t for all t2 [0,T]. A stopping time of the filtrationis measurable with respect to the information available. Hence, it is a feasible exercise date.4The Snell envelope can be interpreted as the discounted value of a continuously reissued (or continuously quoted) Americanclaim. The optimization problem (Equation 5) can also be defined for all t 2 [0, T]. The value function represents the price ofthis continuously quoted American claim.

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(1978). The theory for general stochastic processes can be found in El Karoui (1981; see alsoKaratzas & Shreve 1998 for results in the case of continuous time processes). The relationbetween the Snell envelope and the value of an American claim is derived in Bensoussan (1984)and Karatzas (1988).

3.3. Other Pricing Approaches

Various other approaches have been used to derive and study the prices of American derivativesand the associated exercise policies. Popularmethods include variational inequalities, lattices, andvarious types of exercise policy approximations.

Jaillet, Lamberton & Lapeyre (1990) develop the variational inequalities approach toAmerican option pricing. It can be used to justify the finite difference algorithm proposed byBrennan & Schwartz (1977). Mathematical background on variational inequalities can befound in Bensoussan & Lions (1978) and Kinderlehrer & Stampacchia (1980). Numericalmethods for variational inequalities are described in Glowinski, Lions & Tremolieres (1981) andGlowinski (1984). Relations between the variational inequality approach to American optionpricing, a linear complementarity formulation, and the free boundary approach are clarified byDempster & Hutton (1999). They also examine numerical schemes based on these formulations.Feng et al. (2007) review applications to derivatives pricing and numerical implementations basedon the variational inequalities method.

Lattices such as binomial and multinomial trees provide simple frameworks in which thevaluation of American claims becomes transparent and amenable to numerical implementation.Development of the binomial option pricing model goes back to Cox, Ross & Rubinstein (1979)and Rendleman&Bartter (1979). Both papers describe and implement a backward algorithm forthe valuation of American options. Various extensions and refinements of the method havefollowed. Notable contributions include the accelerated binomial model of Breen (1991); thetrinomial andmultinomial models of Boyle and coauthors (1988, 1989), andKamrad&Ritchken(1991); and Broadie & Detemple’s (1996) binomial model with Black-Scholes correction (BBS).

An American option can also be viewed as the limit of a sequence of Bermudan options.Parkinson (1977) exploits this relation and presents a backward algorithm for this approach.This idea is, in fact, at the core of Breen’s (1991) accelerated binomial model. Variations of themethod can be found inGeske& Johnson (1984); Bunch& Johnson (1992); andHo, Stapleton&Subrahmanyam (1997).

An alternative to restricting the number of exercise dates is to constrain the form of theexercise policy. Simple exercise policies, such as the first hitting time of a constant barrier,have known distributions under certain conditions. Explicit valuation formulas are thenobtained and can be used to derive lower bound approximations (LBA) for American optionprices. They can also be used to construct upper bound approximations (UBA) and mixtureapproximations (LUBA). Broadie & Detemple (1996) describe this approach. Related con-tributions include Omberg (1987); Bjerksund & Stensland (1993b); Ju (1998); and Chung,Hung & Wang (2010).

4. PRICES AND OPTIMAL EXERCISE POLICIES

This section describes the structure of American claim prices and optimal exercise policies. Ex-ercise premium representations of the price (Section 4.1) and an integral equation for the optimalexercise boundary (Section 4.2) are presented. An interesting duality formula is also described

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(Section 4.3). Throughout the section, V(S, Y, t) is reinterpreted as the value of a continuouslyquoted derivative.

4.1. Exercise Premium Representations

There are several decompositions of the American claim price available. The first one relatesthe American claim price to the price of a European claim. Recall that t(v) is the solution ofthe stopping time problem supt2Sð½v,T�ÞE

�v½btgðStÞ� started at time v. The set {t(v) ¼ v} is the

event where immediate stopping (hence immediate exercise of a newly issued derivative) isoptimal.

Theorem 3: The price of the American-style derivative with payoff g(S) has the earlyexercise premium (EEP) decomposition

VðSt,Yt, tÞ ¼ VeðSt,Yt, tÞ þPðSt,Yt, tÞ, ð8Þ

where

VeðSt,Yt, tÞ[E�tbt,TgðSTÞ

, ð9Þ

PðSt,Yt, tÞ[E�t

"Z T

t1ftðvÞ¼vgbt,v

�rvgðSvÞdv� dgðSvÞ

�#. ð10Þ

The quantity Ve(St, Yt, t) is the value of the European claim with maturity dateT and exercise payoff g(ST), and P(St, Yt, t) is the EEP. In the event {t(v) ¼ v},supt2Sð½v,T�ÞE�

v½btgðStÞ� ¼ bvgðSvÞ.

The decomposition in Equations 8–10 expresses the American derivative price as the sum ofthe European price and a premium for exercise prior to the maturity date. The EEP is the presentvalue of the instantaneous gains realized by exercising early. These gains are collected in the event{t(v) ¼ v}, where immediate exercise of a newly issued American-style derivative is optimal.Instantaneous gains consist of interest collected on the realized payoff (rvg(Sv)) net of the expectedchange in the payoff per unit time

�E�v

dgðSvÞ

�dv�. If the payoff function is differentiable in the

immediate exercise region, the instantaneous gains can be written as

rvgðSvÞ � g0ðSvÞ Sv�rðYvÞ � dðSv,YvÞ

,

where g0(×) is the derivative of the payoff function. The expected appreciation of the payoffincludes a loss (resp. gain), due to the payment of dividends on the underlying asset, if g0(Sv) > 0(resp. g0(Sv) < 0). By exercising the contract, the holder recaptures the loss (resp. forgoes thegain) if g0(Sv) > 0 (resp. g0(Sv) < 0).

The second decomposition of the American derivative price emphasizes the premium relativeto the exercise payoff.

Theorem 4: The price of the American-style derivative with payoff g(S) has thedelayed exercise premium (DEP) decomposition

VðSt,Yt, tÞ ¼ gðStÞ þPdðSt,Yt, tÞ, ð11Þ

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where

PdðSt,Yt, tÞ[E�t

"Z tðtÞ

tbt,v

�dgðSvÞ � rvgðSvÞdv

�#ð12Þ

is the DEP.

The DEP representation expresses the American derivative price as the sum of the exercisepayoff and a premium for delaying exercise until optimal. The premium is the present value ofinstantaneous gains for delaying exercise. The gains are collected over the event {t(t) > t} whereimmediate exercise is suboptimal. They consist of the expected change in the payoff per unit time�E�v

dgðSvÞ

�dv�net of the interest forgone by failing to exercise (rvg(Sv)). The instantaneous

gains achieved by waiting to exercise are the negative of the instantaneous gains achieved byexercising early.

The exercise premiumdecompositions in Theorems 3 and 4were derived byKim (1990), Jacka(1991), and Carr, Jarrow & Myneni (1992) in the context of the Black-Scholes market modelwith constant coefficients. The review article by Myneni (1992) provides perspective on thedecomposition. The extension to Ito price processes is in Rutkowski (1994). Diffusion modelsare considered in Kim & Yu (1996) and Detemple & Tian (2002).

4.2. Integral Equations for Optimal Exercise Boundaries

The EEP representation provides a constructive way to identify the immediate exercise region ofan American claim. Let

Eg ¼ fðS,Y, tÞ 2D� ½0,T� :VðS,Y, tÞ ¼ gðSÞgbe the immediate exercise region of the claim with payoff g. The complement Cg is the continua-tion region. The (Y, t)-section, denoted by Eg(Y, t) [ {S 2 Rþ : (S, Y, t) 2 Eg}, is the set of pointsS at which immediate exercise is optimal at time t when the state variable level is Y.

Continuity of the price function can be invoked to argue that the continuation region is an openset. The immediate exercise region is closed. Depending on the contract payoff, each of its sectionscan consist of a single connected set or multiple disconnected sets. It can have one or multipleboundaries. Suppose there are J boundaries, Bg

j , j ¼ 1, . . . , J. Let Bgj ¼ fBg

jt : t 2 ½0,T�g be the jth

boundary of Eg. Bgjt ¼ Bg

j ðYt, tÞ is parametrized by Yt. Each boundary Bgj is a stochastic process.

The value Bgj ðY, tÞ is a boundary point of the section Eg(Y, t).

Theorem 5: The immediate exercise boundaries fBgj : j ¼ 1, . . . , Jg of an American

derivative with payoff g(S) solve the system of recursive integral equations

g�Bgj ðY, tÞ

¼Ve

�Bgj ðY, tÞ,Y, t

þP

�Bgj ðY, tÞ,Y, t

, ð13Þ

where

P�Bgj ðY, tÞ,Y, t

[E�

t

"Z T

t1fSv2EgðYv,vÞgbt,v

�rvgðSvÞdv� dgðSvÞ

#ð14Þ

and the initial value of Sv is Bgj ðY, tÞ, for j¼ 1, . . . , J. This system of backward equations

is subject to the boundary conditions limt→TBgj ðY, tÞ ¼ Bg

j ðYÞ for j ¼ 1, . . . , J.

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The system of integral equations (Equations 13 and 14) for the exercise boundaries is animmediate consequence of the EEP decomposition (Equations 8–10). At any boundary point ofthe exercise region, say S ¼ Bg

j ðY, tÞ, the contract value is the immediate exercise payoff at the

point in question, VðBgj ðY, tÞ,Y, tÞ ¼ gðBg

j ðY, tÞÞ. This explains the left-hand side of Equation

13. The components of the EEP decomposition are evaluated at the same point, giving the right-hand side of Equation 13. Finally, the premium (Equation 14) consists of gains collected in theevent of exercise, on the set {Sv2 Eg(Yv, v)}. Values of the underlying asset at time v are conditionalon the starting value S ¼ Bg

j ðY, tÞ at time t.The boundary conditions that apply, limt→T Bg

j ðY, tÞ ¼ Bgj ðYÞ, depend on the particular

contractual form examined. Bgj ðYÞ is often the combination of an exercise condition at terminal

date T ðgðBgj ðY,TÞÞ ¼ 0Þ and a condition pertaining to the limiting behavior of the integrand in

Equation 14. Section 5 provides examples for specific contractual forms.For some contracts, the system of integral equations (Equations 13 and 14) is coupled. This is

the case if the exercise section {Sv 2 Eg(Yv, v)} depends on more than one boundary. Section 5provides examples illustrating this point.

The recursive integral equation for the optimal exercise boundary was originally derived in thecontext of the Black-Scholes framework. The equation appears in Kim (1990); Jacka (1991); andCarr, Jarrow&Myneni (1992). Peskir (2005) showsuniqueness of the solution. Integral equationsfor general diffusions are derived in Detemple&Tian (2002). A large body of literature is devotedto the development of numerical schemes based on the integral equation. Contributions on thatfront include Kim (1990); Huang, Subrahmanyam & Yu (1996); Sullivan (2000); and Kallast &Kivinukk (2003).

4.3. Duality

The next proposition provides an alternative perspective on the valuation of an American de-rivative. It expresses the American derivative price in terms of a related path-dependent Europeancontract.

Theorem 6: The price of the American-style derivative with payoff g(S) is

VðS0,Y0, 0Þ ¼ infM2H1

0ðQÞE�

"sup

v2½0,T�ðbvgðSvÞ�MvÞ

#, ð15Þ

where H10ðQÞ is the space of Q-martingales that are null at t ¼ 0 and satisfy

supt2[0,T]jMtj 2 L1(Q).

Equation 15 is a duality formula. Instead ofmaximizing the discounted payoffwith respect toa set of feasible stopping times, the price is obtained by minimizing over a class of martingales.The expectation on the right-hand side is the initial value of a path-dependent contract payinggðSyÞ � b�1

y My at the random time s¼ arg supv2[0,T](bvg(Sv)�Mv). The random time s dependson the whole trajectory of the adjusted payoff bvg(Sv)�Mv over the life of the contract. Becauseit depends on information up to the maturity date, time s is not a stopping time of the filtration.The price of the American derivative minimizes the initial value of the path-dependent contractwith respect to the class of martingale adjustment processes described [i.e., the class H1

0ðQÞ].The duality (Equation 15) has important practical ramifications. It immediately implies the

upper bound

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VðS0,Y0, 0Þ�E��

supv2½0,T�

�bvgðSvÞ �Mv

��[UðMÞ ð16Þ

for anymartingaleM2H10ðQÞ. Computation of the upper boundU(M) can be performed by using

Monte Carlo simulation. Minimization over a class of admissible martingales M2A⊆H10ðPÞ

can then be carried out to obtain tighter upper bounds.The duality formula follows from results in Davis & Karatzas (1994) and Rogers (2002).

Rogers (2002) establishes the formula in a general setting where payoff and interest rate areadapted processes defined on a filtered probability space. The literature that follows focuses onBermudanderivatives.Haugh&Kogan (2004) derive a versionof theduality formula for diffusionmodels. They provide a three-step procedure to calculate upper and lower bounds for theBermudan’s price. The first step of this procedure computes an approximation of the price,which is then used to construct the two bounds. Andersen& Broadie (2004) develop a variationof this method, where the price bounds are derived from an approximation of the optimalexercise policy. The procedure involves nested simulations to compute lower price bounds andassociated martingales along simulated trajectories of the underlying variables (simulationwithin simulation). Their model assumes aMarkovian structure for the underlying variable, butallows for a path-dependent payoff and interest rate. Belomestny, Bender & Schoenmakers(2009) devise an approach to calculate upper bounds without the need for nested Monte Carlosimulations. The method is cast in a diffusion setting with Brownian information filtration. Ituses the martingale representation theorem and a simulation-based least squares regression toconstruct amartingale for Equation 16. Further perspective on duality is found inRogers (2010).He provides a construction of the Snell envelope for a Bermudan option that is entirely based onthe dual representation. The procedure uses a backward recursion to construct an increasingsequence that converges to the value of the dual on the left-hand side of Equation 15.

Monte Carlo methods that are not based on the duality formula have also been proposed.Contributions in that area include Bossaerts (1989); Tilley (1993); Barraquand & Martineau(1995); Broadie & Glasserman (1997, 2004); Broadie, Glasserman & Jain (1997); Raymar &Zwecher (1997); Garcia (2003); and Ibanez & Zapatero (2004). Related contributions mixingMonte Carlo simulation and dynamic programming are proposed byCarriere (1996), Tsitsiklis&Van Roy (1999, 2001), and Longstaff & Schwartz (2001).

5. PARAMETRIC OPTION PRICING MODELS

This section examines the valuation of American-style options in specific market models. Itshows the complexity of optimal exercise decisions. The presentation focuses on the Black-Scholes framework (Section 5.1) and the Vasicek stochastic interest rate model (Section 5.2). Itconcludes with comments on the literature dealing with multiasset options (Section 5.3).

5.1. Constant Coefficients

Assume the Black-Scholes settingwith constant coefficients (r, d, s). Two contracts are examined.Pricing is carried out using the EEP decomposition and the associated integral equation.

5.1.1. Derivative with a single exercise boundary: call option. The first example involves anAmerican call option with strike K and expiration date T. In this case, the immediate exerciseregion is the set

Ec ¼ fðS, tÞ 2Rþ � ½0,T� : S�Bc or ST �Kg,

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where Bc is a function of time. Specializing Theorems 3 and 5 leads to the followingcharacterizations:

Corollary 1: The price of the American-style call with payoff g(S) ¼ (S�K)þ has theEEP decomposition C(St, t; B

c) ¼ Ce(St, t) þ P(St, t; Bc), where

CeðSt, tÞ ¼ Ste�dðT�tÞN�dðSt;K,T � tÞ�,

�Ke�rðT�tÞN�dðSt;K,T � tÞ � s

ffiffiffiffiffiffiffiffiffiffiffiT � t

p ,

ð17Þ

PðSt, t;BcÞ ¼ E�t

"Z T

te�rðs�tÞ1fSs�Bc

sgðdSs � rKÞds#¼

Z T

tf�St;B

cv, v� t

�dv, ð18Þ

with

f�St;B

cv, v

�¼ dSte�dvN�d�St;B

cv, v

� � rKe�rvN�d�St;B

cv, v

�� sffiffiffiv

p , ð19Þ

dðS;K, vÞ ¼logðS=KÞ þ

�r� dþ 1

2s2

�v

sffiffiffiv

p , ð20Þ

and where N(×) is the cumulative standard normal distribution. The immediateexercise boundary satisfies the recursive integral equation

Bct � K ¼ Ce�Bc

t , t�þP

�Bct , t;B

c� ð21Þ

for t 2 [0, T), subject to the boundary condition limt↑T1Bct ¼ maxfK, ðr=dÞKg.

The EEP decomposition provides additional insights about the value created by opti-mally exercising an American call early. From Theorem 3, it follows that PðSt, t;BcÞ ¼

E�t

"Z T

te�rðs�tÞ 1fSs�Bc

sgðdSs � rKÞds#. This expression shows that the instantaneous gain consists

of the dividend collected by acquiring the underlying asset net of the interest forgone by paying thestrike.

The formulas in Corollary 1 suggest a valuation approach in two steps. The first step calculatesthe optimal exercise boundary by solving the integral equation (Equation 21). Substituting thesolution in Equations 17–20 and using the EEP formula produce the call price. Similar formulascan be derived for American put options.5

Figure 1 shows the structure of the immediate exercise region for anAmerican call. The exerciseregion expands as the maturity date approaches. It is also connected in the upward direction. Forany fixed date t, if immediate exercise is optimal at some underlying asset price, it remains optimalat any higher asset price.

5An American put is the same as an American call under a new numéraire. This relation is known as American put-callsymmetry. The early literature on this topic includes Grabbe (1983), Bjerksund & Stensland (1993a), Chesney &Gibson (1995), andMcDonald& Schroder (1998). Generalizations are in Schroder (1999) and Detemple (2001). Theseextensions rely on a change of numéraire method introduced in Jamshidian (1992) and Geman, El Karoui & Rouchet(1995).

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Kim (1990), Jacka (1991), and Carr, Jarrow & Myneni (1992) discuss the characterizationsof the American call price and its (unique) exercise boundary in Corollary 1. An alternativeintegral equation for the boundary is derived in Little, Pant & Hou (2000).

5.1.2. Derivative with multiple exercise boundaries: chooser option. The second example dealswith an American chooser option with exercise payoff given by max{C(S, t), P(S, t)} and maturitydate T1. This contract gives the right to choose the best of an American put and an American callupon exercise.6 C(S, t) (resp. P(S, t)) is the price of the underlying American call (resp. put). Theunderlying options have common strike priceK andmaturity dateT2>T1. They have equal pricesif S ¼ S�t , where S�t is the unique solution of the equation CðS�t , tÞ ¼ PðS�t , tÞ for t � T2. At T2,S�T2

¼ K. The function S�t is called the diagonal curve.The immediate exercise region of the American chooser consists of two sets that are discon-

nected for t < T1 and have S�T1as a common point at T1. Specifically,

Ech ¼�ðS, tÞ 2Rþ � ½0,T1� : S�B1 or S�B2 or t ¼ T1

�for time-dependent boundaries B1 and B2, such that B1

t > B2t for t < T1. At time T1, exercise

is always optimal. If ST1 > S�T1(resp. ST1 < S�T1

), it is optimal to choose the American call(resp. put). If ST1 ¼ S�T1

, there is indifference.

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5100

105

110

115

120

125

130

135American call boundary

Time to maturity

Boun

dary

val

ues

Call boundary

Exercise region

Figure 1

Immediate exercise region of an American call option with strike price K ¼ 100 and maturity date T ¼ 0.5.Parameter values are r ¼ d ¼ 0.06 and s ¼ 0.2. The exercise boundary is calculated by discretizing theintegral equation. The number of discretization points is n ¼ 200.

6There are four varieties of chooser options, American on American, American on European, European on American, andEuropean on European.

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The price of the derivative is described next.

Corollary 2: Consider an American chooser with payoff gðSÞ ¼ maxfCðS, tÞ,PðS, tÞg. The price has the EEP decomposition CH

�St, t;B1,B2

� ¼ CHeðSt, tÞ þPch

1

�St, t;B1

�þ Pch2

�St, t;B2

�, where7

CHeðSt, tÞ ¼ E�t

he�rðT1�tÞmax

�CT1

,PT1

�i, ð22Þ

Pch1

�St, t;B

1�¼ E�

t

"Z T1

te�rðs�tÞ1fSs�B1

sgðdSs � rKÞds#

¼Z T1

tf1

�St;B

1s , s� t

�ds, ð23Þ

and

Pch2

�St, t;B

2�¼ E�

t

"Z T1

te�rðs�tÞ1fSs�B2

sgðrK� dSsÞds#

¼Z T1

tf2

�St;B

2s , s� t

�ds, ð24Þ

with

f1

�St;B

1s , v

�[ dSte

�dvN�dU

�St,B

1s , v

� � rKe�rvN

�dU1

�St,B

1s , v

� , ð25Þ

f2

�St;B

2s , v

�[ rKe�rvN

�dL1�St,B

2s , v

� �dSte

�dvN�dL�St,B

2s , v

� , ð26Þ

and

dU�St,B

1s , v

�¼

log�St�B1s

�þ�r� dþ 1

2s2

�v

sffiffiffiv

p , ð27Þ

dU1�St,B

1s , v

�¼ dU

�St,B

1s , v

�� s

ffiffiffiv

p, ð28Þ

dL�St,B

2s , v

�¼ �

log�St�B2s

�þ�r� dþ 1

2s2

�v

sffiffiffiv

p , ð29Þ

dL1�St,B

2s , v

�¼ dL

�St,B

2s , v

�þ s

ffiffiffiv

p. ð30Þ

The immediate exercise boundaries (B1, B2) solve the system of coupled recursiveintegral equations:

7A valuation formula for CHe(St, t) is provided in Detemple & Emmerling (2009).

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B1t � K ¼ CH

�B1t , t;B

1,B2�

K� B2t ¼ CH

�B2t , t;B

1,B2�

8<: ð31Þ

for t 2 [0, T1), subject to the boundary conditions limt↑T1B1t ¼ maxfBc

T1, S�T1

g,limt↑T1B

2t ¼ minfBp

T1, S�T1

g. Here BcT1

(resp. BpT1) is the immediate exercise boundary

of the underlying call (resp. put) at T1.

In the case of an American chooser, the EEP has two components, Pch1 ðSt, t;B1Þ and

Pch2 ðSt, t;B2Þ. The first (resp. second) component is the present value of the gains realized, in

the event of optimal early exercise, if the American call (resp. put) is chosen. Each com-ponent depends on the relevant exercise boundary.

The valuation formula in Corollary 2 shows that the price CH(St, t; B1, B2) depends on the

two exercise boundaries. This follows because the price reflects the two early exercise premia.The dependence on the pair of boundaries implies that the resulting integral equations inEquation 31 are coupled. At any given time t, each exercise boundary component depends onthe future values of both boundary components.

Figure 2 illustrates the structure of the immediate exercise region and the associated boundariesfor the American chooser. Prior to the maturity date, the exercise region consists of two dis-connected sets. The regionwhere it is optimal to exercise the chooser as a call (resp. put) is includedin the exercise region of the underlying call (resp. put). At the maturity date, the exercise regionconsists of the whole vertical axis.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.060

70

80

90

100

110

120

130

140

150American call, put, and chooser boundaries

Time to maturity

Boun

dary

val

ues

Call boundary

Put boundary

Chooser exercise region

Chooser exercise region

Chooser upperboundary

Chooser lowerboundary

Diagonal curve

Figure 2

Immediate exercise region of a chooser option with maturity date T1 ¼ 1 when the underlying Americanoptions have strike price K ¼ 100 and maturity date T2 ¼ 1.2. Time to maturity is measured relative to T1.Parameter values are r ¼ d ¼ 0.06 and s ¼ 0.2. Exercise boundaries are calculated by discretizing theintegral equations. The number of discretization points is n ¼ 100.

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American choosers are studied in Detemple & Emmerling (2009). Exotic choosers, wherethe underlying options have different strikes and maturity dates, are also examined therein.American choosers are options on options. When the underlying options are written on thesame asset, the chooser ultimately depends on a single underlying price. Other examples ofAmerican contracts that are written on a single underlying asset and have multiple exerciseregions and boundaries are strangles and straddles. Chiarella & Ziogas (2005) analyzeAmerican strangles using PDE and Fourier transform methods. Battauz, De Donno & Sbuelz(2014) provide an example where the exercise region is surrounded by two (upper and lower)continuation regions.

5.2. Stochastic Interest Rates

Exercise regions have more complex structures when the model coefficients depend on statevariables. As an illustration, suppose that the interest rate is stochastic and behaves according tothe Vasicek (1977) model. The pair (S, r) evolves according to�

dSt=St ¼ ðrt � dÞdt þ sdWst

drt ¼ kðr� rtÞdt þ srdWrt

ð32Þ

under the risk-neutral measure. The coefficients ðd,s, k, r,srÞ are constants. The Q-Brownianmotions (Ws, Wr) have constant correlation r.

In this setting, the price of a long-term bond with maturity date T is B(t, T) ¼ exp(H(t, T) �rtG(t, T)) with

Hðt,TÞ ¼ �krZ T

t

Z s

te�kðs�vÞdvdsþ 1

2s2r

Z T

t

�1� e�kðT�sÞ

2ds,

Gðt,TÞ ¼Z T

te�kðT�sÞds.

The bond volatility coefficient is sB(t, T) ¼ �srG(t, T). Under the forward measure, the jointdistribution of (sT [ log(ST/St), rT) is bivariate normal with parameters

mrðrt, t,TÞ ¼ rte�kðT�tÞ þ krZ T

te�kðT�vÞdvþ sr

Z T

te�kðT�vÞsBðv,TÞdv,

msðrt, t,TÞ ¼Z T

tmrðrt, t, vÞdv� 1

2s2ðT � tÞ þ

Z T

trssBðs,TÞds,

Ssðt,TÞ2 ¼Z T

t

�s2r Gðv,TÞ2 þ s2 þ 2rssrGðv,TÞ

dv,

Srðt,TÞ2 ¼ s2r

Z T

te�2kðT�vÞdv,

Sr,sðt,TÞ ¼ sr

Z T

te�kðT�vÞ�srGðv,TÞ þ rs

�dv

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and correlation r(t, T) ¼ Sr,s(t, T)/(Sr(t, T)Ss(t, T)). The conditional distribution of sT givenrT is normal with conditional mean and variance

mðr; rt, t,TÞ ¼ msðrt, t,TÞ þ rðt,TÞSsðt,TÞ�r� mrðrt, t,TÞ

Srðt,TÞ�,

sðrt, t,TÞ2 ¼ Ssðt,TÞ2�1� rðt,TÞ2

.

For a call option with strike K and maturity date T, the immediate exercise region is

Ec ¼ fðS, r, tÞ 2Rþ � R� ½0,T� : S�Bcðr, tÞ for t < T or S�K for t ¼ Tg,whereBc(r, t) is a function of the interest rate and time. The exercise boundary is a two-dimensionalsurface. The characterizations in Theorems 3 and 5 result in the following:

Corollary 3: The price of the American-style call with payoff g(S) ¼ (S�K)þ hasthe EEP decomposition C(St, rt, t; B

c) ¼ Ce(St, rt, t) þ P(St, rt, t; Bc), where

CeðSt, rt, tÞ ¼ Ste�dðT�tÞN

�h�

StBðt,TÞ,K, t,T

��

�KBðt,TÞN�h�

StBðt,TÞ,K, t,T

�� Ssðt,TÞ

�, ð33Þ

PðSt, rt, t;BcÞ ¼ dSt

Z T

tBðt, sÞe�dðs�tÞf1

�St, rt, t;Bcð×, sÞ, s

�ds

�KZ T

tBðt, sÞf2

�St, rt, t;Bcð×, sÞ, s

�ds, ð34Þ

with

f1�x, rt, t;Bcð×, sÞ, s�

¼ e12sðrt ,t,sÞ2

Srðt, sÞZ 1

�1emðr; rt ,t,sÞN

�gðr; x,Bcðr, sÞ, rt, t, sÞ

n�r� mrðrt, t, sÞ

Srðt, sÞ�dr, ð35Þ

f2�x, rt, t;Bcð×, sÞ, s�

¼ 1Srðt, sÞ

Z 1

�1rN

�g�r; x,Bcðr, sÞ, rt, t, s

�� sðrt, t, sÞ

n�r� mrðrt, t, sÞ

Srðt, sÞ�dr, ð36Þ

and

hðx, y, t, sÞ ¼logðx=yÞ� dðs� tÞþ 1

2Ssðt, sÞ2

Ssðt, sÞ , ð37Þ

gðr; x, y, rt, t, sÞ ¼ logðx=yÞ� dðs� tÞþmðr; rt, t, sÞsðrt, t, sÞ þ sðrt, t, sÞ. ð38Þ

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The immediate exercise boundary satisfies the recursive integral equation

Bcðrt, tÞ � K ¼ Ce�Bcðrt, tÞ, rt, t�þP

�Bcðrt, tÞ, rt, t;Bc� ð39Þ

for t 2 [0, T), subject to the boundary condition limt↑T Bc(rt, t) ¼ max{K, (rT/d) K}.

The EEP has the standard form with instantaneous gains dSv � rvK in the event of exercise attime v. The stochastic nature of the interest rate is the source of the increased complexity of theintegrands in Equation 34. These integrands, as well as the two components of the option price,depend on the interest rate and the asset price.

The integral equation (Equation 39) for the exercise boundary reflects this complexity. At anygiven time t, the exercise boundary is a curve parametrized by the level rtof the interest rate. Solvingthe integral equation requires solving it for any possible date and any possible value of the interestrate. The computational burden increases for numerical procedures based on Equation 39.

Figure 3 illustrates the shapeof the immediate exercise region in thismodel. The exercise surfacehas a nonlinear dependence on the interest rate. As the maturity date approaches, it convergesto max{K, (rT/d) K}.

The early literature on American options with stochastic interest rate includes Amin &Bodurtha (1995); Ho, Stapleton & Subrahmanyam (1997); Chung (1999); and Menkveld &Vorst (2000). The American option value is approximated by restricting the number of exercisedates (Bermudan option approximation) and using a two-points Richardson extrapolationscheme. Detemple & Tian (2002) derive the integral equation in Equation 39 and developa numerical algorithm based on it. Medvedev & Scaillet (2010) deal with stochastic interest rate

American call boundary

00.02

0.040.06

0.080.1

0

0.1

0.2

0.3

0.450

60

70

80

90

100

110

Interest rateTime to maturity

Boun

dary

val

ues

Figure 3

Immediate exercise boundary of an American call with strike price K ¼ 50 and maturity date T ¼ 0.4 in theVasicek interest rate model. Parameters of the interest rate process are ðk, r,sr, rÞ ¼ ð0:005, 0:06, 0:01, 0:2Þ.Parameters of the asset return process are (d, s) ¼ (0.05, 0.2). The exercise boundary is calculated bydiscretizing the integral equation in time and space. The number of discretization points for (T, r) is 503 50.

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and volatility. They provide an approach based on an asymptotic (near maturity) expansion of thevalue obtained by pursuing a suboptimal exercise policy. The expansion is optimized over aparameter of the suboptimal policy to approximate the American option value. The optimizationstep is similar to the procedure for the LBA introduced in Broadie&Detemple (1996) for a modelwith constant coefficients.

5.3. Multiasset Derivatives

Contracts written on multiple underlying assets give rise to particularly complex exercise deci-sions. Exercise regions often havemultiple exercise boundaries. In some cases, exercise set sectionsat a given date consist of disconnected sets.

Tan&Vetzal (1995) use a numerical approach to derive insights about immediate exerciseregions of various contracts. Gerber& Shiu (1996) examine the valuation of various perpetualoptions on two assets. Geltner, Riddiough & Stojanovic (1996) use perpetual options on themaximum of two assets to price real estate projects. Broadie & Detemple (1997) provide ananalysis of American claims with general payoffs written on multiple assets. Properties ofexercise regions are identified. Special contracts such as calls on the maximum of two assets,puts on the minimum of two assets, and spread options are examined in detail. Integralequations for the exercise boundaries are derived. Villeneuve (1999) proves additionalproperties and provides applications to various contractual forms. A focus of his analysis is thenonemptiness of the exercise region. Detemple, Feng & Tian (2003) deal more specificallywith contracts such as calls on the minimum of two dividend-paying assets. They show thepresence of a local time component in the EEP formula and characterize the immediate exerciseboundary.

A substantial literature dealswith numericalmethods formultiasset American options. Selectedreferences include the tree methods in Boyle, Evnine & Gibbs (1989) and Kamrad & Ritchken(1991); the variational inequality approach in Jaillet, Lamberton&Lapeyre (1990); the quadraturemethod in Andricopoulos et al. (2007); the method of lines approach in Kovalov, Linetsky &Marcozzi (2007); and the Fourier transform method in Chiarella & Ziveyi (2014). Monte Carlosimulation is the method of choice for high-dimensional problems. References in that realm arediscussed in Section 4.3.

6. NONSTANDARD DERIVATIVES

Various contractual features can be the source of unusual properties of optimal exercise decisionsand valuation formulas. This section illustrates some of the complications that can arise. It focuseson the simple and transparent example of an American capped call option. This derivative pro-vides a good example of a contract that does not satisfy the standard version of the EEPdecomposition.

AnAmerican capped call option has exercise payoff (min {S, L}� K)þ [ (S ⋀ L� K)þ for somecap L > K. It can be viewed as a call option on the minimum of two assets, one of which hasa constant priceL. Consider the Black-Scholes financial market framework with a strictly positiveinterest rate. A standard (i.e., uncapped)American call with contractual provisions (K,T) has priceC(S, t) and exercise boundary Bc, as described in Corollary 1. Let CL(S, t) (resp. BL) be the price(resp. exercise boundary) of the capped call with characteristics (L, K, T).

Simple arguments can be invoked to identify the immediate exercise region of the capped call.Consider a point (S, t) 2 Rþ 3 [0, T]. First, note that immediate exercise of the capped call isoptimal if S� L. In this case, the payoff is maximal and cannot be improved upon by waiting (thepresent value of any waiting strategy is less than the immediate exercise payoff).

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Second, suppose that L� S�Bct . The limited upside of the capped call implies that its price is

bounded above by the uncapped call price, CL(S, t) � C(S, t). It follows that immediate exercisemust be optimal for the capped call if it is optimal for the call. If this is not true, there exists awaitingstrategy that improves the value of the call, a contradiction.

Lastly, consider the case minfL,Bctg > S. Assume first that L � (r/d)K. Exercising at S < L is

then suboptimal because the local gains dS� rK are negative. Next, suppose thatL> (r/d)K. Giventhat the exercise boundary of a standard call option is continuous and decreases as the maturitydate approaches, there exists a shorter maturity call, with exercise boundary Bc(T0), such that theinequalities minfL,Bc

tg > Bct ðT0Þ>S hold. As Bc

t ðT0Þ > S, immediate exercise of this call issuboptimal [i.e., C(S, t; T0)> (S�K)þ]. Exercising at the first hitting time of the boundaryfBc

sðT0Þ : s2 ½t,T0�g, or atT0 if no such time exists in [t,T0], is a feasible policy for the capped call.Moreover, as L > Bc

t ðT0Þ, it follows that L>BcsðT0Þ for all s 2 [t, T]. Implementing this exercise

policy produces the same payoff as the uncapped option with maturity date T0. So the cappedoption must be worth at least as much, CL (S, t) � C(S, t; T0). As C(S, t; T0) > (S � K)þ; it nowfollows that CL (S, t) > (S � K)þ. Immediate exercise is suboptimal for the capped option.

To summarize,

Theorem 7: The immediate exercise boundary of the American capped call withpayoff g(S) ¼ (S ⋀L�K)þ is Bc¼min{L, B}.

Figure 4 illustrates the structure of the exercise region of the capped call option.

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Time to maturity

American capped call boundary

Call boundary

Capped call exercise regionCapped call exercise regionCapped call exercise region

Cap

100

105

110

115

120

125

130

135

Boun

dary

val

ues

Figure 4

Immediate exercise region of an American capped call option with strike price K ¼ 100, cap L ¼ 120,and maturity date T ¼ 0.5. Parameter values are r ¼ d ¼ 0.06 and s ¼ 0.2. The exercise boundary iscalculated by discretizing the integral equation. The number of discretization points is n ¼ 200.

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Theorem 7 leads to the following modification of the EEP decomposition:

Theorem 8: The price of the American-style capped call with payoff g(S)¼ (S ⋀ L�K)þ

has the EEP decomposition CL(St, t; BL) ¼ Cae(St, t) þ PL(St, t; B

L), where

CaeðSt, tÞ ¼ E�t

�e�rðtL⋀T�tÞ

�StL⋀T � K

þds�, ð40Þ

PL�St, t;BL� ¼ E�t

"Z tL⋀T

te�rðs�tÞ1fSs�BL

s gðdSs � rKÞds#

ð41Þ

for t � tL ⋀ T, where tL ¼ inf{v 2 [0, T] : Sv ¼ L} or tL ¼1 if no such time exists in[0,T].The component Cae(St, t) is the value of a call optionwith automatic exercise atthe cap;PL(St, t; B

L) is the exercise premium above that benchmark. The immediateexercise boundary BL satisfies the recursive integral equation

BLt � K ¼ Cae�BL

t , t�þPL�BL

t , t;BL� ð42Þ

for t2 [0,T), subject to the boundary condition limt↑T BLt ¼ minfmaxfK, ðr=dÞKg,Lg.

The formula in Theorem 8 expresses the American capped call price as the price of a cappedcall with automatic exercise at the cap, augmented by an EEP. The latter consists of the in-stantaneous gains in the event of exercise prior to theminimumof the hitting time of the cap and ofthe maturity date. The EEP decomposition in the proposition is unusual in that it takes a barrieroption (the option with automatic exercise at the cap) as a benchmark.

The immediate exercise boundary of the capped call option is completely identified in Theorem7. It can also be deduced from the integral Equation 42 associated with Equations 40 and 41. AtT�, the boundary is theminimum ofBc

T� andL. IfBcT�<L, the solution of the integral equation is

BLt ¼ Bc

t as long as Bct < L. At the point t� where Bc

t� ¼ L, the exercise premium evaluated atBLt� ¼ L is null and the capped call with automatic exercise equals L � K. The integral equation

solution is BLt� ¼ L. The same applies at times t prior to t�.

It is also possible to provide an alternative decomposition that measures the exercise premiumrelative to a European capped call option. When that benchmark is selected, the gains from earlyexercise must account for the nondifferentiability of the payoff at the cap. The resulting EEPinherits a (nonstandard) local time component.

The valuation of American capped call options is studied in Broadie & Detemple (1995). Anexplicit expression for Equation 40 is provided therein (see also Rubinstein & Reiner 1991). Inaddition to the case of constant caps, they also examine contracts with growing caps that can haveintricate exercise regions. The integral equation (Equation 42) appears in Broadie & Detemple(1999). Detemple & Tian (2002) provide an extension to stochastic interest rate, dividend yield,and volatility. An EEP representation with a local time component can be derived as in Detemple,Feng & Tian (2003).

Gao, Huang & Subrahmanyam (2000) derive similar decomposition formulas for Americanbarrier options. They present numerical results based on the associated integral equation. Othernumerical methods that can be used to calculate the prices of American barrier options include thetree methods of Boyle & Lau (1994), Reimer & Sandmann (1995), Ritchken (1995), Cheuk &Vorst (1996), and Figlewski & Gao (1999) and the finite difference approaches of Boyle & Tian(1998), Dempster & Hutton (1997, 1999), and Zvan, Vetzal & Forsyth (2000).

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7. JUMP DIFFUSIONS

The study of American derivatives and optimal exercise decisions has been extended beyond thediffusion model described in Equation 1. An important extension considers a setting where theunderlying price follows a jump-diffusion process. This model is of particular relevance in light ofthe empirical evidence regarding the importance of jumps for asset returns.

Consider the evolution, under the risk-neutral measure, of the underlying price

dSt=St� ¼ ðr� dÞdt þ sdWt þZ

gðzÞvðdz, dtÞ, ð43Þ

where v(dz, dt) is a compensated jump measure, g(×) is a function on R, and the coefficients(r, d, s) are constants. The jump measure is driven by a Poisson process N with intensity l. Thejump size z is random and independent of (W, N). Its probability measure is k(dz). The functiong(z)measures the relative size of a jump in the asset price [i.e.,DSt/St�¼ g(z)] in the event of a jump.The compensator of the jump measure is m(dz, dt) [ lk(dz)dt. All quantities and measures in-troduced are under the risk-neutral measure and therefore include an appropriate risk adjustment.

Themodel inEquation 43 is a simple andnatural extension of the standard geometric Brownianmotion specification. Merton (1976) derives a valuation formula for European options in thissetting. A special case is g(z)¼ exp (z)� 1with z⇝Nðmz,s

2z Þ. In this instance, the expected relative

jump in the price, conditional on a jump occurring, isZR

gðzÞkðdzÞ ¼ expðmz � s2z=2Þ � 1.

Theorem 9:The price of an American-style call with payoff g¼ (S�K)þ has the EEPdecomposition C(St, t; B

c) ¼ Ce(St, t) þ Pc(St, t; Bc) þ Pd(St, t; B

c), where

CeðSt, tÞ ¼ E�t ½bt,TðST � KÞþ�, ð44Þ

PcðSt, t;BcÞ ¼ E� Z T

tbt,vðdSv� � rKÞ1fSv��Bc

vgdv�, ð45Þ

PdðSt, t;BcÞ ¼ �E

"Z T

t

ZRbt,vfðSv�, v; zÞ1fSv�ð1þgðzÞÞ<Bc

vg1fSv��Bcvgmðdz, dvÞ

#, ð46Þ

with m(dz, dv) ¼ lk(dz)dv, and

fðSv�, v; zÞ ¼ CðSv��1þ gðzÞ�, vÞ � �Sv�ð1þ gðzÞÞ � K

�. ð47Þ

The integral equation for the exercise boundary is Bct � K ¼ CeðBc

t , tÞ þPcðBct , t;B

cÞ þPdðBc

t , t;BcÞ for t 2 [0, T), subject to the boundary condition limt↑T Bc

t ¼ maxfK,B�g,where B� solves

B� ¼ Krþ l

ZR1fB�ð1þgðzÞÞ<KgkðdzÞ

dþ l

ZRð1þ gðzÞÞ1fB�ð1þgðzÞÞ<KgkðdzÞ

. ð48Þ

In the presence of jump risk, the EEP has two components. The first, in Equation 45, has thestandard form. It counts the gains from collecting the dividend on the underlying asset, net of theinterest expense associatedwith the payment of the strike. The second, in Equation 46, is a negative

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jump premium (i.e., a discount). The jump discount tallies the losses associated with sudden exitsfrom the immediate exercise regiondue todownward jumps. It vanishes if the probability of a jumpldt goes to zero.

The jump premium has an unusual structure in that it depends on the American option price inthe continuation region. This is a significant departure from the EEP decompositions in the pre-vious sections, where the exercise premium is free of the (unknown) option price. The consequenceis that the decomposition is more difficult to implement numerically. The immediate exerciseboundary still satisfies an integral equation, but this equation now depends on the option price atfuture dates. Solving for the boundary requires the resolution of the option price. In effect, the pair(C, Bc) is the solution to a system of coupled integral equations.

The decomposition formula in Theorem 9 is proved in Pham (1997), using a free boundaryapproach. Gukhal (2001) justifies the formula based on limiting arguments and discrete timeapproximations. Chiarella & Ziogas (2009) provide further perspective on the formula. Theyidentify the terminal value of the exercise boundary. They also develop a quadrature scheme forsolving the system of integral equations.

Various other numerical approaches have been proposed for pricing American options onjump-diffusion processes. Amin (1993), Broadie & Yamamoto (2003), and Wu & Dai (2009)implement tree methods. Zhang (1994, 1997) extends the variational inequality approach ofJaillet, Lamberton & Lapeyre (1990) to jump diffusions and studies finite difference schemesfor implementation.Mullinaci (1996) develops an algorithm based on the Snell envelope of thediscounted payoff. Themethod of lines is used inMeyer&van derHoek (1997),Meyer (1998),and Chiarella et al. (2009). Finite difference schemes are also applied by Andersen&Andreasen(2000), Carr &Hirsa (2003), D’Halluin, Forsyth & Labahn (2004), and D’Halluin, Forsyth &Vetzal (2005). The moving boundary approach is extended by Chockalingam &Muthuraman(2010). A Markov chain approximation is proposed by Simonato (2011).

8. CONCLUSION

The past 50 years have witnessed significant advances in the understanding of American-stylederivatives. Dominance arguments have been developed to shed light on the structure of opti-mal exercise decisions. Price decompositions have been derived to explain the sources of value.Numerical approaches have been designed for faster and more accurate implementation. Im-portant gaps in knowledge nevertheless remain. To conclude this review, the following list oftopics, which could provide fertile ground for future research, is suggested.

FUTURE ISSUES

1. Stochastic volatility and jumps: Empirical evidence suggests the relevance of diffusion aswell as jump components in the structure of security returns and volatilities. Valuationmodels incorporating both aspects are at the forefront of current research. Efforts aremainly confined to European derivatives. The impact on the valuation and exercisepolicies of American derivatives remains to be thoroughly studied.

2. Exotic contracts: Financial innovation has brought and will continue to bring a flow ofnew products to the market. Some of the contractual forms already available have earlyexercise provisions that are not that well understood. Others do not carry exerciseprovisions. In both cases, it is important to determine the impact of these clauses.Investors and issuers benefit from improved decision-making, valuation, and risk

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management. Adding well-understood early exercise provisions to exotics withoutsuch features may appeal to certain clienteles, thereby creating newmarket opportunities.

3. High-dimensional problems: Numerical methods for large-scale problems remain in-adequate. For sure, much progress has been realized with the discovery of the dualityapproach. Duality naturally opens the door to the use of Monte Carlo methods, whichare ideally suited for handling large-scale problems.Difficulties nevertheless remain. Thechoice of an appropriate set of martingales over which to optimize remains a challenge.Methods proposed to address the issue are not yet satisfactory. This area is currently theobject of sustained research attention.

4. Unhedgeable risks: Although financial innovation has given investors access to a richset of instruments to manage economic and financial risks, certain exposures remainimperfectly hedgeable (e.g., volatility risk pertaining to individual stocks). Americanderivatives written on unhedgeable risks raise particularly challenging questions. Whenspanning fails, sufficiently different investorswill disagree on the valuationof cash flows.As a result, optimal exercise policies can no longer be described by the formulaspresented above. Ultimately, valuation may require an analysis of supply and demandconditions.

5. Incomplete information: Parameters of underlying return processes are typically notknown or cannot be estimated with certainty. Relevant state variables are often un-observed or observed at discrete dates. Information sets of issuers and holders are notnecessarily identical. Each of these elements complicates the exercise decision process. Abetter understanding of their impact is needed to improve valuations and hedgingpolicies associated with American derivatives.

DISCLOSURE STATEMENT

The author is not aware of any affiliations,memberships, funding, or financial holdings thatmightbe perceived as affecting the objectivity of this review.

ACKNOWLEDGMENTS

I thank Robert Jarrow,Marcel Rindisbacher, Diogo Duarte, Ting Fei and Thu Truong for helpfulcomments.

LITERATURE CITED

Amin K. 1993. Jump diffusion option valuation in discrete time. J. Finance 48:1833–63Amin KI, Bodurtha JN. 1995. Discrete-time valuation of American options with stochastic interest rates.Rev.

Financ. Stud. 8:193–233Andersen A, Andreasen J. 2000. Jump-diffusion processes: volatility smile fitting and numerical methods for

option pricing. Rev. Deriv. Res. 4:231–62Andersen L, Broadie M. 2004. A primal-dual simulation algorithm for pricing multi-dimensional American

options. Manag. Sci. 50:1222–34AndricopoulosAD,WiddicksM,NewtonDP,Duck PW. 2007. Extending quadraturemethods to valuemulti-

asset and complex path dependent options. J. Financ. Econ. 83:471–99

482 Detemple

Ann

u. R

ev. F

in. E

con.

201

4.6:

459-

487.

Dow

nloa

ded

from

ww

w.a

nnua

lrev

iew

s.or

g A

cces

s pr

ovid

ed b

y U

nive

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of

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Lib

rary

on

11/2

2/14

. For

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l use

onl

y.

Barone-Adesi G, Whaley RE. 1987. Efficient analytic approximation of American option values. J. Finance42:301–20

Barraquand J, Martineau D. 1995. Numerical valuation of high dimensional multivariate Americansecurities. J. Financ. Quant. Anal. 30:383–405

Battauz A, De Donno M, Sbuelz A. 2014. Real options and American derivatives: the double continuationregion. Manag. Sci. Forthcoming. http://pubsonline.informs.org/doi/abs/10.1287/mnsc.2013.1891

Belomestny D, Bender C, Schoenmakers J. 2009. True upper bounds for Bermudan products via non-nestedMonte Carlo. Math. Finance 19:53–71

Bensoussan A. 1984. On the theory of option pricing. Acta Appl. Math. 2:139–58Bensoussan A, Lions JL. 1978. Applications des Inequations Variationelles en Contrôle Stochastique. Paris:

DunodBismut JM, Skalli B. 1977. Temps d’arrêt, théorie générale de processus et processus de Markov. Z.

Wahrscheinlichkeit 39:301–13Bjerksund P, Stensland G. 1993a. American exchange options and a put-call transformation: a note. J. Bus.

Finance Account. 20:761–64Bjerksund P, StenslandG. 1993b.Closed-formapproximation ofAmerican options. Scand. J.Manag.9:88–99Black F, Scholes M. 1973. The pricing of options and corporate liabilities. J. Polit. Econ. 81:637–54Bossaerts P. 1989. Simulation estimators of optimal early exercise. Work. Pap., Carnegie Mellon Univ.Boyle PP. 1988. A lattice framework for option pricing with two state variables. J. Financ. Quant. Anal.

23:1–12Boyle PP, Evnine J,Gibbs S. 1989.Valuation of options on several underlying assets.Rev. Financ. Stud. 2:241–50Boyle PP, Lau SH. 1994. Bumping up against the barrier with the binomial method. J. Deriv. 1:6–14Boyle PP, Tian Y. 1998. An explicit finite difference approach to the pricing of barrier options. Appl. Math.

Finance 5:17–43Breen R. 1991. The accelerated binomial option pricing model. J. Financ. Quant. Anal. 26:153–64Brennan M, Schwartz E. 1977. The valuation of American put options. J. Finance 32:449–62Brenner M, Courtadon G, Subrahmanyam M. 1985. Option on the spot and options on futures. J. Finance

40:1303–17Broadie M, Detemple J. 1995. American capped call options on dividend-paying assets. Rev. Financ. Stud.

8:161–91BroadieM,Detemple J. 1996. American option valuation: new bounds, approximations, and a comparison of

existing methods. Rev. Financ. Stud. 9:1211–50BroadieM, Detemple J. 1997. The valuation of American options onmultiple assets.Math. Finance 7:241–85Broadie M, Detemple J. 1999. American options on dividend paying assets. In Fields Institute Communi-

cations, Vol. 22: Topology and Markets, ed. G Chichilnisky, pp. 69–97. Toronto: AMSBroadie M, Glasserman P. 1997. Pricing American-style securities using simulation. J. Econ. Dyn. Control

21:1323–52Broadie M, Glasserman P. 2004. A stochastic mesh method for pricing high-dimensional American options.

J. Comput. Finance 7:35–72BroadieM,GlassermanP, JainG. 1997. EnhancedMonteCarlo estimates forAmericanoption prices. J.Deriv.

5:25–44Broadie M, Yamamoto Y. 2003. Application of the fast Gauss transform to option pricing. Manag. Sci.

49:1071–88Bunch D, Johnson H. 1992. A simple and numerically efficient valuation method for American puts using

a modified Geske-Johnson approach. J. Finance 47:809–16Carr P, Hirsa A. 2003. Why be backward? Forward equations for American options. Risk 16:103–7Carr P, Jarrow R, Myneni R. 1992. Alternative characterizations of American put options. Math. Finance

2:87–106Carriere JF. 1996. Valuation of the early-exercise price for options using simulations and nonparametric

regression. Insur. Math. Econ. 19:19–30ChenX,Chadam J. 2006.Amathematical analysis of the optimal exercise boundary forAmerican put options.

SIAM J. Math. Anal. 38:1613–41

483www.annualreviews.org � Optimal Exercise for Derivative Securities

Ann

u. R

ev. F

in. E

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4.6:

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11/2

2/14

. For

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y.

ChenX,ChadamJ, JiangL,ZhengW.2008.Convexity of the exercise boundaryof theAmericanput optionona zero dividend asset. Math. Finance 18:185–97

Chen X, Cheng H, Chadam J. 2013. Nonconvexity of the optimal exercise boundary for an American putoption on a dividend-paying asset. Math. Finance 23:169–85

Chesney M, Gibson R. 1995. State space symmetry and two-factor option pricing models. Adv. FuturesOptions Res. 8:85–112

Cheuk THF, Vorst TCF. 1996. Complex barrier options. J. Deriv. 4:8–22Chiarella C, Kang B, Meyer GH, Ziogas A. 2009. The evaluation of American option prices under stochastic

volatility and jump-diffusion dynamics using themethodof lines. Int. J. Theor.Appl. Finance12:393–425Chiarella C, Ziogas A. 2005. Evaluation of American strangles. J. Econ. Dyn. Control 29:31–62Chiarella C, Ziogas A. 2009. American call options under jump-diffusion processes—a Fourier transform

approach. Appl. Math. Finance 16:37–79Chiarella C, Ziveyi J. 2014. Pricing American options written on two underlying assets. Quant. Finance

14(3):409–26Chockalingam A, Muthuraman K. 2010. Pricing American options when asset prices jump. Oper. Res. Lett.

38:82–86Chung S. 1999. American option valuation under stochastic interest rates. Rev. Deriv. Res. 3:283–307Chung S, Hung M, Wang J. 2010. Tight bounds on American option prices. J. Bank. Finance 34:77–89Courtadon GR. 1982. Amore accurate finite difference approximation for the valuation of options. J. Financ.

Quant. Anal. 17:697–703Cox JC, Ross SA. 1976. The valuation of options for alternative stochastic processes. J. Financ. Econ.

3:145–66Cox JC, Ross SA, Rubinstein M. 1979. Option pricing: a simplified approach. J. Financ. Econ. 7:229–63Davis MHA, Karatzas I. 1994. A deterministic approach to optimal stopping. In Probability, Statistics and

Optimization: A Tribute to Peter Whittle, ed. FP Kelly, pp. 455–66. Chichester: WileyDempster MAH, Hutton JP. 1997. Fast numerical valuation of American, exotic and complex options. Appl.

Math. Finance 4:1–20Dempster MAH, Hutton JP. 1999. Pricing American stock options by linear programming. Math. Finance

9:229–54Detemple J. 2001. American options: symmetry properties. InHandbooks inMathematical Finance: Topics in

Option Pricing, Interest Rates and Risk Management, ed. J Cvitanic, E Jouini, M Musiela, pp. 67–104.Cambridge, UK: Univ. Press

Detemple J, Emmerling T. 2009. American chooser options. J. Econ. Dyn. Control 33:128–53Detemple J, Feng S, Tian W. 2003. The valuation of American call options on the minimum of two dividend-

paying assets. Ann. Appl. Probab. 13:953–83Detemple J, Tian W. 2002. The valuation of American options for a class of diffusion processes.Manag. Sci.

48:917–37D’Halluin Y, Forsyth PA, Labahn G. 2004. A penalty method for American options with jump diffusion

processes. Numer. Math. 97:321–52D’Halluin Y, Forsyth PA, Vetzal KR. 2005. Robust numerical methods for contingent claims under jump

diffusion processes. IMA J. Numer. Anal. 25:87–112Ekstrom E. 2004. Convexity of the optimal stopping boundary for the American put option. J. Math. Anal.

Appl. 299:147–56El Karoui N. 1981. Les Aspects Probabilistes du Controle Stochastique. Lect. Notes Math. 876:73–238Evans JD, Kuske R, Keller JB. 2002. American options on assets with dividends near expiry. Math. Finance

12:219–37Fakeev AG. 1971. Optimal stopping of a Markov process. Theory Probab. Appl. 16:694–96Feng L, Kovalov P, Linetsky V,Marcozzi M. 2007. Variational methods in derivatives pricing. InHandbooks

inOperations Research andManagement Sciences, Vol. 15: Financial Engineering, ed. JR Birge, VLinetsky,pp. 301–42. Amsterdam: Elsevier

Figlewski S, Gao B. 1999. The adaptive mesh model: a new approach to efficient option pricing. J. Financ.Econ. 53:313–51

484 Detemple

Ann

u. R

ev. F

in. E

con.

201

4.6:

459-

487.

Dow

nloa

ded

from

ww

w.a

nnua

lrev

iew

s.or

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cces

s pr

ovid

ed b

y U

nive

rsity

of

Uta

h -

Mar

riot

Lib

rary

on

11/2

2/14

. For

per

sona

l use

onl

y.

Gao B, Huang JZ, Subrahmanyam M. 2000. The valuation of American barrier options using the de-composition technique. J. Econ. Dyn. Control 24:1783–827

Garcia D. 2003. Convergence and biases of Monte Carlo estimates of American option prices usinga parametric exercise rule. J. Econ. Dyn. Control 27:1855–79

Geltner D, Riddiough T, Stojanovic S. 1996. Insights on the effect of land use choice: the perpetual option onthe best of two underlying assets. J. Urban Econ. 39:20–50

Geman H, El Karoui N, Rochet JC. 1995. Changes of numéraire, changes of probability measure and optionpricing. J. Appl. Probab. 32:443–58

Gerber HU, Shiu ESW. 1996. Martingale approach to pricing perpetual American options on two stocks.Math. Finance 6:303–22

Geske R, Johnson H. 1984. The American put option valued analytically. J. Finance 39:1511–24Glowinski R. 1984. Numerical Methods for Nonlinear Variational Problems. Berlin: SpringerGlowinski R, Lions JL, Tremolieres R. 1981. Numerical Analysis of Variational Inequalities. Amsterdam:

North-HollandGoodman J, Ostrov D. 2002. On the early exercise boundary of the American put option. SIAM J. Appl.

Math. 62:1823–35Grabbe OJ. 1983. The pricing of call and put options on foreign exchange. J. Int. Money Finance 2:239–53Gukhal CR. 2001. Analytical valuation of American options on jump-diffusion processes. Math. Finance

11:97–115Harrison M, Kreps D. 1979. Martingales and arbitrage in multiperiod security markets. J. Econ. Theory

20:381–408Harrison M, Pliska S. 1981. Martingales and stochastic integrals in the theory of continuous trading. Stoch.

Proc. Appl. 11:215–60Haugh M, Kogan L. 2004. Approximating pricing and exercising of high-dimensional American options:

a duality approach. Oper. Res. 52:258–70Ho TS, Stapleton RC, SubrahmanyamMG. 1997. The valuation of American options with stochastic interest

rates: a generalization of the Geske-Johnson technique. J. Finance 52:827–40Huang JZ, Subrahmanyam M, Yu G. 1996. Pricing and hedging American options: a recursive integration

method. Rev. Financ. Stud. 9:277–330Ibanez A, Zapatero F. 2004.Monte Carlo valuation of American options through computation of the optimal

exercise frontier. J. Financ. Quant. Anal. 39:253–75Jacka SD. 1991. Optimal stopping and the American put. Math. Finance 1:1–14Jaillet P, Lamberton D, Lapeyre B. 1990. Variational inequalities and the pricing of American options. Acta

Appl. Math. 21:263–89Jamshidian F. 1992. An analysis of American options. Rev. Futures Mark. 11:72–80Ju N. 1998. Pricing an American option by approximating its early exercise boundary as a multipiece ex-

ponential function. Rev. Financ. Stud. 11:627–46Kallast S, Kivinukk A. 2003. Pricing and hedging American options using approximations by Kim integral

equations. Eur. Finance Rev. 7:361–83Kamrad B, Ritchken P. 1991. Multinomial approximating models for options with k state variables.Manag.

Sci. 37:1640–52Karatzas I. 1988. On the pricing of American options. Appl. Math. Optim. 17:37–60Karatzas I, Shreve S. 1998. Methods of Mathematical Finance. New York: Springer-VerlagKim IJ. 1990. The analytic valuation of American options. Rev. Financ. Stud. 3:547–72Kim IJ, Yu GG. 1996. An alternative approach to the valuation of American options and applications. Rev.

Deriv. Res. 1:61–85Kinderlehrer D, Stampacchia G. 1980. An Introduction to Variational Inequalities and their Applications.

New York: AcademicKovalov P, Linetsky V, Marcozzi M. 2007. Pricing multi-asset American options: a finite element method-of-

lines with smooth penalty. J. Sci. Comput. 33:209–37Kuske RA, Keller JB. 1998. Optimal exercise boundary for an American put option. Appl. Math. Finance

5:107–16

485www.annualreviews.org � Optimal Exercise for Derivative Securities

Ann

u. R

ev. F

in. E

con.

201

4.6:

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of

Uta

h -

Mar

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Lib

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11/2

2/14

. For

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sona

l use

onl

y.

Little T, Pant V, Hou C. 2000. A new integral representation of the early exercise boundary for American putoptions. J. Comput. Finance 3:73–96

Longstaff FA, Schwartz ES. 2001. Valuing American options by simulation: a simple least squares approach.Rev. Financ. Stud. 14:113–48

MacMillan LW. 1986. Analytic approximation for the American put option. Adv. Futures Options Res.1:119–39

McDonald R, Schroder M. 1998. A parity result for American options. J. Comput. Finance 1:5–13McKean HP. 1965. A free boundary problem for the heat equation arising from a problem in mathematical

economics. Ind. Manag. Rev 6:32–39Medvedev A, Scaillet O. 2010. Pricing American options under stochastic volatility and stochastic interest

rates. J. Financ. Econ. 98:145–59Menkveld AJ, Vorst T. 2000. A pricing model for American options with Gaussian interest rates. Ann. Oper.

Res. 100:211–26Merton RC. 1973a. An intertemporal capital asset pricing model. Econometrica 41:867–87Merton RC. 1973b. Theory of rational option pricing. Bell J. Econ. Manag. Sci. 4:141–83MertonRC. 1976.Option pricingwhen underlying stock returns are discontinuous. J. Financ. Econ.3:125–44Meyer GH. 1998. The numerical valuation of options with underlying jumps. Acta Math. 67:69–82Meyer GH, van der Hoek J. 1997. The valuation of American options with the method of lines. Adv. Futures

Options Res. 9:265–85Mitchell D, Goodman J, Muthuraman K. 2014. Boundary evolution equations for American options. Math.

Finance 24(3):505–32Mullinaci S. 1996. An approximation of American option prices in a jump-diffusionmodel. Stoch. Proc. Appl.

62:1–17Muthuraman K. 2008. A moving boundary approach to American option pricing. J. Econ. Dyn. Control

32:3520–37Myneni R. 1992. The pricing of the American option. Ann. Appl. Probab. 2:1–23Nielsen BF, Skavhaug O, Tveito A. 2002. Penalty and front-fixing methods for the numerical solution of

American option problems. J. Comput. Finance 5:69–97Omberg E. 1987. The valuation of American put options with exponential exercise policies. Adv. Futures

Options Res. 2:117–42Parkinson M. 1977. Option pricing: the American put. J. Bus. 50:21–36Peskir G. 2005. On the American option problem. Math. Finance 15:169–81PhamH.1997.Optimal stopping, free boundary, andAmericanoption in a jump-diffusionmodel.Appl.Math.

Optim. 35:145–64Ramaswamy K, Sundaresan SM. 1985. The valuation of options on futures contracts. J. Finance 40:1319–40Raymar S, Zwecher M. 1997. AMonte Carlo valuation of American call options on the maximum of several

stocks. J. Deriv. 5:7–23ReimerM, Sandmann K. 1995. A discrete time approach for European and American barrier options.Work.

Pap., Dep. Stat., Rheinische Friedrich-Wilhelms-Univ.Rendleman RJ, Bartter BJ. 1979. Two-state option pricing. J. Finance 34:1093–110Ritchken P. 1995. On pricing barrier options. J. Deriv. 3:19–28Rogers C. 2002. Monte Carlo valuation of American options. Math. Finance 12:271–86Rogers C. 2010. Dual valuation and hedging of Bermudan options. SIAM J. Financ. Math 1:604–8Rubinstein M, Reiner E. 1991. Breaking down the barriers. Risk 4:28–35Rutkowski M. 1994. The early exercise premium representation of foreign market American options. Math.

Finance 4:313–25Samuelson PA. 1965. Rational theory of warrant pricing. Ind. Manag. Rev. 6:13–32Schroder M. 1999. Changes of numeraire for pricing futures, forwards and options. Rev. Financ. Stud.

12:1143–63Schwartz ES. 1977. The valuation of warrants: implementing a new approach. J. Financ. Econ. 4:79–94Shiryaev AN. 1978. Optimal Stopping Rules. New York: Springer-Verlag

486 Detemple

Ann

u. R

ev. F

in. E

con.

201

4.6:

459-

487.

Dow

nloa

ded

from

ww

w.a

nnua

lrev

iew

s.or

g A

cces

s pr

ovid

ed b

y U

nive

rsity

of

Uta

h -

Mar

riot

Lib

rary

on

11/2

2/14

. For

per

sona

l use

onl

y.

Simonato JG. 2011. Computing American option prices in the lognormal jump–diffusion framework witha Markov chain. Finance Res. Lett. 8:220–26

Sullivan M. 2000. Valuing American put options using Gaussian quadrature. Rev. Financ. Stud. 13:75–94Tan KS, Vetzal KR. 1995. Early exercise region for exotic options. J. Deriv. 3:42–55Tilley JA. 1993. Valuing American options in a path simulation model. Trans. Soc. Actuar. 45:83–104Tsitsiklis JN, Van Roy B. 1999. Optimal stopping of Markov processes: Hilbert space theory, approximation

algorithms, and an application to pricing high-dimensional financial derivatives. IEEE Trans. Automat.Contr. 44:1840–51

Tsitsiklis JN, VanRoy B. 2001. Regressionmethods for pricing complex American-style options. IEEETrans.Neural Netw. 12:694–703

VanMoerbekeP.1976.Onoptimal stoppingand freeboundaryproblems.Arch.Ration.Mech.Anal.60:101–48Vasicek OA. 1977. An equilibrium characterization of the term structure. J. Financ. Econ. 5:177–88Villeneuve S. 1999. Exercise regions of American options on several assets. Finance Stoch. 3:295–322Wu L, Dai M. 2009. Pricing jump risk with utility indifference. Quant. Finance 9:177–86Wu L, Kwok Y. 1997. A front-fixing finite differencemethod for the valuation of American options. J. Financ.

Eng. 6:83–97Zhang X. 1994.Analyse numérique des options américaines dans un modèle de diffusion avec des sauts. PhD

Thesis, Ecole Nationale des Ponts et Chaussées ParisZhangXL. 1997.Numerical analysis of American option pricing in a jump-diffusionmodel.Math.Oper. Res.

22:668–90Zvan R, Vetzal KR, Forsyth PA. 2000. PDE methods for pricing barrier options. J. Econ. Dyn. Control

24:1563–90

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Annual Review of

Financial Economics

Volume 6, 2014Contents

History of American Corporate Governance: Law, Institutions, and PoliticsEric Hilt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Blockholders and Corporate GovernanceAlex Edmans . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

Corporate Takeovers and Economic EfficiencyB. Espen Eckbo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

Payout PolicyJoan Farre-Mensa, Roni Michaely, and Martin Schmalz . . . . . . . . . . . . . . 75

Corporate Liquidity Management: A Conceptual Framework and SurveyHeitor Almeida, Murillo Campello, Igor Cunha,and Michael S. Weisbach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

Corporate Pension PlansJoão F. Cocco . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

Bank Capital and Financial Stability: An Economic Trade-Off or a FaustianBargain?Anjan V. Thakor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

Contingent Capital Instruments for Large Financial Institutions: A Review ofthe LiteratureMark J. Flannery . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225

Counterparty Risk: A ReviewStuart M. Turnbull . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241

The Industrial Organization of the US Residential Mortgage MarketRichard Stanton, Johan Walden, and Nancy Wallace . . . . . . . . . . . . . . . 259

Investor Flows to Asset Managers: Causes and ConsequencesSusan E.K. Christoffersen, David K. Musto, and Russ Wermers . . . . . . . 289

v

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Exchange-Traded Funds: An Overview of Institutions, Trading, and ImpactsAnanth Madhavan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311

Stock Prices and Earnings: A History of ResearchPatricia M. Dechow, Richard G. Sloan, and Jenny Zha . . . . . . . . . . . . . . 343

Information Transmission in FinancePaul C. Tetlock . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365

Insider Trading Controversies: A Literature ReviewUtpal Bhattacharya . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385

Security Market ManipulationChester Spatt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405

Financialization of Commodity MarketsIng-Haw Cheng and Wei Xiong . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419

Forward Rate Curve SmoothingRobert A. Jarrow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443

Optimal Exercise for Derivative SecuritiesJérôme Detemple . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 459

Indexes

Cumulative Index of Contributing Authors, Volumes 1–6 . . . . . . . . . . . . . . 489Cumulative Index of Chapter Titles, Volumes 1–6 . . . . . . . . . . . . . . . . . . . . 491

Errata

An online log of corrections toAnnual Review of Financial Economics articles maybe found at http://www.annualreviews.org/errata/financial

vi Contents

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AnnuAl Reviewsit’s about time. Your time. it’s time well spent.

AnnuAl Reviews | Connect with Our expertsTel: 800.523.8635 (us/can) | Tel: 650.493.4400 | Fax: 650.424.0910 | Email: service@annualreviews.org

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Annual Review of Statistics and Its ApplicationVolume 1 • Online January 2014 • http://statistics.annualreviews.org

Editor: Stephen E. Fienberg, Carnegie Mellon UniversityAssociate Editors: Nancy Reid, University of Toronto

Stephen M. Stigler, University of ChicagoThe Annual Review of Statistics and Its Application aims to inform statisticians and quantitative methodologists, as well as all scientists and users of statistics about major methodological advances and the computational tools that allow for their implementation. It will include developments in the field of statistics, including theoretical statistical underpinnings of new methodology, as well as developments in specific application domains such as biostatistics and bioinformatics, economics, machine learning, psychology, sociology, and aspects of the physical sciences.

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from Observational Studies, David Madigan, Paul E. Stang, Jesse A. Berlin, Martijn Schuemie, J. Marc Overhage, Marc A. Suchard, Bill Dumouchel, Abraham G. Hartzema, Patrick B. Ryan

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Christopher D. Steele, David J. Balding•Using League Table Rankings in Public Policy Formation:

Statistical Issues, Harvey Goldstein•Statistical Ecology, Ruth King•Estimating the Number of Species in Microbial Diversity

Studies, John Bunge, Amy Willis, Fiona Walsh•Dynamic Treatment Regimes, Bibhas Chakraborty,

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