RY-Expected Optimal Exercise Time

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Th e Journal of Financial EngineeringVolume 4Number 1Pages 55-73

Expected Optimal Exercise Time of aPerpetual American Option:A Closed-form SolutionRudy Yaksick

ABSTRACTUsing martingale methods, we find that the expected optimal exercise time of aperpetual, dividend-paying American call option contract is the ratio of thetime-independent stopping boundary to the risk-adjusted drift of the stock priceprocess. This ratio is an analytical expression. Of independent interest is thecomputational simplic ity of our derivation. Specifically, we use only the optionalsampling theorem of martingale theory an d elementary a lgebra . In contrast, thenon-martingale approach requires tedious integration and solution of an ordinarydifferential equation.Ke y Words and Phrases : Brownian motion, first passage time, martingale,American call option, optimal exercise.

R u d y Yaksick; Graduate School of Managem ent , Clark Universi ty, 950 Mai n Street; Worcester, VMassachuset ts 01610.Rudy Yaksi ck is also a Lecturer in the Department of Finance and Economics, Boston Universi tyand Research Associate, Center for the Study of Financial Engineering. The au t hor is part icularlyindebted to Professors I. Karatzas and M. Tamarkin for advice and encouragem ent as wel l as to E.Henning for technical comments. Final ly, this research was par t ial ly supported by the Institute fo rMat hem at i cs and i ts ApplicationsUniversi ty of Minnesota, with funds provided by the U.S.Nat i ona l Science Foundation.

Expected Optimal Exercise Time 55


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I. INTRODUCTIONMartingale methods simplify the solution of stochastic, sequential decisionproblems. For example, Chung (1974, pp. 327-28) uses martingale methods todemonstrate in two lines, an d "without an y computation" (his quotes), that th eprobability of a gambler's ruin is inversely proportional to the gambler's in i t ialcapital.

This note has a similar objective. That is , we aim to demonstrate thatmartingale methods g reatly simp lify the soluti on of two problems. T he first is theclassic problem: What is the expected first passage (stopping) time to a single,fixed boundary of a Brownian motion with drif t? The first passage time is ofinterest since the solution of a wide variety of gambling and investment decisionproblems often involves determining whether and when a key, underlyingstochastic process hits some boundary . For example, in the classic gambler ' s ru inproblem one of the main issues is: What is the expected duration of the game?

This issue can be rephrased in sharper, an a ly t ic terms as: How much t ime wil lelapse, on average, before the gambler ' s d iscounted net gain reaches a boundarylevel of zero?In Section II we demonstrate the relative ease with which one can derivevia

martingale m eth o d s th e Laplace transform of the first passage t ime (to a singleboundary) density function of a Brownian motion with drift. The derivationbegins with a straightforward application of the optional sampling theorem tomartingales associated w ith Bro wni an motio n. Then, elementary algebra is usedto obtain the transform. This is a much less tedious approach than the commonpractice (a s i l lustrated in Srinivasan an d Mehata [1978] section 5.5) of computingth e Laplace transforms of the terms of the backward Kolmogorov equation an dthen solving th e resultant l inear ordinary dif ferential equation in order to obtainth e Laplace t ransform of the first passage density function.Having obtained the Laplace transform, we simply differentiate it to obtainou r main probabilistic result. Th e expected first passage t ime is the ratio of thestopping boundary to the drift of the Brownian motion.Th e second problem (which is addressed in Section III) is of particular

interest to finance practitioners in both the investment and corporate sectors.Hedging, speculation, arbitrage, an d capital budgeti ng decisions often hinge onthe issue: What is the expected optimal exercise (stopping) time of an Americanoption contract that pays dividen ds? An American option is a financial securitywh ich gives the holder the right, but not the obligation, to exercise the option(and thereby obtain th e underlying asset of the option) at any time throughout th elife of the option. W e preface the an alysis of th is issue by reviewing McKean's(1965) seminal analysis of the pricing of a perpetual American warrantasecurity closely related to an American option. Then, we replicate his analysis56 Yaksick

with in the context of Karatzas' (1988, 1989) recent martingale-based frameworkfo r valui ng American options. In this framesvork, whic h is a bit more elaboratethan th e Harrison-Pliska (1981) seminal model, the key economic idea of theabsence of arbitrage opportunities is l inked to the probabilistic concept of amart ingale . Moreover, th is framework is quite general s ince it enables marketpart icipants to consume as well as invest , thus permitt ing a unif ied approach toth e problems of option pricing, consumption an d investment, an d eq u i l ib r iu m ina financial market .Ou r main f inancial result is that (for a perpetual, d ividen d-pay ing Americancall option) th e optima l exercise t ime is the rat io of the stopping boundary (b) toth e drift of the transformed stock price process. Here, b equals th e product of theinverse of the stock volati l i ty t imes the log of the ratio of the optimal exercisestock price to the ini t ia l (time zero) stock price. And the optimal exercise s tockprice is a fu n c t io n of the risk-free rate and the dri f t an d volati l i ty parameters ofth e stock price stochastic process. Thus, th e optimal exercise t ime is easilycomputable since it is an analytical expression.W e conclude (i n Section IV) by suggesting future research tasks that m ayconfirm the co m p u ta t io n a l simplicity of martingale methods under a wid er classof stochastic processes as well as stopping problems h av in g more general , curvedan d time-dependent boundaries. F inally , al l proofs ar e contained in Ap p en d ix A .

II. PROBABILISTIC FOUNDATIONS AND RESULTSTo make the paper self-contained, we def ine some basic concepts from the theoryof stochastic processes.DefinitionsBrownian MotionLet: i) (O,F,P) be a filtered probability space; ii) B = (B(t ) , F,; 0 < t < ) be acontinuous stochastic process on this space; and, iii) u and o denote the drift an ddispersion coefficient parameters of B, respectively. We say that B is aone-dimensional (u, a) Brownian motion with respect to F if:i) B is adapted to the filtration F = (F c , tTc[0,]), where T is thecontinuous time index set, of the measurable sample space (Q,F).ii) B(t) = B(0) + ut + oW(t), t > 0, where W is a s tandard Brow nian motion(Wiener process) an d B(0) is independent o f W.i i i) For 0 S s < t, every increment B(t + s) - B(s) is i n d ep en d en t of Fs andis normally dis tributed with constant mean ut and constant variance crt.

iv) B(0) = 0, a.s.



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First Passage TimeSuppose that i) (fl.F) is a measurable sample space with a filtration F, and ii)(den is a sample point in the outcome space 2. We say that the random variableTb, where Tb: fl - T , is a first passage t ime (with respect to F) to a singleboundary beR if, for every ter

( i n f f t S 0; B (co) = b], if B (co) S b and sign u = sign b , for some t S 0o o , if B t(co) < b and sign u = sign b, for all 1 2 0Main Probabilistic ResultsWe now prove the three probabilistic results. The first is a demonstration of theease with which one can obtain, via martingale methods, the Laplace transformof the first passage time density of B(t) to a single boundary b. To appreciate thesimplicity, the reader is advised to consult Bhattacharya an d W ay mire (1990,section 1.10) for a discussion of the mathematically tedious, non-martingaleapproach. Finally, the first main result rests on the following four lemmas.

Lemma 2.1:Le t I 3 ( t ) be the Brownian motion process defined earlier. Associated with B(t) isth e fol lowing martingale:

(2.1)(t) = exp[XB(t) - 0t]where 0 sXu + XV/2, and\isany arbitrary, real constant.Lemma 2.2: Optional Sampling TheoremLet V(t) be the martingale defined in (2.1); let Tb be a stopping time. If i) Prob(T b < 00} = i, ii) BtfV>.,] = 0,where l ( T b > n i 's an indicator function having a value 1, when Tb > n, and zero,otherwise. Then:

E[V(0)] (2.2)

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where (T bA t) = min(Tb , t) .

Lemma 2.3:Le t V(t), the martingale defined in (2.1), satisfy the hypotheses of Lemma 2.2; letTb be a stopping time. Suppose that the hypotheses of the optional samplingtheorem are satisfied. Then:

l=E[V(TbAt)] (2.3)

Lemma 2.4:Let V(t), the martingale defined in (2.1), satisfy the hypoth eses of Lemma 2.2; letTb be a stopping time. Suppose that the hypotheses of the optional samplingtheorem are satisfied. If i) A , is sufficiently large such that 0 > 0 and, thus, 0 e , ] ) = 0

Then: (2.4)

With these intermediate results, we now obtain our first main probabilisticresult.Theorem 2.1: Laplace Transform of First Passage Time DensitySuppose that i) B(t) is a (u,o) Brownian motion; ii) b#Q is the single, fixedboundary; and iii) Tb is the first time, if any, that B(t) reaches the level b. Thenthe Laplace transform of the first passage time density of B(t) to a singleboundary b is:

E le- J = exp(ub/o2 - |b|[(u2 + 2d 2Q) i]l


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Theorem 2.2: Existence of Finite First Passage TimeRetain th e hypotheses of Theorem 2.1. Then B(t) a tta ins b *0 with probabilityone if and only if u an d b have the same sign.Our final resultthe expected first passage time of a Brownian motionprocess with driftis obtained by differentiating (2.5), the Laplace transform ofthe first passage t ime density function of B(t) to a single boundary.Theorem 2.3: Expected First Passage TimeRetain the hypotheses of Theorem 2.1. Then, the expected first passage t ime ofB(t) is :

b/u, b and u have same sign, b and u have different signs

(2.6)

IntuitionThe unde r ly ing in tuit ion of the result is clearly illustrated in Figure 1. The processB(t) increases deterministically along the straight trend line with slope u. Thus,u measures the expected rate of change in B, per uni t time. That is, u = (E[B(t)]- B( t0))/t. The height of the t ime-independent boundary (line bb) indicates thevertical distancein this case b unitsthat B(t) must travel in order to reach theboundary. Since B(t) is expected to change (increase) in value by u units pe r uni tt ime, the expected time to traverse the height of b units (i.e., reach the boundary)is thus b/u. Of course, the actual time to first passage to the boundary could begreater or less than E[TJ since B(t) will fluctuate around the trend line due to therandom shocks odW.

As a further aid to understanding, the main result can be interpreted in thecontext of a gambling problem. In that problem, our result indicates that theexpected duration of the gambler's game is the ratio of the gambler 's initialwealth to the drift of the net gain process. Finally, to gauge the computationalsimplic ity of the martingale approach to comput ing the expected duration of agambl ing game, the reader is directed to Feller (1968, pp . 348-49). In a randomwalk setting, he employs th e more tedious method involving th e solution ofdifference equations.

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Figure 1Expected First Passage Time (EM [Tb])

TIME[Tb]

IH. EXPECTED EXERCISE TIME: PERPETUALAMERICAN CALL OPTION

Background: McKean's Optimal Exercise ResultsMcKean (1965) was the first to provide a mathematically complete solution to theproblem of va lu ing a perpetual American warrant in a Markovian setting. Heassumed a perfect security market in which three assets are con t inuous ly t r adedand no arbitrage opportunities are present. The first asset is a riskless bond whoseprice D(t) satisfies the linear differential equation:

dD(t) = D(t)rdt (3.1)where r > 0 is a constant, continuously com pounded interest ra te offered to allborrowers and lenders. The underlying asset, on which the perpetual Americancall option is written, is a stock whose price S(t) satisfies the linear, stochasticdifferential equation:



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dS(t) = S(t)[(m(t) + u)dt + odW(t)], S(0) > 0 (3.2)where m(t) is the "drift" or instantaneous expected growth rate of S(t); ue(0,r), thedi v i dend rate, and a are non-negative constants ; and d W(t ) is a StandardBrowni an motion under P. Finally , the th ird asset is a perpetual American callopt ion with an exercise price, q = $1.ResultsIn t h i s co n tex t McKean obtained a closed-form solution for the two u n k n o wn sof the warrant holder ' s v alu a t io n (optimal s topping) problem. These are theo p t i m a l exercise t ime (T'B) and the time-indepen dent exercise (stopping ) bound ary(S'), respectively:

i) T'B = inf{t > 0; S(t) S S*} for all tE[0,]ii) S' = $/( - 1) > 1

(3.3)(3.4)

w h e r e : O a [(52 + 2ro2)5 - Sj/o2 andS s r - u -.So2Replication in Karatzas' FrameworkRepl i ca t i ng McKean's result in Karatzas' f ram ewo rk i s not straightforward. A sKaratzas (1988, p. 54) indicates , to evaluate perpetual options, two technicald i f f i cu l t i e s have to be surmounted. That is , W(t) must now be: i ) a Brownianmotion on the entire se t [0,o], and ii) accompanied by a filtration that measuresall of the processes in the model. Then, to ensure that arbitrage (riskless,se l f - f i nanc i ng) t rading profi ts cannot be reaped, a new , equivalent probabil i tymeasure (P0) is constructed which is also a martingale measure. Under th is ne wequ i va l en t martingale measure (which, i f unique, ensures market completeness),the orig inal Brownian motion, W(t), is re-expressed, via Girsanov's (1960)change-of-measure theorem, as:

iW0(t) = W(t) + J0(s)dS for all te[0, ~] (3.5)

wh ere :0(0 = [m(t) + u - r]/o

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Finally , the optimal exercise (stopping) time (T p* ) is re-expressed as:

I,,' = inf{t > 0; W0(t) + vt 5B} (3.6)

where the exercise boundary (B) is:B = (l/o)log[SVS(0)] (3.7)

and:i) S* = q [ < D / ( < I > - 1)]E(0,~)ii) O = (o2)-'[(82 + Iro2)5 - 8], 1 < < D < r (r - u)- 1in ) 8 = r - u - .So2iv) v = 8/a

These results are derived in Karatzas (1989, section 6).Expected Exercise TimeW e n o w derive th e main result an d three related lemmas. Th e first l emmaspecifies th e relat ionship between th e stock price at t ime zero and the op t ima lexercise price S* that excludes the possibility of a zero exercise tim e. This is anuninterest ing case since it means th e call option should be immediately exercised.Lemma 3.1:Suppose that there exists a solution to the option holder's optimal exercise(s topping) problem. If S(0) < S*, then T, > 0.

Lem m a 3.2 is the analogue of Theorem 2.1.Lemma 3.2:The Laplace transform of TB is:

E[exP(-aT,j)] = exp{[Bv - |B| (v 2 + 2o2a)5]/cr), a > 0 (3.8)Th e th ird lemma specifies th e relat ionship that must hold between th eparameters B and v (as well as the sign of v) for the exercise time to be



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finite-valued with probability one. Thus, this lemma is the analogue of Theorem2.2.Lemma 3.3:Suppose that S* > S(0), an d v, B *0. Then, S(t) reaches S* (equivalently , W 0(t)+ vt attains B) wit h probability one if and only if v and B have the same sign.Finally , we obtain our main financial result.Theorem 3.1: Expected Optimal Exercise Time. Retain the hypotheses ofLemma 3.3. Then, the expected optimal exercise time of a perpetual Americancall option that pays con t inuous d iv idends is :

E(V)[TJ = P/v, i f p a n d v > 0[-, if p > 0, v < 0

(3.9)

In sum, this appears to be a novel result. Th e closest work to this appears tobe C a r m a n ' s (1989) concept of "fugit" (an abbreviation of the Latin phrasetempus fugit, or "time flies"). This concept measures the average lifetime of anAmerican option. However, this concept w as developed in a discrete-time setting.Thus, it yields only an approximate measure of the expected optimal exerciset ime.

IV. IMPLICATIONS AND EXTENSIONSThese results appear to significantly reduce th e computational burden associatedwith stochastic, sequential decision problems. In particular, these resultsdemonstra te that ou r martingale-based methodology can be used to compute th eexpected timing of an optimal decision in a wide variety of stochastic, dynamicdecision problems that can be formulated as a stopping problem.

Consider, fo r example, Martzoukos and Teplitz-Sembitzky's (1992) recentanalysis of a common capital budgeting problem faced by energy planners indeveloping countries. When designing rural electrification programs, plannersm u s t determine the optimal sequencing of decentralized solutions (involvingdiesel-powered generators) an d centralized, grid-based systems. Initially,generators tend to be economic investments since demand is spatially dispersedand loads are relatively low. However, as power demand grows, grid systemsbecome preferred. Given th e uncerta inty about demand growth, planners t hus faceth e problem of optimally timing an irreversible investment in electricity

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transmission lines. Martzoukos an d Teplitz-Sembitzky used martingale results(similar to those used here) to compute the expected optimal timing of th isirreversible, expensive inv estme nt in transmission lines, given uncertainty aboutdemand.-Another cogent example is Collins ' (1992) use of the first passagedensity to estimate the expected timing of foreign exchange realignments in theEuropean Monetary System.Finally , our result is easy to implem ent, relative to the finite-lived o ption casewhich requires numerical methods, because the optimal exercise boundary can befound by analytic means. That is, using the classical optimization methodsemployed by McKean, one can find the optimal exercise boundary by simplymaximizing th e optimal value of the option over the set of boundary points (sincethe "boundary" in this case is a single point), e.g., Karlin and Taylor (1975, pp.364-65).Alternatively, one can take a more elegant analytical approach, such as theone developed by Karatzas (1989, sec. 6). He applied martingale methods to theoption buyer ' s optima l exercise (stopping) problem in order to prove the existenceof an op timal stopp ing time as well as characterize it in terms of an optim alexercise boundary that is identical to that first computed by McKean. 'Several extensions to this research might a id in confirming the computationalsimplicity obtainable via martingale methods. First, the class of stopping problemsmight be widened to include curved and time-dependent boundaries. This would,in turn, permit the comparison of the simplicity of the marting ale approachagainst, fo r example, the complexity of: 1) Durbin's (1985) approximation of thefirst passage time (FPT) density of a co ntinuous Gaussian process to a generalboundary , and 2) Durb in and William s' (1992) recent approximation of the FPTdensity of a Brow nian motion to a curved bou ndary. Their approximation is aconvergent series of multiple integrals of increasing dimensionality. Of course,these densities would subsequently be integrated in order to compute th emoments.Likewise, the mean (and higher moments) for processes other than Brownianmotion with drift should also be computed via martingale methods. Then a basiswould exist fo r comparing th e computational effort of this work against th e resultsof Ricciardi and Sato (1988). They have derived closed-form expressions for allmoments of the first passage time of the unrestricted, conditional Ornstein-Uhlenbeck process with a constant boundary.



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APPENDIXA.I Section II ProofsWe begin by proving four lemmas that are intermediate results required fo r th eproof of Theorem 2.1.PROOF: Lemma2.1.Th e objective of the proof is to demonstrate, via simple computation, thatexp(XB(t)) satisfies th e martingale property. First, th e Markov property- impliesthat:

E[exp(XB(t)IF,)] = exp(XB(s)) E[exp(X(B(t) - B(s)))IFJ= exp(XB(s)) exp{(Xut + XVt/2) - (Xps + XVs/2)}= exp(XB(s)) expHXus + XVs/2)} x exp{(Xfit + X:cri/2)}

CA. l )since E[exp(XB(t))] = exp(Xut + XVt/2) because B(t) - B(s)Is independent of F,and isdistributed N(u(t - s), (^(t - s)). PROOF: Lemma 2.2.Optional Sampling Theorem.Se e Karatzas andShreve (1991, p. 19). PROOF: Lemma 2.3.First, we know tha t E[V (T b A t ) ]=E[V (Tb ) ] (A.2)where T,,At = min{T b,t}, since V[(Tb A t)] = V(TJ whenever Tb < Mid t > Tb.Next, by Lemma 1 and B(0) = 0, we know that:

E[V(0)] = E[exp(XB(0) - 90)]= Etexp(O)], so

E[V(0)] = 1 (A.3)Also, by Lemma 2.2, we know that:

E[V(0)] = EtVOUl (A.4)

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Finally, by combining (A.2) through (A.4), w e conclude that:1 = E[V(0)] = E[V(Tb)] = E[(Tb A t)] (A.5)

PROOF: Lemma 2.4.Passing to the limit as t- > , we observe from the conclusion of Lemma 2.3, that:

1 = l i m E [ V ( T b A t ) ]|_M= E[lim V( Tb A t)]

(A.6)

where the limit can be taken under the expectation operator since (V(T b A t ) } ishypothesized to be uniformly integrable and, thu s, Theorem 5 o f Shiryayev (1984,p. 187) can be invoked. Also:

fexp{Xb - 0Tb). if T bH m V( T b A t) =' |0,

(A.7)i f T k

since V( Tb A t) = V(T,,) whenever Tb < ~ and t > Tb. Since Tb is finite-valued byhypothesis, we can use (A.7) to re-xpress (A.6) as:1 = E[exp(Xb - 0Tb)]1 = e^texpt-eT,,)], or

E[e-6T"] = e-u (A.8)

PROOF: Theorem2.1.By Lemma 2.1 we know that V(t) is the martingale associated with B(t), i.e.V(t) = exp{XB(t) - t) (A.9)

where 8- Xu + XV/2, and X is any teal constant. By Ummas 2 and 3 weconclude that 1 = E


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Passing to the limit as t-+ oo, and using Lemma 2.4,we can conclude that:1= lim jf5exP^B 0, and 0

u - (u 2 + 2o 2 ) 5]/o2, b < 0Substituting (A . 16 ) into the right-hand side of (A.12), we find that theaf*/* fi*anefXfr !Laplace transform is:

| exp{[bu - b( u2 + 2o20)5]/o2}, b, 0 > 0[exp{[bM + b( u2 + 2c T20) 5]/a 2}. b < 0 and 0 > 0

Combining values, we obtain: (A. 17)

(u 2 , 068 Yaksick

PROOF: Theorem 2.2.Following Karlin an d Taylor (1975, p. 362), we know that:(A. 19)

8->U

Using (A . 18), we obtain:Pw[Tk < o o ] = H m exp( [bu - Ib l (u2 + 2o20)3]/crz}, 0 > 0 (A.20)e - > o

After evaluating the limit, we conclude that:

b < ] = e x p ( ( b M - IbuO/o2} (A.21)

Hence:

p W [ T 0

< 1, ub < 0(A.22)

Thus, whenever p and b have different signs, the density of Tb under P * M) isdefective since Tb is infinite with positive pro babil ity. That is:p(f)[T b = oo] = 1 - exp( (bu - > 0 (A.23)

In conclusion, T b is not finite-valued if we suppose that b and u havedifferent signs. PROOF: Theorem2.3.Differentiating (A . 18) wiside by -1, we obtain:

multiplying each



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+ 2a20)5]}exp(ub/a2 - |b|[(u2 + 2c20)5]/o2)(A.24)

Finally , evaluating the derivative as 010, we obtain our main result:E*[TJ =b/ u (A.25)

A.2 Section HI ProofsPROOF: Lemma 3.1.Suppose that S(0) > S'. Then:

Hence:

B = log(S'/S(0))/a < 0

;= inf{t S 0; W0(t) + vt ;> B} = 0(A.26)

(A.27)

since T?* is, by definition, the first time that the P0-Brownian motion W0(t) + v,with dri f t [(r - u)/o] - . 5 < T , hits or exceeds the level B = log(S'/S(0))/o. Hence,EJexp(-rTp')] = 1for S(0) S'. PROOF: Lemma 3.2.Following the argu ment of the proof of Theorem 2.1, one can easily demonstratethat the Laplace transform of T, is:

= exp( [ Bv - j BK v2 + 2cr!a)5]/(r) a > 0 (A.28)PROOF: Lemma 3.3.Following th e arg u m en t of Theorem 2.2, one can easily demonstrate that:

P*V)[T6 < oo] = ex p {(Bv - IOur first conclusion is straightforward. If v < 0, then:

P[T, = ]= 1 - P(T B < oo) > 0

(A.29)

(A.30)

Ou r second major conclusion is:

1,< 1,

> 0 and v > 0< 0 and v > 0

(A.31)

Thus, wh en ev er v and B have different signs, the density of TB u n d er P < v) isdefective since T, is infinite with positive probability. That is :p("[T8 = H = 1 - ex p {(Bv - > 0 (A.32)

In conclusion, T6 is not finite-valued if we suppose either that B and v havedifferent signs or v < 0. Note, for a s l ightly different starting point in the proof, the reader is urged toconsult Karatzas and Shreve (1991, pp. 196-97).PROOF: Theorem 3.1.Different ia t ing th e Laplace transform with respect to a, and mult iplying each sideby -1, we obtain:

E[TB] = B(v 2 + 2o2a)-5E[exp(- 0.

NOTE1. See My n en i (1992) for an i l lustrat ion of th is approach, when applied to thevaluation of a finite-lived American pu t option. Unfortunately , h is approachdoes not lead to an explicit solution for the optimal exercise boundary.Instead, one has to first reformulate the optimal s topping problem as a freeboundary problem. Then, th e optimal exercise boundary < : : m be found (vianumerical methods) as the solution of n nonl inear integnii . ' . . terra) equation.Finally, to understand the l ink between optimal s topping and free b o u n d aryproblems, the reader is urged to consult the seminal references: Mikhalevich

(1958), Chemoff (1961), and Lindley (1961).

70 Yaksick Expected Optim al Exercise Time 71


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'

',I

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