Valuation of. American Continuous-Installment Options

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1 Valuaion of American Coninuous-Insallmen Opions P. Ciurlia Diparimeno di Meodi Quaniaivi Universià degli Sudi di Brescia I. Roko Deparmen of Economerics Universiy of Geneva Ocober 4 Absrac We presen hree approaches o value American coninuous-insallmen calls and pus and compare heir compuaional precision. In an American coninuous-insallmen opion, he premium is paid coninuously insead of up-fron. A or before mauriy, he holder may erminae paymens by eiher exercising he opion or sopping he opion conrac. Under he usual assumpions, we are able o consruc an insananeous riskless dynamic hedging porfolio and derive an inhomogeneous Black-Scholes parial differenial equaion for he iniial value of his opion. This key resul allows us o derive valuaion formulas for American coninuous-insallmen opions using he inegral represenaion mehod and consequenly o obain closed-form formulas by approximaing he opimal sopping and exercise boundaries as mulipiece exponen- We are graeful o Manfred Gilli, Henri Loubergé and Evis Këllezi for encouragemens, suggesions and remarks. 1

2 ial funcions. This process is compared o he finie-difference mehod o solve he inhomogeneous Black-Scholes PDE and a Mone Carlo approach. 1 Inroducion In a convenional American-syle opion conrac, he buyer pays he premium enirely upfron and acquires he righ, bu no he obligaion, o exercise he opion a any ime up o a fixed mauriy ime T. Here we consider an alernaive form of American-syle opion conrac in which he buyer pays a smaller up-fron premium and hen a consan sream of insallmens a a cerain rae per uni ime. However, he buyer can choose a any ime o sop making insallmen paymens eiher by exercising he opion or by sopping he opion conrac. There is lile lieraure on insallmen opions. Davis e al. 1, derive noarbirage bounds for he iniial premium of a discreely-paid insallmen opion and sudy saic versus dynamic hedging sraegies wihin a Black-Scholes framework wih sochasic volailiy. Their analysis is resriced o European-syle insallmen opions, which allows for an analogy wih compound opions, previously considered in Geske 1977 and Selby and Hodges Davis e al. 3 values venure capial using an analogy wih he insallmen opion. Ben-Ameur e al. 4 develops a dynamic-programming procedure o price American-syle insallmen opions and derive some heoreical properies of he insallmen opion conrac wihin he geomeric Brownian moion framework. Their approach is applied o insallmen warrans, which are acively raded on he Ausralian Sock Exchange. Finally, Wysup e al. 4 compares pricing echniques for insallmen opions wrien on exchange raes. The aim of his paper is o presen hree alernaive approaches for valuing American coninuous-insallmen calls and pus and o compare heir compuaional advanages. In Secion, we formulae he American coninuous-insallmens opion valuaion problem as a free boundary-value problem and obain an analyic soluion by uilizing he resuls in Carr e al. 199, Jacka 1991 and Kim In Secion 3, we describe in deail he alernaive approaches. Numerical resuls are compared in Secion 4. Secion 5 concludes. 1 We are graeful o Seward Hodges for suggesing his approach.

3 American Coninuous-Insallmen Opions The paricular feaure of he pricing problem of an American coninuous-insallmen opion is he deerminaion, along wih he iniial premium and he opimal exercise boundary, of a furher boundary called he opimal sopping boundary..1 Black-Scholes PDE for Coninuous-Insallmen Opions We assume he sandard model for perfec capial markes, coninuous rading, no-arbirage opporuniies, a consan ineres rae r >, and an asse paying coninuous proporional dividends δ > wih price S following a geomeric Brownian moion ds = µs d + σs db, 1 where µ = r δ and db is a Wiener process on a risk-neural probabiliy space. The Black-Scholes iniial premium V of a coninuous-insallmen opion V = V S, ; q depends on he curren value of he underlying asse S, ime, and he coninuous insallmen rae q. Applying Iô s Lemma o we obain he dynamics for he iniial value of his opion dv = V + µs V S + 1 σ S V S q V d + σs S db. 3 The only difference in expression 3 relaive o he sandard Black-Scholes framework is he presence of he consan rae q ha has o be paid o say in he opion conrac. We now consruc he replicaing porfolio consising of one coninuous-insallmen opion and an amoun Φ of he underlying asse. The value of his porfolio is Π = V ΦS and is dynamics is given by dπ = dv ΦdS ΦS δd. 3

4 Puing 1 and 3 ogeher, we ge dπ = V µs S Φ + V + 1 σ S V S q Φ S V δ d + σs S Φ db. Seing Φ = V S he coefficien of db vanishes. The porfolio is insananeously riskless and, o avoid arbirage opporuniies, mus yield reurn r. So we mus have r V V S S = V + 1 σ S V S q V S S δ. Rearranging his equaion gives he inhomogeneous Black-Scholes PDE for he iniial premium of a coninuous-insallmen opion V + µs V S + 1 σ S V S rv = q. 4. Valuaion of American Coninuous-Insallmen Calls Consider an American coninuous-insallmen call on S wih srike price K and mauriy ime T. We denoe he iniial premium of his call a ime by CS, ; q, defined on he domain D = {S, [, [ [, T ]}. For each ime [, T ], here exiss an upper criical asse price B above which i is opimal o sop he insallmen paymens by exercising he opion early, as well as a lower criical asse price A below which i is advanageous o erminae paymens by sopping he opion conrac. According o hese upper and lower criical asse prices he iniial premium CS, ; q is CS, ; q = S K + if S [, A ] [B, [ 5 CS, ; q > S K + if S ]A, B [. 6 The sopping and exercise boundaries are he ime pahs of lower and upper criical asse prices A and B, for [, T ], respecively. These boundaries divide he domain D ino a sopping region D 1 = {S, [, A ] [, T ]}, a coninuaion region D = {S, ]A, B [ [, T ]}, and an exercise region D 3 = {S, [B, [ [, T ]}. To ensure ha he fundamenal consrain CS, ; q S K + is saisfied in he domain D, equaion 5 impose ha, in he sopping and exercise regions, he iniial premium 4

5 CS, ; q equals o he opion payoff S K +. By conras, he inequaliy expressed in 6 shows ha, in he coninuaion region, i is advanageous o coninue paying he insallmen premiums since he call is worh more alive han dead. The iniial premium is given by 5 if he asse price sars eiher in D 1 or D 3, so we assume ha he call is alive a he valuaion ime, i.e., A < S < B. The iniial premium CS, ; q of he American coninuous-insallmen call saisfies he inhomogeneous Black-Scholes PDE 4 in D ; ha is, CS, ; q + µs CS, ; q S + 1 σ S CS, ; q S rcs, ; q = q on D. 7 Exending he analysis of McKean 1965, we deermine ha CS, ; q and he sopping and exercise boundaries A and B joinly solve a free boundary-value problem consising of 7 subjec o he following final and boundary condiions: CS T, T ; q = S T K + 8 lim CS, ; q = 9 S A CS, ; q lim = 1 S A S lim CS, ; q = B K 11 S B CS, ; q lim = 1. 1 S B S The value maching condiions 9 and 11 imply ha he iniial premium is coninuous across he sopping and exercise boundaries, respecively. Furhermore, he high conac condiions 1 and 1 furher imply ha he slope is coninuous. Equaions 9 1 are joinly referred o as smooh fi condiions and ensure he opimaliy of he sopping and exercise boundaries. We solve his problem wih he inegral represenaion mehod inroduced in Carr e al. 199, Jacka 1991 and Kim 199. Ziogas e al. 4 presens a survey of he mehods for deriving he various inegral represenaions of American opion prices. Le ZS, e r CS, ; q be he discouned iniial premium funcion of he American coninuous-insallmen call, defined in he domain D. In his domain, he funcion ZS, inheris he properies of he iniial premium funcion CS, ; q, i.e., i is a convex 5

6 funcion in S for all, coninuously differeniable in for all S and a.e. wice coninuously differeniable in S for all. Applying Iô s Lemma o ZS, yields ZS, T σ ZS T, T = ZS, + ds + S S ZS, S + ZS, d. In erms of CS, ; q his means e rt CS T, T ; q = CS, ; q + + e r σ S e r CS, ; q S ds CS, ; q S r CS, ; q + CS, ; q d. From 8 we know ha CS T, T ; q = S T K + and, separaing he iniial premium ino CS, ; q = 1 {A<S <B } CS, ; q + 1 {S B } S K, we have e rt S T K + = CS, ; q + e 1 r CS, ; q {A<S <B } + 1 {S B S } µs d + σs db σ + e r CS, ; q 1 {A <S <B } S S r CS, ; q + CS, ; q d + e r 1 {S B } rs K d. On he coninuaion region, he iniial premium funcion CS, ; q saisfies he inhomogeneous Black-Scholes PDE 7, so he erms muliplying 1 {A<S <B } sum o q. Using his, and aking expecaions, reduces he above equaion o ] c E [e rt S T K + = CS, ; q + q + e r N d S, A, N d S, B, d e r δs e r δ N d 1 S, B, + rkn d S, B, d. By rearranging his expression, we obain he inegral represenaion for he iniial premium of he American coninuous-insallmen call: CS, ; q = c + q δs e δ N d 1 S, B, + q rke r N d S, B, d e r N d S, A, d, 13 6

7 where d 1 x, y, = lnx/y + r δ + σ / σ and d x, y, = d 1 x, y, σ and c is he Black-Scholes/Meron European call pricing formula. Equaion 13 expresses he iniial premium of an American coninuous-insallmen call as he sum of he corresponding European call value, he early exercise premium, and he expeced presen value of insallmen paymens along he opimal sopping boundary. The early exercise premium can be viewed as he value of a coningen claim ha allows ineres earned on he srike price, decreased by he insallmen premium, o be changed for dividends paid by he asse whenever he asse price is above he opimal exercise boundary. The opimal sopping boundary A is implicily defined by he following inegral equaion: = c A, K, T + q δa e δs N d 1 A, B s, s + q rke rs N d A, B s, s ds e rs N d A, A s, s ds. 14 Equaion 14 reflecs he fac ha he iniial premium of an American coninuous-insallmen call a he ime of opimal sopping is equal o he opion payoff, which is zero. Similarly, applying he boundary condiion 11, we obain he inegral equaion saisfied by he opimal exercise boundary B : B K = c B, K, T + q δb e δs N d 1 B, B s, s + q rke rs N d B, B s, s ds e rs N d B, A s, s ds. 15 This suggess ha he iniial premium of American coninuous-insallmen calls should be compued in wo seps. In he firs, 14 and 15 are solved for A and B, respecively. Given he opimal sopping and exercise boundaries, 13 is solved nex. Unforunaely, direc soluions for he inegral equaions 14 and 15 are no possible. According o Kolodner 1956, hese are Volerra inegral equaions and can only be solved numerically. 7

8 In Secion 3.1, we presen a numerical approximaion mehod for solving 13 direcly in closed form..3 Valuaion of American Coninuous-Insallmen Pus For he valuaion of an American coninuous-insallmen pu we proceed in he same way as for he call. We denoe by P S, ; q, defined on he same domain D, he iniial premium funcion of he American coninuous-insallmen pu. For each ime, here mus be a lower criical asse price F below which i is opimal o erminae paymens by exercising he opion, as well as an upper criical asse price G above which i is advanageous o erminae paymens by sopping he opion conrac. The exercise and sopping boundaries, which are he ime pahs of lower and upper criical asse prices F and G, divide he domain D ino an exercise region D 1 = {S, [, F ] [, T ]}, a coninuaion region D = {S, ]F, G [ [, T ]}, and a sopping region D 3 = {S, [G, [ [, T ]}. The iniial premium funcion P S, ; q saisfies he inhomogeneous Black-Scholes PDE in he coninuaion region D ; ha is, P S, ; q + µs P S, ; q S + 1 σ S P S, ; q S rp S, ; q = q on D, 16 subjec o he following erminal and boundary condiions P S T, T ; q = K S T + 17 lim P S, ; q = K F 18 S F P S, ; q lim = 1 19 S F S lim P S, ; q = S G P S, ; q lim =. 1 S G S By applying he resuls of he previous secion, he soluion o he free boundary-value 8

9 problem 16 1 is P S, ; q = p + q q + rk e r N d S, F, S δ e δ N d 1 S, F, d e r N d S, G, d. Using he propery of he normal cdf, we can rewrie he equaion as P S, ; q = p + 1 q + rk 1 e rt S 1 e δt r + S δ e δ N d 1 S, F, q + rk e r N d S, F, d q r 1 e rt + q e r N d S, G, d. 3 Applying he boundary condiions 18 and, we obain he inegral equaions for F and G : K F = p F, K, T + 1 q + rk 1 e rt δt F 1 e r + F δ e δs N d 1 F, F s, s q + rk e rs N d F, F s, s ds q r 1 e rt + q e rs N d F, G s, s ds, 4 = p G, K, T + 1 q + rk 1 e rt δt G 1 e r + G δ e δs N d 1 G, F s, s q + rk e rs N d G, F s, s ds q r 1 e rt + q e rs N d G, G s, s ds. 5 3 Numerical Mehods Here we presen he hree alernaive approaches o value he American coninuous-insallmen opions. Firs we implemen he valuaion formulas derived in Secion using he mulipiece exponenial funcion mehod of Ju Second he finie-difference mehod for solving he inhomogeneous Black-Scholes PDE is presened. Finally, we consider a Mone Carlo mehod. 9

10 3.1 Implemenaion of he Valuaion Formulas by he Mulipiece Exponenial Funcion MEF Mehod Once he inegral equaions defining he opimal sopping and exercise boundaries are solved, he compuaion of he iniial premium simply implies numerical inegraion. Unforunaely, hese inegral equaions canno be solved explicily. However, here is a special feaure of equaions 13 and 3 ha has been invesigaed in he lieraure. Noing ha he exercise boundary appears only as an argumen o he logarihm funcion in he definiions of d 1 and d, Ju 1998 argues ha he inegral equaion for he American pu value does no depend on he exac values of he exercise boundary criically. Making use of his propery and approximaing he boundary as a mulipiece exponenial funcion, he obains a closed-form formula for pricing American-syle opions. To exend he approach in Ju 1998, hereafer called he Mulipiece Exponenial Funcion MEF mehod, we divide he inerval [, T ] ino M equal ime inervals and define j = j T/M, j = 1,,..., M. Le C CI be he approximaed iniial premium of an American coninuous-insallmen call corresponding o he approximaed opimal sopping and exercise boundaries by M-piece exponenial funcions A j e aj and B j e bj, for j = 1,,..., M, respecively. Then C CI is given by if S A M C CI = C M, S, A, B, a, b, φ, ν, T if A M < S < B M S K if S B M, 6 where C j, x, A, B, a, b, φ, ν, τ j = cx, K, τ q I i 1, i, x, A j i+1 e aj i+1t τ, a j i+1, 1, r i=1 j + x δ I i 1, i, x, B j i+1 e b j i+1t τ, b j i+1, 1, δ i=1 + q rk j I i 1, i, x, B j 1+1 e b j i+1t τ, b j i+1, 1, r. i=1 To deermine he coefficiens A j, a j, B j and b j, j = 1,,..., M, we apply he value-mach 1

11 and high-conac condiions 9 1 a each ime sep j. This yields C j, A j e ajt j, A, B, a, b, φ, ν, j = C x j, Aj e a jt j, A, B, a, b, φ, ν, j = C j, B j e b jt j, A, B, a, b, φ, ν, j = B j e b jt j K C x j, Bj e b jt j, A, B, a, b, φ, ν, j = 1, 7 where C x j, x, A, B, a, b, φ, ν, τ = e δτ N d 1 x, K, τ j q I x i 1, i, x, A j i+1 e a j i+1t τ, a j i+1, 1, r + + x i=1 j δi i 1, i, x, B j i+1 e bj i+1t τ, b j i+1, 1, δ i=1 j δi x i 1, i, x, B j i+1 e bj i+1t τ, b j i+1, 1, δ i=1 + q rk j I x i 1, i, x, B j i+1 e b j i+1t τ, b j i+1, 1, r. i=1 The funcions I and I x are defined, respecively, by I i 1, i, x, y, z, φ, ν = 1 e ν i 1 N ν + 1 z1 + 1 e z z 3 z 1 N ν z z1 1 e z z 3 +z 1 ν z 3 z 1 i 1 + z e ν i N z 1 i + z i 1 i z 3 i + z N z 3 i 1 + z i i 1 N z 3 i z N z 3 i 1 z, 8 i i 1 I x i 1, i, x, y, z, φ, ν = 1 e ν i 1 n z 1 i 1 + z e ν i n z 1 i + z 1 ν i 1 i 1 i i σx + 1 z1 + z [ 3 e z z 3 z 1 z 3 z 1 N z 3 i + z N z 3 i 1 + z ν z 3 i i 1 + e z z 3 z 1 n z 3 i + z 1 n z 3 i 1 + z ] 1 1 i i i 1 i 1 σx 1 z1 z [ 3 e zz3+z1 z 3 + z 1 N z 3 i z N z 3 i 1 z ν z 3 i i 1 + e n zz3+z1 z 3 i z 1 n z 3 i 1 z ] 1 1 i i i 1 i 1 σx, 9 See he Appendix for he derivaion of hese funcions. 11

12 where z 1 = r δ z + φ σ /, z = lnx/y σ σ and z 3 = z 1 + ν. To find he coefficiens, we mus solve he sysem of four equaions 7 for j = 1,,..., M. A each sep j, he above sysem is solved using a Newon mehod. The approximaion procedure of American coninuous-insallmen pus proceeds in he same way as for calls. Le P CI be he approximaed iniial premium of an American coninuous-insallmen pu corresponding o he approximaed opimal exercise and sopping boundaries by M-piece exponenial funcions F j e fj and G j e gj, for j = 1,,..., M, respecively. Then P CI is given by K S if S F M P CI = P M, S, F, G, f, g, φ, ν, T if F M < S < G M if S G M, 3 where P j, x, F, G, f, g, φ, ν, τ = px, K, τ + 1 q + rk 1 e rτ x 1 e δτ r j + x δi i 1, i, x, F j i+1 e f j i+1t τ, f j i+1, 1, δ i=1 j q + rk I i 1, i, x, F j i+1 e f j i+1t τ, f j i+1, 1, r q r 1 e rτ + q i=1 j I i 1, i, x, G j i+1 e g j i+1t τ, g j i+1, 1, r. i=1 As for calls, applying he value-mach and high-conac condiions 18 1 a each ime sep j, we can deermine he coefficiens F j, f j, G j and g j, j = 1,,..., M. 3. Solving he Inhomogeneous Black-Scholes PDE wih Finie Differences The valuaion of he iniial premium of an American coninuous-insallmen opion by finie differences is obained wih he Crank-Nicolson mehod. For he call, he inhomogeneous Black-Scholes PDE and he final and boundary condiions have been defined in 7 and 8 1. For he pu, hese are defined in 16 and For discreizaion, a uniform 1

13 grid in space and ime is used. To achieve greaer accuracy, criical poins are fixed midway beween wo grid poins in space. The opimal exercise problem is solved simply by aking he maximum beween he coninuaion value and he opion payoff. This echnique is known as he explici payou mehod. Oher echniques consider a PSOR or a Newon mehod o solve he linear complemenariy problem e.g., Coleman e al.. The opimal sopping problem is solved in a similar way by aking only posiive coninuaion values. 3.3 Valuaion wih a Mone Carlo Mehod We modify he leas-squares Mone Carlo mehod inroduced by Longsaff and Schwarz 1 o accommodae he pricing of he American coninuous insallmen opions. Le us consider a discree-ime sample pah S i, i =, 1,..., M for he price of an underlying asse, wih M = T/, where T is he ime o mauriy and is he ime discreizaion. For European-syle opions he price is given by E e rt fs M, where f denoes he payoff funcion and E he expecaion under he risk-neural measure. When we consider on early exercise, he value of he conrac for each simulaed ime insan i corresponds o he maximum beween he inrinsic value fs i and he expeced coninuaion value. Therefore a ime sep i, he value V i S i of he opion, condiional on S i, is } V i S i = max {fs i, E i e r V i+1 S i+1 S i, where he funcion V is defined recursively for i = M 1, M,...,. The value of V M S M is simply fs M, i.e., he payoff a mauriy. Longsaff and Schwarz 1 approximaes he condiional expecaion of he coninuaion value E i by a linear regression of he presen value of V i+1 S i+1 a i on a se of polynomials of he curren asse price S i. To ge observaions for he regression, we have o replicae he sample pah of he underlying asse price. The jh replicaion for he asse price is denoed by S j i, and correspondingly he jh replicaion of he coninuaion value, which is he presen value of V j i+1 Sj i+1, is 13

14 denoed by y j i. Regressing on a second-order polynomial, he approximaion of yj i is y j i = α 1 + α S j i + α 3 S j i, and he condiional expecaion of he coninuaion value E i y j i is given by ŷj i = ˆα 1 + ˆα S j i + ˆα 3 S j i, where ˆα k, k = 1,, 3, are he esimaed regression coefficiens. In he case of coninuously-paid insallmens a a consan rae q, he coninuaion value y j i becomes e r V j i+1 Sj i+1 q 1 e r, r and we use he same regression for he esimaion of he condiional expecaion. The decision for early exercise a ime i, for a sample j, is aken if fs j i > ŷj i, where j J E i, he se of pahs ha are in-he-money a ime i. The decision for early sopping is aken if ŷ j i <, where j J S i, he se of pahs ha are ou-of-he-money a ime i. The ses J E i and J S i consiue a pariion of he se J of replicaed pahs. I should be noiced ha he condiional expecaion ŷ j i is esimaed separaely on he se J E i and he se J S i. Therefore he iniial value of he opion a ime sep i, condiional on S j i, is V j i Sj i = { } max fs j i, E i e r V j i+1 Sj i+1 Sj i } max {, E i e r V j i+1 Sj i+1 Sj i if j J E i if j J S i. The compuaion of he opion price is now achieved hrough he Algorihm 1, which provides a skeleon for he implemenaion of a compuer code. 14

15 Algorihm 1 1: Generae S R N M : Iniialize T j = M and V j = fs j M, for j = 1,..., N 3: for i = M 1 1 do 4: y j = e r T j i V j q 1 r e r T j i, for j = 1,..., N 5: Compue Ji E = {j fs j i > } 6: Esimae ŷ j JE i = Ey j JE i S j JE i i 7: Compue Ji E = {j j Ji E fs j i > ŷj JE i } 8: Updae T j = i and V j = fs j E i, for j Ji 9: Compue Ji S = {j fs j i = } 1: Esimae ŷ j JS i = Ey j JS i S j JS i i 11: Compue Ji S = {j j Ji S ŷ j JS i < } 1: Updae T j = i and V j =, for j Ji S 13: end for 14: y j = e r T j V j q 1 r e r T j, for j = 1,..., N 15: v = 1 N M j=1 yj Saemens 5 8 consider he case where early exercise has o be checked and saemens 9 1 where sopping has o be checked. The ses J E and J S correspond respecively o he pahs where early exercising or sopping has aken place. Elemen j of array T informs us abou he ime sep where he early exercise or sopping decision has been aken for he jh pah. The inrinsic value of he opion a ime sep T j is given in V j. In saemen 14, he opion value a ime for each pah is saved in y, and, in saemen 15, he average of hese values is compued. The convergence of his mehod is analyzed in Glassermann and Yu 4, where he choice of he order of he polynomial approximaing E is discussed in conjuncion wih he number N of pah replicaions and ime seps M. 4 Numerical Resuls and Discussions In his secion we repor and compare numerical resuls obained wih each of he hree mehods for several values of some relevan parameers. All algorihms have been implemened in Malab 7.xx and he resuls are repored in Table 1. 15

16 FDM MEF Mone Carlo σ S T q M = M = 6 M = 1 s.e / / / / / / Table 1: Iniial premiums of American coninuous-insallmen calls K = 1 and δ =.4. For he finie-difference mehod, we use 6 seps beween and for he asse price and 4 ime seps per quarer of a year. The mulipiece exponenial funcion MEF mehod has been esed for M =, M = 6 and M = 1. The resuls for he Mone Carlo mehod are based on 1 aniheic pahs and a fourh-order Hermie polynomial for he regressions. The number of ime seps used for his mehod is 8 per quarer of 16

17 a year. Following Glassermann and Yu 4, p. 18 hese seings saisfy he condiions for convergence. To esimae he sandard errors, we compue a saisic wih 5 iniial premiums. The values repored in he able are he medians of his saisic. Comparing he resuls obained by he MEF mehod for M = 1 wih he resuls given by he oher wo mehods we see, in Table 1, ha he approximaions coincide from wo o five digis. If he MEF mehod is used wih M =, we ge from one o hree correc digis. In erms of compuaional efficiency, he finie-difference mehod resul o be fases wih a compuaional ime of less han 1 second o calculae he iniial premiums a all grid poins for a 3-monh American coninuous-insallmen call. The opimal sopping and exercise boundaries can be derived from he values on he space-ime grid. The MEF mehod wih M = 1 needs roughly 1 seconds o solve he pricing problem for he same opion and provides he iniial premium for a single value of S, as well as a poinwise approximaion of he boundaries. If we consider M =, he compuaional ime becomes comparable o ha of he finie differences. A ineresing feaure of his mehod is he deerminaion of he hree componens in which he iniial premium has been decomposed via inegral represenaion. A difficuly of he MEF mehod may consis in he appropriae choice of he iniial values when one solving he non-linear sysem 7. The Mone Carlo approach needs approximaively 14 seconds o find he iniial premium. Since he resul is of random naure we need o compue confidence inervals which imply repeaed evaluaions of he iniial premium. An advanage of he Mone Carlo mehod is ha i can be exended easily o exoic payoffs and mulifacor opion. The lef panel in Figure 1 presens he iniial premium funcion CS, ; q and he opimal sopping and exercise boundaries, boh calculaed by finie differences. The righ panel in Figure 1 shows how each mehod approximaes he boundaries. The approximaions of he exercise and sopping boundaries obained by he finie-difference mehod are respecively he solid and he doed lines. The crosses and circles represen he welve-piece exponenial exercise and sopping boundaries, respecively. The clouds of poins along he boundaries are he opimal sopping and exercise decisions for each pah in he Mone Carlo mehod. 17

18 Figure 1: Lef panel: Iniial premium funcion CS, ; q of an American coninuousinsallmen call K = 1, T = 3/1, σ =., r =.5, δ =.4 and q = 8. Righ panel: Opimal sopping and exercise boundaries approximaed by finie differences, he welve-piece exponenial boundaries and he sopping and exercise decisions of he Mone Carlo simulaions. 5 Concluding Remarks We have presened hree alernaive approaches for solving he free boundary-value problem of American coninuous-insallmen opions. Firs we derived he inhomogeneous Black- Scholes PDE for coninuous-insallmen opions using a combinaion of hedging and riskneural valuaion argumens. This resul allows he derivaion of an inegral represenaion for he iniial premium of hese opions, using he resuls in Carr e al. 199, Jacka 1991 and Kim 199. The mulipiece exponenial funcion MEF mehod allows an approximaion in closed form o he valuaion formulas for he American coninuous-insallmen opions. To es he MEF mehod we adaped wo exising numerical mehods o he pricing problem of he nonsandard American opions. All hree mehods produce similar resuls from which we conclude he soundness of our approaches. The focus of his paper is on American coninuous-insallmen calls. However, by presening a mahemaically and compuaionally meaningful way o analyze he premaure sopping of American opions, his sudy enhances applicaions of he coningen-claims approach o invesmen problems in general. For example, invesmens involving periodic paymens ha can be sopped a any ime can be analyzed using he framework developed in his paper. 18

19 Appendix Derivaion of funcions I and I x Le us assume ha for he generic inerval [ i 1, i ] he sopping and exercise boundaries A and B are approximaed by exponenial funcions Ae a and Be b, respecively. To make use of his approximaion, he inegrals in equaions 13 and 3 can be evaluaed in closed form for his inerval. We firs consider he inegral I 1 = i i 1 e r N d S, Ae a, d. Defining x 1 = r δ a σ //σ, x = lns /A/σ, we have ha d S, A, = x 1 1/ + x 1/. Inegraion by pars yields I 1 = 1 r e r i 1 Nx 1 1/ i 1 + x 1/ i 1 e r i Nx 1 1/ i + x 1/ + e x 1x r π i i 1 e 1 x 3 +x 1 x 1 1/ x 3/ d, i where x 3 = x 1 + r. By making use of he following ideniies dnx 3 1/ + x 1/ = e x 3x π e 1 x 3 +x 1 x 3 1/ x 3/ dnx 3 1/ x 1/ = ex3x e 1 x 3 +x 1 x 3 π 1/ + x 3/, we ge I 1 e r i 1 N x 1 i 1 + x = 1 r + 1 x1 + 1 r x x1 1 r x 3 e x x 3 x 1 e x x 3 +x 1 x 1 i + x i e r i N i 1 N x 3 i + x N x 3 i 1 + x i i 1 N x 3 i x N x 3 i 1 x. i i 1 From he above equaion follows immediaely ha I = = 1 r i e r N d S, Be b, d i 1 e ri 1 N x r x x4 1 r x 6 x 4 i 1 + x 5 e ri N i 1 e x5x6 x4 e x 5x 6 +x 4 x 4 i + x 5 i x 6 i + x 5 N x 6 i 1 + x 5 i i 1 N N x 6 i x 5 N x 6 i 1 x 5, i i 1 19

20 where x 4 = r δ + b σ //σ, x 5 = lns /B/σ, and x 6 = x 4 + r. If we define y 1 = r δ +b+σ //σ, y = lns /B/σ, y 3 = y1 + δ, a similar derivaion would yield I 3 = = 1 δ i e δ N d 1 S, Be b, d i 1 e δi 1 N y δ y y1 1 δ y 3 y 1 i 1 + y e δi N i 1 e yy3 y1 e y y 3 +y 1 y 1 i + y i y 3 i + y N y 3 i 1 + y i i 1 N N y 3 i y N y 3 i 1 y. i i 1 Using equaion 8, he inegrals I 1, I and I 3 can be expressed uniquely as I 1 = I i, i 1, x, A, a, 1, r, I = I i 1, i, x, B, b, 1, r, I 3 = I i 1, i, x, B, b, 1, δ. The funcion I x is he firs parial derivaive of 8 wih respec o x. References Ben-Ameur, H., M. Breon and P. François 4. A Dynamic Programming Approach o Price Insallmen Opions. To appear in European Journal of Operaional Research. Carr, P., R. Jarrow and R. Myneni 199. Alernaive Characerizaions of American Pu Opions. Mahemaical Finance, Coleman, T.F., Y. Li and A. Verma. A Newon Mehod for American Opion Pricing. Journal of Compuaional Finance 53, Davis, M., W. Schachermayer and R. Tompkins 1. Pricing, No-Arbirage Bounds and Robus Hedging of Insalmen Opions. Quaniaive Finance 16, Davis, M., W. Schachermayer and R. Tompkins. Insallmen Opions and Saic Hedging. Journal of Risk Finance 3, Davis, M., W. Schachermayer and R. Tompkins 3. The Evaluaion of Venure Capial As an Insalmen Opion: Valuing Real Opions Using Real Opions. To appear in Zeischrif für Beriebswirschaf.

21 Geske, R The Valuaion of Corporae Liabiliies as Compound Opions. Journal of Financial and Quaniaive Analysis 14, Glassermann, P. and B. Yu 4. Number of Pahs Versus Number of Basis Funcions in American Opion Pricing. To appear in The Annals of Applied Probabiliy. Jacka, S.D Opimal Sopping and he American Pu. Mahemaical Finance 1, Ju, N Pricing an American Opion by Approximaing Is Early Exercise Boundary as a Mulpiece Exponenial Funcion. Review of Financial Sudies 113, Kim, I.J The Analyical Valuaion of American Opions. Review of Financial Sudies 34, Kolodner, I.I Free Boundary Problem for he Hea Equaion wih Applicaions o Problems of Change of Phase. Communicaions in Pure and Applied Mahemaics 9, Longsaff, F.A. and E.S. Schwarz 1. Valuing American Opions by Simuaion: A Simple Leas-Square Approach. Review of Financial Sudies 141, McKean, H.P Appendix: A Free Boundary Problem for he Hea Equaion Arising from a Problem in Mahemaical Economics. Indusrial Managemen Review 6, Selby, M.J.P. and S.D. Hodges On he Evaluaion of Compound Opions. Managemen Science 33, Wysup, U., S. Griebsch and C. Kühn 4. FX Insalmen Opions. Working Paper. Ziogas, A, C. Chiarella and A. Kucera 4. A Survey of he Inegral Represenaion of American Opion Prices. Working Paper, Universiy of Techology, Sidney. 1