MESTRADO MATEMÁTICA FINANCEIRA TRABALHO FINAL DE MESTRADO DISSERTAÇÃO NUMERICAL ALGORITHMS FOR THE VALUATION OF INSTALLMENT OPTIONS

Transcription

1 MESTRADO MATEMÁTICA FINANCEIRA TRABALHO FINAL DE MESTRADO DISSERTAÇÃO NUMERICAL ALGORITHMS FOR THE VALUATION OF INSTALLMENT OPTIONS SOFIA SANDE DE ARAÚJO SETEMBRO - 01

2 MESTRADO MATEMÁTICA FINANCEIRA TRABALHO FINAL DE MESTRADO DISSERTAÇÃO NUMERICAL ALGORITHMS FOR THE VALUATION OF INSTALLMENT OPTIONS SOFIA SANDE DE ARAÚJO ORIENTAÇÃO: MARIA DO ROSÁRIO LOURENÇO GROSSINHO SETEMBRO - 01

3 Conens 1 Inroducion... 5 Lieraure Review Theoreical Framework Numerical Mehodology Prior Mehods The De Hoog algorihm Compuaional Resuls Conclusions References... 38

4 Acknowledgmens Firs and foremos, I would like o express my sincere graiude o my advisor Prof. Maria do Rosário Grossinho for he coninuous suppor of my hesis, for her moivaion, enhusiasm and grea knowledge. Besides my advisor, I would like o hank he res of my hesis commiee: Prof. Onofre Alves Simões and Prof. Fernando Gonçalves for heir encouragemen and insrucive commens and quesions. Las bu no he leas I would like o hank my family for supporing me hroughou all my sudies a Universiy, my friends for heir suppor and encouragemen and my fiancé for his personal suppor and grea paience a all imes. I would no have been possible o wrie his hesis wihou he help and suppor of he kind people around me. 3

5 Absrac Insallmen opions are financial derivaives in which par of he iniial premium is paid up-fron and he oher par is paid discreely or coninuously in insallmens during he opion s lifeime. This work deals wih he numerical valuaion of European insallmen opions. Trough he sudy of he coninuous case, we can show ha numerical inversion of Laplace ransform works well for compuing he opion value. In paricular, we will invesigae he De Hoog algorihm and compare i o oher mehods for he inverse Laplace ransformaion, namely Euler summaion, Gaver-Sehfes and Kryzhnyi mehods. Keywords: Insallmen opions, compound opions, Laplace ransform, numerical inversion, sopping boundaries, De Hoog algorihm Resumo Insallmen opions são derivados financeiros cuja pare inicial do prémio é paga anecipadamene e a oura pare é dividida, discreamene ou coninuamene, em parcelas durane o empo de vida do conrao. Ese rabalho esuda a valorização numérica de insallmen opions do ipo Europeu. Esudando principalmene o caso conínuo podemos mosrar que a inversão numérica da ransformada de Laplace é um bom méodo para calcular o valor da opção. Em paricular, vamos invesigar o algorimo conhecido por De Hoog e compará-lo a ouros méodos numéricos, sendo eles conhecidos por Euler summaion, Gaver-Sehfes e méodo de Kryzhnyi. Palavras-chave: Insallmen opions, compound opions, Laplace ransform, numerical inversion, sopping boundaries, De Hoog algorihm 4

6 1 Inroducion Derivaives have been cied as a key facor behind he 008 financial crisis due mainly o heir complexiy and lack of ransparency ha caused capial markes o undervalue credi risk. Opions enjoyed grea populariy wihin his class of financial asses and a paricular ype of opions has recenly emerged: he insallmen opion. Unlike mos basic derivaives, insallmen opions presen a valuaion challenge as analyical mehods like he Black-Scholes model are unable o offer a closed soluion; herefore, a numerical approach is required. There are a number of alernaives in solving he valuaion problems of his ype of opions. The bes pracice known so far is o apply he Laplace ransform mehod and hen use numerical algorihms o inver i. Kimura [1] had success on valuaing hese opions numerically using Euler-summaion and Gaver-Sehfes mehods; he Kryzhnyi mehod was addiionally esed by Ehrhard e al. [14]. The main goal of his work is o es a new mehod known as he De Hoog algorihm and compare i o he previous mehods. This algorihm is a Fourier Series Expansion [6,13] ha uses an acceleraion algorihm for is quoien-difference and has proven o be suiable for he long ime inegraion of dissipaive equaions. In fac, and regarding he compuaional resuls, i appears o be he bes candidae o valuae hese opions numerically. Please noe ha some pars of his work follow closely he work of Kimura [1]. The remainder of his paper is organized as follows: Secion provides he available lieraure of he opic. Secion 3 gives all he heoreical background needed o formulae he valuaion problem of insallmen opions. Secion 4 describes he mehods of he inverse Laplace ransform used for he numerical valuaion. Secion 5 presens he compuaional resuls and finally Secion 6 gives he main conclusions. 5

7 Lieraure Review There has no been much research on insallmen opions, mainly because i is an recen opic. The firs publicaion was an aricle by Karseny and Sikorav [0]. Davis e al. [9,8] derive no-arbirage bounds for he iniial premium of an insallmen opion, which are used no only o se up saic hedges bu also o compare hem wih dynamic hedging sraegies wihin a Black Scholes framework considering sochasic volailiy. Also, i concerns he European discree insallmen opions only, allowing an analogy wih compound opions, as covered by Geske [16] and Hodges and Selby [19]. Ben-Ameur e al. [3] develop a dynamic programming procedure o price American discree insallmen opions and invesigae some heoreical properies of he insallmen opion conrac regarding he geomeric Brownian moion. Griebsch e al. [18] deduce a closed form soluion o he iniial premium of a European discree insallmen opion in erms of mulidimensional cumulaive Normal disribuion funcions. When examining he limiing case of an insallmen opion wih a coninuous paymen plan, i is found o be equivalen o a porfolio consising of a European vanilla opion and an American pu on his vanilla opion wih a imedependen srike. Kimura and Kikuchi [] develop a valuaion of insallmen opions based on he Laplace ransform while Ciurlia and Roko [4] apply he mulipiece exponenial funcion (MEF) mehod o define an inegral form of he value of an American opion. However, his mehod is criically resriced since i generaes a disconinuiy in he opimal sopping and early exercise boundaries. Alobaidi e al. [1] aemp he Laplace ransformaion o solve he free boundary problem for he European case, bu he employed mehod is raher specific and no appropriae for a numerical compuaion. Kimura [1] applies he Laplace ransform approach o solve he valuaion problems of boh American and European coninuousinsallmen opions. 6

8 3 Theoreical Framework While in a convenional opion conrac he buyer pays he oal premium up-fron and acquires he righ, bu no he obligaion, o exercise he opion a mauriy (European ype) or a any ime unil mauriy (American ype), in an insallmen opion he buyer pays a lower up-fron premium, paying is remaining par in a series of insallmens or furher premium paymens, o be paid during he lifeime of he opion up o mauriy. A each insallmen dae he holder has he righ o decide if he coninues o pay for he conrac or he erminaes he paymens, in which case he opion lapses wih no furher paymens from eiher side. The opporuniy o end he conrac a any ime before mauriy urns he valuaion of hese opions ino a free boundary problem. There are wo cases of he insallmen paymens: discree and coninuous. An insallmen opion wih paymens a pre-specified daes is usually referred o as a discree-insallmen opion. The coninuous case means ha he holder pays a sream of he insallmens a a given rae per uni ime. I is like accumulaing he premium sum by a cerain coninuous rae ha will be paid by he holder in he case of exercising. Insallmen opions will appeal o invesors who are willing o pay a lile exra for he opporuniy o erminae paymens and reduce losses if heir invesmen posiion is no working ou. They have been raded acively in acual markes as hey have significan advanage over oher opions: he prevenion of losses hrough possibiliy of erminaion; he low iniial premium ha is easy o schedule in he firm s budge, ec. Typically hese opions are raded in Foreign Exchange markes beween banks and corporaes. Also, many life insurance conracs and capial invesmen projecs can be hough of as insallmen opions (cf. Dixi and Pindyck [11]). Primary invesigaions of he insallmen opions were relaed o compound opions. 7

9 In general, he compound opion is a paricular case of an insallmen opion. Alobaidi e al. [1], Davis e al. [9] and ohers have menioned ha in he case of wo discree insallmen paymens, we have a compound opion, or in oher words, an opion on an opion. Noe ha he compound opions were he iniial poin for furher sudy of insallmen opions. In fac, hese wo ypes of opions look raher similar, so i is imporan o disinguish beween hem. A compound opion is an opion on an opion and as a consequence i has wo srike prices and wo expiraion daes. In he momen of buying he compound opion, he holder pays he iniial premium and on he firs expiraion dae he can choose eiher o buy he opion or no. A his ime he compound opion urns ino a European vanilla opion which can be exercised or no on he second expiraion dae. Noe ha he iniial price of he compound opions is obviously smaller han he vanilla opion price since he premium is spli over ime. Figure.1: The lifeime of a compound opion. is he incepion dae, 0 1 is he firs expiraion dae, T is he ime of mauriy, k0 is he iniial premium, k1 is he firs srike price, K is he srike price a he ime of mauriy. For boh compound and insallmen opions we have ha heir oal premium is always higher han he price of he sandard ones. This is explained by he addiional righ o erminae he conrac wihou paying he whole sum of he premium. In 1984, he so-called sequenial compound opion (SCO) or muli-fold compound opion was inroduced by Geske and Johnson [17]. A muli-fold compound opion is simply he composiion of he European vanilla opions presening an opion on an 8

10 opion on an opion and so on. Each fold opion may be eiher call or pu. Acually a muli-fold compound opion is nohing else han he discree insallmen opion, alhough he firs published paper devoed o his ype of opions was wrien laer in 1993 by Karseny and Sikorav [0]. Figure.: The classificaion of insallmen opions (cf. Ehrhard e al. [14]). 9

11 The Black-Scholes Pricing Model We will now briefly describe he valuaion of discree insallmen opions and for he coninuous case we will definee he problem of valuaing European insallmen opions. The seup is a sandard Black Scholes framework where he price of he underlying asse evolves according o a geomeric Brownian moion. Firs consider he discree case. Le S be he price of he underlying asse following a geomeric Brownian moion described d by he sochasic differenial equaion (SDE) ds S d S dw (3.1) where r, r is he consan risk-free rae and denoes he coninuous dividend yield. Is he volailiy coefficien of he asse price and dw is a sandard Brownian moion on a risk-neural probabiliy space. Figure 3.1: The lifeime of a discree insallmen opion. As we can see in Figure 3.1, 0 is he iniial dae and k0 is he iniial premiumm equal o he iniial value of he opion, V 0. The discree insallmen opion has n insallmen daes 1,..., n. A each of hese daes, he holder has o pay he premium k1, k.., kn 1 in order o coninue he conrac. We wan o compue he iniial value of he insallmen opion o ener he conrac. We know ha he opion payoff a he ime of mauriy T is given by V max s k,0 n n n 10

12 where s S is he price of he underlying asse a T, k T n is he srike price and, as usual, n 1 for he underlying vanilla call opion and n 1 for a pu opion. A ime n 1 he opion value is given by he discoun expecaion of he value V n. Repeaing his procedure, we can define he payoff funcion of his opion. We also know ha a ime i he holder can sop paying he premiums, erminaing he conrac, or pay ki o coninue. In he case of coninuaion, he value of he opion a ime i is given by he backward recursion V i s i1 i 1 r max e V i S S,0 for 1,..., 1 i 1 s k i i i n Vn s for i n Thus, he unique arbirage-free price of he insallmen opion is 1 0 V s k e V S S s 1 0 r (3.) Using he Curnow and Dunne inegral reducion echnique o solve (3.), a closedform soluion o valuae he insallmen opion was derived (cf. Griebsch e al. [18] for deails). There are oher mehods o valuae discree insallmen opions (cf. Ben-Ameur e al. [3]) bu in comparison o hem he presened closed-form formula seems o be he mos suiable way o do ha. Consider now he coninuous case. Le S be he price of he underlying asse following a geomeric Brownian moion described by he sochasic differenial equaion (SDE) described in (3.1). The Black-Scholes iniial premium V of a coninuous insallmen opion V V, S, q (3.3) depends on he ime, he curren value of he underlying asse S, and he coninuous insallmen rae q. In ime d he premium qd has o be paid o say in he opion conrac. 11

13 Applying Iô s Lemma o (3.3) we obain he dynamics for he opion s iniial value (3.4) We now consruc a porfolio consising of one opion and an amoun of he underlying asse and is dynamics is given by V V 1 dv r S V q d S V dw S S S V S d dv ds S d. (3.5) Plugging (3.1) and (3.4) ino (3.5) we obain 1 d r V V S V V q S. d S dw S S S To urn his porfolio riskless we choose V S. Also, o avoid arbirage opporuniies he porfolio has o yield he reurn r, so we mus have V V 1 V V r V S S q S. S S S Finally, by rearranging his equaion, we obain an inhomogeneous Black-Scholes parial differenial equaion (PDE) for he iniial premium of his opion V 1 r S V V S rv. q S S (3.6) We should have q greaer han zero. If i is equal o zero hen we ge he homogeneous Black-Scholes PDE. The Call case Consider a European-syle insallmen opion wih mauriy dae T and srike price K. Le c c, S ; q denoe he value of his call opion a ime defined on he domain, S 0, T 0, D. 1

14 The payoff a he mauriy is max S K, 0 conrac a any ime 0,T T. The addiional opporuniy o end he urns he valuaion of coninuous insallmen opions ino an opimal sopping problem. This is equivalen o finding he poins S which he erminaion of he conrac is opimal. Le S and C denoe he sopping region and coninuaion region respecively. The sopping region is defined in erms of he value funcion c, S ; q for which he opimal sopping ime S S D c S q,, ; 0 saisfies c inf u, T u, S S. c Since he coninuaion region C is he complemen of S in D, we have S c S q C =, D, ; 0. The boundary ha lies beween regions S and C is called sopping boundary and is u by, for defined by S inf S 0, c, S ; q 0, 0, T. Since, ; c S q is nondecreasing in S, he sopping boundary 0, T S is a lower criical asse price below which i is convenien o erminae he conrac by sopping he paymens. In he coninuaion region C he value c, S ; q is obained by solving he inhomogeneous PDE c c 1 c r S S rc q, S S S S wih he boundary condiions c c lim c, S; q 0, lim 0, lim SS SS S S S and he erminal condiion 13

15 , ; max,0 c T S q S K. The following inegral represenaion is he value funcion of he coninuous insallmen call opion, ;, T r u, u, (3.7) c S q c S q e N d S S u du where d 1 log a, b,,,,, 1 1 a b r d a b d a b 1 z 1 v N z e dv, z and c, S c, S ;0 is he value of he European vanilla call opion T r T 1 c, S S e N d S, K, T Ke N d S, K, T. This proof is given in he work of Kimura [1]. From (3.7) we can easily see ha he price of he coninuous insallmen opion is he difference beween he corresponding European vanilla call opion and he expeced presen value of he insallmen premiums along he opimal sopping boundary. Also from (3.7) we immediaely see ha c, S ; q c, S for 0, T, which means ha he paymen of insallmen makes he iniial premium lower han he vanilla counerpar. Furhermore, he opimal sopping boundary 0, T following inegral equaion T ru u S is implicily defined by he c, S q e N d S, S, u du 0 which can be solved numerically for 0, T and Roko [4]. S, e.g., by he MEF mehod as in Ciurlia 14

16 However, in he curren work, o find he values of opions and, herefore, he opimal sopping boundaries, we consider an alernaive approach based on Laplace ransforms, which generaes he ransformed sopping boundary in a closed-form. Regarding ha we will presen he Laplace-Carson ransformaion mehod and inverse Laplace ransformaion mehods. The Pu case We proceed he same way for he valuaion of a coninuous insallmen pu opion. Is value a ime is defined by p p, S ; q For each ime 0, T on he same domaind. here exiss an upper criical asse price above which i is advanageous o erminae paymens by sopping he opion conrac. The sopping boundary S also divides he domain D ino a coninuaion region 0, T and a sopping region The value p p, S ; q C =, S 0, T 0, S S T S, 0, S,. saisfies he inhomogeneous Black-Scholes PDE in he coninuaion regionc, i.e. p p 1 p r S S rp q, S S S S subjec o he following boundary and erminal condiions p p lim p, S; q 0, lim 0, lim SS SS S 0 S S p T S q K S, ; max,0 Again, as in Kimura [1], he value funcion of he coninuous insallmen pu opion is represened by he following inegral T r u p, S ; q p, S q e N d S, Su, u du (3.8) 15

17 where p, S p, S ;0 is he value of he European vanilla pu opion T r T 1 p, S Ke N d S, K, T S e N d S, K, T. A decomposiion of he oal Premium To undersand he original idea of he decomposiion of he oal premium le us reurn o he subjec of he compound opions. Davis e al. [9] recommended an alernaive way of looking a he compound call on a call opion. Acually, he price of he underlying call o be paid a ime 0 is he amoun r1 0, i.e., he sum of he iniial premium and he discouned value of he k k k e 0 1 second premium. A he same ime, he holder has he righ o ge rid of his opion and selling i for he price k a ime 1 1. Hence, he oal premium of he compound call on a call opion can be viewed as he underlying call opion plus a pu on he call wih exercise a ime and srike price k Following he same idea, suppose ha he oal premium of he insallmen opion equals he underlying European vanilla call opion plus he righ o leave a any ime a a pre-deermined rae. 1 0,,,,,, r BS BS call k k e c T S K p S k Griebsch e al. [18] proved his idea considering he limiing case of discree insallmen opions and he risk-neural approach. They observed ha he oal premium of he coninuous insallmen call opion is he European vanilla call opion plus an American pu opion on his European call where, ;,, ; c S q K c S P S q (3.9) c q K 1 e r r T (3.10) 1 BS BS Here c, T, S, K and p, T, S, K price K mauriy a T and underlying spo price S call denoe a European vanilla call and pu on a call opion respecively, wih a srike 16

18 is he discouned sum of he premiums no o be paid if he holder decides o erminae he conrac a ime, and for he se S, T 0, T P S q e k c u S r u c, ; esssupu S max,,0, T u u of sopping imes wih values in F a.s. is he value of he American compound pu opion wih he mauriy a T wrien on he European vanilla call opion. This decomposiion will be used o obain he Greeks formulas showed laer on. The Laplace Transform Nowadays inegral ransforms are a common pracice o solve problems of he mahemaical modeling. Baeman [] was he firs o consider he Laplace ransform as a ool for solving inegral equaions. Definiion 1: (Laplace ransformaion) Assume ha f ( ) is a real valued funcion defined for all posiive in he range Then he Laplace ransform of he funcion f ( ) is defined by 0,. if he inegral 0 e f ( ) d L f ( ) e f ( ) d (3.11) 0 converges. The parameer is a complex number. Applying he Laplace ransform o he Black-Scholes PDE (wih wo variables, ime and asse price) will reduce i o an ordinary differenial equaion (ODE) wih respec o he asse price, which is a much simpler problem. In his work we use a generalizaion of he Laplace ransform called he Laplace-Carson ransform (LCT). 17

19 Definiion : (Laplace-Carson ransformaion) For he same assumpions as above, he Laplace-Carson ransform of he funcion f ( ) is defined by LC f ( ) e f ( ) d (3.1) 0 There is no essenial difference beween hese wo ransformaions, bu he principal reason why LCT is used is ha i generaes relaively simpler formulas for opion pricing problems. From Definiion i follows ha for any wo funcions f ( ) and g( ) condiions of Definiion 1 hen af ( ) bg( ) e af ( ) bg( ) d a f ( ) b g( ) 0 saisfying he LC LC LC (3.13) Lemma 1: Assuming ha f ( ) is coninuous and differeniable and f '( ) is coninuous excep a a finie number of poins in any finie inerval 0,T hen LC f '( ) LC f ( ) f (0) (3.14) The proof can be viewed on Cohen [5]. The Inverse Laplace Transform 1 Denoe by L F he inverse Laplace ransform for a funcion F L f ( ), i.e. Noe ha for funcions f ( ) and g( ) have L 1 F f ( ) ha only differ in a finie se of values of, we L f ( ) = L g( ). This means ha he inverse Laplace ransform canno be unique in he class of piecewise coninuous funcions. 18

20 Hence, for applying he Laplace ransform o our problem we need o be in he area of uniqueness. This leads us o Lerch s heorem. Theorem 1: (Lerch s heorem [5]) If here are wo funcions f ( ) and g( ) ( ) Tg( ) T f wih he same inegral ransform hen a null funcion can be defined by ( ) f ( ) g( ) 0 so ha he inegral 0 ( ) d vanishes for all 0. 0 Now, if we have an ODE soluion for he corresponding ransformed PDE, and an exac 1 formula for deermining L F we can easily produce a coninuous soluion for our PDE. In general, here is an analyical formula for he Laplace ransform inversion, called he Bromwich inegral. Theorem : (The Inversion heorem [5]) Le f ( ) have a coninuous derivaive and le f ( ) ae where and a are posiive consans. Define F L f ( ) for Re. Then where c. ci 1 u f ( ) e F( u) du i ci Noe ha his inegral is oo complex for compuing direcly, hus various numerical mehods are applied for compuing he funcion values from is Laplace ransform. The analyical expressions for ransformed variables Transformed opion values The nex sep is o apply he Laplace ransform on our inhomogeneous Black-Scholes PDE described in (3.6) and solve i in he ransformed variables. As usual, we firs consider he call case. 19

21 For convenience we rever he direcion of ime by changing he variable T and defining c, S; q ct, S; q c, S; q and S S S for 0. From Definiion T he Laplace-Carson ransform (LCT) of hese variables is as follows c S q c S q e c S q d, ; LC, ;, ; LC S S e S d Applying he LCT o (3.6) we will ge an inhomogeneous ODE of he same order. Bu o solve an ODE of his ype i is necessary o solve firs he corresponding homogeneous ODE, so i makes sense o consider he ransformaion of he original Black-Scholes PDE for he vanilla opions, where q is absen. 0 0 Lemma 1: Le c, S LC c, S backward running process. Then be he LCT of he associaed vanilla call for he c, S 1 S, S K S K S, S K r (3.16) where for i 1, we have i S 1 3 and he parameers and depend on and are wo real K r S i r K 1 0 roos of he quadraic equaion 1 1 r r 0 Proof: The original proof can be found in [1]. i Afer changing variables, he Black-Scholes PDE reads c c 1 c r S S rc 0, S 0, S S (3.17) 0

22 supplied wih he boundary condiions dc lim c, S 0, lim, S ds S 0 and he iniial condiion 0, max,0 c S S K S K. Afer ransforming equaion (3.17) and using (3.13) and (3.14), we obain a corresponding ODE 1 d c dc S r S rc S K 0, S 0, ds ds (3.18) wih he boundary condiions lim c, S 0, lim S0 S dc ds Equaion (3.18) is a linear homogeneous ODE of Euler-ype and can be reduced o a linear ODE wih consan coefficiens by subsiuing (3.16). S e y and solved easily yielding Theorem 3: [1] If S S, 1, ;, r 1 S c S q c S q S q r (3.19) oherwise c, S; q 0. The sopping boundary S S is given by S 1 Proof: For S 0, S he resul is obvious. In a similar way in he proof of Lemma 1, we obain he ODE for c, S; q as K q 1 1 K 1 d c dc S r S rc q S K, S S ds ds (3.0) wih he boundary condiions dc dc ds ds lim c, S; q 0, lim 0, lim SS S S S 1

23 I is sraighforward ha he soluion for (3.0) is a sum of soluions for he homogeneous equaion and a paricular soluion of he inhomogeneous equaion. I can be easily seen ha he second par of he formula for c, S; q he corresponding inhomogeneous ODE. For a deailed proof, cf. [1]. is a soluion for The same approach can be used o compue he soluion for he pu case. The following heorem formulaes he resul. Theorem 4: [1] If S S, and p, S; q 0 oherwise, where 1 q S q, ;, r 1 S p S q p S r (3.1) The sopping boundary is given by p, S S S K 1 S, S K r S, S K 1 1 Transformed Greeks Recall he decomposiion of he oal premium of he insallmen opion. Kimura [1] showed ha his decomposiion of he opion in a vanilla call opion and an American compound opion is very valuable when rying o approximae he Greeks of he insallmen opions. Using (3.9) and he inegral represenaion (3.7) we obain K q T r u c 1 K P, S ; q q e N d S, S u, u du. K For coninuous insallmen pu opions, Kimura proved heorems similar o he heorems used in he call case, via he same PDE/LCT approach. However, he properies of he sopping boundary for he pu case are subly differen from he call case. See [1] for deails.

24 Subsiuing N z 1 N z and using (3.10) we obain an inegral represenaion for he American compound opion T r u c q r P, S ; q q e N d S, S u, u du. Regarding he lineariy of he LCT and using i on (3.9) we ge for ime-reversed values which is equivalen o From Theorem 3 we see ha Here, he inverse LCT of LC c, S ; q LC K LC c, S ; q LC P, S ; q c, ;,, ; c S q K c S P S q. c 1 c, ; r 1 S P S q K LC q S q r he erm can be compued analyically T 1 q ru q rt 1 q e du e K r r hus he ransformed value of an American pu on a call is 0 q 1 S c, S; q r 1 S P (3.) Hence, he Greeks of c, S ; q, i.e. dela, gamma and hea, can be expressed by Greeks of he vanilla call and Greeks of he American pu on a vanilla call wih floaing srike price K. where e d, 1 S, K, c S d, 1 S, K, c S e S c 1 LC c, S; q c, S P S c c 1, ;, LC c S q c S P S c c r 1, ;, qe + LC c S q c S P Se c S r d1 S, K,, Se d1 S, K, rke d S, K, c 3

25 Using (3.) we find explici formulas for he ransformed Greeks of American compound opions. Pc Pc q 1 1 S LC P c S S r 1 S S P 1 1 c Pc q 1 S LC 0 P c S S r 1 S S Using he same argumens for he pu case we ge he inegral represenaion T r u p u Pc q 1 S LC Pc S q P c S q Pc S q P c r 1 S S P, S ; q q e N d S,, u du 0, ; 0, ;, ; 0 and is LCT given by Once again we have where 1 q S p, S; q r 1 S P p 1 LC, ;, S p p 1, ;, LC S p p r 1, ;, qe + LC p p S q p S P p S q p S P p S q p S P, e d1 S, K, p S d, 1 S, K, p S e S Se p S r, d1 S, K, Se d1 S, K, rke d S, K, and correspondingly P P q S 1 p p 1 1 LC P p S S r 1 S S 1 P p Pp q S LC 0 P p S S r 1 S S 1 p LC Pp S q Pp S q Pp S q P p r 1 S 0 P q S, ; 0, ;, ; 0 4

26 4 Numerical Mehodology 4.1 Prior Mehods The Euler-summaion mehod The Fourier-Series mehod for numerically invering Laplace ransforms was firs proposed by Dubner and Abae [1]. An acceleraion echnique ha has proven o be effecive in our conex is Euler summaion, proposed by Simon e al. [6]. This mehod is based on he Bromwich conour inversion inegral, which can be expressed as he inegral of a real-valued funcion of a real variable by choosing a specific conour. The mehod is described as follows (cf. C. O Cinneide [4] for deails) in dependence of he parameers A, l, mand n (e.g. A 19, l 1, m 11 and n 38 ) 1. Compue a, k 0,1,, m n : k A l A l e e A ak bk, k 1, a0 b0 F l l l where l A ij ik ij l bk Re F e, k 0 j1 l l s 1 a, j 0,, n m.. Compue j j k 0 3. Approximae f ( ) using The Gaver-Sehfes mehod k k Anoher way o represen he inverse ransform is given in he following resul of Pos and Widder [5,8]. m m f ( ) k0 n m snk Theorem 5: (The Pos-Widder heorem [5]) 5

27 Le f ( ) be a coninuous funcion on he inerval 0, of exponenial order. If he inegral F e f ( ) d converges for every 0 hen n1 ( 1) n ( n) n f ( ) lim F n n! (4.1) The advanage of formula (4.1) lies in he fac ha f is expressed in erms of he value of F and is derivaives on he real axis. However i has he big disadvanage of he convergence o f ( ) being very slow, alhough i can be speeded up using appropriae exrapolaion echniques. Tha is how a group of numerical Laplace ransform inversion mehods called akin o Pos-Widder formula was developed. Davies and Marin [7] give an accoun of he mehods hey esed in heir survey and comparison of mehods for Laplace ransform inversion. In heir lising of mehods which compue a sample 3 hey give he formula 0 In n, u f ( ) du where he funcions, u n converge o he dela funcion as n increases o infinie and hus lim I f ( ). n n The Pos-Widder formula may be hough of as being obained from he funcion n nu nu e n, u n 1! Using similar argumens, Gaver [15] has suggesed he use of he funcions n! n au n, u a1 e e n! n 1! nau where a ln which yields a similar resul o (4.1) bu involves he nh finie difference n F( na), namely n! n f ( ) lim In ( ) lim a F( na) n n n! n 1! As wih he Pos-Widder formula, he convergence of In( ) o f ( ) is slow. 3 Davies and Marin [7] divide up he mehods hey invesigaed ino 6 groups. Mehods which compue a sample are mehods ha were available a ha ime. Pos-Widder wih n 1 and he Gaver-Sehfes mehod are included in his group. 6

28 However, Gaver has showed ha I ( ) f ( ) n can be expanded as an asympoic expansion in powers of 1 n and Sehfes improved he Gaver s mehod [7] giving an algorihm based on approximaing f ( ) by he sum a N n1 K F( na) where n K n ( 1) min n, N N n N k k n1 ( k)!! 1!!! N k k n k k n This algorihm is called he Gaver-Sehfes algorihm. The Kryzhniy mehod In his work we also consider he mehod of inverse Laplace ransform suggesed by Kryzhniy [3], who claims ha he algorihms based on he choice of differen dela convergen sequences can be compared by analyzing he focusing 4 abiliies of he numerical and he exac inverse ransforms of e. Primarily, he produced a soluion in erms of he Mellin ransform of equaion (3.11), which can be invered afer muliplying i by a suiable chosen facor R u sin Rln u 1 u 1 The resul can be expressed by he nex wo equaions where u sin Rln u fr ( ) f ( u) du u 1 u 1 0 f ( ) F( u) R, u du R R as 0 ( is a regularizaion parameer). 0 Here, insead of sopping he compuaion afer some number N we have a value of. some funcion R in poin. Afer some generalizaion we have R, u an arbirary coninuous funcion wih 1 0. sin Rln L where u is For deailed informaion cf. Kryzhniy [3]. 4 Focusing abiliies means how he peakness of a dela approximaing funcion is kep while increasing. 7

29 4. The De Hoog algorihm De Hoog e al. [10] proposed an improved procedure for numerical inversion of Laplace ransform based on acceleraing he convergence of he Fourier series obained from he inversion inegral using rapezoidal rule. The iniial algorihm was proposed by Crump [6] bu was significanly improved by De Hoog e al. [10]. Given a complex-valued ransform F L f ( ), he rapezoidal rule gives he approximaion o he inverse ransform e 1 ik f Re F F e T k 1 T e k Re ak z T k 1 ik T wih 1 ik a0 F, ak F, k 1,,, T and i e T z. This is he real par of he sum of a complex power series in z i e T. The algorihm acceleraes he convergence of he parial sums of his power series by using he epsilon algorihm o compue he corresponding diagonal Pade approximans. The algorihm aemps o choose he order of he Pade approximan o obain he specified relaive accuracy while no exceeding he maximum number of funcion evaluaions allowed. The parameer is an esimae for he maximum of he real pars of he singulariies of F and an incorrec choice of may give false convergence, even in cases where he correc value of is unknown, he algorihm will aemp o esimae an accepable value. In our work we use a sligh modificaion o he De Hoog mehod ha consiss of spliing he ime vecor in segmens of equal magniude which are invered individually, giving a beer overall accuracy. 8

30 5 Compuaional Resuls As we saw in previous secions, he LCT s are useful for numerical compuaion of he values of he opion prices and sopping boundaries by numerical inversion. Since he showed LCT s are so complicaed ha hey canno be analyically invered, numerical inversion is he bes measure we can have for analyzing he real-ime behaviors. A se of MaLab funcions were developed for valuing coninuous insallmen opions and is Greeks via he inverse Laplace ransform mehods. Kimura [1] uses wo algorihms for he Laplace ransform: he Euler summaion and he Gaver-Sehfes mehods. Beyond hese, Ehrhard e al. [14] use he Kryzhniy mehod. In his paper we presen one more algorihm known as he De Hoog algorihm (see secion 4.). We will use hese mehods for invering he LCT s of he sopping boundaries given in Theorems 4 and 5. Therefore our algorihm for valuing he coninuous insallmen opion consiss of he following numerical procedures: i is finding he value of he sopping boundary, hen compue numerically he inegral in (3.7) or (3.8) and finally compue he opion value by using he value of his inegral and he associaed vanilla opion. Our numerical inegraion is made via he MaLab rouine quad, which uses he Simpson formula for he inegraion and deermines inegraion nodes auomaically, evaluaing hen he sopping boundary in each node. In Figure 5.1 we can see some opimal sopping boundaries and heir sensiive o he coninuous insallmen rae q. We firs noice ha he boundary value is an increasing (decreasing) funcion of q for he call (pu) case. This can be easily seen by he inequaliies ds dq 0 or ds dq 0 which are necessary condiions. In addiion, we see from hese figures ha he opimal sopping boundaries are no always monoonic funcions of. 9

31 Figure 5.1: Sopping boundaries of coninuous insallmen opions ( 0, T 1, K 100, 0.03, r 0.0, 0.3 ) Also, in Figure 5. we see he opimal sopping boundaries in dependence of he dividend yield, from whichh we can see ha now he boundary value is an increasing funcion of in boh cases. Figure 5.: Sopping boundaries of coninuous insallmen opions ( 0, T 1, K 100, q 10, r 0.0, 0. ) Noe ha in hese figures he sopping boundary value a mauriy T 1 agrees wih he srike price K 100 as proved in Theorems 4 and 6 in Kimura [1]. The values of he insallmen opion compued by numerical Laplace inversion can be viewed in ables 5.1 and 5. for he call and he pu case respecively. The values used for q and S are he same ha hose used by Kimura [1] and Ehrhard e al. [14] so we can compare he resuls and conclude abou he new mehod used. 30

32 In Kimura [1] we can see he resuls from Euler and Gaver-Sehfes mehod produced by he auhor, which are a lile differen han ours. Paricularly he resuls of he Euler mehod differ significanly from hose produced by Gaver-Sehfes algorihm, which caused he auhor o misrus he las one. This happened because he auhor followed wo differen procedures for each mehod. For wha concerns he Euler mehod, he auhor did no apply i o he inversion of he opion values c, S; q and p, S; q. Therefore he used he same procedure han ours. Bu for he Gaver-Sehfes mehod, he auhor applied i direcly o he opion values, hence geing greaer differences beween mehods. On Ehrhard e al. [14] we can also see he resuls of hree of he mehods ha we are considering, bu once again hese values are slighly differen han ours, perhaps because of a lile misleading in heir MaLab code. As for our resuls i can be seen from he ables ha all he four mehods produce pracically equal values, where he absolue difference beween he values is less han q S Euler-based Gaver-Sehfes Kryzhniy De Hoog ,7071 3,7071 3,7071 3, ,3994 8,3994 8,3994 8, , , , , ,80,80,80, ,6385 6,6385 6,6385 6, ,9687 1,9687 1,9687 1, ,6755 0,6755 0,6755 0, ,746 4,746 4,746 4, ,533 10,533 10,533 10,533 Table 5.1: Values of coninuous insallmen call opions ( 0, T 1, K 100, 0.05, r 0.03, 0. ) 31

33 q S Euler-based Gaver-Sehfes Kryzhniy De Hoog , , , , , , , , ,5703 5,5703 5,5703 5, , , , , ,483 8,483 8,483 8, ,8486 3,8486 3,8486 3, ,153 1,153 1,153 1, ,7647 5,7647 5,7647 5, ,7011 1,7011 1,7011 1,7011 Table 5.: Values of coninuous insallmen pu opions ( 0, T 1, K 100, 0.05, r 0.03, 0. ) Figure 5.3 presens a 3D plo of he call and he pu values in dependence of ime and he asse price S. Figure 5.3: The opion value for he call and he pu ( T 1, K 100, 0.05, r 0.03, 0., q 10 ) In order o es he performance of he numerical ransform inversion for he LCT of he Greeks, we compued he values of he hedged parameers (dela), (gamma) and (hea). Figures 5.4 and 5.5 plo and respecively as funcions of S varying he parameer q and Figure 5.6 plos as a funcion of S varying he parameer insead of q. Boh figures plo also he associaed vanilla opions drawn in a dashed line, as well as he sopping boundaries represened by markers. Unlike he conclusions of Kimura who found ha he Gaver-Sehfes mehod performed very poorly if he posiion is ou-of-he-money, we concluded ha boh mehods behave well in he whole region where he sopping boundary is no reached. 3

34 Figure 5.4: The greek value for he call and he pu case in dependence of q ( 0, T 1, K 100, 0.04, r 0.0, 0. ) Figure 5.5: The greek value for he call and he pu case in dependence of q ( 0, T 1, K 100, 0.04, r 0.0, 0. ) Figure 5.6: The greek value for he call and he pu case in dependence of ( 0, T 1, K 100, 0.04, r 0.0, 0. ) 33

35 A his poin and looking a Tables 5.1 and 5. we sill do no have useful informaion o compare he efficiency of he used mehods. The firs hing ha migh be ineresing o do is o measure he performance of each mehod when compuing he value of one insallmen opion. In Table 5.3 we can find hese resuls. Euler-based Gaver-Sehfes Kryzhniy De Hoog ime (sec),6473 0,449 1,8 0,065 Table 5.3: The average of he ime i akes o compue he value of one insallmen opion per mehod. As we can see he De Hoog mehod seems o be he bes candidae o valuae hese opions numerically so far. Anoher way o compare algorihms used for he inverse Laplace ransform was proposed by Kryzhniy [3]. The mehod is based on invering he funcion F ( ) e whose analyical inverse ransform is he dela funcion on x. The idea is ha finding a mehod ha uses more focusing approximaion o he dela funcion is an eviden way for improving he provided resuls. If he algorihm gives us a good approximaion of he dela funcion while invering e and preserves is peakness while increasing, i will give good approximaions of oher funcions oo. Figure 5.7 presens he resuls of reconsrucing he dela funcion by each mehod. 34

36 Figure 5.7: The reconsrucion of he dela funcion by he various algorihms. As we can see, boh Euler-summaion and De Hoog algorihms give much beer resuls while showing he peaked values. Bu we canno ignore he fas oscillaions of he curves obained by boh mehods, especially by he Euler-summaion algorihm. Trying o approximae a damped oscillaing funcion we can see in Figure 5.8 ha again neiher he Gaver-Sehfes nor he Kryzhniy mehods can compee wih he 35

37 Euler-summaion or he De Hoog algorihms, which values are maching wih he exac soluion. Figure 5.8: The reconsrucion of he damped oscillaing funcion by each mehod. In he curren work, and once we are reconsrucing non oscillaing funcions, boh mehods show good resuls. However, as for compuaional coss, he De Hoog algorihm sands ou from any of he oher mehods. 36

38 6 Conclusions We can spli ou he sudy of insallmen opions in wo cases: he discree and he coninuous one. Griebsch e al. [18] derived a closed form soluion for insallmen call and pu opions in he Black-Scholes model, proving he equivalence of he limiing case of a coninuous insallmen plan. In his work we consider he coninuous case and jus exploi he European ype, facing he sopping boundary problem. We use he LCT of he sopping boundaries, opion values and some hedging parameers of he coninuous insallmen opions, as i was done by Kimura [1]. The Laplace ransform is a powerful mehod for enabling solving differenial equaions in science. However, someimes i leads us o soluions in he Laplace domain ha are no readily inverible o he real domain by analyical means. Numerical inversion mehods are hen used o conver he obained soluion from he Laplace domain ino he real domain. Four inversion mehods were evaluaed in his paper. Alhough here is no reason o misrus any of he mehods, i seems ha hose ones based on he fas fourier ransform (FFT), like he Euler-summaion and he De Hoog algorihms, are he mos powerful. These mehods require complex arihmeic bu have benefis such as handling a broader class of ime behaviors, sill being simple o implemen and only uilizing double precision complex daa ypes. The Gaver-Sehfes and he Kryzhniy algorihms lead o accurae resuls for many problems. However, hese mehods fail o predic funcions such as hose wih an oscillaory response. The resuls obained for opions and for he comparison of he mehods were presened on a previous secion. Relying on compuaional coss we would sugges he use of he De Hoog algorihm o valuae coninuous insallmen opions, since i is he fases one of he presened mehods. 37

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