Pricing Vulnerable American Options. April 16, Peter Klein. and. Jun (James) Yang. Simon Fraser University. Burnaby, B.C. V5A 1S6.

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1 Pricing ulnerable American Opions April 16, 2007 Peer Klein and Jun (James) Yang imon Fraser Universiy Burnaby, B.C. 5A 16 (604)

2 Pricing ulnerable American Opions Absrac We exend he models of Johnson and ulz (1987), Klein (1996) and Klein and Inglis (2001) o price vulnerable American opions. Our model incorporaes a defaul boundary a he ime of mauriy as in Klein and Inglis and a defaul barrier before mauriy which is variable and linked o he payoff on he opion. We compare vulnerable American opions wih non-vulnerable American opions and find some ineresing properies: for example, an increase in he volailiy of he underlying asse price has a mixed effec on he value of a vulnerable American pu. We also find ha he probabiliy of early exercise for vulnerable American opions is comparaively higher han for non-vulnerable American opions; furher, he price of he underlying asse a which early exercise is opimal is sensiive o such facors as he degree of credi risk, he correlaion beween he underlying asse and he asses of he opion wrier, and he volailiy of he asses of he opion wrier. These resuls imply he independence assumpion in he special case of Hull and Whie (1995) is inappropriae for vulnerable American opions. JEL classificaion: G13 Key words: ulnerable American opions; Credi risk; Derivaive securiies; Defaul barrier; Defaul boundary; Pricing; Pu opions. 1

3 1. Inroducion I is well-known ha he value of an over-he-couner opion is affeced by he credi risk of is wrier. Opions ha are vulnerable o credi risk end o have lower values han comparable non-vulnerable opions since invesors require higher expeced payoffs o compensae for he credi risk hey bear. This paper exends he vulnerable opion pricing models of Johnson and ulz (1997), Klein (1996) and Klein and Inglis (2001) o price vulnerable American opions. The model incorporaes a defaul boundary a he ime of expiry which, as in Klein and Inglis (2001), can be divided ino wo componens: a fixed componen sanding for he liabiliies of he opion wrier oher han he opion, and a variable componen represening he poenial payoff of he opion. This paper also incorporaes a defaul barrier ha is applicable a any ime prior o expiry. As in Longsaff and chwarz (1995) financial disress occurs before expiry if he oal value of he asses of he firm drops below a hreshold value. This hreshold also depends on he payoff on he opion. This paper provides numerical examples o compare vulnerable American opions wih non-vulnerable American opions and o illusrae some ineresing findings. For example, an increase in he volailiy of he underlying asse has a mixed effec on he value of he vulnerable American pu; his resul is in conras o he always posiive effec for non-vulnerable American opions. We also find ha he probabiliy of early exercise for vulnerable American opions is 2

4 comparaively higher han for non-vulnerable American opions. Furher, unlike for non-vulnerable American opions, he price of he underlying asse a which immediae exercise occurs is dependen on he degree of credi risk, he correlaion beween he underlying asse and he asses of he opion wrier, and he volailiy of he asses of he opion wrier. A comparison wih he resuls of Hull and Whie (1995) for vulnerable American call opions is also presened. 2. Lieraure Review While he impac of credi risk on he value of deb has been well sudied since he early days of modern opion pricing heory (for example, see Black and choles (1973), Meron (1974), Black and Cox (1976), himko e al. (1993) and Longsaff and chwarz (1995)), i was no unil much laer ha he effec of credi risk on he value of opions was firs analyzed by Johnson and ulz (1987). They develop a model which values European opions subjec o defaul risk a he mauriy of he opion. Under he assumpion ha he opion is he only liabiliy of he opion wrier and he opion holder receives all he asses of he opion wrier if defaul occurs, hey demonsrae ha he effec of credi risk can be significan. They also analyze various addiional consideraions for American opions, such as he effec on he opimaliy of early exercise. They poin ou ha, heoreically, a vulnerable American call on a non-dividend-paying sock may be exercised early while a non-vulnerable American call on non-dividend-paying sock will no. Bu hey do no provide numerical examples o demonsrae his poin and hey do no exend his analysis o vulnerable American pu opions. 3

5 Compared wih Johnson and ulz (1987), Hull and Whie (1995) assume ha in addiion o he opion, oher equal ranking claims on he asses of he opion wrier exis when he opion wrier defauls. They examine he effec of credi risk on boh American and European opions. Defaul can happen a any ime before he opion expires. The defaul boundary is fixed and defaul is assumed o occur whenever he value of he asses of he opion wrier falls below ha fixed level. In he even of defaul, he opion holder receives a proporion of he nominal amoun of he claim. Their numerical examples show ha he impac of credi risk on American opions is much less han for European opions. They give numerical examples of vulnerable foreign currency opions issued by a bank and find ha as he correlaion beween he foreign currency exchange rae and he asses of he bank decreases, he difference beween he percenage price reducion from defaul risk for American and European call opions increases, while he difference for vulnerable pus decreases. The early exercise of vulnerable opions is implici in heir pricing resuls bu is no a principal focus of heir paper. They also provide resuls in he special case when he sae variables deermining he price of he opion are assumed o be independen of he sae variables deermining he occurrence of defaul and he payou rae. Jarrow and Turnbull (1995) consider boh he effec on opion prices of credi risk of he opion wrier and also of he asse underlying he opion. They apply a foreign currency analogy approach o price opions wih credi risk. The payou raio is no linked o he value of he asses of he counerpary, as in Hull and Whie (1995). 4

6 Klein (1996) exends he model of Johnson and ulz (1987) by allowing he opion wrier o have liabiliies oher han opion and by allowing he recovery rae o depend on he value of he asses of he opion wrier. These assumpions are more appropriae when applying he model o many business siuaions. He derives a closed-form soluion for European opions and shows ha he model is easy o calibrae. Klein and Inglis (2001) furher exend he lieraure by allowing boh he claim of he opion holder and he oher liabiliies of he opion wrier o rigger defaul a he mauriy of he opion. They also link he payoff received by he opion holder in he even of defaul o he value of he opion wrier s asses. The papers discussed above mainly deal wih vulnerable European opions, especially call opions. The aim of our paper is o focus on vulnerable American opions, and specifically, vulnerable American pu opions. Our model allows he defaul barrier before mauriy o change wih he underlying asse price. I reains he feaure of Klein (1996) and Klein and Inglis (2001) ha links he payou rae o he value of he opion wrier s asses. Alhough we are unable o obain a closed-form soluion, we use he numerical mehod oulined in Hull and Whie (1990) o value some example opions and o show how hese values are differen from non-vulnerable American opions ha are oherwise idenical. The srucure of he paper is as follows. ecion 3 briefly discusses he assumpions of our model. The parial differenial equaions and boundary condiions are given in ecion 4. In ecion 5, 5

7 we provide numerical examples o invesigae he properies of vulnerable American pu opions. ecion 6 provides a brief conclusion. 3. The assumpions of he model The assumpions of our model are similar o hose in Klein and Inglis (2001) bu modified o apply o American insead of European opions, as oulined below. Assumpion 1. Le denoe he marke value of he asses of he opion wrier. The dynamics of are given by he diffusion process d = μ d + σ dz where μ is he insananeous expeced rae of reurn on he asses of he opion wrier, σ is he insananeous sandard deviaion of he reurn (assumed o be consan) on he asses of he opion wrier and Z is a sandard Wiener process. Assumpion 2. Le represen he marke value of he asse underlying he opion. The dynamics of are given by he diffusion process d = μ d + σ dz where μ is he insananeous expeced rae of reurn on he asse underlying he opion, σ is he insananeous sandard deviaion of he reurn (assumed o be consan) on he asse underlying he opion and Z is a sandard Wiener process. 6

8 Assumpion 3. The insananeous correlaion beween d Z and d Z is ρ. Assumpion 4. The markes are perfec and fricionless, i.e., here are no axes ransacion coss or informaion asymmeries. ecuriies can be raded in coninuous ime. Assumpion 5. In an even of defaul, he nominal claim of he opion holder is he inrinsic value of he opion. Assumpion 6. A he mauriy of he opion, =T, defaul occurs only if he value of he opion wrier s asses a ime T, T, is less han he hreshold value D + max(x- T,0), where max(x- T,0) denoes he nominal claim of he pu opion holder, D represens he defaul level, which could be he value of he oher liabiliies of he opion wrier, X is he srike price of he pu opion and T represens he price of he underlying asse a he mauriy of he opion. Assumpion 7. Before he mauriy of he opion, <T, defaul occurs only if he value of he opion wrier s asses a ime,, is less han he hreshold value D + max(x-,0), where max(x-,0) denoes he nominal claim of he pu opion holder and he underlying asse prior o he mauriy of he opion for <T. represens he price of 7

9 Assumpion 8. A a given ime of defaul,, he opion holder receives (1- w ) imes he nominal claim, where w represens he percenage wrie-down of he nominal claim a ime. Assumpion 9. When defaul happens, he value of he opion wrier s asses is subjec o deadweigh bankrupcy/reorganizaion coss of α which is expressed as a percenage of he value of he asses of he opion wrier. Assumpion 10. The percenage wrie-down on he nominal claim of he opion holder upon defaul is w =1- (1-α) /( D + P ) where he raio n /( D + P ) represens he value of he n opion wrier s asses available o pay he claim expressed as a proporion of oal claims a ime, and n P sands for he nominal claim. Assumpion 11. For simpliciy, assume he asse underlying he pu opion is a sock ha does no pay dividends, i.e. dividend q=0. The erm srucure is fla and he risk free rae is r. One of he disinc feaures of our model is assumpion 7. I allows defaul o occur prior o mauriy and also allows he early defaul barrier o be variable and linked o he price of he asse underlying he opion. This implies ha early exercise of he vulnerable American pu opion may happen no only because he opion is American syle, bu also because early defaul can indeed occur. Because of his assumpion, some ineresing properies can be derived, which will be discussed in he following secions of his paper. 8

10 9 4. The parial differenial equaions of he model Using he no-arbirage approach, he price of a vulnerable pu opion P mus saisfy he parial differenial equaion given by Johnson and ulz (1987): P rp P P q r P P r P s v s v = ) ( σ ρσ σ σ Noe Johnson and ulz assume q=0 in he above equaion. The boundary condiions implied in he assumpions can be expressed as follows: (1) A he mauriy of he pu opion, =T T T T T X D X X + >, T T T T T T X D X X X D + +, ) ( ) 1 ( α 0 oherwise (2) Prior o he mauriy of he pu opion <T 0 ), ( lim = P If defaul does no happen prior o mauriy f f f X D X X P + > =, ) ( ) ( ) ), ( ( 1 ) ), ( ( = f P f X D X + >, ) (

11 where f () is he free boundary a. (3) Prior o he mauriy of he pu opion <T If defaul happens prior o mauriy ( 1 α ) D P X n n, + P D + X where n P is he value of he pu opion assuming no defaul a ime. The boundary condiion se (1) characerizes he payoffs of he pu opion a mauriy. The boundary condiion se (2) is similar o he boundary condiions of he non-vulnerable American pu opion. The boundary condiion se (3) expresses he amoun which he pu opion holder will receive if he opion wrier s asses hi he variable defaul boundary mauriy. D + X prior o The above parial differenial equaion given he paricular boundary condiions does no have an analyic soluion and mus be solved numerically. We employ he hree-dimensional binomial ree approach o solve he parial differenial equaion as suggesed by Hull and Whie (1990). 10

12 5. Numerical examples 5.1 American pus In his secion, we presen some numerical examples o illusrae various properies of vulnerable American pu opions and o compare hem wih oherwise idenical non-vulnerable American pu opions. In he base case 1, he parameers have been chosen o be similar o many business siuaions. The pu opion is a he money. The opion wrier is a highly leveraged firm (90% deb-asse raio). The correlaion beween he value of he opion wrier s asses and he value of he asse underlying he opion is zero. ince he value of a vulnerable opion is affeced by he credi risk of is wrier, one would expec ha vulnerable opions end o have lower values han comparable non-vulnerable opions. Figure 1 confirms his resul; i shows he relaionship beween he pu opion value and he sock price for a vulnerable American pu opion and a non-vulnerable American pu opion 2. The parameers are he base case parameers excep ha he value of is allowed o change. As he sock price decreases, he pu opion becomes more valuable. When he pu opion is deep in-he-money (i.e. <100), he value of vulnerable pu is considerably lower han ha of non-vulnerable pu as expeced; since is sufficienly small, he opion wrier s asses hi he variable defaul boundary D + X and defaul acually occurs a cerain nodes in he ree (i.e., when D + X ). However, when he pu opion is no deep in-he-money (i.e. 120<<160), defaul is less likely o 1 The exac parameers are =200, X=200, =1000, σ =0.2, 2 The exac parameers are =200, X=200, σ =0.2, T=2, r=0.05 σ v =0.2, D=900, T=2, r=0.05, ρ =0.0, α=

13 occur and he effec of credi risk is less; in his case he value of he vulnerable pu is almos he same as ha of he non-vulnerable pu. In hese examples i is someimes opimal o exercise he vulnerable pu immediaely a ime 0, as is he case for he non-vulnerable pu. An ineresing phenomenon is ha he sock price for he vulnerable pu a which exercise is immediae is differen from ha for he non-vulnerable pu. Furhermore, his criical price also depends on he degree of vulnerabiliy, he correlaion beween he underlying asse and he asses of he opion wrier, and he volailiy of he asses of he opion wrier. Our resuls are oulined in Tables 1, 2 and 3. As he degree of vulnerabiliy increases, i.e., D increases from 900 o 960, he sock price a which immediae exercise occurs increases from 165 o 183, while he non-vulnerable pu begins o be exercised immediaely when sock price is smaller han 158. This is because, as D increases, he pu opion holder becomes more concerned abou he possibiliy of defaul in he fuure, and hus has a greaer endency o exercise immediaely. imilarly, as he volailiy he sock price a which immediae exercise occurs increases. σ v increases or he correlaion ρ decreases, The dependence of he criical sock price for immediae exercise on he degree of vulnerabiliy implies ha he probabiliy of early exercise a each non-erminal sep in he ree mus also depend on he degree of vulnerabiliy. As Figure 2 and 3 show, he probabiliy of early exercise a each sep for he vulnerable and non-vulnerable pu increases wih he passage of ime owards mauriy in he ree when is small, bu when is large, he probabiliy decreases. The probabiliy for he vulnerable pu is comparaively higher. As increases, he probabiliy of 12

14 early exercise decreases. Noe a probabiliy equal o one a he firs sep means immediae exercise is opimal a ime zero. The relaionship beween he opion value and he srike price X for vulnerable American pus and non-vulnerable American pus is shown in Figure 4. The parameers are he base case parameers excep ha he value of X is allowed o change. As he sock price X increases, he pu opion will become more valuable. When he pu is deep in-he-money (e.g. X>200), he value of he vulnerable pu is considerably lower han ha of non-vulnerable pu, which means ha he opion wrier s asses hi he variable defaul boundary and defaul acually occurs a cerain nodes in he ree (i.e., when D + X ). In conras, when he pu opion is no deep in-he-money (e.g. 140<X<180), defaul will no happen ( > D + X ) and he value of vulnerable pu is almos he same as ha of non-vulnerable pu. Figure 5 depics he relaionship beween he pu opion value P and he value of he opion wrier s asses for vulnerable American pu opions and non-vulnerable American pu opions. The parameers are he base case parameers excep ha he value of is allowed o change. The higher he iniial value of, he less likely he value of will hi he variable defaul boundary in he fuure. Thus if he value of large enough (i.e., > D + X is always rue) he value of vulnerable pu should be equal o ha of non-vulnerable pu. Figure 6 shows he effec of he opion wrier s deb D on he value of vulnerable American pu opion. The parameers are he base case parameers excep ha he price of sock and opion 13

15 wrier s deb D are allowed o change. As he value of D increases, he possibiliy of defaul increases and he value of will be more likely o hi he variable defaul boundary. Therefore, as D increases, P decreases. A exremes, when D is sufficienly small (e.g. D <700), > D + X ends o be always rue and he price of he vulnerable American opion is almos equal o ha of he non-vulnerable American opion. When D is large, he price of he vulnerable American opion will approach zero. Figure 7 shows he effec of deadweigh coss α on he value of he vulnerable American pu opion. The parameers are he base case parameers excep ha and he deadweigh coss α are allowed o change. When he pu opion is deep in-he-money (i.e. <100), defaul acually occurs a cerain nodes in he ree (i.e., when D + X ) and, as he value of α increases, he payoff o he pu opion holder when he opion wrier defauls decreases. Therefore as α increases P decreases. A exremes, when α=1, he payoff o he pu is zero when defaul occurs, leading o he zero pu price when he pu opion is deep in-he-money. However, when he pu opion is no deep in-he-money (i.e. 120<<160), defaul is less likely o occur ( > D + X ) and he value of he pu opion is no much affeced by he magniude of he deadweigh coss α. An ineresing propery is ha he volailiy of he sock price value of he vulnerable American pu. Figure 8 shows he effec of σ may have a mixed effec on he σ on he value of he vulnerable American pu. The parameers are he base case parameers excep ha D = 960 and he sock price and he volailiy σ are allowed o change. The value of he vulnerable 14

16 American pu increases wih he value of σ if he sock price is large. However, if he pu opion is in-he-money (e.g. = 160), he possibiliy of defaul increases, leading o he negaive relaionship beween he value of he vulnerable American pu and σ. Bu if he pu opion is deep in-he-money (e.g. < 140), he value of he vulnerable American pu is no relaed o σ. imilarly, we can examine he effec of oher four facors (volailiy of he value of he asses of he opion wrier σ, he correlaion ρ, he ime o mauriy T, he risk-free ineres rae r) on he value of vulnerable American pu. Alhough we do no presen numerical examples, he effec of σ is generally he same as he effec of σ on he value of vulnerable American pu, he value of he pu is negaively relaed o ρ or r, and here is a posiive relaionship beween T and he value of he pu. 5.2 American calls o far we have focused on vulnerable American pus alhough our model can easily be exended o price American call opions. Table 4 compares he percenage reducion due o credi risk in our model for foreign currency American call values (i.e. q>0) wih he resuls of Hull and Whie (1995). In general, he percenage reducion in our model is somewha higher han ha for American opions in heir model. There are wo reasons for his. Firs, he payou raio in our model is no exacly he same as p in he Hull and Whie model. Even if he payou raio is assumed o be zero in boh models (i.e., α=1 and p=0), however, he percenage reducion is higher in our model. This is because in our model, he criical sock price a which immediae or 15

17 early exercise is opimal depends on he degree of vulnerabiliy. As shown in our Tables 1 and 2, for American pu opions, his effec is eviden even when he correlaion beween and is assumed o be zero. This implies ha he independence assumpion ha underlies he resuls in Tables 1 and 2 in Hull and Whie may no be appropriae for American opions. 6. Conclusion This paper exends he models of Johnson and ulz (1997), Klein (1996) and Klein and Inglis (2001) o price vulnerable American pu opions. Our model assumes a defaul boundary a he ime of mauriy as in Klein and Inglis, and also a defaul barrier before mauriy which is variable and linked o he underlying asse price. While no closed-form soluion o he parial differenial equaion is obained, we apply he hree-dimensional binomial ree mehod of Hull and Whie (1990) o provide numerical examples o illusrae some ineresing properies of vulnerable American opions. Comparisons beween he vulnerable American opions and non-vulnerable American opions are also presened. The resuls of he model indicae ha he properies of vulnerable American opions are more complicaed han we expeced. In addiion o five facors (T, r,, X, σ ) ha affec he value of non-vulnerable pu, he value of vulnerable pu P also depends on D,, ρ, α and σ. Early exercise possibiliies make he relaionships even more ineresing. For example, he relaionship beween he volailiy of sock price or he value of opion wrier s asses and 16

18 vulnerable American pu price can be negaive or posiive, a propery significanly differen from ha of he non-vulnerable American pu. Furhermore, he probabiliy of early exercise for vulnerable opions is apparenly higher. The price of he underlying asse a which immediae exercise happens for vulnerable opions differs from ha for non-vulnerable opions, and is affeced by he degree of credi risk, he correlaion beween he underlying asse and he asses of he opion wrier, and he volailiy of he asses of he opion wrier. 17

19 References Black, F., and J.C. Cox. aluing corporae securiies: some effecs of bond indenure provisions. Journal of Finance, 31 (1976), Black, F., and M. choles. The valuaion of opions and corporae liabiliies. Journal of Poliical Economy, 8 (1973), Hull, J. Opions, fuures, and oher derivaives, 6h ed. Prenice Hall (2006). Hull, J.C., and A. Whie. aluing derivaive securiies using he explici finie difference mehod. Journal of Financial and Quaniaive Analysis, 25 (1990), Hull, J.C., and A. Whie. The impac of defaul risk on he prices of opions and oher derivaive securiies. Journal of Banking and Finance, 19 (1995), Jarrow, R., and. Turnbull. Pricing derivaives on financial securiies subjec o credi risk. Journal of Finance, 50 (1995), JohnsonH., and R. ulz. The pricing of opions wih defaul risk. Journal of Finance, 42 (1987), Klein, P. Pricing Black-choles opions wih correlaed credi risk. Journal of Banking and Finance, 50 (1996),

20 Klein, P., and M. Inglis. Pricing vulnerable European opions when he opions payoff can increase he risk of financial disress. Journal of Banking and Finance, 25 (2001), Longsaff, F., and E. chwarz. aluing risky deb: a new approach. Journal of Finance (1995), Meron, R.C. On he pricing of corporae deb: he risk srucure of ineres raes. Journal of Finance, 29 (1974), himko e. al. The pricing of risky deb when ineres raes are sochasic. The Journal of Fixed Income, 3 (1993),

21 pu val ue vul ner abl e non- vul ner abl e Figure 1. ulnerable American pu values as a funcion of sock price : comparison beween vulnerable American pu and non-vulnerable American pu. Calculaions of vulnerable pu opion prices are based on he base case parameers. The numerical soluion is based on a hree-dimensional binomial ree using 100 seps. Calculaions of non-vulnerable pu opion prices are also based on he base case parameers and he soluion is obained via a binomial ree using 100 seps. 20

22 Table 1. The sock price a which exercise is immediae as a funcion of D. Calculaion of vulnerable pu opion prices are based on he base case parameers. The numerical soluion is based on a hree-dimensional binomial ree using 100 seps. Calculaions of non-vulnerable pu opion prices are also based on he base case parameers and he soluion is obained via a binomial ree using 100 seps. Prices in bold indicae immediae exercise is opimal. D =900 D =920 ulnerable pu opion price D =940 D =960 Non-ulnerable pu opion price

23 Table 2. The sock price a which exercise is immediae as a funcion of ρ. Calculaion of vulnerable pu opion prices are based on he base case parameers. The numerical soluion is based on a hree-dimensional binomial ree using 100 seps. Calculaions of non-vulnerable pu opion prices are also based on he base case parameers and he soluion is obained via a binomial ree using 100 seps. Prices in bold indicae immediae exercise is opimal. ulnerable pu opion price Non-ulnerable pu ρ=-0.8 ρ=-0.4 ρ=0 ρ=0.4 ρ=0.8 opion price

24 Table 3. The sock price a which exercise is immediae as a funcion of σ v. Calculaion of vulnerable pu opion prices are based on he base case parameers. The numerical soluion is based on a hree-dimensional binomial ree using 100 seps. Calculaions of non-vulnerable pu opion prices are also based on he base case parameers and he soluion is obained via a binomial ree using 100 seps. Prices in bold indicae immediae exercise is opimal. ulnerable pu opion price Non-ulnerable pu σ v =0.05 σ v =0.15 σ v =0.2 σ v =0.25 σ v =0.35 opion price

25 probabiliy of early exercise =120 =157 =184 = s ep Figure 2 The probabiliy of early exercise a each sep as a funcion of seps in he hree-dimensional binomial ree. Calculaion of vulnerable pu opion prices are based on he base case parameers. The numerical soluion is based on a hree-dimensional binomial ree using 100 seps. 24

26 pr obabi l i y of ear l y exer ci se =120 =157 =184 = s ep Figure 3 The probabiliy of early exercise a each sep as a funcion of seps in he binomial ree. Calculaions of non-vulnerable pu opion prices are based on he base parameers. The soluion is based on binomial ree using 100 seps. 25

27 pu val ue vul ner abl e non- vul ner abl e X Figure 4. ulnerable American pu values as a funcion of srike price X: comparison beween vulnerable American pu and non-vulnerable American pu. Calculaions of vulnerable pu opion prices are based on he base case parameers. The numerical soluion is based on a hree-dimensional binomial ree using 100 seps. Calculaions of non-vulnerable pu opion prices are also based on he base case parameers and he soluion is obained via a binomial ree using 100 seps. 26

28 pu val ue vul ner abl e non- vul ner abl e Figure 5. ulnerable American pu values as a funcion of he value of he opion wrier s asses : comparison beween vulnerable American pu and non-vulnerable American pu. Calculaions of vulnerable pu opion prices are based on he base case parameers. The numerical soluion is based on a hree-dimensional binomial ree using 100 seps. Calculaions of non-vulnerable pu opion prices are also based on he base case parameers and he soluion is obained via a binomial ree using 100 seps. 27

29 P D=0 D=700 D=900 D=1100 D= Figure 6. The effec of he opion wrier s deb D (i.e. D in he figure) on he value of vulnerable American pu opion. Calculaions of vulnerable pu opion prices are based on he base case parameers. The numerical soluion is based on a hree-dimensional binomial ree using 100 seps. 28

30 P al pha=0 al pha=0. 25 al pha=0. 5 al pha=0. 75 al pha= Figure 7. The effec of deadweigh coss α (i.e. alpha in he figure) on he value of vulnerable American pu opion. Calculaions of vulnerable pu opion prices are based on he base case parameers. The numerical soluion is based on a hree-dimensional binomial ree using 100 seps. 29

31 P si gma=0. 01 si gma=0. 2 si gma= Figure 8. The effec of he volailiy σ (i.e. sigma in he figure) on he value of vulnerable American pu opion. Calculaions of vulnerable pu opion prices are based on he base case parameers excep ha D =960. The numerical soluion is based on a hree-dimensional binomial ree using 100 seps. 30

32 Table 4 Percenage reducion in call values due o credi risk, compared wih Hull and Whie (1995, Tables 1 and 2). The exac parameers are =1, X=1, =100, σ =0.15, σ v =0.05, T=1, r=0.05, ρ=0.0, q=0.05. The numerical soluion is based on a hree-dimensional binomial ree using 100 seps. D ρ α α= α= HW-American p= HW-American p= α= α= HW-American p= HW-American p= α= α= HW-American p= HW-American p= α= α= HW-American p= HW-American p= α= α= HW-American p= HW-American p=

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