Opimal Early Exercise of Vulnerable American Opions March 15, 2008 This paper is preliminary and incomplee.
Opimal Early Exercise of Vulnerable American Opions Absrac We analyze he effec of credi risk on opimal early exercise policy for American opions. We find ha he criical sock price a which early exercise of a vulnerable American pu opion is opimal can be significanly higher han for an oherwise idenical non-vulnerable American pu. In conras, for American call opions he criical sock price, if i exiss, can be significanly lower. We demonsrae ha premaure early exercise because of a pending credi even can miigae bu no eliminae he effec of credi risk on he value of American opions. Through numerical examples we also demonsrae ha premaure early exercise can accoun for a significan amoun of he oal reducion due o credi risk in he value of American opions, even when here is no wrie-down due o credi loss when premaure early exercise occurs. JEL classificaion: G13 Key words: Vulnerable American opions; Credi risk; Derivaive securiies; Defaul barrier; Defaul boundary; Pricing; Pu opions. 1
1. Inroducion I is well-known ha he value of an opion can be affeced by he credi risk of is wrier. This risk is negligible if he opion is exchange raded bu is an imporan consideraion for opions in he over-he-couner marke. In general, credi risk decreases he value of an opion since he paymen ha is acually received may be less han he payoff sipulaed in he opion conrac. There is a fairly exensive lieraure on he effec of credi risk on he pricing of vulnerable European-syle opions (see, for example, Johnson and ulz (1987), Hull and Whie (1995), Jarrow and Turnbull (1995), Klein (1996) and Klein and Inglis (2001 and 1999)). This effec is ypically analyzed by reducing he payoff from an oherwise idenical non-vulnerable opion by an amoun which represens he expeced credi loss. The percenage wrie-down depends on he crediworhiness of he wrier bu can also vary wih he magniude of he payoff on he opion if he process driving financial disress is no independen of i. The analysis for American-syle opions is more complicaed because he presence of credi risk may also affec he opimaliy of early exercise. For example, if he holder believes defaul is pending, i may be opimal o exercise a vulnerable American-syle opion earlier, and a a differen price for he underlying asse, han would oherwise be he case. If such premaure early exercise is imed correcly i may allow he holder of he opion o avoid he loss ha would ypically arise when nominal claims are wrien down because of he occurrence of financial disress. There is sill an indirec effec on he value of he opion, however, because he iming of and payoff from premaure early exercise are no he same as wha would be opimal in he absence of credi risk. 2
The lieraure on he pricing of vulnerable American-syle opions has analyzed he effec of credi risk on opimal early exercise policy only very generally and in combinaion wih overall pricing resuls (see Johnson and ulz (1987), Hull and Whie (1995), Jarrow and Turnbull (1995) and more recenly Liu and Liu (2005) and Chang and Hung (2006)). These aricles recognize ha credi risk may make early exercise more likely bu hey provide lile concree guidance on how opimal early exercise policy differs from ha for non-vulnerable opions. The lieraure also generally claims ha he risk of credi loss is negligible for American-syle opions. This is based on he assumpion ha he holder of he opion is able o foresee accuraely a pending defaul and exercises accordingly. The loss in value due o premaure early exercise iself, however, has no been direcly analyzed in he lieraure and has no ye been shown o be insignifican. These issues are imporan no only for he academic communiy bu for indusry praciioners as well. Many opions raders sill se early exercise policy based on he assumpion of no credi risk, or under he popular assumpion of independence as proposed by Hull and Whie (1995). If credi risk induces significan differences in opimal early exercise policy he reasonableness of hese assumpions, as well as he exercise policy which hey imply may be in quesion. Properly recognizing he reducion in value due o premaure early exercise as an indirec effec of credi risk even when full payou of he premaure early exercise amoun occurs also has imporan accouning and risk managemen implicaions. The purpose of his paper is o analyze more closely he effec of credi risk on opimal early exercise policy. I finds he underlying asse price a which early exercise is opimal is ofen 3
very differen for a vulnerable American opion as compared o is non-vulnerable win. Furher, he price of he underlying asse a which early 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. This paper also finds he probabiliy of early exercise is higher for vulnerable American opions and ha he change in opimal early exercise policy has a significan effec on value. The ouline of his paper is as follows. ecion 2 reviews he exising lieraure on vulnerable opions. ecion 3 presens resuls based on very general assumpions. ecion 4 inroduces a srucural model under more resricive assumpions in order o allow he general resuls o be illusraed in a series of numerical examples in ecion 5. ecion 6 provides a brief conclusion. 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 expiry 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 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. They also find ha if prices are 4
coninuous he holder will always exercise before defaul and hus will never incur a loss. Bu hey do no provide numerical examples o demonsrae his poin and hey do no exend his analysis o vulnerable American pu opions. 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. 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. This assumpion, which is commonly made in indusry, implies opimal early exercise policy is unaffeced by he credi risk of he opion wrier. 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 5
of he counerpary, as in Hull and Whie (1995). Klein (1996) exends he model of Johnson and ulz (1987) by allowing he opion wrier o have liabiliies oher han he 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 expiry 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. [add brief review of Liu and Liu (2005) and Chang and Hung (2006) which are exensions of Klein and Inglis (2001), and Klein and Yang (2007) which focuses primarily on pricing of vulnerable American opions and no on he effec of premaure early exercise.] [add a brief discussion of lieraure on early exercise policy for non-vulnerable American opions; his will give inuiion o he proofs below. In general, his lieraure says o exercise when sock price saisfies f = X.] 3. General resuls We follow he approach of Johnson and ulz (1987) and firs analyze he effec of credi risk on opimal early exercise policy under very general assumpions ha do no depend on specific 6
disribuions or opion pricing and credi risk models. We will, in a laer secion, provide numerical examples based on a simple se of assumpions for he moion of he underlying asse as well as he process driving financial disress. 3.1 Assumpions and Noaion Le f () denoe he value a ime of a non-vulnerable American-syle opion wrien on underlying asse () wih srike price X and ime of expiry T. Define c as he criical price of he underlying asse a which early exercise is opimal. I is well known ha if f is an American-syle call opion and pays no dividends, early exercise is never opimal and hus his criical value does no exis; oherwise his criical value depends on he various parameers deermining he value of he opion such as erm o expiry, volailiy of he underlying asse, srike price and he riskless ineres rae. Assume an oherwise idenical opion is wrien by a counerpary ha is subjec o credi risk. We make wo very general assumpions concerning he naure of his credi risk. Assumpion A1. The probabiliy of defaul by he wrier of he opion is sricly greaer han zero. Assumpion A2. In he even of defaul, he opion holder s receip is sricly less han i would be if here was no defaul. We noe Assumpion A1 is consisen wih he definiion of vulnerable opions on page 279 in Johnson and ulz (1987), i.e., opions for which he probabiliy of defaul is no zero. 7
Assumpion A2 is needed o ensure he occurrence of defaul is indeed of some imporance for he opion holder. We assume ha all relevan variables are sufficienly well behaved ha we can use he risk neuraliy approach developed by Cox and Ross (1976) and Harrison and Pliska (1981). In order o analyze he effec of credi risk on opimal early exercise policy we need o consider he value of he vulnerable opion in wo differen ways. The firs, henceforh denoed as f, is he value of he vulnerable opion when early exercise is deermined by criical sock price c i.e., when early exercise occurs a he same ime, and only a he same ime, as would be opimal for he non-vulnerable opion. The second way in which he value of he vulnerable opion needs o be considered, henceforh denoed as f ^, assumes he opion holder may aler he iming of early exercise because of he possibiliy of defaul. Define c^ as he new criical price of he underlying asse which can be used o deermine if early exercise of his vulnerable opion is opimal. This criical value depends on he various parameers deermining he value of he opion and also on he process driving financial disress 1. We seek o deermine wheher c^ = c and if f ^ < f. For compleeness, define f ^ as he value of he non-vulnerable derivaive based on he early exercise policy ha is opimal for he vulnerable opion f ^, i.e., based on he criical asse value c^ insead of c. Analyzing f ^ will be useful when conducing an aribuion analysis of he difference in values beween f ^ and f. 1 These are he same as he hea and phi variables in Hull and Whie (1995) 8
3.2 Disribuion and model-free resuls We are now able o derive some general resuls abou he effec of credi risk on opimal early exercise policy. Proposiion 1. Credi risk affecs opimal early exercise policy; he criical sock price c^ a which early exercise is opimal for a vulnerable American-syle opion is no equal o he criical sock price c a which early exercise is opimal for is non-vulnerable win. Proof: Assumpions A1 and A2 imply f < f. Assume c = c^. This implies f = f ^. By definiion c saisfies = f X and c^ saisfies = f ^ - X = f - X. This implies f = f which conradics f < f. Thus c c^. This resul means he independence assumpion in Hull and Whie (1995) can never hold, i.e., ha he phi variables and he hea variables can never be independen. Proposiion 2. Early exercise can miigae bu canno enirely eliminae he effec of credi risk on he value of vulnerable American-syle opions. Value is reduced eiher hrough he probabiliy of credi loss or hrough he reduced payoff from premaure early exercise, or boh. Proof: Proposiion 1 implies f > f ^ and f ^ > f. ince f ^ f ^ we have f > f ^ f ^ > f. A he level of generaliy of assumpions A1 and A2 we canno prove f ^ sricly > f ^. This 9
is because we have no ye imposed any assumpions wih respec o he process driving he payou rae when financial disress occurs. If i is driven by an asse price (presumably he value of he asses of he opion wrier) and if his process (and hence payou rae) is coninuous, Johnson and ulz show i is opimal o exercise a, or jus before he defaul barrier is hi; in such case here is no loss due o wrie-down bu only due o he premaure early exercise and f ^ = f ^. In our model, if he reducion in he opion holder s receip in Assumpion A2 is arbirarily small hen we should obain he same resul as in Johnson and ulz. In pracice, his assumpion is unreasonable, hus we should find a sric inequaliy. Also noe Johnson and ulz did no idenify he effec of he change in opimal early exercise policy on he value of he vulnerable opion. Proposiion 3a. The criical sock price a which early exercise is opimal for vulnerable American pus is higher han i is for non-vulnerable American pus. Proof: Consider an American pu opion and is vulnerable win. Assume c > c^. This implies here exiss a value beween c and c^ for which X f and X < f ^, which violaes f > f ^. ince c c^ from Proposiion 1 we have c < c^. Proposiion 3b. The criical sock price a which early exercise is opimal for vulnerable American calls, if i exiss, is lower han i is for non-vulnerable American calls. Proof: [similar o ha for 3a; o be added] Noe his resul is consisen wih he Johnson and ulz resul ha for calls on non-dividend 10
paying sock early exercise may occur, which implies ha such a criical price may exis. Proposiion 4. A small reducion in value due o credi risk can have a very large effec on he criical sock price a which early exercise is opimal for vulnerable American opions. The raio beween he reducion in value and he change in criical sock price is greaer han 1/(1- Δ ) where Δ is defined in he sandard way as f / and evaluaed a c^. Proof: c saisfies = f X and c^ saisfies = f ^ - X. Thus c^ - c = f ( c ) - f ^ ( c^ ) = f ( c ) f ( c^ ) + α ( c^ ) where α ( c^ ) f ( c^ ) - f ^ ( c^ ) > 0 represens he reducion in value due o credi risk evaluaed a c^. ince Δ is sricly decreasing in, c^ - c > α ( c^ ) / (1 - Δ ) where Δ is evaluaed a = c^. Noe since f is likely o be well in he money a c^ he absolue value of Δ is likely o be closer o 1 han o 0. This implies he change in criical sock price is much greaer han he reducion in credi risk. 4. A srucural model We now inroduce a simple srucural model in order o demonsrae he significance of our general resuls on opimal early exercise policy and valuaion of vulnerable American pu opions. The assumpions of our model are similar o hose in Klein (1996) and Klein and Inglis (2001). Assumpion B1. Le V denoe he marke value of he asses of he opion wrier. The dynamics of V are given by he diffusion process dv = μ V d + σ V dz V V 11
where μ V is he insananeous expeced rae of reurn on he asses of he opion wrier, σ V is he insananeous sandard deviaion of he reurn (assumed o be consan) on he asses of he opion wrier and Z V is a sandard Wiener process. Assumpion B2. 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. Assumpion B3. The insananeous correlaion beween d Z V and d Z is ρ. Assumpion B4. 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 B5. In an even of defaul, he nominal claim of he opion holder is he value of he oherwise idenical non-vulnerable pu opion n P. Assumpion B6. A he expiry of he opion, =T, defaul occurs only if he value of he opion wrier s asses a ime T, V T, is less han he hreshold value D + P, where n T D is a fixed amoun which could correspond o he value of he oher liabiliies of he opion wrier. 12
Assumpion B7. Before he expiry of he opion, < T, defaul occurs only if he value of he opion wrier s asses a ime, V, is less han he hreshold value D + P. n Assumpion B8. 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 B9. 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 B10. The percenage wrie-down on he nominal claim of he opion holder upon defaul is w =1- (1-α) V /( D + P ) where he raio n V /( 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. Assumpion B11. The asse underlying he pu opion pays dividends a coninuous rae q. The erm srucure is fla and he risk free rae is r. 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): 1 2 σ vv 2 2 2 P 2 V + rv P V 1 2 + σ s 2 2 2 2 P P P + ( r q) + ρσ vσ sv 2 V P rp = 13
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 expiry of he pu opion, = T X X V D P T T, T > + n T V ( 1 α ) D T n P n T T X, + PT V T D + P n T 0 oherwise (2) Prior o he expiry of he pu opion, < T lim P(, ) = 0 If defaul does no happen prior o expiry P( X X V D f ( ), ) = f ( ) f ( ), > + n P P ( ( ), ) = 1 ( ) X, V D f > f + P n where f () is he free boundary a. (3) If defaul occurs prior o expiry, < T V ( 1 α ) D P X n n, + P V D + P n The boundary condiion se (1) characerizes he payoffs of he pu opion a expiry. 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 14
receive if he opion wrier s asses hi he defaul boundary D + P prior o expiry. n 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). Noe ha in conras wih vulnerable European opions, he payou rae will always equal 1 α in he limi as Δ 0. 5. Numerical examples 5.1 American pus In his secion, we presen some numerical examples o illusrae he facors which have impac on he price of he underlying asse a which early exercise is opimal, he dependence of he probabiliy of early exercise on he degree of vulnerabiliy, he aribuion of he reducion in value and various oher properies of vulnerable American pu opions. In he base case 2, he parameers have been chosen o be similar o many business siuaions. The pu opion is a he money. The opion wrier is, or has become wih he passage of ime, 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. We firs consider he effec of vulnerabiliy on he criical sock price a which early exercise is opimal. Our resuls are oulined in Tables 1, 2 and 3. As he degree of vulnerabiliy increases, 2 The exac parameers are =200, X=200, V=1000, σ =0.2, 15 σ =0.2, D=900, T=2, r=0.05, ρ =0.0, α=0.25 and q=0. v
i.e., D increases from 900 o 960, he sock price a which immediae exercise occurs increases from 164 o 176, while he non-vulnerable pu 3 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 decreases, he sock price a which immediae exercise occurs also increases. σ v increases or he correlaion ρ The dependence of he criical sock price 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. Figures 1 and 2 show he probabiliy of early exercise a each sep for he vulnerable and non-vulnerable pu, respecively. For example, when = 155, he probabiliy of early exercise is above.6 for a vulnerable opion bu is below.6 for a non-vulnerable opion in Figure 3. Noe a probabiliy equal o one a he firs sep means immediae exercise is opimal a ime zero. As discussed above, premaure early exercise can miigae bu no eliminae he effec of credi risk, and he remaining reducion in value can be aribued o wo sources: premaure early exercise wih full payou; and wrie-downs of conracual amouns because of a credi even. Table 4 illusraes hese findings. In our example, he effec of premaure early exercise, f - f ^, is always lower han he effec of wrie-down upon defaul, f ^ -f ^, and f - f ^ can also be significan. This implies he change in opimal early exercise policy can be an imporan deerminan of value. 3 The exac parameers are =200, X=200, σ =0.2, T=2, r=0.05 and q=0. 16
Figure 3 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. The parameers are he base case parameers excep ha α = 0.75 and 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 n D + P and defaul acually occurs a cerain nodes in he ree (i.e., when V D + ). However, when he pu opion is no deep in-he-money (i.e. 110 < < n P 150), defaul is less likely o occur and he effec of credi risk is less; in his case he value of he vulnerable pu is closer o ha of he non-vulnerable pu. 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 α = 0.75 and he value of X is allowed o change. As he srike price X increases, he pu opion will become more valuable. When he pu is deep in-he-money (e.g. X > 300), he value of he vulnerable pu f ^ is considerably lower han ha of non-vulnerable pu f, 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 V D + ). In conras, when he pu n P n opion is no deep in-he-money (e.g. 260 < X < 280), defaul will no happen ( V > D + ) and he value of vulnerable pu is closer o ha of non-vulnerable pu. P [add discussion of addiional ables and figures which will show effec of changes in oher parameer values on he size and aribuion of he effec of credi risk.] 17
Anoher ineresing propery is ha he volailiy of he sock price σ has a mixed effec on he value of he vulnerable American pu. Figure 8 shows he effec of σ on he value of he vulnerable American pu. The parameers are he base case parameers and he sock price and he volailiy σ are allowed o change. The value of he vulnerable 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. = 100), he possibiliy of defaul increases, leading o he negaive relaionship beween he value of he vulnerable American pu and bu f ^ does no. σ. As expeced, f also has he same propery The effecs of changes in he oher variables are more sraighforward. Figure 5 depics he relaionship beween he pu opion value 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 α = 0.75 and he value of V is allowed o change. The higher he iniial value of V, he less likely he variable defaul boundary will be hi in he fuure. In n he limi, if he value of V large enough (i.e., V > D + is always rue) he value of vulnerable pu equals ha of non-vulnerable pu. P Figure 6 shows he effec of he opion wrier s oher liabiliies D on he value of vulnerable American pu opion. The parameers are he base case parameers excep ha α = 0.75 and he price of sock and opion wrier s deb D are allowed o change. As he value of D increases, he possibiliy of defaul increases and he value of V will be more likely o hi he variable defaul boundary. Therefore, as D increases, P decreases. A exremes, when D is 18
sufficienly small (e.g. D < 700), V > D + ends o be always rue and he price of he n P 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. 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. imilarly, we can examine he effec of oher four facors (volailiy of he value of he asses of he opion wrier σ V, he correlaion ρ, he ime o expiry 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 V 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 our numerical examples have focused on vulnerable American pus alhough our model can easily be exended o price American call opions. Table 5 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 19
percenage reducion is higher in our model. This is because in our model, he criical sock price a which immediae or 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 V 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 We have analyzed he effec of credi risk on opimal early exercise policy for American opions and find ha he criical sock price a which early exercise of a vulnerable American pu opion is opimal can be significanly higher han for an oherwise idenical non-vulnerable American pu. In conras, for American call opions he criical sock price, if i exiss, can be significanly lower. We demonsrae hrough numerical examples based on a specific srucural model ha premaure early exercise because of a pending credi even can miigae bu no eliminae he effec of credi risk on he value of American opions. We also demonsrae ha premaure early exercise can accoun for a significan amoun of he oal reducion due o credi risk in he value of American opions, even when here is no wrie-down due o credi loss when premaure early exercise occurs. 20
References Black, F., and J.C. Cox. Valuing corporae securiies: some effecs of bond indenure provisions. Journal of Finance, 31 (1976), 351-367. Black, F., and M. choles. The valuaion of opions and corporae liabiliies. Journal of Poliical Economy, 8 (1973), 637-659. Cox and Ross (1976) Harrison and Pliska (1981) Hull, J. Opions, fuures, and oher derivaives, 6h ed. Prenice Hall (2006). Hull, J.C., and A. Whie. Valuing derivaive securiies using he explici finie difference mehod. Journal of Financial and Quaniaive Analysis, 25 (1990), 87-99. 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), 299-322. Jarrow, R., and. Turnbull. Pricing derivaives on financial securiies subjec o credi risk. Journal of Finance, 50 (1995), 53-85. 21
Johnson,H., and R. ulz. The pricing of opions wih defaul risk. Journal of Finance, 42 (1987), 267-280. Klein, P. Pricing Black-choles opions wih correlaed credi risk. Journal of Banking and Finance, 50 (1996), 1211-1229. 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), 993-1012. Klein, P., and M. Inglis. Valuaion of European opions subjec o financial disress and ineres rae risk. Journal of Derivaives, 6 (1999), 44-56. Klein, P. and J. Yang, Pricing vulnerable American opions, working paper, imon Fraser Universiy, (2007). Longsaff, F., and E. chwarz. Valuing risky deb: a new approach. Journal of Finance (1995), 789-819. Meron, R.C. On he pricing of corporae deb: he risk srucure of ineres raes. Journal of Finance, 29 (1974), 449-470. himko e. al. The pricing of risky deb when ineres raes are sochasic. The Journal of Fixed Income, 3 (1993), 58-65. 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 Vulnerable pu opion price D =920 D =940 D =960 Non-Vulnerable pu opion price 155 45 45 45 33.58 45 156 44 44 44 32.87 44 157 43 43 43 32.15 43 158 42 42 42 31.44 42.01 159 41 41 41 30.74 41.08 160 40 40 40 30.09 40.17 161 39 39 39 39 39.27 162 38 38 38 38 38.37 163 37 37 37 37 37.49 164 36 36 36 36 36.65 165 35.06 35.03 35 35 35.83 166 34.14 34.11 34 34 35.01 167 33.21 33.18 33 33 34.19 168 32.29 32.26 32 32 33.40 169 31.39 31.35 31 31 32.63 170 30.52 30.45 30 30 31.89 171 29.66 29.59 29 29 31.16 172 28.83 28.75 28 28 30.44 173 28.00 27.92 27 27 29.73 174 27.18 27.10 26 26 29.05 175 26.36 26.28 25 25 28.38 176 25.58 25.46 24.07 24 27.71 177 24.92 24.71 23.41 23.20 27.05 178 24.25 24.05 22.76 22.55 26.40 179 23.59 23.39 22.12 21.92 25.78 180 22.95 22.75 21.50 21.29 25.18 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. Vulnerable pu opion price ρ=-0.8 ρ=-0.4 ρ=0 ρ=0.4 ρ=0.8 Non-Vulnerable pu opion price 155 45 45 45 45 45 45 156 44 44 44 44 44 44 157 43 43 43 43 43 43 158 42 42 42 42 42 42.01 159 41 41 41 41 41.01 41.08 160 40 40 40 40 40.08 40.17 161 39 39 39 39 39.15 39.27 162 38 38 38 38.03 38.23 38.37 163 37 37 37 37.09 37.31 37.49 164 36 36 36 36.15 36.40 36.65 165 35 35 35.06 35.22 35.49 35.83 166 34 34.01 34.14 34.38 34.59 35.01 167 33 33.08 33.21 33.48 33.71 34.19 168 32 32.16 32.29 32.59 32.85 33.40 169 31.0722 31.28 31.39 31.72 32.00 32.63 170 30.2314 30.40 30.52 30.85 31.14 31.89 171 29.4048 29.53 29.66 29.99 30.29 31.16 172 28.5921 28.66 28.83 29.14 29.44 30.44 173 27.8139 27.81 28.00 28.29 28.59 29.73 174 27.0877 26.98 27.18 27.45 27.75 29.05 175 26.387 26.25 26.36 26.60 26.91 28.38 176 25.7009 25.57 25.58 25.77 26.07 27.71 177 25.0464 24.93 24.92 24.94 25.26 27.05 178 24.4321 24.50 24.25 24.19 24.93 26.40 179 23.8392 23.85 23.59 23.53 24.18 25.78 180 23.249 23.23 22.95 22.89 23.44 25.18 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. σ =0.05 v σ =0.15 v Vulnerable pu opion price σ =0.2 σ =0.25 v v σ v =0.35 Non-Vulnerable pu opion price 155 45 45 45 45 45 45 156 44 44 44 44 44 44 157 43 43 43 43 43 43 158 42 42 42 42 42 42.01 159 41.01 41 41 41 41 41.08 160 40.10 40 40 40 40 40.17 161 39.21 39 39 39 39 39.27 162 38.34 38 38 38 38 38.37 163 37.47 37.07 37 37 37 37.49 164 36.60 36.15 36 36 36 36.65 165 35.74 35.26 35.06 35.02 35 35.83 166 34.90 34.37 34.14 34.09 34 35.01 167 34.11 33.49 33.21 33.17 33 34.19 168 33.34 32.62 32.29 32.24 32 33.40 169 32.57 31.76 31.39 31.33 31 32.63 170 31.82 30.94 30.52 30.42 30.35 31.89 171 31.07 30.15 29.66 29.56 29.45 31.16 172 30.34 29.38 28.83 28.72 28.59 30.44 173 29.62 28.61 28.00 27.89 27.74 29.73 174 28.93 27.85 27.18 27.06 26.89 29.05 175 28.26 27.10 26.36 26.24 26.04 28.38 176 27.60 26.36 25.58 25.42 25.20 27.71 177 26.95 25.67 24.92 24.66 24.47 27.05 178 26.31 25.01 24.25 23.99 23.70 26.40 179 25.69 24.37 23.59 23.34 23.04 25.78 180 25.07 23.74 22.95 22.69 22.39 25.18 25
1.2 1 0.8 0.6 0.4 0.2 0 0 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 76 80 84 88 92 96 Probabiliy of early exercise =130 =155 =180 =200 sep Figure 1 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. 26
1.2 1 0.8 0.6 0.4 0.2 0 0 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 76 80 84 88 92 96 Probabiliy of early exercise =130 =155 =180 =200 sep Figure 2 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. 27
Table 4 The reducion in pu values due o credi risk. The exac parameers are =200, X=200, V=1000, σ =0.2, σ v =0.2, T=2, r=0.05 and q=0. The non-vulnerable American pu price is 15.4443. The numerical soluion is based on a hree-dimensional binomial ree using 100 seps. D α =0.5 ρ 900 920 940 960-0.8 f-f^ 2.213 3.029 3.97 4.999 f^-f^ 2.152 2.908 3.881 4.913 f^-f 0.06 0.071 0.088 0.097-0.4 f-f^ 2.728 3.69 4.066 5.464 f-f^ 0.406 0.205 0.377 0.148 f^-f^ 2.321 3.485 3.69 5.316 f^-f 0.452 0.317 0.48 0.239 0 f-f^ 3.092 4.114 4.901 5.768 f-f^ 1.104 0.393 0.4 0.296 f^-f^ 1.988 3.722 4.501 5.472 f^-f 1.09 0.808 0.646 0.549 0.4 f-f^ 4.121 4.495 5.621 6.143 f-f^ 1.22 1.25 0.398 1.153 f^-f^ 2.9 3.245 5.224 4.99 f^-f 1.306 1.462 0.901 0.836 0.8 f-f^ 4.751 5.813 6.377 7.29 f-f^ 1.432 0.925 1.596 0.138 f^-f^ 3.319 4.888 4.782 7.152 f^-f 2.144 1.39 1.07 0.325 α =1 ρ 900 920 940 960 f f-f^ 4.201-0.8 5.81 7.598 9.568 f f-f^ 0.441 0.276 0.7 0.821 f^-f^ 3.76 5.535 6.898 8.747 f f^-f 0.275 0.317 0.432 0.499-0.4 f f-f^ 4.725 6.666 7.193 10.07 f f-f^ 1.14 1.172 1.222 1.025 f^-f^ 3.584 5.495 5.971 9.043 f^-f 1.509 1.247 1.705 1.176 f 0 f-f^ 4.966 7.137 7.479 10.51 f f-f^ 1.193 1.221 3.724 1.066 f f^-f^ 3.774 5.917 3.755 9.444 f^-f 3.203 2.538 3.377 1.899 0.4 f f-f^ 6.568 7.358 10.01 10.92 f f-f^ 5.308 3.249 1.339 4.013 f^-f^ f 1.26 4.109 8.67 6.91 f^-f 4.156 4.365 2.849 2.802 0.8 f-f^ f 6.498 8.157 10.2 11.66 f-f^ 5.129 3.768 7.052 4.548 f f^-f^ 1.369 4.389 3.145 7.114 f f^-f 7.097 6.041 4.463 3.33 f 28
60 50 pu value 40 30 20 10 f^ f f f^ 0 150 160 170 180 190 200 210 220 230 240 250 pu value 160 140 120 100 80 60 40 20 0 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200 210 220 230 240 250 f^ f f f^ Figure 3. Vulnerable American pu values as a funcion of sock price : comparison beween vulnerable American pu f^, f, f^ and non-vulnerable American pu f. Calculaions of vulnerable pu opion prices are based on he base case parameers excep α = 0.75. 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. 29
60 50 pu value 40 30 20 f^ f f f^ 10 0 150 160 170 180 190 200 210 220 230 240 250 X pu value 160 140 120 100 80 60 40 20 0 150 160 170 180 190 200 210 220 230 240 250 260 270 280 290 300 310 320 330 340 350 f^ f f f^ Figure 4. Vulnerable American pu values as a funcion of srike price X: comparison beween vulnerable American pu f^, f, f^ and non-vulnerable American pu f. Calculaions of vulnerable pu opion prices are based on he base case parameers excep α = 0.75. 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. 30
pu value 18 16 14 12 10 8 6 4 2 0 500 600 700 800 900 1000 1100 1200 1300 1400 1500 V f^ f f f^ Figure 5. Vulnerable American pu values as a funcion of he value of he opion wrier s asses V: comparison beween vulnerable American pu and non-vulnerable American pu. Calculaions of vulnerable pu opion prices are based on he base case parameers excep α = 0.75. 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. 31
160 140 120 pu value 100 80 60 D=0 f ^ D=900 f^ D=9000 f^ 40 20 0 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200 210 220 230 240 250 160 140 120 pu value 100 80 60 D=0 f D=900 f D=9000 f 40 20 0 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200 210 220 230 240 250 160 140 120 pu value 100 80 60 D=0 f ^ D=900 f^ D=9000 f^ 40 20 0 50 70 90 110 130 150 170 190 210 230 250 Figure 6. The effec of he opion wrier s deb D on he value of vulnerable American pu opion. Calculaions of vulnerable pu opion prices are based on he base case parameers excep α = 0.75. The numerical soluion is based on a hree-dimensional binomial ree using 100 seps. 32
160 140 120 pu value 100 80 60 alpha=0 f^ alpha=0.75 f^ alpha=1 f^ 40 20 0 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200 210 220 230 240 250 160 140 120 pu value 100 80 60 alpha=0 f alpha=0.75 f alpha=1 f 40 20 0 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200 210 220 230 240 250 160 140 120 pu value 100 80 60 alpha=0 f^ alpha=0.75 f^ alpha=1 f^ 40 20 0 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200 210 220 230 240 250 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. 33
pu value 140 120 100 80 60 40 20 sigma=0.01 f^ sigma=0.2 f^ sigma=0.4 f^ 0 20 60 100 140 180 220 260 300 340 380 140 120 pu value 100 80 60 40 sigma=0.01 f sigma=0.2 f sigma=0.4 f 20 0 20 60 100 140 180 220 260 300 340 380 200 pu value 150 100 50 sigma=0.01 f^ sigma=0.2 f^ sigma=0.4 f^ 0 20 60 100 140 180 220 260 300 340 380 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. The numerical soluion is based on a hree-dimensional binomial ree using 100 seps. 34
Table 5 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, V=100, σ =0.15, σ =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. v D ρ 90 92 94 96-0.8 α =0.5 f^ 0.17 0.18 0.40 3.42 α =0.5 f 0.18 0.40 3.01 14.02 α =0.5 f^ 0.17 0.18 0.39 2.77 HW p=0.5 f^ 0.00 0.00 0.10 1.73-0.4 α =0.5 f^ 0.21 0.33 1.04 4.16 α =0.5 f 0.28 0.69 2.62 8.62 α =0.5 f^ 0.20 0.28 0.58 1.67 HW p=0.5 f^ 0.01 0.09 0.59 3.03 0 α =0.5 f^ 0.41 0.55 1.24 4.14 α =0.5 f 0.44 0.72 1.88 5.98 α =0.5 f^ 0.38 0.43 0.64 1.27 HW p=0.5 f^ 0.03 0.15 0.73 3.24 0.4 α =0.5 f^ 0.24 0.30 0.66 2.43 α =0.5 f 0.25 0.32 0.75 2.81 α =0.5 f^ 0.24 0.25 0.29 0.52 HW p=0.5 f^ 0.01 0.06 0.40 2.07 0.8 α =0.5 f^ 0.26 0.26 0.32 1.13 α =0.5 f 0.26 0.26 0.32 1.14 α =0.5 f^ 0.26 0.26 0.26 0.26 HW p=0.5 f^ 0.00 0.00 0.06 0.89 35
Table 5 (Coninued) D ρ 90 92 94 96-0.8 α =1 f^ 0.17 0.18 0.41 3.67 α =1 f 0.18 0.63 5.84 27.87 α =1 f^ 0.17 0.18 0.40 3.53 HW p=0 f^ 0.00 0.00 0.11 1.78-0.4 α =1 f^ 0.22 0.36 1.33 5.89 α =1 f 0.37 1.20 5.04 17.04 α =1 f^ 0.21 0.31 1.07 3.37 HW p=0 f^ 0.02 0.11 0.72 3.72 0 α =1 f^ 0.42 0.65 1.73 6.38 α =1 f 0.51 1.07 3.38 11.58 α =1 f^ 0.39 0.48 0.91 2.71 HW p=0 f^ 0.04 0.22 1.10 4.93 0.4 α =1 f^ 0.25 0.35 0.98 4.15 α =1 f 0.26 0.40 1.27 5.38 α =1 f^ 0.24 0.26 0.39 1.16 HW p=0 f^ 0.01 0.10 0.68 3.61 0.8 α =1 f^ 0.26 0.27 0.38 2.00 α =1 f 0.26 0.27 0.38 2.01 α =1 f^ 0.26 0.26 0.26 0.27 HW p=0 f^ 0.00 0.01 0.12 1.78 36