Pricing Black-Scholes Options with Correlated Interest. Rate Risk and Credit Risk: An Extension

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1 Pricing Black-choles Opions wih Correlaed Ineres Rae Risk and Credi Risk: An Exension zu-lang Liao a, and Hsing-Hua Huang b a irecor and Professor eparmen of inance Naional Universiy of Kaohsiung and Professor of inance eparmen of Money and Banking Naional Chengchi Universiy b Ph.. Candidae inancial Engineering ivision eparmen of Money and Banking Naional Chengchi Universiy Address: 64, Chih-nan Rd., ec., Mucha, aipei, 116, aiwan; el: ex. 8151; ax: ; liaosl@nccu.edu.w.

2 Pricing Black-choles Opions wih Correlaed Ineres Rae Risk and Credi Risk: An Exension Absrac his aricle provides a closed-form valuaion formula for he Black-choles opions subjec o ineres rae risk and credi risk. No only does our model allow for he possible defaul of he opion issuer prior o he opion s mauriy, bu also considers he correlaions among he opion issuer s oal asse, he underlying sock, and he defaul-free zero coupon bond. We furher ailor-make a specific credi-linked opion for hedging he defaul risk of he opion issuer. he numerical resuls show ha he defaul risk of he opion issuer significanly reduces he opion values, and he vulnerable opion values may be remarkably overesimaed in he case where he defaul can occur only a he mauriy of he opion. Keywords: ulnerable Opions, Credi Risk, Ineres Rae Risk, ime-changed Brownian Moion 1

3 1. Inroducion he radiional Black-choles opion pricing formula is derived under he assumpion ha here is no defaul risk of he opion issuer. or exchange-raded opions, his is reasonable since mos exchanges have been very successful in organizing hemselves o ensure ha heir conracs are always honored. or he over-he-couner (OC) conracs, on he oher hand, he counerpary defaul risk is imporan. In recen years, OC opions have become increasingly popular, and hence he defaul risk of he opion issuer should be considered in he pricing of he OC opions. A vulnerable opion is he opion ha he issuer may defaul. Generally speaking, here are wo pricing mehods for modeling credi risk: he reduced-form approach and he srucural approach. Hull and Whie (1995) consruc a general reduced-form model and analyze he effecs of credi risk on European and American opions, which allows for he possible defaul of he opion issuer prior o he mauriy of he opion. he analyic formula is derived in he case ha he underlying asse of he opion is assumed o be independen of he opion issuer s asse. Jarrow and urnbull (1995) use a foreign currency analogy ogeher wih a pre-specified defaul process which is also independen of he underlying asse of he defaulable claims, and obain he valuaion of defaulable claims in boh he discree-ime and coninuous-ime frameworks. However, he independence assumpion, used o keep he pricing formulas reasonably racable, seems only sensible for he case ha he asse of he opion wrier is well-diversified or fully-hedged. he srucural approach for pricing European vulnerable opions is firs proposed by Johnson and ulz (1987). hey assume he financial disress can occur only a he mauriy of he opion, and when i is he case, he opion holders ake over all he asses of he opion issuer. heir model also allows for he correlaion beween he value of he

4 opion issuer s oal asse and ha of he underlying sock, which is an imporan deerminan of he vulnerable opion prices. Klein (1996) exends Johnson and ulz (1987) o consider oher liabiliies in he capial srucure of he opion issuer. ubsequenly, Klein and Inglis (1999) (hereafer, KI (1999)) generalize Klein (1996) wih he asicek (1977) sochasic ineres rae model. aking he correlaions among he underlying sock price, he risk-free ineres rae, and he opion issuer s oal asse value ino consideraion, hey derive he closed-form pricing formula of he vulnerable opion via he parial differenial equaion (PE) approach. Recenly, Klein and Inglis (001) employ numerical mehods o invesigae he generalized defaul barrier which is he sum of he variable defaul barrier of Johnson and ulz (1987) and he fixed defaul barrier of Klein (1996). All srucural models menioned above suppose ha he defaul of he opion issuer only occurs a he mauriy of he opion, which seems o be a defec of hose models. In his paper, we consruc a srucural model o value he Black-choles vulnerable opions, considering no only he possible defaul of he opion issuer prior o he mauriy of he opion, bu also he correlaions among he oal asse value of he opion wrier, he value of he underlying sock, and he value of he defaul-free zero coupon bond. Moreover, we apply he forward risk-neural pricing approach ogeher wih he echnique of he ime-changed Brownian moion o derive he closed-form valuaion of he vulnerable opion under sochasic ineres raes. No only does our model provide a framework o overcome he weakness of he previous srucural models which he opion issuer can defaul only a he mauriy (e.g., Klein (1996) and KI (1999)), bu also resolves he limiaion of he reduced-form approach which he defaul of he opion issuer and he underlying sock prices are assumed o be independen (e.g., Hull and Whie (1995) and Jarrow and urnbull (1995)). he presen 3

5 paper also ailor-makes a specific credi-linked opion o hedge he defaul risk of he vulnerable opion. he numerical resuls show ha he defaul risk of he opion issuer is criical for he valuaion of he opion, and he vulnerable opion prices may be significanly overesimaed when he defaul can occur only a he mauriy of he opion. inally, he model can be furher exended o allow he value of he opion issuer s oal liabiliy o be sochasic, and can also be applied o he valuaion of a vulnerable zero-coupon bond opion. he remainder of his paper is organized as follows. In ecion, we se up he basic assumpions and ouline he pricing framework. ecion 3 is devoed o he derivaion of he closed-form valuaion of he Black-choles vulnerable opions and he corresponding hedging sraegy under sochasic ineres raes. Nex, we presen some numerical analysis and explore he properies of he defaul risk of he opions issuer in ecion 4. Besides, wo applicaions of our model are provided in ecion 5. inally, ecion 6 summarizes he aricle and makes concluding remarks.. he aluaion ramework In his secion, he radiional assumpions of he Black-choles economy under sochasic ineres raes are employed, and moreover, we exend he economy o allow for he possible defaul of he opion issuer prior o he mauriy of he opion. he main assumpions of our model are inroduced and discussed as follows. Assumpion 1: Le ( ) denoe he underlying sock price a ime. he process of * ( ) on he (spo) risk-neural filered probabiliy space ( Ω,, Q,( ) 0 ) =, where * 0 < and * is he erminaion dae of his economy, is given by 4

6 d( ) = r( ) d + σ ( ) dw ( ), (1) ( ) where r( ) denoes he insananeous risk-free ineres rae, σ ( ) is he ime-varying volailiy of he insananeous rae of reurn of ( ), and process on he same space. W is a Wiener Assumpion : Le B( ) denoe he value of he savings accoun (or money marke accoun) a ime. If one iniially saves B (0), defined o be one dollar, he/she can obain he amoun of B( ), which is equal o exp r( s) ds, a ime > 0. 0 Assumpion 3: Le B(, ) denoe he ime value of he defaul-free zero coupon bond (ZCB) which pays one dollar a ime, for 0. he dynamics of B(, ) on he risk-neural filered probabiliy space is given by db(, ) r( ) d b(, ) dwb ( ) B(, ) = +, () where b(, ) is he ime-varying volailiy of he insananeous rae of reurn of B(, ) and W B is a Wiener process on he same space. Assumpion 4: Le ( ) denoe he oal asse value of he opion issuer a ime. he evoluion of ( ) on he risk-neural filered probabiliy space is given by d ( ) = r( ) d + σ ( ) dw ( ), (3) ( ) where σ ( ) is he ime-varying volailiy of he insananeous rae of reurn of ( ), and W is a Wiener process on he same space. Moreover, he consan insananeous 5

7 correlaion coefficiens among he above hree Wiener processes are given by d E dw ( ) dw ( ), for i, j =,, B and i j, where he expecaion is aken ij i j under risk neuraliy. Assumpion 5: rading akes place coninuously in ime, and unresriced borrowing and lending of funds is possible a he same insananeous risk-free ineres rae. urhermore, he markes of hese radable asses, including ( ), B( ), B(, ), and ( ), are fricionless, i.e., here are no ransacion coss or axes, and no limiaion agains he shor sales. Assumpion 6: here is a pre-specified sochasic defaul hreshold of he opion issuer s oal asse value, which is defined as B(, ), where is a consan represening he expeced marke value of he opions issuer s oal liabiliy a. Once he oal asse value of he opion issuer goes down and ouches his hreshold a he ime of defaul τ, defined by inf{ < s : ( s) B( s, )}, he opion issuer will declare resrucuring or bankrupcy. Assumpion 7: If resrucuring or bankrupcy occurs, he opion issuer s asse is immediaely liquidaed and he scrap value is (1 α) ( τ ), where α is assumed o be a consan showing he raio of bankrupcy or resrucuring coss of he issuer s asse ( 0 α 1 ). he recovery rae of he opion holder herefore equals (1 α) ( τ ) * a τ, where ( τ, ) denoes he expeced marke value * B of he opion issuer s oal liabiliy a τ. Upon defaul, he opion holder can direcly receive he recovery value (in dollar amoun), which is equal o he recovery rae 6

8 muliplied by he nominal claim, he marke value of he non-vulnerable opion on he same underlying asse a τ. Assumpions 1-3 and Assumpion 5 are he sandard assumpions in he Black-choles economy wih sochasic ineres raes under risk neuraliy. Assumpion 1 represens he uncerainy source of he underlying sock under risk neuraliy. Assumpion provides he economy an insananeously risk-free asse (he savings accoun). hen, Assumpion 3 exhibis he uncerainy source of he price of he defaul-free ZCB under risk neuraliy, which is equivalen o model he uncerainy of ineres raes in he same risk-neural world. Here we do no specify any sochasic ineres rae model bu assume he sochasic behavior of ZCB prices direcly insead. Many sochasic ineres rae models, such as asicek (1977) and Heah e al. (199), can be used in our framework by defining he appropriae form of b(, ). Assumpion 4 shows he uncerainy source of he opion issuer, which will be uilized o model he possible defaul of he opion issuer, and he correlaions among he hree uncerainy sources are also defined. Nex, Assumpion 5 is imposed o ensure ha our economy can work ideally as he radiional Black-choles environmen. Assumpion 6 supposes he ime when he opion issuer will go bankrup in our model and finally, Assumpion 7 expresses he recovery value received by he opion holder when he opion issuer defauls. o highligh he seup of he opion issuer s defaul risk in our framework, we explain Assumpions 6 and 7 in more deails. Assumpion 6 means ha as long as he oal asse value of he opion issuer falls below he sochasic defaul hreshold B(, ), he bankrupcy or resrucuring akes place immediaely. In erms of he forward price, i is equivalen o say ha once he forward value of he opion issuer s 7

9 oal asse a ime, (, ) = ( ) B(, ), goes down and ouches he consan barrier, he opion issuer hen goes bankrup. Here can also be explained as he expeced defaul rigger of he forward value of he issuer s oal asse. As a resul, he ime of defaul τ is he firs passage ime ha he value of he issuer s oal asse falls below he sochasic defaul rigger B(, ), or equivalenly, he forward value of he issuer s oal asse goes down and ouches he consan defaul barrier. An imporan implicaion of his assumpion is ha defaul occurs on all deb obligaions simulaneously. his is realisic since when a firm defauls on an obligaion, i ypically defauls on oher obligaions as well due o he normal cross-defaul provision. Noice ha a he mauriy of he opion, he hreshold of he issuer s oal asse value is equal o, which is consisen wih he fixed defaul boundary of KI (1999). In Assumpion 7, he recovery rae a he ime of defaul is (1 α) ( τ ) * = 1 α (by Assumpions 6 and 7), where (1 α) ( τ ) represens he scrap value of he opion issuer s oal asse a he ime of defaul. aking he vulnerable call opions as an example, he recovery value (in dollar amoun) of he opion holder a he ime of defaul is equal o (1 α) C( τ ) = ( ) + B K τ. By he no-arbirage argumen, he equivalen P (1 α) ( τ, )E ( ) dollar amoun of he opion holder s he recovery value a he mauriy is, herefore, P equal o (1 α )E ( ( ) ) + K τ which depends on he informaion se a τ, i.e., i is τ -measurable. he equivalen dollar value is hen used o price he vulnerable opions in he laer secion, and he approach adoped here is similar o ha of Briys and de arenne (1997), which invesigaes he valuaion of risky corporae debs. 8

10 3. aluing ulnerable Opions wih ochasic Ineres Raes In his secion, we firs provide he radiional Black-choles formulas under sochasic ineres raes. Nex, we obain he pricing formulas for he Black-choles vulnerable opions where he opion issuer can defaul only a he mauriy by our pricing mehodology and hen make a connecion o KI (1999). ubsequenly, we derive he pricing formulas for he Black-choles vulnerable opions where he opion issuer can defaul hroughou he remaining life of he opion. or comparison purpose he radiional non-defaul Black-choles pricing formulas under sochasic ineres raes are summarized below in Proposiion 1. Proposiion 1: enoe (, ) = ( ) B(, ) as he forward price of he underlying sock. he ime values of he European call opion C( ) and pu opion P( ) wihou he opion issuer s defaul risk are respecively given by C( ) = ( ) Φ( d ) KB(, ) Φ ( d ), (4) 1 P( ) = ( ) Φ( d ) + KB(, ) Φ( d ), (5) 1 where K is he srike price, is he mauriy dae, Φ( ) denoes he cumulaive sandard normal disribuion, (, ) 1 ln + ( u, ) du K σ 1 σ ( u, ) du d = = d + σ ( u, ) du, and σ (, ) = σ ( ) σ ( ) b(, ) + b (, ). B irs of all, if he risk-free ineres rae is a consan, i.e., B(, ) = ( r ) exp ( ), and σ ( ) is also a consan, hen he formula will reduce o he 9

11 radiional Black-choles pricing formula. By means of he forward price of he underlying sock, wo uncerainy sources, including he underlying sock price and he defaul-free ZCB price, can be simplified as one uncerainy source, namely, he forward price of he underlying sock. Consequenly, he pricing formulas only involve one-dimensional sandard normal disribuions which show he probabiliies ha he opions will be in-he-money. Nex, we use Assumpions 1-5 o derive he closed-form pricing formulas for he vulnerable calls and pus which only suffer from he defaul risk of he opion issuer a he mauriy of he opion. Le C ˆ( ) and P ˆ( ) denoe he values of hese vulnerable opions and heir corresponding final payoffs are given as follows: + (1 α) ( ) + Cˆ( ) = ( ( ) K ) 1 { ( ) > } + ( ( ) K ) 1 { ( ) }, (6) + (1 α) ( ) + Pˆ( ) = ( K ( ) ) 1 { ( ) > } + ( K ( ) ) 1 { ( ) }, (7) + where ( x) max ( x,0) and 1 {B} denoes he indicaor funcion wih value 1 if even B occurs and wih value zero oherwise. he firs erms of Equaions (6) and (7) represen he payoffs of he opions when he opion issuer does no defaul a he mauriy of he opion; namely, a he mauriy he value of he opion issuer s oal asse ( ) is greaer han he expeced defaul rigger, and hus he defaul does no occur. he second erms of boh he equaions are he recovery values of he opions a he mauriy when he defaul occurs a he mauriy. he recovery value is, herefore, equal o he nominal claim of he opion, ( ( ) ) K +, muliplied by he recovery rae (1 α) ( ). he closed-form pricing formulas of he vulnerable call and pu in his case are given in he following proposiion. 10

12 Proposiion : he ime values of a vulnerable Black-choles call opion C ˆ( ) and pu opion P ˆ( ) which only suffer from counerpary defaul risk a he mauriy of he opion are respecively given as follows: { ( ) ( ) C ˆ( ) = ( ) Φ e, e, KB (, ) Φ e, e, ( u, ) du σ * e e 3 e 7 (1 α) (, ) + ( ) Φ,, ( ) (1 α) ( ) K Φ * ( e4, e8, ), (8) { ( ) ( ) P ˆ( ) = ( ) Φ e, e, + KB (, ) Φ e, e, ( u, ) du σ * e e 3 e 7 (1 α) (, ) ( ) Φ,, ( ) (1 α) ( ) + KΦ * ( e4, e8, ), (9) where Φ(,, ) denoes he bivariae cumulaive sandard normal disribuion, e = d, e = d, 1 1 (, ) 1 ln + ( u, ) du (, ) u du K σ + σ 3 = = 4 + σ ( u, ) du e e σ ( u, ) du, (, ) 1 ln ( u, ) du (, ) u du σ + σ 5 = = 6 + (, ) (, ) σ σ ( u, ) du e e X u du, (, ) 1 ln ( u, ) du (, ) u du σ σ 7 = = 8 (, ) (, ) σ σ ( u, ) du e e X u du, 11

13 X (, ) = σ ( u, ) du σ ( u, ) du, σ (, ) = σ ( ) σ ( ) b(, ) + b (, ), B = σ ( u, ) du u du σ σ (, ) ( u, ) du, and σ = σ σ σ σ +. (, ) ( ) ( ) (, ) ( ) (, ) ( ) (, ) b B b B b Proof ee Appendix A. When he diffusion erms of Equaions (1) and (3) are consans and he risk-free ineres rae follows asicek (1977), hen he pricing formulas for he vulnerable call and pu can be reduced o hose of KI (1999). he inuiions of Equaions (8) and (9) are as follows. irs, alhough here are hree risk facors in his model, including he underlying sock price, he opion issuer s oal asse value and he defaul-free ZCB price, i can be reduced via forward prices (i.e., by aking he ZCB as a numeraire) o a wo-facor case which involves he forward price of he underlying sock and he forward value of he issuer s oal asse. As a resul, Equaions (8) and (9) jus correlae o he above wo facors, heir reurn volailiies, covariance and correlaion coefficien. In addiion, here are four bivariae cumulaive sandard normal disribuions in Equaions (8) and (9), respecively. In conras wih he radiional Black-choles pricing formula, hese four disribuions exhibi he join probabiliies of which he opion will be in-he-money and wheher he opion issuer will declare bankrupcy or no a he mauriy of he opion. he firs wo bivariae disribuions of Equaions (8) and (9) show he join probabiliies of which he opion will be in-he-money and he opion issuer will no defaul a he mauriy. he 1

14 las wo bivariae disribuions of Equaions (8) and (9), on he oher hand, represen he join probabiliies of which he opions will be in-he-money and he opion issuer will defaul a he mauriy. In wha follows, we exend he above model via Assumpions 6 and 7 o incorporae he possible defaul of he opion issuer prior o he mauriy of he opion ino he pricing of he vulnerable opions. Le C * ( ) and P * ( ) denoe he values of he pah-dependen vulnerable call and pu opions a ime prior o on he se { τ > }, which means he opion issuer has no been bankrup before ime. he final payoffs are respecively given as follows: * (1 α) ( τ ) P + + C ( ) = E * ( ( ) K ) τ 1 { } + ( ( ) K ) 1 < τ { < τ }, (10) * (1 α) ( τ ) P + + P ( ) = E * ( K ( ) ) τ 1 { } + ( K ( )) 1 < τ { < τ }. (11) We firs clarify he relaionship beween he ime of defaul τ and min ( s, ), < s which is he minimum forward price of he opion issuer s oal asse over he ime period (, ]. he even of { } τ > is equivalen o ha of { min ( s, ) } < s >, and boh of he evens exhibi ha he opion issuer does no defaul during he remaining life of he opion. he firs erms of Equaions (10) and (11) are he recovery value of he vulnerable opions a he mauriy when he opion wrier riggers he defaul a ime τ and τ <, i.e., he minimum forward price of he opion issuer s oal asse is less han or equal o he expeced defaul rigger over he ime period (, ]. On he oher hand, he second erms of Equaions (10) and (11) are he values of vulnerable opions when he opion issuer does no defaul during he remaining life of he opion. rom Assumpion 6 and Assumpion 7, we have ( τ, ) and * B 13

15 ( τ ) = B( τ, ). ubsiuing hem ino Equaions (10) and (11), we can rewrie hem as follows: ( ) ( ) C K K * P + + ( ) = (1 α)e ( ) τ 1 + ( ) 1 { min ( s, ) } min ( s, ) > < s < s ( ) ( ) { } P K K. * P + + ( ) = (1 α)e ( ) τ 1 + ( ) 1 { min ( s, ) } min ( s, ) > < s < s { }, and By he forward risk-neural pricing mehod and he law of ieraed expecaion, we have + ( ) (, )E ( ) (, )(1 α)e ( ( ) ) 1 * P * P C = B C = B K { min ( s, ) } + B ( K ) B C { min ( s, ) > } < s P + P ** (, )E ( ) 1 (, )E ( ), and + ( ) (, ) ( ) (, )(1 α)e ( ( )) 1 * P * P P = B P = B K < s { min ( s, ) } < s + B ( K ) B P { min ( s, ) > } < s P + P ** (, )E ( ) 1 (, ) ( ). In views of he above equaions, when pricing he vulnerable opions a he ime < τ, wihou loss of generaliy, we can replace he final payoff C * ( ) and P * ( ) wih C ** ( ) and P ** ( ) in calculaing C * ( ) and P * ( ), respecively, where C ** ( ) and P ** ( ) can be wrien as follows. ( ) α ( ) ** + + C K K ( ) = ( ) ( ) 1, and (1) ( ) α ( ) ** + + P K K { min (, ) } s < s ( ) = ( ) ( ) 1. (13) { min (, ) } s < s rom Equaions (1) and (13), he characerisics of C ** ( ) and P ** ( ) are similar o hose of American pu opions. he value of an American pu opion can be decomposed ino he value of a European pu opion plus he early exercise premium due o he sraegic flexibiliy of he opion owner. In our framework, he value of he 14

16 vulnerable opion can be decomposed ino he value of he sandard Black-choles (non-vulnerable) opions minus he issuer defaul discoun because of he implici defaul flexibiliy of he opion issuer. Moreover, he value of he vulnerable call opion can be decomposed as he sum of a non-vulnerable call opion and a shor posiion in α unis of down-and-in ouside barrier call opions according o he definiion of he ouside barrier opions provided by Heynen and Ka (1994). Also, Equaion (13) shows ha a vulnerable pu opion can be viewed as a non-vulnerable pu opion ogeher wih a shor posiion in α unis of down-and-in ouside barrier pu opions. o derive he pricing formulas of he vulnerable call and pu opions in our framework, we furher ransform he down-and-in opions ino he down-and-ou opions via he in-ou pariy of he barrier opions. A long posiion in a down-and-ou ouside barrier call and a down-and-in ouside barrier call is equivalen o a non-vulnerable call opion, which can be expressed as ( ) { } ( ) { } min ( s, ) > min ( s, ) < s < s ( ) ( ) K 1 + ( ) K 1 ( ) K. herefore, a vulnerable call opion can also be replicaed by a long posiion in (1 α) unis of he non-vulnerable calls and α unis of he down-and-ou ouside barrier calls. By he same oken, a vulnerable pu opion can also be replicaed by a long posiion in (1 α) unis of he non-vulnerable pus and α unis of he down-and-ou ouside barrier pus. Accordingly, he pricing formulas of he vulnerable call and pu opions are presened below. Proposiion 3: he ime values of he vulnerable Black-choles call opion C * ( ) and pu opion P * ( ), on he se { τ > }, which means he opion issuer has no been bankrup before ime, are respecively given as follows: 15

17 ( ) * C = α Φ h1 KB Φ h ( ) (1 ) ( ) ( ) (, ) ( ) { ( ) h1, h5, KB(, ) h, h6, ( ) ( ) + α Φ Φ X (, ) (, ) ( ) Φ 3, 7, (, ) ( h h ) (, ) + KB(, ) Φ ( h4, h8, ), (14) ( ) * P = α Φ h1 + KB Φ h ( ) (1 ) ( ) ( ) (, ) ( ) { ( ) h1, h5, KB(, ) h, h6, ( ) ( ) + α Φ + Φ X (, ) (, ) + ( ) Φ 3, 7, (, ) ( h h ) (, ) KB(, ) Φ ( h4, h8, ), (15) where h1 = d1, h = d, X (, ) (, ) 1 ln + ( u, ) du ln K σ + (, ) 3 4 σ ( u, ) du h = = h + σ ( u, ) du, (, ) 1 ln ( u, ) du (, ) u du σ + σ 5 = = 6 + (, ) (, ) σ σ ( u, ) du h h X u du, (, ) 1 ln ( u, ) du (, ) u du σ + σ 7 = = 8 + (, ) (, ) σ σ ( u, ) du h h X u du, and oher noaions are defined as before. Proof ee Appendix B. 16

18 Equaions (14) and (15) will reduce o he Black-choles pricing formula under sochasic ineres raes when = 0, i.e., here is no opion issuer s defaul risk under he condiion (0) > 0. If = 0, hen our pricing formula becomes very similar ha of o Hull and Whie (1995). A he mauriy of he opion, Equaions (14) and (15), as expeced, are respecively equivalen o Equaions (8) and (9), which only consider he defaul a he mauriy. In addiion, he consan value of he opion issuer s oal liabiliy will be exended o follow a sochasic process 1 in ecion 5. he inuiions of Equaions (14) and (15) are as follows. Alhough Equaions (14) and (15) consider he possible defaul of he opion issuer during he remaining life of he opion, he desired formulas only consis of one-dimensional and wo-dimensional cumulaive sandard normal disribuions. hese formulas, herefore, are no more complex han he pricing formulas which allow for he defaul only a he mauriy. In Equaions (14) and (15), he one-dimensional disribuions can be viewed as hose of he radiional Black-choles pricing formulas under sochasic ineres raes, i.e., hey are he probabiliies ha he opion will be in-he-money. On he oher hand, he wo-dimensional disribuions of Equaions (14) and (15) shows he join probabiliies ha he opion will be in-he-money and wheher he opion issuer will declare defaul or no during he remaining life of he opion, which can be derived by applying some echniques of probabiliy heory, as shown in Appendix B. Afer deriving he desired formulas, we launch ino he hedging sraegies of he vulnerable opions. In view of Equaion (1), we can hold a vulnerable call opion ogeher wih a long posiion in α unis of down-and-in ouside barrier call opions o replicae a non-vulnerable call opion. or hose hedged fund managers or insiuional 1 or example, Ammann (001) assumes i follows a geomeric Brownian moion. 17

19 invesors who hold a grea posiion on calls and pus issued by a single opion wrier, hey can reques oher financial insiuions o issue he desired ailor-made down-and-in ouside barrier call and pu opions. Afer hedging he defaul risk, he opion holders can direcly apply he sandard hedging sraegies of he Black-choles opions. Noice ha he ailor-made down-and-in ouside barrier opion can be viewed as some kind of credi-linked opions, whose holders obain an opion on he underlying sock a he mauriy when he pre-specified defaul even occurs, or else ge nohing back. In our model, he defaul even can be specified as ha he forward value of he opion issuer s oal asse has ever gone down and ouched during he remaining life of he opion. Moreover, he value of a down-and-in ouside barrier opion is, in fac, equal o he price difference beween he non-vulnerable opion and he vulnerable call opion. 4. Numerical Examples In his secion, we provide he numerical analyses abou he values of he non-vulnerable calls and he vulnerable calls of KI (1999) and our model. hen we also poin ou some properies of he credi-linked call opion employed o hedge he defaul risk of he opion issuer. imilar o he Black-choles opion pricing formula under sochasic ineres raes, he value of he vulnerable call opion is dependen on he underlying sock price, ( ), he srike price, K, he defaul-free ZCB price, B(, ), he ime-o-mauriy,, he reurn volailiies of he underlying sock and he defaul-free ZCB, σ and σ B, and he correlaion coefficien beween he raes of reurn of he defaul-free ZCB We hereby assume ha here is no issuer s defaul risk on he down-and-in ouside barrier opion. 18

20 and he underlying sock, B. Moreover, our formula also depends on he opion issuer s oal asse value, ( ), he expeced marke value of he opion issuer s oal liabiliy,, he reurn volailiy of he issuer s oal asse, σ, he correlaion coefficien beween he raes of reurn of he defaul-free ZCB and he issuer s oal asse, B, he correlaion coefficien beween he raes of reurn of he issuer s oal asse and he underlying sock,, and he proporional bankrupcy coss, α. As he same as he sandard Black-choles model, our model shows ha higher risk-free ineres rae (lower defaul-free ZCB price) increases he value of vulnerable calls and decreases he value of vulnerable pus, and higher σ increases boh he values of vulnerable calls and pus. Moreover, higher σ decreases he value of vulnerable call and pu opions due o he increase of he opion issuer s defaul risk. inally, he effec of bankrupcy coss on he value of he vulnerable opions, as expeced, exhibi ha higher α decreases he value of he vulnerable calls and pus. Nex, we focus on he vulnerable call opion. igure 1 illusraes he inrinsic values of he call, C In, he Black-choles call prices under sochasic ineres raes (Equaion (4)), C B, he vulnerable call prices of KI (1999) (Equaion (8)), C KI, and he vulnerable call prices of our model (Equaion (14)), C LH, as a funcion of he prices of he underlying sock. In common wih he Black-choles call opion formula, C KI, and C LH are increasing as he price of he underlying sock increases, and are shaped in he same way. As expeced, C B (non-vulnerable) is greaer han C KI which considers he defaul risk only a he mauriy, and moreover, C KI is greaer han C LH which allows for he possible defaul during he remaining life of he opion. All he parameers, aken from KI (1999), are provided in he foonoe of igure 1, and are served as he base case in able 1. As noed in KI (1999), he deep-in-he-money vulnerable opions may 19

21 be worh less han he inrinsic value of he opions since boh he ime value and he inrinsic value of he vulnerable opions are suffered from he defaul risk of he opion wrier. able 1 represens a number of numerical examples for C B, C KI and C LH, which are provided o show he effecs on he opion values under various correlaion coefficiens among he raes of reurn of he underlying asse, he defaul-free ZCB, and he opion issuer s oal asse. Panel 1, he base case of which he parameers follow KI (1999), illusraes ha he value of he vulnerable call of KI (1999) is underesimaed by abou.1% (relaive o our model) due o he neglec of he possibiliy ha he opion issuer defauls prior o he mauriy. Compared o Panel 1, Panel represens ha he effec of he ineres rae risk on he wo vulnerable call values is he same as ha on he non-vulnerable call value, ha is, he presence of ineres rae risk will raise all he call prices. Panel 3 exhibis he effecs of on he values of he vulnerable calls. Inuiively, he values of he non-vulnerable call are unchanged due o he independence of he oal asse value of he opion issuer. Moreover, higher value of induces greaer values of boh he vulnerable calls. his is because he higher underlying sock price and opion issuer s oal asse value will boh increase he value of he vulnerable call. Consequenly, when rises, he vulnerable calls will suffer from smaller defaul risk. In addiion, one may be surprised ha when = 0.5 in Panel 3 of able 1, he vulnerable call value of our model is greaer han ha of KI (1999). o be more concree o explain, we highligh he mos imporan disincion beween KI (1999) and our model, and show ha one can no direcly infer ha our model suffers from more defaul risk of he issuer han KI (1999). Our model akes he possibiliy of early defaul ino consideraion, while he recovery rae is a consan a he mauriy of he 0

22 opion. On he oher hand, he recovery rae of KI (1999) is posiively relaed o ( ), he inverse of he deb raio of he opion issuer a he mauriy, 3 whereas defaul may occur only a he mauriy. When he deb raio of he opion issuer is rising, he recovery rae of KI (1999) is decreasing, showing ha he opion holder suffers from increasing defaul risk. However, his effec will no appear in our framework because our recovery rae is a consan. We provide able and igure o furher explore he above surprising resul. able exhibis he price differences in percenage of he vulnerable calls beween KI (1999) and our model. Under higher deb raio of he opion issuer, longer ime o mauriy of he opion, and highly negaive correlaion beween he raes of reurn of and, he surprising resul appears, i.e., C LH is greaer han C KI. or example, when ( ) = 0.9, = 0.8 and = 3, C KI is underesimaed by abou 8.8% relaive o our model. We can provide a more clear-cu illusraion in igure, which shows he values of he wo vulnerable calls under various deb raios and values of when he ime o mauriy of he opion equals o hree years. In igure, we can find ha eiher greaer deb raio or highly negaive correlaion beween he reurns of and leads o he reducion of boh he wo vulnerable call values. When he deb raio is higher and he value of is smaller, he decreasing speed of C KI is greaer han C LH, and C KI falls below C LH as he value of is lower han 0.5. he main reason is ha he recovery rae of KI (1999) is negaively relaed o he deb raio of he opion issuer. herefore, when he deb raio increases from 0.5 o 0.9, C KI falls sharply and becomes lower han C LH. In summary, his surprising phenomenon appears especially when he deb raio is high, he ime o mauriy is long and is highly 3 herefore, he opion issuer s asse of KI (1999) is allowed o recover from defaul by mauriy. 1

23 negaively correlaed. Panel 4 in able 1 represens he effec of B on call values. Cerainly, he non-vulnerable call value is also unchanged because is value is independen of he value of he opion issuer s oal asse. Higher value of B, oher hings being equal, resuls in a slighly negaive impac on C KI and a posiive impac on C LH, which are boh small. igure 3 furher shows he values of he wo vulnerable calls under various deb raios of he opion issuer and values of B when he ime o mauriy of he opion equals o hree years. In view of igure 3, he effec of B on he price differences of he wo vulnerable calls is small, bu he price differences beween he wo vulnerable calls and he non-vulnerable call are sill significan, especially when he deb raio is high. Moreover, C LH is always less han C KI under various values of B even if he deb raio of he opion issuer is high. Panel 5 in able 1 exhibis he effec of B on he values of he non-vulnerable call and he wo vulnerable calls. I is clear ha he higher he value of B, he lower he values of he non-vulnerable call and he wo vulnerable calls. In igure 4, he impacs of B on he prices of all he call opions are in he same manner, i.e., he slopes of all he call opions are similar. igure 4 also shows ha higher deb raio (higher defaul risk of he issuer) will resul in greaer reducions of he wo vulnerable call values. Again, noice ha C LH is always less han C KI under various values of B even hough he deb raio of he opion issuer is high. Panels 6 and 7 in able 1 represen he effecs of various combinaions of, B and B. he impacs of hese correlaions on he value of he non-vulnerable call and he wo vulnerable calls are he same as he above analyses. As expeced,

24 seems o be he mos imporan facor in valuing vulnerable opions while he oher wo correlaions also have significan impac on he values of vulnerable opions paricularly when he deb raio of he opion wrier is high and he opion is ou of money. able 3 illusraes he values of credi-linked call opions under various correlaion coefficiens and volailiies of he raes of reurn. he value of a credi-linked call increases as he price of he underlying sock or he deb raio of he opion wrier increases. In addiion, higher reurn volailiy of raises credi-linked call values; higher correlaion coefficiens beween he raes of reurn of and will also make he credi-linked call value greaer. herefore, he credi-linked calls can be used o hedge he defaul risk of he vulnerable calls. Moreover, he effecs of B and B on he prices of he credi-linked call opion are unsable and small, however. inally, one may be surprised ha in some cases, conrary o he case of Black-choles call opion, he higher he volailiy of he rae of reurn of he underlying sock, he lower he value of a credi-linked call opion. or example, we can observe in able 3 when B = 0.5, B = 0.5, = 0.5 and mos imporan, σ = 0.1, higher volailiy lower credi-linked call values whose values are very small. 5. Exensions In his secion, we generalize our model and allow he value of he opion issuer s oal liabiliy o be sochasic. Nex, we will apply our pricing framework o value a vulnerable call opion on he defaul-free zero coupon bond. o allow he value of he opion issuer s oal liabiliy o be sochasic, we direcly assume ha he inverse of he deb raio of he opion wrier, R( ) ( ) ( ), 3

25 follows a geomeric Brownian moion, which replaces he seup of ( ) in Assumpion 4. Namely, we assume ha he dynamics of he inverse of he opion wrier s deb raio under risk-neuraliy are given as: ( σ R R ) dr( ) = R( ) r( ) d + ( ) dw ( ), where σ ( ) denoes he ime-varying volailiy of he insananeous rae of reurn of R R( ), which represens he uncerainy sources of boh he issuer s oal asse value and oal liabiliy value. 4 In wha follows, we modify he hreshold of he opion issuer s oal asse value in Assumpions 6 and 7 o he hreshold of he inverse of he deb raio of he issuer, defined by γ, where γ is a posiive consan. he ime of defaul is hus changed o = inf { < : ( ) 1 γ } inf { s : γ ( s) ( s) } τ s R s = <. In oher words, he opion issuer will defaul when he issuer s deb raio, ( ) ( ), is greaer han or equal o γ. In addiion, he corresponding recovery rae of he vulnerable call a he mauriy of he opion is alered o (1 α) γ. o price he vulnerable call opion in his case, we firs provide he final payoff of he vulnerable call wih consideraion of he sochasic value of he opion issuer s oal liabiliy as follows: (1 α) = +. (16) ( ) ( ) * + + C ( ) ( ) K 1 ( ) K γ min R( s) min R( s) > < s γ < s γ Comparing o Equaion (10) wih he consan value of he issuer s oal liabiliy, we jus need he following wo subsiuions: (1 α) (1 α) γ, 1 γ and (, ) ( ) B(, ) R( ). Hence, he ime value of he vulnerable Black-choles call opion wih sochasic value of he opion issuer s oal liabiliy, on 4 Here we implicily assume R( ) is a coninuously radable asse. 4

26 he se { τ > }, can be direcly obained by modifying Equaion (14) wih he above subsiuions. Nex, we apply our model o price a vulnerable call opion on a defaul-free zero coupon bond. We firs assume ha he dynamics of he underlying ZCB price B(, 1) under risk-neuraliy, wih mauriy 1 which is larger han, are given by: db(, 1) = B(, 1 )( r( ) d + σ B ( ) dwb ( ) ). hen, all we have o do is o replace he 1 1 forward price of he underlying sock, (, ) = ( ) B(, ), wih he forward price of he underlying defaul-free ZCB, = B(, ) B(, ), in Equaion (14). Again, wih B 1 1 some modificaions, he pricing formula of a vulnerable ZCB call opion can be derived. 6. Concluding Remarks his aricle derives he closed-form pricing formulas for he Black-choles vulnerable opions under sochasic ineres raes by he forward risk-neural pricing approach wih a ime-changed Brownian moion. Our model improves he reduced-form models (Hull and Whie (1995) and Jarrow and urnbull (1995)) o consider he correlaions among he raes of reurn of he underlying sock, he opion issuer s oal asse and he defaul-free zero coupon bond. We also exend he previous srucural models (Klein (1996) and KI (1999)) o incorporae he possible defaul of he opion issuer during he remaining life of he opion, including he mauriy of he opion. In addiion, we provide he specific credi-linked opion which can be used o hedge he defaul risk of he vulnerable opions in our model. In general, he numerical resuls show ha he values of he non-vulnerable opions are overesimaed by abou 10%-15% relaive o he vulnerable opion values of 5

27 our model in mos of he cases. he vulnerable call values of KI (1999), which only consider he defaul risk a he mauriy of he opion, are also overesimaed relaive o our model for he mos par. I is reversed, however, when he deb raio of he opion issuer is higher, he ime o mauriy is longer, and he underlying sock price and he issuer s asse value are highly negaively correlaed. In addiion, he essenial deerminans of he vulnerable opion values in our model, which are absen in he Black-choles model, are he deb raio of he opion issuer, he reurn volailiy of he issuer s oal asse, and he correlaion coefficiens among he raes of reurn of he opion issuer s oal asse, he underlying sock and he defaul-free zero coupon bond. In paricular, he correlaion coefficien beween he raes of reurn of he issuer s oal asse and he underlying sock is he criical facor in he pricing of vulnerable opions. We generalize he model o ake he randomness of he opion issuer s oal liabiliy ino consideraion and apply he formula o price a vulnerable call opion on a defaul-free zero coupon bond. Our pricing mehodology of he vulnerable opions is analogous o hose of he risky corporae bonds (such as Longsaff and chwarz (1995) and Briys and de arenne (1997)). herefore, one can direcly employ our framework o price he risky derivaives which are allowed o be decomposed ino he bond par and he opion par, such as bonds wih warran and equiy-linked bonds. 6

28 Appendix A Proof of Proposiion Under he forward risk-neural probabiliy measure, i.e., B(, ) is aken as he numeraire, he forward prices of he underlying sock and he opion issuer s oal asse, (, ) and (, ), are maringales wih respec o he forward risk-neural filered * probabiliy space (,,P,( ) 0 ) = Ω. heir dynamics are respecively given by d (, ) P = σ (, ) dw ( ), (A1) (, ) where W P ( ) and d (, ) P = σ (, ) dw ( ), (A) (, ) P ( ) * W are wo Wiener processes on (,,P,( ) 0 ) = Ω wih he insananeous correlaion coefficien, = σ ( u, ) du u du σ σ (, ) ( u, ) du. imilarly, we can employ ( ) as he numeraire and change o he corresponding equivalen probabiliy measure * probabiliy space (,,P,( ) 0 ) = respecively given as follows: P by Girsanov heorem. Consequenly, under he new filered Ω, he dynamics of (, ) and (, ) are d (, ) P = σ (, ) d + σ (, ) ( ) dw, and (A3) (, ) d (, ) P = σ (, ) (, ) ( ) d + σ dw. (A4) (, ) According o Equaions (A1)-(A4), ( ) P E ( ) K 1 { ( ) K, ( ) > } 7

29 ( ) ( ) = K > K K > (, ) P (, ), (, ) P (, ), (, ) ( ) ( ) = (, ) Φ e, e, KΦ e, e,. (A5) By he same way, d (, ) P = σ (, ) (, ) ( ) d + σ dw, (A6) (, ) d (, ) P = σ (, ) d + σ (, ) ( ) dw, (A7) (, ) d (, ) P = ( σ (, ) (, )) (, ) ( ) + σ d + σ dw, and (A8) (, ) d (, ) P = ( σ (, ) (, )) (, ) ( ) + σ d + σ dw. (A9) (, ) According o Equaions (A6)-(A9), ( ) K P E ( ) ( ) 1 * { ( ) K, ( ) } σ ( u, ) du (, ) (, ) P (, ), (, ) ( ) = e K < ( ) K K < (, )P (, ), (, ) σ ( u, ) du = (, ) (, ) e Φ e, e, K (, ) Φ e, e,. (A10) ( ) ( ) rom Equaions (A5) and (A10), we can apply he forward risk-neural pricing mehod o derive he pricing formula of he vulnerable call opion C ˆ( ). inally, by means of he pu-call pariy, we can obain he valuaion formula of he vulnerable pu opion P ˆ( ). 8

30 Appendix B Proof of Proposiion 3 o prove Proposiion 3, we rewrie he final payoff of he vulnerable call opion as ( ) α ( ) * + + C α K K ( ) = (1 ) ( ) ( ) 1. (B1) ( s) min > < s B ( s, ) We firs summarize some basic resuls concerning he funcional of a sandard Brownian moion wih a consan drif erm. he firs well-known resul, which is commonly referred o as an applicaion of he reflecion principle of a sandard Brownian moion (see Harrison (1985), P.15, or Musiela and Rukowski (1997), P.470), is provided in he following lemma. Lemma 1: Le Y1 ( ) = ν1 + σ1w 1( ) be a sandard Brownian moion wih a consan drif, where ν 1 R and σ 1 > 0, and define τ = inf( 0 : Y1 ( ) < 0) and Z( ) = min{ Y ( ) : 0 }. hen for wo consans y > z, we have 1 ν 1 y + ν 1 z y + z + ν1 σ Pr { Z( ) > z, Y1 ( ) y} = Φ e Φ σ σ. (B) 1 1 Nex, for anoher correlaed Brownian moion wih a consan drif, defined by Y ( ) = ν + σ W ( ), where denoes he correlaion coefficien beween hese wo sandard Brownian moions, we can make he following orhogonal ransformaion: W1 W and W W 1 W +, where W and W denoe he wo independen sandard Brownian moions on he same filered probabiliy space. ince Y( ) is independen of he admixure of Z( ) and Y1( ), given Y1( ) = y, we have 9

31 Pr[ Z( ) > z, Y ( ) x] = Pr[ Y ( ) x Y1 ( ) = y] Pr[ Z( ) > z, Y ( ) dy]. 1 z iffereniaing (B) wih respec o y yields Pr[ Z( ) > z, Y1 ( ) dy] and σ x + v + ( y v1 ) σ 1 Pr[ Y ( ) x Y1 ( ) = y] = Φ. Again, by Girsanov heorem, (1 ) we have he following resul: 1 α σ + αv αy ( ) z + ( v 1 + ασ1σ ) x + ( v + ασ ) E e 1 { Z ( ) > z, Y ( ) x} = e Φ,, σ1 σ σ 1 ( v1 + ασ1σ ) z x ( v σ z) α σ + αv σ 1 z + ( v1 + ασ1σ ) σ 1 e Φ,,. (B3) σ1 σ ince he coefficiens of Es of our model are all ime-varying, we employ he ime-changed Brownian echnique o ransform hem ino ime-independence. or example, define 1 P σ σ 0 0, and we can see ha if Y ( ) = ( u, ) du + ( u, ) dw ( u) σ (, ) saisfies he usual condiions (such as σ ( u, ) du < ), hen 0 P σ ( u, ) dw ( ) u is a maringale, 0 P a.s. We can define anoher sopping ime as { } Τ = inf > 0 : A( ), where ( ) σ (, ) 0 A u du, and apply he ime change Brownian moion echnique (see eele (001), P.03, or Karazas and hreve (1991), P.174); consequenly, he ime-changed process Y ˆ( ) Y ( A 1 ( ) ) can be represened as he Brownian moion wih he consan drif erm and diffusion erm, i.e., 30

32 1 Y ˆ( ) = ˆ 1 ( ) + W, [0, A ( )], where W ˆ 1 is a sandard Brownian moion wih respec o ˆ, where ˆ A 1 ( ). By he same way, if we define X ( ) = 1 P σ σ 0 0, hen we have ˆ ( ) ( u, ) du + ( u, ) dw ( u) X = 1 ˆ ( ) + W. Under he forward probabiliy measure and applying he ime change Brownian echnique o Equaion (B3) wih α = 1, we have P P X ( ) B(0, )E ( )1 (0)E e 1 { ( ) K, min ( s, ) > } = K X ( ) ln, min Y ( s) ln 0< s > (0, ) 0< s ( s, ) X (0, ) = (0) Φ h, h, (0, ) (0) Φ h, h, ( ) ( ) (0, ) where h1, h3, h5, h 7, and are evaluaed a = 0. imilarly, when α = 0, we have P P B(0, )E K1 KB(0, )E 1 { ( ) K, min ( s, ) > } = K X ( ) ln, min Y ( s) ln 0< s > (0, ) 0< s ( s, ), (B4) (0, ) = KB(0, ) Φ ( h, h6, ) (0, ) ( 4, 8, ) KB Φ h h, (B5) where h, h 4, h 6, h 8 and are evaluaed a = 0. inally, combining Equaions (B4) and (B5), we can generalize he pricing formula o he value of a vulnerable call opion a ime by he Markov propery of Wiener process. he valuaion formula of a vulnerable pu opion can be derived in a similar way, and hus i is omied. 31

33 References 1. Ammann, M., 001, Credi risk valuaion: mehods, models, and applicaions, pringer-erlag, Berlin, Heidelberg, New York.. Black,., choles, M., 1973, he pricing of opions and corporae liabiliies, Journal of Poliical Economy, 81, Briys, E., de arenne,., 1997, aluing risky fixed rae deb: An exension, Journal of inancial and Quaniaive Analysis, 3, Harrison, J.M., Kreps,.M., 1979, Maringales and arbirage in muliperiod securiies markes, Journal of Economic heory, 0, Harrison, J.M., Pliska,.R., 1981, Maringales and sochasic inegrals in he heory of coninuous rading, ochasic Processes and heir Applicaions, 11, Harrison, J.M., 1985, Brownian moion and sochasic flow sysems, Wiley, New York. 7. Heah,.C., Jarrow, R.A., Moron, A., 199, Bond pricing and he erm srucure of ineres raes: A new mehodology for coningen claim valuaion, Economerica, 60, Heynen, R., Ka, H., 1994, Crossing barriers, Risk, 7(6), Hull, J., Whie, A., 1995, he impac of defaul risk on he prices of opions and oher derivaive securiies, Journal of Banking and inance, 19, Jarrow, R.A., urnbull,.m., 1995, Pricing derivaives on financial securiies subjec o credi risk, Journal of inance, 50, Johnson, H., ulz, R., 1987, he pricing of opions wih defaul risk, Journal of inance, 4, Karazas, I., hreve,., 1991, Brownian moion and sochasic calculus, nd ediion, pringer-erlag, Berlin, Heidelberg, New York. 13. Klein, P., 1996, Pricing Black-choles opion wih correlaed credi risk, Journal of Banking and inance, 0, Klein, P., Inglis, M., 1999, aluaion of European opions subjec o financial disress and ineres rae risk, Journal of derivaives, 6, Klein, P., Inglis, M., 001, Pricing vulnerable European opion when he opion s payoff can increase he risk of financial disress, Journal of Banking and inance, 5,

34 16. Longsaff,.A., chwarz, E.., 1995, A simple approach o valuing risky fixed and floaing rae deb, Journal of inance, 50, Meron, R. C., 1973, he heory of raional opion pricing, Bell Journal of Economics and Managemen cience, 4, Musiela, M., Rukowski, M., 1997, Maringale mehods in financial modelling, pringer-erlag, Berlin, Heidelberg, New York. 19. asicek, O., 1977, An equilibrium characerizaion of he erm srucure, Journal of inancial Economics, 5,

35 Call alues 40 C B C KI 30 C LH 0 C In igure 1. he values of he non-vulnerable and he vulnerable call opions under various values of he underlying sock his figure compares our model (C LH ) o he inrinsic value of he European call (C In ), he sandard Black-choles call opion (C B ) and he vulnerable call of KI (1999) (C KI ). he parameers are aken from KI (1999), referred o as he base case. hey are as follows: ( ) = 100, = 90, K = 40, B(, ) = , α = 0.5, = 3, σ ( ) = 0., σ ( ) = 0., b(, ) = 0.03, = 0, = 0 and = 0. Noice ha in he following numerical analyses, all parameers are he same as he base case, unless oherwise noed. B B 34

36 eb Raio C B C KI C LH eb Raio C B C KI C LH 7.6 eb Raio C B C KI C LH 6.5 igure. alues of he Black-choles calls and he wo vulnerable calls under various deb raios of he opion issuer and values of his figure compares our model (C LH ) o he sandard Black-choles call opion (C B ) and he vulnerable call of KI (1999) (C KI ). 35

37 eb Raio C B C KI C LH B eb Raio C B 8. C KI C LH B eb Raio B C B C KI 7.6 C LH 7.4 igure 3. alues of he Black-choles calls and he wo vulnerable calls under various deb raios of he opion issuer and values of B his figure compares our model (C LH ) o he sandard Black-choles call opion (C B ) and he vulnerable call of KI (1999) (C KI ). 36

38 eb Raio C B C KI B C LH eb Raio B C B C KI C LH eb Raio C B 7.5 C KI B C LH igure 4. alues of he Black-choles calls and he wo vulnerable calls under various deb raios of he opion issuer and values of B his figure compares our model (C LH ) o he sandard Black-choles call opion (C B ) and he vulnerable call of KI (1999) (C KI ). 37

39 able 1 alues of he European call opions under various correlaion coefficiens * Non- ulnerable ulnerable vulnerable call values of call values of Panel Case call values KI (1999) our model 1 Base case ** (10.7%) (1.8%) σ B = (10.9%) (1.9%) 3 = (3.6%) (7.7%) = (0.6%) (17.9%) 4 B = (10.7%) 7.486(1.6%) = 0.5 B (10.6%) (1.9%) 5 B = (1.1%) (13.6%) = 0.5 B (9.6%) (1.1%) 6 = 0.5; = 0.5 B B (1.%) (13.5%) B = 0.5; = 0.5 B (9.5%) (11.7%) B = 0.5; = 0.5 B (11.9%) (13.6%) B = 0.5; = 0.5 B (9.6%) (1.3%) 7 = = = 0.5 B B (4.1%) (7.7%) = B = B = (18.%) (16.7%) * We assume he call is a-he-money, and he values in parenheses are he price differences in percenage, defined by he non-vulnerable call value minus he vulnerable call value and divided by he non-vulnerable call value. ** he parameers of he base case are given in igure 1. 38

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