VALUATION OF PUT OPTIONS ON LEVERAGED EQUITY

Transcription

1 ALUATION OF PUT OPTIONS ON LEERAGED EQUITY Dr Marco Realdon Department of Economic and Related Studie Helington York YO10 5DD UK 15/1/005 Abtract Thi paper preent new cloed form olution for the valuation of European put option and of "down-an-in" barrier option written on leveraged equity. Unlike in pat literature (Toft and Prucyk, 1997) and in keepingwithempiricalevidence,themodelallowequitytoretainvalue even after the firm default and reorganiation. Thi tylied fact can ignificantly alter the valuation of equity put and "down-and-in" option a bankruptcy cot, bargaining power of equity holder, debt maturity and other firm parameter change. The value of "in-the-money" put often decreae in the firm aet volatility. The model can produce a variety of realitic implied equity volatility "kew". Keyword: equity put option, leveraged equity, default and reorganiation, barrier option, "down-and-in" option. JEL claification: G13; G33. 1 INTRODUCTION Thi paper tudie the valuation of put option and "down-and-in" barrier option written on leveraged equity, whereby equity i itelf a claim on the firm aet and i ubject to default rik. New cloed form olution for valuing put option are provided on the aumption that the underlying equity may retain value even after the firm default and reorganiation. Such aumption (hereafter the "recovery aumption") differentiate thi paper from previou contribution, notably Toft and Prucyk (1997). The focu on the valuation of put option and "down-and-in" option i due to the fact that the "recovery aumption" affect thee 1 option more than other.

2 The "recovery aumption" i upported by ubtantial empirical evidence a reported in Gilon, Long and Lang (1990), Wei (1990), Frank and Torou (1994) and other. Such evidence tell of abolute priority rule violation during firm reorganiation following default, either during private debt renegotiation or during formal bankruptcy proceeding. So equity holder keep a valuable claim on the aet of the firm even after default and even if debtor have not been paid in full. Moreover, even after the firm ha defaulted and it tock ha been delited, the tock can keep trading "over the counter". Thi paper how how thee tylied fact can be reflected in the valuation of equity put option and "down-and-in" barrier option, for uch option appreciate preciely when equity value decreae and the firm approache ditre. Hence cloed form olution for thee option are provided on the aumption that equity can be valuable even after default and that option holder cloe out their poition at the time of default. The latter aumption i buttreed by the proviion of the ISDA 00 mater agreement, which pertain alo to over-the-counter equity derivative, and by the regulation of ome option exchange, in that option poition are "cloed out" and liquidated when the underlying tock i delited following the firm default and bankruptcy. Then the "cloe out" value of the option depend on the firm equity value after bankruptcy. A central reult i that the "recovery aumption", bankruptcy cot and the bargaining power of equity holder during reorganiation can ignificantly affect the valuation of put option and "down-and-in" barrier option. One conequence i that the value of an equity put can decreae in the firm aet volatility a the put i deep in the money and the firm approache default. Moreover, the value of a put decreae in the firm debt average maturity. Thee reult are more material for the valuation of long term put option, uch a "Leap" or over the counter option, ince the longer i the time to expiry, the more likely i the default of the underlying equity and the more critical i the "recovery aumption". The "recovery aumption" a well a debt maturity eem capable to explain a variety of pattern for the volatility "kew" that are implied by oberved equity option price. The paper i organied a follow. After a reference to the mot relevant literature, the equity put valuation model i preented and comparative tatic are performed. Then the analyi move to the implied equity volatility "kew" that the put valuation model can explain and to the valuation of "down-and-in" option. The concluion follow. 1.1 Literature Thi paper view and value equity put option and "down-and-in" option a compound claim, i.e. claim written on equity, whereby equity i itelf a claim on the firm aet. The valuation of equity option a compound claim tarted with Geke (1979)

3 and ha recently developed through the work of Toft (1994), Toft and Prucyk (1997) and Ericon (00). Geke and Ericon view the equity underlying an option a a claim of finite maturity on the firm aet. Intead in Toft and Prucyk and in thi paper equity i viewed a a time independent claim of indefinite maturity, which implifie the option valuation model and eem realitic. Geke, Ericon, Toft and Prucyk concentrate on the valuation of "European" equity call option, wherea thi paper conider the valuation of equity put option and of "down-and-in" barrier option, for which the accurate modelling of financial ditre i much more critical than for the valuation of call option. For example, Toft and Prucyk aumed that equity become worthle and that the equity call option i lot upon the firm default, which i reaonable becaue default implie that the call will almot certainly not be exercied. But an equity put option i not expected to be lot upon default and it eem inaccurate to aume that equity i worthle after default, ince the tock and the put option can keep trading after default even if the tock i delited. The literature on "trategic" tructural model of credit rik, e.g. Anderon, Sundarean and Tychon (1996), Mella-Barral and Perraudin (1997), Fan and Sundarean (000), ha already recognied that in financial ditre equity retain value when debt can be renegotiated. So thi paper bring the reult of uch literature to bear for the valuation of equity option. PUT OPTIONS ALUATION WHEN EQUITY IS STILL ALUABLE AFTER DEFAULT Thi ection preent a valuation model for put option written on leveraged equity, in the pirit of Toft and Prucyk (1997). The firm whoe equity underlie the put option can default and can be reorganied. But unlike in Toft and Prucyk, here the underlying equity retain value even after default and reorganiation, which i a tylied fact featuring alo in variou recent tructural model of credit rik and confirmed by the empirical literature. In particular, the put valuation model follow Fan and Sundarean (000) in auming that the firm default and i reorganied a oon a the firm aet value drop to alowerbarrier. Firm reorganiation avert bankruptcy and the aociated bankruptcy cot. A in Fan and Sundarean, reorganiation take the form of a debt-equity wap, whereby debt holder exchange their debt claim for an equity claim on the firm aet and previou equity holder retain a diluted equity claim. Thi formulation of reorganiation doe not eem retrictive ince a debt-equity wap may alo proxy the payoff to debt and equity holder that are aociated with other form of debt renegotiation, uch a trategic debt ervice or negotiation in a formal bankruptcy proceeding. The uual aumption of tructural model of credit rik underlie alo thi model. In particular perfect market, abence of arbitrage and dynamic market completene are 3

4 aumed. The rik neutral proce of the value of the firm aet follow a geometric Brownian motion uch a d = (r b) dt + dz (1) where b i the aet pay-out rate, r i the default free hort interet rate that i aumed contant over time, i the aet volatility and dz i the differential of a Wiener proce. The model for equity value E ( ) treat debt of finite average maturity a per Leland (1998). So at any time the firm generated net cah flow for equity holder are equal to cf = b C (1 t x )+m(d() F ) () where b i the aet generated cah flow, C are the coupon payment to debt holder, m i the yearly debt "roll-over" rate a in Leland (1998), D ( ) i total debt value, F i total debt face value, t x i the corporation tax rate. In any hort interval dt,a fraction m dt of debt i retired at face value m dt F and ubtituted by newly iued debt worth m dt D ( ), o that the nominal amount of outtanding debt i contant over time. Coupon payment generate a tax hield equal to Ct x. Uing tandard valuation argument, we can deduce that the value of the firm equity E ( ) mut atify the following equation and boundary condition: 1 d E ( ) d de ( ) + (r b) re ( )+cf =0 (3a) d E ( ) + Ct x C + mf (3b) r m + r E ( )=ya (3c) de ( ) = ya (3d) d = where a denote the fraction of aet value that would be lot in cae of aet liquidation and where y i a coefficient compried between 0 and 1 that capture the bargaining power of equity holder in the reorganiation proce that follow default. Condition 3b i a no-bubble condition. Condition 3c tate the payoff to equity holder upon reorganiation at. Condition 3d i a "mooth pating" condition, which implie that equity holder chooe the reorganiation barrier endogenouly o a to maximie equity value E ( ). In thi model firm reorganiation alway prevent liquidation and liquidation cot are never incurred. The y parameter i key. When t x =0, y can be thought of a the fraction of the bankruptcy cot aved through reorganiation that i attributed to equity holder. The olution to equation 3a for equity and the endogenou reorganiation barrier are 4

5 E ( )= C + mf m + r µ C + mf m + r T x ( )= Ct x r C+mF m+r µ qm + (1 ya) + T x ( ) (4) µ µ q 1 = q m Ct x r q (6) (1 ya)(q m 1) with q m = (r b 1 ) (r b 1 ) +(r+m) and q = (r b 1 ) T x ( ) i the value of the tax hield aociated with coupon payment. The balance heet identity i E ( )+D()= + T x ( ),where D ( )= C + mf m + r µ + C + mf m + r (5) (r b 1 ) +r. µ qm + (1 ya). (7) Some remark about thee reult are fitting. The parameter y can capture either the effect of conceion from debt holder to equity holder when debt i privately renegotiated or the effect of violation to the abolute priority rule during a formal bankruptcy proceeding. So the above pricing model eem uitable even when reorganiation tale place within a bankruptcy proceeding. The model jut require an etimate of ya,whereay and a need not be known individually. Note that if y =0, m =0and t x =0,then = C+mF m+r qm Ctx r q reduce to = Cq r(q 1) (1 ya)(q m 1), which i pecial cae of the endogenou default barrier in Leland (1994a). If a =0and t x =0,then = = C+mF m+r q m Ct x r q (1 ya)(q m 1) reduce to = C+mF m+r q m (q,which m 1) i a pecial cae of Leland (1994b). Having preented the model for the claim on the underlying firm, we can now value a put written on the firm equity..1 The put option Standard valuation argument imply that the value of a "European" put option P (,t) on the firm equity E ( ) mut atify the following equation and condition: dp (,t) + 1 d P (,t) dt d dp (,t) + (r b) r P (,t) =0 d (8) P (,T) =max(x E ( ), 0) (9) P (,t) 0 (10) P (,t)=p di (ya,t, ) (11) 5

6 where P di (ya,t, ) i the value of a "down-and-in" put option on ya, with trike X, time to expiry T and with "down-and-in" barrier et equal to. Without lo of generality, today date i t =0o that T i the expiry date and alo meaure the reidual life of the option. Condition 9 i the put option payoff at maturity T. Condition 10 tate that the put i approximately worthle a the firm aet and hence the firm equity E ( ) become very valuable. Condition 11 tate that, a = for ome t T,thefirm i reorganied, equity holder receive ya and the value of the equity put become equal to the value of a "down-and-in" put on ya that i "knocked-in" preciely a and when =. In other word, condition 11 tate that, a the firm i reorganied, the nature of the equity claim on the firm aet irreveribly change from E ( ) to ya and that after reorganiation the put on equity no longer i a claim on E ( ) but a claim on ya. The meaning of and the olution to equation 8 and repective condition are clearer if we write the equity put value a P (,t) =P do (E ( ),t, )+P di (ya,t, ) (1) where P do (E ( ),t, ) i the value of a "down-and-out" put option on E ( ), with trike X, time to expiry T and with "down-and-out" barrier equal to. So, when the underlying firm can default and be reorganied, the equity put P (,t) can be viewed and valued a the um of a "down-and-out" put option on equity value before default E ( ) plu a "down-and-in" put option on equity value after default ya.the "down-and-in" and "down-and-out" barrier are the ame and are equal to the default and reorganiation barrier. Thi put valuation model acknowledge that the put and the equity claim are not lot when the firm default. After default equity i till valuable and the put i till "alive". More preciely, ince upon default and reorganiation equity value i E ( )=ya, after default equity value i ya. And if, a in Toft and Prucyk (1997), we aume that equity i worthle after default or equivalently that y =0, we implicitly aume that P di (0,t, )=0. When valuing call equity option, it may be afe to aume that defaulted equity i worthle, but not o when valuing put option or "down and in" option, whoe value heavily depend on the "down ide" of equity. The comparative tatic below confirm thi point. From tandard reference (e.g. Wilmott page 0, 1998) we know that the value of a "down-an-in" European put option on ya with time to expiry T, trike X and "in" barrier i 6

7 P di (ya, t, )=Xe rt (13) µ µ µ (r b) 1 1 N d, 1 + (N (d 1 ) N (d )) (14) µ µ ya 1 N d, 1 µ (r b) 1 T + (N (d 3 ) N (d 4 )) where d 1 = ln X ( ) ya r 1 T T d = ln r + 1 T (16) T X ln ( d 3 = r + 1 ) ya T (17) T d 4 = ln r 1 T. (18) T Then appendix A how that the olution for the "down-and-out" put on E ( ) i (15) where P do (E ( ),t, )=O (X, t, ) O (E ( ),t, ) (19) O (X, t, )=e rt X (0) µ µ µ N d, 1 µ µ T N d, 1 T + x µ 1 (r b) à µ µ e rt X N d, 1 à Ã! ( ) T N d x, 1!! T O (E ( ),t, )=O (,t)+o (T x ( ),t) O (D ( ),t) (1) O (,t, )=e bt N µ (r b) à e bt N µ µ µ µ d, 1 N d µ d µ, 1 N, 1 + () x à Ã!!! ( ) d x, 1 7

8 µ O (D ( ),t, )=e ( r+n(qm))t C + mf m + r µ 1 qm µ qm Ω 1 (,T) à µ µ rt C + mf + e N d, 1 m + r à rt C + mf e m + r N n(qm) qm + (1 ya) (3) ( ) qm Ω (,T) + µ 1 r b µ µ T N d, 1 à µ µ d, 1 µ 1 r b T N x d à ()! T! x, 1!! T O (T x ( ),t, )= e ( r+n(q))t Ct µ q x 1 (4) r µ n(q) q q Ω 3 (,T) ( ) q Ω 4 (,T) + + e rt Ct x r e rt Ct x r à N à N µ µ d, 1 µ 1 r b µ µ T N d, 1 à µ µ d, 1 µ 1 r b T N x d à ( )! T! x, 1!! T ³ ³ ³ ³ with Ω 1 (,T) =N d,q m N d x,q m, Ω (,T) =N d,q m ³ ³ ³ ³ ³ ³ N d (),q x m, Ω 3 (,T)=N d,q N d x,q, Ω 4 (,T)=N d,q ³ ³,q N d ( ), d (z,w) = w ln(z)+(n(w)+ 1 w )T x w and n (w) =(r b)w+ 1 T w (w 1). x need to be found numerically and i uch that E ( x )=X. Equation 0, 1,, 3, 4 provide the value of claim that pay repectively X, E ( ),, D ( ) and T x ( ) at maturity T if at T equity value i ya <E( ) <X, and only if aet value ha not reached the "down-and-out" barrier before T. Having preented the put valuation model, the comparative tatic of uch model follow.. Comparative tatic We conider a bae cae cenario with the following bae cae parameter: X =50, b =3%, = 0%, r =4%, F =50, m =0, C =0.05 F, a = 0%, y =1, t x =0, T =1. Figure 1 how how put value in the bae cae when y i either 1 or 0, whereby y i a proxy for the bargaining power of equity holder during reorganiation. Figure 1 how that the range of variation of put value can be wide a the bargaining power of 8

9 equity holder varie. Put value decreae in y, ince equity value E ( ) increae in y. Put value exhibit a kink when equal (notice that decreae a y decreae). The kink i due to default and reorganiation of the equity claim. After reorganiation the equity value function E ( ) become le teep in. The kink ugget that E ( ) i concave in the firm aet value a approache from above. It follow that oon before default and reorganiation put value decreae if the firm aet volatility increae. Thi feature i abent in pat compound option model. Figure 1 how that a increae and the put get more "in-the-money", a rie i y produce a greater abolute decreae in put value. The reaon i that a rie in y produce a greater abolute increae in equity value E ( ) when i low and that the the abolute value of the put "delta" i greater when i low. Intead, a rie in y caue a greater percentage drop in put value when i low and the put i le "out of-themoney". Since in the put valuation formula the bargaining power parameter y i everywhere multiplied by the bankruptcy cot parameter a and vice vera, the effect on put value of a given percentage change in y i the ame a the effect on put value of that ame percentage change in a. So all that ha been aid about the effect of y on put value P (,t) i valid alo for the effect of a on P (,t). Thu the aumption about the recovery value of equity after default, the bargaining power of equity holder and potential bankruptcy cot can ignificantly affect equity put option value. Thi i even more the cae a T increae. Figure how how put value P (,T) change around the bae cenario value a, ceteri paribu, aet volatility, aet payout rate b and debt average maturity m change in turn. Put value decreae in aet payout b, incee ( ) increae in b. Putvaluemay decreae a well a increae in the firm aet volatility. Figure how that, a uual, put value increae in aet volatility when the put i not "deep-in-the-money", but decreae in volatility when the firm approache default and the put become "deepin-the-money". In fact, epecially when approache, equity value E ( ) increae in volatility. Moreover, a approache, Figure 1 how how put value i a kinked concave function of the firm aet value. Unlike in Toft and Prucyk (1997), thi put valuation model encompae the cae in which the firm debt ha finite average maturity. Figure how how put value generally increae in m and debt average maturity 1 m decreae. The reaon i that, ceteri paribu, horter debt maturity decreae equity value E( ) when approache the default barrier and the option i "in-the-money". Moreover, increaing m increae. When i very high, horter maturity increae equity value, but the option i "out-of-the money". A the put get very "far from the money", change in m produce ignificant proportional change in put value. Put value decreae in r becaue equity value rie in r and becaue the preent value of the final payoff decreae in r. Athefirm leverage rie, i.e. a F and C rie, the default barrier rie, o that reorganiation become a more likely propect and bankruptcy cot increae too ince uch cot are equal to a. A a reult put 9

10 value become more enitive to the propect of firm reorganiation and to y and a. Put value uually increae a time to expiry T get longer, but not o when the put i "deep-in-the-money" and the firm approache default, ince put value after default decreae in T. Finally, unreported imulation howed that alo in the preent model early exercie of the "American" put can be optimal even in the abence of dividend. We can conclude that the difference between the above comparative tatic and thoe of ordinary put confirm the ignificant impact that firm reorganiation can have on put value..3 When market are incomplete The model above aume dynamic market completene, which allow u to regard the value of the firm aet a the price of a traded aet and to aume that it rik neutral proce i a in equation 1. But even if the market i incomplete the propoed model i till valid if only (r b) i ubtituted with (n λ),wheren i the real drift of and λ i the market price of -rik. The reaon for thi adjutment i that market incompletene caue the rik neutral proce for to be no longer a per equation 1 but d = (n λ) dt + dz. (5) Market incompletene eem more appropriate an aumption when the firm tock i not traded in the tock market or when the tock ha been delited after default. In thi regard, Ericon (1998) argue that, a long a the firm tock i traded, the value proce of the firm aet can be replicated by trading in the tock. But replication i more unlikely after deliting becaue the tock would not be trading in the tock market any more. So we may want to aume market incompletene jut after default. In uch cae (n λ) hould ubtitute (r b) jut in equation 13, 15, 16, 17 and Calibrating the model and the implied volatility kew The above preented put valuation model depend on more and different parameter than the Black and Schole put model. Some parameter like m, F and C can be etimated from balance heet data, ome like r and t x from the financial environment and ome like b,, a and y can be inferred from (the time erie of) equity price and put price. The parameter b,, a and y can be imply "calibrated" to preent equity and put price or etimated through a maximum likelyhood method from time erie of price a in Ericon and Reneby (001). Such parameter offer more "flexibility" in calibrating the put valuation model than the Black and Schole model doe. If the above model i ued in ubtitution to the Black and Schole model, need to employ implied volatility kew to explain the oberved price of put option with different trike and expiry date can be eliminated. Moreover, the calibration of the above model provide parameter etimate that can be ued to value alo the firm debt. In other word the propoed model etablihed a link between equity put option and pread on the firm debt. 10

11 A virtue of the propoed put valuation model i that it can predict variou pattern of implied equity volatility "kew" a y, a, m, F and b change a hown in Table 3. Overall the model eem quite capable of explaining the type of volatility kew that can be etimated from oberved option price. The firt two ection of table 3 how that, even when the firm i far from default ( = 00 and F = 50) implied volatility decreae a bankruptcy cot and the bargaining power of equity holder (i.e. a a and y) increae. The third ection how how implied volatility rie a debt average maturity ( 1 m ) decreae and the fourth how how volatility increae a leverage increae. A volatility kew can be detected in all the ection of the table, whereby implied equity volatility i high when the trike price of the put i lower, in keep with empirical evidence and with the fact that leveraged equity i more volatile when the firm approache default. After analying how the firm default and reorganiation affect the valuation of plain put option, the following ection turn to the valuation of barrier option. 3 ALUATION OF BARRIER OPTIONS Like put option, "down-and-in" barrier option on leveraged equity are here of interet becaue their value i more enitive than the value of other option to the "recovery aumption" for defaulted equity. Like put option, "down-and-in" option are amenable to cloed form olution. Given the above framework and the "recovery aumption" for equity value after default, the formula for a "down-and-in" put with "in" barrier at E ( u )=U uch that x > u > i: P di (,t,u) =P (,t) P do (,t,u) (6) where P (,t) i a before given by equation 1, 19, 0, 1,, 3, 4 and where P do (,t,u) i the value of a "down-an-out" put on equity E ( ) with "out" barrier at E ( u )=U, with trike X and reidual life T. It follow that P do (,t,u) =O (X, t, U) O (E ( ),t,u) (7) where, if only i ubtituted with u, the formula for O (X, t, U) i the ame a the right hand ide of equation 0 and the formula for O (E ( ),t,u) i the ame a the right hand ide of equation 1. Notice that, ince u > the value of the "down and out" put option P do (,t,u) i not affected by default and the "recovery aumption", ince the put i "knocked out" before default. Intead the value of the plain put P (,t) i clearly depend on default and the "recovery aumption" a apparent from equation 1, 19, 0, 1,, 3 and 4. So from equation 6 it follow that the value of the "down-and-in" put on leveraged equity P di (,t,u) alo depend on default and the "recovery aumption". Moreover, any percentage change in P (,t) produced by a change in the recovery parameter y or a will correpond to an even greater percentage change in P di (,t,u), preciely becaue y or a affect P (,t) but do not affect P do (,t,u). 11

12 Similar formula and argument are valid alo for "down-and-in" call option a hown in appendix B. Thee argument highlight that the precie modelling of equity during financial ditre i particularly important for both "down-and-in" put and call option, but not for "down-and-out" option. 4 CONCLUSIONS Thi paper ha preented new cloed form olution for the valuation of equity put option and "down-and-in" barrier option on leveraged equity. The baic aumption ("recovery aumption") i that uch option are not lot and that equity retain value even after the firm default, in keeping with empirical evidence on the recovery value of the firm equity after default. The "recovery aumption" ditinguihe thi paper from pat literature, in particular from Toft and Prucyk (1997). The analyi ha hown that the equity put option i very enitive to the "recovery aumption", to bankruptcy cot, to the bargaining power of equity holder during reorganiation and to debt average maturity. Unlike i the Black and Schole model, equity put value decreae in aet volatility when the put i "deep-in-the-money" and the firm approache ditre. Thee reult are more material when the life of the option i longer."down-and-in put" option are much more enitive then plain put to the modelling of ditreed equity and to the "recovery aumption". The "recovery aumption" enable the put valuation model to predict a richer et of hape of implied volatility kew of the type oberved in the equity option market. Thee reult alo ugget that equity call option price rather than equity put option price hould be ued to imply equity volatility, ince the former, unlike the latter, are relatively inenitive to the "recovery aumption" of ditreed equity. Finally future reearch can employ oberved equity price, put option price and the cloed form olution here preented to etimate the firm aet value, drift, volatility, bankruptcy cot and bargaining power parameter to be ued in pricing the firm debt. Appendix A. Derivation of the Put option formula Thi appendix derive equation 19, 0, 1,, 3 and 4. ³ We can write D ( )= C+mP r+m + A q m, with A = C+mP r+m +(1 ay) ( ) q m. Then by applying Ito lemma it follow that where d (D ( )) = d (A q m )=n (q m ) A qm dt + A qm q dz n (w) =(r b)w + 1 w (w 1). So the term (A q m ) follow a geometric Brownian motion. (A-1) 1

13 Then the preent value of a claim Q (D( ), x,t) that pay D ( )= C+mP r+m + A qm at time T if T x and that pay nothing otherwie i: µ Q (D( ), x,t)=q(a q m C + mp, x,t)+q r + m, x,t (A-) where Q (A qm, x,t) = A qm e ( r+n(qm))t N = e rt C + mp µ r + m N d µ C + mp Q r + m, x,t µ µ d µ x, 1,q m (A-3) x T (A-4) and where N (u) i the cumulative of the tandard normal denity with u a the upper limit of integration and with d (z,w) = w ln (z)+ n (w)+ 1 w T w. (A-5) T Uing reult for valuing "down-and-out" barrier option (ee e.g. Wilmott (1998) at page 19), the value of a claim Q (D ( ),, x,t) that pay at time T only if T x and only if t > for any time t<t, and that pay nothing otherwie i: µ Q (D ( ),, x,t)=q(a q m C + mp,, x,t)+q r + m,, x,t (A-6) where Q (A q m,, x,t)=a e ( r+n(q m))t (7) µ µ µ n(qm) à Ã!! q m qm N d,q ( m ( ) qm ) N d,q m x x and µ C + mp Q r + m,, x,t = C + mp r + m e rt (8) à µ µ N d, 1 µ 1 r b à Ã! T ( ) N d, 1!! T. x x Then we can write 13

14 O (D ( ),t)=q(d(),,,t) Q (D ( ),, x,t) which give equation 3. Similarly we can derive (A-9) O (,t) =Q (,,,T) Q (,, x,t) O (T x ( ),t)=q(t x ( ),,,T) Q (T x ( ),, x,t) O (X, t) =Q (X,,,T) Q (X,, x,t). (A-10) (A-11) (A-1) Finally, boundary condition 9, 10, 11 and the linearity of partial differential equation 8 allow u to write equation 1 and P do (E ( ),T)=O (,T)+O (T x ( ),T) O (D ( ),T). (A-13) Appendix B. The valuation of barrier call option on leveraged equity For completene the value of a "down-and-in" barrier option on leveraged equity i here reported. The value of a "down-and-in" call C di (,t,u) on E ( ) with "in" barrier at E ( u )=U uch that x > u >, with maturity T and trike X i: C di (,t,u) =C (,t) C do (,t,u) where C (,t) i the value of a call option on leveraged equity E ( ) given that equity after default and reorganiation i worth ayand where C do (,t,u) i a "down-and-out" call on E ( ) with "out" barrier E ( u )=U. C (,t) and C do (,t,u) alo have trike X and maturity T. We can derive a cloed form olution for C (,t), ince before default put-call parity allow u to write: C (,t)+xe rt = P (,t)+c 0 (,t),wherec 0 (,t) i the value of call imilar to C (,t) but which ha 0 trike price. C 0 (,t) i equivalent to the right to receive equity E ( ) at time T. Then, employing argument imilar to the one in appendix A, we can write C do (,t) =O c (E ( ),t) O c (X, t) (B-1) with O c (X, t) =e rt X N µ µ µ 1 (r b) Ã Ã!! d, 1 (u) N d x u x, 1 (B-) 14

15 O c (E ( ),t)=o c (,t)+o c (T ( ),t) O c (D ( ),t) (B-3) O c (,t) =e bt O c (D ( ),t)= qm N N µ C + mf m + r µ µ d,q m x µ µ µ (r b) à Ã!! d, 1 (u ) N d x u x, 1 (B-4) µ + (1 ya) e r+n(q)t (5) u n(qm) à Ã!! qm ( ( u ) qm u ) N d x,q m à + C + mf µ µ m + r e rt N d, 1 + µ 1 r b à Ã! T (u ) N d x u x, 1 +!! T. O c (T x ( ),t)= Ct x r e r+n(q)t (6) µ µ µ n(q) à Ã!! q q N d,q ( u ) q ( u ) N d x x,q + Ct x r e rt à N BIBLIOGRAPHY u µ µ d, 1 + µ 1 r b à Ã! T (u ) N d x u x, 1 +!! T. [1] Anderon R., Sundarean S. and Tychon P., 1996, "Strategic analyi of contingent claim", European Economic Review 40, [] Black F. and Schole M., 1973, The pricing of option and corporate liabilitie, Journal of political economy [3] Ericon J. and Reneby J., 1998, "On the tradeability of firm aet", Working paper 89 Stockholm School of Economic. [4] Ericon J., 000, "Stock option a barrier contingent claim", SSI/EFI Working paper erie in Economic and Finance n

16 [5] Ericon J. and Reneby J., 001, "The valuation of corporate liabilitie: theory and tet", European Finance Aociation 001. [6] Fan H. and Sundarean S., 000, Debt valuation, renegotiation and optimal dividend policy, Review of financial tudie 13, n.4, [7] Frank J. and Torou W., 1989, "An empirical invetigation of US firm in reorganization", Journal of finance 44, n.3, [8] Frank J. and Torou W., 1994, "A comparion of financial recontracting in ditreed exchange and in Chapter 11 reorganization" Journal of financial economic 35, [9] Geke R., 1979, "The valuation of compound option", Journal of financial economic 7, n.1, [10] Gilon S. and Long K. and Lang L., 1990, "Troubled debt retructuring: an empirical invetigation of private reorganiation of firm in default, Journal of Financial Economic 7, [11] Goldtein R.,Ju N. and Leland H., 001, "An EBIT-Baed Model of Dynamic Capital Structure", Journal of Buine 74, n.4. [1] Leland H., 1994a, Corporate debt value, bond covenant and optimal capital tructure, Journal of finance 49, n.4, [13] Leland H., 1994b, Bond price, yield pread, and optimal capital tructure with default rik, Univerity of California at Berkeley Reearch Program in Finance working paper. [14] Leland H., 1998, Agency cot, rik management and capital tructure, Journal of finance 53, n.4, [15] Mella-Barral P. and Perraudin W., 1997, Strategic debt ervice, Journal of finance 5, n., [16] Mella-Barral P. and Tychon P., 1999, Default Rik in Aet Pricing, Forthcoming in Finance. [17] Toft K.B. and Prucyk B., 1997, "Option on leveraged equity: theory and empirical tet", Journal of Finance 5, n.3,

17 Figure 1: alue of put P (,T) with bae cae parameter a y change from 0 to 1. [18] Wei L., 1990, "Bankruptcy reolution: direct cot and violation of priority of claim", Journal of Financial Economic 7, [19] Wilmott P., 1998, "Derivative: the theory and practice of financial engineering", Wiley. 17

18 Figure : Difference in put value P (,T) from the bae cae cenario a, ceteri paribu, = 30%, b =6%and m = 40%. 18

19 Figure 3: The table how the effect on implied volatility of departure from the bae cae cenario in which = 00, X =50, b =3%, = 0%, r =4%, F =50, m =0, C =0.05 F, a = 0%, y =1, t x =0, T =1. 19