The Valuation of Option Subject to Default Risk

Transcription

1 he aluation of Option ubject to efault Rik hen-yuan Chen epartment of Finance Ming Chuan Unierity No. 5 Chung-han N. Rd. ec. 5 aipei aiwan el: ext ychen@mcu.edu.tw Abtract here i no ufficient margin ettlement mechanim for unlited option thu the credit rik of the option writer mut be conidered when inetor ealuate the price of unlited option. hi paper intend to extend option pricing model and derie a path-dependent aluation model for option ubject to counterparty default rik. Numerical example how that our model can effectiely identify the impact of counterparty credit rik on ulnerable option alue. he mot important reult of enitiity analyi i that the effect of the correlation between counterparty aet alue and underlying tock price on ulnerable option alue i obcured. Keyword: option option pricing credit rik default rik path-dependent

2 I. Introduction A the need to hedge financial rik increae a ariety of non-exchange option or oer-the-counter option are widely traded between financial intitution and their intitutional cotumer. In contrat to exchange lited option market there i no clearing organization uch a Option Clearing Corporation in the oer-the-counter market requiring that the option writer pay an initial margin and be daily reettled. hu it i unreaonable to aume that option inetor are not expoed to poible default rik from their writer. Johnon & tulz (987) called thi type of option a ulnerable option. he Black & chole (973) option pricing model i not adequate to ealuate the option which are ulnerable to counterparty credit rik. eeral tudie hae focued on the default rik of ulnerable option for example Johnon & tulz (987) Hull & White (995) Jarrow & urnbull (995) and Klein (996). Howeer thee paper conider the credit rik in their pricing model only on the option expiration date. hat i at the expiration date if the aet alue of the option writer i greater than it debt the option writer will make the neceary payment to the in-the-money option holder. On the contrary if the option writer i inolent the in-the-money option holder can only claim a proportion of option payoff on the counterparty aet alue. Under thi aumption the default rik whether the option writer i olent or not i only examined at the date of maturity and the path-dependent proce of aet alue i ignored. o if the aet alue of the option writer i eer below it debt (namely default) prior to expiration date but end with aet alue aboe debt then the in-the-money option holder till can claim full payoff from the writer. In fact thi i eldom the real utiation. Hence the alue of ulnerable option i oerpriced

3 by the Klein (996) model. he purpoe of thi paper i to extend the ulnerable option pricing model in Klein (996) and preent a path-dependent aluation model for option ubject to counterparty default rik. We alo compare the pricing reult of our model with Black & chole (973) default-free option model and Klein (996) pathindependent ulnerable option model. he ret of thi paper i organized a follow. he next ection reiew the literature about the credit rik of option. ection III preent the methodology for pricing path-dependent ulnerable option. ection I analyze the difference of our model with thoe of Black & chole (973) and Klein (996). In ection enitiity analyi will be conducted to illutrate the impact of key ariable on ulnerable option alue. he final ection conclude the paper. II. Literature Reiew efault rik in deriatie ecuritie ha been dicued in preiou reearch for example forward contract in Kane (98) and wap in Cooper & Mello (99). Howeer Johnon & tulz (987) wa the firt paper coering default rik in option. hey conidered many option with writer default rik are written by priate financial intitution that hae limited aet intead of organized exchange and there are no guarantee by a third party. herefore Johnon & tulz (987) aume the option i the ole debt claim of the writer and if the option writer i unable to make a promied payment the ulnerable option holder receie all the aet of the option writer. hey derie cloed form olution for the European option price in ome different cae by auming tochatic procee both for the aet alue of the option writer and the alue of the underlying aet. Johnon & tulz (987) alo examine the impact of credit rik on option alue and how that 3

4 the alue of option ubject to default rik are different from thoe of default-free option. Following Johnon & tulz (987) Hull & White (995) alo preent a model for aluing oer-the-counter option when option holder are expoed to potential default rik. hey extend the Johnon & tulz (987) model and allow the counterparty to hae other equal ranking liabilitie. When the counterparty default option holder are aumed to obtain only a proportion of it no-default claim. Hull & White (995) not only propoe a model for ulnerable option but alo numerically imulate and compare the difference between European call American call and ordinary call option under normal default rik. Allowing both a tochatic proce for the default-free term tructure and the term tructure for riky debt Jarrow & urnbull (995) proide a new technique for aluing and hedging deriatie ecuritie ubject to credit rik. hey derie a rik-neutral pricing procedure by applying arbitrage-free dynamic for thee term tructure. hi methodology can alo be applied to oer-the-counter option corporate debt and other ecuritie. For implicity both Hull & White (995) and Jarrow & urnbull (995) are auming independence between the aet underlying the option and the default rik of the counterparty. On the contrary Klein (996) indicate that thi aumption i not realitic in many buine ituation. he default rik of ulnerable option occur when the aet alue of the counterparty decline a the alue of the option itelf increae a well. hu Klein (996) aume the credit rik of the option writer and the aet underlying option to be correlated and derie a pricing formula which allow for the counterparty to hae other debt claim and the exitence of capital forbearance. Moreoer the proportion of nominal claim for option holder 4

5 in the cae of the counterparty default i endogenou to the model. he Klein (996) model i more practical than thoe of Hull & White (995) and Jarrow & urnbull (995); howeer the author till ignore the poibility that the counterparty may default prior to the option expiration date. On the expiration date once the counterparty aet alue exceed it liabilitie the in-the-money option holder can receie their option intrinic alue no matter whether the aet alue of the counterparty wa eer lower than it debt claim during the option lifetime. Obiouly Klein (996) aume the ulnerable option price i path- independent thu the default rik can be examined only on the expiration date. In next ection thi paper extend the Klein (996) model and derie a more accurate path-dependent pricing model for ulnerable option ubject to counterparty credit rik. III. he Pricing Model for ulnerable Option Follow Klein (996) it i aumed that both the tock price () underlying option and the aet alue () of the option writer follow geometric Brownian motion. Uing the Cox & Ro (976) rik neutrality approach the appropriate procee for and can be hown a: d = rdt + dw () d = rdt + dz () where r denote the rikfree rate and w and z follow tandard Wiener procee. and repreent the intantaneou tandard deiation of underlying tock and option writer aet repectiely. Let ( ) and ( ) be the alue of the underlying tock (option writer aet) at initial time and at maturity date repectiely. hee tochatic procee imply that ln and ln are normally 5

6 ditributed with mean of and ( r.5 ) and tandard deiation of ( r.5 ) and repectiely. Moreoer the joint ditribution for ln and ln are biariate normal a: n ( ln ( r.5 ) ln + ( r.5 ) ρ) + (3) t t where n i the probability denity function of tandard biariate normal ditribution. i the correlation coefficient between ln and ln. If the option writer aet alue t fall below ome amount anytime throughout the option lifetime then we aume the option holder get nothing een though the option i in-the-money. If t neer fall below prior to maturity date and at the expiration date i greater than or equal to then the in-the-money option holder can fully claim their intrinic alue from the counterparty. Howeer if at the expiration date i maller than then the option writer default and the in-the-money option holder can only get a proportion of their claim. Under thee condition the alue of a ulnerable option (C) can be tated a: r C = e E{max( K )[( t t ) + (( α) / t < and < or expreed a: C = e r { max( + max( E max( K )Prob( K ) K )Prob( t ) < [( α) / ][ Prob(the firt time < occur at maturity ] t t < ) )]} (4) (5) intead of r C = e E[max( K )[( ) + (( α ) / < )] (6) in Klein (996) where E denote rik neutral expectation oer and K i the A in dealing with problem financial intitution we allow to be maller than the option writer outtanding liabilitie. In other word there i capital forbearance for the counterparty. 6

7 7 trike price of the option; and i bankruptcy cot expreed a a percentage of the aet alue of the option writer. Equation (4) mean that option holder can t get full claim from the counterparty at maturity day if default occur. hi i an important difference between a ulnerable option and a non-ulnerable option. Beide Equation (4) differ from equation (6) in modeling the poibility of default eent. Equation (6) examine the default condition only on the option expiration date and doen t conider the default poibility prior to maturity. On the other hand the econd term of equation (5) i ignored by equation (4) and the third term of equation (5) i alo different from the econd of equation (4). In fact equation (6) i the alue of a path-independent ulnerable option and equation (4) or (5) i the alue of a path-dependent ulnerable option. Following Rubintein and Reiner (99a 99b) and Briy Bellalah Mai and arenne (998) equation (5) can be oled a follow: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ) ( ) ( ) ( ) ( ) ( ) ( h d d KN h c c N e y y N Ke x x N b b N Ke a a N C r r r ρ α ρ α ρ ρ ρ ρ ρ µ µ + + = + (7) + + = + + = r a r K a ρ ln ln where r b r K b + = + = ln ln [ ] [ ] r x r K x ) ( ln ) ( ln ρ + + = + + = ] ) ( [ln ] ) ( [ln r y r K y + = + =

8 c ln = K + r + + ρ c ln = + r + + ρ d = + + = ln + r + ρ d ln r K ln h( ) = π exp{ [ln( ) ( r ) ] /( )} In equation (7) N i biariate tandard normal cumulatie ditribution function and h() i the denity function for the firt time when the option writer aet alue fall below. Note that the alue of ulnerable option C i not only influenced by the price of underlying ecurity () the olatility of ecurity return ( ) the trike price (K) the lifetime of the option () and rikfree rate (r) but alo the aet alue () the aet olatility ( ) and the liability () of the option writer and the correlation coefficient ( ) between the underlying ecurity and the option writer aet. Comparing equation (7) with the Klein (996) model we can find that the firt line of equation (7) i the ame a the latter but the lat two line of the former are different from the latter. he econd line of equation (7) and h() of the third line in equation (7) are ignored in the latter thu reulting in the Klein (996) model oeretimating the alue of the option with counterparty credit rik. In the next ection we will compare the path-dependent ulnerable option pricing model (7) with Black & chole (973) model and Klein (996) path-independent model. I. Model Comparion In order to compare our model with thoe of Black & chole (973) and Klein (996) we take ame numerical example a in Klein (996). he calculation 8

9 reult are hown in able. All calculation of option alue in able are baed on following parameter alue: =.3 =.3 =.5 r =.4833 =.3333 K = 4 = 4 = 5 = 5 = 5 unle otherwie noted. Black & chole (973) alue and alue calculated by Klein (996) are alo reported in the lat three column for comparion. able alue of ulnerable option and non-ulnerable option Cae = =.5 B & Klein ( = ) Klein( =.5) Bae cae = = = = = = = = K = K = = = = = r =.833 # r = #: he option alue in the lat two row are not reported in able of Klein (996). In order to emphaize hi model can identify the effect of default rik on option alue Klein (996) pecially let the bae alue of counterparty aet be ee Briy et al. (998). 9

10 equal to it debt alue = = 5; namely the counterparty i at the bankruptcy margin when elling the ulnerable option. Hence the probability of counterparty default i extremely high and Klein (996) can differentiate hi model from Black & chole (973) model. For example in the bae cae the alue of Black & chole and Klein are and 3.49 repectiely. (ee the forth and the fifth column in able ) When conidering a more realitic cae for example let the bae alue of counterparty aet be intead of 5 we can ee that both price of Klein model and Black & chole model are equal. In other word when the counterparty i olent enough to protect option holder while writing the oer-the-counter option then Klein (996) model i unable to effectiely differentiate credit rik from Black & chole (973) model due to the low poibility of defaulting to make promied payment by the counterparty. On the other hand in bae cae after deducting the alue of default poibility prior to expiration date which i ignored by Klein (996) our model how the alue of path-dependent ulnerable option with credit rik to be which i ubtantially lower than 3.49 for Klein (996) model. A able how except for = 3 and cae our model conitently indicate that once = = 5 the ulnerable option alue are zero while Klein (996) model till report the option are worthy no matter the change in each ariable. In fact under the bae cae the default poibility of counterparty before option maturity date i ery high the ulnerable option hould be worthle. hu Klein (996) path-independent pricing model will oeretimate the ulnerable option alue. imilarly in = cae our model indicate the ulnerable option alue i 3.69 intead of both in Black & chole and Klein. hi mean when the probability of counterparty default i relatiely low the model preented in thi

11 paper till can point out the credit rik effect in ulnerable option while Klein (996) model doen t work. ake another example = 3 cae in able we can find the ulnerable option alue hould be 3.69 intead of.873 reported in Klein (996). hi pecial cae repreent the counterparty to be inolent in the beginning then the reaonable alue of ulnerable option hould be worthle een negatie not poitie a in Klein. Jut becaue we want to conider the credit rik embedded in ulnerable option no rational inetor will buy inolent option and the option alue would be minu. hi cae how that the reaonable pricing model for option ubject to credit rik mut conider the default probability of the counterparty within the option lifetime. For illutrating the difference between our model and Klein (996) we further calculate ome numerical example reported in able. From able we can eaily find that both price in the lat two column are alway equal in each cae. he reult indicate if the counterparty capital i adequate ( = and = 5 ) there will be no difference between Klein (996) path-independent ulnerable option pricing model and Black & chole (973) non-ulnerable option pricing model. hi i becaue Klein (996) ignore the default rik occurring prior to option maturity and hence oerprice the ulnerable option. In contrat our model how the alue of path-dependent ulnerable option are lower than thoe of Klein (996) and Black & chole (973) except for the cae of =. which mean the default poibility of counterparty i extremely low. he aboe reult can be reconfirmed in able 3 which we let the counterparty aet alue be 8 intead of and other ariable be the ame a in able. Again the price difference between Klein (996) and Black & chole

12 (973) are either zero or ery mall. On the contrary our model till can effectiely differentiate the credit rik from the non-ulnerable option pricing model and Klein model. In next ection we will perform enitiity analyi for the path-dependent ulnerable option pricing model (7). able alue of ulnerable option and non-ulnerable option ( =) Cae = B & Klein Bae cae # = = = = = = = = K = K = = = r = r = #: All calculation of option alue baed on ame parameter alue a in able except for =. Comparing with Klein (996) model Klein ignore the poibility that the counterparty may default within the option lifetime and hence oerprice the ulnerable option with credit rik.

13 able 3 alue of ulnerable option and non-ulnerable option ( =8) Cae = B & Klein Bae cae # = = = = = = = = K = K = = = r = r = #: All calculation of option alue baed on ame parameter alue a in able except for =.. enitiity Analyi In thi ection we analyze the key ariable in model (7) how to affect the path-dependent ulnerable option price. he reult of enitiity analyi are preented in Figure to 7 which all option alue are calculated with the ame parameter alue in able except for i 8 intead of 5 and i. In Figure when initial aet alue ( ) of option writer increae lead both the counterparty debt ratio and default rik to decreae therefore the path-dependent ulnerable option alue increae. At the ame time the price difference between our model and thoe of Black & chole (973) and Klein (996) are getting maller due to the lower default probability. 3

14 We can find that in Figure a the olatility of counterparty aet alue increae the alue of path-dependent ulnerable option decreae. hi i due to the fact that larger aet olatility implie higher default poibility of counterparty which will reduce the ulnerable option alue. Figure he impact of counterparty aet alue on ulnerable option alue Figure he impact of tandard deiation of counterparty aet alue on ulnerable option alue 4

15 he analyi reult of i preented in Figure 3. Our model indicate that the effect of the correlation between counterparty aet alue and underlying tock price on ulnerable option alue i indeterminate. In the cae of negatie correlation between counterparty aet alue and underlying tock price the increae in will caue an increae in ulnerable option alue. On the other hand in the cae of poitie correlation a i increaing the ulnerable option alue decreae. hi finding i ubtantially different from the reult of Klein (996) model which report that the higher the i the higher the ulnerable option alue i. o confirm our finding we alo calculate alternatie numerical example and get the robut reult. 3 Figure 3 he impact of coefficient of correlation between and on ulnerable option alue A in the propertie of non-ulnerable call option Figure 4 to 7 how the 3 We hae changed the parameter alue of K r and repectiely and conitently obtain the indeterminate effect of on ulnerable option alue that i a parabolic 5

16 ulnerable option alue to be poitiely correlated with following ariable: the price of underlying aet the olatility of underlying tock the lifetime and the rikle rate. In other word the higher the price of underlying aet the greater the olatility of underlying tock the higher the rikle rate and the longer the lifetime will reult in the higher the ulnerable option alue. All thee reult are conitent with Black & chole (973) option pricing model. Figure 4 he impact of underlying tock price on ulnerable option alue From Figure to 7 we can alo clearly find the fact that the price gap between Klein (996) model and Black & chole (973) model are alway ery mall een equal to zero no matter how the alue of key ariable change. hee reult how that once under the aumption of well capital adequacy for the option writer Klein (996) model can t truly reflect the effect of counterparty credit rik on ulnerable option. In contrat to Klein (996) model the model in thi paper can relation between ulnerable option alue and a hown in Figure 3. 6

17 effectiely differentiate the default rik from non-ulnerable option in all numerical example except for the cae of change in underlying tock price. Figure 5 he impact of tandard deiation of underlying aet alue on ulnerable option alue Figure 6 he impact of rikle rate on ulnerable option alue 7

18 Figure 7 he impact of lifetime on ulnerable option alue I. Concluion Conidering the credit rik of the option writer thi paper derie a path -dependent ulnerable option aluation model which can effectiely improe Klein (996) pricing formula quality and can truly reflect the impact of counterparty credit rik on ulnerable option alue. After putting the poibility of default rik prior to maturity into pricing model we can find that Klein (996) model oeretimate the ulnerable option alue. he theoretical alue for a path -dependent ulnerable option i lower than thoe of Black & chole (973) model and Klein (996) model. From enitiity analyi we find that a counterparty aet alue rie the ulnerable option alue alo rie. Howeer the increae in the olatility of the counterparty aet alue will lead to a decreae in ulnerable option alue. On the other ide the mot important reult i that the effect of the correlation between 8

19 counterparty aet alue and underlying tock price on ulnerable option alue i uncertainty which i different from the reult of the poitie relation in Klein (996) model. In according with the propertie of non-ulnerable call option we alo ee that the price of underlying aet the olatility of underlying tock the rikle rate and the option lifetime hae poitie effect on the ulnerable option alue. Reference Allen L. and A. aunder Forbearance and aluation of epoit Inurance a a Callable Put Journal of Banking and Finance pp Black F. and M. chole he Pricing of Option and Corporate Liabilitie Journal of Political Economic May-June 973 pp Briy E. M. Bellalah H. M. Mai and F. de arenne Option Future and Exotic eriatie John Wiley & on 998. Cooper I. A. and A.. Mello he efault Rik of wap Journal of Finance pp Cox J. C. and. A. Ro he aluation of Option for alternatie tochatic Procee Journal of Financial Economic 976 pp Hull John C Aeing Credit Rik in a Financial Intitution Off-Balance heet Commitment Journal of Financial and Quantitatie Analyi pp Hull John C. and Alan White he Impact of efault Rik on the Price of Option and other eriatie ecuritie Journal of Banking and Finance 995 pp Jarrow Robert A. and tuart M. urnbull Pricing eriatie on Financial ecuritie ubject to Credit Rik Journal of Finance 995 pp

20 Johnon H. and R. tulz he Pricing of Option with efault Rik Journal of Finance 987 pp Kane E. J. Market Incompletene and iergence between Forward and Future Interet Rate Journal of Finance pp.-34. Appearance and Reality in epoit Inurance: he Cae for Reform Journal of Banking and Finance 986 pp Kaufman G. G. ome hortcoming of Analyzing epoit Inurance a a Put Option: A Communication Journal of Financial Engineering 99 pp Klein Peter Pricing Black chole Option with Correlated Credit Rik Journal of Banking and Finance 996 pp. -9. Rubintein M. and E. Reiner Breaking own the Barrier Rik 99 pp Rubintein M. and E. Reiner Uncrambling the Binary Code Rik 99 pp