Local Volatility Enhanced by a Jump to Default

Transcription

1 Local Volailiy Enhanced by a Jump o Defaul Peer Carr Bloomberg LP and Couran Insiue, New York Universiy Dilip B. Madan Rober H. Smih School of Business Universiy of Maryland College Park, MD Ocober 25, 27 Absrac A local volailiy model is enhanced by he possibiliy of a single jump o defaul. The jump has a hazard rae ha is he produc of he sock priceraisedoaprespecified negaive power and a deerminisic funcion of ime. The empirical work uses a power of 1.5. I is shown how one may simulaneously recover from he prices of credi defaul swap conracs and equiy opion prices boh he deerminisic componen of he hazard rae funcion and revised local volailiy. The procedure is implemened on prices of credi defaul swaps and equiy opions for GM and FORD over he period Ocober 24 o Sepember Inroducion The developmen of liquid markes in credi defaul swaps has made i clear ha some of he value of deep downside pu opions mus be due o he probabiliy of defaul rendering equiy worhless. Sock price models ha assume a sricly posiive price for equiy poenially oversae downside volailiies o compensae for he assumed absence of he defaul even in he model. To recify his siuaion one needs o recognize he defaul possibiliy in he sock price model. This could be done in a variey of parameric models (Davis and Lischka (22), Andersen and Buffum (23), Albanese and Chen (25), Linesky (26), Alan and Leblanc (25)) and one could hen seek o infer he parameers of he model from raded opion prices. Having done so one would exrac in principle a risk neural defaul probabiliy from opion prices ha could hen be compared o a similar probabiliy obained from he prices of credi defaul swaps. Dilip Madan acknowledges he Financial Inegriy Research Nework, An ARC nework in parially supporing he work on his paper during his erm in Ausralia as a FIRN visior. 1

2 Alernaively one could recognize ha opion prices consiue an indirec assessmen of defaul while he credi defaul swap is clearly more direcly focused on his even. This suggess ha we joinly employ daa on credi defaul swaps and equiy opions o simulaneously infer he risk neural sock dynamics in he presence of defaul as a likely even. Such an approach is all he more called for in he conex of local volailiy sock price dynamics (Dupire (1994), Derman and Kani (1994)) ha inroduce a wo dimensional local volailiy surface describing he insananeous volailiy of he innovaions in he sock price as a deermininsic funcion of he sock price and calendar ime. All he equiy opion prices are hen needed o infer he local volailiy surface and one needs oher raded asses o infer he parameers relaed o he defaul likelihood. Given he populariy of local volailiy in assessing equiy risks we consider enhancing his model o include a defaul even. For oher enhancemens ha have recenly been considered we refer he reader o Ren, Madan and Qian (27). One could seek o perform such an enhancemen by allowing for he sock o diffuse o zero and o hen be absorbed here. There are such models referenced above ha lack a local volailiy srucure and one may seek o enhance hese wih a local volailiy formulaion. However, i has long been recognized ha a purely diffusive formulaion has difficulies wih credi spreads especially a he shorer mauriies. Hence he populariy of reduced form credi models (Jarrow and Turnbull (1995)) ha incorporae jumps in he asse price. In he ineress of parsimony we inroduce jus a single jump o defaul wih a hazard rae or insananeous defaul probabiliy ha is jus a funcion of he sock price and ime. Unlike local volailiy and is eye on he wo dimensional surface of implied volailiies we recognize ha our addiional asses are a bes he one dimensional coninuum of credi defaul swap quoes and so we model he hazard rae as a produc of wo funcions, one a deerminisic funcion of calendar ime while he oher is jus given by he sock price raised o a negaive power. The laer recognizes he inuiion ha defaul ges more likely as equiy values drop. Wih his enhancemen we show how one may use credi defaul swap and equiy opion prices joinly o infer he deerminisic funcion of calendar ime in he hazard rae and he local volailiy surface of his defaul exended local volailiy model for a prespecified level of he elasiciy of hazard raes o he sock price. The procedures developed are implemened on credi defaul swap and equiy opion prices for FORD and GM covering he period Ocober 24 o Sepember 27. I is observed ha on average he elasiciy of he proporion of radiional local volailiy aribued o he defaul componen declines wih respec o srike and is.4768 and.3967 for GM and FORD respecively. This proporion is posiively relaed o mauriy wihelasiciies of.646 and.54 for GM and FORD respecively. Addiionally he deerminisic componen of he hazard funcion is generally increasing wih mauriy and has semielasiciies and elasiciies of.136 and.2163 for GM and.975 and.2641 for FORD. The ouline of he res of he paper is as follows. Secion 2 presens he 2

3 formal model for he sock price evoluion. In secion 3 we develop he equaions for he deerminisic componen of he hazard rae and he defaul enhanced local volailiy surface. Secion 4 presens he mehods employed o evaluae runcaed power prices in he embedded defaul free dynamics ha are needed for he local volailiy consrucion. Secion 5 presens he applicaion of he mehods o FORD and GM and summarizes he resuls. Secion 6 concludes. 2 Local Volailiy enhanced by a Single Jump o Defaul We enhance a local volailiy formulaion for he risk neural evoluion of he sock price by including a single jump o defaul. The jump occurs a random ime given by he firs ime a couning process N() jumps by uniy. Thereafer we se he arrival rae of jumps o zero and he process N() remains frozen a uniy. Prior o he jump in he process N() i has an insananeous arrival rae λ() of a jump given by he produc of a deerminisic funcion of ime f() and he sock price raised o he power p, ha will be a negaive number in our applicaions, λ() = 1 N( _ ) f()s(_) p. Le σ (S, ) denoe he volailiy in he sock price when he sock price is a level S a ime and defaul has no ye occurred. The sock price dynamics may hen be wrien as ds = (r() q())s(_)d + σ(s(_),)s(_)dw () (1) S(_)[dN () (1 N(_))f()S(_) p d] where W () is a Brownian moion and r(),q() are deerminisic coninuously compounded ineres raes and dividend yields respecively. Equaion (1) may be explicily solved in erms of he produc of wo maringales, one coninuous, M(), and he oher ha is coninuous unil i jumps o zero and hen i says here, J(). The sock price may also be wrien as S() = A()M()J() µz A() = S() exp (r(u) q(u))du µz Z M() = exp σ(s(u_,u))s(u_)dw (u) 1 σ 2 (S(u_,u))S(u_) 2 du 2 µz J() = exp (1 N(u_))f(u)S(u_) p du (1 N()) The only pah o defaul or a zero equiy value is he jump in he process N() o he level 1. The dynamics is risk neural and he sock growh rae is he ineres rae less he dividend yield. 3

4 Embedded in he sock evoluion subjec o he possibiliy of defaul is he defaul free sock price model, or he law of he sock price condiioned on no defaul or equivalenly on being posiive. This is a useful process and we shall use i in reconsrucing he local volailiy funcions in paricular from quoed opion prices. Hence we now proceed o describe his process ha we henceforh refer o as he defaul free process. We begin by noing ha he probabiliy of surviving unis of ime is given by µ Z P (no defaul o ) = V () = E exp f(u)(1 N(u_))S(u_) p du (2) Consider now any pah dependen claim paying a, F (S(u),u ) if here is no defaul ill ime, and zero oherwise. The ime forward price, w, of his claim is µ Z w = E exp λ(u)du F (S(u),u ). (3) Le Qe be he defaul free law or he law of he sock condiional on no defaul o ime. I follows by definiion ha Q w = V ()E Q [F (S(u),u )]. (4) Combining equaions (??) and (4)weobserve ha ³ R dq e exp λ(u)du = h ³ R i. dq E exp λ(u)du Define by p(s, ) he densiy of he sock a ime under he measure Q. e By consrucion his densiy inegraes o uniy over he posiive half line and i may be exraced from daa on defaul free prices or opion prices under he law Q. e Of paricular ineres is he growh rae of he sock under he law Q. e This is deermined on evaluaing he expecaion of S() under Q. e This is given by ³ R exp λ(u)du Q E [S()] = E Q h ³ R is() E exp λ(u)du ³ ( R exp λ(u)du) exp (r(u) q(u))du ³ E[exp( λ(u) du )] R = S()E Q R 1 exp σ(s(u_,u))s(u_)dw (u) 1 σ 2 (S(u_,u))S(u_) 2 du ³ 2 R exp λ(u)du ³ R E Q [exp (r(u) q(u))du = S() V () 4

5 as under Q e, N() =. Le us now wrie µ Z V () =exp h(u)du where by consrucion log V () h() =. We hen obain ha µz E QQ [S()] = S() exp (r(u) (q(u) h(u))du. We hen see ha under he Qe measure we have a dividend yield adjusmen o q a () =q() h(). (5) The exposure o he defaul hazard appears in he defaul free model as a negaive dividend yield of h(). This is quie inuiive as an operaional way o ge defaul free is o buy insurance agains i a he premium flow h() ha is an expense o be paid ou of he dividend sream. Hence he reduced dividend flow. Defaul free opion prices or prices of opions under he measure Q e may be consruced by a simple ransformaion. Suppose we have esimaed V () he survival probabiliy curve from CDS prices. We can hen consruc prices of defaul free call and pu opions by C(K, ) ec(k, ) = (6) V () P (K, ) Ke r(u)du (1 V ()) ep(k, ) = (7) V () We also have by definiion of p(s, ) ha hese prices saisfy he equaions Z r(u)du e C(K, ) = e (S K)p(S, )ds K Z K r(u)du e P (K, ) = e (K S) p(s, )ds. We may evaluae he pu call pariy condiion for hese prices and observe ha C ec(k, ) P e (K, ) = V () P Ke r(u)du (1 V ()) V () S()e q(u)du r(u)du = Ke. V () 5

6 We see again ha for hese defaul free prices he value of he forward sock is as shown earlier S() e q(u) du E[S()] e = V () We may define erm dividend yields and hazard raes by and wrie R q(u)du qe() = R h(u)du η() = Ee[S()] = S()e (qq() η()) We may now apply any sandard defaul free model o he prices Ce(K, ), Pe (K, ) wih he dividend yield adjused o q a () = qe() η() o recover he defaul free densiies p(s, ) and we shall shorly see how we use his densiy o build he deerminisic componen of he hazard rae f() and he local volailiy funcion. 3 Recovering hazard and volailiy funcions from CDS and Opion Markes The firs sep is o recover he survival funcion V () from CDS quoes and for his we employ he Weibull model for he life curve and he mehods described in Madan, Konikov and Marinescu (26). We nex observe on differeniaing ln V () wih respec o ha 1 V () = Q f()e [S() p ] (8) V () Hence he funcion f() may be recovered from V () and he prices of opions under Q. e We see immediaely how we will use he law of S() under Qe or he densiy p(s, ) o recover he funcion deerminisic componen of hazard raes or he funcion f(). In fac he characerisic funcion of ln S() under Qe evaluaed a ip gives us direcly he defaul free power price E QQ [S() p ] from which we may consruc f() in accordance wih equaion (8). For he recovery of he local volailiy funcion wih deerminisic ineres raes we proceed as follows. By definiion marke call prices are given by µ Z µ Z C(K, ) =exp r(u)du E exp λ(u)du (S() K) + 6

7 Differeniaion wih respec o K yields µ Z r( u)du C K = e E exp λ(u)du 1 S()>K µ Z C KK = C = e KK r(u)du E exp λ(u)du 1 S(_)=K µ Z C KC K = E exp λ(u)du S()1 S()>K Applying he Meyer Tanaka formula (Meyer (1976), Dellacherie-Meyer (198), and Yor (1978)) o he call price payoff yields Z (S() K) + = (S() K) + + (1 N(u_))1 S(u_)>K ds(u) Z 1 + (1 N(u_)) 1 S(u )=K σ 2 (S(u),u)S(u) 2 du 2 Z +K 1 S(u_)>K (1 N(u_))dN (u) Taking expecaions we ge Z µ Z u e r(u)du C(K, ) = (S() K) + + E exp λ(v)dv S(u_)(r(u) q(u)) du Z µ Z u 1 + E exp λ(v)dv 1 S(u_)=K σ 2 (K, u)k 2 du 2 Z µ Z u +K f(u)e exp λ(v) S(u_) p 1 S(u_)>K du Differeniaing wih respec o and muliplying by he discoun facor we ge ha µ Z r()c(k, )+C = e r(u)du E exp λ(u)dv S(_)(r() q())1 S(_)>K µ Z 1 + e r(u)du E exp λ(u)du 1 S(_ )=K σ 2 (K, )K 2 2 µ Z +Ke r(u)du f()e exp λ(u)du S(_) p 1 S(_)>K I follows ha 1 r()c(k, )+C = (r() q()) (C KC K )+ K 2 CKK σ 2 (K, ) 2 +Kf()V ()e r(u)du E Q S(_) p 1 S(_)>K 7

8 We define he runcaed power price under Qe as ζ(k, ) Hence we consruc ζ(k, ) =e r(u)du E Q S(_) p 1 S(_)>K (9) 2 C + q()c +(r() q())kc K Kf()V ()ζ(k, ) σ (K, ) =2. (1) K 2 C KK Boh he funcions f() and σ(k, ) may he be recovered from he prices of opions under Q. e We consider in he nex secion he procedures for consrucing he runcaed power prices (9) under Q. e We may usefully rewrie equaion (1) in he form 2 2 Kf()V ()ζ(k, ) C + q()c +(r() q())kc K K σ (K, )+ = 2 (11) C KK = σ 2 LV (K, T ) C KK where he righ hand side of equaion (11) is he radiional local dollar variance formulaion of Dupire (1994) and Derman and Kani (1994). We hen see ha he new local variance is reduced by he survival probabiliy imes he expeced hazard in he region S() >K from where we can jump o zero and earn K dollars on he call, relaivized by he densiy a K. We shall repor on he relaive magniudes of he diffusion componen and he jump componen in he pariioning of Dupire local variance. 4 Truncaed Power Prices We use CDS quoes o consruc Weibull densiy parameers for he survival funcion V (). We hen ransform marke prices o defaul free prices using equaions (6,7). The dividend yields are adjused using equaion (5). One may now esimae on his daa wih hese adjused dividend yields any defaul free model of opion prices. We employ for he purpose he V GSSD model repored on in Carr, Geman, Madan and Yor (27). We hereby esimae he characerisic funcion of he logarihm of S() for each under he law Q. e We now describe he explici consrucion of runcaed power prices of equaion (9) from he esimaed V GSSD defaul free parameers. We seek he value of µ Z W (K) =exp r(u)du E QQ S() p 1 S()>K We have he characerisic funcion of x = ln(s()) ha we denoe by φ x (u). We are ineresed in Z w(k) =e r (e xp e k )f(x)dx k 8

9 where r is now he erm discoun rae. Consider he Fourier ransform Z r Z (α+iu)k γ(u) = e e e xp f(x)dxdk Z r k Z x = e dx f(x) e Z (α+p+iu)x e = e r dx f(x) α + iu r φ x(u i(α + p)) = e α + iu xp+(α+iu)k dk We obain he runcaed power prices by Fourier inversion as e αk Z w(k) = e iuk γ(u)du. 2π These may hen be subsiued ino (1) o obain he local volailiy surface. We compuehese a agridofsrikes andmauriies for whichwe seek helocal volailiies σ(k, T ). Once we have esimaed he parameers for he no defaul process we may compue C(K, e T ) a a grid of srikes and mauriies. We hen use our esimaed survival funcion o define C(K, T ) = V (T )Ce(K, T ) These prices are hen used in he expression (1) along wih he no defaul runcaed power prices compued as per secion (4) o find he local volailiy surface σ(k, T ). For he funcion f() we use (8) and he characerisic funcion for log prices evaluaed a ip. 5 Sample Compuaions for GM and FORD We now illusrae hese procedures for daa on GM and FORD from Ocober 24 o Sepember 27. We esimaed he Weibull parameers for 795 and 784 days respecively ou of 185 and 178 days for GM and FORD respecively. The resuls are summarized in Table 1. TABLE 1 Weibull Parameers from CDS Curves GM FORD c a c a mean sd max min

10 We hen used he implied Weibull survival funcions o adjus marke prices o defaul free prices along wih adjusing dividend yields o ge he righ defaul free forwards. For hese prices and dividend yields we fied he VGSSD model o ge he parameers for he defaul free opion prices. These are summarized in Tables 2 and 3 for GM and FORD respecively. TABLE 2 GM Defaul Free Surface σ ν θ γ mean sd max min TABLE 3 FORD Defaul Free Surface σ ν θ γ mean sd max min The nex sep is he compuaion of runcaed power prices wih he no defaul dynamics and he adjused dividend yields. These are consruced for an expanding srike range as we raise mauriy from.5 o 1. We used he srike range kd = p/1.1.2 p sqr( min ) ku = p p sqr( min ) e On his grid we consruc runcaed power prices under he Q measure by he Fourier inversion described above. We used a power of 1.5. We now have he e power price under Q as well as he runcaed power prices and we can use he Weibull cdf and he equaion for f() o build he funcion f(). e In he he nex sep we consruc Q call prices using he adjused dividend yields and he V GSSD parameers ha have been esimaed on his grid. These prices are convered o marke call prices via C(K, ) =V ()Ce(K, ) where V () comes from he Weibull model. We now have all he ingrediens o implemen he revised local volailiy consrucion. The final oupu consiss of a local volailiy funcion, and he funcion f(). These are graphed for GM and FORD over 4 o 5 subses pariioning levels of he parameer c in he survival funcion and he level of aggregae volailiy of he defaul free surface. 1

11 f() high c medium c low c low v low c med v low c high v gm deerminisic hazard funcion ime in years 8 6 high c medium c low c low v low c high v ford deerminisic hazard funcion f() ime in years Figure 1: Deerminisic componens of hazard rae funcions for GM and FORD 11

12 .5 gm local volailiy funcion a 4 monhs ford local volailiy funcion a 4 monhs year.4 local volailiy local volailiy local volailiy.4.3 local volailiy srike in moneyness srike in moneyness gm local volailiy funcion a 8 monhs ford local volailiy funcion a 8 monhs.8.5 local volailiy local volailiy srike in moneyness srike in moneyness gm local volailiy funcion a one year ford local volailiy funcion a one year srike in moneyness srike in moneyness Figure 2: Implied Local Volailiies for GM and FORD 12

13 For FORD we had four subses while for GM we had five subses. Wih a view o sumarizing he resuls we compued he proporion of he radiional local volailiy ha is allocaed o he defaul componen or he fracion Kf()V ()ζ(k, ) σ 2 def (K, ) = CKK σ 2 LV (K, ) This generally varies wih srike and mauriy and we compued he elasiciies of his proporion by regressing he logarihm of his raio on he logarihm of he srike and mauriy. Addiionally we also regressed he logarihm of f() on mauriy and logarihm of mauriy. To avoid he effecs of cases where hese summary funcional forms did no fi well we repor summary saisics of hese regressions only when he R 2 exceeded 9% along wih he propoion of imes ha his crierion was me. Table 4 provides he resuls for he defaul proporion of local volailiy while Table 5 presens he resuls for he deerminisic hazard funcion f(). TABLE 4 log defaul proporion regressions GM (R 2 crierion saisfacion 79.89%) consan log(k/1) log() mean sd min max FORD (R 2 crierion saisfacion 71.43%) mean sd min max We see from Table 4 ha he elasiciy wih respec o srike of he defaul proporion is.4768 and.3967 for GM and FORD respecively. The corresponding elasiciies wih respec o mauriy are.646 and.54 respecively. Table 5 log f() GM (R 2 crierion saisficaion 88.77%) consan mauriy log(mauriy) mean sd min max FORD (R 2 crierion saisfacion 94.33%) mean sd min max

14 We see from Table 5 ha he hazard funcion is generally increasing wih mauriy wih semielasiciies and elasiciies of.136 and.2163 for GM wih.975 and.2641 being he corresponding values for FORD. 6 Conclusion We enhance a local volailiy model by he addiion of he possibiliy of a single jump o defaul wih a hazard rae ha is a deerminisic funcion of ime scaled by he sock price raised o a presepcified negaive power. Our empirical work uses he prespecified power of 1.5. We show in his conex how one may simulaneously recover prices of credi defaul swap conracs and he equiy opion prices boh he deerminisic componen of he hazard rae funcion and revised local volailiy. The procedure requires one o consruc afer esimaing he survival probabiliies o various mauriies, he prices of defaul free opions o which one fis a sandard defaul free model wih revised dividend yields o accoun for he paymen of premia necessary o ge defaul free in a defaulable world. This defaul free model is criically used o infer he prices of powers of he sock price runcaed o be above srike levels for a variey of mauriies. These runcaed power prices are needed in consrucing he revised local volailiy funcion from a grid of defaulable call prices ha may be inferred from he defaul free model coupled wih he survival funcion. The enire procedure was implemened on prices of credi defaul swaps and equiy opions for GM and FORD over he period Ocober 24 o Sepember 27. We found ha he revised local volailiy mus be reduced o accomodae he possibiliy of defaul by a proporion ha is dependen on boh he srike and mauriy. On average he elasiciy of he defaul proporion of local volailiy is.4768 and.3967 for GM and FORD respecively. The corresponding elasiciies wih respec o mauriy are posiive a.646 and.54. The deerminisic componen of he hazard funcion is genrally increasing wih respec o mauriy wih semi-elasiciies and elasiciies of.136,.2163 for GM and.975,.2641 for FORD. References [1] Albanese, C. and O. Chen (25), Pricing Equiy Defaul Swaps, Risk, 6, [2] Andersen,L.and D. Buffum (23/4), Calibraion and Implemenaion of Converible Bond Models, Journal of Compuaional Finance, 7,2. [3] Ayache, E., P. A. Forsyh and K. R. Vezal (23), The Valuaion of Converible Bonds wih Credi Risk, Journal of Derivaives, 9, [4] Alan, M. and B. Leblanc (25), Hybrid Equiy-Credi Modelling, Risk, 8. 14

15 [5] Carr, P., H. Geman, D. Madan and M. Yor (27), Self decomposabiliy andopionpricing, Mahemaical Finance, 17, [6] Davis, M and F. Lischka (22), Converible Bonds wih marke risk and credi risk, In Applied Probabiliy, Sudies in Advanced Mahemaics, American Mahemaical Sociey, [7] Derman, E. and I. Kani (1994), Riding on a smile, Risk 7, [8] Dupire,B.(1994), Pricing wih a smile, Risk 7, [9] Linesky, V. (26), Pricing Equiy Derivaives subjec o Bankrupcy, Mahemaical Finance, 16, 2, [1] Jarrow,R.A.and S. Turnbull(1995), Pricing Opions on Financial Securiies subjec o Defaul Risk, Journal of Finance, 5, [11] Madan, D., M. Konikov and M. Marinescu (26), Credi and Baske Defaul Swaps, Journal of Creid Risk, 2, [12] Dellacherie, C. and P. Meyer (198), Probabiliés e Poeniel, Theorie des Maringales, Hermann, Paris. [13] Meyer, P. (1976), Un Courssur lesinégralessochasiques, Séminaire de Probabiliés X, Lecure Noes in Mahemaics, 511, Springer-Verlag, Berlin. [14] Ren, Y., D. Madan and M. Qian (27), Calibraing and Pricing embedded local volailiy models, Risk, 9. [15] Yor, M. (1978), Rappels e Préliminaires Généraux, in: Temps Locaux, Sociéé Mahémaique de France, Asérisque, 52-53,