Standard derivatives pricing theory (see, for example, Hull,

Transcription

1 Cuing edge Derivaives pricing Funding beyond discouning: collaeral agreemens and derivaives pricing Sandard heory assumes raders can lend and borrow a a risk-free rae, ignoring he inricacies of he repo and collaeralisaion markes. Here, Vladimir Pierbarg shows ha hese force adjusmens o discouning, forward prices and implied volailiies, depending on he pariculars of collaeral posing 42 Sandard derivaives pricing heory (see, for example, Hull, 2006 relies on he assumpion ha one can borrow and lend a a unique risk-free rae. he realiies of being a derivaives desk are, however, raher differen hese days, as hisorically sable relaionships beween bank funding raes, governmen raes, Libor raes, ec, have broken down. he pracicaliies of funding, ha is, how dealers borrow and lend money, are of cenral imporance o derivaives pricing, because replicaing naurally involves borrowing and lending money and oher asses. In his aricle, we esablish derivaives valuaion formulas in he presence of such complicaions saring from firs principles, and sudy he impac of marke feaures such as sochasic funding and collaeral posing rules on values of fundamenal derivaives conracs, including forwards and opions. Simplifying considerably, we can describe a derivaives desk s aciviies as selling derivaives securiies o cliens while hedging hem wih oher dealers. Should he desk defaul, a clien would join he queue of he bank s crediors. he siuaion is a bi differen for rading among dealers where, o reduce credi risk, agreemens have been pu in place o collaeralise muual exposures. Such agreemens are based on he so-called credi suppor annex (CSA o he Inernaional Swaps and Derivaives Associaion maser agreemen, so we ofen refer o collaeralised rades as CSA rades. As collaeral is used o offse liabiliies in case of a defaul, i could be hough of as an essenially risk-free invesmen, so he rae on collaeral is usually se o be a proxy of a risk-free rae such as he fed funds rae for dollar ransacions, Eonia for euro, ec. Ofen, purchased asses are posed as collaeral agains he funds used o buy hem, such as in he repo marke for shares used in dela hedging. Secured borrowing will normally arac a beer rae han unsecured borrowing. In a bank, funding funcions are ofen cenralised wihin a reasury desk. he unsecured raes ha he reasury desk provides o he rading desks are generally linked o he unsecured funding rae a which he bank iself can borrow/lend, a rae ypically based on he bank credi raing, ha is, is perceived probabiliy of defaul. he money ha a derivaives desk uses in is operaions comes from a muliude of sources, from he collaeral posed by counerparies o funds secured by various ypes of asses. We show in his aricle how o aggregae hese raes o come up wih he value of a derivaives securiy given he rules for collaeral posing and repo raes available for he underlying. Noe ha some desks may be required o borrow a raes differen from hose ha hey can lend a a complicaion we avoid in his aricle as our formalism does no exend readily o he non-linear parial differenial equaions ha such a se-up would require. Having derived an appropriae exension o he sandard no-arbirage resul, we hen look carefully a he differences in value of CSA (ha is, collaeralised and non-csa (no collaeralised versions of he same derivaives securiy. his is imporan as dealers ofen calibrae heir models o marke-observed prices of derivaives, which ypically reflec CSA-based valuaions, ye hey also rade a large volume of non- CSA over-he-couner derivaives. We demonsrae ha a number of ofen significan adjusmens are required o reflec he difference beween CSA and non-csa rades. he firs adjusmen is o use differen discouning raes for CSA and non-csa versions of he same derivaive. he second adjusmen is a convexiy, or quano, adjusmen and affecs forward curves such as equiy forwards or Libor forward raes as hey urn ou o depend on collaeralisaion used. his is a consequence of he sochasic funding spread and, in paricular, of he correlaion beween he bank funding spread and he underlying asses. he hird adjusmen ha may be required is o volailiy informaion used for opions in paricular, he volailiy smile changes depending on collaeral. We show some numerical resuls for hese effecs. Preliminaries We sar wih he risk-free curve for lending, a curve ha corresponds o he safes available collaeral (cash. We denoe he corresponding shor rae a ime by r C (; C here sands for CSA, as we assume his is he agreed overnigh rae paid on collaeral among dealers under CSA. I is convenien o parameerise erm curves in erms of discoun facors; we denoe corresponding risk-free discoun facors by P C (,, 0 <. Sandard Heah-Jarrow-Moron heory applies, and we specify he following dynamics for he yield curve: _AR_0310_Cuing_edge.ind /3/10 12:20:59

2 asiarisk.com.hk Cuing edge Derivaives pricing / P C (, r C dp C, d σ C, dw C ( where W C ( is a d-dimensional Brownian moion under he risk-neural measure P and s C is a vecor-valued (dimension d sochasic process. In wha follows, we shall consider derivaives conracs on a paricular asse, whose price process we denoe by S(, 0. We denoe by r R ( he shor rae on funding secured by his asse (here R sands for repo. he difference r C ( r R ( is someimes called he sock lending fee. Finally, le us define he shor rae for unsecured funding by r F (, 0. As a rule, we would expec ha r C ( r R ( r F (. he exisence of non-zero spreads beween shor raes based on differen collaeral can be recas in he language of credi risk, by inroducing join defauls beween he bank and various asses used as collaeral for funding. In paricular, he funding spread s F r F ( r C ( could be hough of as he (sochasic inensiy of defaul of he bank. We do no pursue his formalism here (see, for example, Gregory, 2009, or Burgard & Kjaer, 2009, posulaing he dynamics of funding curves direcly insead. Likewise, we ignore he possibiliy of a counerpary defaul, an exension ha could be developed easily. Black-Scholes wih collaeral Le us look a how he sandard Black-Scholes pricing formula changes in he presence of a CSA. Le S( be an asse ha follows, in he real world, he following dynamics: ds ( / S ( µ S ( d + σ S ( dw ( Le V(, S be a derivaives securiy on he asse; by Iô s lemma i follows ha: dv d + ( ds LV ( where L is he sandard pricing operaor: and D is he opion s dela: L + σ ( S 2 S S 2 V ( Le C( be he collaeral (cash in he collaeral accoun held a ime agains he derivaive. For flexibiliy, we allow his amoun o be differen 1 from V(. o replicae he derivaive, a ime we hold D( unis of sock and g( cash. hen he value of he replicaion porfolio, which we denoe by Π(, is equal o: S V ( Π( ( S ( + γ ( he cash amoun g( is spli among a number of accouns: n Amoun C( is in collaeral. n Amoun V( C( needs o be borrowed/len unsecured from he reasury desk. n Amoun D(S( is borrowed o finance he purchase of D( socks. I is secured by sock purchased. n Sock is paying dividends a rae r D. 1 In wha follows we use (3, (5 wih eiher C 0 or C V. However, hese formulas, in heir full generaliy, could be used o obain, for example, he value of a derivaive covered by one-way (asymmeric CSA agreemen, or a more general case where he collaeral amoun racks he value only approximaely (1 (2 he growh of all cash accouns (collaeral, unsecured, socksecured, dividends is given by: dγ r C C r R ( + r F ( ( V ( C ( S ( + r D ( ( S d On he oher hand, from (2, by he self-financing condiion: which is, by Iô s lemma: hus we have: dγ ( dv ( ( ds ( dv ( ( ds ( ( LV ( d 2 + σ S 2 S 2 2 S 2 V 2 + σ S 2 S 2 2 S 2 V ( d r C ( C ( + r F ( ( V ( C ( + ( r D ( r R ( V which, afer some rearrangemen, yields: V + r R r F ( V S S + σ S 2 ( r C ( C ( r D ( V r F 2 S 2 2 V S 2 S S he soluion, obained by essenially following he seps ha lead o he Feynman-Kac formula (see, for example, Karazas & Shreve, 1997, heorem 4.4.2, is given by: E e r F u V du + e r F v u V dv ( r F ( u r C ( u C u du in he measure in which he sock grows a rae r R ( r D (, ha is: d + σ S ( dw S ds ( / S ( r R ( r D ( (3 (4 Noe ha if our probabiliy space is rich enough, we can ake i o be he same risk-neural measure P as used in (1. We noe ha his derivaion validaes he view of Barden (2009 (who also cies Hull, 2006 ha he repo rae r R ( is he righ risk-free rae o use when valuing asses on S(. By rearranging erms in (3, we obain anoher useful formula for he value of he derivaive: V ( E e r ( C u du V ( (5 E e u r ( C vdv r ( F ( u r C ( u ( V ( u C ( u du We noe ha: E ( dv ( r F ( V ( ( r F ( r C ( C ( d ( r F ( V ( s F ( C ( d ( _AR_0310_Cuing_edge.ind /3/10 12:21:10

3 Cuing edge Derivaives pricing 44 So, he rae of growh in he derivaives securiy is he funding spread r F ( applied o is value minus he credi spread s F ( applied o he collaeral. In paricular, if he collaeral is equal o he value V hen: E ( dv ( r C V ( d, V du E e r C u (7 V and he derivaive grows a he risk-free rae. he final value is he only paymen ha appears in he discouned expression as he oher paymens ne ou given he assumpion of full collaeralisaion. his is consisen wih he drif in (1 as P C (, corresponds o deposis secured by cash collaeral. On he oher hand, if he collaeral is zero, hen: E ( dv ( r F V ( d (8 and he rae of growh is equal o he bank s unsecured funding rae or, using credi risk language, adjused for he possibiliy of he bank defaul. We show laer ha he case C V could be handled by using a measure ha corresponds o he risk-free bond P C (, E (e r C(udu as a numéraire and, likewise, he case C 0 could be handled by using a measure ha corresponds o he risky bond (, E (e r F(udu as a numéraire. Before we proceed wih valuing derivaives securiies in our se-up, le us commen on he porfolio effecs of he collaeral. When wo dealers are rading wih each oher, he collaeral is applied o he overall value of he porfolio of derivaives beween hem, wih posiive exposures on some rades offseing negaive exposures on oher rades (so-called neing. Hence, poenially, valuaion of individual rades should ake ino accoun he collaeral posiion on he whole porfolio. Forunaely, in he simple case of he collaeral requiremen being a linear funcion of he exac value of he porfolio (he case ha includes boh he no-collaeral case C 0 and he full collaeral case C V, he value of he porfolio is jus he sum of values of individual rades (wih collaeral aribued o rades by he same linear funcion. his easily follows from he lineariy of he pricing formula (3 in V and C. Zero-srike call opion Probably he simples derivaives conrac on an asse is a promise o deliver his asse a a given fuure ime. he conrac could be seen as a zero-srike call opion wih expiry. In he sandard heory, of course, he value of his derivaive is equal o he value of he asse iself (in he absence of dividends. Le us see wha he siuaion is in our case. he payou of he derivaive is given by V( S( and he value, a ime, assuming no CSA, is given by: V zsc ( E e r F u On he oher hand, if r D ( 0, hen: E e r R u S du du S S as follows from (4 and, clearly, S( V zsc (. he difference in values beween he derivaive and he asse are now easily undersood, as he zero-srike call opion carries he credi risk of he bank, while he asse S( does no. Or, in our language of funding, he asse S( can be used o secure funding which is refleced in he discoun rae applied while V zsc canno be used for such a purpose. Forward conrac We now consider a forward conrac on S(, where a ime he bank agrees o deliver he asse a ime, agains a cash paymen a ime. n Wihou CSA. A no-csa forward conrac could be seen as a derivaive wih he payou S( (, a ime, where (, is he forward price a for delivery a. As he forward conrac is cos-free, we have by (3 ha: so we ge: 0 E e r F u du (, Going back o (9, le us define: ( S ( (, E e r F u du E e r F u E e r F u du S du Noe ha his is essenially a credi-risky bond issued by he bank. hen we can rewrie (9 as: (, E % S ( where he measure P ~ is defined by he numeraire P (, as: F e r ( F u du 0 (, E e r F u 0 du is a P-maringale. Finally we see ha (, is a P ~ - maringale. We noe ha he value of an asse under no CSA a ime wih payou V( is given, by (8, o be: E e r F u V du V (, E % ( V ( so i could be calculaed by simply aking he expeced value of he payou in he risky -forward measure. n Wih CSA. Now le us consider a forward conrac covered by CSA, where we assume ha he collaeral posed C is always equal o he value of he conrac V. Le he CSA forward price (, be fixed a, hen he value, from (5, is given by: 0 V ( E e r ( C u du V ( so ha: E e r C u (, du ( S ( (, E e r C u du E e r C u S du Comparing his wih (9, we see ha in general: (,, By he argumens similar o he no-csa case, we obain: (9 ( _AR_0310_Cuing_edge.ind /3/10 12:21:19

4 asiarisk.com.hk Cuing edge Derivaives pricing ( (, E S where he measure P is he sandard -forward measure, ha is, a measure defined by P C (, E (e r C(udu as a numeraire. We noe ha he value of an asse under CSA a ime wih payou V( is given, by (7, o be: E e r C u V du V P ( C, E V ( so i could be calculaed by simply aking he expeced value of he payou in he (risk-free -forward measure. n Calculaing CSA convexiy adjusmen. Le us now calculae he difference beween CSA and non-csa forward prices. We have: where: ( (, E % S E e r C u E P C,, is a P -maringale, as: E e r ( F u du S (, du e ( r ( F u r ( C u du M, E M, M, We noe ha, rivially: so: 1 M, (,, Cov (, e s ( F u du S ( S S P ( F, sf ( u du (, e 0 (12 P C M (, E e s ( F u du 0 M, E M, 1 M (, E M (, E M (, M (, ( S ( F ( CSA, (13 ( M (,, (, o obain he acual value of he adjusmen we would need o posulae join dynamics of s F (u and S(u, u. We presen a simple model below where we carry ou he calculaions. n Relaionship wih fuures conracs. A firs sigh, a forward conrac wih CSA looks raher like a fuures conrac on he asse. Recall ha wih fuures conracs, he (daily difference in he fuures price ges credied/debied o he margin accoun. In he same way, as forward prices move, a CSA forward conrac also specifies ha money exchanges hands. Bu here is an imporan difference. Consider he value of a forward conrac a >, a conrac ha was enered a ime (so V( 0. hen: E V By (10: E e r ( C u du e r ( C u du V ( V ( E ( S ( (, S e r ( C u du E e r ( C u du,, ( (, so he difference in conrac values on and ha exchanges hands a is equal o he discouned (o difference in forward prices. For a fuures conrac, he difference will no be discouned. herefore, he ype of convexiy effecs we see in fuures conracs are differen from wha we see in CSA versus no-csa forward conracs, a conclusion differen from ha reached in Johannes & Sundaresan (2007. European-syle opions Consider now a European-syle call opion on S( wih srike K. Depending on he presence or absence of CSA, we ge wo prices: V CSA ( E e r F u du ( E e r C u du ( S ( K + ( S ( K + (where for he CSA case we assumed ha he collaeral posed, C, is always equal o he opion value, V CSA. By he same measure-change argumens as in he previous secion: V CSA ( (, E % (( S ( K + ( P C (, E (( S ( K + he difference beween measures P ~ and P no only manifess iself in he mean of S( as already esablished in he previous secion bu also shows up in oher characerisics of he disribuion of S(, such as is variance and higher momens. We explore hese effecs in he nex secion. n Disribuion impac of convexiy adjusmen. Le us see how a change of measure affecs he disribuion of S(. In he spiri of (11, we have: M, ( (, E M, ( S ( K + where M(, is defined in (12. hen, by condiioning on S(, we obain: ( (, E α,,s ( ( ( S ( K + where he deerminisic funcion a(,, x is given by: M (, E M (, S ( x α,, x (14 Inspired by Anonov & Arneguy (2009, we approximae he funcion a(,, x by a linear (in x funcion: α (,, x α 0 (, + α 1 (, x and obain a 0 and a 1 by minimising he squared difference (while _AR_0310_Cuing_edge.ind /3/10 12:21:31

5 Cuing edge Derivaives pricing 1 Hisorical credi spread/ineres raes and credi spread/equiy correlaion calculaed wih a rolling one-year window % May 20, 2005 Aug 30, 2005 using he fac ha E (M(, /M(, 1 and E(S( F (, : CSA M (, E ( M (, α 1 (, S ( F ( CSA, Var ( S ( (, 1 α 1 (, α 0 We recognise he erm: Credi/raes correlaion Credi/equiy correlaion Dec 13, 2005 M, E M, S (, as he convexiy adjusmen of he forward beween he no-csa and CSA versions (see (13, and rewrie: α 1 Mar 27, 2006 Jul 6, 2006 Oc 16, 2006 Jan 31, 2007 (,, Var F ( CSA, ( S ( Differeniaing (14 wih respec o K wice, we obain he following relaionship beween he probabiliy densiy funcions (PDFs of S( under he wo measures: %P ( S ( dk α 0 (, + α 1, ( K P S ( dk (15 so he PDF of S( under he no-csa measure is obained from he densiy of S( under he CSA measure by muliplying i wih a linear funcion. I is no hard o see ha he main impac of such a ransformaion is on he slope of he volailiy smile of S(. We demonsrae his impac numerically below. Example: sochasic funding model Le us consider a simple model ha we can use o esimae he impac of collaeral rules on forwards and opions. We sar wih an asse ha follows a lognormal process: ds ( / S ( O ( d + σ S dw S ( and funding spread ha follows dynamics inspired by a simple onefacor Gaussian model of ineres raes 2 : ds F ( ℵ F ( θ s F ( d + σ F dw F wih dw S (, dw F ( rd. Here r is he correlaion beween he asse 2 While a diffusion process for he funding spread may be unrealisic, he impac of more complicaed dynamics on he convexiy adjusmen is likely o be mued May 14, 2007 Aug 22, 2007 Dec 6, 2007 Mar 19, 2008 Jun 27, 2008 Oc 7, 2008 Jan 21, 2009 May 1, 2009 A. Relaive differences beween non-csa and CSA forward prices wih s S 30%, s F 1.50%, ℵ F 5.00% and he funding spread. We also assume for simpliciy ha r C (, r R ( are deerminisic, while r D ( 0. hen: and: ime/r 30% 20% 10% 0% 10% % 0.04% 0.02% 0.00% 0.02% % 0.17% 0.09% 0.00% 0.09% % 0.39% 0.19% 0.00% 0.19% % 0.68% 0.34% 0.00% 0.34% % 1.04% 0.52% 0.00% 0.52% % 1.48% 0.74% 0.00% 0.73% % 1.99% 0.99% 0.00% 0.98% % 2.56% 1.27% 0.00% 1.26% % 3.20% 1.59% 0.00% 1.56% % 3.91% 1.94% 0.00% 1.90% d ( (, E S (, /, σ S dw S ( wih W S ( being a Brownian moion in he risk-neural measure P. On he oher hand: where: d (, /, O ( d σ F b 1 e ℵ F b ℵ F dw F As M(, is a maringale under P (since r C ( is deerminisic, he measures P and P coincide, we have from (12 ha: dm (, / M (, σ F b( dw F ( Also boh M(, and (, are maringales under P. We hen have: d M (, (, (, Recall ha: so ha: / ( M, ρd + O ( dw ( σ S σ F b ( 0, ( 0, E M (, M ( 0, ( F ( CSA, F ( CSA 0, ( 0, 0, and, in he case ℵ F 0: ( 0 exp σ S σ F b( ρd ℵ F ( 0, exp σ S σ F ρ b ( 0, ( 0, 0, ( 1 exp σ S σ F ρ 2 / 2 ( _AR_0310_Cuing_edge.ind /3/10 12:21:44

6 Cuing edge Derivaives pricing 48 2 Difference in CSA v. non-csa implied disribuion for European opions using (15, expressed in implied vol across srikes, for differen levels of correlaion r % Original (CSA implied volailiies Adjused (non-csa implied volailiies, corr 30% Adjused (non-csa implied volailiies, corr 10% Adjused (non-csa implied volailiies, corr 10% Srike Noe: 10 years, FCSA(0, 100, σf 1.50%, ℵF 5.00% We noe ha he adjusmen grows as (roughly 2. A similar formula was obained by Barden (2009 using a model in which funding spread is funcionally linked o he value of he asse. Le us perform a couple of numerical experimens. We sar wih an equiy-relaed example. Le us se s F 30%, a number roughly in line wih implied volailiies of opions on he S&P 500 equiy index (SPX. We esimae he basis-poin volailiy of he funding spread o be s F 1.50% and mean reversion o be ℵ F 5% by looking a hisorical daa of credi spreads on US banks. Figure 1 shows a rolling hisorical esimae of correlaions beween credi spreads and he SPX (as well as credi spread and ineres raes in he form of a five-year swap rae. From his graph, we esimae a reasonable range for he correlaion r o be [ 30%, 10%]. In able A, we repor relaive adjusmens: 0, ( 0, ( 0, for differen values of correlaions and for differen from one o 10 years. Clearly, he adjusmens could be quie significan. Nex we look a he difference in implied volailiies for CSA and non- CSA opions. We look a opions expiring in 10 years across differen srikes, wih (0, 100. We assume ha he marke prices of CSA opions are given by he 30% implied volailiy (for all srikes, so ha he CSA disribuion of he asse is lognormal wih 30% volailiy. hen we express he disribuion of he underlying asse for non-csa opions as given by (15 in erms of implied volailiies (using pu opions and he original value of he forward, 100, o ensure fair comparison. Figure 2 demonsraes he impac non-csa opions have lower volailiy (lower pu opion values, and he volailiy smile has a higher (negaive skew. Finally, le us look a CSA convexiy adjusmens o forward Libor raes. able B presens absolue differences (ha is, (0, (0, in non-csa versus CSA forward Libor raes fixing in one o 30 years over a reasonable range of possible correlaions. We use he same parameers for he funding spread as above ogeher wih recen marke-implied caple volailiies and forward Libor raes. Again, he differences are no negligible, especially for longer-expiry Libor raes. B. Absolue differences beween non-csa and CSA forward Libor raes, using marke-implied caple volailiies and sf 1.50%, ℵF 5.00% ime/r 20% 0% 20% 40% % 0.00% 0.00% 0.00% % 0.00% 0.01% 0.01% % 0.00% 0.01% 0.02% % 0.00% 0.02% 0.04% % 0.00% 0.03% 0.05% % 0.00% 0.05% 0.10% % 0.00% 0.09% 0.18% % 0.00% 0.18% 0.37% % 0.00% 0.30% 0.60% % 0.00% 0.42% 0.84% % 0.00% 0.54% 1.07% Conclusions In his aricle, we have developed valuaion formulas for derivaive conracs ha incorporae he modern realiies of funding and collaeral agreemens ha deviae significanly from he exbook assumpions. We have shown ha he pricing of non-collaeralised derivaives needs o be adjused, as compared wih he collaeralised version, wih he adjusmen essenially driven by he correlaion beween marke facors for a derivaive and he funding spread. Apar from raher obvious differences in discouning raes used for CSA and non-csa versions of he same derivaive, we have exposed he required changes o forward curves and, even, he volailiy informaion used for opions. In a simple model wih sochasic funding spreads we demonsraed he ypical sizes of hese adjusmens and found hem significan. n Vladimir Pierbarg is head of quaniaive research a Barclays Capial. He would like o hank members of he quaniaive and rading eams a Barclays Capial for houghful discussions, and referees for commens ha grealy improved he qualiy of he aricle. vladimir.pierbarg@barcap.com Anonov A and M Arneguy, 2009 Analyical formulas for pricing CMS producs in he Libor marke model wih he sochasic volailiy SSRN elibrary Barden P, 2009 Equiy forward prices in he presence of funding spreads ICBI Conference, Rome, April Burgard C and M Kjaer, 2009 Modelling and successful managemen of credi-counerpary risk of derivaive porfolios ICBI Conference, Rome, April References Gregory J, 2009 Being wo-faced over counerpary credi risk Risk February, pages Hull J, 2006 Opions, fuures and oher derivaives Prenice Hall Johannes M and S Sundaresan, 2007 Pricing collaeralized swaps Journal of Finance 62, pages Karazas I and S Shreve, 1997 Brownian moion and sochasic calculus Springer _AR_0310_Cuing_edge.ind /3/10 12:22:09