A Regularization Approach to Hedging Collateralized Mortgage Obligations

Transcription

1 Proceedngs of the World Congress on Engneerng 2008 Vol II A Regularzaton Approach to Hedgng Collateralzed ortgage Oblgatons Yury Gryazn, chael andrgan Abstract The paper presents a novel regularzaton approach to the hedgng of Collateralzed ortgage Oblgatons (CO. Our method s related to well nown Opton-Adjusted Spread (OAS methodology, but provdes a better way to account for embedded optonalty. In ths method, the constructon of the optmal hedgng portfolo of lqud mar nstruments s consdered as an essental part of the valuaton procedure. To ensure the unqueness of the soluton of the resultng ll-condtoned optmzaton problem, the standard Thonov type regularzaton technque s appled. The developed numercal optmzaton technque s based on the combnaton of the the Broyden-Fletcher-Goldfarb-Shanno (BFGS and ewton methods. The numercal results on the hedgng of the portfolos of CO and European swaptons based on onte Carlo smulaton are presented. Index Terms Collateralzed ortgage Oblgatons (CO, Opton-Adjusted Spread (OAS, onte-carlo smulaton, Thonov regularzaton. I. ITRODUCTIO Collateralzed ortgage Oblgatons (CO can have a hgh degree of varablty n cash flows. Because of ths, t s generally recognzed that a yeld to maturty of statc spread calculaton s not a sutable valuaton methodology. Snce the 1980 s Opton-Adjusted Spread (OAS has become a ubqutous valuaton metrc n the CO maret. There have been many crtcsms of OAS methodology, and some nterestng modfcatons have focused on the prepayment sde of the analyss, e.g., [2, 3]. One of the problems wth usng OAS analyss s the lac of nformaton about the dstrbuton of the ndvdual spreads, whch n turn leads to the dffcultes n the constructon of the hedgng portfolo for CO. To mprove the CO valuaton methodology and to develop a robust procedure for the constructon of the optmal hedge for CO, we ntroduce a combnaton of two new metrcs. We start wth the term-structure/prepayment model approach of OAS and go on to use the path-by-path structure of the cash flows of a CO n much more detal. Our methodology s to desgn a portfolo of swaptons whch mnmzes the varance n the ndvdual spreads as much as possble,.e., we are mnmzng the spread varance. In dong so, we desgn an optmal hedge; at least t s optmal anuscrpt receved arch 15, The wor of frst author was supported n part by excan Consejo aconal de Cenca y Tecnologa (COACYT under Grant # CB-2005-C F. Dr. Y. Gryazn s wth the Department of athematcs, Idaho State Unversty, Pocatello, ID USA(phone: ; fax: ; e-mal: gryazn@ su.edu. Dr.. andrgan was wth Rmroc Captal anagement, C, San Juan Capstrano, CA USA (e-mal: chagoopa@gmal.com. from the standpont of the probablty dstrbuton more or less mpled by swap rates and swapton prces. It should be emphaszed that when calculatng spreads we are dong so on the portfolo of CO and swaptons, thus we are ncludng the cost of hedgng n our valuaton. Our two man outputs are the mean and the standard devaton of the ndvdual spreads. Ths new spread varance mnmzaton (SV methodology can lead to qute dfferent conclusons about CO than OAS does. In partcular, n comparng two bonds, the more negatvely convex bond may loo cheaper on an OAS bass, but rcher accordng to our analyss. Ths s not smply a dfference n opnon: In contrast to OAS analyss, we fully value embedded optons. The man dffculty n mplementng our new methodology s n the mnmzaton of our spread-varance functonal. The dffculty s partly because the optmzaton problem s ll-condtoned, and n many stuatons, ths can be overcome by ntroducng a regularzaton term. Our approach s to use the standard Thonov regularzaton Thonov [4], whch has the strong ntutve appeal of lmtng the szes used n hedgng. A word about statc versus dynamc hedgng may be n order. Our methodology s to set up a statc hedge. It may be argued that an essentally perfect hedge may be created dynamcally. However, even f one s dynamcally hedgng, then one can t loc n a postve OAS. For a typcal CO dynamc hedgng wll cost money. Those costs should be dscounted flat and thus wll decrease a postve spread.. A dynamcally-hedged portfolo wll not have the OAS as a spread. oreover, ones hedgng costs wll be related to the amount of volatlty n the future so can be qute uncertan. Our methodology greatly reduces dynamc hedgng costs by settng up an optmzed statc hedge, and thus reduces the uncertanty n dynamc hedgng costs. The rest of the paper s organzed as follows. In the second secton we brefly revew OAS and pont out n more detal the problems we see wth t. In the thrd secton we explctly defne our hedgng methodology. In the forth secton we gve a bref descrpton of the regularzed numercal method. In the ffth secton we present some detals on the term-structure model used; on our prepayment assumptons and summarze numercal results from our analyss. II. OPTIO-ADJUSTED SPREAD AAYSIS Standard OAS analyss s based on fndng a spread at whch the expected value of the dscounted cash flows wll be equal to the maret value of a CO. Ths s encapsulated n eqn. (1: ISB: WCE 2008

2 Proceedngs of the World Congress on Engneerng 2008 Vol II = E % d(t, s c(t. (1 =1 Here s the maret value of CO, E % denotes expectaton wth respect to the rs neutral measure, d(t, s s the dscount factors to tme t, = 1,..., wth a spread of s and c(t s the cash flow at tme t. ote that n the OAS framewor, a sngle spread term s added to the dscounted factors to mae ths formula a true equalty, ths spread, s, s referred to as the OAS. The goal of OAS analyss s to value a CO relatve to lqud mar nterest rate dervatves, and, thus, the rs-neutral measure s derved to prce those mars accurately. To calculate expected values n practce one uses onte-carlo smulaton of a stochastc term-structure model. We use the two-factor Gaussan short-rate model G2++ as n Brgo and ercuro [1] calbrated to U.S. swapton prces, but other choces may be sutable. In terms of numercal approxmaton, t wll gve us eqn. (2: V = 1 t cf (n,t + Err. (2 n=1 =1 t =1 1 +Δt r(n,t Here Δt = t t 1, = 1,...,, V, = 1,..., are the maret values of the mars, cf (n,t, n = 1,...,, = 1,...,, = 1,..., are the future cash flows of the mars, s the number of generated trajectores, s the number of tme ntervals untl expraton of all mars and CO, and s the number of mars n the consderaton. The last term Err represents the error term. Though, the detaled consderaton of calbraton procedure s outsde the scope of ths presentaton, t s worth to menton that the absolute value of the Err term s bounded n most of our experments by fve bass ponts. The second step n the OAS analyss of COs s to fnd the spread term from the eqn. (3: = 1 cf (n,t. (3 n=1 =1 j =1 1+Δt j (r(n,t j + s Here cf (n,t, n = 1,...,, = 1,...,, are cash flows of the CO. These cash flows come from the structure of the CO, a perfectly nown quantty, and a prepayment model, a more subjectvely nown quantty. The parameter s, the OAS, s an ndcator as to whether the CO s underprced or overprced: If the OAS s postve then the CO s underprced, f t s negatve then the CO s overprced. ot only ts sgn, but also the magntude of the OAS commonly quoted as a measure of cheapness of a CO. An mportant ssue s managng a portfolo of COs s how to hedge the portfolo by usng actvely traded mars. We return to ths queston lttle bt later but for now let's assume that we somehow have found the lst and correspondng weghts of mars that would provde a suffcent hedge for our portfolo. To evaluate ths portfolo, we wll extend the OAS analyss to the valuaton of portfolo of CO and the mars. The straghtforward extenson of OAS approach wll result n eqn. (4: + w V = =1 1 cf (n,t + w cf. (4 =1 =1 + s Ths consderaton maes some serous drawbacs and flaws of OAS analyss apparent. Some of them are: It matches a mean to V, but provdes no nformaton on the dstrbuton of the ndvdual dscounted values. One can use t to calculate some standard rs metrcs, but t gves no way to desgn a refned hedge. It s senstve to postons n your mars. Suppose the OAS on a CO s postve. ow suppose you have a short poston n group of mars. Then necessarly the OAS of your portfolo ncreases. Ths s smply because of eqn. (5: V = 1 cf (n,t =1 r(n,t j (5 and so > 1 =1 cf (n,t V < j =1 1 +Δt j + s OAS T 1 cf (n,t cf (n,t. (6 t =1 + s OAS Ths shows that OAS s senstve to leveragng by shortng a combnaton of swaptons to ncrease the s OAS. III. THE SPREAD VARIACE IIIZATIO(SV APPROACH In our approach, nstead of usng the same spread for all paths, we are loong at the ndvdual spreads for every path of the portfolo of CO and mars. We try to fnd a portfolo of mars so that we mnmze the varaton n the ndvdual spreads. The spread for path n s the value s(n such that eqn. (7 satsfed: + w V = =1, (7 1 cf (n,t + w cf (n,t =1 =1 j =1 1 +Δt + s(n where the w, =1,, are the weghts of the ndvdual mars (a negatve ndcates a short poston. The goal s to apply an optmzaton algorthm to fnd weghts so that the varance of the s(n s a mnmum. In ths form the problem s not well defned; the weghts may exhbt some nstablty and tend to drft to nfnty: Buy larger and larger amounts of the entre portfolo of mars and your ndvdual spreads wll all approach zero, so your spread varance wll approach zero. A common approach to the soluton of ths type of problem s to add regularzaton term to the target functonal. Instead of mnmzng var(s(n, mnmze var(s(n+ α [w ] 2 Thonov [4], where α s small, on the order of Thonov regularzaton maes the problem well defned and a soluton method stable. From the hedgng pont of vew, such regularzaton provdes the bounded vector of optmal weghts for the mars and so ntroduces practcaltes n the mplementaton of an actual hedge. ISB: WCE 2008

3 Proceedngs of the World Congress on Engneerng 2008 Vol II IV. UERICA ETHOD As t was mentoned before, we are consderng that for every nterest rate trajectory the ndvdual spread depends on the weghts for mars n the portfolo. Then the target functonal s defned n eqn. (8: f (w 1 = 1 s(n, w 1,...,w n 2 μ 2, (8 where 1 μ = s(n, w 1. The Jacoban and Hessan of the functon are gven by eqn. (9: f w = 2 (9 n = 1 (s(n,,...,w μ s(n, w,..., w 1 n, = 1,...,, w1 n w 2 f = 2 s s 2 s + (s μ n = 1 w w wl w l w w l 1 s 2 w 1 s,,l = 1,...,. w l (10 Usng mplct dfferentaton one can fnd s 2 s,, w w w l, l = 1,...,. But ths functonal n general s ll-condtoned. To ensure the convergence of the optmzaton method, we ntroduce standard Thonov regularzaton. The modfed target functonal could be presented as n eqn. (11: %f (w 1 = f (w 1 + α w 2 2. (11 In most stuatons ths guarantes the convergence of the numercal method to the unque soluton. In our approach we are usng the optmzaton technque based on the combnaton of the Broyden-Fletcher-Goldfarb-Shanno (BFGS Vogel [5] and ewton methods. The BFGS approach s appled on the early teratons. When the l 2 norm of the gradent of the target functon becomes less then we apply the ewton method assumng that the approxmaton s already close enough to the soluton and the quadratc convergence rate of the ewton method can be acheved. As we already mentoned, ths regularzaton eeps the value of the target functonal small and stablzes the optmzaton method by preventng the weghts of the mars n the portfolo w from becomng large through the penalty term α w 2 2. To eep the condton number of the Hessan bounded, one has to use farly large regularzaton parameter. On the other hand, n order to eep the regularzed problem reasonably close the orgnal one, we expect that α needs to be small. In addton to these pure mathematcal requrements, n the case of managng a portfolo, one has to tae nto consderaton the cost of the hedgng. Snce regularzaton term prevent the optmal weghts drft to nfnty, the regularzaton parameter becomes a desrable tool n eepng hedgng cost under control. In our experments we found that the parameter α = 10 9 represents a good choce for the regularzaton n most numercal experments. V. UERICA RESUTS To llustrate our proposed methodology we consder hedgng an unstructured trust IO (nterest only strp wth a baset of European swaptons of dfferent stres and expraton dates, all wth ten-year tenor. To generate the trajectores of the short-rate, we use the two-addtve-factor Gaussan model G2++ Brgo and ercuro [1]. The dynamcs of the nstantaneous-short-rate process under the rs-neutral measure are gven by r(t = x(t + y(t + ϕ(t, r(0 = r 0, where the processes {x(t:t 0} and {y(t:t 0} satsfy (12: dx(t = ax(tdt + σdw 1 (t, x(0 = 0, (12 dy(t = by(tdt + ηdw 2 (t, y(0 = 0. Here (W 1,W 2 s a two-dmensonal Brownan moton wth nstantaneous correlaton ρ : dw 1 (tdw 2 (t = ρdt,. The parameters a,b,σ,η, ρ are defned n the calbraton procedure to match the prces of a set of actvely traded European swaptons. The l 2 norm of the dfference vector of the model swapton prces and the maret prces n our experments was bounded by fve bass ponts. In our experments, we wll use the standard devaton of the spread as an ndcator of the qualty of the hedge. The bond under consderaton has the followng parameters: WAA = 60 months, WAC = 6%, coupon = 5.5%, and a prce of $ We are usng very smple prepayment model defned by lnear nterpolaton of the data from Table 1. Interest rates(% CPR(% Interest rates(% CPR(% Table 1: Prepayment rates For optmzaton we use a baset of 64 payer and recever European swaptons wth yearly expraton dates from years 1 to 16, ncludng at the money (AT and +/- 50 out of the money. The regularzaton parameter used s10 9, and the number of trajectores n all experments was 500. The SV numercal method was mplemented n atlab wth usng the BFGS standard mplementaton avalable n the atlab optmzaton toolbox. The computer tme for the atlab optmzaton routne was 42 sec on the standard PC wth 2Hz processor frequency. ISB: WCE 2008

4 Proceedngs of the World Congress on Engneerng 2008 Vol II Fgure 1: Convergence hstory. The Fg. 1 represents the convergence hstory of the teratons n the optmzaton method. Fg. 2 shows the square root of the target functonal, whch s approxmately the standard devaton of the spread dstrbuton. One can see that as a result of the applcaton of the new methodology, the standard devaton of the spread dstrbuton reduced sgnfcantly, whch n turn could be consdered as reducton n rs of the portfolo of CO and mars. Fgure 2: Convergence of the target functonal. Fg. 3 presents the evoluton of weghts n our portfolo. Fgure 3: Evoluton of weghts. In the next seres of experments, we llustrate the qualty of the hedge constructed usng dfferent numbers of swaptons. Table 2 presents the results of these experments. In our experments we used the IO strp n combnaton wth dfferent numbers of swaptons. For reference we nclude results wth no hedge at all; ths s the common OAS analyss (but we also nclude the mean spread. For our SV approach, we frst use just two AT swaptons, one payer and one recever, both wth expraton date n one year. Then we use eght swaptons wth expraton dates of 1 and 2 years. For each of these expratons we nclude an AT payer, an AT recever, an AT+50 bps payer, and an AT-50 bps recever. The test wth twenty optons uses expraton dates 1 to 5 years, agan wth four swaptons per expraton date. In the last experment we use swaptons wth expraton dates 1 to 16. In all of these experments we used one unt of the trust IO wth the cost of $ As we can see from the frst lne of the table, the cost of the hedge s an ncreasng functon of the number of optons. But the most mportantly, we manage to sgnfcantly decrease the standard devaton of the spread dstrbuton by usng spread varance mnmzaton methodology. umber of Swaptons Cost of hedge($ σ (bass ponts(bps ean spread (bps Table 2: Hedgng results OAS (bps otce that, when usng just two optons, the cost of our hedgng portfolo s essentally zero, we are buyng the AT recever swapton and sellng an equal amount of the AT payer swapton. Ths s effectvely enterng nto a forward rate agreement, and so ths case could be consdered as a hedgng strategy based on only the duraton of portfolo. The experments wth the larger set of swaptons s a refnement ths strategy and tae nto account more detal about the structure of the cash flows of the bond. It proves to be a very successful approach to the hedgng of the path-dependent bond. As we can see from the frst lne of the table, the cost of the hedge s an ncreasng functon of the number of optons. But the most mportantly, we manage to sgnfcantly decrease the standard devaton of the spread dstrbuton by usng spread varance mnmzaton methodology. otce that, when usng just two optons, the cost of our hedgng portfolo s essentally zero, we are buyng the AT recever swapton and sellng an equal amount of the AT payer swapton. Ths s effectvely enterng nto a forward rate agreement, and so ths case could be consdered as a hedgng strategy based on only the duraton of portfolo. The experments wth the larger set of swaptons s a refnement ths strategy and tae nto account more detal about the structure of the cash flows of the bond. It proves to be a very successful approach to the hedgng of the path-dependent bond. ISB: WCE 2008

5 Proceedngs of the World Congress on Engneerng 2008 Vol II The last two rows of the table present two dfferent metrcs: The mean spread and the OAS of the portfolo of bond and hedges. We can see that they become close as standard devaton decreases. In fact, they would be the same f the standard devaton becomes zero. Wth no swaptons, or wth very few of them, these metrcs can result n drastcally dfferent conclusons about the cheapness of the bond. Contrary to a common nterpretaton of OAS, t does not represent the mean spread of an unhedged bond. In fact, t can be expected to be close to the mean spread only when consderng a refned hedgng portfolo of swaptons. Though the dscusson of the advantages of the SV analyss s outsde of the scope of ths paper, t s worth mentonng that t cares sgnfcant nformaton about the path-dependent bond and s worth tang nto account alongsde the standard OAS parameter. VI. COCUSIO In ths paper we presented a regularzaton approach to the constructon of an optmal portfolo of CO and swaptons. The standard Thonov regularzaton term serves as an mportant tool for preventng the weghts of the mars n the optmal portfolo drft to the nfnty and so eeps the hedgng procedure practcal. The optmzaton method demonstrates excellent convergent propertes and could be used n practcal applcatons for hedgng a portfolo of CO. The numercal results demonstrate the effectveness of the proposed methodology. The future development may nclude the constructon of new target functonal based on the combnaton of the spread varance and cost functon. Ths modfcaton mght mprove the effcency of the developed hedgng strategy. There appears to be a wde varety of applcaton of our SV approach, and n general of systematc path-by-path analyss of securtes wth stochastc cash flows. We plan to apply our analyss to other CO structures drectly. In addton, our analyss lends tself to comparng structured CO wth lqud strp nterest only and prncpal only bonds. REFERECES [1] D. Brgo & F.ercuro, Interest Rate odels: Theory and Practce. Berln Hedelberg, Sprnger-Verglag, 2001, pp [2] Cohler, G., Feldman,. & ancaster, B., Prce of Rs Constant (PORC: Gong Beyond OAS. The Journal of Fxed Income, 6(4, 1997, pp [3] evn, A. & Davdson, A., Prepayment Rs and Opton-Adjusted Valuaton of BS. The Journal of Portfolo anagement, 31( 4, [4] Thonov, A.., Regularzaton of ncorrectly posed problems. Sovet athematcs Dolady, 4, 1963, pp [5] Vogel C.R., Computatonal methods for nverse problems. SIA, Phladelpha, ISB: WCE 2008