Modeling autocallable structured products

Transcription

1 Orgnal Artle Modelng autoallable strutured produts Reeved (n revsed form): 22nd September 2011 Geng Deng PhD, FRM, s Dretor of Researh at Seurtes Ltgaton and Consultng Group, In. Dr Deng holds a BS n Appled Mathemats from Tsnghua Unversty n Chna, and an MA n Mathemats, an MS n Statsts and a PhD n Operatons Researh, all from the Unversty of Wsonsn-Madson. He has publshed papers n varous journals and onferene proeedngs, nludng Mathematal Programmng and Journal of Investng. Joshua Mallett s Researh Assoate at US Seurtes and Exhange Commsson. Mr Mallett holds a BS n Aountng and a Master of Aountany from Brgham Young Unversty, as well as an MA n Busness Admnstraton from the Unversty of Mhgan. Crag MCann PhD, CFA, s Prnpal at Seurtes Ltgaton and Consultng Group, In. Dr MCann reeved a BA and an MA n Eonoms from the Unversty of Western Ontaro and a Dotorate degree n Eonoms from the Unversty of Calforna, Los Angeles. He has publshed n the Journal of Investng, the Journal of Index Investng, the Journal of Legal Eonoms, the Journal of Appled Corporate Fnane and the Harvard Busness Revew. Correspondene: Geng Deng, Seurtes Ltgaton and Consultng Group, In Far Rdge Dr., Sute 250, Farfax, VA 22033, USA E-mal: GengDeng@slg.om ABSTRACT Sne frst ntrodued n 2003, the number of autoallable strutured produts n the Unted States has nreased exponentally. The autoall feature auses the produt to be redeemed f the referene asset s value rses above a pre-spefed all pre. Beause an autoallable strutured produt matures mmedately f t s alled, the autoall feature redues the produt s duraton and expeted maturty. In ths artle, we present a flexble Partal Dfferental Equaton framework to model autoallable strutured produts. Our framework allows for produts wth ether dsrete or ontnuous all dates. We value the autoallable strutured produts wth dsrete all dates usng the fnte dfferene method, and the produts wth ontnuous all dates usng a losed-form soluton. In addton, we estmate the probabltes of an autoallable strutured produt beng alled on eah all date. We demonstrate our models by valung a popular autoallable produt and quantfy the ost to the nvestor of addng ths feature to a strutured produt. Journal of Dervatves & Hedge Funds (2011) 17, do: /jdhf ; publshed onlne 3 November 2011 Keywords: strutured produts; PDE; autoallable; allable & 2011 Mamllan Publshers Ltd Journal of Dervatves & Hedge Funds Vol. 17, 4,

2 Modelng autoallable strutured produts INTRODUCTION Autoallable strutured produts 1,2 have beome nreasngly ommon n reent years. The frst autoallable strutured produt on reord n the Unted States was ssued by BNP Parbas on 15 August Fgures 1(a) and (b) plot the number and aggregate fae value of autoallable strutured produts ssued between 2003 and As the fgures ndate, the number of ssues nreased sharply n 2007 and has ontnued to grow through 2010 at a 40 per ent annual growth rate. In just the frst 6 months of 2010 there were more than 2500 autoallable produts ssued. The aggregate fae value of newly ssued autoallable strutured produts follows the same pattern, wth a surge n 2007 and ontnued growth sne then. One reason for the rapd expanson of autoallable strutured produts s the ease wth whh the all feature an be attahed to exstng types of strutured produts. 3 6 The all feature auses the strutured produt to be redeemed f the referene asset s pre reahes or exeeds a predefned level (the all pre) on a all date. In ths artle we desrbe the all feature, explan how to value t and show an example of the valuaton methodology. We use ths example to dsuss the ost ths feature an add to a struture produt. We value autoallable strutured produts usng a general Partal Dfferental Equaton (PDE) approah. We set up the PDE usng the Blak-Sholes equaton and add boundary ondtons representng the produt s features, nludng the autoall feature. 7,8 We dvde the autoallable strutured produts nto two ategores: produts that have dsrete all dates ( dsrete autoallables ) and produts that have ontnuous all dates ( ontnuous autoallables ). Fgures 2(a) and (b) demonstrate graphally the dfferene between dsrete and ontnuous autoallables. Both fgures plot the same underlyng stok pre over tme. The ontnuous autoallable strutured produt s alled mmedately upon rossng the all pre C, whlethe dsrete autoallable must wat untl t 3 before t s alled. If the underlyng stok pre had dropped bak below C on t 3,thedsrete autoallable strutured produt would not have been alled. Thus, holdng all else equal, a ontnuous autoallable strutured produt s more lkely to be alled than a dsrete one. a b Number of Autoallables Issued Aggregate Autoallable Issue Sze Fgure 1: Number and total ssue sze of autoallable strutured produts, January 2003 June & 2011 Mamllan Publshers Ltd Journal of Dervatves & Hedge Funds Vol. 17, 4,

3 Deng et al a b Dsrete Callable Contnuous Callable Fgure 2: An autoall event. Although we only onsder onstant all pres n the artle, the methodologes are expandable to exponentally nreasng all pres. Closed-form solutons are also avalable. The extenson s analogous to valung a barrer opton wth an exponentally varyng barrer. 9,10 An autoallable strutured produt s fundamentally smlar to a reverse-onvertble 11 that pays a hgh oupon n exhange for exposng the nvestor to the downsde rsk of the referene seurty. Although autoallable strutured produts tend to be ssued for longer terms than reverse-onvertbles, autoallable strutured produts an have shorter effetve duratons beause of the embedded all feature. See Arza, 12 Chemmanur et al 13 and Chemmanur and Smonyan 14 for a dsusson of why nvestment banks ssue mandatory onvertbles and why nvestors purhase them. For example, a ommon autoallable strutured produt would have the followng payoffs: If the referene asset s pre s above the all pre on one of the all dates, t s alled, and pays a pre-spefed fxed-rate return. If the referene asset s pre s below the all pre on every all date, the produt s never alled. In suh a ase, the nvestors reeve the produt s fae value at maturty, f the fnal pre of the referene asset s above a predetermned threshold. If the fnal pre s below the threshold, nvestors reeve the same negatve perentage return as the referene asset. The artle proeeds as follows: In the next seton, we explan our valuaton framework. The frst subseton dsusses autoallable strutured produts wth dsrete all dates, and the seond subseton presents autoallable strutured produts wth ontnuous all dates. The followng seton mplements our valuaton framework for an example of strutured produt. We onlude n the last seton. In the appendx we explan the man features of popular autoallable strutured-produts. AUTOCALLABLE STRUCTURED PRODUCT VALUATION MODELS There are three man haratersts of the all feature that wll affet the value of the strutured produt: the tmng of the all dates, the probablty of beng alled on eah all date and the determnaton of the payoff at maturty. In ths seton we set up the valuaton of autoallable strutured produts 328 & 2011 Mamllan Publshers Ltd Journal of Dervatves & Hedge Funds Vol. 17, 4,

4 Modelng autoallable strutured produts as a PDE problem. The PDE problem s general enough to be used on both dsrete and ontnuous autoalls. Modelng autoallable strutured produts usng PDE Our valuaton model follows the Blak-Sholes framework wth rsk-neutral assumptons. The referene asset s pre s a generalzed Brownan moton ds t ¼ðr qþs t dt þ ss t dw t ; ð1þ where r s the rsk-free rate, q s the dvdend yeld, and s s the volatlty of the pre proess. (Throughout ths artle, we assume r, q and s are onstant and ontnuously ompounded over the produt s term [0, T ]. For smplty, we omt the subsrpt t from S t.) If we assume the pre of the strutured produt V(S, t) s a funton of tme ta[0, T ] and the referene asset s pre SA[0, N). The Blak-Sholes formula mples that a strutured produt s dynam value an be expressed as the followng PDE: qv qt þ 1 2 s2 S 2 q2 V qv þðr q Þ S qs2 qs ðr þ C DSÞV ¼ 0; ð2þ where CD S s the redt default swap (CDS) spread of the ssuer. (Strutured produts are unseured debt seurtes, and hene lose value f the ssuer defaults. It s therefore essental to nlude the ssuer s redt rsk CD S n the PDE to alulate the strutured produt s present value 6,15 ). Many dfferent strutured produt features an be modeled as varatons on equaton (2). For example, when the strutured produt s not alled, the payoff at maturty f (S T )s typally a funton of the value of the referene asset at maturty: V ðs T ; TÞ ¼f ðs T Þ: For smplty and wthout loss of generalty, we assume the ntal prnpal of a strutured produt s equal to the referene asset s ntal value S 0. Embedded all and put optons and the autoall feature an all be modeled as boundary ondtons. 6 The autoall feature s boundary ondton s V ðc; tþ ¼P t ; for t 2 T C ; ð3þ where C s the tme-ndependent all pre, P t s the fnal payoff f the note s alled, and T C s a set of dsrete or ontnuous all dates. One the autoall s trggered, the strutured produt matures mmedately and the fnal payout s P t. Called strutured produts typally pay out a fxed rate of return. Therefore, the payoff follows P t ¼ He Bt ; where B s the rate of return, and H s a onstant. Valung autoallable strutured produts wth dsrete all dates Most dsrete autoallables do not have a losedform soluton. Instead, the PDE s solved and the produts are valued va numeral methods suh as the fnte-dfferene method. 16 For dsrete autoallable strutured produts, the boundary ondtons of the PDE are the equatons V ðc; tþ ¼P t for all t 2 T C ; V ð0; tþ ¼f ð0þe ðrþc DSÞðTtÞ : The frst ondton requres that the produt s value never exeeds the autoall payout on a all & 2011 Mamllan Publshers Ltd Journal of Dervatves & Hedge Funds Vol. 17, 4,

5 Deng et al date. The seond ondton guarantees that f the referene asset s pre hts 0 t wll reman 0. For tratablty, we defne t usng a general funton f (0) ¼ 0. Ths boundary ondton s neessary as t guarantees that the strutured produt annot ever be alled f the referene asset beomes worthless. The frst step n solvng the PDE s to smplfy the omplex notaton and transform the equaton nto a standard heat equaton. Usng a dmensonless hange of varables smlar to Wlmott et al 8 and Hu, 17 we transform the varables {S, t, V(S, t)} nto {x, t, u(x, t)} as follows S ¼ Ce x ; t ¼ T s 2 ; V ðs; tþ ¼Ce axþbt uðx; tþþf ð0þ e ðrþc DSÞðTtÞ ; where the onstants are k 1 ¼ 2ðr qþ s 2 ; a ¼ 1 2 ðk 1 1Þ; b ¼a 2 2ðr þ C DSÞ s 2 : After the hange of varables, the Blak- Sholes equaton s redued to a heat equaton qu qt ¼ q2 u ; for 1oxo0; t40; ð4þ qx2 the boundary ondtons beome uð1; tþ ¼0; uð0; tþ ¼ C 1 e bt P t f ð0þe ðrþc DSÞ s 2 for T s 2 2 T C and the ntal ondton beomes (the hange of varables onverts the fnal ondton nto an ntal ondton) uðx; 0Þ ¼C 1 e ax ðf ðce x Þfð0ÞÞ; for 1oxo0: To further smplfy notaton we denote h 1 ðtþ ¼C 1 e bt P t f ð0þe ðrc DSÞ s 2 and h 2 ðxþ ¼C 1 e ax ðf ðce x Þfð0ÞÞ; redung the boundary ondtons and ntal ondton to uð1; tþ ¼0; uð0; tþ ¼h 1 ðtþ for T s 2 2 T C; uðx; 0Þ ¼h 2 ðxþ: ð5þ The fnte dfferene method allows us to dsretze the doman of the funton u(x, t), whh s a plane (x, t)a(n,0] [0, Ts 2 /2]. We dsretze the plane nto an N M grd, where the sze of eah grd blok s dx dt. Beause x has no lower bound, we an assgn x an arbtrarly large mnmum value of Ndx. The bounds on t requre that dt satsfy Mdt ¼ Ts2 2 : Generally speakng, the auray of the valuaton nreases as dx and dt get smaller. dt s typally set to orrespond to one tradng day, suh that dt ¼ 2dt=s 2 ¼ 1=250 of a year. There are three fnte dfferene methods: the explt fnte dfferene method, the mplt fnte dfferene method and the Crank-Nolson method. The methods dffer n how they approxmate the dervatves qu/t and q 2 u/x 2. In ths example, we use the explt fnte dfferene method, whh 330 & 2011 Mamllan Publshers Ltd Journal of Dervatves & Hedge Funds Vol. 17, 4,

6 Modelng autoallable strutured produts approxmates the dervatves as qu t q 2 u x 2 umþ1 n u m n ; dt um nþ1 2um n þ um n1 ðdxþ 2 : Solvng equaton (4) wth the ondtons n equaton (5) s equvalent to solvng u mþ1 n u m n dt 0onoN; wth the ondtons ¼ um nþ1 2um n þ um n1 ðdxþ 2 ; 0omoM u 0 n ¼ h 2ðndxÞ; 0pnpN; u m 0 ¼ 0; 0omoM; u m N ¼ h 1ðmdtÞ; for T 2ðmdtÞ=s 2 2 T C : The formula updatng mdt to (m þ 1)dt s therefore u mþ1 n ¼ u m n þ dt ðdxþ 2 um nþ1 2um n þ um n1 ; 0onoN; 0omoM: The soluton s derved teratvely from m ¼ 0-M, whh orresponds to t ¼ T-0. For the onvergene and stablty of the explt fnte dfferene method, we requre that dt/(dx) 2 p1/2. One all of the u n M, for n ¼ 1, 2,y, N are derved, we an approxmate u(x, Ts 2 /2) for every x. By reversng the hange of varables, we an use u(x, Ts 2 /2) to fnally solve the orgnal funton V(S, t) att ¼ 0. An alternatve probablty approah to valung dsrete autoallables Another way to estmate the value of a dsrete autoallable strutured produt s by alulatng the probablty of the autoall beng exersed on eah all date, and then use the probablty at eah date to value the strutured produt. Let p, ¼ 1,y, n be the probablty of the all beng exersed at tme t. The probablty of the all never beng exersed s then 1 P n ¼ 1p, where eah p s ondtonal on the all not beng exersed at any prevous all date (t 1, y, t 1 ). Reall that the ondtonal dstrbuton of S t js t 1 follows a lognormal dstrbuton p ffff ¼ S t 1 e ðrq1 2 s2 ÞDt þsd t W ; S t where Dt s the tme between all dates Dt ¼ t t 1 and W, ¼ 1,y,n are..d. standard normal varables. To smplfy notaton, we use X ¼ðr q ð1=2þs 2 ÞDt p þ sd ffffff t W to represent the ontnuously ompounded return from t 1 to t. Ths means the endng stok pre S T an be wrtten as S T ¼ S 0 e P n ¼1 ðrq 1 2 s2 ÞDt þsd p ffff t W ¼ S 0 ep n Beause of the pre s Markov property, the X s are parwse ndependent. Furthermore, f Dt s a onstant, the X s are..d. normal varables. The probablty of the all beng exersed at tme t an now be wrtten as p ¼ Prob S t j oc; j ¼ 1; 2;...; 1; and S t XC ¼ Prob Xj X k o log C ; j ¼ 1; 2;...; 1; S k¼1 0! and X ¼ ¼1 X : X k X log C S k¼1 0 Z Z gðx 1 ;...; x n Þdx 1 dx 2 dx n ; P j C x k o log S0 ; j¼1;2;...;1; k¼1 P C x k X log S0 k¼1 where g(x 1,y, x n ) s the jont probablty densty funton (PDF) of X 1,y, X n. Beause the X s & 2011 Mamllan Publshers Ltd Journal of Dervatves & Hedge Funds Vol. 17, 4,

7 Deng et al are ndependent, the jont PDF an be expressed as the produt of eah X s ndvdual PDF. We an now estmate the produt s present value as the dsounted expeted ash flows, where the ash flow probabltes are the p we just alulated. autoallable strutured produts wth dsrete all dates, yeldng the heat equaton qu qt ¼ q2 u ; for 1oxo0; t40; ð8þ qx2 wth the boundary ondtons uð1; tþ ¼0; uð0; tþ ¼ h 1 ðþ t for t40 V ðs 0 ; 0Þ ¼ Xn ¼1 ¼ Xn ¼1 e ðrþc DSÞt p P t þ e ðrþc DSÞT e ðrþc DSÞt p P t þ e ðrþc DSÞT P j k¼1 P j k¼1 Z x k o log Z x k o log Z C S0 Z C S0 ; j¼1;2;...;n ; j¼1;2;...;n f ðs T Þ gðx 1 ;...; x n Þdx 1 dx n f S 0 e P n x ¼1 gðx 1 ;...; x n Þdx 1 dx n ð6þ If the strutured produt s payoff at maturty s onstant f(s T ) ¼ P T, the equaton an be further redued to and the ntal ondton uðx; 0Þ ¼h 2 ðxþ for 1oxo0: V ðs 0 ; 0Þ ¼ Xn ¼1 e ðrþc DSÞt p P t ðrþc DSÞT þ e 1 Xn ¼1 p!p T : ð7þ Valung autoallable strutured produts wth ontnuous all dates For a ontnuous autoallable strutured produt, the boundary ondtons of the PDE are ontnuous equatons V ðc; tþ ¼P t ; V ð0; tþ ¼f ð0þe ðrþc DSÞðTtÞ : We apply the hange of varables and smplfatons from the seton Valung The next step s to onvert the two boundary ondtons so that they are both zero boundares (homogenous boundares). To do ths we ntrodue the transformaton v(x, t) ¼ u(x, t)y(x, t), where y(x, t) ¼ e x h 1 (t). Usng ths transformaton, the Blak-Sholes equaton beomes qv qt ¼ q2 v qx 2 þ ex ðh 1 ðtþh 0 1 ðtþþ; for 1oxo0; t40: ð9þ The new, homogenous boundary ondtons are vð1; tþ ¼0; and the new ntal ondton s vðx; 0Þ ¼h 2 ðxþe x h 1 ð0þ; vð0; tþ ¼0; for t40 for 1oxo0: 332 & 2011 Mamllan Publshers Ltd Journal of Dervatves & Hedge Funds Vol. 17, 4,

8 Modelng autoallable strutured produts The transformed ontnuous autoall PDE problem s thus a standard nhomogeneous PDE problem wth homogeneous boundary ondtons. By further smplfyng notaton, the PDE problem an be solved usng methods n Evans. 18 Spefally, let h 3 (x) ¼ h 2 (x)e x h 1 (0) and h 4 (x, t) ¼ e x (h 1 (t)h 1 0 (t)). The PDE problem s then the followng general form qv qt ¼ q2 v qx 2 þ h 4ðx; tþ; for 1oxo0; t40; vð1; tþ ¼0; vð0; tþ ¼0; and vðx; 0Þ ¼h 3 ðxþ: The soluton to ths PDE s vðx; tþ ¼ p 1 2 ffffffffff pt Z 0 1 h 3 ðsþ e ðxsþ2 =4t e ðxþsþ2 =4t ds Z t þ 0 Z 0 1 h 4 ðs; rþ pffffffffffffffffffffffffffffffff 2 pðt rþ e ðxsþ2 =4ðtrÞ e ðxþsþ2 =4ðtrÞ dsdr: ð10þ Autoallable Optmzaton Seurtes wth Contngent Proteton. More than US$1.4 bllon n fae value of these produts were ssued n See the Appendx for more detals of the dfferent brands of autoallable strutured produts and ther man features.) If the referene asset has a umulatve postve return on any autoall date, the strutured produt s alled and nvestors wll reeve a postve, pre-spefed yeld. If the produt s not alled, at maturty the payoff wll be: V ðs; TÞ ¼f ðsþ ¼ I ¼ S 0; S4L; S; otherwse; ð11þ where I s the strutured produt s fae value, S 0 s the referene asset s ntal value, S s the referene asset s fnal value and L s the threshold pre. If the strutured produt s not alled, nvestors wll reeve a 0 per ent or a negatve return. Fgure 3 llustrates the autoallable strutured produt s payoff at maturty f t s not alled. One v(x, t) s solved, we an now solve our transformaton u(x, t) ¼ v(x,t) þ e x h 1 (x). One ths funton s solved we an fold bak and fnd the value of V(S, t). EXAMPLE OF AN AUTOCALLABLE STRUCTURED PRODUCT As an example of our valuaton methodology we desrbe a smple autoallable strutured produt. (Ths example s smlar n ts features to one of the more popular brands, the Fgure 3: Maturty payoff f the autoallable strutured produt s not alled. & 2011 Mamllan Publshers Ltd Journal of Dervatves & Hedge Funds Vol. 17, 4,

9 Deng et al To demonstrate the applaton of our models, we value three stylzed types of autoallable strutured produts. The frst example, our benhmark ase, does not have an autoall feature, but has a onstant oupon payment. The payoff struture resembles a plan vanlla reverse onvertble strutured produt. The seond type has an autoall feature wth monthly all dates, and the thrd type has an autoall feature wth ontnuous all dates. For all three examples we assume that the referene asset s ntal stok pre S 0 and the fae value of the note I are both $100, the all pre s $102, the rsk-free rate r s 5 per ent, the volatlty s of the referene asset s 20 per ent, the dvdend yeld q of the referene asset s 1 per ent, the ssuer s CDS spread CD S s 1 per ent, the ontrat length T s 1 year, and the threshold L s $80. If the referene asset s pre s over the all pre on an all date (that s, S t XC ¼ 102), the produt wll be alled and wll pay a 9.2 per ent annualzed return (that s, P t ¼ He Bt ¼ 100e 0.09 ). (Ths ase s our benhmark ase, hene we use a 9.2 per ent oupon rate that makes ths example frst type non-autoallable note a par value note, that s, prnpal ¼ $100.) Many autoallable produts have a all pre dental to the pre of the stok (C ¼ S 0 ); however, our assumpton C4S 0 s wthout loss of generalty. (In a ontnuous ase, f the all pre were dental to the stok pre the produt would lkely be mmedately alled at ssuane, defeatng the pont of suh a all provson). Case 1: Benhmark Not autoallable In ths ase, the valuaton of autoallable strutured produt s relatvely straghtforward. Beause the referene asset s fnal pre follows a lognormal dstrbuton S T ¼ S 0 e ðrq1 2 s2 ÞTþs ffffff T p W ; the value of the strutured produt s the dsounted expeted ash flow V ðs ; 0Þ ¼e ðrþc DSÞT 0 1 Z f ðs T Þ gðs T ÞdS T þ S 0 TBA: 0 where g( ) s the PDF of S T.Wesetthe produt s ssue date value to be $ per $ fae value by our hoe of parameters. As many have shown (see for example Henderson and Pearson 5 ) reverse onvertble strutured produts tend to be overpred, that s, that they are ssued on average at a pre that exeeds the present value of ther expeted future ash-flows. We use ths as a benhmark example and hene set t artfally to be pred at fae value. Case 2: Autoallable at dsrete all dates Generally, autoallable strutured produts have dsrete autoall dates. We assume that the produt n ths example s allable monthly. We frst mplement the explt fnte dfferene method to alulate the produt value. We set the range of x to be [5, 0] and the range of t to be [0, (Ts 2 /2]. The resultng n m grd has ¼ bloks. Followng equaton (6), the value of the produt s $98.39 per $ fae value. We also alulate the monthly probabltes of the all beng exersed p, and show the results n Table & 2011 Mamllan Publshers Ltd Journal of Dervatves & Hedge Funds Vol. 17, 4,

10 Modelng autoallable strutured produts Table 1: The probablty of the produt beng alled on eah monthly all date, ondtonal on not beng alled at an earler date Month p Case 3: Contnuously autoallable If the all dates are ontnuous, we an follow the steps n the seton Valung autoallable strutured produts wth ontnuous all dates to get the losed-form soluton. After the frst phase of hange of varables, we get a homogeneous heat equaton qu qt ¼ q2 u ; for 1oxo0; t40; qx2 uð1; tþ ¼0; uðx; 0Þ ¼C 1 e ax f ðce x Þ; uð0; tþ ¼ C 1 e bt P t ; Usng our notaton, h 1 (t) ¼ C 1 e bt P t and h 2 (x) ¼ C 1 e ax f(ce x ). Applyng the seond phase of hange of varables to make the boundary ondtons equal zero. Let y(x, t) ¼ e x h 1 (t) ¼ C 1 e xbt P t and a new funton v(x, t) ¼ u(x, t) þ y(x, t), then the PDE hanges to an nhomogeneous equaton qv qt ¼ q2 v qx 2 þ C1 e xbt P t 1 þ b þ 2B s 2 ; for 1oxo0; t40; vð1; tþ ¼0; vð0; tþ ¼ 0; vðx; 0Þ ¼C 1 e ax f ðce x Þ H C exþbt : Here h 3 ðxþ ¼C 1 e ax f ðce x ÞðH=CÞe xþbt and h 4 ðx; tþ ¼C 1 e xbt P t ð1 þ b þð2b=s 2 ÞÞ. Applyng equaton (10), the soluton s vðx; tþ ¼ S 0 tax x þ 2at N pffffffffff C ea2 N D 1 x pffffffffff S 0 tþax N x þ ffffffffff 2at p C ea2 N D 1 þ x pffffffffff þ e D2þð1aÞx N D 3 x pffffffffff e D2ð1aÞx N D 3 þ x pffffffffff H x C ebtþxþt N pffffffffff þ H x C ebtxþt N pffffffffff þ H 2B ð1 þ b þ C s 2 Þ Z t 0 N e D4þx N! x D 5 pffffffffffffffffffffffffffffff H 2ðt rþ C ed4x! x D 5 þ pffffffffffffffffffffffffffffff dr; 2ðt rþ where the parameters are D 1 ¼ logðl=cþþ2at p ffffffffff ; D 2 ¼ tða 1Þ 2 ; logðl=cþþða1þ D 3 ¼ p ffffffffff ; D 4 ¼ BT ðbþ 2B þ t rþ; s2 p D 5 ¼ ffffffffffffffffffffffffffffff 2ðt rþ: & 2011 Mamllan Publshers Ltd Journal of Dervatves & Hedge Funds Vol. 17, 4,

11 Deng et al The value of the of the produt s V(S, t) valued at (S 0, 0), where the form s V ðs; tþ ¼Ce axþbt ðvðx; tþþyðx; tþþ: The value V(S,0) s $ Comparng the three ases We an now ompare the values n the three dfferent ases: $100.00, $98.39 and $99.54, respetvely. Investments wth the autoall feature are worth less than ther non-allable benhmark. The reason for t s farly ntutve. A non-allable nvestment essentally guarantees a oupon payment untl maturty. Wth an autoall feature, the oupon may be pad for a shorter perod or may not be pad at all. Sne both nvestments share the same downsde rsk, addng the all feature (wthout adjustng the pre or the oupon) lowers the value of the nvestment. In ths example, the ontnuously autoallable strutured produt s more valuable than the dsrete autoallable strutured produt. Although ths s not neessarly always true. One the oupon payment of a plan vanlla reverse onvertble s replaed wth an autoallable feature, the nvestment has a hgher value f t s alled and the longer t takes to get alled. A dsrete autoallable feature s less lkely to be alled, but holdng all else equal may be alled later f t s alled. Hene, t s more lkely for a ontnuous feature to be more valuable than a dsrete one but ths does not have to be always the ase. Real-lfe example We alulate the produt value of a real Autoallable Optmzaton Seurtes wth Contngent Proteton note ssued by UBS. (The CUSIP for the produt s 90267C136. See the produt s prng supplement at /178916_ b2.htm.) The note s lnked to the stok of Bank of Amera. It was ssued on 26 Marh 2010 and had a maturty of 1 year. The referene asset s pre on the ssue date was S 0 ¼ $ The dvdend yeld q and mpled volatlty of the underlyng stok s were per ent and per ent, respetvely. UBS s 1-year CDS spread was per ent. On the ssue date, the 1-year ontnuously ompounded rsk-free rate was per ent. The all pre C equaled the ntal pre S 0. If the note were alled, nvestor would reeve a return of 16.1 per ent, and f t were not alled, the ontngent proteton level was L ¼ 0.7S 0. Applyng our methods, we get a produt value of $97.73 per $ nvested. CONCLUSION An autoallable strutured produt s alled by the ssuers f the referene asset s pre exeeds the all pre on a all date. The feature has been embedded n many dfferent types of strutured produts, nludng Absolute Return Barrer Notes and Optmzaton Seurtes wth Contngent Proteton. We provde a general PDE framework to model autoallable strutured produts. We solve the PDE for autoallable strutured produts wth dsrete all dates, for whh there s typally not a losed-form soluton, by usng the fnte dfferene method. For ontnuous autoallables, we derve the losed-form soluton. We llustrate our modelng approahes wth an example. We then quantfy the nremental ost of 336 & 2011 Mamllan Publshers Ltd Journal of Dervatves & Hedge Funds Vol. 17, 4,

12 Modelng autoallable strutured produts addng an autoall feature to a plan-vanlla reverse-onvertble. We also show the dfferene between the value of an autoall feature wth ontnuous all dates and one wth dsrete all dates. The Seurtes and Exhange Commsson, as a matter of poly, dslams responsblty for any prvate publaton or statement by any of ts employees. REFERENCES 1 Fres, C. and Josh, M. (2008) Condtonal Analyt Monte-Carlo Prng Sheme of Auto-allable Produts. Workng paper. Avalable at SSRN: ssrn.om/abstrat= Georgeva, A. (2005) The Use of Strutured Produts: Applatons, Benefts and Lmtatons for the Insttutonal Investor. Workng paper. 3 Baule, R., Entrop, O. and Wlkens, M. (2008) Credt rsk and bank margns n strutured fnanal produts: Evdene from the German seondary market for dsount ertfates. Journal of Futures Markets 28(4): Bergstresser, D.B. (2008) The Retal Market for Strutured Notes: Issurane Patterns and Performane, Henderson, B. and Pearson, N. (2010) The dark sde of fnanal nnovaton: A ase study of the prng of a retal fnanal produt. The Journal of Fnanal Eonoms 100(2): Deng, G., Mallett, J. and MCann, C. (2010) On the Valuatons of Strutured Produts. SLCG workng paper. 7 Blak, F. and Shole, M. (1973) The prng of optons and orporate labltes. Journal of Poltal Eonomy 81(3): Wlmott, P., Dewynne, J. and Howson, S. (1994) Opton Prng: Mathematal Models and Computaton. Oxford: Oxford Fnanal Press. 9 Kuntomo, N. and Ikeda, M. (1992) Prng optons wth urved boundares. Mathematal Fnane 2(4): L, A. (1999) The prng of double barrer optons and ther varatons. Advanes n Futures and Optons Researh 10: Hernández, R., Lee, W.Y. and Lu, P. (2007) An Eonom Analyss of Reverse Exhangeable Seurtes: An Opton-prng Approah. Unversty of Arkansas. Workng Paper. 12 Arza, E.R. (1997) PERCS, DECS, and other mandatory onvertbles. Journal of Appled Corporate Fnane 10(1): Chemmanur, T.J., Nandy, D. and Yan, A. (2006) Why Issue Mandatory Convertbles? Theory and Empral Evdene. EFA 2003 Glasgow. Avalable at SSRN: 14 Chemmanur, T.J. and Smonyan, K. (2010) What drves the ssuane of putable onvertbles: Rsk-shftng, asymmetr nformaton, or taxes? Fnanal Management 39(3): Hull, J. (2011) Optons, Futures and Other Dervatves, 8th edn. New York: Prente-Hall. 16 Zvan, R., Vetzal, K.R. and Frosyth, P.A. (2000) PDE methods for prng barrer optons. Journal of Eonom Dynams & Control 24(11): Hu, C.H. (1996) One-touh double barrer bnary opton values. Appled Fnanal Eonoms 6(4): Evans, L.C. (2010) Partal Dfferental Equatons: Seond Edton. Provdene, Rhode Island: Ameran Mathematal Soety. 19 Deng, G., Guedj, I., Mallett, J. and MCann, C. (2011) The Anatomy of Absolute Return Barrer Note. SLCG. Forthomng n the Journal of Dervatves, APPENDIX Desrptons of selet exstng autoallable strutured produts Autoallable optmzaton seurtes wth ontngent proteton Autoallable Optmzaton Seurtes wth Contngent Proteton have been ssued by several nvestment banks, nludng Royal Bank of Canada, UBS, JPMorgan and HSBC. Payout f the produt s alled Autoallable Optmzaton Seurtes wth Contngent Proteton generally have monthly or quarterly all dates, wth the fnal all date beng at the produt s maturty. In general, the produt s alled f the referene asset has a postve umulatve return on the all date. When the produt s alled, nvestors reeve the produt s fae value plus a pre-spefed annual yeld. & 2011 Mamllan Publshers Ltd Journal of Dervatves & Hedge Funds Vol. 17, 4,

13 Deng et al Payout f the produt s not alled The payout f not alled vares by ssuane between yeldng a 0 per ent return, or a negatve return ted to the stok return of the referene asset. Most produts ompare the referene asset s fnal pre to a threshold. If the fnal pre s at or above the threshold, nvestors reeve the produt s fae value at maturty. If the fnal pre s below the threshold, nvestors reeve the same negatve perentage return as the referene asset. Some produts ompare the referene asset s lowest pre durng the produt s lfe to a threshold. If the lowest pre s at or above the threshold, nvestors reeve the produt s fae value at maturty. If the lowest pre s below the threshold, nvestors reeve the same negatve perentage return as the referene asset. Autoallable absolute return barrer notes (Autoallable ARBNs) Autoallable ARBNs are ontnuously allable strutured produts, ssued by Lehman Brothers and UBS. A non-autoallable ARBN s analyzed n depth n Deng et al. 19 Payout f the produt s alled An autoallable ARBN generally has ontnuous all dates. The produt s alled whenever the referene asset s pre rosses ether an upper or a lower barrer, advanng the return of the strutured produt s fae value. Payout f the produt s not alled The note s not alled f the referene asset s pre stays wthn the barrers. At maturty, the note pays nvestors the absolute return of the referene asset, whh s a return bounded by the sze of the barrers. (Sem-) Annual revew notes wth ontngent prnpal proteton Sem-annual and Annual Revew Notes wth Contngent Prnpal Proteton have been ssued by several nvestment banks, nludng JPMorgan, Credt Susse and HSBC. Payout f the produt s alled Revew Notes wth Contngent Prnpal Proteton have dsrete all dates. The frequeny of the all dates vares, wth Annual Revew Notes havng annual all dates and Sem-annual Revew Notes havng sem-annual all dates. In both ases, the fnal all date s generally at the produt s maturty. In general, the produt s alled f the referene asset has a postve umulatve return on a all date. However, some produts exerse the autoall f the referene asset s umulatve return s postve or not too negatve (for example, 10 per ent). Regardless of the autoall trgger, exersng the autoall enttles nvestors to reeve the produt s fae value plus a pre-spefed annual yeld. Payout f the produt s not alled If the produt s not alled, the payoff at maturty s guaranteed to be no more than the fae value of the produt. All of the Revew Notes we examned have one of three knds of loss buffers, whh we refer to as standard buffers, ontngent buffers and fadng buffers. Regardless of the buffer style, nvestors reeve the produt s fae value at maturty as long as the referene asset s umulatve return s above the buffer (for example, 20 per ent). If the referene asset s return s below the buffer, nvestors wll lose money. Revew Notes wth standard buffers expose nvestors to any loss n the referene asset beyond the buffer. Thus, a 20 per ent standard buffer wll offset a 23 per ent return on the referene asset so the nvestor only loses 3 per ent. 338 & 2011 Mamllan Publshers Ltd Journal of Dervatves & Hedge Funds Vol. 17, 4,

14 Modelng autoallable strutured produts Reverse Exhangeable Notes (ssued by JPMorgan) both make oupon payments. Fgure 4: Standard buffers, ontngent buffers and fadng buffer. Revew Notes wth ontngent buffers expose nvestors to all of the referene asset s losses f the loss s greater than the buffer. For example, f the produt has a 20 per ent buffer and the referene asset has a 23 per ent return, nvestors wll reeve a 23 per ent return. However, those same nvestors would reeve a 0 per ent return f the referene asset had a 19 per ent return. Revew Notes wth fadng buffers provde a standard buffer that dmnshes as the referene asset s return gets worse. Fgure 4 graphs the relatonshp between the proteton offered by a fadng buffer and the referene asset s return. The graph shows that the nvestor begns wth a 20 per ent buffer, but the buffer beomes smaller as the referene asset s return beomes more negatve. In the extreme, a 0 per ent buffer orrespondng to a referene asset return of 100 per ent. Thus, an referene asset return of 23 per ent would equate to a per ent loss for the nvestor. Autoallable reverse onvertble notes Autoallable Reverse Convertble Notes (ssued by Eksportfnans and HSBC) and Autoallable Payout f the produt s alled Autoallable Reverse Convertble Notes generally have a sngle, dsrete all date early n the lfe of the produt. If the produt s referene asset has a postve umulatve return on the all date, the produt s alled and nvestors reeve any arued oupon payments and the fae value of the note. Autoallable Reverse Exhangeable Notes are smlar, but tend to have multple dsrete all dates or ontnuous all dates after an ntal non-allable perod. Payout f the produt s not alled If the produt s not alled, the payout at maturty s smlar to a non-autoallable Reverse Exhangeable Note or a Reverse Convertble Note. If the referene asset s pre ever rosses a barrer set below ts ntal pre, nvestors reeve the oupon payments plus the produt s fae value redued by the lesser of a 0 per ent return or the referene asset s perentage return at maturty. If the referene asset s pre never rosses the barrer, nvestors reeve the oupon payments plus the fae value of the produt. Some produts ompare the referene asset s fnal value, rather than ts lowest value, to the barrer return. If the referene asset s fnal value s below the barrer, the nvestor s exposed to the referene asset s losses. Otherwse, the produt returns ts fae value. Ether way, nvestors reeve the oupons. Strateg aelerated redempton seurtes Bank of Amera, Merrll Lynh and Eksportfnans have all ssued Strateg Aelerated Redempton Seurtes. & 2011 Mamllan Publshers Ltd Journal of Dervatves & Hedge Funds Vol. 17, 4,

15 Deng et al Payout f the produt s alled These autoallable strutured produts have dsrete all dates, wth the fnal all date beng at the produt s maturty. The produt s alled f the referene asset has a non-negatve return on the all date. When the produt s alled, nvestors reeve a pre-spefed yeld on ther nvestment. Payout f the produt s not alled If the produt s not alled, the payout at maturty s smlar to a Reverse Exhangeable Note, exept that Strateg Aelerated Redempton Seurtes do not pay oupons. If the referene asset s endng value s not below a threshold, nvestors reeve the fae value of the produt. If the endng value of the referene asset s below the threshold, nvestors lose a multple of the referene asset s negatve return. Although the multple an theoretally be less than 1 or greater than 1, all of the produts we saw had a multple of 1. Some produts do not have a threshold. Instead, nvestors are guaranteed to lose a multple of the referene asset s negatve return f the produt s not alled. Bear market strateg aelerated redempton seurtes Bank of Amera and the Norwegan redt nsttuton Eksportfnans have also ssued Bear Market Strateg Aelerated Redempton Seurtes. These strutured produts are the same as regular Strateg Aelerated Redempton Seurtes, exept that nvestors lose money f the referene asset s return s too hgh and earn a pre-spefed yeld f the referene asset loses value. Premum mandatory allable equty-lnked seurtes (PACERS) PACERS, ssued by Ctgroup, pay oupons and have a set of dsrete all perods. Eah perod s a setoftwoorthreeontnuousalldates.ifthe referene asset s value on any all date s equal to or greater than ts ntal value, the produt s alled and nvestors reeve a pre-spefed yeld n addton to the arued oupons. If the produt s not alled, the payout s smlar to that of an Autoallable Reverse Convertble Note. Spefally, nvestors reeve the same perentage return as the referene asset f the referene asset s endng value s below a threshold (for example, 25 per ent). Otherwse, nvestors reeve the fae value of the note. Regardless of the referene asset s value, nvestors reeve the oupons f the produt s not alled. Also smlar to Autoallable Reverse Convertble Notes, some PACERS ompare the referene asset s lowest value, rather than the fnal value, to the threshold to determne whether nvestors reeve the fae value of the PACERS or the same return as the referene asset. Dslamer The vews expressed heren are those of the author and do not neessarly reflet the vews of the Commsson or of the author s olleagues upon the staff of the Commsson. 340 & 2011 Mamllan Publshers Ltd Journal of Dervatves & Hedge Funds Vol. 17, 4,