U.U.D.M. Projec Repor 2009:4 Srucured producs: Pricing, hedging and applicaions for life insurance companies Mohamed Osman Abdelghafour Examensarbee i maemaik, 30 hp Handledare och examinaor: Johan Tysk Mars 2009 Deparmen of Mahemaics Uppsala Universiy
Acknowledgemen I would like o express my appreciaion o Professor Johan Tysk my supervisor, no only for his excepional help on his projec, bu also for he courses (Financial Mahemaics and Financial Derivaives) ha he augh which graned me he undersanding opions heory and he necessary mahemaical background o come wrie his hesis. I would also like o hank him because he is he one who inroduced me o he Financial Mahemaics Maser a he iniial sage of my sudies. Also hanks o he res of he professors in he Financial Mahemaics and Financial Economics Programme who provided insrucion, encouragemen and guidance, I would like o say Thank you o you all. They did no only each me how o learn, hey also augh me how o each, and heir excellence has always inspired me. Finally, I would like o hank my Faher, Ramadan for his financial suppor and encouragemen, my moher, and my wife Nellie who for heir paience and coninuous suppor, when I was sudying and wriing his hesis.
Inroducion Chaper Financial derivaives. Wha is he srucured produc?.. Equiy-linked srucured producs..2 Capial-Guaraneed Producs.2 Financial Derivaive opics.2 Fuures and Forward conracs pricing and hedging.2.2 The fundamenal exposure ypes.2.3 European ype Opions.2.4 American ype opions.2.5 Bermudian Opions.2.6 Asian opion ypes.2.7 Clique opions Chaper 2 ineres rae srucured producs 2. Floaing Rae Noes (FRNs, Floaers) 2.2 Opions on bonds 2.3 Ineres Rae Caps and Floors 2.4 Ineres rae swap (IRS) 2.5 European payer (receiver) swapion 2.6 Callable/Puable Zero Coupon Bonds 2.7 Chaper 3 Srucured Swaps 3. Variance swaps 2
Chaper Inroducion In recen years many invesmen producs have emerged in he financial markes and one of he mos imporan producs are so-called srucured producs. Srucured producs involve a large range of invesmen producs ha combine many ypes of invesmens ino one produc hrough he process of financial engineering. Reail and insiuional invesors nowadays need o undersand how o use such producs o manage risks and enhance heir reurns on heir invesmen. As srucured producs invesmen require some derivaives insrumens knowledge. The auhor will presen some derivaive inroducion and opics ha will be used in he main conex of srucured producs. Srucured invesmen producs are ailored, or packaged, o mee cerain financial objecives of invesors. Typically, hese producs provide invesors wih capial proecion, income generaion and/or he opporuniy o generae capial growh. So he auhor will presen he use of such producs and heir payoff and analyse he use of differen sraegies. In fac, hose producs can be considered ready-made invesmen sraegy available for invesors so he invesor will save ime and effor o esablish such complex invesmen sraegies. In he pricing models and hedging, he auhor will ackle mainly he basic models of underlying equiies and ineres rae derivaives and he will give some pricing examples. Srucured producs end o involve periodical ineres paymens and redempion (which migh no be proeced). A par of he ineres paymen is used o buy he derivaives par. Wha ses hem apar from bonds is ha boh ineres paymens and redempion amouns depend in a raher complicaed fashion on he movemens of for example baske of asses, baske of indices exchange raes or fuure ineres raes. Since srucured producs are made up of simpler componens, I usually break hem down ino heir inegral pars when I need o value hem or assess heir risk profile and any hedging sraegies. 3
This approach should faciliae he analysis and pricing of he individual componens. For many produc groups, no uniform naming convenions have evolved ye, and even where such convenions exis, some issuers will sill use alernaive names. I use he marke names for producs which are common; a he same ime, I ry o be as accurae as possible. Commonly used alernaive names are also indicaed in each produc s descripion.. Wha are srucured producs? Definiion: Srucured producs are invesmen insrumens ha combine a leas one derivaive conrac wih underlying asses such as equiy and fixed-income securiies. The value of he derivaive may depend on one or several underlying asses. Furhermore, unlike a porfolio wih he same consiuens he srucured produc is usually wrapped in a legally complian, ready-o-inves forma and in his sense i is a packaged porfolio. Srucured invesmens have been par of diversified porfolios in Europe and Asia for many years, while he basic concep for hese producs originaed in he Unied Saes in he 980s. Srucured invesmens 'compee' wih a range of alernaive invesmen vehicles, such as individual securiies, muual funds, ETFs (exchange raded fund) and closed-end funds. The recen growh of hese insrumens is due o innovaive feaures, beer pricing and improved liquidiy. The idea behind a srucured invesmen is simple: o creae an invesmen produc ha combines some of he bes feaures of equiy and fixed income namely upside poenial wih downside proecion. This is accomplished by creaing a "baske" of invesmens ha can include bonds, CDs, equiies, commodiies, currencies, real esae invesmen russ, and derivaive producs. 4
This mix of invesmens in he baske deermines is poenial upside, as well as downside proecion. The usual componens of a srucured produc are a zero-coupon bond componen and an opion componen. The payou from he opion can be in he form of a fixed or variable coupon, or can be paid ou during he lifeime of he produc or a mauriy. The zero-coupon bond componen serves as buffer for yield-enhancemen sraegies which profi from acively acceping risk. Therefore, he invesor canno suffer a loss higher han he noe, bu may lose significan par of i. The zero-coupon bond componen is a floor for he capial-proeced producs. Oher producs, in paricular various dynamic invesmen sraegies, adjus he proporion of he zero-coupon bond over ime depending on a predeermined rule... Equiy-linked srucured producs The classificaion refers o he implici opion componens of he produc. In a firs sep, I disinguish beween producs wih plain vanilla and hose wih exoic opions componens. While in a second sep, exoic producs can be uniquely idenified and named, a similar differeniaion wihin he group of plain-vanilla producs is no possible. Their paymen profiles can be replicaed by one or more plain-vanilla opions, whereby he opion ypes (call or pu) and posiion (long or shor) is produc-specific. Therefore, I assign erms o some producs ha bes characerize heir paymen profiles. A classic srucured produc has he basic characerisics of a bond. As a specialfeaure, he issuer has he righ o redeem i a mauriy eiher by repaymen of isnominal value or delivery of a previously fixed number of specified shares. Mos srucured producs can be divided ino wo basic ypes: wih and wihou coupon paymens generally referred o as reverse converibles and discoun cerificaes. 5
In order o value srucured producs, I decompose hem by means of duplicaion, i.e., he reconsrucion of produc paymen profiles hrough several single componens. Thereby, I ignore ransacions coss and marke fricions, e.g., ax influences...2 Capial-Guaraneed Producs Capial-guaraneed producs have hree disinguishing characerisics: Redempion a a minimum guaraneed percenage of he face value (redempiona face value (00%) is frequenly guaraneed). No or low nominal ineres raes. Paricipaion in he performance of underlying asses The producs are ypically consruced in such a way ha he issue price is as close as possible o he bond s face value (wih adjusmen by means of he nominal ineres rae). I is also common ha no paymens (including coupons) are made unil he produc s mauriy dae. The invesor s paricipaion in he performance of he underlying asse can ake an exremely wide variey of forms. In he simples varian, he redempion amoun is deermined as he produc of he face value- and he percenage change in he underlying asse s price during he erm of he produc. If his value is lower han he guaraneed redempion amoun; he insrumen is redeemed a he guaraneed amoun. This can also be expressed as he following formula: R=N(+max(0,S T -S 0 )) 6
S 0 = N + N. max(0,s T -S 0 )) S 0 where R: redempion amoun N: face value S 0 : original price of underlying asse S T : Price of underlying asse a mauriy. Therefore, hese producs have a number of European call opions on he underlying asse embedded in hem. The number of opions is equal o he face value divided by he iniial price (cf. he las erm in he formula). The insrumen can hus, be inerpreed as a porfolio of zero coupon bonds (redempion amoun and coupons) and European call opions. The possible range of capial-guaraneed producs comprises combinaions of zero coupon bonds wih all conceivable ypes of opions. This means ha he number of differen producs is huge. The mos imporan characerisics for classifying hese producs are as follows: () Is he bonus reurn (bonus, ineres) proporionae o he performance of he underlying asse (like call and pu opions), or does i have a fixed value once a cerain performance level is reached (like binary barrier opions)? (2) Are he srike prices or barriers known on he dae of issue? Are hey calculaed as in Asian opions or in forward sar opions? (3) Wha are he characerisics of he underlying asse? Is i an individual sock, 7
an index or a baske? (4) Is he currency of he srucured produc differen from ha of he underlying asse? In he secions ha follow, a small bu useful selecion of producs is presened. As here are no uniform names for hese producs, hey are named afer he opions embedded in hem..2 Derivaive inroducion and opics Derivaives are hose financial insrumens whose values derive from price of he underlying asses e.g. bonds, socks, meals and energy. The derivaives are raded in wo main markes: ETM and OTC. ) The Exchange raded marke is a marke where individual s rade sandardized derivaive conracs. Invesmen asses are asses held by significan numbers of people purely for invesmen purposes (examples: bonds,socks ) 2) Over he couner (OTC) is he imporan alernaive o ETM. I is elephone and compuer linked nework of dealers,who do no physically mee. This marke became larger han ETM and srucured produc are raded in he OTC marke alhough his marke has a huge number of ailored derivaive conrac. One of he disadvanages of he OTC markes is ha such markes suffer from grea exposure o credi risk. 8
.2. Fuures and Forward conracs pricing and hedging Forward conracs are paricularly simple derivaives. I is an agreemen o buy or o sell an asse a cerain ime T for a cerain price K. The pay-off is (S T - K ) for long posiion and (K - S T ) for shor posiion. A fuure price K is delivery price in a forward conrac which is updaed daily and F 0 is forward price ha would apply o he conrac oday. The value of a long forward conrac, ƒ, is ƒ = (F 0 K)e rt Similarly, he value of a shor forward conrac is (K F 0 ) e rt Forward and fuures prices are usually assumed he same. 2 When ineres raes are uncerain hey are, in heory, slighly differen: 3 A srong posiive correlaion beween ineres raes and he asse price implies he fuures price is slighly higher han he forward price 4 A srong negaive correlaion implies he reverse Fuures conracs is sandardized forward conac and raded in exchange markes for fuures. Selemen price: he price jus before he final bell each day Open ineres: he oal number of conracs ousanding Ways Derivaives are used To hedge risks To speculae (ake a view on he fuure direcion of he marke) To lock in an arbirage profi To change he naure of a liabiliy and creaing synheic liabiliy and asses To change he naure of an invesmen and change he exposure o asses saus wihou incurring he coss of selling. 9
Now I will inroduce some imporan hedging and rading sraegies ha Srucured produc depend on. Shor selling involves selling securiies you do no own. Your broker borrows he securiies from anoher clien and sells hem in he marke in he usual way, a some sage you mus buy he securiies back so hey can be replaced in he accoun of he clien. You mus pay dividends and oher benefis he owner of he securiies. by Oher Key Poins abou Fuures They are seled daily 2 Closing ou a fuures posiion involves enering ino an offseing rade 3 Mos conracs are closed ou before mauriy If a conrac is no closed ou before mauriy, i usually seled by delivering he asses underlying he conrac. $00 received a ime T discouns o $00e -RT a ime zero when he coninuously compounded discoun rae is r If r is compounded annually F 0 = S 0 ( + r ) T (Assuming no sorage coss) If r is compounded coninuously insead of annually F 0 =S 0 e rt For any invesmen asse ha provides no income and has no sorage coss when an invesmen asse provides a known yield q (r q )T F 0 = S 0 e where q is he average yield during he life of he conrac (expressed wih Coninuous compounding) 0
Valuing a Forward Conrac assume ha sock index ha pays dividends income on he index he paymen is fixed and known in advance. Can be viewed as an invesmen asse paying a dividend yield 2 The fuures price and spo price relaionship is herefore (r q )T F 0 = S 0 e where q is he dividend yield on he porfolio represened by he index For he formula o be rue i is imporan ha he index represen an invesmen asse. In oher words, changes in he index mus correspond o changes in he value of a radable porfolio. Index Arbirage When F 0 >S0e (r-q)t an arbirageur buys he socks underlying he index and sells fuures When F 0 <S 0 e (r-q)t an arbirageur buys fuures and shors or sells he socks underlying he index Index arbirage involves simulaneous rades in fuures and many differen socks Occasionally (e.g., on Black Monday) simulaneous rades are no possible and he heoreical no-arbirage relaionship beween F 0 and S 0 does no holds so F 0 S 0 e (r+u )T, where u is he sorage cos per uni ime as a percen of he so he equaliy should hold. Oherwise here will be an arbirage opporuniy.
How o hedge using fuures A proporion of he exposure ha should opimally be hedged is h= ρ* (σ S / σ F ) where σ S is he sandard deviaion of ds, he change in he spo price during he hedging period, σ F is he sandard deviaion of df, he change in he fuures price during he hedging period ρ is he coefficien of correlaion beween ds and df. To hedge he risk in a porfolio he number of conracs ha should be shored is where P is he value of he porfolio, β is is bea, and A is he value of he asses. In pracice regression echniques are employed o hedge equiy opion by using equiy index fuures (he auhor is working in his field). This echnique implemened also in dynamic hedging sraegies..2.2 The fundamenal exposure ypes The fundamenal exposure ypes are he generic opion payoffs. Combining hese wih a long zero coupon bond gives he primal srucured producs, some of which have no failed o go ou of fashion. The following Figure shows clearly he ineracion beween invesmen view and payoff. 2
.2.3 European ype Opions Le he price process of he underlying asse be S (), [0,T]. European opions give he holder he righ o exercise he opion only on he expiraion dae T. Hence he holder receives he amoun (S(T)), where ϕ is a conrac funcion. Moreover, here are wo basic ypes of European opions namely European call opions and European pu opions. 3
European Call opion: a derivaive conrac ha gives is holder he righ o buy he underlying asses by cerain dae a cerain srike price. European Pu opion: a derivaive conrac ha gives is holder he righ o sell he underlying asses by cerain dae a cerain srike price. Pricing of European opion Black and Scholes derived a boundary value parial differenial equaion (PDE) for he value F(, s) of an opion on a sock. This value F(, s) solves he Black&Scholes PDE Under risk neural measure for one underlying asse only. F(, s) F(, s) 2 + r S + σ S s 2 F(, s) = Φ( s) 2 2 F(, s) r s (, Fs ) = 0 in [0 T ] R+. Here r is he ineres rae; σ is he volailiy of he underlying assumed fixed parameers. Asse S and Φ(s) = max(s k,0) is he conrac funcion. According o he Feynman-Kac heorem PDE soluion can represened as an expeced value F(,s)=e r(t-) [ ( s, )] E, s Φ T where he underlying sock S( ) follows he dynamics s(u)=r s(u) u+s(u)σ (u,s(u)) W(u) This price process is called geomeric Brownian moion. Here W is a Wiener process where S sars in s a ime 0. For he purpose of opion pricing I hus should assume ha he underlying sock follows his dynamics even if in realiy we do no expec he value of he sock o grow wih he ineres rae r. The American version of hose wo opions is he same excep ha i can be exercised earlier han exercise dae. 4
.2.4 An American opion gives he owner he righ o exercise he opion on or before he Expiraion dae T before he expiraion, dae (also called early exercise). The holder of an American opion needs o decide wheher o exercise immediaely or o wai. If he holder decides o exercise a say T, hen he receives Φ(S()) where Φ is he appropriae conrac funcion. Similarly, his opion can also be classified ino wo basic ypes: American call opions which give he owner he righ o buy an underlying asse for a given srike price on or before he expiraion dae, and American pu opion which gives he owner he righ o sell an underlying asse for a cerain srike price on or before he expiraion dae. If he underlying sock pays no dividends, early exercise of an American call opion is no opimal. On he oher hand early exercise of an American pu opion can be opimal even if he underlying sock does no pay dividends. An American opion is worh a leas as much as an European opion. To compare by examples here are wo examples how he wo prices compares For example Prices of he following opions long plain vanilla call opion non dividend share for 3 monhs o expiry dae opion he wo price funcions (European and American plain vanilla opion) are ploed here for he same srikes of 00 curren share price 20 5
Risk free rae of 0 % Volailiy of 40. Figure. is showing he price funcion of European opion using Black and Scholes formula. Figure.2 is showing he price funcion of he American opion using Bjerksund & Sensland approximaion.for more deails abou his approximaion see he Bjerksund & Sensland approximaion 2002. The able used o generae he 3 d graph for he American opion using Bjerksund approximaion & Sensland approximaion. Time o mauriy days Asse price 0.00 30.88 5.76 72.65 93.53 4.4 35.29 56.8 77.06 50.00 50.2736 50.8432 5.4228 52.0323 52.6754 53.3462 54.0368 54.7405 55.457 45.00 45.2736 45.8445 46.4380 47.0762 47.7554 48.4640 49.92 49.9288 50.672 40.00 40.2736 40.8484 4.4678 42.490 42.8763 43.636 44.408 45.780 45.9544 35.00 35.2736 35.8592 36.5246 37.2678 38.0568 38.8680 39.687 40.5056 4.384 30.00 30.2737 30.887 3.630 32.4580 33.3226 34.978 35.0704 35.9335 36.7836 25.00 25.2742 25.9552 26.890 27.7556 28.7079 29.6527 30.5807 3.4882 32.3744 20.00 20.2792 2.06 22.456 23.206 24.2569 25.277 26.2528 27.203 28.94 5.00 5.332 6.4396 7.6873 8.8874 20.0238 2.05 22.277 23.092 24.056 0.00 0.4857 2.0799 3.5459 4.8645 6.0723 7.955 8.254 9.2524 20.2072 05.00 6.94 8.280 9.840.2294 2.477 3.64 4.6736 5.6747 6.6255 00.00 2.7763 5.0530 6.6920 8.0687 9.2905 0.4065.4440 2.420 3.3462 95.00 0.8696 2.7262 4.907 5.4529 6.5875 7.639 8.6080 9.5299 0.4073 90.00 0.638.2453 2.3693 3.49 4.400 5.324 6.2009 7.0380 7.843 85.00 0.059 0.464.806.9564 2.7338 3.4970 4.242 4.9657 5.67 80.00 0.0007 0.38 0.5035.0009.5555 2.355 2.7253 3.367 3.9055 75.00 0.0000 0.0273 0.774 0.4466 0.795.937.6237 2.0735 2.5355 70.00 0.0000 0.0038 0.0494 0.684 0.3564 0.5989 0.8823.96.5325 65.00 0.0000 0.0003 0.003 0.056 0.359 0.263 0.4283 0.6253 0.8487 60.00 0.0000 0.0000 0.005 0.022 0.0424 0.098 0.807 0.2894 0.428 55.00 0.0000 0.0000 0.000 0.002 0.003 0.0297 0.0640 0.50 0.832 50.00 0.0000 0.0000 0.0000 0.0002 0.008 0.0069 0.082 0.0377 0.067 6
Figure. European call Figure.2 American call Bjerksund 7
A Trinomial ree has been se up for he American opion in case of he American opion. A 500 seps rinomial ree is consruced wih marix of underlying price is as follows. The following diagram shows how he firs node is calculaed also I will menion here how we calculae he relevan probabiliies of up and down probabiliies and here is par of algorihm d is he ime sep n is number of seps v is he volailiy pu is he up probabiliy Pd is he down probabiliy d = T / n u = Exp(v * Sqr(2 * d)) d = / u pu = (Exp(r * d / 2) - Exp(-v * Sqr(d / 2))) ^ 2 / (Exp(v * Sqr(d / 2)) - Exp(-v * Sqr(d / 2))) ^ 2 pd = (Exp(v * Sqr(d / 2)) - Exp(r * d / 2)) ^ 2 / (Exp(v * Sqr(d / 2)) - Exp(-v * Sqr(d / 2))) ^ 2 pm = - pu pd 8
9
20 Calculaions of able used o generae 3-D graph Time o mauriy in days Asse price 0.0 0 30.8 8 5.7 6 72.6 5 93.5 3 4. 4 35. 29 56.8 77. 06 28. 82 239. 7 260. 59 28. 47 302. 35 323. 24 344. 2 3 50.00 50. 369 50.4 222 50.7 070 50.9 944 5.2 892 5.5 945 5.9 8 52. 240 2 52.5 795 53.2 823 53.6 432 54.0 076 54.3 775 54.7 508 55. 223 55.5 009 5 45.00 45. 369 45.4 222 45.7 080 46.0 003 46.3 059 46.6 269 46.9 632 47. 32 5 47.6 73 48.4 94 48.8 0 49. 892 49.5 774 49.9 679 50.3 604 50.7 523 5 40.00 40. 369 40.4 223 40.7 09 4.0 39 4.3 380 4.6 833 42.0 467 42. 423 8 42.8 4 43.6 26 44.0 208 44.4 286 44.8 404 45.2 52 45.6 630 46.0 700 4 35.00 35. 369 35.4 228 35.7 95 36.0 437 36.3 985 36.7 787 37. 799 37. 592 5 38.0 53 38.8 803 39.3 38 39.7 493 40. 842 40.6 74 4.0 462 4.4 784 4 30.00 30. 369 30.4 252 30.7 427 3. 069 3.5 07 3.9 44 32.3 874 32. 845 4 33.3 088 34.2 449 34.7 064 35. 730 35.6 292 36.0 867 36.5 394 36.9 85 3 25.00 25. 369 25.4 36 25.8 025 26.2 357 26.7 093 27.2 038 27.7 0 28. 220 6 28.7 273 29.7 378 30.2 338 30.7 226 3.2 095 3.6 83 32. 547 32.6 227 3 20.00 20. 370 20.4 775 20.9 442 2.4 825 22.0 492 22.6 23 23. 932 23. 760 3 24.3 55 25.3 976 25.9 250 26.4 35 26.9 46 27.4 448 27.9 303 28.4 040 2 5.00 5. 404 5.6 42 6.2 555 6.9 340 7.6 075 8.2 652 8.9 042 9. 57 8 20. 226 2.2 695 2.8 28 22.3 553 22.8 769 23.3 940 23.8 969 24.3 87 2 0.00 0. 877 0.9 996.8 786 2.7 056 3.4 827 4.2 93 4.9 2 5. 575 0 6.2 035 7.4 033 7.9 707 8.5 84 9.0 487 9.5 70 20.0 822 20.5 802 2 05.00 5.55 6 6.90 85 8.00 59 8.95 3 9.80 37 0.5 8.3 05. 988 5 2.6 383 3.8 499 4.4 99 4.9 70 5.5 027 6.0 97 6.5 279 7.0 230 00.00 2.04 77 3.68 63 4.84 87 5.8 6 6.66 89 7.44 38 8.6 2 8.8 337 9.47 00 0.6 569.2 55.7 547 2.2 767 2.7 835 3.2 763 3.7 566 95.00 0.39 9.57 85 2.55 87 3.4 67 4.8 70 4.89 85 5.55 80 6. 87 6.77 96 7.89 62 8.42 23 8.93 05 9.42 28 9.90 0 0.3 663 0.8 227 90.00 0.03 08 0.50 40.3 3.76 40 2.37 47 2.95 62 3.52 30 4.0 567 4.58 38 5.57 04 6.04 68 6.5 9 6.96 29 7.40 2 7.82 8 8.24 46 8 85.00 0.00 07 0. 03 0.40 00 0.77 75.8 66.6 33 2.03 99 2.4 694 2.89 24 3.7 8 4.2 53 4.5 95 4.90 93 5.29 9 5.67 9 6.05 03 6 80.00 0.00 00 0.0 5 0.0 7 0.28 0 0.5 05 0.77 48.06 44.3 67.67 64 2.3 73 2.63 93 2.96 0 3.28 49 3.60 5 3.92 08 4.24 3 4 75.00 0.00 00 0.00 0.02 04 0.07 9 0.8 09 0.3 86 0.48 62 0.6 75 0.88 2.32 75.56 48.80 82 2.05 39 2.30 44 2.55 87 2.80 78 3 70.00 0.00 00 0.00 00 0.00 25 0.0 66 0.05 09 0.0 85 0.8 84 0.2 895 0.40 69 0.68 76 0.84 64.0 26.8 56.36 87.55 26.74 59 65.00 0.00 00 0.00 00 0.00 02 0.00 24 0.0 07 0.02 9 0.06 0 0. 045 0.6 5 0.3 48 0.40 74 0.50 85 0.62 7 0.74 3 0.86 49 0.99 98 0.00 0.00 0.00 0.00 0.00 0.00 0.0 0.0 0.05 0.2 0.7 0.22 0.28 0.35 0.43 0.5 0
60 50 40 30 20 260.59 77.06 Time o mauriy 93.53 0.00 50 55 60 65 70 75 80 85 90 95 00 05 0 5 20 25 30 35 40 45 50 Asse price 0 0 As we can see here ha he rinomial mehod is value he American opion han he approximaion bu i will converge as he number of seps increase. 2
.2.5 Bermudan Opion This ype of opions lies beween American and European. They can be exercised a cerain discree ime poins for any discree ime < < < =T. Therefore he Bermudan opions being a hybrid of European and American opions, he value of a Bermudan is greaer han or equal o an idenical European opion bu less han or equal o is equivalen American opion. I will price some of Bermudan ype opion like equiy Clique opion..2.6 Asian opion ypes This ype of opion depends on he average value of he underlying asse over a ime, Therefore, an Asian opion is pah dependen. Asian opions are cheaper relaive o heir European and American counerpars because of heir lower volailiy feaure The are broadly hree caegories: ) Arihmeic average Asians, 2) Geomeric average Asians 3) Combinaion of and 2 The pay-off can be averaged on a weighed average basis, whereby a given weighs is applied o each sock being averaged. This can be useful for aaining an average on a sample wih a highly skewed sample populaion. There are no known closed form analyical soluions arihmeic opions, due o he a propery of hese opions under which he lognormal assumpions collapse so i is no possible o analyically evaluae he sum of he correlaed lognormal random variables. 22
A furher breakdown of hese opions concludes ha Asians are eiher based on he average price of he underlying asse, or alernaively, here is he average srike ype. The payoff of geomeric Asian opions is given as: n / n Payoff Asian call =max 0, S i X i= n / n Payoff Asian pu =max 0, X S i i= Kemna & Vors (990) pu forward a closed form pricing soluion o geomeric averaging opions by alering he volailiy, and cos of carry erm. Geomeric averaging opions can be priced via a closed form analyic soluion because of he reason ha he geomeric average of he underlying prices follows a lognormal disribuion as well, whereas wih arihmeic average rae opions, his condiion collapses. The soluions o he geomeric averaging Asian call and pus are given as: C G =S e (b-r)(t-) N(d )-X e -r(t-) N(d 2 ) and, P G = X e -r(t-) N(-d 2 )- S e (b-r)(t-) N(-d ) where N(x) is he cumulaive normal disribuion funcion of: d =ln(s/x)+(b+0.5σ A )T σ A T 2 d 2 =ln(s/x)+(b-0.5σ A )T σ A T 2 23
The adjused volailiy and dividend yield are given as: σ A = σ / 3 b=/2(r-d-σ /6) 2 The payoff of arihmeic Asian opions is given as Payoff Asian call =max(0,( Si /n)-x) n i= Payoff Asian pu= max(0,x-( Si /n) n i= Here I will menion one of he approximaions o calculae he price of a srucured produc ha has an Asian srucured produc. ) The zero coupon bonds pars are valuaed using he relevan spo ineres raes. 2)The Asian opion for which paymens are based on a geomeric average are relaively easy approximaions have been developed by Turnbull and Wakeman (99), Levy (992) and Curran (992). In Curran s model, he value Of an Asian opion can be approximaed using he following formula: 24
Here is an example of capial guaraneed srucured produc ha has Asian pay off. On he FTSE 00 index using Curran s model. Average calculaed quarerly and he ineres rae used are annual compounded and volailiy is used are annual rae. The main parameers used are as follows Asse price ( S ) 95.00 Average so far ( SA ) 00.00 Srike price ( X ) 00.00 Time o nex average poin () 0.25 Time o mauriy ( T ) 5.00 Number of fixings n 4.00 Number of fixings fixed m 0.00 Risk-free rae ( r ) 4.50% Cos of carry ( b ) 2.00% Volailiy ( σ ) 26.00% Value 0.7396 25
20.0000 00.0000 80.0000 60.0000 40.0000 20.0000 323.24 0.0000 50.00 65.00 80.00 95.00 0.00 Asse price 25.00 40.00 55.00 70.00 85.00 200.00 28.82 4.4 Time o mauriy 0.00 The frequency wih which he value of he underlying asse is sampled varies widely from produc o produc. The averages are usually calculaed using daily, weekly or monhly values. Depending on wheher an Asian call or pu opion is embedded, he redempion amoun is calculaed using one of he following formulas: =Zero coupon bond + Asian opion value. 26
.2.7 Clique opions Clique are opion conracs, which provide a guaraneed minimum annual reurn in exchange for capping he maximum reurn earned each year over he life of he conrac. Applicaions: Recen urmoil in financial markes has led o a demand for producs ha reduce risk while sill offering upside poenial. For example, pension plans have been looking a aaching Guaranees o heir producs ha are linked o equiy reurns. Some plans, also in VA life producs such as hose described. Pricing Clique opions The Pricing framework here will be in he deerminisic volailiy model. Clique opions are essenially a series of forward-saring a-he-money opions wih a single premium deermined up fron, ha lock in any gains on specific daes. The srike price is hen rese a he new level of he underlying asse. I will use he following form, considering a global cap, global floor and local caps a predefined reseing imes i (i =,..., n). n S S m m i mna F mx a Ci, x i n P=exp(-r n )N.E Q i= Si i, F C i i, where N is he noional, C is he global cap, F is he global floor, F i, i =... n he local f floors, C i, i =,..., n are he local caps, and S is he asse price following a geomeric Brownian moion, or a jump-diffusion process. Under geomeric Brownian moion wih only fixed deerminisic annual rae of ineres 27
I can use he binomial mehod (CRR) binomial ree o price Clique opion. This binomial clique opion valuaion model which mainains he imporan propery of flexibiliy, can be used o price European and American cliques. The seings for his model are he same as hose described in he previous secion: I have he Cox-Ross-Rubinsein (CRR) binomial ree wih σ σ U=e and D = e- The adjused risk-neural probabiliy for he up sae is P = e σ -D U-D In addiion (-p) for he downsae probabiliy. This ime, insead of calculaing he probabiliy of each payoff, I use he backward valuaion approach described in Hull (2003), Haug (997)), adjusing i o Clique opions wih no cap or floor applied. The adjusmen is as follows: For each node ha falls under he rese dae m, he new srike price is deermined. If he sock price a m is above he original srike, he pu will rese is srike price equal o he hencurren sock price. For call opions: if he sock price m is below he original srike, he call will rese is srike price equal o he hen-curren sock price. Pricing example 28
Curren sock price = 00 Exercise price = 00 Time o mauriy =20 year Time o rese = 0 year Risk-free ineres rae = 4,5% Dividend yield =2% Sigma = 20%. In addiion, here is comparison beween plan vanilla European call and European Clique opion prices for various sock prices 29
0 00 90 80 70 60 50 40 clique price Plan vanila CRR 30 20 0 0 50.00 70.00 90.00 0.00 30.00 50.00 70.00 90.00 20.00 230.00 250.00 And here is comparison beween plan vanilla American call and European Clique opion prices for various sock prices 40 30 20 0 00 90 80 70 60 50 40 30 20 0 0 50.00 70.00 90.00 0.00 30.00 50.00 70.00 90.00 20.00 230.00 250.00 clique price CRR vanilla As you can see from boh chars ha he price is differen only when he sock price is less han 00 srike price for boh he American and European opion. 30
Chaper 2 ineres rae srucured producs 2.. Floaing Rae Noes (FRNs, Floaers) Floaing rae noes does no carry a fixed nominal ineres rae. The coupon paymens are linked o he movemen in a reference ineres rae (frequenly money marke raes, such as he LIBOR) o which hey are adjused a specific inervals, ypically on each coupon dae for he nex coupon period. A ypical produc could have he following feaures: The iniial coupon paymen o become due in six-monhs ime corresponds o he 6-monh LIBOR as a he issue dae. Afer six monhs he firs coupon is paid ou and he second coupon paymen is locked in a he hen curren 6-monh LIBOR. This procedure is repeaed every six monhs. The coupon of an FRN is frequenly defined as he sum of he reference ineres rae and a spread of x basis poins. As hey are regularly adjused o he prevailing money marke raes, he volailiy of floaing rae noes is very low. Replicaion Floaing rae noes may be viewed as zero coupon bonds wih a face value equaing he sum of he forhcoming coupon paymen and he principal of he FRN. Because heir regular ineres rae adjusmens guaranee ineres paymens in line wih marke condiion. 2.2 Opions on bonds Bond opions are an example for derivaives depending indirecly (hrough price movemens of he underlying bond) on he developmen of ineres raes. I is common o embed bond opions ino paricular bonds when hey are issued o make hem more aracive o poenial purchasers. A callable bond, for example, allows he issuing pary o buy back he bond a a predeermined price in he fuure. A puable bond, on he oher hand, allows he holder o sell he bond back o he issuer a a cerain fuure ime for a specified price. 3
Pricing bond opions The well-known Black-Scholes equaion was derived for he pricing of opions on sock prices and i was published in 973. Shorly aferwards, he model has been exended o accoun for he valuaion of opions on commodiy conracs such as forward conracs. In general, his model describes relaions for any variable, which is log normally disribued and can herefore be used for opions on ineres raes as well. The main assumpion of he Black model for he pricing of opions on bonds is ha a ime T he value of he underlying asse V T follows a lognormal disribuion wih he Sandard deviaion. S[ln V T ]=σ T. Furhermore, he expeced value of he underlying a ime T mus be equal o is forward price for a conrac wih mauriy T, since oherwise, arbirage would be possible. E[V T ]=F 0 E[max(V-K),0]=E[V]N(d)-KN(d2) E[max(K-V),0]=KN(-d2)-E[V]N(-d) where he symbols d and d2 are d = ln (E[V]/K)+s 2 /2 s d2= d = ln (E[V]/K)-s 2 / 2 =d-s s This is also he main resul of Black's model which, for he firs ime, allowed an Analyical approach o he pricing of opions on any log normally disribued underlying. 32
The symbol N(x) denoes he cumulaive normal disribuion. For a European call opion on a zero-coupon bond his leads o he well-known resul for he value of he opion. The call price is given by C= P(0,T)(F 0 N(d)-KN(d2)) where he value a ime T is discouned o ime 0 using P(0;T) as a risk free deflaor. The value of he corresponding pu opion is P= P(0,T)( KN(-d2) -F 0 N(-d))) Here is pricing example of European bond call opion and pu opion using he Black model and he following parameer. Bond Daa Term Srucure Time (Yrs) Rae (%) Principal: 00 Coupon Frequency: 0.5 4.500% Bond Life (Years): 5 Quarerly 5.000% Coupon Rae (%): 6.000% 2 5.500% Quoed Bond Price (/00): 98.80303 3 5.800% 4 6.00% Opion Daa 5 6.300% Pricing Model: Black - European Imply Volailiy Srike Price (/00): 00.00 Opion Life (Years): 3.00 Yield Volailiy (%): 0.00% Quoed Srike Call Pu Calculae 33
This is he graph of he call opion price agains he srike 4.5 4 3.5 Opion Price 3 2.5 2.5 0.5 0 95.00 97.00 99.00 0.00 03.00 05.00 Srike Price This is graph of he pu opion price agains he srike 7 6 5 Opion Price 4 3 2 0 95.00 97.00 99.00 0.00 03.00 05.00 Srike Price 34
2.3 Ineres Rae Caps and Floors Ineres rae caps are opions designed o provide hedge agains he rae of ineres on a floaing-rae noe rising above a cerain level known as cap rae. A floaing rae noe is periodically rese o a reference rae, eg. LIBOR. If his rae exceeds he cap rae, The cap rae applies insead. The enor denoes he ime beween rese daes. The Individual opions of a cap are denoed as caples. Noe ha he ineres rae is always se a he beginning of he ime period, while he paymen mus be made a he end of he period. In addiion o caps, floors and collars can be defined analogously o a cap, a floor Provides a payoff if he LIBOR rae falls below he floor rae, and he componens of a floor are denoed as floorles. A collar is a combinaion of a long posiion in a cap and a shor posiion in a floor. I is used o insure agains he LIBOR rae leaving an ineres rae range beween wo specific levels. Consider a cap wih expiraion T, a principal of L, and a cap rae of RK. The rese daes are, 2,., n, and n+ = T. The LIBOR rae observed a ime k is se for he ime Period beween k and k+, and he cap leads o a payoff a ime k+ which is Lδ k Max(F k -R K,0) where δ k = k+ - k. If he LIBOR rae F k is assumed lognormal disribued wih volailiy σ k, each caple can be valued separaely using he Black formula. The value of a caple becomes C=Lδ k P(0, k+ ) (F k N(d)- R K N(d2)) 35
wih 2 d= ln(f k / R K )+ σ k k /2 σ k k 2 d2= ln(f k / R K )- σ k k /2 σ k k For he pricing of he whole cap or floor, he values of each caple or floorle have o be discouned back using discoun facor as he numeraire: for N number of floorle and caples N C ip (, i) C i= 0 oal= N F ip (, i) F i= 0 oal = A Swap is an agreemen beween wo paries o exchange cash flows in he fuure. 2. Ineres rae swap(irs) A company agrees o pay a fixed ineres rae on a specific principal for a number of years and, in reurn, receives a floaing ineres rae on he same principal (pay fixed receive floaing). The floaing ineres rae is usually he LIBOR rae. Such 'plain vanilla' ineres rae swaps are ofen used o ransform floaing rae o fixed-rae loans or vice versa. A swap agreemen can be seen as he exchange of a floaing-rae (LIBOR) bond wih a fixed-rae bond. The forward swap rae S α,β () a ime for he ses of imes T and year fracions τ is he rae in he fixed leg of he above IRS ha makes he IRS a fair conrac a he presen ime. 36
S α,β () = P(;T α )- P(;T β ) β i= α + τi P(,Ti) Applicaion Life insurance companies use he hedge ineres rae risk and exend heir asse duraion in order o say mached wih heir long duraion liabiliies. 2.5 European payer (receiver) swapion is an opion giving he righ (and no obligaion) o ener a payer(receiver) IRS a a given fuure ime, he swapion mauriy. Usually he swapion mauriy coincides wih he firs rese dae of he underlying IRS. The underlying-irs lengh (T T 2 in our noaion) is called he enor of he swapion. Someimes he se of rese and paymen daes is called he enor srucure. I can wrie he discouned payoff of a payer swapion by considering he value of he underlying payer IRS a is firs rese dae T, which is also assumed o be he swapion mauriy. Such a value is given by changing sign in formula. Black s model is used frequenly o value European swapion, - x m C= ( + F / m) rt F e [ F * N( d) X ( Nd 2) ] x m P= ( + F / m) rt F e [ X * N( d 2) F ( Nd ) ] 37
2 d= ln(f /X )+ σ k /2 σ T d2 =d - σ T where F is he srike swap rae and X is he curren implied forward swap rae for which is here he mauriy of he opion elemen of he swapion and sar ime of he swap and ime 2 is he ime when he swap conrac erminae T= 2- Pricing and applicaions Here is example of pricing receiver swapion ha life insurer use o hedge heir ineres rae exposure in guaraneed annuiy opion. Swap / Cap Daa Term Srucure Underlying Type: Time (Yrs) Rae (%) Swap Opion 3.96% Selemen Frequency: 2 3.879% Principal : 00 Semi-Annual 3 3.853% Swap Sar (Years):.00 4 3.928% Swap End (Years): 30.00 5 3.992% Swap Rae (%):.82% Imply Breakeven Rae 6 4.8% 7 4.203% Pricing Model: 8 4.288% Black - European 9 4.406% 0 4.68% Volailiy (%): 5.00% Imply Volailiy 4.586% 2 4.482% Rec. Fixed 3 4.376% Pay Fixed Price: DV0 (Per basis poin): Gamma0 (Per %): Vega (per %):.38E-08 -.25E-09.72E-08 7.45E-08 Calculae 38
25 20 Opion Price 5 0 5 0.00% 2.00% 3.00% 4.00% 5.00% 6.00% 7.00% 8.00% 9.00% 0.00% Swap Rae 39
2.6 Callable/Puable Zero Coupon Bonds Callable (puable) zero coupon bonds differ from zero coupon bonds in ha he Issuer has he righ o buy (he invesor has he righ o sell) he paper premaurely a a specified price. There are hree ypes of call/pu provisions. European opion: The bond is callable/puable a a predeermined price on one specified day. American opion: The bond is callable/puable during a specified period. Bermuda opion: The bond is callable/puable a specified prices on a number of predeermined occasions. A call provision allows he issuer o repurchase he bond premaurely a a specified price. In effec, he issuer of a callable bond reains a call opion on he bond. The invesor is he opion seller. A pu provision allows he invesor o sell he bond premaurely a a specified price. In oher words, he invesor has a pu opion on he bond. Here, he issuer is he opion seller. Call provision The issuer has a Bermuda call opion which may be exercised a an annually changing srike price. Replicaion This insrumen breaks ino callable zero coupon bonds down ino a zero coupon bond and a call Opion. callable zero coupon bond = zero coupon bond + call opion 40
where + long posiion - Shor posiion The decomposed zero coupon bond has he same feaures as he callable zero coupon bond excep for he call provision. The call opion can be a European, American or Bermuda opion. Variance swaps Variance swaps are insrumens, which offer invesors sraighforward and direc exposure o he volailiy of an underlying asse such as a sock or index. They are swap conracs where he paries agree o exchange a pre-agreed Variance level for he acual amoun of variance realised over a period. Variance swaps offer invesors a means of achieving direc exposure o realised variance wihou he pah-dependency issues associaed wih dela-hedged opions. Buying a variance swap is like being long volailiy a he srike level; if he marke delivers more han implied by he srike of he opion, you are in profi, and if he marke delivers less, you are in loss. Similarly, selling a variance swap is like being shor volailiy. However, variance swaps are convex in volailiy: a long posiion profis more from an increase in volailiy han i loses from a corresponding decrease. For his reason variance swaps normally rade above ATM volailiy. 4
Marke developmen Variance swap conracs were firs menioned in he 990 s, bu like vanilla opions only really ook off following he developmen of robus pricing models hrough replicaion argumens. The direcness of he exposure o volailiy and he relaive ease of replicaion hrough a saic porfolio of opions make variance swaps aracive insrumens for invesors and marke-makers alike. The variance swap marke has grown seadily in recen years, driven by invesor demand o ake direc volailiy exposure wihou he cos and complexiy of managing and dela hedging a vanilla opions posiion. Alhough i is possible o achieve variance swap payoffs using a porfolio of opions, he variance swap conrac offers a convenien package bundled wih he necessary dela-hedging. This will offer invesors a simple and direc exposure o volailiy, wihou any of he pah dependency issues associaed wih dela hedging an opion. Variance swaps iniially developed on index underlings. In Europe, variance swaps on he Euro Soxx 50 index are by far he mos liquid, bu DAX and FTSE are also frequenly raded. Variance swaps are also radable on he more liquid sock underlings especially Euro Soxx 50 consiuens, allowing for he consrucion of variance dispersion rades. 42
Variance swaps are radable on a range of indices across developed markes and increasingly also on developing markes. Bid/offer spreads have come in significanly over recen years and in Europe hey are now ypically in he region of 0.5 vegas for indices and vegas for single-socks alhough he laer vary according o liquidiy facors. Example : Variance swap p/l An invesor wan o gain exposure o he volailiy of an underlying index (e.g, Dow Jones FTSE 00 ) over he nex year. The invesor buys a -year variance swap, and will be delivered he difference beween he realised variance over he nex year and he curren level of implied variance, muliplied by he variance noional. Suppose he rade size is 2,500 variance noional, represening a p/l of 2,500 per poin difference beween realised and Implied variance. If he variance swap srike is 20 (implied variance is 400) and he subsequen variance realised over he course of he year is(5%) 2 = 0.0225 (quoed as 225), The invesor will make a loss because realised variance is below he level bough. Overall loss o he long = 437,500 = 2,500 x (400 225). The shor posiion will profi by he same amoun..: Realised volailiy 43
Volailiy measures he variabiliy of reurns of an underlying asse and in some sense provides a measure of he risk of holding ha underlying. In his noe I am concerned wih he volailiy of equiies and equiy indices, alhough much of he discussion could apply o he volailiy of oher underlying asses such as credi, fixed-income, FX and commodiies. Figure 3 shows he hisory of realised volailiy on he Dow Jones Indusrial Average over he las 00 years. Periods of higher volailiy can be observed, e.g. in he early 930 s as a resul of he Grea Depression, and o a lesser exen around 2000 wih he build-up and unwind of he docom bubble. Also noiceable is he effec of he 987 crash, mosly due o an excepionally large single day move, as well as numerous smaller volailiy spikes. Summary of he equiy volailiy characerisics The following are some of he commonly observed properies of (equiy marke) volailiy: Volailiy ends o be ani-correlaed wih he underlying over shor ime periods Volailiy can increase suddenly in spikes Volailiy can be observed o experience differen regimes Volailiy ends o be mean revering (wihin regimes) 44
This lis suggess some of he reasons why invesors may wish o rade volailiy: as a parial hedge agains he underlying. Especially for a volailiy spike caused by a sudden marke sell-off; as a diversifying asse class; o ake a macro view e.g. or a poenial change in volailiy regime; for o rade a spread of volailiy beween relaed insrumens. Pricing model and hedging Firs le us undersand he cash flow srucure he following diagram explain he cash flow exchanged by looking o he following diagram 45
Volailiy swaps are series of forward conracs on fuure realized sock volailiy, variance. Swaps are similar conrac on variance, he square of he fuure volailiy. Boh hese insrumens provide an easy way for invesors o gain exposure o he fuure level of volailiy. A sock's volailiy is he simples measure of is risk less or uncerainy. Formally, he volailiy σ R(S). σ R (S) is he annualized sandard deviaion of he Sock s reurns during he period of ineres, where he subscrip R denoes he observed or "realized" volailiy for he sock. The easy way o rade volailiy is o use volailiy swaps, someimes Called realized volailiy forward conracs, because hey provide pure exposure To volailiy (and only o volailiy). A sock volailiy swap is a forward conrac on he annualized volailiy. Is payoff a expiraion is equal o N( σ 2 R (S)-K var ) Where σ R(S)) is he realized sock volailiy (quoed in annual erms) over he life of he conrac. 46
T ( σ 2 R(S) =/T 0 σ 2 (S) ds K var is he delivery price for variance, and N is he noional amoun of he swap in dollars per annualized volailiy poin squared. The holder of variance swap a expiraion receives N dollars for every poin by which he sock's realized variance has exceeded he variance delivery price K var. Therefore, pricing he variance swap reduces o calculaing he realized volailiy square. Valuing a variance forward conrac or swap is no differen from valuing any oher derivaive securiy. The value of a forward conrac P on fuure realized variance wih srike price Kvar is he expeced presen value of he Fuure payoff in he risk-neural world: P=E(e -rt ( σ 2 R (S)-K var ) where r is he risk-free discoun rae corresponding o he expiraion dae T (Under he assumpion of deerminisic risk free rae)and E denoes he expecaion. Thus, for calculaing variance swaps we need o know only E [( σ 2 R (S)] Namely, mean value of he underlying variance. Approximaion (which is used he second order Taylor expansion for funcion px) where E[ σ 2 R (S)] E (V ) - Var(V) 47
8 E(V) 3/2 Where V = σ 2 R (S) In addiion, Var(V) 8 E(V) 3/2 his he erm of he convexiy adjusmen. Thus, o calculae volailiy swaps ineed he firs and he second erm his variance has unbiased esimaor namely: Var n (S)=n/(n-)*/T * log 2 n i= V=Var(S)= lim Var n (S) n S S - Where we negleced by /n log 2 n i= S S - For simpliciy reason only. Inoe ha iuse Heson (993) model: 2 Log S = ( r σ / 2) d + σ dw S - 48
49 E(var n (S))= n ) (lo g 2 = n S S E (n-)t snd E( log 2 S S )= ) ( d r 2 _ ) ( d r d E 2 ) (σ + 4 s d E 2 2 σ σ -E( d E 2 ) (σ dw σ )+ d E 2 ) (σ
Appendix Variance and Volailiy Swaps for Heson Model of Securiies Markes Sochasic Volailiy Model. Le (;F;F ; P) be probabiliy space wih filraion F; [0; T]: Assume ha underlying asse S in he risk-neural world and variance follow he following model, Heson (993) model: σ ds =r d+ dw s dσ 2 =K(θ 2 -σ 2 )d+ γ σ 2 dw where r is deerminisic ineres rae, σ 0 and θ are shor and long volailiy, k > 0 is a reversion speed, γ > 0 is a volailiy (of volailiy) parameer, w and w2 are independen sandard Wiener processes. The Heson asse process has a variance ha follows Cox-Ingersoll- Ross (985) process, described by he second equaion. If he volailiy follows Ornsein-Uhlenbeck process (see, for example, Oksendal (998)), hen Io's lemma shows ha he variance follows he process described exacly by he second equaion. 50
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