Chaper 2 The opion pricing framework The opion markes based on swap raes or he LIBOR have become he larges fixed income markes, and caps (floors) and swapions are he mos imporan derivaives wihin hese markes. Thereby, a cap (floor) can be inerpreed as a porfolio of opions on zero bonds. Hence, pricing a cap (floor) is very easy, if we have found an exac soluion for he arbirage-free price of a caple (floorle) (see e.g. Briys, Crouhy and Schöbel 11. On he oher hand, a swapion may be inerpreed as an opion on a porfolio of zero bonds 1. Therefore, even in he simples case of lognormal-disribued bond prices, he porfolio of he bonds would be described by he disribuion of a sum of lognormal-disribued random variables. Unforunaely, here exiss no analyic densiy funcion for such a sum of lognormal-disribued random variables. Hence, using a muli-facor model wih Brownian moions or Random Fields 2 as he sources of uncerainy, i seems unlikely ha exac closed-form soluions can be found for he pricing of swapions. The characerisic funcion of he random variable X(T 0, {T i }) = log u c ip(t 0,T i ) wih he coupon paymens c i a he fixed daes T i {T 1,...,T u } canno be compued in closed-form. Oherwise, we are able o find a closed-form soluion for he momens of he underlying random variable V (T 0, {T i }) = u c ip(t 0,T i ) a he exercise dae T 0 of he swapion. Hence, using he analyic soluion of he momens wihin our Inegraed Edgeworh Expansion (IEE) enables us o compue he T i -forward measure exercise probabiliies Π T i K = E T i 1V (T0,{T i })>K (secion (5.3.3 )). Reasonable carefulness has o be paid for he fac ha he characerisic funcion of a lognormal-disribued 1 The owner of a swapion wih srike price K mauring a ime T 0, has he righ o ener a ime T 0 he underlying forward swap seled in arreas. A swapion may also be seen as an opion on a coupon bearing bond (see e.g. Musiela and Rukowski 61). 2 Eberlein and Kluge 29 find a closed-form soluion for swapions using a Lévy erm srucure model. A soluion for bond opions assuming a one-facor model has been derived by Jamishidian 42. 7
8 2 The opion pricing framework random variable canno be approximaed asympoically by an infinie Taylor series expansion of he momens (Leipnik 53). As a resul of he Leipnikeffec we runcae he Taylor series before he expansion of he characerisic funcion ends o diverge. In conrary o he compuaion of opions on coupon bearing bonds via an IEE, we can apply sandard Fourier inversion echniques for he derivaion zero bond opion prices. Applying e.g. he Fracional Fourier Transform (FRFT) echnique of Bailey and Swarzrauber 4 is a very efficien mehod o compue opion prices for a wide range of srike prices. This can eiher be done, by direcly compuing he opion price via an Fourier inversion of he ransformed payoff funcion or by separaely compuing he exercise probabiliies Π T i k. Running he firs approach has he advanage ha we only have o compue one inegral for he compuaion of he opion prices. On he oher hand, someimes we are addiionally ineresed in he compuaion of single exercise probabiliies 3. Therefore, we prefer he laer as he opion price can be easily compued by summing over he single probabiliies 4. 2.1 Zero-coupon bond opions In he following, we derive a heoreical pricing framework for he compuaion of opions on bond applying sandard Fourier inversion echniques. Saring wih a plain vanilla European opion on a zero-coupon bond wih he srike price K, mauriy T 1 of he underlying bond and exercise dae T 0 of he opion, we have ZBO w (,T 0,T 1 ) = we Q = we Q wke Q r(s)ds (P(T 0,T 1 ) K)1 wx(t0,t 1 )>wk r(s)ds+x(t 0,T 1 ) 1 wx(t0,t 1 )>wk r(s)ds 1 wx(t0,t 1 )>wk, (2.1) wih w = 1 for a European call opion and w = 1 for a European pu opion 5. We define he probabiliy Π Q,a k given by 3 Noe ha he FRFT approach is very efficien. Hence, he compuaion of single exercise probabiliies runs nearly wihou any addiional compuaional coss and wihou geing an significan increase in he approximaion error (see e.g. figure (5.1)). 4 Furhermore, we wan o be consisen wih our IEE approach, where he price of he couponbond opions can only be compued by summing over he single exercise probabiliies Π T i K 5 In his hesis, we mainly focus on he derivaion of call opions (w = 1), keeping in mind ha i is always easy o compue he appropriae probabiliies for w = -1 via E Q r(s)ds+ax(t 0,T 1 ) 1 X(T0,T 1 )<k = 1 Π,a Q k.
2.1 Zero-coupon bond opions 9 Π,a Q k E Q r(s)ds+ax(t 0,T 1 ) 1 X(T0,T 1 )>k (2.2) for a = {0,1}, wih X (T 0,T 1 ) = logp(t 0,T 1 ) and he (log) srike price k = logk. Armed wih his, we are able o compue he price of a call opion via ZBO 1 (,T 0,T 1 ) = Π Q,1 k K Π Q,0 k and accordingly he price of a pu opion via ( ) ( ZBO 1 (,T 0,T 1 ) = K 1 Π Q,0 k 1 Π Q,1 ). k Finally, defining he ransform Θ (z) E Q r(s)ds+zx(t 0,T 1 ), (2.3) for z C we obain he risk-neural probabiliies by performing a Fourier inversion 6 Π,a Q k = 1 2 + 1 Θ (a + iφ)e iφk Re dφ. π iφ 0 Noe ha we obain a Black and Scholes -like opion pricing formula if he Fourier inversion can be derived in closed-form (see e.g. secion (5.2.1)). Assuming more advanced models, like a muli-facor HJM-framework combined wih unspanned sochasic volailiy (USV), he opion price ofen can be derived by performing a FRFT (see e.g. secion (7.2)). Then, given he exercise probabiliies Π,a Q k we easily obain he price of he single caples (floorles). Finally, we ge he price of he ineres rae cap (floor), by summing over he single caples (floorles) for all paymen daes {T i } = {T 1,..., T N }. The final payoff of a caple (floorle) seled in arreas wih he mauriy T 1 and a face value of one is defined by le w (T 1 ) max{w(l(t 0,T 1 ) CR),0}, wih = T 1 T 0, he cap rae CR and he LIBOR L(T 0,T 1 ) in T 0. Hence, we obain he payoff le w (T 0 ) = 1 + L(T 0,T 1 ) max{w(l(t 0,T 1 ) CR),0} { ( = max w 1 1 + CR ) },0, 1 + L(T 0,T 1 ) a he exercise dae T 0, where he las erm equals a zero-coupon bond paying he face value 1 + CR a ime T 1. A las, he payoff is given by 6 See for example Duffie, Pan and Singleon 28.
10 2 The opion pricing framework ogeher wih he zero-coupon bond le w (T 0 ) = max{w(1 P(T 0,T 1 )),0}, P(T 0,T 1 ) = 1 + CR 1 + L(T 0,T 1 ). This implies ha he payoff of a caple cle(,t 0,T 1 ) = le 1 (T 0 ) is equivalen o a pu opion on a zero-coupon bond P(,T ) wih face value N = 1 + CR and a srike price K = 1. Therefore, we obain he dae- price of a caple cle(,t 0,T 1 ) = ZBO 1 (,T 0,T i ( ) ( ) = K 1 Π Q,0 k 1 Π Q,1 k and accordingly he price of a floorle f le(,t 0,T 1 ) = ZBO 1 (,T 0,T 1 ) = Π Q Q,1 k KΠ,0 k. As such, we can easily compue he price of a European cap Cap(,T 0,{T i }) = and he price of he equivalen floor Floor(,T 0,{T i }) = N N ZBO 1 (,T 0,T i ) ZBO 1 (,T 0,T i ), by summing over all caples (floorles) for all paymen daes T i for i = 1,...,N. 2.2 Coupon bond opions Now, applying he same approach as in secion (2.1) we derive he heoreical opion pricing formula for he price of a swapion based on he Fourier inversion of he new ransform Ξ (z) E Q r(s)ds+zlogv (T 0,{T i }). Saring from he payoff funcion of a European opion on a coupon bearing bond we can wrie he opion price a he exercise dae T 0 as follows
2.2 Coupon bond opions 11 CBO w (,T 0,{T i }) = we Q r(s)ds (V (T 0,{T i }) K)1 wv (T0,{T i })>wk = we Q wke Q r(s)ds+ X(T 0, {T i }) 1 w X(T 0, {T i })>wk r(s)ds 1 w X(T 0, {T i })>wk. (2.4) Togeher wih and V (T 0, {T i }) = u c i P(T 0,T i ) we have CBO w (,T 0,{T i }) = we Q X(T 0, {T i }) = logv (T 0, {T i }) = log wke Q ( u c i P(T 0,T i ) ) r(s)ds+ X(T 0, {T i }) 1 w X(T 0, {T i })>wk r(s)ds 1 w X(T 0, {T i })>wk for all paymen daes {T 1,...,T u }. By defining he probabiliy Π,a Q k E Q, r(s)ds+a X(T 0,{T i }) 1 X(T 0, {T i })>k we direcly obain he price of a zero-coupon bond call opion CBO 1 (,T 0,{T i }) = Π Q Q,1 k KΠ,0 k and respecively he price of he pu opion ( ) ( CBO 1 (,T 0,{T i }) = K 1 Π Q,0 k 1 Π Q,1 ). k Noe ha he payoff funcion of a swapion 7 wih exercise dae T 0 and equidisan paymen daes T i for i = 1,...,u is given by 8 { } S w (T 0,{T i }) = max w u,, (SR L(T 0,T i 1,T i ))P(T 0,T i ),0, (2.5) 7 The owner of a payer (receiver) swapion mauring a ime T 0, has he righ o ener a ime T 0 he underlying forward payer (receiver) swap seled in arreas (see e.g. Musiela and Rukowski 61) 8 The payoff funcion can be defined easily for non equidisan paymen daes i.
12 2 The opion pricing framework wih = T i T i 1 for i = 2,...,u. Again, w = 1 equals a receiver swapion and w = -1 a payer swapion. Now, plugging he swap rae ogeher wih he forward rae SR = 1 P(,T u) u P(,T i), L(T 0,T i 1,T i ) 1 in equaion (2.5) finally leads o { ( S w (T 0,{T i }) = max or more easily w S w (T 0,{T i }) = max SR { w u ( ) P(T0,T i 1 ) P(T 0,T i ) 1 P(T 0,T i ) + P(T 0,T u ) 1 ( u c i P(T 0,T i ) 1 where he coupon paymens for i = 1,...,u 1 are given by ogeher wih he final paymen c i = SR, c u = 1 + SR. ),0 } ),0 }, (2.6) Now, we direcly see ha a swapion 9 in general can be seen as an opion on a coupon bond wih srike K = 1 and exercise dae T 0 paying he coupons c i a he paymen daes {T i } = {T 1,..., T u }. Armed wih his, we obain he price of a receiver swapion S 1 (,T 0,{T i }) = CBO 1 (,T 0,{T i }) (2.7) = Π Q,1 0 Π Q,0 0 and respecively he price of a payer swapion S 1 (,T 0,{T i }) = CBO 1 (,T 0,{T i }) ( ) ( = 1 Π Q,0 0 1 Π Q,1 ), 0 9 In he following we use he erm swapion and opion on a coupon bond opion inerchangeably. Neverheless, keeping in mind ha a swapion is only one special case of an opion on a coupon bond.
2.2 Coupon bond opions 13 given he (log) srike price k = 0. Now, ogeher wih he ransform Ξ (z) E Q r(s)ds+z X(T 0,{T i }), (2.8) wih z C we heoreically could compue he risk-neural probabiliies Π Q,a k by performing a Fourier inversion via Π,a Q k = 1 2 + 1 Re π 0 Ξ (a + iφ)e iφk iφ dφ. Unforunaely, here exiss no closed-form soluion for he ransform Ξ (a + iφ). This direcly implies ha we need a new mehod for he approximaion of he single exercise probabiliies Π,a Q k assuming a muli-facor model wih more han one paymen dae. On he oher hand, he ransform Ξ (n) can be solved analyically for nonnegaive ineger numbers n. This special soluions of Ξ (z) can be used o compue he n-h momens of he underlying random variable V (T 0, {T i }) under he T i forward measure. Then, by plugging hese momens in he IEE scheme we are able o obain an excellen approximaion of he single exercise probabiliies (see e.g. secion (5.3.3) and (5.3.4)). Recapiulaing, we have derived heoreically a unified seup for he compuaion of bond opion prices in a generalized muli-facor framework. In general, he opion price can be compued by he use of exponenial affine soluions of he ransforms Θ (z), for z C applying a FRFT and Ξ (n), for n N performing an IEE. The ransforms Θ (z) and Ξ (z), by iself can be seen as a modified characerisic funcion. Unforunaely, here exiss no closed-form of he ransform Ξ (z), meaning ha he sandard Fourier inversion echniques can be applied only for he compuaion of opions on discoun bonds. On he oher hand, he ransform Ξ (n) can be used o compue he n-h momens of he underlying random variable V (T 0,{T i }). Then, by plugging he momens (cumulans) in he IEE scheme he price of an opion on coupon bearing bond can be compued, even in a muli-facor framework.
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