PRICING OF SHIBOR-BASED FLOATING-RATE BONDS IN CHINA

Transcription

1 PRICING OF SHIBOR-BASED FLOATING-RATE BONDS IN CHINA CHEN YANGYIFAN (B.Sc., NJU, China) A THESIS SUBMITTED FOR THE DEGREE OF MASTER OF SCIENCE DEPARTMENT OF MATHEMATICS NATIONAL UNIVERSITY OF SINGAPORE 2016

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3 DECLARATION I hereby declare ha his hesis is my original work and i has been wrien by me in is enirey. I have duly acknowledged all he sources of informaion which have been used in he hesis. This hesis has also no been submied for any degree in any universiy previously. CHEN YANGYIFAN 15-Dec-2016

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5 Acknowledgmens I am very graeful for my supervisor Professor Dai Min a Naional Universiy of Singapore for his supervision. I can no be hankful enough for his aspiring guidance and invaluably consrucive criicism. I feel very forunae o sudy and work wih him during he las year. I wan o express my graiude o everyone in my office room, who suppored me during he course of his Maser program. I am sincerely graeful o hem for sharing insighful views on many issues on boh research and life. I also wan o express my warm hanks o Mr. Zhang Yaquan for his suppor and advice on his hesis. I am graeful ha he universiy provides me he scholarships o complee he degree, as well as a wonderful environmen for research. Finally, I would like o express my special hanks o my parens for heir uncondiional love. Thank you, Chen Yangyifan 5

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7 Conens 1 Inroducion Produc and Payoff Inroducion of Shibor Marke Noaion Shibor-based Floaing Rae Bonds Shibor-Based Ineres Rae Swaps Lieraure Review Conribuion of Our Thesis Preliminaries CIR Model Hull-Whie Model Pricing Shibor-based Floaing-Rae Bond Model for Shibor The Model Calibraion o The Iniial Term Srucure Explici Expression for Price of Shibor-based Floaing-Rae Bond Explici Expression of Presen Value of he Coupons Explici Expression for Shibor3M-based Floaing Rae Bond Liquidiy Risk Empirical Pricing Performance Daa and Calibraion Sripping he forward Shibor from IRS Calibraion for Hull-Whie model Calibraion for Shor-erm Spread and Correlaion Parameer k Calibraion for Liquidiy Inensiy Resuls

8 5 Summary and Conclusion 37 Bibliography 39 A Malab Codes 41 A.1 Codes for calculaing risk-free zero-coupon bond price A.2 Codes for calculaing η() A.3 Codes for sripping he forward Shibor from Shibor3M IRS A.4 Codes for calculaing C(, T ) A.5 Codes for liquidiy adjusmen A.6 Codes for valuaion of fricionless floaing-rae bond

9 Absrac Shibor(Shanghai Inerbank Offered Rae) is playing a more and more imporan role in Chinese bond marke, as one of he mos widely used benchmark rae. In his hesis, we use a shor-rae model for he insananeous risk-free spo rae, as well as he shor-erm spread beween Shibor and risk-free rae. By carefully inroducing a deerminisic shif, our model can fi o he iniial erm srucure. Since we only use Markovian process in our model, closed-form expressions for discouned value of Shibor-based coupon paymens are available, which allows us o implemen he model easily in pracice. Closed-form expressions for he value of Shibor3M-based floaing rae bond in China are also derived, and compared o he real marke price. We find ha he model presened in his hesis can reproduce he price of Shibor3M-based floaing-rae bonds properly, especially afer aking he liquidiy risk ino accoun. Keywords: Shibor; Floaing-rae bond; Muliple curve seing; Shor-rae Models. 9

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11 Chaper 1 Inroducion 1.1 Produc and Payoff Inroducion of Shibor The Shibor is an daily calculaed reference rae based on he ineres raes for unsecured funds lending in he Shanghai inerbank money marke. I is calculaed from raes quoed by 18 banks for enors ranging from overnigh o one year. In Table 1.1, we lis he Shibor wih differen mauriies on Dec China s Cenral Bank has been rying o promoe Shibor ino a benchmark ineres rae since i was launched in In 2007, ineres rae swap based on SHIBOR accouns for abou 13% of he oal swaps. In 2008, he same raio goes up by 22%. Afer 2009, all forward rae agreemen is based on Shibor. Moreover, many Shibor-based floaing-rae bonds are issued afer he launch of Shibor. As China s Libor, Shibor plays an imporan role during ineres rae markeizaion in China. Libor is very similar o Shibor, which is calculaed by he Briish Bankers Associaion. Before he financial crisis a few years ago, Libor rae was regarded as a risk-free rae, which was a common pracice a ha ime. Afer 2007, however, a basic relaion ha should hold rue for risk-free rae was violaed for Libor. More specifically, we always expec risk-free rae o saisfy: 1 + F (; T, T + ) = p(, T ) p(, T + ) (1.1) where F (; T, T + ) is he discreely compounded forward rae, and p(, T ) denoes he price of a risk-free zero coupon bond price wih mauriy T. I is no surprising ha his relaion does no hold rue for Shibor as well. In Figure 1.1, we show ha, during 2015, here is obvious discrepancy beween he forward Shibor L(T ; T + 3M, T + 6M) sripped from he Shibor3M-based ineres rae swaps 1 and he sandard spo replicaion calculaed by using he raio of he bond prices p(, T ), as in (1.1). Here, p(, T ) is calculaed hrough he spo Shibor a ime L(;, T ) p(, T ) = (T )L(;, T ) 1 Deails abou he way we implemen his will be discussed in following pars. (1.2) 11

12 Terms O/N 1W 2W 1M 3M 6M 9M 1Y Shibor(%) Table 1.1: Shibor on Dec sandard spo replicaion Sipped from Shibor-based IRS Figure 1.1: Discrepancy of forward Shibor beween he sandard spo replicaion from (1.1) and he one sripped from Shibor-based IRS /01/ /01/ /01/ /01/ /01/2016 If Shibor were risk-free, he spread beween forward raes implied by hese wo mehods should be negligible. However, he spread in Figure 1.1 indicaes ha Shibor is no a risk-free rae in he curren Chinese bond marke, as similar o Libor/Euribor. This phenomenon is firs noed by Morini (2009)[8], where economical explanaions are given for hese spreads. In our hesis, we ry o model he Shibor as a sum of risk-free rae r and a shor-rae spread s, which will be discussed fully in laer chapers Marke Noaion In his secion, we assume a fricionless marke, which is free of arbirage opporuniies. We denoe shor rae risk-free rae as r. Given he risk-free rae r, we may define he risk-free bank accoun as: B = e 0 rudu We denoe he sandard maringale measure as Q, under which all raded asses are (local) maringales afer being discouned by B as numéraire. Then he price of risk-free zero coupon bond p(, T ) can be generaed as: p(, T ) = E Q [ B ] = E Q B [e r u du ] (1.3) T (1.3) implies ha he process p(,t ) B = E Q [ 1 B T ] is a Q-maringale. By aking i as densiy funcion for an equivalen measure, we can ge he sandard forward measure Q T as: dq T dq F = p(, T ), [0, T ] (1.4) B p(0, T ) In his case, under he forward measure Q T, he raded asses discouned by p(, T ) are maringales, and he relaionship beween wo forward measure is given by dq T dq S F = p(, T ) p(0, S) p(, S) p(0, T ) (1.5) 12

13 Table 1.2: Bond Informaion. The daa are obained from China Foreign Exchange Trading Sysem and Wind Financial Terminal. Name GK Face Value 100 Issuer China Developmen Bank Issuance Dae Expiraion Dae Volume(Bn) 150 Paymen Frequency Quarerly Benchmark Floaing Rae Shibor3M-5D Rese Frequency Quarerly Spread(Bp) Shibor-based Floaing Rae Bonds A floaing rae bond is a bond wih variable coupon, which is based on he sum of a fixed quoed spread and a floaing benchmark rae, Shibor in our case. In oher words, i offers o is holder a sream of fuure paymens, depending on he reference floaing rae in he fuure. We denoe by 0 T 0 < T 1 < < T n a discree enor srucure wih a fixed = T k T k 1 and by N he face value. If he quoed spread is zero, he coupon paymens a ime T k can be expressed as N L(T k 1 ; T k 1, T k ), where L(T k 1 ; T k 1, T k ) is he spo Shibor a T k 1 for he period [T k 1, T k ] wih k = 1,..., n. In Table 1.2, we lis he general informaion of a ypical Shibor-based floaing-rae bonds, including he issuance daes, expiraion daes, redempion daes, and he spread. If we define he forward Shibor as L(; T, T + ) := E QT + [L(T ; T, T + )], according o he definiion, he presen value of a fricionless floaing rae bond a ime T 0 is given by P floa (, T n ) = = n k=1 N p(, T k )E QT k [L(T k 1 ; T k 1, T k )] + Np(, T n ) n N p(, T k )L(; T k 1, T k ) + Np(, T n ) k=1 Obviously, he key o pricing fricionless Shibor-based floaing rae bond is he forward Shibor L(; T k 1, T k ). If Shibor were risk-free rae, we have L(; T k 1, T k ) = p(,t k 1) p(,t k ), and he price of Shibor-based floaing rae bond is given by (1.6) P floa (, T n ) = = n N p(, T k )L(; T k 1, T k ) + Np(, T n ) k=1 n k=1 = Np(, T 0 ) N p(, T k ) 1 [ p(, T k 1) p(, T k ) 1] + Np(, T n ) (1.7) (1.7) indicaes ha if he benchmark floaing rae were risk-free, he value of he corresponding bond is worh par righ afer each coupon paymen day, i.e. p floa (T k, T n ) = N. 13

14 Figure 1.2: Closed Marke Mid Curves for Shibor3M-based IRS Closed Marke Mid Swap Curves Closed marke mid swap /08/ /09/ /10/ Mauriy (Year) Shibor-Based Ineres Rae Swaps Anoher imporan Shibor-relaed produc in Chinese bond marke is Shibor-based ineres rae swap. In general, ineres rae swap is a liquid financial derivaive insrumen beween wo paries, in which a sream of floaing rae based fuure paymens is exchanged for a fixed one. In China, he floaing reference rae is ofen aken o be he Shibor. The swap is sared a ime T 0 0, and a series of paymen daes are denoed by T 1 < T 2 < < T n, where T k T k 1 = is fixed for all k = 1, 2,... n. In a payer swap, he fixed rae coupon is paid and he floaing rae coupon is received. Oherwise, he swap is called a receiver swap. Therefore, he value of he payer swap based on Shibor3M rae L(T ; T, T + 3M) a ime is given by P Swap (; T 0, T n, R, N) = N n k=1 p(, T k )E QT k [L(T k 1 ; T k 1, T k ) R] (1.8) Recall ha we define he forward Shibor as L(; T, S) := E QS [L(T ; T, S)]. Therefore, (1.8) can be wrien as n P Swap (; T 0, T n, R, N) = N p(, T k )[L(; T k 1, T k ) R] (1.9) k=1 The swap rae R(; T 0, T n ) in a fricionless marke should be he rae ha makes he value of he swap equal o zero, which can be easily calculaed by seing P Swap (; T 0, T n, R, N) = 0 in (1.8). Therefore, we have he following equaion R(; T 0, T n ) = n k=1 p(, T k)l(; T k 1, T k ) n k=1 p(, T k) (1.10) Figure 1.2 graphs he closed marke mid price for Shibor3M-based IRS on hree differen daes. Noice ha (1.10) allows we o srip he forward Shibor L(; T, T + ) from he marke swap raes R(; T 0, T k ) 1 k n, once we can find he risk-free zero coupon bond price p(, T ). In fac, by a simple manipulaion on (1.10), we can ge he forward Shibor implied by he raes of Shibor3M ineres rae swaps 14 L(; T j 1, T j ) = k=j 1 k=1 [R(; T 0, T j ) L(; T k, T k )] + R(; T 0, T j )p(, T j ) p(, T j ) (1.11)

15 1.2 Lieraure Review On op of Shibor s populariy in he financial indusry, Shibor has also drawn much academic aenion. Ma, Liu, and Lan (2014)[7] invesigae he erm srucure of Shibor wih he wo-facor Vasicek model, where principle componen analysis (PCA) is used o calibrae he coefficiens involved. Zhang (2014)[13] uses single-facor shor rae model o analyze he dynamics of Shanghai Inerbank Offered Rae, where hey focus on maximum likelihood esimaion for Meron models and CIR models. Li and He (2009)[10] apply ime series heories, such as TARCH, GARCH and PARCH model, o invesigae he erm srucure of Shibor. However, few aenion has been paid o pricing Shibor-based floaing-rae bonds in China. A ypical Shibor-based floaing rae bond in Chinese bond marke is a deb insrumen wih he variable benchmark rae o be 5 day average of Shibor3M rae. In his hesis, we follow he shor-rae models given in he book of Grbac and Runggaldier (2015)[3], where pos-crisis dual curve modeling are concerned. Here, dual curve discouning refers o he pracice of projecing he cash flows by one ineres rae and discoun hem by anoher. Acually, he shor-rae approach in he dual curve seing is firsly inroduced by Kenyon (2010) [5], where he risk-free zero-coupon bond price p(, T ) is modeled as: and he forward Shibor as: p(, T ) := E Q [ B ] = E Q B [e r udu ] T where he ficiious bond price p(, T ) are defined by L(; T, T + ) := 1 ( p(, T ) 1) (1.12) p(, T + ) p(, T ) := E Q [ B ] = E B Q [exp( (r u + s u )du)] T where r and s are wo mean revering process. s here sands for he spread process. According o he definiion, L(; T, T + ) = E QT + [L(T ; T, T + )] should be a Q T + -maringale. However, he definiion in (1.12) does no necessarily give a Q T + -maringale, which means he model is no arbirage-free. Morino and Runggaldier (2014)[9] bring a model similar o Kenyon s model, wih one common Vasicek process ϕ 1 and wo idiosyncraic CIR processes ϕ 2, ϕ 3, all independen: r = ϕ 1 + ϕ 2, s = kϕ 1 + ϕ 3 Insead of defining he forward Libor rae, hey only use he ficiious bond price p(, T ) o define he spo Libor as: L(;, + ) := 1 1 ( 1) (1.13) p(, + ) In his case, he forward Libor rae can be generaed as a Q T + -maringale: L(; T k 1, T k ) = E QT k {L(T k 1 ; T k 1, T k )}, which means ha his model is arbirage free. In heir model, however, he iniial erm srucures T p(0, T ) and T L(0; T, T + ) are oucomes of he model, insead of inpus of he model, as hey should be in order o fi he iniial erm srucures. In order o solve his problem, Grasselli and Migliea (2016)[2] inroduce deerminisic shifs in he risk-free rae r and he shor-erm spread s, respecively. As shown in heir hesis, he corresponding shor-rae model can always fi o he iniial erm srucures wih some properly chosen deerminisic shifs. 15

16 1.3 Conribuion of Our Thesis Our aricle gives a simple shor-rae approach o modeling he forward Shibor in Chinese bond marke. We base ourselves on he model given by Grbac and Runggaldier (2015). Insead of using Vasicek model for risk-free rae in heir book, we choose o use Hull-Whie model o model r, which makes sure ha our model fis o he iniial erm srucures for risk-free rae. We will hen apply he shor-rae model o price he Shibor3M-based floaing-rae bonds. Closed-from valuaion expressions for presen value of fuure Shibor-based coupon paymen and Shibor-based floaing-rae bonds will be derived, as a resul of he fac ha all he involved sochasic process are markovian process. We also incorporae he deerminisic shif echnique brough by he paper from Grasselli and Migliea (2016), which allows us o make our model fi o he iniial erm srucure of Shibor-rae, which is sripped from he Shibor3M ineres rae swap raes. In addiion, he closed form expressions also allows us o apply he liquidiy adjusmen easily, based on he framework brough by Cui, Dai, Xu and Kou (2016)[12]. We show ha he discouned valuaion funcion of Shibor-based coupon paymens is always an increasing funcion of shor-erm spread s in our model. We also invesigae he relaionship beween he discouned value of Shibor-based coupon paymens and he spo risk-free rae, as well as he correlaion inensiy parameer k involved. By he empirical pricing es, we sugges ha he model is effecive in reproducing he marke price, especially afer we ake he liquidiy risk ino accoun. The res of his hesis proceeds as follows. In chaper 2, we quickly review some echnical preliminaries abou he shor-rae models, which will be used heavily laer. In chaper 3, we derive he model and apply i o value a Shibor-based floaing rae bond in a fricionless marke. Afer ha, we also give he explici expression for he liquidiy adjused floaing rae bond price. In chaper 4 we es he model wih a cerain Shibor3M-based floaing rae bond and check he empirical pricing performance. In chaper 5, we shall summarize he aricle and conclude. 16

17 Chaper 2 Preliminaries In his chaper, we shall have a quick review for he shor-rae models used in his hesis. In a shorrae model, he fuure evoluion of ineres raes are described hrough characerizing he evoluion of insananeous spo rae. For example, in a shor-rae model, he sochasic sae variables is aken o be he insananeous risk-free rae r for describing he evoluion of risk-free rae. In his hesis, we use Hull-Whie model for risk-free rae, and CIR model for he shor-erm spread. Here, we quickly review some well known resuls of hese wo models. Proof for he lemma below can be found in he book of Lamberon and Lapeyre (2007)[6], as well as many oher exbooks abou sochasic calculus. 2.1 CIR Model The Cox Ingersoll Ross(CIR) model describes he evoluion of shor erm ineres raes by an one facor model. As an exension of he Vasicek model, CIR model is firs inroduced by Cox, Ingersoll and Ross (1985) [1], which assumes ha he shor erm ineres rae follows he following sochasic differenial equaion: ds = (a bs )d + σ s dw where a, b, σ are consans, and W is a Weiner process. I is a mean-revering model, and says posiive if 2a σ 2. Lemma Consider a CIR ype process ds = (a bs )d + σ s dw For K {k R : E{e ks T < }} and γ > 0, we have E {exp( γs u du Ks T )} = exp[a s (, T ) B s (, T )s ] (2.1) where, he coefficiens A s (, ) and B s (, ) only depend on {a, b, σ, γ, K} 17

18 B s (, T ; a, b, σ, γ, K) = K(e(T )h (h b) + h + b) + 2γ(e (T )h 1) Kσ 2 (e (T )h 1) + h b + e (T )h (h + b) (2.2) A s (, T ; a, b, σ, γ, K) = a B s (s, T )ds = 2a σ 2 log( where h = b 2 + 2γσ 2. (h+b)(t ) 2he 2 Kσ 2 (e h(t ) 1) + e h(t ) (h + b) + h b ) (2.3) 2.2 Hull-Whie Model The firs Hull-Whie model is inroduced by John C. Hull and Alan Whie in 1990 [4], which is sill popular nowadays. Hull and Whie exend he Vasicek model in he sense ha a ime-dependen parameer η is incorporaed dr = (η ar )d + σdw (2.4) where η is a deerminisic funcion, a, σ are consans, and W is a Winer process. Firs, we recall some well known resuls abou his model. Before we go furher wih his model, we have o menion he Vasicek model firs. Lemma Consider he Vasicek ype process x dx = (a bx )d + σdw where a, b, σ are consans, and W is a Weiner process. γ, K R, we have he following E {exp( γx u du Kx T )} = exp[a x (, T ) B x (, T )x ] (2.5) where he coefficiens A x (, ) and B x (, ) only depend on {a, b, σ, γ, K}, and are given by B x (, T ; a, b, σ, γ, K) = γ [ bk ( b γ + 1)e b(t ) 1 ] A x (, T ; a, b, σ, γ.k) = a Lemma Consider he Hull-Whie ype process B x (s, T )ds + σ2 2 B 2 x(s, T )ds = a (γ bk)eb( T ) + b(k + γ(t )) γ b 2 + σ 2 b2 K 2 (γ bk) 2 e 2b( T ) 4γ(bK r)e b( T ) 4b 3 + σ 2 2bKγ 2br2 + 2bγ 2 T 3γ 2 4b 3 (2.6) dr = (η ar )d + σdw If shor-erm risk-free rae follows his model, he value of zero-coupon bonds is given by 18 E [exp( r s ds)] = exp[a h (, T ) B h (, T )r ] (2.7)

19 where he coefficiens saisfy B h (, T ; η( ), a, σ) = 1 a (1 e a(t ) ) A h (, T ; η( ), a, σ) = η(s)b h (s, T ; η, a, σ)ds + σ2 2a 2 [T + 2 a e a(t ) 1 2a e 2a(T ) 3 2a ] As well known, he ime-dependen parameer η can be idenified from he sandard echnique of yield curve fiing. Recall ha he forward rae is given by f(0, T ) = T (2.8) ln p(0, T ) (2.9) We have ln p(0, T ) = A(0, T ) + r 0 B(0, T ) where A(0, T ) and B(0, T ) is he coefficiens given in Lemma By subsiuing p(0, T ) in (2.9), we ge: f(0, T ) = Differeniaing w.r.. T, we ge: 0 η(s)b T (s, T )ds σ2 σ2 B(0, T ) + 2a 2a B(0, T )B T (0, T ) + B T (0, T )r 0 (2.10) f T (0, T ) = η(t )+ η s B T T (s, T )ds σ2 0 2a B T (0, T ) By Subsiuing formula (2.8) for B(, T ), we ge + σ2 2a [(B T (0, T )) 2 + B(0, T )B T T (0, T )] + B T T (0, T )r 0 η(s) = σ2 f(0, s) + af(0, s) + T 2a 2 (1 e 2as ) (2.11) Noice ha, in (2.11), we need differeniaed erm srucure T f(0, T ) o deermine η, which may amplify error from observaion. Indeed, we can ge he zero-coupon bond price only wih f(0, T ), if we decompose r ino r = x() + α(), where α() is deerminisic and x() solves which is a Vasicek ype process, and α() hus saisfies dx() = ax()d + σdw, x(0) = 0. (2.12) { da() = dr dx() = (η aα())d a(0) = r(0) (2.13) which gives α() = r 0 e a + 0 e a( s) η(s)ds By simple calculaion, we ge α() = f(0, ) + σ2 2a 2 (1 e a ) 2, and p(, T ) can be expressed hrough p(, T ) = E [exp( a(s) + x(s)ds)] (2.14) 19

20 Proposiion Consider he Hull-Whie ype process For any K R and γ R, we have he following E {exp( γr u du Kr T )} dr = (η ar )d + σdw =exp [ A x (, T ; 0, a, σ, γ, K) x()b x (, T ; 0, a, σ, γ, K) ] exp [ where he coefficiens saisfy x() = r() f(0, ) σ2 2a 2 (1 e a ) 2 α() = f(0, ) + σ2 2a 2 (1 e a ) 2 γα(u)du Kα(T ) ] (2.15) and he A x (, T ; p 1, p 2, p 3, γ, K) and B x (, T ; p 1, p 2, p 3, γ, K) are he coefficiens given in lemma wih he original parameers a, b, σ replaced by p 1, p 2 and p 3. Proof. If we decompose r ino r = α() + x() like (2.12), we can ge E {exp( γr u du Kr T )} =E {exp( γx u du Kx T )} exp ( γα(u)du Kα(T ) ) (2.16) Since x() is a Vasicek process wih x(0) = 0, a x = 0, b x = a, and σ x = σ, he firs erm in (2.16) can be solved by lemma E {exp( γx u du Kx T )} = exp [ A x (, T ; 0, a, σ, γ, K) x()b x (, T ; 0, a, σ, γ, K) ] (2.17) where x() = r() a() = r() f(0, ) σ2 (1 e a ) 2, and A 2a 2 x (, T ; p 1, p 2, p 3, γ, K) and B x (, T ; p 1, p 2, p 3, γ, K) are he coefficiens given in lemma The second erm exp ( γα(u)du Kα(T ) ) iself is a deerminisic funcion, and we leave i unouched here. 20

21 Chaper 3 Pricing Shibor-based Floaing-Rae Bond In his Chaper, we shall use he shor-rae model o derive valuaion expressions for Shibor-based floaing-rae coupon paymens in a fricionless marke. The presen value of a floaing-rae bond can hen be calculaed by he sum of discouned values of he floaing-rae coupon paymens and he discouned value of he erminal principal paymen. 3.1 Model for Shibor The Model In his secion, we base ourselves on he models discussed in las chaper, and incorporae he deerminisic shifs in he model from Grasselli and Migliea (2016), in an aemp o make our model fi o he iniial erm srucure of he Shibor. We consider he risk-free shor rae r and an independen shor rae spread s. We assume ha he spo risk-free rae is a Hull-Whie process dr = (η a 1 r )d + σ 1 dw 1 hen he price of risk-free zero-coupon bond is given by Lemma (2.2.2) p(, T ) = E [exp( r s ds)] = exp[a r (, T ) B r (, T )r ] (3.1) where he coefficiens saisfy B r (, T ) = 1 (1 e a 1(T ) ) a 1 A r (, T ) = η(s)b r (s, T )ds + σ2 1 2a 2 [T + 2 e a 1(T ) 1 e 2a 1(T ) 3 ] 1 a 1 2a 1 2a 1 (3.2) For he Shibor, we use ϕ o represen he shor-erm spread beween Shibor-rae and risk-free rae, which incorporaes all he risk among he inerbank marke, ha affecs he Shibor. More specificly, we assume 21

22 ha ϕ u = kr u + s u, and define he ficiious bond price as follows: where s u is a CIR process, and saisfies p(, T ) :=E Q {exp[ (r u + ϕ u )du]} (3.3) =E Q {exp[ ( ) (1 + k)ru + s u du]}. ds = (a 2 b 2 s )d + σ 2 s dw 2, (3.4) and k R represen he inensiy of he correlaion beween he risk-free rae r and he shor-erm spread ϕ. Proposiion Under he assumpions above, he forward Shibor is given by where L(; T, T + ) = 1 [ C(, T ) p(, T + ) 1] C(, T ) =E Q [e r u du p(t, T + ) p(t, T + ) ] =E Q [e + r udu 1 p(t, T + ) ] Proof. According o he definiion of forward Shibor, we have L(; T, T + ) =E By he change of measure in (1.4), we ge QT + = 1 + EQT [L(T ; T, T + )] [ 1 p(t, T + ) 1] L(; T, T + ) = EQT [ p(t, T + ) 1] = 1 ( 1 p(, T + ) EQ [e r udu p(t, T + ) p(t, T + ) ] 1) = 1 ( 1 + p(, T + ) EQ [e r u du 1 p(t, T + ) ] 1) Calibraion o The Iniial Term Srucure As we have discussed, he iniial erm srucure of forward Shibor is an oupu of he model currenly. However, in erms of applicaion, i is always desirable ha he model fis o he iniial erm srucure. Based on he mehod brough by Grasselli and Migliea (2016), we solve he problem by adding a deerminisic shif θ() o ϕ in order o mach he iniial erm srucures L marke (0; T, T + ). Now we define he calibraed shor-erm spread ϕ c as ϕ c := ϕ + θ() = kr + s + θ() (3.5) where k R represens he correlaion inensiy, s is sill he CIR process defined in (3.4), and θ() is a deerminisic funcion. 22

23 Proposiion The calibraed ficiious bond price p c (, T ) and he calibraed forward Shibor L c (; T, T + ) are given by he following equaions p c (, T ) = p(, T ) exp( θ(u)du) (3.6) where L c (; T, T + ) = 1 [ C c (, T ) 1] (3.7) p(, T + ) C c (, T ) =C(, T )e + T θ(u)du =E Q + [e r udu 1 p(t, T + ) ]e T + T Proof. From he definiion of p c (, T ), we direcly wrie i as θ(u)du On he oher hand p c (, T ) :=E Q {exp[ (r u + ϕ c u)du]} (3.8) =E Q {exp[ (r u + ϕ u + θ(u))du]} (3.9) = p(, T ) exp( θ(u)du) (3.10) L c (; T, T + ) =E QT + = 1 + EQT [L c (T ; T, T + )] [ = 1 + EQT = 1 [EQ [e 1 p c (T, T + ) 1] T + [ e θ(u)du p(t, T + ) 1] + r u du 1 T + p(t,t + ) ]e θ(u)du p(, T + ) = 1 [C(, T T + )e θ(u)du 1] p(, T + ) = 1 [ C c (, T ) p(, T + ) 1] 1] As proven in he paper of Grasselli and Migliea (2016) for a similar case, he shor rae model can fi o he iniial erm srucures by choosing proper deerminisic shif funcions. Here, we give he similar condiion for θ( ) o guaranee ha our model also fis o he iniial erm srucure Shibor. Theorem The model fis o iniial forward Shibor erm srucure L marke (0; T, T + ), if θ(u) saisfies he following equaion ( + exp T ) θ(u)du = [ 1 + L marke (0; T, T + ) ] p(0, T + ) C(0, T ) (3.11) 23

24 Proof. Proposiion (3.1.2) gives he modeled forward Shibor a ime 0 as If θ(u) saisfies (3.11), we can rewrie (3.12) as L c (0; T, T + ) = 1 [C(0, T T + )e θ(u)du 1] (3.12) p(0, T + ) L c (0; T, T + ) = 1 [1 + Lmarke (0; T, T + ) 1] = L marke (0; T, T + ) (3.13) Noice ha he heorem indicaes ha exp ( + T θ(u)du ) is uniquely deermined by he iniial forward Shibor erm srucure, once we have he parameers A = {a 1, σ 1, a 2, b 2, σ 2, k}. Even hough we can no idenify he value of θ( ), we can sill implemen he pricing procedure wih exp ( + T θ(u)du ) iself. 3.2 Explici Expression for Price of Shibor-based Floaing-Rae Bond In his secion, we derive explici expression for price of Shibor3M-based floaing-rae bond. We need o menion ha, in Chinese bond marke, he benchmark floaing rae is ofen se o be 5-day average of Shibor3M rae. In ha case, he coupon paymen a ime T k is N 4 5 i=0 L(T k 1 i ; T k 1 i, T k i ), where L(Tk 1 i ; T k 1 i, T k i) denoes he spo Shibor3M rae on i-h ransacion day before T k 1. In his case, he presen value of he coupon paymen o be received a ime T k is given by N 5 p(, T k)e QTk [ 4 L(Tk 1 i ; T k 1 i, T k i )] i=0 All he following resuls can be generalized for his case. For he simpliciy of documenaion, we only give he version for Shibor3M-based floaing-rae bond. We firs give he explici expression for discouned value of fuure coupon paymens, and hen derive he expression of Shibor3M-based floaing rae bond by adding up he presen value of all fuure cash flows Explici Expression of Presen Value of he Coupons Proposiion In our model, he presen value of he coupon paymen a ime T k is a funcion of he spo shor-erm risk-free rae r, shor-erm Shibor spread ϕ, deerminisic shif θ( ) and he iniial risk-free forward rae f(0, ). I is given by he following formula V (r, ϕ, f(0, ), θ( ), T k ) =Np(, T k )[ CeA x()b 4 s()b 5 1] p(, T k ) =N[Ce A x()b 4 s()b 5 e Ar(,T k) B r(,t k )r ] (3.14) 24

25 where and A 3 = A x (T k 1, T k ; 0, a 1, σ 1, 1, 0) B 3 = B x (T k 1, T k ; 0, a 1, σ 1, 1, 0) A 1 = A s (T k 1, T k ; a 2, b 2, σ 2, 1, 0) B 1 = B s (T k 1, T k ; a 2, b 2, σ 2, 1, 0) A 2 = A x (T k 1, T k ; 0, a 1, σ 1, 1 + k, 0) B 2 = B x (T k 1, T k ; 0, a 1, σ 1, 1 + k, 0) A 4 = A x (, T k 1 ; 0, a 1, σ 1, 1, B 3 B 2 ) B 4 = B x (, T k 1 ; 0, a 1, σ 1, 1, B 3 B 2 ) A 5 = A s (, T k 1 ; a 2, b 2, σ 2, 0, B 1 ) B 5 = B s (, T k 1 ; a 2, b 2, σ 2, 0, B 1 ) A = A 3 + A 4 + A 5 [ k 1 C = e α(u)du+ k T k 1 kα(u)du A 1 A 2 + k T k 1 θ(u)du α() = f(0, ) + σ2 1 2a 2 (1 e a1 ) 2 1 x() = r() α() s() = ϕ kr ] (3.15) he coefficiens A x, B x, A s, B s, A r, B r here are defined in he same way as in proposiion 2.2.3, lemma 2.1.1, and (3.2), respecively. Proof. The presen value of he coupon paymen a ime T k is given by he following formula V (r, ϕ, f(0, ), T k ) =N p(, T k )E QT k [L c (T k 1 ; T k 1, T k )] =Np(, T k )[ Cc (, T k 1 ) p(, T k ) 1] =Np(, T k ) [ C(, T k 1 ) p(, T k ) e k T k 1 θ(u)du 1 ] (3.16) where C(, T k 1 ) =E Q [e k r u du 1 p(t k 1, T k ) ] =E Q since s and x are independen, (3.17) can be wrien as { e k r u du E Q T k 1 [ exp ( k T k 1 (1 + k)r u + s u du )] } (3.17) C(, T k 1 ) =E Q { e k r u du E Q T k 1 [ exp ( k T k 1 (1 + k)r u du )] E Q T k 1 [ exp ( k T k 1 s u du )] } (3.18) According o lemma 2.1.1, E Q [ ( Tk T k 1 exp s u du )] = exp[a 1 B 1 s(t k 1 )] (3.19) T k 1 25

26 where A 1 =A s (T k 1, T k ; a 2, b 2, σ 2, γ = 1, K = 0) (3.20) B 1 =B s (T k 1, T k ; a 2, b 2, σ 2, γ = 1, K = 0) If we decompose r ino r = x() + α() like we did in (2.13), where x() saisfies he following SDE dx = a 1 x()d + σ 1 dw 1, x(0) = 0 which is a Vasicek ype process, and α() hus saisfies { da() = dr dx() = [η a 1 α()]d a(0) = r(0) (3.21) which gives α() =r 0 e a e a 1( s) η(s)ds =f(0, ) + σ2 1 2a 2 (1 e a1 ) 2 1 According o proposiion 2.2.3, afer decomposing r, we can derive k ET Q k 1 {exp( (1 + k)r u du)} T k 1 =exp [ A 2 x()b 2 ] exp [ k T k 1 (1 + k)α(u)du ] (3.22) where A 2 and B 2 is given by he A x, B x defined proposiion A 2 =A x (T k 1, T k ; 0, a 1, σ 1, 1 + k, 0) B 2 =B x (T k 1, T k ; 0, a 1, σ 1, 1 + k, 0) (3.23) Combining (3.19), (3.22) and (3.18), we can ge C(, T k 1 ) = E Q = E Q { { e k r udu E Q T k 1 [ exp ( k T k 1 (1 + k)r u du )] E Q T k 1 [ exp ( k T k 1 s u du )] e k x(u)du exp[a 1 B 1 s(t k 1 )] exp[a 2 x(t k 1 )B 2 )] = C 1 E Q [e k x udu+b 2 x(t k 1 ) ]E Q [eb 1s(T k 1 ) ] } exp [ k } T k 1 (1 + k)α(u)du k α(u)du ] (3.24) where C 1 = exp [ k 1 α u du + k ] T k 1 kα(u)du A 1 A 2, which is a consan. The res pars in (3.24) can be solved easily by proposiion and lemma 2.1.1: E Q [e k x u du+b 2 x(t k 1 ) ] =E Q k 1 {e x u du+b 2 x(t k 1 ) E Q T k 1 [e k T x(u)du k 1 ]} =E Q {e k 1 x u du+b 2 x(t k 1 ) e A 3 B 3 x(t k 1 ) } (3.25) =e A 3+A 4 x()b 4 26

27 and E Q [eb 1s(T k 1 ) ] =e A 5 s()b 5 (3.26) where A 3 = A x (T k 1, T k ; 0, a 1, σ 1, 1, 0) B 3 = B x (T k 1, T k ; 0, a 1, σ 1, 1, 0) A 4 = A x (, T k 1 ; 0, a 1, σ 1, 1, B 3 B 2 ) B 4 = B x (, T k 1 ; 0, a 1, σ 1, 1, B 3 B 2 ) A 5 = A s (, T k 1 ; a 2, b 2, σ 2, 0, B 1 ) B 5 = B s (, T k 1 ; a 2, b 2, σ 2, 0, B 1 ) (3.27) The presen value of he coupon paymen a ime T k is herefore given by V (r, ϕ, f(0, ), T k ) =N [ k T θ(u)du C(, T k 1 )e k 1 p(, Tk ) ] k =N[C 1 e A 3+A 4 +A 5 x()b 4 s()b 5 T θ(u)du e k 1 p(, Tk )] =N[C 1 e A 3+A 4 +A 5 x()b 4 s()b 5 T θ(u)du e k 1 e A r (,T k ) B r (,T k )r ] k T θ(u)du If we denoe C = C 1 e k 1, A = A3 + A 4 + A 5, we can ge he formula given in he proposiion. k The closed-form expression in Proposiion involves nohing more complicaed han exponenial funcion, and depend on he parameers A = {a 1, σ 1, a 2, b 2, σ 2, k}. Even hough he coefficiens are defined recursively, i is relaively easy o program his expression o ge he discouned presen value of Shibor-based coupon paymen. Proposiion indicaes ha V s = DB 5 e s()b 5, where D > 0 does no vary wih s(). I is easy o observe from he formula in lemma ha B 1 = B s (T k 1, T k ; a 2, b 2, σ 2, 1, 0) > 0, and B 5 = B s (, T k 1 ; a 2, b 2, σ 2, 0, B 1 ). In fac, we have 2(e (T k T k 1 )h 1) B 1 = h b 2 + e (T k T k 1 )h (h + b 2 ) 2B 1 b 2 B 5 = B 1 σ2 2(e(T k 1 )b 2 1) + 2b 2 e (T k 1 )b 2 (3.28) This means ha if B 5 > 0 wih he parameers given, he discouned value of he fuure Shibor-based coupon paymen is always an increasing funcion of he spread s(). This is inuiive since he higher he spread s() is, he bigger he expecaion of he fuure Shibor will be. Figure 3.1 shows his propery, wih he iniial risk-free forward rae o be defined as f(0, ) = r 0 + (1 e a 1 )(0.25 r 0 ). When i comes o r(), i is no easy o deermine how will he discouned value change wih respec o r(). This is mainly because r() plays wo roles in he valuaion of Shibor-based floaing rae bond. Specifically, an increase of r() resul in a smaller discouned risk-free bond price p(, T ), as well as a higher forward Shibor L(; T, T + ). Figure 3.2 shows ha, wih he parameers given, an increase in r will decrease he discouned value of coupon paymens wih shor mauriies, and increase he discouned value of long erm coupon paymens. 27

28 Figure 3.1: Discouned value of Shibor-based coupon paymen as a funcion of he mauriy.the parameers used are a 1 = 12, σ 1 = 0.04, a 2 = 0.02, b 2 = 2, σ 2 = 0.01, ϕ 0 = 0.04, k = φ 0 =0.02 φ 0 =0.04 φ 0 = Discouned Value of Coupon Paymen a 1 =12, σ 1 =0.04, a 2 =0.02, b 2 =2, σ 2 =0.01 r 0 =0.25, k= Mauriy Figure 3.2: Discouned value of Shibor-based coupon paymen for differen r 0. The parameers used are a 1 = 12, σ 1 = 0.04, a 2 = 0.02, b 2 = 2, σ 2 = 0.01, ϕ 0 = 0.04, k = 0 Discouned value of Coupon Paymens r 0 =0.25 r 0 =0.35 r 0 = Mauriy Figure 3.3: Discouned value of Shibor-based coupon paymen for differen correlaion inensiy k. The parameers used are a 1 = 12, σ 1 = 0.04, a 2 = 0.02, b 2 = 2, σ 2 = 0.01, ϕ 0 = 0.04, r 0 = k=0.00 k=-0.03 k= Discouned value of Coupon Paymens Mauriy 28

29 Figure 3.3 graphs he relaionship beween discouned value of coupon paymen and he correlaion inensiy parameer k. As shown, he effec of correlaion inensiy is quie significan. The discouned value of he coupon paymen wih mauriy 5y increase by 16 basis poins as k increase from o Explici Expression for Shibor3M-based Floaing Rae Bond Since we have he presen value of all fuure coupon paymens according o proposiion 3.2.1, we can derive he explici expression for Shibor3M-based floaing rae bond now. Proposiion In our model, he value of he Shibor3M-based floaing rae bond a ime is given by P (, N) = where n V (r, ϕ, f(0, ), θ( ), T k ) + Np(, T n ) (3.29) k=1 V (r, ϕ, f(0, ), θ( ), T k ) =Np(, T k )[ CeA x()b 4 s()b 5 1] p(, T k ) (3.30) =N[Ce A x()b 4 s()b 5 e Ar(,T k) B r(,t k )r ] all he coefficiens here are defined in he same way as in Proposiion Proof. The proposiion can be easily derived by he combinaion of proposiion and (1.6) Liquidiy Risk So far, we have only discussed he price of a floaing rae bond under he assumpion ha here is no credi or liquidiy risk. Since he bond ha we are considering is raded only once a week in some periods, we have o ake he liquidiy risk involved ino accoun. Here, we base ourselves on he framework brough in Cui, Dai, Xu and Kou (2016), in order o ake ino accoun he liquidiy risk, and model he liquidiy adjused Shibor-based floaing rae bond price as n P risk (, N) = E Q k [e r u+h ul udu ]N L(; T k 1, T k ) + NE Q n [e r u+h ul udu ] (3.31) k=1 In his model, he risk-free rae shor-erm ineres r is replaced by he liquidiy-adjused shor rae process R = r + h L, where h L is he risk-neural mean loss rae. Since we don have enough informaion o separaely idenify he hazard rae h and he fracional loss L, we choose o model he mean loss λ = h L by a deerminisic funcion. More specifically, we approximae he λ by a second order polynomial funcion of. Here, we also assume ha λ is sable in a shor period, so ha we can calibrae he coefficiens in he polynomial funcion by he recen price hisory on each pricing day. In his case, (3.31) can be wrien as n P risk (, N) = E Q k [e r u+λ udu ]N L(; T k 1, T k ) + NE Q n [e r u+λ udu ] = = k=1 n k=1 n k=1 E Q k [e e k r u du ]e k λ u du N L(; T k 1, T k ) + NE Q [e n λ u du V (r, ϕ, f(0, ), θ( ), T k ) + e n λ u du Np(, T n ) r u du ]e n λ u du (3.32) 29

30 where V (r, ϕ, f(0, ), θ( ), T k ) =Np(, T k )[ CeA x()b 4 s()b 5 1] p(, T k ) =N[Ce A x()b 4 s()b 5 e A r(,t k ) B r (,T k )r ] all he coefficiens here are defined in he same way as proposiion (3.33) 30

31 Chaper 4 Empirical Pricing Performance 4.1 Daa and Calibraion In his secion, we implemen our model o price a specific Shibor3M-based floaing rae bond. We firs choose a period, during which we would like o es our model. Afer ha, we srip he forward Shibor from Shibor3M IRS raes a he beginning of he pricing period, in order o make our model fi o he iniial erm srucure of T L marke (0; T, T + ). Then we calibrae he res of parameers in our model Sripping he forward Shibor from IRS Recall ha he forward Shibor is defined as: L(; T k 1, T k ) := E QT k [L(T k 1 ; T k 1, T k )] And he he price for a Shibor3M rae swap is given by: P Swap (; T 0, T n, R, N) =N =N n k=1 p(, T k )E Qk {L(T k 1 ; T k 1, T k ) R} (4.1) n p(, T k )[L(; T k 1, T k ) R] (4.2) k=1 The swap rae R(; T 0, T n ) in a fricionless marke should be he rae ha makes he value of he swap P Swap (; T 0, T n, R, N) equal o zero. By seing P Swap = 0, we derive R(; T 0, T n ) = n k=1 p(, T k)l(; T k 1, T k ) n k=1 p(, T k) (4.3) As discussed in (1.8), he marke forward Shibor saisfies L(; T j 1, T j ) = k=j 1 k=1 [R(; T 0, T j ) L(; T k, T k )] + R(; T 0, T j )p(, T j ) p(, T j ) (4.4) 31

32 From he linear sysem given by (4.4), we can ge he forward Shibor L marke (0; T, T + ) implied by he iniial Shibor3M IRS raes R(0; T 0, T j ). Afer his, we can derive our deerminisic shif erm exp ( + T θ(u)du ) according o (3.11): where exp ( + T θ(u)du ) = [ 1 + L marke (0; T, T + ) ] p(0, T + ) C(0, T ) C(0, T ) =E Q [e 0 r udu p(t, T + ) p(t, T + ) ] =C 1 e A 3+A 4 +A 5 s()b 5 where he coefficiens A 3, A 4, A 5, B 4, B 5 and C 1 are given in Proposiion Equaion (4.5) indicaes ha we can derive he deerminisic shif, once we have he parameers A = {a 1, σ 1, a 2, b 2, σ 2, k}, which are discussed in he following pars. (4.5) Calibraion for Hull-Whie model We obain he daily risk free yield curve from Chinabond, and calculae he η(s) and α(s) from he yield curve, as menioned in (2.11). η(s) = T f(0, s) + af(0, s) + σ2 1 2a 2 (1 e 2a1s ) 1 (4.6) α() = f(0, ) + σ2 1 2a 2 (1 e a1 ) 2 1 Noice ha we also need o deermine he a 1 and σ 1 in he Hull-Whie model. According o he approach proposed in Q. Meng e.al. (2013)[11], (a 1, σ 1 ) can be esimaed hrough minimizing he following objecive funcion: g(a, σ) = j 1 M 2 j ( sd[log p(i, i + M j ) log( p( i 1, i + M j ) )] ) 2 V arm j (4.7) p( i 1, i ) where M j s denoes he specrum of mauriies of he yields, sd denoes sandard deviaion operaor wih i aken over he sampling period, and V arm j = 1 a 2 (1 e am j ) 2 σ2 2a (1 e 2aδ ) We use he yield curve daa from o o esimae a and σ, since we will implemen our model during he period from o The esimaion we ge is (a 1, σ 1 ) = ( , ) Calibraion for Shor-erm Spread and Correlaion Parameer k Calibraion of parameers for spread s can be done once we have calibraed he parameers (a 1, σ 1 ) in dr = (η a 1 r ) + σ 1 dw 1 Since L(; T k 1, T k ) = 1 (C(, T k 1) p(, T k ) 1), (4.8) 32

33 and C(, T ) =E Q [e r u du p(t, T + ) p(t, T + ) ] =C 1 e A 3+A 4 +A 5 x()b 4 s()b 5 where he coefficiens A 3, A 4, A 5, B 4, B 5 and C 1 are given in Proposiion Equaion (4.8) indicaes ha L(; T k 1, T k ) is indeed a funcion of A = {a 1, σ 1, a 2, b 2, σ 2, k}, according o Proposiion We can calibrae {a 2, b 2, σ 2, k} according o he pas hisory of marke forward Shibor L marke (s; T, T + ) by minimizing L(s; s + T, s + T + ) L marke (s; s + T, s + T + ) 2 s,t where s [S 0, S 1 ], which is before our pricing period, and T denoes differen mauriies for he Shibor3M-based IRS. Based on he marke daa beween and , we ge he following esimaions k = 0.12, a 2 = 0.025, b 2 = , σ 2 = Calibraion for Liquidiy Inensiy Afer we have A = {a 1, σ 1, a 2, b 2, σ 2, k} and he iniial erm srucure modifier e + T θ(s)ds, we can have he heoreical fricionless price of he corresponding floaing rae bond by Proposiion Now we ake liquidiy risk ino accoun, and derive he liquidiy adjused bond price P risk (, N) as in (3.31) where P risk (, N) = n k=1 e k λ udu V (r, ϕ, f(0, ), θ( ), T k ) + e n λ udu Np(, T n ) (4.9) V (r, ϕ, f(0, ), θ( ), T k ) =Np(, T k )[ CeA x()b 4 s()b 5 p(, T k ) 1] =N[Ce A x()b 4 s()b 5 e A r(,t k ) B r (,T k )r ] (4.10) We calibrae he coefficiens α = {α 0, α 1, α 2 } in he mean loss rae λ = α 0 + α 1 + α 2 2 by minimizing differences beween he modeled liquidiy adjused prices and he real prices in he previous 5 ransacions days T i, i = 1, 2, 3, 4, 5: 4.2 Resuls α = arg min α 5 P risk (T i, N) P marke (T i, N) 2 (4.11) i=1 In his secion, we use our model and he parameers given above o price a Shibor3M-5D based floaing rae bond (GK130217), which is one of he mos frequenly raded bond among Shibor-based floaing rae bonds, during he pas year. Tabular 4.1 shows he general informaion abou his bond, including he issuers names, he face value, he issuance daes, he expiraion daes, he paymen frequency, he oal volume (in billion), ec. The deerminisic shif erm is deermined by he parameers A = {a 1, σ 1, a 2, b 2, σ 2, k} and he Shibor3M-IRS raes on , which is he las rading day before our pricing period. In Figure 4.1, we plo he forward Shibor3M raes sripped from he Shibor3M IRS raes on

34 Table 4.1: Bond Informaion. The daa are obained from China Foreign Exchange Trading Sysem and Wind Financial Terminal. Name GK Face Value 100 Issuer China Developmen Bank Issuance Dae Expiraion Dae Volume(Bn) 150 Paymen Frequency Quarerly Benchmark Floaing Rae Shibor3M-5D Rese Frequency Quarerly Spread(Bp) Figure 4.1: Forward Shibor3M rae Sripped from Shibor3M IRS raes. The parameers used are a 1 = , σ 1 = , k = 0.12, a 2 = 0.025, b 2 = , σ 2 = The IRS rae is he closed marke mid swap raes on 2015/08/28 forward Shibor3M L marke (0,T,T+ ) forward Time T Figure 4.2: θ() calibraed o he marke dae on The parameers used are a 1 = , σ 1 = , k = 0.12, a 2 = 0.025, b 2 = , σ 2 = θ calibraed o he marke daa on 2015/08/ θ Mauriy 34

35 Figure 4.3: Effec of calibraion wih θ on he bond price The parameers used are a 1 = , σ 1 = , a 2 = 0.02, b 2 = 2, σ 2 = 0.01, k = 0.12, a 2 = 0.025, b 2 = , σ 2 = No calibraion Calibraed wih θ Bond Price /01/ /01/ /01/ /01/ /01/ /01/ /01/2017 Figure 4.4: Comparison beween he modeled fricionless bond price and he price hisory of GK Hisory Price Modeled Price Price /01/ /01/ /01/ /01/ /01/ /01/ /01/2017 Dae As menioned, we can only access exp( + θ u du). On he oher hand, if we assume ha θ(u) could be well approximaed by 1 presened in Figure T T θ u du, we would be able o plo he values of θ(u), which is In Figure 4.3, we compare he fricionless bond price P (, N) modeled wih calibraion o θ u and he bond price modeled wihou calibraion o θ u. As can be seen, wihou calibraion o θ u, he modeled bond price is significanly lower han he bond price implied by he marke price of Shibor3M-IRS a he beginning of our pricing period. This is mainly due o he fac ha our parameers a 2, b 2, σ 2 are calibraed hrough he pas hisory of L marke (s; T, T + ), which may no necessarily fi o he forward Shibor3M rae in he curren marke. Figure 4.4 graphs he modeled fricionless bond price P (, N), as well as he hisory of he real rading price of GK The resul is quie accepable, since he fricionless bond price shares almos he same rend of he real price, and he gap beween he modeled price and he real price is quie sable during he whole pricing period. 35

36 Figure 4.5: Calibraion o marke price hrough Liquidiy Adjusmen 103 Price afer Liquidiy Adjusmen Real Price Price before Liquidiy Adjusmen Pruce /01/ /01/ /01/ /01/ /01/ /01/ /01/2017 Dae The las sep is o calibrae he mean loss rae brough by he liquidiy risk, as explained in secion Figure 4.5 shows ha, o a large exen, he liquidiy adjused model is effecive in reproducing he Shibor-based floaing rae bond prices during he whole pricing period, wih he average pricing error under 20 cens. 36

37 Chaper 5 Summary and Conclusion In his hesis, we use a shor-rae model o price he Shibor3M-based floaing-rae bonds, based on he model given by Grbac and Runggaldier (2015). Closed-from valuaion expressions for presen value of Shibor-based floaing-rae bonds are derived, as a resul of he fac ha all he involved sochasic process are markovian process. Our model can also fis o he iniial erm srucure of Shibor, as long as we carefully choose a deerminisic shif θ, which is firs inroduced by Grasselli and Migliea (2016) for a similar model. In addiion, he closed form expressions also allow us o easily apply he liquidiy adjusmen, which is based on he bond pricing framework brough by Cui, Dai, Xu and Kou (2016). We show ha, he discouned valuaion funcion of Shibor-based coupon paymens is always an increasing funcion of shor-erm spread s in our model. We also invesigae he relaionship beween he discouned value of Shibor-based coupon paymens and he spo risk-free rae, as well as he correlaion inensiy parameer k involved. The empirical resuls sugges ha, o a large exen, he model can reproduce he marke price well, especially afer we ake he liquidiy risk ino accoun. We admi ha our sudy is limied in several ways. For he ease of implemenaion, we assume deerminisic liquidiy inensiy. I would be more reasonable if we can use a sochasic inensiy model, as discussed in he original paper of Duffie and Singleon (1999). We also need o menion ha, we use a one facor Hull-Whie model for he risk-free rae, and a wo facor model for shor-erm spread ϕ in his hesis. We can alernaively use wo facor models for risk-free rae as well. In ha case, anoher deerminisic shif can be added o make he new model sill fi o he iniial erm srucure for risk-free rae. Finally, due o he lack of oher Shibor-relaed producs, we heavily rely on he price of Shibor3M- IRS during he calibraion procedure. Some missing daa is esimaed hrough spline inerpolaion. The calibraion would be done more accuraely if we have access o more Shibor-based producs, like basis swap. We leave hese for furher research. 37

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39 Bibliography [1] John C Cox, Jonahan E Ingersoll Jr, and Sephen A Ross. A heory of he erm srucure of ineres raes. Economerica: Journal of he Economeric Sociey, pages , [2] Marino Grasselli and Giulio Migliea. A flexible spo muliple-curve model. Quaniaive Finance, pages 1 13, [3] Zorana Grbac and Wolfgang J Runggaldier. Ineres Rae Modeling: Pos-Crisis Challenges and Approaches. Springer, [4] John Hull and Alan Whie. Pricing ineres-rae-derivaive securiies. Review of financial sudies, 3(4): , [5] Chris Kenyon. Pos-shock shor-rae pricing. Risk, 23(11):79, [6] Damien Lamberon and Bernard Lapeyre. Inroducion o sochasic calculus applied o finance. CRC press, [7] Chaoqun Ma, Jian Liu, and Qiujun Lan. Sudying erm srucure of shibor wih he wo-facor vasicek model. In Absrac and Applied Analysis, volume Hindawi Publishing Corporaion, [8] Massimo Morini. Solving he puzzle in he ineres rae marke. Available a SSRN , [9] Laura Morino and Wolfgang J Runggaldier. On mulicurve models for he erm srucure. In Nonlinear Economic Dynamics and Financial Modelling, pages Springer, [10] Li Ning and He Dong-ping. Empirical research on erm srucure of shibor based on arch and pgarch model. In Elecronic Commerce and Securiy, ISECS 09. Second Inernaional Symposium on, volume 1, pages IEEE, [11] A. Levy Q. Meng, A. Kaplin. Esimaing parameers in he single facor hull-whie model. Technical Repor, Moodys Analyics. [12] Jing Xu Wei Cui, Min Dai and Seven Kou. A framework for hybrid bond pricing in chinese marke. Working Paper,

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41 Appendix A Malab Codes A.1 Codes for calculaing risk-free zero-coupon bond price f u n c i o n s s = pt (, T, a, sg, e a, yc ) pp ( ) =exp ( newat ( e a, a, sg, 0, ) yc ( 1 ) newbt ( a, 0, ) ) ; i f T<0.05 s s = 1 / ( yc ( 1 ) T+1) ; end i f T==10 s s =pp ( ) ; end i f T<10 l b = f l o o r ( T / ) +1; ub= l b +1; n l b = l b ; nub=ub ; pp ( ub ) =exp ( newat ( e a, a, sg, 0, ub 0.05) yc ( 1 ) newbt ( a, 0, ub 0.05) ) ; pp ( l b ) =exp ( newat ( e a, a, sg, 0, lb 0.05) yc ( 1 ) newbt ( a, 0, lb 0.05) ) ; s s =(T n l b ) / ( nub n l b ) pp ( ub ) +( nub T ) / ( nub n l b ) pp ( l b ) ; end end A.2 Codes for calculaing η() 41