Towards a high-fidelity risk-free interest rate

Transcription

1 Projec Number: IQP - ZZ1-AA79 Towards a high-fideliy risk-free ineres rae An Ineracive Qualifying Projec Repor Submied o The Faculy of Worceser Polyechnic Insiue In parial fulfillmen of he requiremens for he Degree of Bachelor of Science by Xinzhe Zhang May 2017 Approved: Professor Zhongqiang Zhang This repor represens he work of WPI undergraduae sudens submied o he faculy as evidence of compleion of a degree requiremen. WPI rouinely publishes hese repors on is websie wihou ediorial or peer review. For more informaion abou he projecs program a WPI, please see hp://

2 Conens 1 Inroducion 4 2 Term Srucures of bonds Preliminaries No Arbirage Pricing Model Calibraion Assumpions Daa Resource and Inerpreaions Models Vasicek Model Hull-Whie Model Resuls The Vasicek - Affine Term Srucure Model The Hull Whie - Affine Term Srucure Model Conclusion and Discussion 22 Appendices 24 A Yield Marix 25 B Some useful formula 28 B.1 Exac Soluion o he Hull-Whie Model B.2 Proof of Hull-Whie Bond Pricing C Malab Code 30 C.1 Funcion for he Variable A(, T ) C.2 Funcion for he Variable B(, T ) C.3 Funcion for he Bond s Price C.4 Funcion for Res i C.5 Funcion o ransform yield marix o price marix C.6 Calibraion Funcion C.7 Hull-Whie Model

3 Absrac The risk-free ineres rae is no only an essenial parameer in financial marke bu also a key indicaor in economy. To esimae he risk-free ineres rae, we use he reurn raes of reasury bonds, which is an imporan derivaive of risk-free ineres rae. In his projec, we will use several shor rae models and affine erm srucure o calibrae he parameers in hese models as well as in in bond pricing model. 2

4 Acknowledgmen I would like o hank my projec advisor Professor Zhongqiang Zhang and Professor Xuwei Yang a Worceser Polyechnic Insiue. The door o Prof. Zhang and he door o Prof. Yang were always open when I me roubles wih my projec or wriing. Prof. Zhang showed me wha was he appropriae way o wrie a paper and he improvemens while I was doing his projec. Thank Professor Zhongqiang Zhang for his grea help in wriing his paper. Prof. Yang gave me he direcion wha I should do in his projec as well as helped me o undersand he mah ideas behind he projec. Thank Professor Xuwei Yang for his grea help in explaining he mah ideas and how o apply hose ideas. Prof. Zhang and Prof. Yang always helped me adjus he direcion of he projec when necessary. Wihou heir paien help, his repor could no be compleed successfully. 3

5 Chaper 1 Inroducion The risk-free ineres rae is no only an essenial parameer in financial marke bu also a key indicaor in economy. However, he risk-free ineres rae is hidden under he marke and hus is required o be quanified. In his projec, we will quanify he risk-free ineres hrough daa from he bond marke. The bond marke is where deb securiies are issued and raded. The bond marke primarily includes governmen-issued securiies and corporae deb securiies, and i faciliaes he ransfer of capial from savers o he issuers or organizaions ha requires capial for governmen projecs, business expansions and ongoing operaions. A bond is a deb invesmen in which an invesor loans money o an eniy (ypically corporae or governmenal) which borrows he funds for a defined period of ime a a variable or fixed ineres rae. Bonds are used by companies, municipaliies, saes and sovereign governmens o raise money and finance a variey of projecs and aciviies. Owners of bonds are deb holders, or crediors, of he issuer Bond is one of he larges consiuens in financial marke and i can be mainly divided ino wo pars by differen issuers. One is governmen bond and he oher is cooperae bond. The more powerful he organizaion is, he less risk of he bond which is issued by ha organizaion. For example, he U.S. governmen bond has less risk han he counry which is warring in he mid-eas Asia. I is also one of he mos imporan financial commodiy o invesors. For he people who worry abou economic recession or slower economic growh, bond is he only hing ha will promise a fixed reurn in he fuure. Thus bond is always used o form fix-income porfolios. Mos of bonds can no be redeemed before he mauriy dae bu he invesors can rade heir bonds before maerniy daa. Bond is imporan o invesors as well as issuers. Each company have differen credi lines in differen banks. Someimes, such kind of companies will need a large amoun of money which exends he credi lines and hey could no ge money from bank anymore. However, hey can issue bonds o raise money. Moreover, when he company is no saisfied wih he ineres rae ha he bank gives o i, his company can also issue bond wih a lower ineres rae. Pricing is one of he fundamenal opics in financial indusry. I is a process o deermine he value of a financial asse. When a company wans o issue a bond or oher financial derivaives, i firs needs o decide he price of he derivaive. If he company prices he asse a a higher level, here will be less people in he marke willing o buy he asse. However, if he asse s price is oo low, he issuer will lose money. So, how o price he asse wih a fair price is a really imporan hing. Furhermore, bonds or reasury bonds are he larges consiues in financial markes. For example, he US governmen has issued more han nineeen-rillion dollars. Properly pricing he bond is hus exremely imporan o manage such a large moun of money. In his projec, we are going o use shor rae models and bond pricing models. In Chaper 2, we inroduce some basic conceps in finance and some relaed mahemaics ideas. We also elaborae how 4

6 o inerpre he given daa. In Chaper 3, we presen he mahemaical models we will use o esimae he risk-free ineres rae, including wo shor rae models and some bond pricing models. In Chaper 4, we presen some numerical resuls from wo differen mahemaical models and our calibraions. We hen conclude in Chaper 5. 5

7 Chaper 2 Term Srucures of bonds 2.1 Preliminaries In his chaper, we inroduce some preliminary erminologies used in his projec. Arbirage pricing heory was firs proposed by Sephen Ross in 1976 [1]. In arbirage pricing, he reurns of a porfolio can be prediced by doing a linear combinaion of independen single asses. Suppose h is a porfolio and denoe V h () is he value of he porfolio h a ime. An arbirage possibiliy of h on a financial marke refers o V h (0) = 0, P(V h (T ) 0) = 1, and P(V h (T ) > 0) > 0. In oher words, he arbirage opporuniy is when he probabiliy of a porfolio value will increase in he fuure compared o is iniial value. In plain language, an arbirage possibiliy is he possibiliy of making a posiive amoun of money ou of nohing wihou aking any risk. Arbirage means a free lunch on he financial marke. The marke is arbirage free if here are no arbirage possibiliies. We always assume ha our model and daa are from an arbirage free marke. The ineres rae is he amoun charged, expressed as a percenage of principal, by a lender o a borrower for he use of asses. Ineres raes are ypically noed on an annual basis, known as he annual percenage rae. The asses borrowed can be cash, consumer goods, large asses, ec. The ineres rae is essenially a renal, or leasing charge o he borrower, for he use of asses. Bond is a deb of he bond issuer o he holders. A bond should indicae ha when is he due day for he deb (called mauriy dae) and how much should he bond issuer pays o he bond holders (called face value.) [2]. Bond are mainly divided ino wo differen ypes. One is zero coupon bond, defined below. The oher one is coupon paying bond. The holder of coupon paying bond will receive no only he face value a mauriy dae bu also he coupons before he mauriy dae. A zero-coupon bond is a bond bough a a price lower han is face value, wih he face value repaid a he ime of mauriy [3]. Moreover, zero-coupon bond is a bond ha won issue any coupon paymens before he mauriy dae. So, i s a bond ha can be calculaed easily and also he larges componen ype in bonds. The noaion p(, T ) means he bond s price wih mauriy T a ime. Zero coupon bonds play an elemenary role in fixed income marke. Oher more compleed fixed income insrumens, such as coupon bonds and ineres rae swaps, can be represened as a linear funcion of zero coupon bonds. Thus in his paper we focus on sudying zero coupon bonds. In his projec, we consider zero coupon bond wih face value $1, ha is, p(t, T ) = 1. 6

8 2.2 No Arbirage Pricing In his projec, we consider bond price as a funcion of risk free ineres rae, also called shor rae. Shor rae is he insananeous reurn of money accoun ha is commonly regarded as risk free. We define he money accoun process B() as { } B() = exp r(s)ds, 0, 0 which akes he following differenial form { db() B(0) = 1. = r()b()d, We suppose ha he shor rae r() in real world follows he dynamics dr() = µ(, r())d + σ(, r())dw (), where W () is a sandard Brownian moion under physical probabiliy measure P. We ake each zero coupon bond p(, T ) price as a funcion of shor rae model r() We assume furhermore ha, for every T, he price of a T -bond has he form p(, T ) = F (, r(); T ), (2.2.1) where F is a smooh funcion of hree variables. We can also denoe F (, r(); T ) by F T (, r()) = F (, r(); T ), (2.2.2) We consruc a porfolio consising of bonds having differen imes of mauriy. We fix wo imes of mauriy S and T. By Iô s formula, we have he dynamics of F T and F S, where df T = F T α T d + F T σ T dw, df S = F S α S d + F S σ S dw, α T = F T + µf T r σ2 F T rr F T, σ T = σf T r F T, α S = F S + µf S r σ2 F S rr F S, σ S = σf S r F S. Consider a relaive porfolio (u S, u T ) of S-bond and T -bond wih u T + u S = 1, so ha he porfolio value V is given by V = u S F S + u T F T. By Iô s formula, he relaive porfolio value V saisfies he dynamics dv = V (u T α T + u S α S ) d + V (u T σ T + u S σ S ) dw. 7

9 By leing he relaive porfolio risk free, i.e. u T σ T + u S σ S = 0, which ogeher wih he condiion u T + u S = 1 gives And hen he dynamics of V becomes risk free No arbirage condiion gives σ S u T =, u S = σ T, σ T σ S σ T σ S dv = V (u T α T + u S α S )d. α S σ T α T σ S = r(), σ T σ S α S() r() = α T () r(). (2.2.3) σ S () σ T () The above equaliy (2.2.3) holds for all mauriy imes S and T. Proposiion Assume ha he bond marke is free of arbirage. Then here exiss a process λ() such ha he relaion α T () r() σ T () = λ(), (2.2.4) hold for all and for every choice of mauriy ime T. The parameer λ() is called he marke price of risk is he reurn α T () in excess of risk free rae r() per uni risk measured in erms of volailiy σ T (). By subsiuing he represenaion of α T and σ T ino he marke price of risk formula (2.2.4), we obain he parial differenial equaions ha he bond price funcion p(, T ) = F (, r(); T ) saisfies. Proposiion (Term srucure equaion). In an arbirage free bond marke, F T will saisfy he erm srucure equaion { F T + (µ λσ)fr T σ2 Frr T rf T = 0, (2.2.5) F T (T, r) = 1. By Feynmann-Kac formula, he pricing parial differenial equaion (2.2.5) admis a soluion of he following probabilisic form Proposiion (Risk neural valuaion). Bond prices are given by he formula p(, T ) = F (, r(); T ) where [ F (, r; T ) = E Q e ] T r(s)ds 1 r() = r. (2.2.6) Here under he risk neural measure Q and he he dynamics of r() saisfies dr(s) = (µ λσ)ds + σdw, r() = r. s 8

10 Furher more, for any coningen claim on ineres rae r() wih payoff funcion Φ(r(T )), he pricing formula is given by he following proposiion Proposiion (General erm srucure equaion). Le X be a coningen T -claim of he form X = Φ(r(T )). In an arbirage free marke he price Π(; Φ) will be given as Π(; Φ) = F (, r()), where F solves he boundary value problem { F + (µ λσ)f r σ2 F rr rf = 0, F (T, r) = Φ(r). Then F has he sochasic represenaion [ F (, r) = E Q e ] T r(s)ds Φ(r(T )) r() = r. Here under he risk neural measure Q and he he dynamics of r() saisfies dr(s) = (µ λσ)ds + σdw, r() = r, s where W is sandard Brownian moion under maringale measure Q. The erm srucure, as well as he prices of all oher ineres rae derivaives, are compleely deermined by specifying he dynamics of shor rae r() under he maringale measure Q. Insead of specifying dynamics of r() under objecive probabiliy measure P, we specify he dynamics r() under he maringale measure Q. This procedure is known as maringale modeling, and he ypical assumpion will hus be ha r under Q has dynamics given by dr() = µ(, r())d + σ(, r())dw, (2.2.7) where µ and σ are given funcions, and W is a sandard Brownian moion under he measure Q. Then according o proposiion and proposiion 2.2.3, he price funcion p(, T ) = F (, r()) saisfies he parial differenial equaion { F T + µfr T σ2 Frr T rf T = 0, (2.2.8) F T (T, r) = 1, which admis he following probabilisic formula [ F (, r) = E Q e ] T r(s)ds Φ(r(T )) r() = r. Affine Term Srucures To find he price funcion explicily, we assume ha p(, T ) = F (, r(); T ) akes he following form p(, T ) = F (, r(); T ) = e A(,T ) B(,T )r(), (2.2.9) called affine erm srucure, where A and B are deerminisic funcions. If F (, r(); T ) akes he affine erm srucure form (2.2.9), hen by subsiuing ino he erm srucure equaion (2.2.5) we obain F T + µf T r σ2 F T rr r()f T = 0, 9

11 {A (, T ) B (, T )r()} F T + µ( B(, T ))F T σ2 ( B(, T )) 2 F T rf T = 0, A (, T ) [1 + B (, T )]r µ(, r)b(, T ) σ2 (, r)b 2 (, T ) = 0. (2.2.10) The erminal condiion F (T, r; T ) 1 implies Assume µ and σ have he form A(T, T ) = 0, B(T, T ) = 0. µ(, r) = α()r + β(), σ(, r) = γ()r() + δ() (2.2.11) hen he affine erm srucure parial differenial equaion (2.2.10) becomes A (, T ) + β()b(, T ) + 1 [ 2 δ()b2 (, T ) 1 + B (, T ) + α()b(, T ) 1 ] 2 γ()b2 (, T ) r() = 0, which leads o differenial equaions of A(, T ) and B(, T ) { B (, T ) + α()b(, T ) 1 2 γ()b2 (, T ) = 1, { B(T, T ) = Model Calibraion A (, T ) = β()b(, T ) 1 2 δ()b2 (, T ), A(T, T ) = 0. We need o ge informaion abou he drif erm µ and diffusion erm σ in he Q-dynamics (2.2.7). We will follow he seps below o calibrae he unknown parameers. ˆ Le α be he parameer vecor o be esimaed, hen he Q dynamics of r() is wrien as dr() = µ(, r(); α)d + σ(, r(); α)dw () ˆ Solve, for every possible ime of mauriy T, he erm srucure equaion { F T + µfr T σ2 Frr T rf T = 0, F T (T, r) = 1. In his way we have compued he heoreical erm srucure as p(, T ; α) = F T (, r; α). ˆ Collec price daa {p (0, T ); T 0} from bond marke, where = 0 means oday. Denoe his empirical erm srucure by {p (0, T ); T 0}. 10

12 ˆ Chooses parameer vecor α in such a way ha he heoreical curve {p(0, T ; α); T 0} fis he empirical curve {p (0, T ; α); T 0} as well as possible (according o some objecive funcion). This gives our esimaed parameer vecor α. ˆ Inser α ino µ and σ o pin down exacly he erms µ = µ(, r(); α ) and σ = σ(, r(); α ) under he maringale measure Q. This procedure is equivalen o pinning down he exac maringale measure Q deermined by he marke for pricing. Wih he parameers calibraed according o he seps above, we can furher price financial derivaives on bonds and ineres raes. For arbirary ineres rae derivaives X = Γ(r(T )), he price process G(, r()) can be obained by solving he erm srucure equaion { G + µ G r (σ ) 2 G rr rg = 0, 2.4 Assumpions G(T, r) = Γ(r). In his projec, we made some assumpions which made he models compuable. We assume ha ˆ he bond s price a mauriy dae would be 1. ˆ he risk free ineres rae would be a consan han we could find on he marke. ˆ he noion was 0 when we applied he affine erm srucure models so ha we can price he bonds a curren ime. ˆ There exiss a (fricionless) marke for T -bonds for every T > 0. Which means he mauriy daes on each bonds are coninuous ˆ For each fixed, he bond price P (, T ) is differeniable w.r. ime of mauriy T. Such kind of bonds is a conrac which guaranees he holder 1 dollar o be paid on he dae T. The price a dae of a bond wih mauriy dae T is denoed by p(, T ), T. 2.5 Daa Resource and Inerpreaions In his projec, we consider he values of differen mauriy bonds on each day. The bond daa marix comes from figure 2.1. To apply he affine erm srucure on pricing bonds, we need use some shor rae models. For he risk free ineres rae, we will use he money marke accoun ineres rae r = , which came from he Bank of America. In addiion, here are 11 bonds wih differen mauriies (from 1 monh o 30 years) ha we can find on he U.S. DEPARTMENT OF THE TREASURY websie [4]. Yield is one par of he oal reurn of holding a securiy. Normally, yield is an annualized daa. Suppose ha we are holding a bond, whose mauriy dae is T, a ime. The relaion beween he yield Y and reurn rae R on his bond is: R(, T ) = e (T ) Y (,T ). The bond yield marix is a daa se ha conains differen mauriy dae bonds(which is also he column of he marix, T = 1, 1, 1, 1, 2, 3, 5, 7, 10, 20, 30) on differen days (which is also he row of he marix. From Jan, 3rd, o Mar, 31h, 2017). The bond s price obain he formula: P (, T ) = 1. The able 2.1 is he R(,T ) price marix ha we will use in he calibraion. 11

13 Figure 2.1: Those is he daily yield daa marix ha we can access on he U.S.Deparmen of he Treasury websie. Table 2.1: An illusraion of he bond price marix P ( i, T j ). Here 1 represens 2017/01/03, 2 represens 2017/01/04, and 3 represens 2017/01/05. T 1 T 2 T 3 T 4 T 5 T 6 T 7 T 8 T 9 T 10 T For bond price marix, P i,j = P ( i, T j ). We are going o price he T j bond s value a ime i, which is bond P i,j or P ( i, T j ). The face value of a bond a mauriy dae T is 1 dollar, which means P (T, T ) = 1. 12

14 Chaper 3 Models Applying he affine erm srucure models can be realized in he following four seps: 1. Choose a shor rae model. We will use several differen shor rae models o build differen bond pricing models. The shor rae models we will use are ˆ Vasicek Model ˆ Hull-Whie Model 2. Plug he shor rae model ino he affine erm srucure and solve he sochasic differenial equaion. 3. Minimize he 2 norm of he difference beween he heoreical affine erm and he bonds prices, P i,j, in he marke. 4. Calibrae he unknown coefficiens in he model. Here we use Malab o achieve he minimizaion in 2-norm. 3.1 Vasicek Model We firs consider a simple model: Vasicek Model. The Vasicek Model describes ineres raes, proposed by Oldrich Vasicek in 1977 [5]. Nowadays, Vasicek Model is widely used in financial indusry since i is one of he easies shor-rae model. Definiion Vasicek Model is an one-facor linear model, which is wrien as dr() = (b ar())d + σdw, where r() is he ineres rae and a, b, σ are consans ha we need o calibrae from real daa. Also, W is a Brownian Moion. We plug he Vasicek Model ino he Affine Term Srucure and he sysem (2.2.12) becomes B (, T ) ab(, T ) = 1, B(T, T ) = 0. The soluion of his ordinary differenial equaion is B(, T ) = 1 a [ 1 e a(t ) ]. (3.1.1) 13

15 The sysem (2.2.12) becomes whose soluion is A (, T ) = bb(, T ) 1 2 σ2 B 2 (, T ), A(T, T ) = 0, A(, T ) = σ2 2 By (3.1.1) and (3.1.2), we obain { [ ] B(, T ) = 1 a 1 e a(t ) ( ) A(, T ) = 1 (B(, T ) T + ) ab σ2 a 2 2 B 2 (s, T )ds b B(s, T )ds. (3.1.2) σ2 B 2 (,T ) 4a (3.1.3) and he affine erm srucure p(, T ) = e A(,T ) B(,T )r(). Here a, b, σ are he unknown coefficiens o be calibraed. To calibrae hese parameers, we minimize he 2 norm a each ime. log(p (, T )) log(p(, T ; a, b, σ)) 2, all mauriy ime T where P (, T ) s are he observed marke bond s prices wih mauriy dae T a ime. p(, T ) is he heoreical price equaion wih mauriy T a ime. All of he a curren ime were 0. Specifically, we minimize, for our given daa in Appendix A, 11 Res i = log(p i,j (a, b, σ)) log(p i,j ) 2, j=1 where P i,j s are he observed marke bond s prices wih mauriy dae T j a ime i and p i,j (a, b, σ) is he heoreical price equaion wih mauriy T j a ime i, depending on a, b, σ. Leing a Res i = 0, b Res i = 0 and σ Res i = 0, we have Here 11 j=1 11 j=1 11 j=1 a log(p i,j (a, b, σ))[log(p i,j (a, b, σ)) log(p i,j )] = 0, 1 a (B i,j T j + i )[log(p i,j (a, b, σ)) log(p i,j )] = 0, [ σ a 2 (B i,j T j + i ) σ 2a B i,j][log(p i,j (a, b, σ)) log(p i,j )] = 0. a log(p i,j (a, b, σ)) = [( b a 2 + σ2 a 3 )(B i,j T j + i )+ 1 a 2 ab i,j (ab )]+ σ2 σ2 2 4a 2 B2 i,j σ2 B i,j 2a ab i,j r i a B i,j. Simplifying he nonlinear sysem above and use he second equaion in he nonlinear sysem, we ge 11 j=1 [ 1 ( a ab 2 i,j ab σ2 2 ) + σ2 4a 2 B2 i,j σ2 B i,j 2a ab i,j r i a B i,j ][log(p i,j (a, b, σ)) log(p i,j )] = 0, 14

16 11 j=1 (B i,j T j + i )[log(p i,j (a, b, σ)) log(p i,j )] = 0, 11 j=1 B i,j [log(p i,j (a, b, σ)) log(p i,j )] = (3.1.4) 0. Recall ha a B i,j = 1 a B i,j + e a(t j i ) = (1 a 1 a )B i,j. Using he hird equaion in (3.1.4), we hen obain ha 11 j=1 11 j=1 B 2 i,j[log(p i,j (a, b, σ)) log(p i,j )] = 0, (T j i )[log(p i,j (a, b, σ)) log(p i,j )] = 0, 11 j=1 B i,j [log(p i,j (a, b, σ)) log(p i,j )] = 0. Now we use MATLAB o find an approximae soluion o his resuling sysem. problem, we follow he following seps. To solve his 1. Compue he price equaion p(, T ; a, b, σ) firs. 2. Then combine he price equaion and observed daa ogeher o form he 2-norm: 11 Res i = log(p i,j (a, b, σ)) log(p i,j ) 2, j=1 3. Use global ool box minimize he Res i for each i. 4. Plo he coefficiens a, b, σ and Res i. All Malab code are aached in Appendix C. 3.2 Hull-Whie Model Hull-Whie model was firs inroduced by John C. Hull and Alan Whie in 1990 [6]. Hull-Whie model is an ineres rae model and can be separae ino one-facor model as well as wo-facor model. I s an exension of he Vasicek model. Definiion One-Facor Hull-Whie Model. The Hull-Whie model is dr() = (Θ() ar)d + σdw (), (3.2.1) where Θ() is a deerminisic funcion of which can be deermined by Θ(T ) = f(0, T ) T + af(0, T ) + σ2 2a (1 e 2a ) (3.2.2) 15

17 Where f(0, T ) is he forward ineres rae from ime 0 o ime T ha can also be observed online. f(, T ) has he formula: log p(, T ) f(, T ) = T = B(, T ) r() A(, T ) T T (3.2.3) Here a, σ are he consans o be calibraed. W () is a Brownian Moion. Since Θ is ime dependen, i can be considered as an exension of Vasicek Model. Here r() follows he normal disribuion, r() N ( e a r(0) + 0 ) Θ(s)e a( s) ds, σ2 2a (1 e 2a ). In he general shor rae model (2.2.7), we hen have from he Hull-Whie model Then equaions for A and B become The soluions for he sysem (3.2.4) are µ(, r) = Θ() ar(), σ(, r) = σ. B (, T ) = ab(, T ) 1, B(T, T ) = 0, (3.2.4) A (, T ) = Θ()B(, T ) 1 2 σ2 B 2 (, T ), A(T, T ) = 0. B(, T ) = 1 ( ) 1 e a(t ), a [ ] 1 A(, T ) = 2 σ2 B 2 (s, T ) Θ(s)B(s, T ) ds. The bond price a ime wih mauriy dae T is ] p(, T ) = p (0, T ) [B(, p (0, ) exp T )f (0, ) σ2 4a B2 (, T )(1 e 2a ) B(, T )r(). (3.2.5) In his model, we minimize he following he 2 norm a each ime, i.e., P (, T ) p(, T ; a, σ) 2. all mauriy ime T The residues are differen from wha we used in he Vasicek model. Specifically, for each i 11 Res i = p i,j (a, σ)) Pi,j 2, j=1 where P i,j s are he observed marke bond s prices wih mauriy dae T j a ime i and p i,j (a, σ) is he heoreical price equaion wih mauriy T j a ime i, depending on a, σ. All of he a curren ime were 0. 16

18 Chaper 4 Resuls 4.1 The Vasicek - Affine Term Srucure Model We firs presen he resuls using he Vasicek model. To calibrae he parameers, a, b, σ, we minimize he 2 norm a each ime, i.e., log P (, T ) log p(, T ; a, b, σ) 2. Specifically, for each i all mauriy ime T 11 Res i = log(p i,j (a, b, σ)) log Pi,j 2, j=1 where P i,j s are he observed marke bond s prices wih mauriy dae T j a ime i and p i,j (a, b, σ) is he heoreical price equaion wih mauriy T j a ime i, depending on a, b, σ. All of he a curren ime were 0. Since we wan a posiive b, so we le b = β 2 in our minimizaion, i.e., we minimize he residues Res i wih respec o a, β, σ insead of a, b, σ. To solve he minimizaion problem, we use he Global Search Tool Box in Malab, see he aached code in he main funcion. The Res i s are implemened as opions = opimopions(@fmincon, 'Algorihm', 'inerior-poin'); problem = creaeopimproblem('fmincon', 'objecive', P, 'x0', [1 1 1], 'opions', opions); Here we used he inerior poin mehod opion. The iniial guess is [1, 1, 1]. The solver is implemened in Malab as gs = GlobalSearch; [abs,fval] = run(gs,problem); We presen he values of Res i, i.e, he 2-norm error on differen daes i in Figure 4.1. We observe ha he Res i s are indeed very small, 0.02% 0.07%, compared o he values of Pi,j (a he magniude of 0.3 1). In Figure 4.2, we plo he values of a, b, σ on differen days. Here σ is relaively small, which means he noise level is low. To se one value for each of a, b, σ, we ake he averages over all he days. The averaged values are 17

19 Figure 4.1: Residues Res i = 11 j=1 log(p i,j(a, b, σ)) log P i,j 2 on differen days. Figure 4.2: Values of a, b, σ. Top: a on differen days; Middle: b on differen days: Boom: σ on differen days. a = , wih sandard deviaion , 18

20 b = , wih sandard deviaion , σ = wih sandard deviaion We hen obain he calibraed Vasicek - affine erm srucure model as follows. 1 B(0, T ) = ( e T ), 1 A(0, T ) = [(B(0, T ) T ) ] B 2 (0, T ), (4.1.1) P (0, T ) = e A(0,T ) B(0,T ). In Equaion (4.1.1), P (0, T ) is he final bond price a 0 wih he mauriy dae T. The shor rae model will be dr() = ( r())d dW, r() = r(0)e (1 e ) e e s dw s r() is he shor rae a ime. 4.2 The Hull Whie - Affine Term Srucure Model As in he Vasicek model, we perform he leas square mehod. To calibrae he parameers, a, σ, we minimize he 2 norm a each ime, i.e., P (, T ) p(, T ; a, σ) 2. Specifically, for each i all mauriy ime T 11 Res i = p i,j (a, σ)) Pi,j 2, j=1 where P i,j s are he observed marke bond s prices wih mauriy dae T j a ime i and p i,j (a, σ) is he heoreical price equaion wih mauriy T j a ime i, depending on a, σ. All of he a curren ime were 0. We se April 20 in 2016 as he ime 0 (Iniial Time). We need P (0, T ), P (0, ) o price he value for he bond wih mauriy T and curren ime. Here P (0, T ) is obained from he yield marix as described in Chaper 2.5. Here we always se he ime o be one day. Since he bond is only one day before he mauriy dae, we approximaely se he bond price P 1 (0, ) P (0, 0) = 1. Here 252 is 252 he number of rading days in one year. We also need forward ineres rae f (0, ), which we downloaded from Quandl [7]. We are now ready o implemen he leas square mehod and obain he calibraed model. The solver and implemenaion are similar o hose in he Vasicek model. The daa is from he bond yield marix in A.1, A.2 from April 20, 2016 o March, 31h, In Figures 4.3, 4.4, we show he values of a, σ on differen daes. There are several days wih large values of a, σ, several orders of magniudes larger han he residues on differen days. I is no clear o us why his happen as we received a warning message from Malab ha Warning: Marix is singular o working precision. Due o he ime limi of he projec, we 19 0

21 Figure 4.3: Residues Res i = 11 j=1 p i,j(a, σ)) P i,j 2 on differen daes. Figure 4.4: The op figure is he coefficiens σ s on differen daes. The boom figure is he coefficiens a s on differen daes. don invesigae why here are such ouliers and we will consider his laer on. Bu we here don presen he calibraion on hese days. 20

22 To se up one value for each of a, σ, we ake he averages over all he days. The averaged values are a = , wih sandard deviaion , σ = wih sandard deviaion We hen obain he calibraed Hull Whie - affine erm srucure model as follows. { B(, T ) = (1 e (T ) ), p(, T ) = P (0,T ) P (0,) eb(,t )f (0,) 1.008b 2 (,T )(1 e ) B(0,T ). (4.2.1) In Equaion (4.2.1), P (0, T ) is he final bond price a 0 wih he mauriy dae T. The shor rae model will be dr() = (Θ() r)d dW () Where Θ() follows he formula in equaion (3.2.2) Θ(T ) = f(0, T ) T f(0, T ) (1 e ) f(, T ) has he formula in equaion and f(0, T ) can also be observed in marke. All daa are in Appendix A.3. 21

23 Chaper 5 Conclusion and Discussion We have applied affine erm srucure model and shor rae models o esimae risk-free ineres rae. We used he daa from US Deparmen of he Treasury from 2016 o Performing proper leassquare mehods, we obained all he parameers in he shor rae models. Once we go he esimaed parameers wih respec o differen raes in he shor rae models, we averaged hese parameers o have some sense of he values over a relaively long ime. From bond s yield daa, we firs ransformed he bond s yield o he price of he bond whose face value is one dollar. We used wo shor rae models. The firs is he Vasicek model. To reduce compuaional cos, we ook log for he daa in he price marix. Furhermore, we obained he coefficiens a i, b i, σ i which minimized he Res i and hen ook average one hem. The second one is he Hull-Whie model. We used a similar approach and we fi he daa iself insead of fiing a logarihm of he daa. The mehodology in his projec requires minimizaion on each day and some aggregaion of esimaed parameers for overall undersanding is required. To obain each parameer working for all ime, one way is o minimize he residues over all he ime. Bu such a minimizaion can be expensive. To avoid a high compuaional cos, we can apply Kalman filering in furher sudy, which is a leas square mehod for ime series and requires a global opimizaion of parameers. Furher applicaion of he mehodology for predicion will validae he calibraed models and may require furher calibraion(s). However, we believe ha curren resuls have moved us owards a high-fideliy risk-free ineres rae, which is one of he key indicaors of curren financial marke and economy. 22

24 Bibliography [1] Sephen A. Ross. The arbirage heory of capial asse pricing. Journal of Economic Theory, 13, [Online; accessed 31 - March ]. [2] O Sullivanm Arhur and Seven M. Sheffrin. Economics: Principles in acion. Pearson Prenice Hall, Upper Saddle River, NJ, 07458, [3] Frederic S Mishkin. The economics of Money, Banking, and Financial Markes (Alernae Ediion). Addison Wesley (July 24, 2006), Boson, MA. [4] U.S. DEPARTMENT OF THE TREASURY. Daily reasury yield curve raes, [Online; accessed 31 - March ]. [5] Oldrich Vasicek. An equilibrium characerizaion of he erm srucure. journal of Financial Economics, 5(2): , [6] John Hull and Alan Whie. Using hull-whie ineres rae rees. Journal of Derivaives, 3(3), [Online; accessed 31 - March ]. [7] Quandl. Us reasury insananeous forward rae curve. hps:// SVENF-US-Treasury-Insananeous-Forward-Rae-Curve. [Online; accessed 31 - March ]. [8] Peer Carr and Rober Kohn. Ineres rae models - hull whie. hps:// ~benarzi/slides10.3.pdf, [Online; accessed 31 - March ]. [9] Xuwei Yang. Financial mahemaics, [Online; accessed 12h -April ]. 23

25 Appendices 24

26 Appendix A Yield Marix Table A.1: Yield Table - 1 Dae 1 Mo 3 Mo 6 Mo 1 Yr 2 Yr 3 Yr 5 Yr 7 Yr 10 Yr 20 Yr 30 Yr Mauriy Lengh /1/ /1/ /1/ /1/ /1/ /1/ /1/ /1/ /1/ /1/ /1/ /1/ /1/ /1/ /1/ /1/ /1/ /1/ /1/ /1/ /2/ /2/ /2/ /2/ /2/

27 Table A.2: Yield Table /2/ /2/ /2/ /2/ /2/ /2/ /2/ /2/ /2/ /2/ /2/ /2/ /2/ /2/ /3/ /3/ /3/ /3/ /3/ /3/ /3/ /3/ /3/ /3/ /3/ /3/ /3/ /3/ /3/ /3/ /3/ /3/ /3/ /3/ /3/ /3/ /3/

28 Table A.3: Forward Rae Marix [7] Dae 1-day Forward Rae

29 Appendix B Some useful formula B.1 Exac Soluion o he Hull-Whie Model we can solve he shor rae r() from (3.2.1), see e.g. [8]. e a dr = (Θ() ar)e a d + σe a dw (). Moving he erm are a d o he lef side of he equaion leads o d(e a r) = e a dr + ae a rd = Θ()e a d + e a σdw, e a r() = r(0) + r() = e a r(0) + Then r() follows he normal disribuion ( r() N e a r(0) Θ(s)e as ds + σ 0 e as dw (s), Θ(s)e a( s) ds + σe a e as dw (s). 0 ) Θ(s)e a( s) ds, σ2 2a (1 e 2a ). B.2 Proof of Hull-Whie Bond Pricing The following proof for he Hull-Whie bond pricing model is from [9]. Proof. p(, T ) = exp {A(, T ) B(, T )r()} { [ ] } 1 = exp 2 σ2 B 2 (s, T ) Θ(s)B(s, T ) ds B(, T )r() { [ ] 1 = exp 2 σ2 B 2 (s, T ) [ft (0, s) + g (s) + a (f (0, s) + g(s))] B(s, T ) { 1 T = exp 2 σ2 B 2 (s, T )ds ft (0, s)b(s, T )ds g (s)b(s, T )ds af (0, s)b(s, T )ds 28 0 } ds B(, T )r() } ag(s)b(s, T )ds B(, T )r()

30 { = exp { = exp = exp { = exp ( = exp 1 2 σ2 B 2 (s, T )ds [ f (0, s)b(s, T ) s=t s= af (0, s)b(s, T )ds ] f (0, s)b (s, T )ds g (s)b(s, T )ds } ag(s)b(s, T )ds B(, T )r() 1 2 σ2 B 2 (s, T )ds + f (0, )B(, T ) + f (0, s)b (s, T )ds g (s)b(s, T )ds } af (0, s)b(s, T )ds ag(s)b(s, T )ds B(, T )r() 1 2 σ2 B 2 (s, T )ds + f (0, )B(, T ) + 1 T 2 σ2 B 2 (s, T )ds + f (0, )B(, T ) } {{ } p (0,T ) p (0,) f (0, s) [B (s, T ) ab(s, )] ds }{{} 1 [g (s) + ag(s)] B(s, T )ds B(, T )r() }{{} (σ 2 /2a)(1 e 2as ) f (0, s)ds } σ 2 2a (1 e 2as )B(s, T )ds B(, T )r() ) [ ] σ f (0, s)ds exp f 2 (0, )B(, T ) + 2 B2 (s, T ) σ2 2a (1 e 2as )B(s, T ) ds B(, T )r() }{{} σ2 4a B2 (,T )(1 e 2a ) 29

31 Appendix C Malab Code C.1 Funcion for he Variable A(, T ) Lising C.1: This is he Malab code for he par A in Vasicek Model funcion coefficiena = AffVA(a,b,,T,sigma) coefficiena = (1./a.ˆ2).*(AffVB(a,,T) - T +).*(a.*b.ˆ2-sigma./2) - ((sigma.*affvb(a,,t)).ˆ2 C.2 Funcion for he Variable B(, T ) Lising C.2: This is he Malab code for he par B in Vasicek Model funcion coefficienb = AffVB(a,,T) coefficienb = (1./a).*(1-exp(-a.*(T-))); C.3 Funcion for he Bond s Price Lising C.3: This is he Malab code for bond price in Vasicek Model funcion AffineVasicek = AffVar(a,b,,T,sigma) AffineVasicek = (AffVA(a,b,,T,sigma)- AffVB(a,,T).*0.005); C.4 Funcion for Res i funcion PriceError = PriceErr(a,b,sigma,YieldMarix, RaeMarix, i) PriceError = ((RaeMarix(i,1) - AffVar(a,b,0,YieldMarix(1,1),sigma)).ˆ2 + (RaeMarix(i,2) - AffVar(a,b,0,YieldMarix(1,2),sigma)).ˆ2 + (RaeMarix(i,3) - AffVar(a,b,0,YieldMarix(1,3),sigma)).ˆ2 + (RaeMarix(i,4) - AffVar(a,b,0,YieldMarix(1,4),sigma)).ˆ2 + (RaeMarix(i,5) - AffVar(a,b,0,YieldMarix(1,5),sigma)).ˆ2 + 30

32 (RaeMarix(i,6) - AffVar(a,b,0,YieldMarix(1,6),sigma)).ˆ2 + (RaeMarix(i,7) - AffVar(a,b,0,YieldMarix(1,7),sigma)).ˆ2 + (RaeMarix(i,8) - AffVar(a,b,0,YieldMarix(1,8),sigma)).ˆ2 + (RaeMarix(i,9) - AffVar(a,b,0,YieldMarix(1,9),sigma)).ˆ2 + (RaeMarix(i,10) - AffVar(a,b,0,YieldMarix(1,10),sigma)).ˆ2 + (RaeMarix(i,11) - AffVar(a,b,0,YieldMarix(1,11),sigma)).ˆ2); C.5 Funcion o ransform yield marix o price marix funcion PriceFuncion = PriceFunc(a,b,,T,sigma) [m,n] = size(yieldmarix); RaeMarix = zeros(m,n); MinVariable = zeros(m-1,3); for j = 1:n for i = 2:m RaeMarix(i,j) = exp(yieldmarix(1,j)*yieldmarix(i,j)*0.01); end end PriceFunc = (RaeMarix(i,:) - AffVar(a,b,0,T,sigma))ˆ2; C.6 Calibraion Funcion Lising C.4: This is he Malab code for calibraing he coefficiens in Vasicek Model % Vasicek - Affine erm srucure YieldMarix = xlsread('bounds Yield'); T = YieldMarix(1,:); Minvariable = zeros(1,4); [m,n] = size(yieldmarix); RaeMarix = ones(m,n); P = 0; Coefficiens = zeros(4,m-1); for j = 1:n for i = 2:m RaeMarix(i,j) = log(1/(exp(yieldmarix(1,j)*yieldmarix(i,j)*0.01))); end end for i = 1:62 P RaeMarix, i+1); opions = opimopions(@fmincon, 'Algorihm', 'inerior-poin'); problem = creaeopimproblem('fmincon', 'objecive', P, 'x0', [1 1 1], 'opions', opions); gs = GlobalSearch; [abs,fval] = run(gs,problem); Coefficiens(1,i) = abs(1); Coefficiens(2,i) = (abs(2))ˆ2; 31

33 Coefficiens(3,i) = abs(3); Coefficiens(4,i) = fval; end C.7 Hull-Whie Model Lising C.5: This is he Malab code for he coefficien B in Hull-Whie Model funcion coefficienb = AffVBHullWhie(a,,T) coefficienb = (1./a).*(1-exp(-a.*(T-))); Lising C.6: This is he Malab code for he pricing equaion in Hull-Whie Model funcion AffineHullWhie = AffVarHullWhie(a,,T,sigma, pt, p, f) AffineHullWhie = (pt./p).*exp(affvbhullwhie(a,,t).*f-((sigma.ˆ2)./(4.*a)).*(1 - exp(-2.*a. AffVBHullWhie(a,,T)*0.0005); Lising C.7: This is he Malab code for he Term Sruure in Hull-Whie Model YieldMarix = xlsread('boundsyield-hullwhie'); T = YieldMarix(1,:); Minvariable = zeros(1,4); [m,n] = size(yieldmarix); RaeMarix = ones(m-1,n); P = 0; Coefficiens = zeros(3,m-1); Forward = xlsread('forwardrae'); ForwardMarix = Forward/100 ; for j = 1:n for i = 2:m RaeMarix(i-1,j) = (1/(exp(YieldMarix(1,j)*YieldMarix(i,j)*0.01))); end end for i = 1:300 P RaeMarix, i, ForwardMarix); opions = opimopions(@fmincon, 'Algorihm', 'inerior-poin'); problem = creaeopimproblem('fmincon', 'objecive', P, 'x0', [1 2], 'opions', opions); gs = GlobalSearch; [abs,fval] = run(gs,problem); Coefficiens(1,i) = abs(1); Coefficiens(2,i) = abs(2); Coefficiens(3,i) = fval; end 32