University of Cape Town

Transcription

1 The copyrigh of his hesis vess in he auhor. No quoaion from i or informaion derived from i is o be published wihou full acknowledgemen of he source. The hesis is o be used for privae sudy or noncommercial research purposes only. Published by he UCT in erms of he non-exclusive license graned o UCT by he auhor.

2 An Invesigaion of Shor Rae Models and he Pricing of Conigen Claims in a Souh African Seing Chris Jones 29 November, 2010 Presened for he Degree of Masers of Science in Mahemaics of Finance in he Deparmen of Saisical Science, Universiy of Cape Town Supervisor: Professor Ronald Becker

3 Plagiarism Declaraion 1. I know ha plagiarism is wrong. Plagiarism is o use anoher s work and preend ha i is one s own. 2. Each significan conribuion o, and quoaion in his disseraion from he work of oher people has been aribued, and has been cied and referenced. 3. This disseraion is my own work. 4. I have no allowed, and will no allow, anyone o copy my work wih he inenion of passing i off as his or her own work. 2

4 Absrac This disseraion invesigaes he dynamics of ineres raes hrough he modelling of he shor rae he spo ineres rae ha applies for an infiniesimally shor period of ime. By modelling such a rae via a diffusion process, one is able o characerize he enire yield curve and price plain vanilla opions. The aim is o invesigae which of he more popular shor rae models is bes suied for pricing such opions, which are acively raded in he marke. Thus one can hen use such models o price more exoic opions, as such opions are ypically less frequenly raded in he marke. Alhough much lieraure exiss wih regards o he heoreical side of shor rae models, discussion abou he pracical implemenaion of such models is limied, paricularly from a Souh African perspecive. This paper inends o no only cover he heory behind shor rae models, bu also o describe he unique Souh African marke pracicaliies ha exis, providing a framework o pricing plain vanilla opions using he more popular shor rae models, before jusifying which models appear mos appropriae o use in he Souh African environmen. I would like o hank Professor Becker for all his suppor and guidance hroughou his hesis. His commens and suggesions proved mos helpful. 3

5 Conens 1 Inroducion Disseraion Ouline Inroducion o Bond Markes Bank Accoun Pure Discoun Bond Spo Raes Arbirage Forward Rae Shor Rae LIBOR and JIBAR Opions on Bonds Ineres Rae Caps Ineres Rae Floors Ineres Rae Swaps Ineres Rae Swapions No-Arbirage Pricing and Numeraire Change Basic Seup Equivalen Maringale Measure Change of Numeraire Girsanov s Theorem and Io s Lemma Pu-Call Pariy Bond Opions Caps and Floors Pricing of Caps and Floors Pricing of Swaps Pricing of Swapions One Facor Shor Rae Models Muli-Facor Shor Rae Models

6 CONTENTS 5 4 Pricing of Opions using Black s Model Opions on Bonds Caps and Floors Swapions Consisency in Pricing Formulas Suiabiliy of Black s Model for Pricing Ineres Rae Opions 28 5 Marke Pracicaliies Shor Rae Pure Discoun Bonds Available Daa Boosrap Mehod Caps and Floors Individual Caples and Floorles Affine Term Srucure Models Bond Opions Applicaion o Bond Opions Vasicek Model Pure Discoun Bond Cap and Floors Ho-Lee Model Pure Discoun Bond Shor Rae Dynamics Revisied Cap and Floors Cox-Ingersoll-Ross Model Pure Discoun Bond Cap and Floors Black-Karasinski Model Simplifying he Shor Rae Equaion Momens of he Shor Rae Revisied Pure Discoun Bond Cap and Floors Two Facor Vasicek Model Pure Discoun Bond Cap and Floors Two-Facor Cox-Ingersoll-Ross Model Pure Discoun Bond Cap and Floors

7 6 CONTENTS 13 Calibraing Shor Rae Models Parameerizaion Daa Forms of Shor Rae Models Compared Comparison of Models Tesing Non-Nesed, Non-Linear Regression Models Empirical Resuls - Fi o Yield Curve Fi o Pure Discoun Bond Prices and he Yield Curve Goodness-of-Fi Parameer Esimaes under each shor rae Model Example of Fi o he Yield Curve Davidson and MacKinnon s P-Tess Fi o Cap and Floor Prices Goodness-of-Fi Fi o Cap and Floor Prices afer Parameers Recalibraed Conclusion 108

8 Lis of Figures 10.1 Black-Karasinski: Branching of Non-Sandard Trinomial Tree Black-Karasinski: Tree Noaion Calibraion Errors of he Shor Rae Models Average Absolue Value of Error Term Sandard Deviaion of he Error Term Ho-Lee Parameer Values Black-Karasinski Parameer Values CIR One Facor Parameer Values Vasicek One Facor Parameer Values CIR Two Facor Process X Parameer Values CIR Two Facor Process Y Parameer Values Vasicek Two Facor Shor Rae Parameer Values Vasicek Two Facor Shor Team Mean Reversion Level Parameer Values Yield Curve on 18 Ocober Sandard Deviaion of he Error Term for Cap Prices Sandard Deviaion of he Error Term for Cap Prices

9 Lis of Tables 15.1 Goodness-of-Fi Saisics Resuls of Pairwise P-Tess Goodness-of-Fi Saisics of Cap Prices Goodness-of-Fi Saisics of Cap Prices

10 Chaper 1 Inroducion The earlies aemp o model ineres raes was published by Vasicek 1977, whereby he shor rae was used as he facor driving he enire yield curve. This paper led o he developmen of various alernaive models e.g., Cox- Ingersoll-Ross, Hull-Whie, Ho-Lee, Brennan and Schwarz each aemping o bes explain he underlying ineres rae process. Par of he appeal of shor rae models is he simpliciy of he models and he ease in solving hese models numerically. The fac ha principal componen analysis shows ha 80% 90% of he price variaion in he bond marke can be explained by a single facor, adds credence o he use of shor rae models, a fac highlighed by Ouwehand However, deracors emphasize he facs ha he shor rae does no exis in realiy and he yield curve derived hrough such models is ypically consrained in how i can evolve as reasons why such models should no be used. Neverheless, an undersanding of shor rae models and he principles underlying hese models is essenial in providing one wih a firm foundaion for he modelling of ineres raes. This paper builds on he iniial groundwork laid by Svoboda 2002 in her paper An invesigaion of various ineres rae models and heir calibraion in he Souh African marke. Svoboda s paper inroduces he more popular shor rae models, as well as explains how one can calibrae such models o he yield curve in he Souh African marke. From he perspecive of a Souh African reader, his paper will exend Svoboda s work by: i. explaining he marke pracicaliies ha exis in he Souh African ineres rae marke; ii. invesigaing furher shor rae models such as he wo facor forms of he Vasicek and Cox-Ingersoll-Ross models; iii. showing how one can price caps and swapions when he shor rae is assumed o follow cerain dynamics; iv. and examining which shor 9

11 10 CHAPTER 1. INTRODUCTION rae models are mos effecive a pricing plain vanilla ineres rae opions over an exended period of ime, when comparing he model-implied price o he raded marke price. 1.1 Disseraion Ouline Chaper 2 inroduces he basic insrumens which one encouners in he bond marke before Chaper 3 lays down he foundaions for he arbiragefree pricing of coningen claims and provides approaches o price various ineres rae opions in an arbirage-free world. Chaper 4 invesigaes he use of Black s model o price ineres rae opions marke convenion is o use his model when pricing such opions. Chaper 5 provides he reader wih a basic undersanding of he marke pracicaliies ha one will encouner in he Souh Africa ineres rae marke whils Chaper 6 inroduces a family of shor rae models, ermed Affine Term Srucure Models. Chapers 7 o 12 inroduce some of he more widely used shor rae models, showing one: i. how o derive he shor rae from he diffusion process; ii. price a pure discoun bond when he shor rae follows cerain dynamics; iii. and jusify he various formulae for pricing caps, assuming he shor rae follows cerain dynamics. Chaper 13 explains how one can calibrae a shor rae model o he yield curve, hus obaining values for he various parameers. Chaper 14 builds on hese resuls by providing a framework o compare various models, each of which purpors o explain he same dependen variable. Chaper 15 compares he fi of he shor rae models covered in he paper o pure discoun bond prices, using goodness-of-fi ess Davidson and MacKinnon s C Tes. The chaper hen goes on o invesigae he abiliy of each of he shor rae models covered o generae prices for a-he-money caps which are close o hose observed in he Souh African ineres rae marke. One would be scepical of using a shor rae model o price exoic opions if he prices generaed by ha model for plain vanilla opions were far from consisen wih hose acually observed in he marke. Chaper 16 highlighs he key resuls from he paper, and provides recommendaions as o which shor rae models appear mos appropriae for pricing coningen claims in he Souh African marke.

12 Chaper 2 Inroducion o Bond Markes This chaper will provide an inroducion o he more common insrumens found in he bond marke and erms used in relaion o he bond marke. 2.1 Bank Accoun B represens he value a ime of R1 invesed in he bank accoun a ime zero. The bank accoun is assumed o evolve under he following differenial equaion db = r B d, where r is he insananeous ineres rae a ime, known as he shor rae. Thus he value of he bank accoun a ime is B = exp r s ds. 2.2 Pure Discoun Bond P, T denoes he value a ime of a pure discoun bond, a conrac whereby i is agreed ha he issuer will pay he holder R1 a ime T, wih no inermediary cashflows beween ime and ime T. For he purpose of his hesis, hese bonds are assumed o have zero defaul risk. 2.3 Spo Raes The coninuously-compounded spo rae a ime for mauriy T, denoed R, T, is he consan rae which an invesmen a ime of P, T would need o earn in order o yield R1 a ime T. i.e., 0 R, T := lnp, T. T 11

13 12 CHAPTER 2. INRODUCTION TO BOND MARKETS Arbirage In an arbirage-free world, he spo rae can never be negaive. If his were o occur, hen a pure discoun bond would rade a a value greaer han is par value, resuling in an arbirage opporuniy. 2.4 Forward Rae The coninuously-compounded forward rae a ime, denoed F, T, S, is he consan rae which applies beween imes T and S, where T S. i.e., F, T, S := 1 P, T ln S T P, S. 2.5 Shor Rae The shor rae r is defined as he insanenous ineres rae a ime applicable for nex momenary period d. Thus 2.6 LIBOR and JIBAR r = lim R, T = R,. T LIBOR, he London Iner Bank Offer Rae, is defined by Jorion 2007 as a benchmark cos of borrowing for highly raed AA credis. Hull 2006 defines i as he rae a which a bank is willing o lend o oher banks. This rae is published daily for a range of borrowing periods. JIBAR, or Johannesburg Iner Bank Agreed Rae, is he Souh African equivalen of LIBOR. 2.7 Opions on Bonds A European call opion on a pure discoun bond provides he holder wih he righ o purchase his bond a an agreed poin in he fuure for an agreed price. A European pu opion on a pure discoun bond provides he holder wih he righ o sell he bond a an agreed poin in he fuure for an agreed price. 2.8 Ineres Rae Caps Hull 2006 describes an ineres rae cap as an opion ha provides a payoff when a specified ineres rae is above a cerain level. The ineres rae is a floaing rae ha is rese periodically. Thus, such an insrumen

14 2.9. INTEREST RATE FLOORS 13 may be used as insurance agains he possibiliy ha he underlying ineres rae rises above a cerain level. Example. You wish o proec agains he risk ha LIBOR raes saring in hree and six monhs ime rises above 6%. This level above which he ineres rae mus rise in order for you o recieve a payoff is known as he cap rae or srike rae, denoed r K. In his example he ineres rae is rese every hree monhs i.e., in hree and six monhs ime hus he period inbeween reses is hree-monhs and is known as he enor, denoed τ i. If he underlying ineres rae is above he cap rae on he rese dae hen he cap paymen for ha period is he produc of he he face of he insrumen ermed he principal amoun, he enor and he difference beween he underlying rae and cap rae. i.e., Nτ i [maxr i r K, 0] where N is he Principal amoun. Suppose you enered ino a nine-monh cap on hree-monh LIBOR wih a cap rae of 6% and a enor of hree monhs and he underlying ineres rae on he rese dae in 3 and 6 monhs ime is 8% and 6% respecively. Then your paymens would be N 1 4 8% 6% = 0.5%N and R0. In order o simplify maers, day coun issues have been ignored; however such issues will be inroduced a a laer sage. A cap can be viewed as a porfolio of a series of opions on he underlying ineres rae, wih each opion known as a caple. Thus when valuing a cap, one can sum he discouned values of each of he caples. i.e., Cap = := n P, T i Nτ i [maxr i r K, 0] i=1 n P, T i Cpl i. i=1 Hull 2006 noes ha caps are usually defined so ha he iniial LIBOR rae, even if i is greaer han he cape rae, does no lead o a payoff on he firs rese dae. 2.9 Ineres Rae Floors An ineres rae floor, as defined by Hull 2006, is an opion which provides a payoff when an ineres rae is below a cerain level. Thus a floor can be used as insurance agains he possibiliy ha he ineres rae falls below a cerain level. The floor will provide he following payoff a each rese dae: Nτ i [maxr K r i, 0].

15 14 CHAPTER 2. INRODUCTION TO BOND MARKETS Similar o a cap, a floor can be viewed as a porfolio of a series of opions on he underlying ineres rae wih each opion known as a floorle. One can also value he floor by simply summing he discouned values of each of he floorles. i.e., Floor = := n P, T i Nτ i [maxr K r i, 0] i=1 n P, T i Flr i. i=1 As is he case wih caps, floors are usually defined so ha he iniial LIBOR rae, even if i is less han he floor rae, does no lead o a payoff on he firs rese dae Ineres Rae Swaps Hull 2006 defines an ineres rae swap as an agreeemen beween wo paries o exchange cash flows in he fuure based on he fuure value of an ineres rae. The agreemen defines he daes when he cash flows are o be paid and he way in which hese cash flows are o be calculaed. The mos common ineres rae swap is a plain vanilla ineres rae swap where one pary agrees o pay cash flows based on a fixed rae and receive cash flows based on a floaing ineres rae, ermed Pay-Fixed Swap. Pay- Floaing Swap is he pary on he opposie side which pays cash flows based on a floaing ineres rae and receives cash flows based on he fixed rae. These paymens are boh based on he same noional principal amoun. This noional principal is no exchanged. In realiy he fixed and floaing paymens are need-off agains one anoher hus only one of he paries makes a paymen a each predeermined dae. In Souh Africa, he fixed and floaing legs are ypically exchanged quarerly. This is no neccessarily he case in oher jurisdicions, as noed by Brigo and Mercurio 2006, where ypically he fixed leg involves semi-annual or annual paymens, and he floaing leg involves quarerly or semi-annual paymens Ineres Rae Swapions Ineres rae swaps are opions on ineres rae swaps whereby, as Hull 2006 explains, hey give he holder he righ o ener ino a cerain ineres rae swap a a cerain ime in he fuure. Swapions can be used o provide insurance agains ineres raes moving beyond a cerain level, whils allowing he holder o benefi from favourable ineres rae movemens.

16 2.11. INTEREST RATE SWAPTIONS 15 A payer swapion is an opion giving he holder he righ o ener ino a pay-fixed swap whils a receiver swapion is an opion giving he holder he righ o ener ino a pay-floaing swap.

17 Chaper 3 No-Arbirage Pricing and Numeraire Change The purpose of he firs half of his chaper is o lay he foundaions for he arbirage-free pricing of conigen claims, and is based around he books of Pelsser 2004 and Brigo and Mercurio 2006 and he lecure noes of Ouwehand The second half of he chaper will focus on he relaionship beween opions and he pricing of such opions in a world where arbirage does no exis. This chaper will highligh he key conceps required, however for more a more formal and rigorous handling of his subjec, one is referred o Musiela and Rukowski 2007 or Hun and Kennedy Basic Seup Throughou his paper we consider a marke where coninuous rading occurs, no ransacion coss are incurred, markes are sufficienly liquid for every securiy, shor sales are allowed and here is perfec divisibiliy of asses. The rading inerval is limied o a finie period [0, T ]. The marke model is he uple M = Ω, F, P, F 0, S 0,..., S N 0, where Ω, F, P is a probabiliy space wih Ω denoing he sample space wih elemens ω Ω, F denoing a σ-algebra on Ω, and P denoing a probabiliy measure on Ω, F. The uncerainy is resolved over [0, T ] according o a filraion F saisfying he usual condiions. S = S 0,..., S N is an N + 1-dimensional adaped cadlag semi-maringale. We assumes ha Ω comes wih a K-dimensional Brownian moion W = W 1,..., W K which generaes he filraion F. Furhermore, we assume ha here exis asses which are raded in he marke, called markeed asses and ha he prices, S, of hese markeed asses can be modelled via 16

18 3.1. BASIC SETUP 17 Iô processes ds = µ, S d + σ, S dw, where µ, S, σ, S are deerminisic. Under hese condiions, he markeed asse price process is srong markov. The asse indexed by 0 is a bank accoun. Is price evolves under he following process ds 0 = r S 0 d, where S 0 0 = 1 and r is he shor rae a ime. For he remainder of his paper, S 0 = B. Definiion 1. A rading sraegy or porfolio is a predicable process φ = φ 0,..., φ N which is inegrable wih respec o he semi-maringale S where φ n denoes he holdings in asse S n a ime. The value of he porfolio a ime is N V φ = φ.s = φ n S n. hus n=0 One requires he rading sraegies o be self-financing: dv = dφ.s = φ.ds + S.dφ + d[s, φ] = φ.ds, V φ = V 0 φ + 0 φ u ds u = V 0 φ + G φ, where he gains process G φ is defined as 0 φ uds u. Hence he value of he porfolio a ime equals he iniial porfolio value plus he gain or loss over he period. No addiional funds are added or removed. Definiion 2. A rading sraegy φ is self-financing if and only if G φ = 0 φ uds u i.e., if and only if dφ.s = φ.ds. A European coningen claim, C, is a derivaive which, a some fuure ime T has a payoff which is a known Borel-measurable funcion of asse prices a ime T. i.e., C T = fs T. Thus C T is an F T -measurable random variable. Definiion 3. A European coningen claim, C, is said o be aainable if and only if here exiss a self-financing sraegy φ such ha C T = V T φ. where T is he exercise dae of he conigen claim. Then φ is called a replicaing porfolio for C.

19 18CHAPTER 3. NO-ARBITRAGE PRICING AND NUMERAIRE CHANGE Definiion 4. A financial marke is complee if and only if every coningen claim is aainable. If he marke is arbirage-free, and since i is assumed ha rading coss are no incurred, hen he value of a replicaing porfolio a ime gives a unique value for he coningen claim, C. Hence one can value he coningen claim by valuing he replicaing porfolio, a process called pricing by arbirage, see e.g., Pelsser If he condiions are me under which a marke is arbirage-free and complee, hen all coningen claims can be priced by arbirage. 3.2 Equivalen Maringale Measure Any price process, N, which has sricly posiive prices for all [0, T ] is ermed a numeraire. Ouwehand 2008 describes a numeraire as a uni ino which oher asses are ranslaed. Thus, if S is he price of S in money, hen Ŝ = S N is he price of S in unis of N. Typically he bank accoun B = S 0 is chosen as he numeraire. In such a case, S = S B is he discouned value of S a ime. Definiion 5. Suppose N is a numeraire. A measure Q on Ω, F is an Equivalen Maringale Measure EMM for numeraire N if and only if 1. Q is equivalen o P. i.e., boh measures have he same null-ses, 2. Ŝ = S N is a Q-maringale. An Equivalen Maringale Measure associaed wih he bank accoun is called a risk-neural measure. Theorem 1. If an EMM Q exiss for some numeraire N, hen here are no arbirage opporuniies. Ouwehand, 2008 Theorem 2 Risk-Neural Valuaion. Suppose ha X is an aainable coningen claim, and ha Q is an EMM for numeraire N. Then, ˆX = E Q [ ˆX T F ], i.e., X = N E Q [ XT N T ] F. Ouwehand, 2008

20 3.3. CHANGE OF NUMERAIRE Change of Numeraire Geman e al 1995 noed ha he EMM Q is no necessarily he mos convenien measure o use when pricing a coningen claim. In such insances, one is beer suied o use a change of numeraire. Jamshidian 1989 illusraes his poin, using a change of numeraire when he shor rae is sochasic in order o price a bond opion. I is criical o undersand he impac of a change in numeraire on a selffinancing porfolio and a coningen claim which is aainable: Proposiion 1. A self-financing porfolio remains self-financing under a change of numeraire. Ouwehand, 2008 Corollary 1. If a coningen claim is aainable in a given numeraire, i is also aainable in any oher numeraire, and he replicaing porfolio is he same. Ouwehand, 2008 This secion ends wih a proposiion which is an exension of he Risk- Neural Valuaion heorem and provides one wih a fundamenal ool for he pricing of coningen claims. Proposiion 2. Assume ha here exiss a numeraire N and a probabiliy measure Q N, equivalen o he iniial Q 0, such ha he price of any raded asse X wihou inermediae paymens relaive o N is a maringale under Q N. i.e., [ ] X XT = E N Q N F. Le U be an arbirary numeraire. Then here exiss a probabiliy measure Q U, equivalen o he iniial Q 0, such ha he price of any aainable claim Ŷ = Y U is a maringale under QU [ ] Y YT = E U Q U F. Moreover he Radon-Nikodym derivaive defining he measure Q U is given by dq U dq N = U T N 0. U 0 N T Brigo and Mercurio, 2006 N T 3.4 Girsanov s Theorem and Io s Lemma Girsanov s heorem provides one wih an undersanding of he impac of a change in numeraire on he sochasic differenial equaion of an asse, paricularly how he drif of an asse will change due o a change in numeraire. U T

21 20CHAPTER 3. NO-ARBITRAGE PRICING AND NUMERAIRE CHANGE Theorem 3 Girsanov s Theorem. For any sochasic process λ such ha 0 λ 2 sds <, wih probabiliy one, consider he Radon-Nikodym derivaive dq dp = ζ given by ζ = exp λ s dw s λ 2 sds, where W is a Brownian moion under he measure P. Under he measure Q he process Ŵ = W is also a Brownian moion. Pelsser, λ s ds, A key poin o noe from his resul is ha a change in numeraire only resuls in a change in he drif of he underlying asse, no in a change in he volailiy. Anoher key resul from sochasic calculus is Io s Lemma. Lemma 1 Io s Lemma. Suppose ha X is a one-dimensional sochasic process given by he differenial equaion dx = µ, ωd + σ, ωdw, and a funcion f, X of he process X exiss, hen provided ha f is sufficienly differeniable f, X f, X df = + µ, ω X σ, ω2 2 f, X X 2 d+σ, ω Pelsser, Pu-Call Pariy f, X X dw. Pu-Call pariy describes he relaionship beween opions and he underlying asses where he characerisics of he various opions are idenical Bond Opions Consider he following wo porfolios: 1. Long one Call Opion and K unis of a pure discoun bond expiring a T. i.e., C + KP, T.

22 3.5. PUT-CALL PARITY Long one Pu Opion and one pure discoun bond expiring a S. i.e., P + P, S. Noe: The erms of he opions are idenical, wih a srike price of K and he underlying a pure discoun bond expiring a S i.e., P, S. The pure discoun bonds P, T and P, S have face values of 1. A ime T, if he pure discoun bond expiring a ime S is greaer han he srike price, K, porfolio 1 will receive P T, S K + K = P T, S whils porfolio 2 will receive 0 + P T, S = P T, S. If he pure discoun bond expiring a ime S is less han he srike price, K, porfolio 1 will receive K whils porfolio 2 will receive P T, S + k P T, S = K. Since he cash flows of he wo porfolios are equivalen a ime T, he wo porfolios mus be equivalen in value a any poin in ime. i.e., C + KP, T Caps and Floors Consider he following wo porfolios: 1. Long one Cap. = P + P, S. 2. Long one Floor and one Pay-Fixed Swap. The cap and floor are assumed o have he same erm, enor, srike rae and underlying ineres rae LIBOR. The fixed rae of he swap is assumed o be he same as he srike rae whils he floaing rae of he swap is assumed o be he same as he underlying ineres rae of he cap and floor. The paymen daes of he swap are assumed o be equivalen o hose of he cap and floor. For any period, if LIBOR is greaer han he srike rae, porfolio 1 will receive LIBOR r K whils porfolio 2 will receive 0 + LIBOR r K. If LI- BOR is less han he srike rae, porfolio 1 will no receive any paymen, whils porfolio 2 will receive r K LIBOR and will pay r K LIBOR i.e., he ne paymen is zero. Since he paymen daes of he wo porfolios are equivalen, and he cash flows of each of he porfolios are equiavlen a any paymen dae, assuming no-arbirage, he wo porfolios mus be equivalen in value a any poin in ime. i.e., Cap = Floor + Pay-Fixed Swap. Noe ha a plain vanilla ineres rae swap wih he same erm and periods beween rese daes as a cap or floor will involve one more paymen dae

23 22CHAPTER 3. NO-ARBITRAGE PRICING AND NUMERAIRE CHANGE han he cap or floor, occurring a he firs rese dae. Thus he above Pay- Fixed Swap is no a plain vanilla ineres rae swap, bu insead can be valued a ime 0 as where τ 1 Plain Vanilla Pay-Fixed Swap 1 + e R 1τ 1 + Rα 1 e R 1τ 1, is he period 0 o 1, he firs rese dae, R 1 is he coninuously compounded rae applicable over he period 0 o 1, α 1 is he lengh of he period from 0 o 1, and hus equal o τ Pricing of Caps and Floors The price of a cap can be shown o be equivalen o a series of European pu opions on pure discoun bonds. This equivalence proves exremely useful in he pricing of caps under shor rae models. The payoff of a caple a T n is [LT n 1, T n K] + τ n which is equivalen o [LT n 1, T n K] + τ n P T n 1, T n a T n 1. Thus he caple payoff a T n 1 is: = [1 + LT n 1, T n τ n 1 + Kτ n ] + P T n 1, T n [1 + LTn 1, T n τ n ]P T n 1, T n = 1 + Kτ n P T n 1, T n 1 + Kτ n 1 + = 1 + Kτ n 1 + Kτ n P T n 1, T n. Thus, a caple wih payoff a T n is equivalen o 1 + Kτ n many pus on P, T n wih Srike 1 1+Kτ n and mauriy T n 1. Similarly a floorle wih payoff [K LT n 1, T n ] + τ n a T n is equivalen 1 o 1 + Kτ n many calls on P, T n wih Srike 1+Kτ and mauriy T n n 1. Finally, he value of a cap or floor can be found by summing up he prices of he respecive caples or floorles Pricing of Swaps A plain-vanilla swap can be valued by seperaely valuing he series of cashflows based on he fixed rae and hose based on he floaing rae. One of wo approaches may be used o value he swap.

24 3.8. PRICING OF SWAPTIONS 23 The firs approach follows he marke pracicaliy in ha he noional principals are no exchanged. As Wes 2008 explains, he value of he cash flows based on he fixed rae, R, where he swap consiss of n paymens, is relaively easy o deermine. i.e., where n V fix = R α i e R iτ i, i=1 R i is he coninuously compounded rae for he period zero o i, τ i is he period zero o i, α i is he lengh of he ih hree-monh period, from period i 1 o i. The value of he cash flows based on he floaing ineres rae is slighly more difficul o deermine. Wes s approach in his 2008 paper is o maniupae he cashflows ino an equivalen form in such a way ha he value of his equiavlen form is far eaiser o deermine. Hence, he value of he floaing-leg of he swap a is iniiaion is shown o be V floa = 1 e Rnτn. Since he fixed rae is se such ha hese values of he wo series of cashflows equae one anoher a incepion R = 1 e Rnτn n i=1 α ie R iτ i. 3.1 The second approach assumes ha he principals are exchanged a mauriy of he swap. Thus, he series of fixed rae cashflows can be valued as a fixed rae bond wih coupon size R. i.e., V fix = R n α i e R iτ i + e Rnτn. i=1 The series of floaing rae cashflows can be valued as a floaing rae bond wih coupon based on he floaing rae. The value of such a bond a is incepion or immediaely afer a coupon has been paid is equal o he principal amoun of he swap. i.e., V floa = Pricing of Swapions When valuing a swapion i is convenien o use he laer approach in he previous secion. Thus, one can view a swapion as an opion o exchange

25 24CHAPTER 3. NO-ARBITRAGE PRICING AND NUMERAIRE CHANGE a fixed rae bond for he principal amoun underlying he swap. Hence, a payer swapion can be regarded as a pu opion on a coupon paying bond wih he srike price equal o he principal amoun. A receiver swapion can be viewed as a call opion on a coupon bearing bond wih he srike price equal o he principal amoun One Facor Shor Rae Models If he shor rae is modelled using a one facor model hen, as Hull 2006 explains, one is able o express he price of a coupon paying bond as he sum of European opions on pure discoun bonds. This is a generalizaion of Jamshidian s 1989 decomposiion. i.e., where C K,τ, r = i Y i C K i,τ,t i, r, 3.2 C K,τ, r is he price a ime of a call opion on a coupon-paying bond, B, whereby he call opion expires a ime τ and has srike Kand he bond expires a ime T N, Y i is he size of he cashflow a ime T i, C K i,τ,t i, r is he price a ime of a call opion on a pure discoun bond, P, T i, whereby he call opion expires a ime τ and has srike K i and he pure discoun bond expires a ime T i. Hull 2006 describes he process in order o value he opion on he coupon paying bond using he above formula: 1. Calculae r, he value of he shor rae such ha he value of he coupon paying bond is equal o he srike price of he opion a he opion expiraion dae, τ. Under one facor shor rae models, he value of a pure discoun bond is a decreasing funcion of he shor rae, hus r is unique. 2. Calculae he price of each of he pure discoun bonds a he opion expiraion dae, τ, where he shor rae is r. 3. Se he srike price, K i, of each of he opions on he pure discoun bonds equal o he value of he pure discoun bonds derived in he previous sep. 4. Value each of he opions on he pure discoun bonds a ime, given he acual shor rae, r. 5. Use formula 3.2 o derive he value a ime of he opion on he coupon paying bond.

26 3.8. PRICING OF SWAPTIONS Muli-Facor Shor Rae Models Jamshidian s 1989 decomposiion for coupon paying bonds, and hence swapions, is no applicable for muli-facor models as he value of he pure discoun bond is no necessairly a decreasing funcion of he shor rae. Thus, one needs o use an alernaive approach o value swapions under such shor rae models, such as Mone Carlo simulaion.

27 Chaper 4 Pricing of Opions using Black s Model Marke convenion is o price bond opions, ineres rae caps and floors and swapions using he Black model. Black 1976 was he firs o show how one can value European fuures opions by exending he Black-Scholes model, which was firs published in Opions on Bonds If one assumes ha he underlying bond price P, T a he mauriy of he opion, ime T, is disribued lognormally, ha F B is he forward bond price wih volailiy σ B, and he srike price of he opion is K hen he call opion on he bond a ime 0 can be priced using Black s formula. i.e., C 0 = P 0, T [F B Nd 1 KNd 2 ]. Similarly, he formula o value a pu opion on he bond a ime 0 is where P 0 = P 0, T [K1 Nd 2 F B 1 Nd 1 ], 4.2 Caps and Floors d 1 = ln F B K σ2 B T σ B T, d 2 = d 1 σ B T. If one assumes ha he fuure underlying ineres rae r i is disribued lognormally wih volailiy σ i. Then he ineres rae opion can be priced using Black s formula. i.e., Cpl i 0 = Nτ i P 0, i+1 [f i 0; i, i+1 Nd 1 r K Nd 2 ], 26

28 4.3. SWAPTIONS 27 and where Flr i 0 = Nτ i P 0, i+1 [r K 1 Nd 2 f i 0; i, i+1 1 Nd 1 ], d 1 = ln f i0; i, i+1 r K σ2 i i, σ i i d 2 = d 1 σ i i, f i ; i, i+1 is he forward rae for he period i o i+1 a ime. 4.3 Swapions Hull 2006 shows ha if one assumes ha he underlying swap rae a he mauriy of he opion s T is disribued lognormally wih volailiy σ, hen one is able o price he swapion using Black s formula. In order o derive he price, consider a payer swapion where he holder has he righ o pay he fixed rae s K and receive JIBAR on a swap ha consiss of m paymens per year for n years and sars in T years wih noional N i.e., he fixed paymen is s kn m. A ime T, if he holder of he opion decides o exercise he opion, hen he holder can immediaely hedge away ineres rae risk by enering ino an equal bu opposie posiion by purchasing a par-floaing swap a he swap rae s T. Obviously he holder will only exercise his opion if he opion is of value. i.e., if s T is greaer han s K. Oherwise he holder of he opion will simply le he opion lapse. Thus, he holder will receive he following payoff m imes per year for n years N m max s T s K, 0. Since he swap rae s T is lognormally disribued wih volailiy σ and assuming ha he paymens occur a ime T i where i = 1, 2, 3,..., nm, hen he value of he paymen a ime zero for he cashflow a ime T i is where N m P 0, T i+1[s 0 Nd 1 s K Nd 2 ], d 1 = ln s 0 s K σ2 T σ T d 2 = d 1 σ T.,

29 28 CHAPTER 4. PRICING OF OPTIONS USING BLACK S MODEL Thus, he value of he payer swapion a ime 0 is where P S0 = nm i=1 N m P 0, T i+1[s 0 Nd 1 s K Nd 2 ] = NA m [s 0Nd 1 s K Nd 2, ] A = nm i=1 P 0, T i+1. Similarly, one can find he value of he receiver swapion a ime 0 which is RS0 = NA m [s K1 Nd 1 s 0 1 Nd 2 ]. 4.4 Consisency in Pricing Formulas Black s formula has been used o value an opion on a bond, a cap and a swapion. On each occassion a differen variable has been assumed o be lognormally disribued he opion on a bond assumes ha he fuure bond price is lognormally disribued, a cap assumes ha he fuure ineres rae is lognormally disribued whils he swapion assumes ha he fuure swap rae is lognormally disribued. Alhough each of hese formulas are consisen by hemselves, Hull 2006 poins ou ha hese formulas are no consisen wih one anoher as only one of he underlying variables can be assumed o be lognormally disribued. One can price he caple or floorle direcly using Black s formula if one assumes ha he ineres rae underlying each of he caples or floorles is lognormally disribued. Alernaively, since a caple floorle can be viewed as a number of pus calls on a pure discoun bond as was shown in he previous secion one can price he pure discoun bond and hence derive he price of he caple or floorle. Such an approach assumes ha he bond price is lognormally disribued. Since hese wo assumpions are no consisen wih one anoher, he values derived under he wo differen approaches will differ wih Wes 2009 noing ha he difference beween hese wo answers ypically differs a he 5h decimal place. 4.5 Suiabiliy of Black s Model for Pricing Ineres Rae Opions Black s model relies on wo approximaions, which Ouwehand 2008 poins ou are paricularly quesionable when pricing ineres rae opions. The

30 4.5. SUITABILITY OF BLACK S MODEL FOR PRICING INTEREST RATE OPTIONS29 key assumpion of Black s model is ha a marke variable, X, is disribued lognormally under he risk-neural measure. Throughou his secion, we will consider a call opion wih mauriy T and srike K. Thus, he value of such an opion a ime zero is [ C 0 = E Q e T 0 rd X T K +]. The firs approximaion is ha E Q [ e T 0 rd X T K +] = E Q [ e T 0 rd] E Q [ XT K +] = P 0, T E Q [ XT K +]. Such an approximaion is only rue if he ineres rae is independen of he marke variable. This is obviously no rue for ineres rae opions, where he marke variable is dependen on he ineres rae. The second approximaion is ha he forward price of X is a Q-maringale. i.e., F T = E Q X T = F 0 he forward price of X a ime 0. Under he risk-neural measure, he expeced value of X T is he fuures price. Since he ineres rae is sochasic, his is no equivalen o he forward price. Ouwehand 2008 goes on o show ha hese wo approximaions are valid for ineres rae insrumens such as bond opions, caps, floors and swapions, as long as cerain assumpions are made abou he disribuion of various ineres rae insrumens under he appropriae measures. Thus Black s model can be used o price ineres rae opions such as bond opions, caps and floors, and swapions.

31 Chaper 5 Marke Pracicaliies This secion aims o provide he reader wih an undersanding of how o apply he heory and conceps discussed in his paper hereby providing a linkage beween he heoreical and pracical worlds. 5.1 Shor Rae The shor rae is defined as he insanenous ineres rae a ha poin in ime however such a rae is purely a heoreical concep. In Souh Africa he shores rae ha is available is he overnigh rae. However, as Wes 2008 noes and Wes 2009 discusses in deail, using such a rae as a proxy for he shor rae is probably no advisable. Cuchiero 2007 agrees wih Wes, reasoning ha such a rae ypically is highly volaile and has a low correlaion wih oher yields. She saes ha he one-monh or hree-monh spo raes are beer proxies, wih one of he reasons for his being he liquidiy of hese raes. Chapman e al 1998 shows ha, from a US conex, using a hree-monh rae as a proxy for he shor rae does no inroduce economically significan biases for one facor affine shor rae models. Thus, he one-monh JIBAR will be used as a proxy for he shor rae whils he hree-monh JIBAR will be used as a proxy for he shor-erm mean-reversion level for he wo facor Vasicek model. 5.2 Pure Discoun Bonds In order o find he value of a pure discoun bond or he ineres rae over a specific period, one requires a yield curve. Since he pure discoun bond marke in Souh Africa is relaively small, wih approximaely R60bn in ousanding deb according o ASSA 2010, one canno derive hese yields solely from hese raded pure discoun bonds. Thus, one is required o 30

32 5.2. PURE DISCOUNT BONDS 31 boosrap he yield curve, a echnique which uses available daa and cerain rules o derive he yield curve Available Daa One is faced wih he choice of using swaps or governmen bonds in order o derive he yield curve hrough boosrapping. Alhough governmen bonds iniially appear he obvious choice, and are he insrumen used in oher counries such as he US, his is no he case in Souh Africa. The disadvanages of using governmen bonds in Souh Africa is he lack of liquidiy of all bu a few issues resuling in a lack of daa poins and a need for subjecive liquidiy adjusmens. The lack of liquidiy is exacerbaed by Basel I s recogniion for capial adequacy. Under Basel I, SA governmen bonds wih a mauriy of less han hree years were fully recognised, hence banks had a preference for shor-erm bonds resuling in very lile rade in bonds as soon as heir mauriies decreased below hree years. Alhough Basel II does no disinguish beween shor- and long-erm bonds for capial recogniion, major Souh African banks sill end o prefer shor-daed bonds, as highlighed by ASSA The disadvanage of using swaps is ha he credi risk is also priced ino he yield alhough wih plain-vanilla swaps, his credi risk is minimal since he noional is no acually swapped. However, according o Wes 2009, he sophisicaion and liquidiy of he swap marke in Souh Africa means ha swap raes are ypically used nowadays when consrucing he risk-free yield curve. PWC s 2008 Long-Term Insurance survey highlighed he fac ha, in Souh Africa, swap raes are preferred bu ha here is no definiive answer, wih 50% of insurers surveyed using swap raes, 33% bonds and 17% undecided. Thus, swap rae daa will be used. Two poins o noe: Firsly, he use of swaps or bonds is muually exclusive. This is paricularly relevan when boosrapping he real yield curve where few insrumens rade. The combined used of such insrumens would almos cerainly resul in one obaining a yield curve which presens arbirage opporuniies. Secondly, he use of he erm swap in his secions incorporaes he overnigh rae, one-monh and hree-monh JIBAR raes, FRA raes ou o wo years and swap raes wih mauriies ranging from wo years o 30 years Boosrap Mehod In order o fill-in he missing daa poins, one requires an inerpolaion mehod. A wide variey of inerpolaion mehods exis. However he mehod

33 32 CHAPTER 5. MARKET PRACTICALITIES used in his model is he Monoone Convex Inerpolaion Mehod, as derived by Hagan and Wes 2006 and Hagan and Wes This mehod has been specifically designed wih ineres rae inerpolaion a hand, hence i is he only mehod according o Hagan and Wes 2006 ha displays all of he following characerisics: Produces forward raes which are always posiive Negaive forward raes for a nominal yield curve would resul in arbirage. The inerpolaion mehod is local hence a change o a rae a one poin in he curve will no affec he enire curve s shape. Produces coninuous forward raes. Produces sable forward raes hus a change o a rae a one poin in he curve does no resul in a significan change in he forward curve. The hedges of financial insrumens are local hus when using his boosrap approach for risk managemen, dela risk is assigned o insrumens close o he given erm of he risky cashflow o be hedged. Wes 2008 describes he approach which one should ake in order o boosrap he yield curve once one has chosen a boosrap mehod using he available daa, including how one can overcome he problem of holes in he erm surcure. 5.3 Caps and Floors Cerain marke pracicaliies exis wih regard o caps and floors, some of which are unique o he Souh African marke. A generalized ineres rae opion n-year opion wih enor period of m-monhs - is used hroughou his secion o beer illusrae he poins. 1. The exac rese daes will occur m, 2m, 3m,..., 12n monhs from oday and will follow he Modified Following Rule. Modified Following Rule. In order o deermine he exac dae n monhs from oday one needs o apply he following wo rules, ermed he Modified Following Rule, and as described by Wes 2008: a The day has o be in he monh which is exacly n-monhs from he curren monh, b The day should be he firs business day on or afer he dae wih he same day number as he curren day. However, if his conradics he previous rule, he day is he las business day of he monh n-monhs from he curren monh.

34 5.4. INDIVIDUAL CAPLETS AND FLOORLETS The opions are seled in arears. e.g., The opion dependen on he ineres rae beween i and i+1 will only be seled a ime i The opions usually do no include proecion agains he iniial LI- BOR rae. i.e., The underlying ineres rae of he firs opion will cover monhs m o 2m ime 1 o 2 and a cap floor will consis of 12n m 1 caples floorles. 4. The opions usually quoed are A-The-Money opions and are quoed as a single volailiy value. 5. The srike rae of an A-The-Money opion is he rae which equaes he value of he cap and he floor. In order o find his rae, one can eiher ry o equae hese wo values or use pu-call pariy: Cap = Floor + Pay-Fixed Swap. Since his srike rae equaes he value of he cap wih he value of he floor, one is ulimaely solving for he rae which makes he swap s value equal o zero, where he swap s characerisics rese daes, underlying ineres rae, ec exacly mach hose of he cap and he floor i.e., he swap is no a ypical plain vanilla ineres rae swap. Thus, he formula o find he srike rae fixed-rae needs o be slighly adaped from ha shown in 3.1. Thus, if one wishes o deermine he srike rae, R K, for an A-The- Money cap wih a enor of hree-monhs and a erm of n-quarers i.e., 3n monhs hen, a ime zero where R K = e R 1τ 1 e Rnτn n i=2 α ie R, iτ i R i is he coninuously compounded rae for he period zero o i, τ i is he period zero o i, α i is he lengh of he ih hree-monh period, from period i 1 o i. 5.4 Individual Caples and Floorles Individual caples and floorles have slighly differen marke pracicaliies as compared o caps and floors. A generalized form of such an opion is a nxm opion, which has he following feaures 1. The sar dae of he opions is n-monhs from he curren dae whils he end-dae is n-m monhs from he sar dae. Boh of hese

35 34 CHAPTER 5. MARKET PRACTICALITIES daes follow he Modified Following Rule. This poin is especially imporan as n-m monhs from he sar dae which is m-monhs from he curren dae is no neccesarily equal o n-monhs from he curren dae, due o he Modified Following Rule. 2. The opion is seled in advance. e.g., The opion dependen on he ineres rae beween i and i+1 will be seled a ime i, as soon as he applicable ineres rae over he life of he opion is known. 3. Since his opion is seled in advance, he paymen is discouned. The payoff is hus: Cpl = Nτ imaxr i r K, r i τ i, and Flr = Nτ imaxr K r i, r i τ i 4. The opions usually quoed are A-The-Money opions and are quoed as a volailiy value. 5. The srike rae for an A-The-Money opion is simply he forward rae.

36 Chaper 6 Affine Term Srucure Models A shor rae model is said o possess affine erm srucure ATS if he price of a pure discoun bond is given by P, T = e A,T B,T r where A, T and B, T are deerminisic funcions. No all shor rae models are affine erm srucure models - he Black-Karasinski model is one such example which is no affine. However he fac ha an ATS model guarenees an explici formula for he price of a pure discoun bond is one of he reasons why his class of shor rae models has gained populariy. Assume a shor rae model wih he following risk neural dynamics: dr = µ, rd + σ, rdw. This class of shor rae models requires boh he drif and he volailiy squared o be affine funcions of he shor rae. i.e.,: µ, r = αr + β, σ 2, r = γr + δ, where α, β, γ and δ are deerminisic funcions of. One is able o obain a sysem of differenial equaions, allowing one o solve for A, T and B, T. i.e.,: dp, T P, T = A rb Bµ, r σ, r2 B 2 d σ, rbdw, where A represens A, T differeniaed wih respec o, and B represens B, T differeniaed wih respec o. 35

37 36 CHAPTER 6. AFFINE TERM STRUCTURE MODELS Bu under he risk neural dynamics, he drif of P, T is r hus A rb Bµ, r σ2, rb 2 = r A rb Bαr Bβ γrb δb2 = r A Bβ δb2 = r1 + B + Bα 1 2 γb2. The lefhand side of equaion 6.1 is independen of r whereas he righ hand side of he equaion conains r. This can only occur if boh sides are idenical o zero, allowing one o find he coupled sysem of differenial equaions: { A, T = B, T β 1 2δB, T 2 AT, T = 0 { B, T = αb, T γb, T 2 1 BT, T = 0 When aemping o solve his series of equaions, one should solve for B, T firs as his equaion does no conain A. The soluion for B, T can hen be plugged ino he equaion for A, T which is solved by inegraing boh sides beween and T. 6.1 Bond Opions Theorem If Ŝ = S P,T is an Iô process of he form dŝ Ŝ = µd + σdw, and if σ is deerminisic, hen he value of a European call C wih mauriy T and srike K on underlying S is given by where C 0 = S 0 Nd 1 KP 0, T Nd 2, σ av = 1 T σ T 2 d, 0 d 1 = ln S 0 KP 0,T σ2 avt σ av T, d 2 = d 1 σ av T. Pu-call pariy can be used o derive he price of he corresponding European pu opion P 0 = KP 0, T 1 Nd 2 S 0 1 Nd 1, wih d 1, d 2 and σ av defined as above. 6.1

38 6.1. BOND OPTIONS 37 Proof. dŝ Ŝ = µd + σdw, d lnŝ = 1 Ŝ dŝ 1 2Ŝ 2 σ2 Ŝ 2 d = µd + σdw 1 2 σ2 d = µ 1 2 σ2 d + σdw, lnŝ = lnŝ0 + µs σ2 s ds + [ Ŝ = Ŝ0 exp µs 1 2 σ2 s ds σsdw s, ] σsdw s. However, given he EMM Q T, Ŝ is a maringale under he Q T measure. Thus [ Ŝ = Ŝ0 exp 1 ] 2 σ2 sds + σsdŵs. 0 Thus lnŝ = X is normally disibued wih he following mean and variance [ ] 1 E Q T lnŝ = lnŝ0 0 2 σ2 sds := M, [ ] Var Q T lnŝ = σ 2 sds := V 2. Hence one can calculae he expeced payoff a ime T E Q T [ maxe X K, 0 ] = = = = 0 ln K ln K M V 1 2πV 2 1 2πV 2 ln K M V 0 x M2 e 2V 2 x M2 e 2V 2 maxe x K, 0dz e x Kdz 1 2πV 2 e 1 2 x2 K e M+V x dx 1 2πV 2 em+v x 1 2 x2 dx ln K M V = e M+ 1 2 V 2 K ln K M V ln K M V 1 2πV 2 e 1 2 x2 Kdx 1 2πV 2 e 1 2 V 2 +V x 1 2 x2 dx 1 2πV 2 e 1 2 x2 dx

39 38 CHAPTER 6. AFFINE TERM STRUCTURE MODELS = e M+ 1 2 V 2 K ln K M V = e M+ 1 2 V 2 K ln K M V ln K M V 1 2πV 2 e 1 2 x V 2 dx 1 2πV 2 e 1 2 x2 dx ln K M V 2 V 1 x2 e 2 dx 2πV 2 1 2πV 2 e 1 2 x2 dx = e M+ 1 2 V 2 M ln K + V 2 M ln K N KN V V ln Ŝ 0 = Ŝ0N K σ2 ln Ŝ 0 K 0 σ KN σ Thus he expeced value of he Call a ime zero is where d 1 = ln S 0 = S 0 Nd 1 KP 0, T Nd 2, KP 0,T σ2 0 σ Applicaion o Bond Opions, d 2 = d σ2 σ 2. Consider a European call P on a pure discoun bond P, S wih srike K and maruiy T where S > T. For ATS models, wih risk-neural dynamics dr = µ, rd + σ, rdw, i has been shown ha he bond prices have he following dynamics under he risk-neural measure: Le ˆP = P,S P,T. Thus ˆ dp ˆP = dp, S P, S dp, T P, T = rd σ, rb, T dw. dp, T P, T + dp, T 2 P, T dp, S P, S dp, T P, T = σ 2, rb, T [B, T B, S]d + σ, rb, T B, SdW.. Thus P 0 = KP 0, T 1 Nd 2 P 0, S1 Nd 1,

40 6.1. BOND OPTIONS 39 where σ av = 1 T σ T 2, rb, T B, S 2 d, 0 d 1 = ln S 0 KP 0,T σ2 avt σ av T, d 2 = d 1 σ av T.

41 Chaper 7 Vasicek Model Vasicek 1977 models he insanenous spo rae as a Ornsein-Uhlenbeck process: where k, φ, σ, r 0 are posiive consans. Inegraing equaion 7.1, one can solve for r : dr = kφ r d + σdw, 7.1 dr e k = kφe k d r ke k d + σe k dw, dr e k = kφe k + σe k dw, r e k = r s e ks + kφ s e kτ dτ + σ r e k = r s e ks + φe k e ks + σ s r = r s e k s + φ1 e k s + σ s e kτ dw τ, e kτ dw τ, s e k τ dw τ. 7.2 Thus r, condiional on F s, is disribued normally wih he following momens: Er F s = r s e k s + φ1 e k s, Varr F s = E [r Er F s 2] ] = E [σ 2 e 2k τ dτ = σ2 2k [1 e 2k s ]. 40 s

42 7.1. PURE DISCOUNT BOND 41 The long-erm mean and variance can be found by aking he limi as ends owards infiniy: lim Er F s = φ, lim Varr F s = σ2 2k. One can hus clearly see several key aribues of he Vasicek model: 1. The shor rae experiences mean reversion, and over he long-erm will end owards φ. 2. There is no cerainy ha he shor rae will always be posiive. In realiy, nominal ineres raes can never be negaive, however he Vasicek model can no ensure ha such a siuaion does no occur - a major drawback of he model. 7.1 Pure Discoun Bond dr = kφ kr d + σdw hus, in erms of he parameers for he Generalized Affine Term Srucure Model, α = k, β = kφ, γ = 0 and δ = σ 2, wih all parameers consan. The sysem of differenial equaions ha mus be solved is herefore: { A = kφb 1 2 σ2 B 2 AT, T = 0 { B = kb 1 BT, T = 0 B, T is a Riccai equaion. However, since he coefficien of B 2 is 0, he Riccai equaion simplifies o become a linear differenial equaion, which can be solved as follows: 1 k db kb 1 = d, lnkb 1 = + C, kb 1 = C e k, B, T = 1 k + C e k,

43 42 CHAPTER 7. VASICEK MODEL bu BT, T mus mee he boundary condiion. i.e., BT, T = 0, 1 k + C e kt = 0, C = 1 k e kt, B, T = 1 k 1 e kt. 7.3 Subsiuing B, T ino A, one can solve for A,T: A = φ1 e kt σ2 2k 2 1 e kt 2, A, T = AT, T = T T A s ds [φ1 e kt s σ2 2k 2 1 e kt s 2 ]ds = φt + φ k 1 e kt + σ2 σ2 T 2k2 k 3 1 e kt + σ 2 4k 3 1 e 2kT = φt + φb + σ2 σ2 T + 2k2 4k e kt + 1 e 2kT = φt + φb + σ2 σ2 T + 2k2 4k [ 1 k 2 1 2e kt + e 2kT + 21 e kt ] = φt + φb + σ2 2k 2 T σ2 B 2 4k Bσ2 2k 2 = σ2 B 2 4k + B T k2 φ σ2 2 k Thus, he price of a pure discoun bond, P, T = e A,T B,T r, has been found for he Vasicek model, wih he values for A, T and B, T represened by 7.4 and 7.3 respecively. Since he Vasicek model consiss of only hree parameers and hese parameers are ime independen, when fiing his model o he observed erm srucure, he model is over-deermined and can only fi exacly o hree bonds. 7.2 Cap and Floors In order o price a caple, one mus firs find he value of a European pu opion on a pure discoun bond and hen use he relaionship beween he

44 7.2. CAP AND FLOORS 43 pu opion and he caple. Since σ is consan and B, T = 1 k 1 e kt, σ 2 avt = T And he price of he pu opion is where 0 σ 2 k 2 e ks e kt 2 e 2k d = σ2 2k 3 e ks e kt 2 e 2kT 1, P 0 = KP 0, T 1 Nd 2 P 0, S1 Nd 1, d 1 = P 0,S ln KP 0,T σ2 avt σ av T, d 2 = d 1 σ av T. Example 1. If he shor rae is assumed o possess he following risk-neural dynamics dr = kφ r d + σdw wih k = 0.1, φ = 0.05 and σ = 0.1 and a caple has he payoff 0.25L a expiry in 1 year, where L is he hree-monh spo LIBOR rae in nine monhs ime, r 0 is 5% and P 0, T is a pure discoun bond wih face value 1 hen K = K Caple τ = = , σ av = = σ 2 2k 3 e ks e kt 2 e 2kT 1 T e 0.11 e e = , 0.75 B0, T = 1 k 1 e kt = e =

45 44 CHAPTER 7. VASICEK MODEL A0, T = σ2 B 2 4k + B T k2 φ σ2 2 k 2 = = P 0, T = e A0,T B0,T r 0 = e = B0, S = 1 k 1 e ks = e 0.11 = A0, S = σ2 B 2 4k + B Sk2 φ σ2 k 2 = = P 0, S = e A0,S B0,Sr 0 2 = e = P 0,S ln KP 0,T σ2 avt d 1 = σ av T = d 2 = d 1 σ av T = P 0 = KP 0, T 1 Nd 2 P 0, S1 Nd 1 = N N = ,

46 7.2. CAP AND FLOORS 45 and hus he value of he caple is Cpl = 1 + K Caple τp 0 = =

47 Chaper 8 Ho-Lee Model Ho and Lee 1986 proposed he firs no-arbirage model of he erm srucure. The model was presened in he form of a binomial ree of bond prices and consised of wo parameers - he sandard deviaion of he shor rae and he marke price of risk of he shor rae. I has since been shown ha he coninuos ime limi of he Ho-Lee model is: dr = φd + σdw, 8.1 where σ, r 0 are posiive consans and φ is a deerminisic funcion. Inegraing equaion 8.1, one can solve for r : r = r Pure Discoun Bond 0 φsds + σw. dr = φd + σdw hus, in erms of he parameers for he Generalized Affine Term Srucure Model, α = 0, β = φ, γ = 0 and δ = σ 2, wih σ consan and φ deerminisic. The sysem of differenial equaions ha mus be solved is herefore: { A = φb 1 2 σ2 B 2 AT, T = 0 { B = 1 BT, T = 0 46

48 8.1. PURE DISCOUNT BOND 47 One can easily solve for B,T: B + 1 = 0, B + + C = 0, bu BT, T = C + T = 0, C = T, B, T = T. 8.2 Subsiuing B, T ino A, one can solve for A, T : A = φt σ2 T 2, 2 A, T = AT, T = = T T T A s ds [φst s]ds + T [ σ 2 T s 2 ] ds 2 [φst s]ds + σ2 T Thus, he price of a pure discoun bond, P, T = e A,T B,T r, has been found for he Ho-Lee model, wih he values for A, T and B, T represened by 8.3 and 8.2 respecively. Since φ is a funcion of ime and no a consan, one is able o choose φ such ha he model fis he iniial erm srucure. ln P 0, T = A0, T + B0, T r bu = T f, T = 0 [φst s]ds σ2 T 3 + T r 0, 6 P, T, T T f0, T = φt T T + φsds σ2 T 2 + r T = φsds σ2 T 2 + r 0, 2 f T 0, T = φt σ 2 T, φt = f T 0, T + σ 2 T,

49 48 CHAPTER 8. HO-LEE MODEL i.e.,: φ = f T 0, + σ 2, where F T 0, = f0, T T T =, 8.4 and f 0, is he insananeous forward price for a mauriy observed a ime zero. Subsiuing 8.4 ino 8.3, one can see ha A, T is a funcion of: he insananeous forward rae as seen a ime zero; observed bond prices a ime zero; volailiy of he shor rae; curren ime ; and ime o mauriy. i.e.,: A, T = = T T [φst s]ds + σ2 T 3 [f T 0, st s]ds T 6 T [σ 2 st s]ds + σ2 T 3 = [f 0, st s] T f 0, sds σ2 T 2 2 T [ = f 0, T P ] 0, s ds σ2 T 2 T 2 = f 0, T + ln P 0, T ln P 0, σ2 T 2, 2 where P 0, is he pure discoun bond price wih mauriy observed a ime zero. Thus P, T = P 0, T P 0, ef σ 0,T 2 T 2 T r Shor Rae Dynamics Revisied r = r φsds + σw = r 0 + f 0, + σ2 2 2 r 0 + σw = f 0, + σ σw.

50 8.3. CAP AND FLOORS 49 Thus r, condiional on F s, is disribued normally wih he following momens: Er F s = f 0, + σ2 2 2, Varr F s = σ 2. The long-erm mean and variance can be found by aking he limi as ends owards infiniy: lim F s, lim F s. One can hus clearly see several key aribues of he Ho-Lee model: 1. The shor rae does no experience mean reversion, and over he longerm will end owards under he risk-neural measure. Such a feaure is highly unrealisic. 2. There is no cerainy ha he shor rae will always be posiive. In realiy, nominal ineres raes can never be negaive, however he Ho- Lee model can no ensure ha such a siuaion does no occur. This is a major drawback of he model. 8.3 Cap and Floors In order o price a caple, one mus firs find he value of a European pu opion on a pure discoun bond and hen use he relaionship beween he pu opion and he caple. Since σ is consan and B, T = T, σ 2 avt = And he price of he pu opion is where T 0 σ 2 T S 2 d = σ 2 T S 2 T. P 0 = KP 0, T 1 Nd 2 P 0, S1 Nd 1, d 1 = ln P 0,S KP 0,T σ2 S T 2 T σs T T, d 2 = d 1 σs T T.

51 50 CHAPTER 8. HO-LEE MODEL Example 1. If he shor rae is assumed o possess he following risk-neural dynamics dr = φd + σdw, wih φ = 0.01 and σ = 0.1 and a caple has he payoff 0.25L a expiry in 1 year, where L is he hree-monh spo LIBOR rae in nine monhs ime, r 0 is 5% and P 0, T is a pure discoun bond wih face value 1 hen K = K Caple τ = = , σ av = σs T = = 0.025, B0, T = T = 0.75 A0, T = φt σ2 T 3 6 = = P 0, T = e A0,T B0,T r 0 = B0, S = S = 1 A0, S = φs2 2 + σ2 S 3 6 = =

52 8.3. CAP AND FLOORS 51 P 0, S = e A0,S B0,Sr 0 = P 0 = KP 0, T 1 Nd 2 P 0, S1 Nd 1 = N N = , and hus he value of he caple is Cpl = 1 + K Caple τp 0 = =

53 Chaper 9 Cox-Ingersoll-Ross Model One of he major disadvanages wih he Vasicek model is ha he model allows he shor rae o be negaive. Cox, Ingersoll and Ross proposed an alernaive model in 1985 whereby he shor rae is always posiive: where k, φ, σ, r 0 are posiive consans. dr = kφ r d + σ r dw, 9.1 The sandard deviaion of he change in he shor rae over an insananeous period is proporional o he square roo of he shor rae. Thus, as he shor rae decreases owards zero, is sandard deviaion decreases. Inegraing equaion 9.1, one can solve for r : dr e k = kφe k d r ke k d + e k σ r dw, dr e k = kφe k + e k σ r dw, r e k = r s e ks + kφ s e kτ dτ + σ r e k = r s e ks + φe k e ks + σ s r = r s e k s + φ1 e k s + σ s e kτ r τ dw τ, e kτ r τ dw τ, s e k τ r τ dw τ.9.2 Alhough i is sraighforward o compue he expecaion of he shor rae, he variance requires some manipulaion: Er F s = r s e k s + φ1 e k s, 52

54 53 Le X = e k r and subsiue his ino 9.2: dx = kφe k + e k σ r dw = kφe k + e k 2 σ X dw, dx 2 = 2X dx + dx 2 = 2X kφe k + 2X 3 2 X 2 = X 2 s + 2kφ + σ 2 EX 2 F s = X 2 s + 2kφ + σ 2 e k 2 σdw + e k σ 2 X d, s = r 2 se 2ks + 2kφ + σ 2 = rse 2 2ks + 2kφ + σ2 k e kτ X τ dτ + 2σ e kτ EX τ dτ s s s e kτ 2 X 3 2 τ dw τ, e kτ r s e ks + φe kτ e ks dτ r s φe k+s e 2ks + 2kφ + σ2 φe 2k e 2ks, 2k Er 2 F s = rse 2 2k s + 2kφ + σ2 r s φe k s e 2k s k + 2kφ + σ2 φ1 e 2k s, 2k Varr F s = rse 2 2k s + 2kφ + σ2 r s φe k s e 2k s k + 2kφ + σ2 φ1 e 2k s r s e k s + φ1 e k s 2 2k = rse 2 2k s + 2kφ + σ2 r s φe k s e 2k s k + 2kφ + σ2 φ1 e 2k s r 2 2k se 2k s 2r s φe k s e 2k s φ 2 1 e 2k s 2 = r sσ 2 k e k s e 2k s + φσ2 2k 1 2e k s + e 2k s. The long-erm mean and variance can be found by aking he limi as ends owards infiniy: lim Er F s = φ, lim Varr F s = φσ2 2k. Similar o he Vaiscek model, he shor rae experiences mean reversion, and over he long-erm will end owards φ. The Cox-Ingersoll-Ross model however does no follow he normal disribuion. Svoboda 2002 shows r, condiional on F s, insead follows a non-cenral chi-quared disribuion. Svoboda 2002 also shows ha, provided he iniial shor rae is posiive

55 54 CHAPTER 9. COX-INGERSOLL-ROSS MODEL and 2kφ σ 2, hen he shor rae is assured of always remaining posiive - a highly preferable feaure of any shor rae model. 9.1 Pure Discoun Bond dr = kφ kr d + σ r dw hus, in erms of he parameers for he Generalized Affine Term Srucure Model, α = k, β = kφ, γ = σ 2 and δ = 0, wih all parameers consan. The sysem of differenial equaions ha mus be solved is herefore: { A = kφb AT, T = 0 { B = kb σ2 B 2 1 BT, T = 0 Ouwehand 2008 shows ha, in order o solve he Riccai equaion for B, T, one can ry a soluion of he form: B, T = B = X cx + d X cx + d cxx cx + d 2. Subsiuing back ino he equaion for B, T : X cx + d cxx cx + d 2 = kx cx + d + 1 X 2 2 σ2 1 cx + d X cx + d cxx = kxcx + d σ2 X 2 cx + d 2 0 = dx + X 2 kc σ2 c 2 + Xkd 2cd d 2. If one chooses c such ha kc+ 1 2 σ2 c 2 = 0 i.e., c = k+ k 2 +2σ 2 2 or k k 2 2σ 2 2 and ses κ = 2c k = k 2 + 2σ 2 hen he differenial equaion reduces o he following: X κx + d 1 κ = 1, lnκx + d = + C, κx + d = C e κ, X = d κ + C e κ,

56 9.2. CAP AND FLOORS 55 bu BT, T = 0, allowing one o solve for C XT = d κ + C e κt = 0, 9.3 C = d κ eκt. 9.4 Subsiuing 9.4 back ino 9.3 allows one o solve for X and hus B, T X = d κ B, T = = e κt 1, d κ eκt 1 cd κ eκt 1 + d e κt 1 ce κt 1 + κ e κt 1 = 1 2 κ + keκt 1 + κ 2e κt 1 = κ + ke κt 1 + 2κ. 9.5 Subsiuing B, T ino A, one can solve for A, T : A = kφb, A, T = AT, T = T T A s ds kφbs, T ds. Svoboda 2002 solves for A, T o find: [ ] A, T = 2kφ σ 2 ln 2κe 1 k+κt 2 2κ + k + κe κt Thus, he price of a pure discoun bond, P, T = e A,T B,T r, has been found for he Cox-Ingersoll-Ross model, wih he values for A, T and B, T represened by 9.6 and 9.5 respecively. 9.2 Cap and Floors In order o price a caple, one mus firs find he value of a European pu opion on a pure discoun bond and hen use he relaionship beween he pu opion and he caple. Since he sandard deviaion of he change in he shor rae is no independen of he shor rae, Theorem can no be used o find he value of his opion.

57 56 CHAPTER 9. COX-INGERSOLL-ROSS MODEL Cox, Ingersoll and Ross 1985 show ha he value of a call opion on a pure discoun bond is: C = P, Sχ 2 2 r[ρ + ψ + BT, S]; 4kφ σ 2 ; KP, T χ 2 2ρ 2 r e κt ρ + ψ + BT, S 2 r[ρ + ψ]; 4kφ σ 2 ; 2ρ2 r e κt ρ + ψ, where AT,S K ln r = BT, S, ψ = k + κ σ 2, ρ = 2κ σ 2 e κt 1, and χ 2.; υ; λ is he non-cenral chi-squared disribuion wih υ degrees of freedom and non-cenral parameer λ. The value of he corresponding pu opion can be found hrough Pu-Call Pariy Example 1. If he shor rae is assumed o possess he following risk-neural dynamics dr = kφ r d + σ r dw, wih k = 0.1, φ = 0.05 and σ = 0.1 and a caple has he payoff 0.25L a expiry in 1 year, where L is he hree-monh spo LIBOR rae in nine monhs ime, r 0 is 5% and P 0, T is a pure discoun bond wih face value 1 hen C 0 = P 0, Sχ 2 2 r[ρ + ψ + BT, S]; 4kφ σ 2 ; 2ρ 2 r 0 e κt ρ + ψ + BT, S KP 0, T χ 2 2 r[ρ + ψ]; 4kφ σ 2 ; 2ρ2 r 0 e κt ρ + ψ

58 9.2. CAP AND FLOORS 57 = χ [ ]; e ; χ [ ]; ; e = , P 0 = C 0 P 0, S + KP 0, T = = , and hus he value of he caple is Cpl = 1 + K Caple τp 0 = =

59 Chaper 10 Black-Karasinski Model Black and Karasinski 1991 proposed a shor rae model ha guareneed posiive ineres raes by assuming ha he insananeous change in he shor rae is he exponenial of an Ornsein-Uhlenbeck process. i.e.,: d lnr = [φ a lnr ]d + σdw, where r 0 is a posiive consan and φ, a, σ are deerminisic funcions. For he remainder of his chaper, we will assume ha a = a, σ = σ i.e., hese wo parameers are consan. Thus d lnr = [φ a lnr ]d + σdw Inegraing equaion 10.1, one can solve for r : dlnr e a = φe a d lnr ae a d + σe a dw, dlnr e a = φe a + σe a dw, lnr e a = lnr s e as + lnr = lnr s e a s + s φτe aτ dτ + σ s s e aτ dw τ, φτe a τ dτ + σ r = e lnrse a s + s φτe a τ dτ+σ s e a τ dw τ. s e a τ dw τ, 10.2 Le lnr = X. Thus X, condiional on F s, is disribued normally wih he following momens: EX F s = lnr s e a s + VarX F s = σ2 [ 1 e 2a s] := V 2. 2a 58 s φτe a τ dτ := M,

60 10.1. SIMPLIFYING THE SHORT RATE EQUATION 59 If Y = e X, he mean and variance of Y can be found as follows: EY = Ee X = = = e M e x 1 e x M2 2V 2 dx 2πV 2 e M+z 1 2πV 2 e z2 2V 2 dz = e M = e M+ V 2 2, EY 2 = Ee 2X = = = e 2M 1 e 2zV 2 z 2 2V 2 dz 2πV 2 1 e z V 2 2 2V 2 + V 2 2 dz 2πV 2 e 2x 1 e x M2 2V 2 dx 2πV 2 e 2M+z 1 e z2 2πV 2 = e 2M = e 2M+2V 2, VarY = EY 2 EY 2 = e 2M+V 2 e V V 2 dz 1 e 4zV 2 z 2 2V 2 dz 2πV 2 1 e z 2V 2 2 2V 2 2πV 2 +2V 2 dz Since r = Y, r, condiional on F s, is disribued lognormally wih he following momens: Er F s = e lnrse a s + s φτe a τ dτ+ σ2 4a [1 e 2a s], Varr F s = e 2 lnrse a s +2 s φτe a τ dτ+ σ2 2a [1 e 2a s ] e σ2 2a [1 e 2a s] Simplifying he Shor Rae Equaion One is able o break-down equaion 10.2 ino a deerminisic funcion and a sochasic equaion. Define r and α as follows: r = ln r α, 10.3 α = lnr 0 e a + 0 φτe a τ dτ. 10.4

61 60 CHAPTER 10. BLACK-KARASINSKI MODEL Thus r can be found as follows: r = lnr α = lnr s e a s + = αse a s + r se a s + = lnr 0 e a + = lnr 0 e a + s 0 s +r se a s + σ = r se a s + σ 0 s φτe a τ dτ + σ φτe a τ dτ + s s s φτe a τ dτ + σ s e a τ dw τ φτe a τ dτ + r se a s + σ e a τ dw τ α φτe a τ dτ α s s e a τ dw τ α e a τ dw τ α e a τ dw τ One can easily o show ha dr = ar d + σdw is a soluion o equaion i.e., Thus where dr e a = ae a r + σe a dw dr e a = σe a dw r e a = r se as + σ r = r se a s + σ s e aτ dw τ ln r = r + α, α = lnr 0 e a + 0 s e a τ dw τ, 10.6 dr = ar d + σdw. φτe a τ dτ, 10.2 Momens of he Shor Rae Revisied Er F s = e lnrse a s + s φτe a τ dτ+ σ2 lim Er F s = exp σ2 lim α + 4a. 4a [1 e 2a s],

62 10.3. PURE DISCOUNT BOND Pure Discoun Bond In conras o he shor rae models previously reviewed, he Black-Karasinski model does no provide an analyically racable soluion for he bond price. Thus, one is required o use mehods such as consrucing a ree or Mone- Carlo simulaion in order o price pure discoun bonds. A ree is a discree-ime represenaion of he sochasic process whereby one assumes ha he ineres rae for each discree ime-period has he same dynamics as ha of he shor rae. As Hull 2006 poins ou, one usually consrucs a rinomial ree, as opposed o a binomial ree, due o he exra degree of freedom. When consrucing a ypical rinomial ree, he ineres rae is allowed o eiher increase, decrease or say consan over each discree ime-period Figure 10.1a. However, as discussed previously, ineres raes have been shown o experience mean-reversion. Thus, Hull e al 1993 and Hull e al 1994a proposed he use of a non-sandard branching ree. Under such a mehod, if he ineres rae is excepionally low hen he rae can eiher say consan, increase or increase significanly Figure 10.1b. Similarly, if he ineres rae is significanly high hen he rae can eiher say consan, decrease or decrease significanly Figure 10.1c. i.e., The non-sandard branching approach incorporaes a form of mean-reversion. For he remainder of his chaper, we will use a non-sandard branching ree. Figure 10.1: Non-Sandard Branching Trinomial Tree When consrucing he ree, one needs o fix he ime horizon T and se he discree ime period i = i+1 i. Noe ha his ime period is ime dependen hus i need no be consan an imporan feaure when confroned wih he pracicaliy of day couns.

63 62 CHAPTER 10. BLACK-KARASINSKI MODEL In erms of noaion, i is used o denoe he ime period i.e., i = 0, 1, 2,..., N whils j informs he heigh of he node. Figure 10.2: Black-Karasinski: Tree Noaion As explained by Brigo and Mercurio 2006 and Hull 2006, he consrucion of he rinomial ree consiss of a wo-sep process. Firsly, one needs o consruc a ree for a process r. The risk-neural dynamics of he coninuous version of his process are: where dr = ar d + σdw, r = ln r α, where r0 = 0. The second sep involves displacing r such ha pure discoun bonds priced hrough he rinomial ree exacly mach he marke prices of hese bonds. α, as defined in 10.4, is used o displace he pure discoun bond prices, wih r deermined using equaion i.e., r = e α+r. Under he firs sep, one requires boh he expeced change in r and variance of he change in r : Er i+1 r i = r i,j = r i,je a i =: M i,j, Varri+1 r i = ri,j = σ2 1 e 2a i =: V 2 2a i,j. ri is he space beween ineres raes a a discree ime period i wih ri,j = j r i. Brigo and Mercurio 2006 recommends one se r i = V i 1 3

64 10.3. PURE DISCOUNT BOND 63 wih Hull 2006 noing ha such a choice proves o be good in erms of error minimizaion. A each discree ime period, r i,j can proceed o r i+1,k+1, r i+1,k or r i+1,k 1 wih probabiliies p u, p m and p d respecively. Since he middle node is r i+1,k, one chooses k such ha his node is as close as possible o M i,j, as defined in Thus, k is se as Mi,j k = round ri+1. The probabiliies for he possible pahs a each poin in ime are chosen o mach he expeced change in r and variance of he change in r over he nex ime period i. Since hese probabiliies mus sum o uniy, one is able o solve he hree simulaaneous equaions. One should noe ha, due o he probabiliies being chosen according o he values of M i,j and V i,j, hese probabiliies will be dependen on he form of non-sandard branching. where p u = η2 j,k p m = 2 3 η2 j,k + η j,k 6Vi 2 2 3V i 3Vi 2 p u = η2 j,k 6Vi 2 η j,k = M i,j x i+1,k. η j,k 2 3V i As Brigo and Mercurio 2006 poin ou, hese probabiliies are guareneed o be non-negaive. Thus, one has a fully deermined rinomial ree for r, including he minimum and maximum values for he heigh of he nodes j min i and j max i respecively. The second sep of he process involes shifing he nodes of he ree a each discree ime period so ha he heoreical pure discoun bond prices mach he observed pure discoun bond prices, hrough he use of α. Le Q i,j denoe he presen value of he opion which pays R1 if, and only if, he node i, j is reached and zero oherwise. Le P 0, be he price observed in he marke of a pure discoun bond which pays R1 a ime. qh, j is defined as he probabiliy of moving from node i, h o node

65 64 CHAPTER 10. BLACK-KARASINSKI MODEL i + 1, j. P 0, 1 = e r 0,j 0 = e eα0+r 0 0 = e eα0 0, lnp 0, 1 α0 = ln. 0 A recursive relaionship beween Q i+1,j and αi follows. One can use he following generalized formula o solve for Q i,j : Q i+1,j = h Q i,h qh, je eαi+h r i i. And hen use he derived values for Q i+1,j o solve for αi where αi is he value which ses he following equaion equal o zero: p 0, i+1 j maxi j=j min i Q i,j e eαi+j r i i. One can use he Newon-Raphson mehod o solve for such a value. In such an insance, one requires he derivaive of he funcion. i.e., j maxi j=j min i Q i,j e eαi+j r i i e αi+j r i i. Once one has solved for αi for all discree ime periods, one can use he values derived in sep one for r i,j o solve for r i,j hrough he use of he discree form of equaion 10.5: 10.4 Cap and Floors r i,j = e α+r i,j. In order o price a caple, one should consruc he rinomial ree as described previously. From he ree one can deermine he payoff for each pah and muliply his payoff by he probabiliy of aking ha pah. The value of he caple is hen he sum across all possible pahways of he presen value of his number. Example 1. If he shor rae is assumed o possess he following risk-neural dynamics d lnr = [φ a lnr ]d + σdw,

66 10.4. CAP AND FLOORS 65 wih a = 0.1, σ = 0.1 and α = where α is defined as: α = lnr 0 e a + 0 φτe a τ dτ, and a caple has he payoff 0.25L a expiry in 1 year, where L is he hree-monh spo LIBOR rae in nine monhs ime, r 0 is 5%, i = one monh and P 0, T is a pure discoun bond wih face value 1 hen Cpl =

67 Chaper 11 Two Facor Vasicek Model An exension of he Vasicek model is he wo facor Vasicek model, where he second sochasic facor allows he mean-reversion level for he shor rae o follow a mean-revering sochasic process, wih he aim for his model of beer fiing empirical daa, and more realisically modelling he dynamics of shor raes. Hibber, Mowbray and Turnbull 2001 discuss he fac ha his model can also be viewed as a special case of he wo facor Hull-Whie model, firs described in heir 1994 paper. The risk-neural dynamics of he shor rae under his model are: dr 1, = k 1 r 2, r 1, d + σ 1 dw 1,, 11.1 dr 2, = k 2 φ r 2, d + σ 2 dw 2,, 11.2 where r 1 is he shor rae a ime, r 2 is he mean-reversion level of he shor rae a ime, k 1, k 2, φ, σ 1, σ 2, r 1,0, r 2,0 are posiive consans and W 1, and W 2, are Brownian moions which are independen of one anoher. Inegraing equaion 11.2, one can solve for r 2,, as shown in chaper 7, Hence r 2, = r 2,s e k2 s + φ1 e k2 s + σ 2 e k2 τ dw 2,τ.11.3 In order o solve for he shor rae, one needs o subsiue equaion 11.3 in equaion 11.1 and inegrae he equaion: dr 1, = k 1 r 2, r 1, d + σ 1 dw 1, = k 1 [r 2,s e k2 s + φ1 e k2 s + σ 2 e k2 τ dw 2,τ r 1, ]d + σ 1 dw 1, = k 1 r 2,s e k2 s d + k 1 φ1 e k2 s d + k 1 σ 2 e k2 τ dw 2,τ d k 1 r 1, d + σ 1 dw 1,, 66 s s s

68 67 e k1 dr 1, = k 1 r 2,s e k 2 s+k 1 d + k 1 φe k1 k 1 φe k 2 s+k 1 d +k 1 σ 2 e k 1 e k2 τ dw 2,τ d k 1 e k1 r 1, d + σ 1 e k1 dw 1,, de k1 r 1, = k 1 r 2,s e k 2 s+k 1 d + k 1 φe k1 d k 1 φe k 2 s+k 1 d +k 1 σ 2 e k 1 e k2 τ dw 2,τ d + σ 1 e k1 dw 1,, e k 1 r 1, = e k 1s r 1,s + k 1 r 2,s e k 2s +k 1 σ 2 s s s s e k 2 k 1 τ e k 2 k 1 τ dτ + k 1 φ e k1τ dτ k 1 φe k 2s e k 2 k 1 τ dτ s s τ e k2m dw 2,m dτ + σ 1 e k1τ dw 1,τ, s r 1, = r 1,s e k1 s + k 1r 2,s e k 2s k 1 e k 2 k 1 e k 2 k 1 s + φ1 e k1 s k 1 k 2 k 1φe k 2s k 1 e k 2 k 1 e k 2 k 1 s k 1 k 2 +k 1 σ 2 e k 1 e k 2 k 1 τ s s τ e k2m dw 2,m dτ + σ 1 e k1 τ dw 1,τ s = r 1,s e k1 s + k 1 r 2,s φ e k2 s e k 1 s + φ1 e k1 s k 1 k 2 m= τ= +k 1 σ 2 e k 1 e k 2m e k 2 k 1 τ dτ dw 2,m + σ 1 e k1 τ dw 1,τ m=s τ=m = r 1,s e k1 s + k 1 r 2,s φ e k2 s e k 1 s + φ1 e k1 s k 1 k 2 +k 1 σ 2 e k 1 e k 2m e k 2 k 1 e k 2 k 1 m dw 2,m + σ 1 e k1 τ dw 1,τ s k 1 k 2 s = r 1,s e k1 s + k 1 r 2,s φ e k2 s e k 1 s + φ1 e k1 s k 1 k 2 e k 2 +k 1 σ 2 e k2m e k 1 dw 2,m k 1 σ 2 e k1m dw 2,m k 1 k 2 s k 1 k 2 s +σ 1 e k1 τ dw 1,τ s s s Thus he shor-erm mean-reversion level, r 2,, condiional on F s, is disribued normally wih he following momens: Er 2, F s = r 2,s e k 2 s + φ1 e k 2 s, Varr 2, F s = σ2 2 2k 2 [1 e 2k 2 s ].

69 68 CHAPTER 11. TWO FACTOR VASICEK MODEL The shor rae, r 1,, condiional on F s, is disribued normally wih he following momens: Er 1, F s = r 1,s e k1 s + k 1 r 2,s φ e k2 s e k 1 s + φ1 e k1 s, k 1 k 2 Varr 1, F s = E[r 1, F s Er 1, F s ] 2 k 1 σ 2 2 = k 1 k 2 2 e 2k 2 e 2k2m dm 2e k 1+k 2 e k 1+k 2 m dm s s = +e 2k 1 s e 2k 1m dm + σ 2 1 s e 2k 1 τ dw τ k 1 σ e 2k2 s 2 k 1 k e k 1+k 2 s 2k 2 k 1 + k e 2k1 s + σ2 1 1 e 2k 1 s. 2k 1 2k 1 The long-erm mean and variance of boh he shor rae and he shor-erm mean reversion level can be found by aking he limi as ends owards infiniy: Shor Rae: lim Er 1, F s = φ, lim Varr k 1 σ , F s = k 1 k 2 2 2k 2 k 1 + k k 1 Shor-Term Mean-Reversion Level: lim Er 2, F s = φ, lim Varr 2, F s = σ2 2 2k. + σ2 1 2k 1. The wo facor Vasicek model displays see several key aribues which he one facor Vasicek model displays: 1. The shor rae experiences mean reversion, and over he long-erm will end owards φ. 2. There is no cerainy ha he shor rae will always be posiive. In realiy, nominal ineres raes can never be negaive however he Vasicek model can no ensure ha such a siuaion does no occur. This is a major drawback of he model.

70 11.1. PURE DISCOUNT BOND Pure Discoun Bond P, T = E Q e T r 1,u du F Thus, in order o price a pure discoun bond under he wo facor Vasicek model, one needs o evaluae he inegral of he shor rae. Brigo and Mercurio 2006 show how one can compue his inegral for a similar wo facor model. Compuing he pure discoun bond price for he Vasicek model is more complicaed, however he mehodology will be based on Brigo and Mercurio s approach. As Brigo and Mercurio 2006 explain, sochasic inegraion by pars allows one o manipulae he inegral of he shor rae ino a more manageable equaion. i.e., T T r 1,u du = T r 1,T r 1, udr 1,u = Subsiuing equaion 11.1 in equaion 11.5: T T udr 1,u + T r 1, = T T T udr 1,u + T r 1,. T uk 1 r 2,u r 1,u du + One can hen subsiue equaions 11.3 and 11.4 in equaion 11.6 hus T r 1,u du = T T + + T uk 1 r 2, e k 2u du T T T T uk 1 φ1 e k 2u du u T uk 1 σ 2 e k2u s dw 2,s du T uk 1 r 1, e k 1u du T 11.5 k 1 T uk 1 k 1 k 2 r 2, φ e k2u e k 1u du T uσ 1 dw 1,u + T r 1,. 11.6

71 70 CHAPTER 11. TWO FACTOR VASICEK MODEL + + T T T T T T uk 1 φ1 e k 1u du T uk1σ 2 e k 2u u 2 e k2m dw 2,m du k 1 k 2 T uk1σ 2 e k 1u u 2 e k1m dw 2,m du k 1 k 2 u T uk 1 σ 1 e k1u s dw 1,s du T uσ 1 dw 1,u +T r 1, Equaion 11.7 can be solved by seperaing he various componens of he equaion, and using inegraion by pars or inerchanging he inegrals o solve each of hese componens. i.e., T T T uk 1 r 2, e k2u T du = r 2, k 1 T k 2 T e k 2u k 2 du T = r 2, k 1 + e k2t 1 r 2, k 1, k 2 k T uk 1 r 1, e k 1u du = r 1, T e k 1T 1 k 1 r 1,, T T uk 1 φ1 e k2u du = T uk 1 φdu = T 2 k 1 φ 2 T T k 1 φ 11.9 T uk 1 φe k 2u du T k 2 e k 2T 1 k2 2 k 1 φ, T T T uk 1 φ1 e k1u 2 du = k 1 φ 2 T φt 2 + e k 1T 1 k 1 φ, 11.11

72 11.1. PURE DISCOUNT BOND 71 T T k 1 T uk 1 k 1 k 2 r 2, φ x e k2u e k 1u du = = k1 2 T k 1 k 2 r 2, φ T ue k2u du T T ue k1u du k1 2 T k 1 k 2 r 2, φ + e k2t 1 k 2 T e k1t 1 k 1 k1 2 u u=t T uk 1 σ 2 e k2u s dw 2,s = k 1 σ 2 T ue k 2u T T u= s=u s= k 2 2, e k 2s dw 2,s du s=t u=t = k 1 σ 2 T ue k2u du e k2s dw 2,s s= u=s T T se k 2 s = k 1 σ 2 + e k2t e k2s k 2 k2 2 e k2s dw 2,s T T s = k 1 σ 2 + e k2t s 1 k 2 k2 2 dw 2,s, u T T uk 1 σ 1 e k1u s dw 1,s = σ 1 T s + e k 1T s 1 dw 1,s, k 1 T uk1σ 2 e k 2u u 2 e k2m dw 2,m du = k2 1 σ 2 k 1 k 2 k 1 k 2 x u=t u= m=u = k2 1 σ 2 k 1 k 2 m= T + e k 2T s 1 k T ue k 2u e k 2m dw 2,m du T s k 2 dw 2,s, 11.15

73 72 CHAPTER 11. TWO FACTOR VASICEK MODEL T T uk1σ 2 e k 1u u 2 e k1m dw 2,m du = k 1 k 2 = k1 2σ 2 k 1 k 2 x σ 2 k 1 k 2 u=t u= m=u m= T T ue k 1u e k 1m dw 2,m du k 1 T s +e k 1T s 1 dw 2,s Thus one can subsiue equaions 11.8 o ino equaion 11.7: T e k 1T 1 r 1,u du = r 1, r 2,k 1 T + e k 2T 1 k 1 k 1 k 2 k 2 + r 2,k 1 T + e k 1T 1 k 1 k 2 k 1 k 1 φ 1 k 1 T + e k2t 1 k 1 k 2 k 2 k2 2 +φt 1 k 1 k 1 k 2 +φ 1 k 1 e k 1T 1 k 1 k 2 k1 2 k T 1σ 2 T s + e k 2T s 1 dw 2,s k 1 k 2 k 2 σ T 2 + k 1 T s + e k 1T s 1 dw 2,s k 1 k 2 T e k 1T s 1 σ 1 dw 1,s k 1

74 11.1. PURE DISCOUNT BOND 73 e k 1T 1 k 1 e k 2T 1 = r 1, r 2, e k1t 1 k 1 k 1 k 2 k 2 k 1 e k 1T 1 +φ + φt k 1 + k 1φ e k 2T 1 e k1t 1 k 1 k 2 k 2 k 1 k 1 σ T 2 e k 2T s 1 dw 2,s k 2 k 1 k 2 σ T 2 + e k 1T s 1 dw 2,s k 1 k 2 σ T 1 e k 1T s 1 dw 1,s. k 1 In order o evaluae he price of a pure discoun bond, one is required o evaluae he firs wo momens of he inegral of he shor rae. i.e., T e k 1T 1 k 1 e k 2T 1 M, T = E r 1,u du F = r 1, r 2, e k1t 1 k 1 k 1 k 2 k 2 k 1 e k 1T 1 +φ + φt T V, T = Var r 1,u du F = k 1 + k 1φ e k 2T 1 e k1t 1, k 1 k 2 k 2 k 1 k 1 σ e 2k 2T k2 2k 1 k e k2t + T 2k 2 k 2 σ k 1 k σ2 1 1 e 2k 1T k e k1t 2k 1 k 1 2k 1 σ2 2 1 e k 1+k 2 T +T k 2 k 1 k 2 2 k 1 + k 2 1 e k 2T k 2 + T Where he variance has been calculaed using he following facs T Var e k 2T s F 1 dw 2,s = = T e 2k 2T s 2e k 2T s + 1 ds 1 e 2k 2T 21 e k2t + T, 2k 2 k 2. 1 e k1t k 1

75 74 CHAPTER 11. TWO FACTOR VASICEK MODEL = = [ T Covar e k 2T s 1 dw ] T 2,s e k 1T s F 1 dw 2,s T e k 1+k 2 T s e k 1T s e k 2T s + 1 ds 1 e k 1+k 2 T 1 e k1t 1 e k2t + T. k 1 + k 2 k 1 k 2 Since he inegral of he shor rae is normally disribued wih mean M, T and variance V, T : P, T = E Q e T where r 1,u du F = e M,T + V,T 2 = e AT r 1,B 1 T r 2, B 2 T, B 1 T = 1 e k 1T, k 1 B 2 T = k 1 1 e k 2T 1 e k1t k 1 k 2 k 2 k 1, AT = B 1 T T φ σ2 1 2k1 2 + B 2 T φ σ2 1 B 1T 2 4k 1 + σ2 2 T 2 k2 2 2B 1T + B 2 T k e 2k 1T k 1 k 2 2 2k 1 2k 11 e k 1+k 2 T k 2 k 1 k 2 2 k 1 + k 2 + k2 1 1 e 2k2T k2 2k 1 k 2 2 2k Cap and Floors In order o price a caple, one mus firs find he value of a European pu opion on a pure discoun bond and hen use he relaionship beween he pu opion and he caple. The price a ime of his European pu opion wih mauriy T and srike K wrien on a pure discoun bond wih payoff R1 a mauriy S is: P = E [e ] T r sds K P T, S + F. Compuing his expecaion requires knowledge abou he join disribuion of he coningen claim X T = K P T, S + and he bank accoun process B T = e T r sds. As Jamshidian 1989 explains, in order o compue his expecaion, one needs o change he probabiliy measure. Thus, he numeraire is changed o P, T wih Q T he corresponding Equivalen

76 11.2. CAP AND FLOORS 75 Maringale Measure. P P, T P B 0 = E [ K P T, S + B T ] F, [ K P T, S + ] = E Q T F, P T, T [ P = P, T E Q T K P T, S + ] F. In order o undersand he dynamics of he shor rae under he EMM and how he marke price of risk has changed under he change in numeraire, one needs o compue he Radon-Nikodym process and hen use Girsanov s heorem. i.e., dq T dq = B 0P T, T B T P 0, T T = e 0 rsds P 0, T = e T = exp k 1 σ 2 k 2 k 1 k 2 0 rsds+m0,t 1 2 V 0,T T + σ 1 k 1 T 0 0 e k 2T s σ 2 1 dw 2,s k 1 k 2 e k 1T s 1 dw 1,s. 1 2 V 0, T The Girsanov heorem implies ha he wo processes W1, T and W 2, T defined as W1, T = W 1, σ 1 e k 1T 1 d, k 1 W2, T k 1 σ 2 = W 2, e k 2T σ 2 1 d + e k 1T 1 d, k 2 k 1 k 2 k 1 k 2 are wo independan Brownian moion under he Q T measure. The dynamics of he shor rae process r 1, and he shor-erm mean-reversion level r 2, under Q T are hus dr 1, = k 1 r 2, r 1, d + σ 1 e k 1T 1 k 1 dr 2, = k 2 φ r 2, d + T k 1 σ 2 k 2 k 1 k 2 0 e k 1T s 1 dw 2,s d + σ 1 dw T 1,, e k 2T 1 d Inegraing equaion 11.18, one can solve for r 2, and hen inpu his soluion ino equaion Inegraing his formula, one can hus solve for σ 2 e k 1T 1 d + σ 2 dw T k 1 k 2 2,

77 76 CHAPTER 11. TWO FACTOR VASICEK MODEL r 1,. Thus, he shor-erm mean-reversion level, r 2,, condiional on F s, is disribued normally wih he following variance: Var Q T r 2, F s = σ2 2 2k 2 [1 e 2k 2 s ]. The shor rae, r 1,, condiional on F s, is disribued normally wih he following variance: Var Q T r 1, F s = k 1 σ e 2k2 s 2 k 1 k e k 1+k 2 s 2k 2 k 1 + k e 2k1 s + σ2 1 1 e 2k 1 s. 2k 1 2k 1 The covariance beween he shor rae and he shor-erm mean-reversion level is Covar Q T r 1,, r 2, F s = k 1σ 2 2 e 2k 2 = k 1 k 2 s k 1 σ2 2 1 e 2k 2 s k 1 k 2 2k 2 e 2k 2m dw 2,m k 1σ 2 2 e k 1+k 2 k 1 k 2 1 e k1+k2 s k 1 + k 2 Le P T, S = e z, s. e k 1+k 2 m dw 2,m z = AS T r 1,T B 1 S T r 2,T B 2 S T. Thus z is disribued normally wih mean M p and variance V 2 p. M p = AS T B 1 S T E Q T r 1,T B 2 S T E Q T r 2,T, Vp 2 = B 1 S T 2 Var Q T r 1,T + B 2 S T 2 Var Q T r 2,T +2B 1 S T B 2 S T Covar Q T r 1,T, r 2,T.

78 11.2. CAP AND FLOORS 77 Thus one is able o evaluae he price of a European pu opion on a pure discoun bond: E Q T [ K e z + F ] = = = = ln K 1 2πVp ln K Mp Vp ln K Mp Vp = K = K = K = K 1 2πVp z Mp 2 2V e p 2 2 z Mp 2 2V e p 2 2 K e z + dz K e z dz 1 2πVp 2 e 1 2 x2 K e Mp+Vpx dx 1 2πVp 2 e 1 2 x2 Kdx ln K Mp Vp ln K Mp Vp e Mp+ 1 2 V 2 p ln K Mp Vp e Mp+ 1 2 V 2 p ln K Mp Vp e Mp+ 1 2 V 2 p 1 N 1 2πVp 2 emp+vpx 1 2 x2 dx 1 2πVp 2 e 1 2 x2 dx ln K Mp Vp 1 2πVp 2 e 1 2 V 2 p +V px 1 2 x2 dx 1 2πVp 2 e 1 2 x2 dx ln K Mp Vp 1 2πVp 2 e 1 2 x Vp2 dx 1 2πVp 2 e 1 2 x2 dx ln K Mp V p 2 Vp Mp ln K V p 1 2πVp e Mp+ 1 2 V p 2 2 e x2 2 dx 1 N Mp ln K + Vp 2. V p Under he EMM Q T X P, T [ X ] T = E Q T F. P T, T Thus subsiuing P τ, S for X τ [ P, S P T, S ] = E P, T Q T F P T, T = E Q T [P T, S F ].

79 78 CHAPTER 11. TWO FACTOR VASICEK MODEL Bu previously i was shown ha ln P T, S is disrubued normally wih mean M p and variance V 2 p, hus P, S P, T = E Q T [P T, S F ] = e Mp+ 1 2 V 2 p. Thus M p ln K V p = and M p ln K + Vp 2 = V p P, S M p = ln P, T ln ln P,S KP,T P,S KP,T 1 2 V 2 p, 1 2 V p 2, V p V 2 p V p. Hence he price of a European pu opion on a pure discoun bond is where P = P, T [ ] K 1 N d 2 e Mp+ 1 2 V p 2 1 N d 1 = KP, T 1 N d 2 P, S 1 N d 1, ln P,S KP,T V p 2 d 1 =, V p d 2 = d 1 V p, Vp 2 = B 1 S T 2 Var Q T r 1,T + B 2 S T 2 Var Q T r 2,T +2B 1 S T B 2 S T Covar Q T r 1,T, r 2,T. The value of he corresponding call opion can be found hrough Pu-Call Pariy. Example 1. If he shor rae, r 1,, and he shor-erm mean-reversion level, r 2,, are assumed o possess he following risk-neural dynamics dr 1, = k 1 r 2, r 1, d + σ 1 dw 1,, dr 2, = k 2 φ r 2, d + σ 2 dw 2,, wih k 1 = 0.1, φ = 0.05, σ 1 = 0.1, k 2 = 0.05 and σ 2 = 0.05 and a caple has he payoff 0.25L a expiry in 1 year, where L is he hree-monh spo LIBOR rae in nine monhs ime, r 1,0 is 5%, r 2,0 = 4.5% and P 0, T

80 11.2. CAP AND FLOORS 79 is a pure discoun bond wih face value 1 hen B 1 T = 1 e k 1T B 2 T = k 1 = 1 e = = k 1 k 1 k = AT = e k 2 T k 2 1 e k1t k 1 1 e P 0, T = e AT B 1T r 1,0 B 2 T r 2,0 = B 1 S = 1 e k 1S B 2 S = k 1 = 1 e = = 1 e k 1 1 e k 2 S 1 e k1s k 1 k 2 k 2 k e e = AS = P 0, S = e AS B 1Sr 1,0 B 2 Sr 2,0 = P 0 = KP 0, T 1 Nd 2 P 0, S1 Nd 1 = N N = ,

81 80 CHAPTER 11. TWO FACTOR VASICEK MODEL and hus he value of he caple is Cpl = 1 + K Caple τp 0 = =

82 Chaper 12 Two-Facor Cox-Ingersoll-Ross Model An exension of he Cox-Ingersoll-Ross model is he wo facor model whereby he shor rae is he sum of wo independen processes, wih each process modelled using a one facor CIR model. i.e., r = x + y, dx = k x φ x x d + σ x x dw,x, dy = k y φ y y d + σ y y dw,y, where k x, φ x, σ x, x 0, k y, φ y, σ y, y 0 are posiive consans. Since hese wo processes are independen of one anoher, one can easily solve for he shor rae as well as he expecaion and variance of he shor rae, based on he resuls of he one facor CIR model in Chaper 9. i.e., x = x s e kx s + φ x 1 e kx s + σ x e kx τ x τ dw τ,x, y = y s e ky s + φ x 1 e ky s + σ y e ky τ y τ dw τ,y, r = x s e kx s + y s e ky s + φ x 1 e kx s + φ y 1 e ky s +σ x e kx τ x τ dw τ,x + σ y e ky τ y τ dw τ,y. s The expecaion and variance of he shor rae are hus: Er F s = x s e kx s + y s e ky s + φ x 1 e kx s + φ y 1 e ky s, Varr F s = x sσx 2 e kx s e 2kx s + y sσy 2 e ky s e 2ky s k x k y s s s + φ xσ 2 x 2k x 1 2e kx s + e 2kx s + φ yσ 2 y 2k y 1 2e ky s + e 2ky s. 81

83 82 CHAPTER 12. TWO-FACTOR COX-INGERSOLL-ROSS MODEL The long-erm mean and variance can be found by aking he limi as ends owards infiniy: lim Er F s = φ x + φ y, lim Varr F s = φ xσx 2 + φ yσy 2. 2k x 2k y As is he case wih he one facor model, he wo facor CIR model exhibis mean-reversion. The second preferable feaure of he shor rae under hese dynamics is ha he shor rae is guareneed o remain posiive provided ha he iniial processes are boh posiive and 2k x φ x σ 2 x and 2k y φ y σ 2 y. Noe ha he independence of he wo processes are key in he analyical racabiliy of he shor rae. Brigo and Mercurio 2006 discuss he fac ha such a propery limis he shape of he curve of he absolue volailiies of he insananeous forward raes, hus such a model may be deemed inappropriae under cerain marke condiions when calibraing o marke daa. Allowing for correlaion beween hese wo processes more specifically, beween he Brownian moions of hese wo processes would no longer limi he shape of he curve of he absolue volailiies of he insananeous forward raes o such a degree however, as he join disribuion of hese wo processes is no known, he loss in racabiliy of he shor rae far ouweighs any poenial benefis Pure Discoun Bond The price of a pure discoun bond under he wo facor Cox-Ingersoll-Ross model can be decomposed ino wo seperae expecaions, due o he independence of he wo processes. These expecaions have already been calculaed in Chaper 9. i.e., P, T = E Q e T r udu F = E Q e T xu+yudu F = E Q e T x udu F E Q e T y udu F = e e Ax,T Bx,T x Ay,T By,T y = e Ax,T +Ay,T Bx,T x By,T y,

84 12.2. CAP AND FLOORS 83 where, for i = x, y B i, T = A i, T = 2k iφ i ln κ i = 2e κ it 1 κ i + k i e κ it, 1 + 2κ i [ σ 2 i k 2 i + 2σ2 i Cap and Floors 2κ i e 1 2 k i+κ i T 2κ i + k i + κ i e κ it 1 In order o price a caple, one mus firs find he value of a European pu opion on a pure discoun bond and hen use he relaionship beween he pu opion and he caple. As discussed in Jamshidian 1989 and shown in he previous chaper, one can use a change in measure o he Equivalen Maringale Measure Q T in order o simplify he calculaion of he expecaion. i.e., P = E [e ] T r sds K P T, S + F [ = P, T E Q T K P T, S + ] F. In order o compue his expecaion, one requires knowledge of he dynamics of he wo processes under Q T. The dynamics for x are shown below, as derived in Brigo and Mercurio Those for y can be similarly displayed. where dx = k x φ x [k x + B x, T σ 2 x]x d + σ x x dw T,x, dw T,x = dw,x + B x, T σ x x d, wih a non-cenral chi-squared disribuion funcion given by where p T x x s z = q x, sp χ 2 ν x,δ x,sq x, sz, ν x = 4k xφ x σx 2, ψ x = k x + κ x, σ 2 x q x, s = 2[ρ x s + ψ x + B x, T ], δ x, s = 4ρ x s 2 x s e κx s, q x, s 2κ x ρ x s = σxe 2 κx s 1. ],

85 84 CHAPTER 12. TWO-FACTOR COX-INGERSOLL-ROSS MODEL Thus P = [ P, T E Q T K P T, S + ] F = P, T 0 0 K P T, S + p T x x s z 1 p T y y s z 2 dz 1 dz 2. Chen and Sco 1992 reveal how one can reduce he mulivariae inegrals o a univariae inegral, allowing one o calculae various opions prices including ha of a European pu opion. Using heir approach, he price of a European call opion is found o be: where C = P, T 0 0 P T, S K + p T x x s z 1 p T y y s z 2 dz 1 dz 2 = P, Sχ 2 L x, L y; ν x, ν y, δ x T,, δ y T, χ 2 L x, L y ; ν x, ν y, δ x, δ y = P, T Kχ 2 L x, L y ; ν x, ν y, δ xt,, δ yt,, Ly 0 F L x L xz y ; ν x, δx f z y ; ν y, δydz y, where F and f are he sandard non-cenral chi-squared probabiliy disribuion funcion and densiy funcion respecively, and L y C = ln A xt, SA y T, S, K C yx = B x T, S, L x = 2[ρ x T + ψ x + B x T, S]yx, L x = 2[ρ x T + ψ x ]y x, κxt δ x T, = 2ρ xt 2 x s e. ρ x T + ψ x Example 1. If he shor rae is assumed o possess he following risk-neural dynamics r = x + y, dx = k x φ x x d + σ x x dw,x, dy = k y φ y y d + σ y y dw,y, wih k 1 = 0.2, φ 1 = 0.05, σ 1 = 0.15, k 2 = 0.005, φ 2 = 0.03 and σ 1 = and a caple has he payoff 0.25L a expiry in 1 year, where L is he hree-monh spo LIBOR rae in nine monhs ime, x 0 is 2%, y 0 is 3% and P 0, T is a pure discoun bond wih face value 1 hen and hus he value of he caple is P 0 = , Cpl =

86 Chaper 13 Calibraing Shor Rae Models The previous chapers in his hesis have focused on he heory behind some of he more common shor rae models, including how hese models can be used o value coningen claims such as caps, floors or swapions. This chaper aims o provide he reader wih an undersanding of how one calibraes a shor rae model. This will provide he foundaion for he following hree chapers which review he resuls of empirical comparisons of he shor rae models discussed in his hesis Parameerizaion In order o es he abiliy of a shor rae model o price coningen claims, one needs o calibrae ha model o marke prices. This process consiss of fiing values of insrumens or indicaors deermined hrough he shor rae model o hose observed in he marke. For example, one could deermine parameer values by fiing he shor rae model o cap and floor prices or o he yield curve. The chosen parameers are hose which enable he heoreical values o fi as closely as possible o he observed values in he marke, where closeness is a measure defined by he user. Examples of measures for closeness include: Squared difference beween heoreical and observed values n i=1 V i V i β 2. Absolue value of he relaive difference beween heoreical and ob- 85

87 86 CHAPTER 13. CALIBRATING SHORT RATE MODELS where served values n i=1 V i V i β V, Vi = is he observed value of insrumen i, V i β = is he heoreical value of insrumen i given he vecor of parameers, β. One of he major disadvanages of he firs approach is ha i is biased by he size of he value of he insrumen. i.e., he closeness measure is far more sensiive o pricing errors when he value of he insrumen or indicaor is large. Thus, if one were using his measure when fiing he model o prices of pure discoun bonds, he fi would no neccesarily be as close for long-erm mauriies as for values for shorer-erm mauriies. The abiliy of he heoreical values derived by he shor rae model o fi values observed in he marke depends on he number of parameers in he model versus he number of primary insrumens o which he model is fied. If here are more insrumens or indicaors han parameers, he model is over-deermined and hus i is highly unlikely ha he heoreical prices will mach he observed prices exacly for all insrumens or indicaors. Models such as he Ho-Lee model and Black-Karasinski overcome his problem hrough he use of parameers which are deerminisic funcions as opposed o consans. This allows hese models o fi he values of a series of insrumens or indicaors exacly and allow one o fi he model o anoher series of insrumens or indicaors using he remaining parameers. The disadvanage of such models is ha he deerminisic parameers are harder o jusify inuiively. Example. The risk-neural dynamics of he shor rae under he Ho-Lee model are: dr = φd + σdw. Since φ is a deerminisic funcion, one can fi he Ho-Lee model exacly o he yield curve and hen choose a value for σ which fis he model as close as possible o caps prices. The approach aken in his hesis is o fi each of he shor rae models o he yield curve by comparing he absolue value of he relaive difference in he heoreical and observed values of pure discoun bonds. i.e., he parameers chosen are hose which minimized he following equaion: n Pi, T P i, T ; β P,, T i=1

88 13.2. DATA 87 where Pi, T = is he observed value a ime of pure discoun bond Bond i wih mauriy T, P i, T ; β = is he heoreical value a ime of pure discoun bond i wih mauriy T given he vecor of parameers, β. Since his equaion is dependan on muliple variables, an algorihm o minimize mulivariae funcions is required. One such algorihm is Nelder-Mead which has gained populariy due o is powerfulness. Rouah e al 2007 noe ha his algorihm is easy o implemen, and i converges very quickly regardless of which saring values are used. Thus Nelder-Mead will be used in his hesis for calibraing he shor rae models, wih all programming compleed in Visual Basic Daa Two differen sources were used o obain he daa: Ineres rae daa was obained from I-Ne Bridge. This daa consiss of one-monh and hree-monh JIBAR; 3x6, 6x9, 9x12, 12x15, 15x18 and 18x21 FRAs; and hree-year, four-year, five-year, six-year, seven-year, eigh-year, nine-year, en-year, welve-year and fifeen-year Swaps. Cap and floor daa was obained from Rand Merchan Bank. This daa consiss of he mid-volailiy of bid and offer quoes for A-he-Money opions of caps wih mauriies ranging from one-year o en-years wih hree-monh enors. 480 daa poins were considered for he period 27 March 2006 o 27 February As discussed in Chaper 5, he Monoone Convex Inerpolaion Mehod has been used o complee he yield curve. I am exremely graeful o Perus Bosman a Prescien Securiies for providing he cap and floor daa Forms of Shor Rae Models Compared The shor rae dynamics for wo of he models discussed in his hesis consis of deerminisic funcions for a leas one of he parameers in ha model. This allows hese models o fi he yield curve exacly, however i does no allow for an effecive comparison across all of he models of heir abiliy o fi he yield curve. Thus, special cases of hese wo shor rae models will be considered whereby he deerminisic funcion is insead consrained o be a consan. Thus he dynmics of each of he models considered are:

89 88 CHAPTER 13. CALIBRATING SHORT RATE MODELS Vasicek One Facor Model dr = kφ r d + σdw, Ho-Lee Model dr = φd + σdw, Cox-Ingersoll-Ross One Facor Model dr = kφ r d + σ r dw, Black-Karasinski Model Vasicek Two Facor Model d lnr = [φ a lnr ]d + σdw, dr 1, = k 1 r 2, r 1, d + σ 1 dw 1,, dr 2, = k 2 φ r 2, d + σ 2 dw 2,, Cox-Ingersoll-Ross Two Facor Model r = x + y, dx = k x φ x x d + σ x x dw,x, dy = k y φ y y d + σ y y dw,y.

90 Chaper 14 Comparison of Models This chaper will provide he foundaion for he comparison of he empirical abiliy of each of he shor rae models o fi he yield curve, by inroducing a variey of approaches o conduc his comparison. Pelsser 2004 poins ou ha one can use goodness-of-fi crieria such as R 2, adjused-r 2 or he sandard error of he regression; or one can use a more formal esing procedure o compare each shor rae model agains one anoher. Goodness-of-fi crieria are easy o calculae and are ypically found o be fairly inuiive saisics. e.g., R 2 is he proporion of variabiliy in he dependen variable ha can be explained by variabiliy in he independen variables. However, a criical disadvanage wih using such measures o compare models is ha goodness-of-fi crieria do no ake ino consideraion he losses associaed wih choosing an incorrec model, as highlighed by Judge e al 1982, and discussed by Pelsser e.g., When comparing wo models which are boh unsuiable, goodness-of-fi crieria will no inform one ha neiher model is appropriae for he ask a hand. Davidson and MacKinnon and Greene 2003 provide alernaive approaches o esing differen economic heories or models which purpor o explain he same dependen variable. A group of hese approaches which will be used for his hesis are he ess for non-nesed, non-linear regression models. Non-nesed describes he fac ha he models being compared are no special cases of one anoher. Non-linear describes he fac ha he dependen variable - he yield of a pure discoun bond wih mauriy T - is no a linear combinaion of he independen variables - he shor rae and ime o mauriy of he pure discoun bond. Noe: Parameer consrains on he wo facor CIR model preven he one facor CIR model from being described as a special case of he wo facor model. 89

91 90 CHAPTER 14. COMPARISON OF MODELS 14.1 Tesing Non-Nesed, Non-Linear Regression Models No concensus approach exiss for he esing of non-nesed, non-linear regression models. Fisher and McAleer 1981, Mizon and Richard 1986 and Greene 2003 all discuss some of he possible approaches which one can use. Pelsser 2004, when comparing he abiliy of shor rae models o fi cap and floor prices, prefers he approach discussed by Davidson and MacKinnon This hesis will adop he same approach. Under Davidson and MacKinnon s approach, one considers a model whose validy one wishes o es. i.e., where y i X i β ɛ 0,i H 0 : y i = fx i, β + ɛ 0,i, is he ih observaion of he dependen variable, is a vecor of he ih observaions of he explanaory variables, is a vecor of parameers o be esimaed, is he error erm of he ih oberservaion of he model and is assumed o be normally and independanly disribued NID wih a mean of 0 and a variance of σ 2 0. One is faced wih an alernaive model which also purpors o explain he same dependen variable. i.e. where y i Z i γ ɛ 1,i H 1 : y i = gz i, γ + ɛ 1,i, is he ih observaion of he dependen variable, is a vecor of he ih observaions of he explanaory variables, is a vecor of parameers o be esimaed, is he error erm of he ih oberservaion of he model and is assumed o be normally and independanly disribued NID wih a mean of 0 and a variance of σ 2 0. Since hese wo models are non-nesed, he ruh of H 0 implies ha he alernaive hypohesis, H 1, is false, and vice versa. In order o perform he ess on H 0 and H 1, one is required o embed he

92 14.1. TESTING NON-NESTED, NON-LINEAR REGRESSION MODELS91 wo compeing models ino an arificial compound model hrough he use of an arificial parameer α. i.e., H C : y i = 1 αfx i, β + αgz i, γ + ɛ i, 14.1 If H 0 is rue, hen he rue value of α is 0. In he case ha α is significanly differen from 0 and 1, he arificial compound model can be used as a specificaion es, providing an indicaion in which direcion one needs o search for a beer model, a significan advanage as highlighed by Pelsser One of he problems wih he arificial compound model in 14.1 is ha, in mos cases, he model can no be esimaed as some of he parameers will no be separaely idenifiable, a poin highlighed in boh Davidson and MacKinnon 1993 and Pelsser Davidson and MacKinnon 1981 sugges a possible soluion o his problem whereby one replaces he parameer vecor of H 1 wih esimaes of hese parameers, assuming H 1 is rue. Thus one can replace γ wih an esimae ˆγ, derived hrough non-linear leas squares, and hus obain he model where ĝ = gz i, ˆγ. H C : y i = 1 αfx i, β + αĝ + ɛ i, 14.2 Two possible approaches exis o es he null hypohesis: If H 0 is linear hen one can perform he J-Tes whereby α and β are esimaed joinly, wih he -saisic used o es he null hypohesis ha α = 0. If H 0 is non-linear, hen one can conduc he P-Tes. By using he Taylor expansion of f i ˆf i + ˆF i β ˆβ, where ˆF i is he marix of parial derivaives of f wih respec o β evaluaed a ˆβ, he nonlinear regression 14.2 can be linearized o obain he Gauss-Newon regression H C : y i ˆf i = αĝ i ˆf i + ˆF i b + ɛ i, 14.3 where one can use he -saisic for α = 0 from his regression. Davidson and MacKinnon 1993 poin ou ha he J-Tes and he P-Tes are asympoically equivalen under H 0. Thus, if one of hese ess is asympoically valid, boh of hem mus be alhough hey may yield differen resuls in small samples.

93 92 CHAPTER 14. COMPARISON OF MODELS The -saisics from he J-Tes and P-Tes are based on he assumpion ha H 0 is rue. Thus, one canno use hese equaions o es H 1 and insead is required o reverse he roles of H 0 and H 1 before repeaing he es. Thus, four possible oucomes exis: Accep boh H 0 and H 1, Accep H 0 and rejec H 1, Rejec H 0 and accep H 1, Rejec boh H 0 and H 1.

94 Chaper 15 Empirical Resuls - Fi o Yield Curve Six shor rae models, in he form discussed in Chaper 13.3, were calibraed and esed on a daily basis across he enire daase of 480 observaion daes in order o undersand he empirical abiliy of each model o fi he yield curve. This secion reviews he resuls of hese ess Fi o Pure Discoun Bond Prices and he Yield Curve The calibraion of hese models involved minimizing he sum across 60 mauriy daes of he absolue value of he relaive difference in pure discoun bond prices, as elaboraed on in Chaper 13. Figure 15.1 shows his minimum value for each model a each daa poin. One should noe ha cerain consrains were implemened during his calibraion process - e.g., he volailiy was consrained o only posiive numbers; 2kφ was consrained o being greaer han σ 2 in he Cox-Ingersoll-Ross one facor model in order o ensure ha he shor rae remains posiive. Figure 15.1 clearly illusraes ha he Ho-Lee model is he poores fi of he six models whils he Cox-Ingersoll-Ross one facor model sruggles o fi he observed values owards he beginning of he daase May Ocober The Vasicek wo facor model generally appears o be he bes fiing model across he enire daa se. Alhough his value which has been minimized provides he reader wih a firm undersanding of he abiliy of each model o fi he daa, i is inuiively more appealling o view he model fi in erms of he annualised 93

95 94 CHAPTER 15. EMPIRICAL RESULTS - FIT TO YIELD CURVE Figure 15.1: Sum of absolue value of relaive difference beween heoreical and observed PDB values yield of pure discoun bonds. i.e., where y i,τ P, T ; β τ P, T ; β y i,τ = ln, τ is he model implied annualized yield of pure discoun bond wih mauriy τ a observaion i, is he heoreical value a ime i of a pure discoun bond wih mauriy T given he vecor of parameers, β, is he erm o mauriy of he bond, T -i. One can compare his model implied annualized yield o he annualzied yield observed in he marke. i.e, where y i,τ ɛ y i,τ = y i,τ + ɛ i, is he observed annualized yield of pure discoun bond wih mauriy τ a observaion i, is he error erm a observaion i which is assumed o be disribued normally wih a mean of 0 and a sandard deviaion of σ ɛ.

96 15.1. FIT TO PURE DISCOUNT BOND PRICES AND THE YIELD CURVE Goodness-of-Fi Two saisics which provide he reader wih an indicaion of he goodnessof-fi of he annualized yields of each of he shor rae models are he average absolue value of he error erm across he differen mauriies a each observaion dae and he sandard deviaion of he error erm, σ ɛ. Table 15.1 conains summary saisics for hese wo goodness-of-fi indicaors, including he minimum, median, maximum, average and quariles. Noe: Q1 and Q3 denoe he firs and hird quarile respecively. Figure 15.2 shows he average of he firs indicaor across he differen mauriies, for each daapoin and for all six models whils Figure 15.3 shows he values for he second indicaor across hese observaion daes. Table 15.1: Summary saisics of annualized yields compued over he 480 observaion daes As shown previously in Figure 15.1, he Ho-Lee model has a significanly poorer fi of he yield curve as compared o he oher five models. This view is suppored when one views he wo goodness-of-fi parameers for he Ho-Lee model. The Black-Karasinski model also has a poor fi in erms of he sandard deviaion of he error erm, averaging 0.422%. Over he iniial period March November 2006, he Vasicek one facor model and Cox-Ingersoll-Ross wo facor model appear o fi he daa he bes, alhough he laer model has a poor fi during June For he remaining observaion daes he bes fiing model is he Vasicek wo facor model. As expeced, he wo facor Vasicek model fis he daa beer han he one facor model, as is he case for he Cox-Ingersoll-Ross models. Ineresingly, however, is he fac ha he maximum value of he average absolue error erm over he 480 observaion daes is larger for boh of he wo facor models as compared o heir respecive one facor models. Generally, he Vasicek models appear o fi he daa beer han he Cox-Ingersoll-Ross models.

97 96 CHAPTER 15. EMPIRICAL RESULTS - FIT TO YIELD CURVE Figure 15.2: Average Absolue Value of Error Term Figure 15.3: Sandard Deviaion of he Error Term

98 15.1. FIT TO PURE DISCOUNT BOND PRICES AND THE YIELD CURVE97 Figure 15.4: Ho-Lee Parameer Values Based on he firs goodness-of-fi measure - he average absolue value of he error erm - he Vasicek wo facor model fis he observed values bes. Based on he second goodness-of-fi measure - he sandard deviaion of he error erm - he Cox-Ingersoll-Ross wo facor model and he wo Vasicek models all fi he daa relaively well Parameer Esimaes under each shor rae Model The individual parameer esimaes for he six shor rae models are shown in Figure 15.4 o Figure The Ho-Lee model has been shown o fi he observed yield curve poorly. One possible reason for his is ha he risk-neural dynamics of he model do no allow for mean reversion of he shor rae. Wu and Zhang 1996 found ineres raes o be mean-revering when esing across a range of OECD counried, whils Pelsser 2004 noes ha on he basis of economic heory, here are compelling argumens for he mean-reversion of ineres raes. Figure 15.4 shows he drif parameer o consisenly remain close o zero, hus one is essenially rying o fi he model o he observed prices using only one parameer, σ.

99 98 CHAPTER 15. EMPIRICAL RESULTS - FIT TO YIELD CURVE Figure 15.5: Black-Karasinski Parameer Values Figure 15.6: CIR One Facor Parameer Values

100 15.1. FIT TO PURE DISCOUNT BOND PRICES AND THE YIELD CURVE99 Figure 15.7: Vasicek One Facor Parameer Values The parameer values in he Cox-Ingersoll-Ross one facor model, as shown in Figure 15.6, appear difficul o jusify inuiively. The volailiy of he model is σ r. Since σ is close o zero for exended periods over he se of observaions, he volailiy of he model over hese periods will be exremely close o zero, resuling in he Brownian moion having minimal impac on he value of he shor rae. The Vaicek one facor model experiences he same issue wih regards o he volailiy of he model, wih σ close o zero for exended periods over he se of observaions, as shown in Figure The Cox-Ingersoll-Ross wo facor model is appealling heoreically due o he fac ha one can model shor-erm flucuaions in he shor rae whils sill allowing for exended periods of eiher high or low ineres raes. Under he model, one of he independen processes, X, is expeced o have high values for he speed of reversion and volailiy parameers whils he oher process, Y, would have low values for hese wo parameers. Figure 15.8 and Figure 15.9 show he speed of reversion o be consisen wih his heory however he volailiy o be inconsisen wih his heory. Since he volailiy of he processes are σ X X and σ Y Y respecively and σ x and σ Y are close o zero for exended periods of ime, he volailiy for each of he processes is low.

101 100 CHAPTER 15. EMPIRICAL RESULTS - FIT TO YIELD CURVE Figure 15.8: CIR Two Facor Process X Parameer Values Figure 15.9: CIR Two Facor Process Y Parameer Values

102 15.1. FIT TO PURE DISCOUNT BOND PRICES AND THE YIELD CURVE101 Figure 15.10: Vasicek Two Facor Shor Rae Parameer Values Figure 15.11: Vasicek Two Facor Shor Team Mean Reversion Level Parameer Values

103 102 CHAPTER 15. EMPIRICAL RESULTS - FIT TO YIELD CURVE Under he Vasicek wo facor model, he shor rae revers o a shor-erm mean-reversion level which in erm revers o a long-erm mean-reversion level. This feaure is appealling as i allows for shor erm flucauaions in he shor rae whils sill allowing for exended periods of eiher high or low ineres raes. Figure reveals ha over exended periods of ime he shor rae has a high speed of reversion owards he shor-eam meanreversion level wih a low volailiy parameer for his process. Over he remaining periods, he speed of reversion is far lower, wih higher volailiy parameers resuling in he shor rae revering o he shor-eam mean reversion levels a a far slower rae. Figure shows ha he shor-eam mean-reversion process ypically revers slowly o he long-erm risk-neural mean-reversion level alhough his level appears highly volaile over he middle and end observaion daes and appears excessively high over he end observaion daes Example of Fi o he Yield Curve In order o provide he reader wih an indicaion of he fi of he six models, Figure shows he observed yield curve on 18 Ocober 2007, as well as he yield curve derived from he six Shor Rae models. A his dae he yield curve is invered, forming a convex shape. i.e., he yield of deb insrumens of a longer-erm mauriy have a lower yield han ha of shor-erm insrumens of he same qualiy. A characerisic of his curve differeniaing i from a ypical invered yield curve is ha he yield iniially increases slighly wih he erm before decreasing and forming he invered shape. This characherisic resuls in he shor rae models sruggling o fi he observed prices. I is immediaely apparen ha boh he Ho-Lee model and Black-Karasinski models are unable o produce an invered convex form. The Ho-Lee model s resulan concave shape is a poor fi of he observed prices whils he Black- Karasinski model fis he observed prices exremely poorly a he shor erm of he curve, as i aemps o fi a humped curve o he observed prices. The Cox-Ingersoll-Ross one facor and wo facor models, as well as he Vasicek one facor model are unable o fi he observed prices as heir curvaure is oo shallow. The Vasicek wo facor model closely maches he observed prices, excep a he shor-end of he curve where he Vasicek model is unable o reproduce he increasing hen decreasing characerisic of he observed yield curve Davidson and MacKinnon s P-Tess Alhough he goodness-of-fi measures provide one insigh ino which models should be preferred, i provides lile indicaion of he acual suiabiliy of

104 15.1. FIT TO PURE DISCOUNT BOND PRICES AND THE YIELD CURVE103 Figure 15.12: Yield Curve on 18 Ocober 2007 each model. Davidson and MacKinnon s P-Tes, as discussed in Chaper 14, allows one o more formally es he abiliy of he various shor rae models o fi he observed prices. Table 15.2 provides he resuls of he various pairwise P-Tess which have been performed, summarizing he number of imes each of he four possible oucomes occurred across he 480 observaion daes for each pairwise es. None of he six models are overwhelmingly acceped as being able o suiably fi he observed yield curve. Over he 15 pairwise ess, boh models being esed were rejeced over a minimum of 58% of he 480 observaion daes. The Vasicek wo facor models appears more suiable han he res of he shor rae models as under each pairwise es, he Vasicek model always ouperms he alernaive model. The Cox-Ingersoll-Ross one facor model and Vasicek one facor model appear o have a similar suiabiliy in fiing he daa, wih boh models being accpeed under he pairwise es over 17.9% of he observaion daes.

105 104 CHAPTER 15. EMPIRICAL RESULTS - FIT TO YIELD CURVE Table 15.2: Resuls of Pairwise P-Tess 15.2 Fi o Cap and Floor Prices The parameers deermined when calibraing he shor rae models o he observed values of pure discoun bonds, as described in Chaper 15.1, were used o invesigae he abiliy of hese shor rae models o price coningen claims such as caps and floors. The raionale behind using he same parameer values, even hough such values were unlikely o be hose which would minimize he difference beween he observed and heoreical cap and floor prices, was o invesigae wheher a shor rae model which fied he yield curve relaively well would be able o provide realisic values when pricing coningen claims Goodness-of-Fi Table 15.3 conains summary saisics for he wo goodness-of-fi indicaors which have previously been used. Figure shows he second of hese indicaors - he sandard error of he regression - across he differen mauriies, for each daapoin and for all six models. One can clearly see ha hese shor rae models sruggle o provide values for a-he-money caps and floors which are close o hose observed in he marke. The bes fiing model is he Ho-Lee model - a model which

106 15.3. FIT TO CAP AND FLOOR PRICES AFTER PARAMETERS RECALIBRATED105 Table 15.3: Summary Saisics of Fi o Cap Prices Compued over he 480 Observaion Daes Figure 15.13: Sandard Deviaion of he Error Term for Cap Prices had a fairly poor fi o he observed annualized yields and pure discoun bond. The Vasicek wo facor model was one of he beer fiing models o he observed annualized yields and pure discoun bond prices however, when using hese same parameers, his model is an exremely poor fi o he observed cap and floor prices Fi o Cap and Floor Prices afer Parameers Recalibraed Since he shor raes models calibraed o he yield curve do no provide realisic values when pricing caps and floors, he shor rae models have

107 106 CHAPTER 15. EMPIRICAL RESULTS - FIT TO YIELD CURVE been recalibraed. On his occasion, he models were calibraed so as o minimize he pricing difference beween he heoreical and observed value of an a-he-money cap wih erm en years and enor hree monhs. Table 15.4 conains summary saisics for he wo goodness-of-fi indicaors which have previously been used, as well as for he average error. This las measure will allow one o see wheher he shor rae model is ypically overpricing or underpricing he cap. If his measure is negaive, hen he acual price is smaller han he heoreical price hence he model is overpricing he insrumen. Figure shows he second of hese indicaors - he sandard error of he regression - across he differen mauriies, for each daapoin and for all six models. One can clearly see ha he Cox-Ingersoll-Ross fis he observed prices he Table 15.4: Summary Saisics of Fi o Cap Prices Compued over 100 Observaion Daes closes over he 100 observaion daes. The Vasicek wo facor model and Black-Karasinski model also appear o provide suiable prices for he caps. Ineresingly, he Black-Karasinski model is he only model which does no consisenly overprice he caps. The reason for his is he underlying disribuions of he models as here is a significan difference in skewness beween he normal and chi-squared disribuions and he lognormal disribuion.