Applications of Interest Rate Models

WDS'07 Proceedings of Conribued Papers, Par I, 198 204, 2007. ISBN 978-80-7378-023-4 MATFYZPRESS Applicaions of Ineres Rae Models P. Myška Charles Universiy, Faculy of Mahemaics and Physics, Prague, Czech Republic. Absrac. Main purpose of his paper is o describe he background of ineres rae modeling, i.e. inroduce main division of ineres rae models, explain wha is he insananeous ineres rae and wha is is suiable represenaive. We will deeply focus on Vašíček and Hull-Whie model and analyze problem of he calibraion of hese models. Inroducion I is well known ha ineres raes and heir srucure modeling is very imporan for financial engineers, acuaries, ec. In recen decades here were developed many models which ry o describe he behavior of yield curve and which are based on he heory of probabiliy and of sochasic processes. Afer some general division of ineres rae wo of he one-facor models will be discussed in pariculary. In nex par we will focus on he insananeous ineres rae which is he modeled value in mos of models. A he end we analyze he problem of model calibraion because correc parameer esimaion is crucial for model s applicaion and menion some open problems. Imporan variables In his chaper we define following imporan variables: R(,T) - he coninuously compounded spo rae a ime for he mauriy T P(,T) - he zero-bond price a ime for he mauriy T The relaionship beween P(,T) and R(,T) is according o inuiion as follows: P(,T) = e R(,T)(T ) (1) r() - he insananeous ineres rae a ime, defined as lim T R(,T) I can be derived ha under some assumpions P(,T) is P(,T) = E Q exp where Q is he risk-neural probabiliy measure. T r(s)ds F, (2) Models of ineres raes Generally here are wo basic ypes of processes ha describe dynamic of ineres raes: One-facor and wo-facor models. One-facor models work wih only one source of uncerainy which is represened by he ineres for some infiniely shor period - ofen is also herefore called as an insananeous ineres rae. Such rae according o he model describes whole yield curve and herefore i is imporan o find is suiable represenaive in pracical usage. Examples of hese models are: Rendleman-Barer model (which is acually idenical wih radiional model for sock), Vašíček model (wih mean reversion o one value), Hull-Whie model (exended 198

Vašíček model including reversion no only o one value, bu o some deermined erm srucure), Ho-Lee model (similar o Hull-Whie bu wihou mean reversion). Generally one-facor models have following form: dr() = (ϑ() a()r())d + σ()r() γ dw() (3) Second basic ype of ineres rae models are wo-facor models. These are more complex and work no only wih he insananeous ineres rae. Examples are Brennan-Schwarz model or Heah-Jarrow-Meron model. We shall look closer o Vašíček and Hull-Whie model: Vašíček model In conrary o he model for equiy prices, ineres raes have values in some limied range, so here is a need of anoher ype of model. We ge his propery applying Ornsein-Uhlenbeck processes ha work wih so called mean reversion. These models have endency o reurn o some reversion value (Vašíček model) or some reversion funcion which argumen is ime (Hull-Whie model). Main equaion ha describes Vašíček model can be wrien as follows: dr() = k(θ r())d + σdw() (4) Remark ha Vašíček model is derived from (3) by seing ϑ(), a() and σ() as consans and γ is equal o 0. The reversion value is θ, parameer k describes he velociy of meanreversion. Parameer σ is an absolue volailiy of ineres rae. For mos of applicaions for his model i is necessary o calculae price of zero-bond P(,T). Inegraing equaion (4) we obain for each s : ( r() = r(s)e k( s) + θ 1 e k( s)) + σ which implies ha r() is normally disribued, so we can wrie P(,T) = E Q exp( Afer some longer calculaions we ge: T s e k( u) dw(u), (5) T r(s)ds is also normally disribued and r(s)ds) F T = exp E Q r(s)ds + 1 2 var Q T r(s)ds F where P(,T) = A(,T)e B(,T)r(), (6) ) } A(,T) = exp {(θ σ2 2k 2 [B(,T) T + ] σ2 4k B(,T)2 B(,T) = 1 k [1 e k(t ) ] 199

I should menioned here are wo disadvanages of his model: 1.) possibiliy ha he rae can be negaive and 2.) reversion o only one value. These disadvanages are almos oally removed in he Hull-Whie model: Hull-Whie model Hull and Whie assumed ha he insananeous shor-rae process evolves under he riskneural measure according o dr() = (ϑ() ar())d + σdw() (7) Mean reversion values varying in ime ϑ() are chosen so as o exacly fi he erm srucure of ineres raes being currenly observed in he marke - he modeled values should in he expeced value copy he acual forward curve. Similar o Vašíček model we can derive he price of zero-bond P(,T) which could be wrien in he same form as in (6), bu obviously wih anoher A(,T) and B(,T). I can also be shown ha probabiliy r() < 0 is very small. Insananeous ineres rae As menioned one-facor models work wih such rae ha should be insananeous. For pracical purposes i is necessary o deermine which rae could represen his insananeous rae. Firs idea is o choose he shores rae ha exiss - overnigh O/N rae. Problem is if we could admi his rae as a suiable represenaive of whole yield curve. We can ge he answer by applying of cluser analysis which ranks among mehods of mulivariae saisical analysis. Cluser analysis Cluser analysis is a mehod ha sors hrough he raw daa and groups hem ino clusers. A cluser is a group of relaively homogenous cases or observaions. We firs define variable c ij as a correlaion of inraday changes of enors i and j. We choose a rae of dissimilariy d ij : d ij = 2(1 c ij ) (8) We apply single linkage mehod for EUR raes (4 years hisory). We can see on Figure 1 ha he dendrogram proves overnigh rae shows some anomaly compared o he res of yield curve which makes choosing overnigh rae as a represenaive of whole curve as unsuiable. Remark ha for hreshold = 0.9 here are 3 clusers - overnigh rae, raes of money marke and raes of capial marke which among ohers confirms heory of separaed markes. I should be menioned ha oher mehods of cluser analysis produce similar resuls. Cluser analysis was applied for he CZK yield curve curve as well. On Figure 2 we can see dendrogram produced using he complee linkage mehod. According o his dendrogram we can make same conclusions as were discussed for EUR raes. Similar resuls are produced using anoher mehods of cluser analysis. Conclusion Cluser analysis proves ha overnigh rae canno be regarded as a suiable represenaive of yield curve, because i is oo dissimilar compared o anoher enors. I is much more appropriae o use one-week, wo-week, one-monh, hree-monh or even six-monh rae as a pracical represenaive of insananeous ineres rae. Calibraion For an applicaion of models i is crucial o be able o correcly esimae he parameers: Generally i is possible o perform i by wo main ways: 200

Dendrogram for a single linkage mehod (EUR) 26Y 27Y 28Y 29Y 21Y 22Y 23Y Heigh 0.0 0.2 0.4 0.6 0.8 1.0 1.2 40Y 20Y 30Y 25Y 3Y 2Y 24Y 50Y 1Y O/N 16Y 17Y 18Y 19Y 11Y 15Y 12Y 13Y 14Y 7Y 10Y 8Y 9Y 5Y 6Y 4Y 5M 6M 9M 4M 2M 3M 1M 1W Figure 1. Cluser analysis of EUR yield curve EUraes hclus (*, "single") Dendrogram for a complee linkage mehod (CZK) 8Y 9Y Heigh 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 O/N 1W 1M 2M 3M 2Y 3Y 4Y 12Y 15Y 5Y 6Y 7Y 10Y 1YCM 6M 9M 1YMM Figure 2. Cluser analysis of CZK yield curve CKraes hclus (*, "complee") 1. Dynamic mehod of calibraion - This approach esimaes parameers from hisorical values. Such mehodology is equivalen o hisorical volailiy calculaion in modeling of equiy prices. 2. Saic mehod of calibraion - This approach esimaes parameers from he acual marke daa (i.e. from he acual yield curve and from he acual bond prices). Such mehodology is equivalen o implied volailiy approach used in modeling of equiy prices. Dynamic mehod of calibraion We shall give an example of he dynamic calibraion of Vašíček model, as he insananeous rae we choose 3-monh rae and we use maximum-likelihood mehod: From formula (5) we can see ha he variable r(), condiional on F s, is normally disribued wih mean r(s)e k( s) + θ(1 e k( s) ) and variance V 2 = σ2 2k (1 e 2k( s) ). 201

Choosing s as some consan ime inerval (e.g. one working day) and subsiuing α = e k( s), we can wrie ha r i+1 = αr i + θ(1 α) + e i,e i N(0,V 2 ) (9) Suppose we have n+1 values of hisorical raes r 0,r 1,...,r n. Then he likelihood funcion L((r 0,...,r n ),α,θ,v 2 ) is: n ( )1 1 L((r 0,...,r n ),α,θ,v 2 2 ) = exp ( (r i αr i 1 θ(1 α)) 2 ) 2πV 2 2V 2 (10) i=1 logl = n ( ) 1 2 log 2πV 2 1 n 2V 2 (r i αr i 1 θ(1 α)) 2 (11) We calculae parial derivaions, equal hem o zero and afer some algebra we ge: k = log( α) s = 1 n n r i r i 1 n n r i r i 1 s log i=1 i=1 i=1 n i=1r n ( n ) 2 i 1 2, r i 1 i=1 n [r i αr i 1 ] 2k n [r i αr i 1 β(1 α)] 2 i=1 θ =, σ 2 i=1 = n(1 α) 1 e 2k( s) (12) On he firs graph of Figure 3 we can see 11-years hisory of 3-monh CZK rae wih esimaed parameers. On he second graph is he same for 3-monh EUR rae (7-years hisory): i=1 CZK 3M rae EUR 3M rae Rae 0.02 0.03 0.04 0.05 0.06 k = 0.264 hea = 0.021 sigma = 0.0063 Rae 0.020 0.025 0.030 0.035 0.040 0.045 0.050 k = 0.132 hea = 0.025 sigma = 0.0027 0 2 4 6 8 10 0 1 2 3 4 5 6 7 Time (in years) Time (in years) Figure 3. 3-monh CZK and EUR ineres rae On Figure 4 see he simulaions of 3-monh CZK rae using Vašíček model. Remark mean reversion of he simulaed raes o he value of parameer θ: 202

Simulaion of CZK rae Rae 0.01 0.02 0.03 0.04 0 1 2 3 4 5 Years Figure 4. Simulaion of 3-M ineres rae Saic mehod of calibraion This approach esimaes parameers from he acual shape of yield curve, i.e. generally from he acual marke daa. This is analogical o he implied volailiy approach ha is mainly used in he modeling of equiy prices. Such mehod does no ake ino accoun he hisory a all which is is big disadvanage. Anoher disadvanage is he credibiliy of esimaed parameers. For example assuming fla yield curve, where all raes are equal o 4%, we ge following values for Vašíček model (using program Solver in Excel): θ = 0.00079, k = 0.92766, σ = 0.00000081. These values are apparenly senseless. Problems For more complicaed models i is very difficul o esimae he parameers by he dynamic mehod, e.g. for Hull-Whie model - see (7) - i is impossible o calculae he esimaors according o he maximum likelihood mehod. Bu afer some algebra we can ge following equaion for price of bond: db(,t) = r()b(,t)d σ()c(k,,t)b(,t)dw(), (13) where C(k,,T) is a funcion of ime and mean reversion parameer k. If we suppose σ does no depend on ime, hen i be should possible o esimae i from he hisorical yield curve, bu he quesion is how exacly o perform i. Nex problem is o deermine if he parameers esimaed using he Saic mehod are enough credible (we saw problem wih Vašíček model esimaion in previous chaper). Conclusion There was discussed a main division of ineres rae models wih examples in he paper. Mos of hem work wih a so called insananeous ineres rae. Applying mehod of cluser analysis we showed he bes represenaive of his rae is 2W, 1M, 3M or 6M rae and no he shores rae O/N or 1W rae. On Vašíček model he dynamic mehod of calibraion was numerically illusraed. The furher goal is o precisely esimae more advanced models. 203

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