7 pages 1. Hull and White Generalized model. Ismail Laachir. March 1, Model Presentation 1

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1 7 pages 1 Hull and Whie Generalized model Ismail Laachir March 1, 212 Conens 1 Model Presenaion 1 2 Calibraion of he model Fiing he iniial yield curve Fiing he caple implied volailiy surface Trinomial Tree 5 4 Example of calibraion 5 Premia 14 1 Model Presenaion This par is mosly aken from [3], chaper 13. Hull and Whie model is a shor-rae model. One of is main characerisics is is abiliy o mach he iniial yield curve by using ime-varying parameer. A one facor version of his model was firs proposed in [1] (already implemened in Premia. I s described by he he sde : dr( = [θ( a r( ] d + σ dw(, where he funcion θ( is used o fi he ineres rae erm sruure.

2 7 pages 2 In his projec we implemen an exenion of his model, by considering he parameer σ ime-dependan. Hence we have he following evoluion for he shor rae r(: dr( = [θ( a r( ] d + σ( dw( This funcion σ( may be used o fi he marke prices of basic ineres rae derivaives like caple/floorle and/or swapion. The process r( can be wrien as : r( = α( + x( where : α( = x( = exp(ar( + exp(a( yθ(ydy exp(a( yσ(y dw y Hence, he price of a Zero coupon bond can be compued hrough he same mehodology as in [1], for < < T : Where P (, T = exp ( 1 ea(t B(, T =, A(, T = 1 a 2 Then he discoun bond saisfy : α(udu A(, T B(, Tx( dp (, T P (, T = r( d σ(b(, T dw and he bracke of he forward discoun bond F( = df( = σ( 2 [B(, S B(, T] 2 F( Then, he average volailiy from o T of F( is : B(y, T 2 σ(y 2 dy P (, S P (, T is : σ avg = T df( d F( 1 = [B(, S B(, T] T σ( 2 e 2a d

3 7 pages 3 This quaniy can be plugged in he following formula o ge he price of a caple wih rese dae T, payemen dae S, caple rae K and τ = S T (cf [3], ch.13, for deails on he compuaion: where : caple price = P (, T N(d 2 (1 + Kτ P (, S N(d 1 d 1 = log ( (1 + Kτ P (,S P (,T σ2 avg T σ avg T d 2 = d 1 σ avg T 2 Calibraion of he model The use of ime-dependan parameers in he model allow more flexibiliy o fi marke daa : yield curve and caple implied volailiy. The funcion α( (similarly θ( is seleced so ha he model fis he iniial erm srucure. The funcion σ( is chosen o fi he marke prices of a se of acively raded ineres-rae opions. In our case we will use only caple. 2.1 Fiing he iniial yield curve Suppose here ha he volailiy funcion σ( has been chosen and we will compue he funcion α( so ha we exacly fi he marke price of zero coupon bonds P M (, T for T >. We recall he price of a discoun facor : P (, T = exp ( α(udu + B(, Tα( A(, T B(, Tr( So, o have P (, T = P M (, T for all T >, we should choose : α( = P M (, A(, Hence, afer some compuaions, we ge : B(, r(

4 7 pages 4 P (, T = P M ( (, T P M (, exp B(, Tf M (, B(, Tr( B(, T 2 e 2a(u 2 σ(u 2 du, where f M (, = P M (, is he insananeous forward rae. 2.2 Fiing he caple implied volailiy surface Suppose we have a vecor of implied volailiies, of a-he-money caples, from he marke. Furhermore, we assume ha all he caples have he same ime o mauriy. This means ha we consider a vecor of daes [T, T 1,..., T N1 ] of caple s rese daes and heir implied volailiies [σimpl, σ1 impl,..., σn1 impl ]. Each caple is payed a T i + τ. For exemple we can use he 6-monhs caples for mauriies from 1year o 2years. Since he volailiy funcion σ( inervene direcly in he expression of he average volailiy σ avg of he forward discoun bond F(, we firs ranslae he caple s volailiies σ impl ino average volailiies σ avg, by a biscecion mehod for example : [σ impl, σ 1 impl,..., σ N1 impl ] [σ avg, σ 1 avg,..., σ N1 avg ] Then, he simples approach is o ake he funcion σ( o be piecewise consan. We use he same daes T i o define he volailiy funcion : ]T i1, T i ], σ( = σ i where σ i is a posiive consan. (we se T 1 =. We recall ha σ( is relaed o σ avg by : i σ( 2 e 2a d = T i [ σ i avg B(, T i + τ B(, T i ] 2 hen : i1 j= σj+1 2 e 2aT j+1 e 2aT j 2a = T i [ aσavg i ] 2 e at i (1 e aτ

5 7 pages 5 So, for i > 1: σ 2 i = ( 2a aσ e 2aT i e 2aT i1 T i 2 ( avg aσ i1 2 avg i T e at i (1 e aτ i1 e at i1 (1 e aτ and for i = 1 : σ 1 = aσ 1 avg e at 1 (1 e aτ 2aT1 e 2aT Trinomial Tree The mehodology o consruc a recombining rinomial ree is almos he same in he case of consan volailiy (described in [1]. Iniialy we will assume ha he volailiy parameers σ( and a have been chosen in he way described above o mach he caple volailiy surface. Then, he only difference beween he wo cases (consan and ime dependan volailiy is in he expression of he second momen of he process x( (recall ha r( = α( + x(. In fac we use he approximaion : V i = Var [x( i+1 x( i ] σ( i 2 ( i+1 i For furher specificaion abou he ree consrucion, see [2]. 4 Example of calibraion In his secion, we calibrae he following daa : iniial discoun facor curve and 1-year ATM caple volailiy curve for maruriies : 1y, 2y,..., 19y. 1 To es our mehod, we use he calibraion procedure of he volailiy funcion σ described in he coninous model. Then we price he same caples using he rinomial ree. The able 1 repors he resuls of he calibraion and ree pricing. Model Volailiy is he caple volailiy compued by rinomial ree, o be compared wih he Marke Volailiy. We used a ree wih 1 seps per year. We also repor he percen error. 1 The caple volailiy curve is aken from he book [4].

6 7 pages 6 Mauriy Marke Volailiy Model Volailiy Nb Time Seps Percen Error % % % % % % % % % % % % % % % % % % % Table 1: Calibraion resuls. The resuls of he calibraion are prey much saisfacory, knowing ha he calibraion of he volailiy funcion is done he coninous model, hen he oupu funcion is used in he discree model rinomial ree. References [1] J.Hull, A.Whie, Numerical procedures for implemening erm srucure models I, Journal of Derivaives, Fall , 2, 5 [2] J.Hull, A.Whie, The General Hull-Whie Model and Supercalibraion. Financial Analyss Journal, Vol. 57, No. 6, November/December 21 5 [3] Kerry Back, A Course in Derivaive Securiies: Inroducion o Theory And Compuaion (Springer Finance 1, 3 [4] D. Brigo, F. Mercurio, Ineres Rae Models - Theory and Pracice: Wih Smile, Inflaion and Credi (Springer Finance 5

7 7 pages 7 References