Hull-White one factor model Version
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1 Hull-Whie one facor model Version Inroducion This plug-in implemens Hull and Whie one facor models. reference on his model see [?]. For a general 2 How o use he plug-in In he Fairma user inerface when you creae a new sochasic process you will find he addiional opion Hull-Whie (one facor). The sochasic process is defined by he parameers shown in he following able. The sochasic process is defined by he parameers shown in he following able. Fairma noaion alpha sigma adjused drif Documenaion noaion α σ d 3 Implemenaion Deails 3.1 Process dynamics The Hull-Whie one facor model is used o describe he evoluion of he shor rae. I is defined by he following sochasic differenial equaions dr() = (θ() αr() + d)d + σdw (1) where r is he shor rae process. The parameers have he following inerpreaion α is relaed o he shor rae mean reversion speed, he greaer is value, he sronger is he mean reversion 1
2 3 Implemenaion Deails σ is he shor rae volailiy d is an opional adjusmen which can be added o he drif. This adjusmen may be usefull o calculae he quano adjusmen. 3.2 Fiing he model o iniial yield curve If F (0, T ) is he funcion of insananeous forward rae, ha is F (0, T ) = ln[p (0, T )] (2) T i can be shown ha in he Hull-Whie model i is given by T F (0, T ) = e αt r(0) + e α(t u) θ(u)du σ2 [ αt ] 2 0 2α 2 so ha θ(t ) can be wrien as (3) θ(t ) = αf (0, T ) + σ2 [ F (0, T ) + 2αT ] (4) T 2α The funcion θ(t ) is hen calculaed from he iniial zero rae yield curve. 3.3 Soluion o he SDE, mean and variance of he process Knowing he process a ime, he process a ime + τ is given by r( + τ) = r()e ατ + e α(+τ) e αu θ(u)du + σe α(+τ) e αu dw (u) (5) = r()e ατ + a( + τ) e ατ a() + σe α(+τ) e αu dw (u) (6) where we have used he formula relaing θ(t ) and F (0, T ) and where he funcion a() is defined as a() = F (0, ) + σ2 2α 2 [ α ] 2 (7) From hese formulas we know ha r() is normally disribued and we can calculae he expeced value and he variance for r. E[r( + τ) r()] = r()e ατ + a( + τ) e ατ a() (8) Var[r( + τ) r()] = σ2 [ 2ατ ] (9) 2α 2
3 4 Calibraion Defining 1 = and 2 = + τ, formula 6 can be rewrien as 2 r( 2 ) = r( 1 )e α(2 1) + a( 2 ) e α(2 1) a( 1 ) + σe α2 e αu dw (u) = r( 1 )e α(2 1) + F (0, 2 ) e α(2 1) F (0, 1 )+ + σ2 α 2 e α2 [cosh(α 2 ) cosh(α 1 )] + σe α e αu dw (u) (10) which is useful in simulaing Mone Carlo pah for Hull-Whie model, in fac r( 2 ) is disribued as r( 2 ) r( 1 )e α(2 1) + F (0, 2 ) e α(2 1) F (0, 1 )+ + σ2 α 2 e α2 [cosh(α 2 ) cosh(α 1 )] + + σ 2α 2α(2 1) N(0, 1). (11) 3.4 Bond Price formula The price of a zero coupon bough a ime which pays a uni amoun a ime T is given by P (, T ) = A(, T )e B(,T )r() (12) where B(, T ) = 1 [ α(t )] (13) α A(, T ) = P (0, T ) [ P (0, ) exp ln[p (0, )] B(, T ) σ2 ( e αt 4α 3 e α) 2 ( e 2α 1 ) ] (14) 4 Calibraion The plug-ins conains also an exension for calibraing Hull and whie one facor model from a marix of cap volailiies. 4.1 Objecive Funcion The esimaor ries o minimize he differences beween he Black-cap prices and he prices of caps calculaed hrough he model. More explicily he opimizaion uses he following objecive funcion n i=1 ( HW 1 i (α 1, σ 1 ) Black i) 2, (15) 3
4 4 Calibraion where HW 1 i (α 1, σ 1 ) is he price of he i h -cap by one facor H&W model, Black i he price of he i h cap by he Black model, and n he number of all caps ino he Cap-Volailiy marix. 4.2 Daa for calibraion The cap based calibraion uses only some specific fields of he file Ineres rae marke daa xml. In paricular: Marke: a sring describing he marke o which daa refer. Possible choices are EU for Europe US for USA UK for Unied Kingdom SW for Swizerland GB for Unied Kingdom JP for Japan Dae: is he dae o which daa refer, he forma is ddmmyyyy ZRMarke: vecor wih zero coupon raes (coninuously compounded raes) ZRMarkeDaes: vecor of mauriies corresponding o ZRMarke CapTenor: is he year fracion beween wo paymens of he cap opions CapMauriy: vecor of cap mauriies (i.e. he rows of he following CapVolailiy marix) CapRae: vecor of cap srikes (i.e. he columns of he following CapVolailiy marix) CapVolailiy: marix of cap-volailiies, i.e. he elemen CapVolailiy[i, j] is he Black model volailiy of cap wih mauriy equal o CapMauriy[i] and srike equal o CapRae[j] An example of an Ineres rae marke daa xml conaining only hese fields is he following <IneresRaeMarkeDaa> <Marke>EU</Marke> <Dae> </Dae> <ZRMarke> </ZRMarke> <ZRMarkeDaes> </ZRMarkeDaes> <CapTenor>0.5</CapTenor> <CapMauriy> </CapMauriy> <CapRae> </CapRae> <CapVolailiy>
5 References </CapVolailiy> </IneresRaeMarkeDaa> References 5
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