Compuaions in he Hull-Whie Model Niels Rom-Poulsen Ocober 8, 5 Danske Bank Quaniaive Research and Copenhagen Business School, E-mail: nrp@danskebank.dk
Specificaions In he Hull-Whie model, he Q dynamics of he spo rae is given by he following sochasic differenial equaion SDE) also know as he Ohrnsein-Uhlenbeck process dr) = Θ) r)) d + σdw) ) where Θ) is he long erm level o which he spo rae, r), is moving, is he rae a which he spo rae rae is pushed owards he long erm level and W) is a Brownian moion under Q. σ is he consan volailiy of changes in he spo rae. σ is assumed consan in his noe. Using he spo rae defined by ), we consruc a money marke accoun by A) = e Ê rudu ) da) = r A)d 3) The specificaion of he spo rae, means ha he Hull-Whie model belongs o he affine class of ineres rae models and hus prices of zero coupon bonds a ime for he ime T mauriy have he following form The T-yield a ime, y, T) is defined as P, T) = e α,t)+β,t)r 4) y, T) = lnp, T) T We generally wan he model o be calibraed o he marke oday meaning ha model prices of zero coupon bonds oday, P, T) T, is equal o he prices observed in he marke. This can be achieved by choosing Θ) in equaion ) so ha he iniial yield curve is mached. Closed orm Soluion for Prices of Zero Coupon Bonds We will now find explici formulas for he funcions α, T) and β, T) in 4) and hus closed form soluions for zero coupon bonds in he Hull-Whie model. irs, however, we derive he fundamenal parial differenial equaion for zero coupon prices in he Hull- Whie model. Sar by finding he dynamics of zero coupon prices by employing Io s lemma. dp, T) = P P d + r dr) + P dr)) r Insering he spo rae dynamics ) yields ) dp, T) α, T) β, T) = + r) d + β, T)dr) + P, T) β, T)dr)) α, T) β, T) = + r) + β, T)Θ) β)r + ) β, T)σ d +β, T)σdW)
We now use he dynamics of he money marke accoun given by 3) and he dynamics of he zero coupon bonds in 5) o find he dynamics of deflaed zero coupon bond prices {}}{ da)dp, T) = A)dP, T) + P, T)dA) + da)dp, T) Again insering wha is know we ge = da)dp, T) A)P, T) = α, T) + β, T) r) + β, T)Θ) β)r + β, T)σ r ) d + β, T)σdW) Under he equivalen maringale measure, Q, deflaed prices are maringales. According o he maringale represenaion heorem we mus hus have ha he d-erm mus be equal o zero, and his holds for all and r. Thus α, T) + β, T)Θ) + β, T)σ = β, T) αt, T) = β, T) = βt, T) = 5) 6) 7) 8) We solve he wo ordinary differenial equaions by firs posulaing a soluion for β, T) β, T) = ) e T ) 9) I is easy o check ha he soluion in 9) in fac solves he ODE in 7) subjec o he boundary condiion in 8). Since he derivaive of α, T) only depends on β, T) simple inegraion of αu,t) u beween and T is a soluion o 5). Recall ha and hus we have α, T) du = αu, T)] T = αt, T) α, T) ) α, T) = βu, T)Θu)du + β u, T)σ du ) As menioned above we wan he model o mach curren zero coupon prices. This is done by choosing Θu) in ) so ha he iniial yield curve is mached. Insead of calibraing he model o zero coupon yields direcly, we calibrae he model o he erm srucure of forward raes. orward raes are defined as rom 9) we ge f M lnp, T), T) = α, T) β, T)r β, T) = e T ) )
Using Leibniz s rule for differeniaing inegrals we have from ), 6) and 8) T α, T) = βt, T)ΘT) + β T, T)σ + σ βu, T)Θu)du βu, T) βu, T)du Insering 9) and ) yields T α, T) = = Puing hings ogeher we ge e T u) Θu)du σ e T u) Θu)du σ ) e T u) e T u) du e T ) e T)] f M, T) = e T u) Θu)du 3) + σ e T ) e T)] + e T r 4) To isolae ΘT) we differeniae wih respec o T f M, T) Using 3) his can be wrien as And hus f M, T) = ΘT) e T u) Θu)du + σ e T e T) e T r = ΘT) f M, T) + σ e T)) e T))] +e T r + σ e T e T) e T r = ΘT) f M, T) σ ΘT) = fm, T) e T ) + f M, T) + σ e T ) 5) Now ha we know ΘT) we can plug i ino ) o find an expression for α, T). We compue firs he inegral σ = σ = σ β u, T)du e T u) ) du e T )) + T ) e T ))] 6) 3
Nex we compue he inegral Insering 5) yields βu, T)Θu)du = T + σ = f e T u) M, u) u ) e T u) Θu)du e T u) Θu)du Compuing he las inegral yields σ e T u) ) e u ) du = We now have σ 3 ) + f M, u) du Θu)du f M, u) f M, u)du u e T u) ) e u ) du 7) e T ) + e T e T+) + ] e σ T ) + σ 3 σ βu, T)Θu)du = e T u)fm, u) T du + e T u) f M, u)du u f M, u) ] T f M, u)du e T ) + e T e T+) + ] e T ) 8) Now use he inegraion by pars formula on he firs erm on he righ hand side in equaion 8) T f e T u) M ), u) du = ] T e T u) f M, u) u e T u) f M, u)du o ge + βu, T)Θu)du = ] T e T u) f M, u) + σ 3 e T u) f M, u)du e T u) f M, u)du f M, T) f M, ) ) f M, u)du e T ) + e T e T+) + ] e σ T ) 9) 4
Now simplifying gives + σ 3 βu, T)Θu)du = f M, )β, T) f M, u)du e T ) + e T e T+) + ] e σ T ) ) Combining he wo inegrals 6) and ) we ge α, T) = Simplifying yields βu, T)Θu)du + σ β, T)du = f M, )β, T) f M, u)du + σ 3 e T ) + e T e T+) + ] e σ + σ e T )) + T ) T e ))] T ) We also have ha α, T) = f M, )β, T) f M, u)du + σ 4 β, T) e ) ) P, T) = f,u)du which leads o and hus and we have ln lnp, T) = f, u)du ) P, T) T = f, u)du P, ) α, T) = f M, )β, T) + ln ) P, T) P, ) + σ 4 β, T) e ) ) 3 Solving he Sochasic Differenial Equaion The soluion o he SDE in equaion ) can be found by employing Io s lemma o find he dynamics of e r) and hen inegraing up. This yields he soluion o ) r T = e T ) r + e T u) Θu)du + σ e T u) dw u 5
Since E Q σ e T u) dw u ] =, we have Noice ha E Q r T] = e T ) r + e T u) Θu)du Var Q r T] = E Q e T u) Θu)du = Liebniz s rule for differeniaing inegral gives ) ] r E Q r T] = σ e T u) du βu, T)Θu)du = βt, T)ΘT) + βu, T)Θu)du βu, T)Θu)du where he firs erm on he righ hand side is equal o zero according o 8). Thus we have e T u) Θu)du = Differeniaing ) wih respec o T we ge βu, T)Θu)du e T u) Θu)du = f M, ) β, T) + f M, u)du σ ) e T ) e T + e T+) = f M, )e T ) + f M, T) fm, u) }{{} σ ) e T ) e T + e T+) Now add and subrac σ e kt ) and simplify o ge where e T u) T ) Θu)du = γt) γ)e γ) = f M, ) + σ e k) Thus he condiional expecaion of he fuure spo rae can now be wrien as And he condiional varians is given by E Q r T] = e T ) r + γt) γ)e T ) 3) Var Q = σ e T )) 4) = 6
4 loaers and Caples In his secion he Hull-Whie model is used o value a fuure sochasic rae oday. We wan o compue he price of a paymen received a ime T A. This paymen covers ineres over he period from ime T A o T A. The paymen is no known unil ime T where i is fixed as he simple rae over he period T o T. T T A T A T The following noaion is used T : Sar dae of fixing period T : End dae of fixing period T A : Sar dae of accrual period T A : End dae of accrual period : Accrual period in years T A T A ) : ixing period in years T T ) A ime T he spo LIBOR rae over he fixing period is given by LT,T ) = ) PT,T ) 5) Since he accrual period is he paymen received a ime T A is equal o LT,T ). The value a ime of he unknown paymen is given by V ) = E Q e Ê ] T A r u du LT,T ) 6) = E Q = E Q A T r u du LT,T )E Q T ] r u du PT,T A ) LT,T ) ]] r u du Changing from he risk-neural measure o he forward measure wih he zero-coupon bond mauring a ime T as numeraire yields V ) = P, T )E T 7) 8) PT, T A ) LT,T ) ] 9) which is he same as V ) = P, T ) E T PT, T A )] ) PT,T ) PT,T A ) { = P, T )E T PT, T A )] } ) PT,T P, T A ) ) In he Hull-Whie model, he raion PT,T A )/PT,T ) is equal o PT,T A ) PT,T ) = eαt,ta ) αt,t )+βt,ta ) βt,t ))r T 3) 7
Which is used o ge { V ) = P, T )e αt,ta ) αt,t ) E T The only unknown objec in Equaion 3) is E T ] } e βt,ta ) βt,t ))r T P, T A ) 3) ] e βt,ta ) βt,t ))r T, bu his expecaion can easily be compued since r T is normally disribued. )] E T exp βt,t A ) βt,t ))r T = exp βt,t A ) βt,t ))E T r T ] + ) βt,t A ) βt,t )) Var T r T ] which can be insered ino 3) o ge a closed form soluion for he value of a single floa paymen. ixing a an Average Rae V ) = E Q = M A r u du M i= E Q M A ] M LT i, T i) i= r u du LT i, T i) The las expression has exacly he same form as he expression found in he previous secion and i s value is hus know in closed form. Caple Pricing wih Unnaural Time Lag We will now find he value of a caple ha caps he simple compounded ineres rae given in Equaion 5). The rae is fixed over he period from T o T bu i is paid ou a ime T A. cpl, T,T,T A ) = E Q = P, T A )E TA = P, T A )E TA = P, T A )E TA A r u du LT, T ) K ) ] + = P, T A )e αt,t ) E TA LT, T ) K ) + ] ) ] + PT,T ) + K) ) ] + e αt,t ) βt,t )r T + K) ] 3) ) ] + e βt,t )r T e αt,t ) + K) Since r T ΦM r, V r ), where M r and V r is he mean and variance of r T respecively and Φ is he sandard cumulaive normal disribuion funcion, we have βt,t )r T Φ βt,t )M r, βt,t ) Vr ). rom Brigo and Mercurio ) we have he following for a lognormally disribued sochasic variable X wih mean M and variance V E X K) +] ) ) = e M+ V M lnk) + V M lnk) Φ KΦ 33) V V 8
The caple value can now be compued wih M = βt,t )M r and V = βt,t )V r which yields cpl, T,T,T A ) = P, T A )e αt,t ) e βt,t )Mr+ βt,t ) V r Φd ) ) e αt,t ) + K) Φ d βt,t ) ) )V r 34) where d = βt,t )M r lne αt,t ) + K)) + βt,t ) Vr βt,t 35) )V r 9
References Brigo, D. and. Mercurio ): Springer, firs ediion. Ineres Rae Models Theory and Pracice,