s Praciioner Course: Ineres Rae Models March 29, 2009
In order o value European-syle opions, we need o evaluae risk-neural expecaions of he form V (, T ) = E [D(, T ) H(T )] where T is he exercise dae, and he payoff is H(T ), which is F T - measurable. Terminal Correlaion When he payoff is ineres-sensiive, evaluing his involves no only he expeced values P(, T ) = E [D(, T )] and E [H(T )] as in Black-Scholes analysis for sock opions, bu also he erminal correlaion beween he payoff and he sochasic discoun facor. s V (, T ) = P(, T ) E [H(T )] + ρ var [D(, T )] var [H(T )]
Change of Alernaively, we can inroduce a convenien change of measure, given by he Radon-Nikodym derivaive whereby dq T = D(, T ) Q P(, T ) V (, T ) = P(, T ) E T [H(T )] This is called he T -forward measure, which reflec he fac ha E T [r(t )] = f (, T ); i.e. he forward rae is he risk-neural expecaion of he spo rae under he T -forward measure. s In he s, his change of measure is merely a formaliy, because we sar off in he forward measure and never look back.
s Simple Spo & Forward For boh pracical and echnical reasons, we need o sar our reamen of he s in a discree seing. We have already defined he simple spo rae L(, T ). P(, T ) = 1 1 + τ(, T ) L(, T ) Similarly, he simple forward rae beween daes T and S > T is defined by P(, S) P(, T ) = 1 1 + τ(t, S) F (; T, S) and F (T ; T, S) = L(T, S)
s Paymen in Arrears Consider a European-syle derivaive wih exercise dae T and an F T -measurable payoff H(T ) = h(l(t, S)) which pays in arrears on dae S. The value of his derivaive oday is V (, T, S) = E [D(, S) h(l(t, S))] = P(, S) E S [h(f (T ; T, S))] (1) In order o evaluae (1), we need o firs specify he process for F (; T, S) under he S-forward measure.
s Since we can wrie F (; T, S) as he value of an asse in erms of he numéraire associaed wih he S-forward measure P(, S), F (; T, S) = 1 τ(t,s) P(, T ) 1 τ(t,s) P(, S) P(, S) we know ha F (, T, S) mus be a maringale under he S-forward measure. For example, we could impose df (; T, S) = σ S () F (; T, S) dw S T (2) ensuring ha he forward rae is non-negaive and ha he evoluion is Markovian.
s The model in (2) can be inegraed o yield log L(T, S) F ( N log F (, T, S) 1 2 T under he S-forward measure. T ) σs 2 ( ) d, σs 2 ( ) d Black Formulæ And since he payoffs for caples and floorles are of he form h(l(t, S)) = τ(t, S) (L(T, S) K) + reminiscen of calls and pus, he lognormal process leads direcly o he Black formulæ.
Caps & Floors A cap (floor) is jus a porfolio of caples (floorles). Each can be valued separaely under he corresponding forward measure. Consider a sequence of fixing and paymen daes, T = {T 0 =, T 1,..., T n = T }. The value of a cap is V cap (, T, K) = n P(, T i ) τ(t i, T i 1 ) E T i (L(T i 1, T i ) K) + i=1 or, subsiuing in he Black formula in he obvious fashion, s n V cap (, T, K) = P(, T i ) τ(t i, T i 1 ) i=1 Ti 1 Bl call K, F (, T i 1, T i ), σt 2 i ( ) d
Caps & Floors Cap Volailiy Typically he marke will quoe a sor-of average implied Black volailiy σ T solving V cap (, T, K) = n P(, T i ) τ(t i, T i 1 ) i=1 Bl call (K, F (, T i 1, T i ), σ T T ) i 1 s and i will be lef o he modeler o deermine how o srip hese ino he individual volailiies of he forward raes. Here, he differen measures and poenial correlaions do no presen a problem; bu parsing ou he ime-dependence beween he differen forward raes require will require modeling choices as in 6.3.1.
Insananeous Forward Rae Volailiy Le us briefly review he calibraion problem for caps & floors. Le us sar by assuming ha each forward rae is piece-wise consan. Tha is, Ti 1 i 1 σt 2 i ( ) d = σi,j 2 (T j T j 1 ) j=1 If we have daa on N differen caps, here are N differen forward rae and 1 2N (N + 1) possible insananeous volailiies (, T 1 ] (T 1, T 2 ]... (T N 1, T N ] F (T 1, T 2 ) σ 1,1 n/a... n/a F (T 2, T 3 ) σ 2,1 σ 2,2 n/a............ n/a F (T N, T N+1 ) σ N,1 σ N,2... σ N 1,N In pracice, we will need o sele on a scheme o reduce his coun o O(N) or ideally O(1). s
s Since 1. caps ell us nohing abou he (insananeous) correlaions amongs raes, and since 2. we have so far only been able o value swapions using he Jamshidian decomposiions wih is assumpion of perfec correlaions The LFM would seem o offer lile help in valuing swapions. Bu recalling an imporan resul abou he value of a swap T V swap (, T, K) = (K s(, T )) P(, T ) dτ(t ) gives us a way o proceed in he spiri of he LFM, which we will call he lognormal swap model (LSM).
Consider a European-syle derivaive wih exercise dae T and F -measurable payoff H(T ) which depends on he swap rae s(t, S) for some erm S > T according o H(T ) = h(s(t, S)). Under he change of measure dq T,S dq = B() S B(T ) T P(T, T ) dτ(t ) S T P(, T ) dτ(t ) s we have ha he value of he derivaive is S ] V (, T ) = E [D(, T ) H(T ) P(T, T ) dτ(t ) = S T T P(, T ) dτ(t ) E T,S [h (s(t, S))]
Swap Furhermore, he forward swap rae s(; T, S) is a maringale under Q T,S. We can see his because he forward swap rae is he value of an asse in erms of he new forward annuiy facor numéraire. s(; T, S) = P(, T ) P(, S) S T P(, T ) dτ(t ) s Swap Therefore, in coninuing analogy o he LFM we can impose a process such as ds(; T, S) = σ T,S () s(; T, S) dw S,T T wih s(t ; T, S) = s(t, S).
Swap Again his can be inegraed o yield log s(t, S) F ( N log s(, T, S) 1 2 T under he T, S-forward swap measure. T ) σt 2,S ( ) d, σt 2,S ( ) d Swapions For example, we can immediaely ge resuls for swapions values. s V RTR (, T, S, K) = T P(, T ) dτ(t ) Bl pu K, s(; T, S), σt 2,S ( ) d S T
Swapion Volailiy Translaions The LFM and LSM are fundamenally incompaible, and neiher are compaible wih racable shor-rae models. There are approximae schemes for ranslaing volailiies, such as Rebonao s formula. Bu forward rae volailiies alone are no sufficien o value swapions, because caps ell us nohing abou correlaions. Addiional complexiy is also apparen when we consider ha if here are N rese daes, here are poenially 1 2N (N 1) swapions. Correlaions So even if we are successful in fiing insananeous forward volailiies wih O(N) or O(1) parameers, we sill need o deal wih poenially 1 2N (N 1) insananeous correlaions 6.9 discusses various schemes for his. s