INTEREST RATES AND FX MODELS

INTEREST RATES AND FX MODELS 5. Shor Rae Models Andrew Lesniewski Couran Insiue of Mahemaics New York Universiy New York March 3, 211

2 Ineres Raes & FX Models Conens 1 Term srucure modeling 2 2 Vasicek s model and is descendans 3 2.1 Modeling mean reversion of raes................. 3 2.2 One-facor Hull-Whie model.................... 5 2.3 Two-facor Hull-Whie model................... 6 3 Zero coupon bonds 7 4 Opions on a zero coupon bond 8 5 Pricing under he Hull-Whie model 9 6 Applicaion: Eurodollar / FRA convexiy correcion 9 1 Term srucure modeling The real challenge in modeling ineres raes is he exisence of a erm srucure of ineres raes embodied in he shape of he forward curve. Fixed income insrumens ypically depend on a segmen of he forward curve raher han a single poin. Pricing such insrumens requires hus a model describing a sochasic ime evoluion of he enire forward curve. There exiss a large number of erm srucure models based on differen choices of sae variables parameerizing he curve, number of dynamic facors, volailiy smile characerisics, ec. Time permis us o discuss erm srucure modeling only in is crudes ouline, and we focus on wo approaches: (a) Shor rae models, in which he sochasic sae variable is aken o be he insananeous forward rae. Hisorically, hese were he earlies successful erm srucure models. We shall focus on a racable Gaussian model, namely Vasicek s model and is descendens. (b) LIBOR marke model, in which he sochasic sae variable is he enire forward curve represened and as a collecion of benchmark LIBOR forward raes. These, more recenly developed, models are descendans of he HJM model and have been popular among he praciioners.

Shor Rae Models 3 Shor raes models use he insananeous spo rae r () as he basic sae variable. The sochasic differenial equaion describing he dynamics of r () is usually saed under he spo measure, and has he form dr () = A (r (), ) d + B (r (), ) dw (), (1) where A and B are suiably chosen drif and diffusion coefficiens, and W is he Brownian moion driving he process. Models of his ype are referred o as one-facor models, as here is only one sochasic drivers; models wih muliple sochasic drivers are called muli-facor models. Various choices of he coefficiens A and B lead o differen dynamics of he insananeous rae. You should consul he lieraure cied a he end of hese noes for a complee caalog of choices available in he reperoire. We shall focus on he Vasicek model and is descenden, he Hull-Whie model. 2 Vasicek s model and is descendans 2.1 Modeling mean reversion of raes The simples erm srucure model of any pracical significance is Vasicek s model. Under he spo measure Q, is dynamics is given by: ogeher wih he iniial condiion: dr () = λ (µ r ()) d + σdw (), (2) r () = r. (3) Originally, his process has been sudied in he physics lieraure, and is known as he Ornsein - Uhlenbeck process. A special feaure of Vasicek s model is ha he sochasic differenial equaion (2) has a closed form soluion. In order o find i we uilize he mehod of variaions of consans. The homogeneous equaion dr () = λr () d has he obvious soluion: r () = Ce λ, (4)

4 Ineres Raes & FX Models wih C an arbirary consan. Seeking a paricular soluion o he inhomogeneous equaion in he form of (4) wih he consan C replaced by an unknown funcion ψ (), r 1 () = ψ () e λ, we find readily ha ψ () has o saisfy he ordinary differenial equaion: Consequenly, dψ () = λµe λ d + σe λ dw (). ψ () = µe λ + σ e λs dw (s). The soluion o our problem is he sum of he soluion (4) wih C = r µ (in order o enforce he iniial condiion) and he paricular soluion r 1 (): r () = r e λ + µ ( 1 e λ) + σ e λ( s) dw (s). (5) To undersand beer he meaning of his soluion, we noe ha he expeced value of he insananeous rae r () is E Q [r ()] = X e λ + µ ( 1 e λ), (6) while is variance is Var [r ()] = σ2 ( ) 1 e 2λ. (7) 2λ This means ha, on he average, as, X () ends o µ, and his limi is approached exponenially fas. This propery is referred o as mean reversion of he shor rae. The rae of mean reversion is equal o λ, and he ime scale τ on which i akes place is given by he inverse of λ, τ = 1/λ. Random flucuaions inerfering wih he mean reversion are of he order of magniude σ/ 2λ. This ends o zero, as λ, and hus srongly mean revering processes are characerized by low volailiy. Elegan and simple as i is, he Vasicek model has a number of serious shorcomings: (a) I is impossible o fi he enire forward curve as he iniial condiion. (b) There is one volailiy parameer only available for calibraion (wo, if you coun he mean reversion rae). Tha makes fiing he volailiy srucure virually impossible.

Shor Rae Models 5 (c) The model is one-facor, meaning ha here is only one sochasic driver of he process. (d)) Wih non-zero probabiliy, raes may become negaive (ypically, his probabiliy is fairly low). Some of hese shorcomings can be easily overcome by means of a sligh exension of he model. 2.2 One-facor Hull-Whie model A suiable generalizaion of he Ornsein-Uhlenbeck process (2) is a process which mean revers o a ime dependen level µ () raher han a consan µ. Such a process is given by ( ) dµ () dr () = + λ (µ () r ()) d + σ () dw (), (8) d where we have also allowed σ o be a funcion of ime. The presence of he ime derivaive of µ () in he drif is somewha surprising. However, solving (8) (using again he mehod of variaion of consans) yields and hus r () = r e λ + µ () µ () e λ + e λ( u) σ (u) dw (u), (9) E Q [r ()] = r e λ + µ () e λ µ (), (1) Var [r ()] = Tha shows ha E [r ()] µ (), as. Noe ha (9) implies ha r () = r (s) e λ( s) + µ () µ (s) e λ( s) + e 2λ( u) σ (u) 2 du. (11) s e λ( u) σ (u) dw (u), (12) for any s <. We shall use his fac in he following. Le us now choose µ () = r (), i.e. ( ) dr () dr () = + λ (r () r ()) d + σ () dw (), (13) d

6 Ineres Raes & FX Models where r () = r () = r. This process is called he exended Vasicek (or Hull- Whie) model. From (12), r () = r () + e λ( s) (r (s) r (s)) + s e λ( u) σ (u) dw (u), (14) and so he insananeous rae is represened as a conribuion from he curren yield curve plus a random perurbaion. This represenaion of r () implies ha E Q s [r ()] = r () + e λ( s) (r (s) r (s)). (15) In paricular, choosing s = in (14) we obain r () = r () + 2.3 Two-facor Hull-Whie model e λ( u) σ (u) dw (u). (16) In he wo-facor Hull-Whie model, he insananeous rae is represened as he sum of (a) he curren rae r (), and (b) wo sochasic sae variables r 1 () and r 2 (). In oher words, r () = r () + r 1 () + r 2 (). A naural inerpreaion of hese variables is ha r 1 () conrols he levels of he raes, while r 2 () conrols he seepness of he forward curve. We assume he sochasic dynamics: dr 1 () = λ 1 r 1 () d + σ 1 () dw 1 (), dr 2 () = λ 2 r 2 () d + σ 2 () dw 2 (), (17) where σ 1 () and σ 2 () are he insananeous volailiies of he sae variables r 1 () and r 2 (), respecively. The wo Brownian moions are correlaed, E [dw 1 () dw 2 ()] = ρ d. (18) The correlaion coefficien ρ is ypically a large negaive number (ρ.9) reflecing he fac ha seepening curve moves end o correlae negaively wih parallel moves.

Shor Rae Models 7 3 Zero coupon bonds The key o all pricing is he abiliy o compue he forward prices of zero coupon bounds P (, T ). Recall ha P (, T ) = E Q [e ] T r(u)du, where he subscrip indicaes condiioning on F. Wihin he Hull-Whie model (and hus, by exension, in he Vasicek model), his expeced value can be compued in closed form! Le us consider he one-facor case. We proceed as follows: where E Q [e T = E Q r(u)du ] [e T (r (u)+e λ(u ) (r() r ())+ ] u e λ(u s) σ(s)dw (s))du = e T r (u)du h λ (T )(r() r ()) E Q [e T u ] e λ(u s) σ(s)dw (s)du, h λ () = 1 e λ. λ Inegraing by pars we can ransform he double inegral in he exponen ino a single inegral T u e λ(u s) σ (s) dw (s) du = Finally, using he fac ha [ E e ] T φ(s)dw (s) T = e 1 2 h λ (T s) σ (s) dw (s). T φ(s) 2 ds, (19) we obain he following expression for he price of a zero coupon bond: where P (, T ) = A (, T ) e h λ(t )r(), (2) A (, T ) = e T r (u)du+h λ (T )r ()+ 1 2 Noe in paricular ha he discoun facor P (, T ) has he form T h λ (T s) 2 σ(s) 2 ds. (21) P (, T ) = P (, T ) = e T r (s)ds+ 1 2 T h λ(t s) 2 σ(s) 2 ds. (22)

8 Ineres Raes & FX Models The generalizaion of formula of (2) o he case of he wo-facor Hull-Whie model reads: where now P (, T ) = A (, T ) e h λ 1 (T )r 1 () h λ2 (T )r 2 (), (23) A (, T ) = e T e 1 2 r (u)du T (h λ1 (T s) 2 σ 1 (s) 2 +2ρh λ1 (T s)h λ2 (T s)σ 1 (s)σ 2 (s)+h λ2 (T s) 2 σ 2 (s) 2 )ds. (24) 4 Opions on a zero coupon bond Using he above expressions for he zero coupon bond, i is possible o derive explici, closed form expressions for valuaion of European opions on such bonds. The calculaions are elemenary, if a bi edious, and we shall defer hem o he nex homework assignmen. We focus on he one facor Hull-Whie model; i is sraighforward o exend hese calculaions o he wo facor model. Consider a call opion sruck a K and expiring a T on a zero coupon bond mauring a T ma > T. Then, is price is equal o where wih PV call = P (, T ma ) N (d + ) KP (, T ) N (d ), (25) d ± = 1 σ log P (, T ma ) P (, T ) K ± σ 2, (26) ( T 1/2 σ = e 2λ(T s) σ (s) ds) 2 h λ (T ma T ). (27) Similarly, he price of a pu sruck a K is given by PV pu = KP (, T ) N ( d ) P (, T ma ) N ( d + ). (28) Since floorles and caples can be hough of as calls and pus on FRAs, hese formulas can be used as building blocks for valuaion of caps and floors in he Hull-Whie model.

Shor Rae Models 9 5 Pricing under he Hull-Whie model A erm srucure model has o be calibraed o he marke before i can be used for valuaion purposes. All he free parameers of he model have o be assigned values, so ha he model reprices exacly (or close enough) he prices of a seleced se of liquid vanilla insrumens. In he case of he Hull-Whie model, his amouns o (a) Maching he curren forward curve, which is accomplished by choosing r () o mach he curren forward curve. (a) Maching he volailiies of seleced opions. This is a bi more difficul, and we proceed as follows. We choose he insananeous volailiy funcion σ () o be locally consan. Tha means ha we divide up he ime axis ino, say, 3 monh period [T j, T j+1 ) and se σ () = σ j, for [T j, T j+1 ). Now, we selec sufficienly many calibraing insrumens, so ha heir number exceeds he number of he σ s. Nex, we opimize he choice of σ s and λ, requiring ha he suiable sum of pricing errors is minimal. Despie he simple srucure of he Hull-Whie model, mos insrumens canno be priced by means of closed form expressions such as hose for caps and floors of he previous secion. One has o resor numerical echniques. Among hem, wo are paricularly imporan: (a) Tree mehods. (a) Mone Carlo mehods. Time does no permi us o discuss hese numerical echniques, and I defer you o lieraure cied a he end of hese noes. 6 Applicaion: Eurodollar / FRA convexiy correcion As a simple applicaion of he Hull-Whie model, we shall now derive a closed form expression for he Eurodollar / FRA convexiy correcion discussed in Lecure 4.

1 Ineres Raes & FX Models We know from Lecures 2 and 3 ha he (currenly observed) forward rae F (T 1, T 2 ) is he expeced value of ( ) 1 1 δ P (T 1, T 2 ) 1 (29) under he T 2 -forward measure Q T2. This is, indeed, almos he definiion of he T 2 -forward measure! Consequenly, F (T 1, T 2 ) is given by: F (T 1, T 2 ) = 1 ( ) 1 δ P (T 1, T 2 ) 1 = 1 ( ) P (, T 1 ) δ P (, T 2 ) 1. (3) I is easy o calculae i wihin he Hull-Whie model. Le us firs consider he one-facor case. Using (22), we find ha F (T 1, T 2 ) = 1 ( T2 T r e (s)ds+ 1 1 2( T 2 h λ (T 2 s) 2 σ(s) 2 ds T 1 ) h λ (T 1 s) 2 σ(s) 2 ds) 1. δ (31) On he oher hand, he rae F fu (T 1, T 2 ) implied from he Eurodollar fuures conrac is given by he expeced value of (29) under he spo measure Q. We have explained his fac in Lecure 4, aribuing i o he pracice of daily 1 margin accoun adjusmens by he Exchange. In order o calculae his expeced value we proceed as in he calculaion leading o he explici formula for P (, T ): E Q [ ] T2 T r()d e 1 [ = E Q T2 T e 1 (r ()+ ] e λ( s) σ(s)dw (s))d T2 T r = e ()d 1 E Q T2 T r = e ()d 1 E Q [ e T 2 h λ (T 2 s)σ(s)dw (s) T 1 h λ (T 1 s)σ(s)dw (s) [ T1 e (h λ(t 2 s) h λ (T 1 s))σ(s)dw (s)+ ] T 2 T h λ (T 2 s)σ(s)dw (s) 1 T2 T r = e ()d+ 1 1 2( T 1 (h λ(t 2 s) h λ (T 2 s)) 2 σ(s) 2 ds+ T 2 T h λ (T 2 s) 2 σ(s) 2 1 ds), ] 1 which we model as coninuous

Shor Rae Models 11 and so F fu (T 1, T 2 ) = 1 ( T2 e δ = F (T 1, T 2 ) T 1 r ()d+ 1 2( T 1 (h λ(t 2 s) h λ (T 2 s)) 2 σ(s) 2 ds+ T 2 + 1 δ (1 + F (T 1, T 2 )) T 1 h λ (T 2 s) 2 σ(s) 2 ds) 1 ) ( e T 1 ) [h λ (T 2 s) 2 h λ (T 2 s)h λ (T 1 s)]σ(s) 2 ds 1. Consequenly, he Eurodollar / FRA convexiy adjusmen is given by ED / FRA (T 1, T 2 ) = 1 δ (1 + F (T 1, T 2 )) 1 δ T1 ( e T 1 ) [h λ (T 2 s) 2 h λ (T 2 s)h λ (T 1 s)]σ(s) 2 ds 1 [ hλ (T 2 s) 2 h λ (T 2 s) h λ (T 1 s) ] σ (s) 2 ds. (32) In he case of a consan insananeous volailiy, σ () = σ, he las inegral can be evaluaed in closed form, and he resul is: ED / FRA ( σ2 (1 e 2λT 1 )( ) 1 e λ(t 2 T 1 ) 2 2λ 3 δ + ( 1 e λ(t 2 T 1 ) )( 1 e λt 1 ) 2 ). (33) This formula is very easy o implemen in compuer code. The calculaions in he case of he wo-facor Hull-Whie model are similar, if a bi more edious. The corresponding formula reads: ED / FRA (T 1, T 2 ) = 1 δ T1 ρ jk h λj (T 2 s) 1 j,k 2 [h λk (T 2 s) h λk (T 1 s)] σ j (s) σ k (s) ds, (34) where ρ 11 = ρ 22 = 1, ρ 12 = ρ 21 = ρ. Noe ha for any real value λ, h λ (s) is non-negaive and monoone increasing. Therefore, he convexiy adjusmens implied by he Hull-Whie model are always posiive (as hey should be!).

12 Ineres Raes & FX Models References [1] Andersen, L., and Pierbarg, V.: Ineres Rae Modeling, Vol. 2, Alanic Financial Press (21). [2] Brigo, D., and Mercurio, F.: Ineres Rae Models - Theory and Pracice, Springer Verlag (26). [3] Hull, J.: Opions, Fuures and Oher Derivaives Prenice Hall (25).