LECTURE 7 Interest Rate Models I: Short Rate Models

Transcription

1 LECTURE 7 Ineres Rae Models I: Shor Rae Models Spring Term 212 MSc Financial Engineering School of Economics, Mahemaics and Saisics Birkbeck College Lecurer: Adriana Breccia abreccia@emsbbkacuk Lecure noes will be posed a hp://wwwemsbbkacuk/for_sudens/msc_fineng/pricing_emms14p/index_hml This noe is par of a se of lecure noes, wrien by Raymond Brummelhuis, PRICING II: MARTINGALE PRICING PricingII-Lecure Noes 27)pdf available a hp://econ19econbbkacuk/fineng/rb_pricing/ 1

2 PRICING II: MARTINGALE PRICING 55 5 Lecure V: Ineres Rae Models I: Shor Rae Models The earlies ineres rae models ook as heir saring poin a sochasic model for he shor rae, or insananeous ineres rae, r defined as he rae of ineres for he infiniesimal) inerval [, + d]: 16) r d = oal ineres gained in [, + d] In pracice, one akes he yield in a 1 monh US Treasury bill, or a comparable sor-mauriy bond, as a proxy for he sor rae Saring poin of hese sor rae models is a SDE for r : 17) dr = αr, )d + σr, )dw, for given coefficiens αr, ) and σr, ) Examples 51 Imporan examples of shor rae models are: i) The Vasicek model hisorically he firs): 18) dr = αθ r )d + σdw Observe ha he Vasicek-model is mean revering since i is simply an Ornsein-Uhlenbeck process: cf Mah Mehods I), which is economically reasonable ineres raes should flucuae along a long erm mean equilibrium rae, deermined by economic equilibrium beween demand and supply), and a common feaure of all shor rae models A disadvanage of he Vasicek model is ha ineres raes can become negaive, which is economically undesirable why?), alhough i does seem o happen every once and hen ii) The Cox, Ingersoll and Ross model, or CIR-model: 19) dr = αθ r )d + σ r dw The CIR-model has he advanage ha he shor rae will say posiive ), wih probabiliy 1 iii) The Longsaff model: 11) dr = αθ r )d + σ r dw, aka he double square roo process iv) Hull and Whie s generalized Vasicek model, in which one allows he coefficiens o be deerminisic) funcions of ime: 111) dr = α)θ) r )d + σ)dw A disadvanage of he earlier models is ha hey have oo lile parameers o be able o closely fi he iniial yield curve a a given ime Making he coefficiens ime-dependen even one of hem, usually θ) enables one o have an exac fi One can similarly inroduce imedependen coefficiens in he CIR or in Longsaff s model

3 56 PRICING II: MARTINGALE PRICING iv) The lognormal model, Black, Derman and Toy, Sandmann and Sondermann): 112) d log r = α)θ) log r )d + σ)dw, wih deerminisic ime-dependen coefficiens α), θ) and σ) Ineres raes clearly say posiive, and are mean-revering, bu prices of cerain insrumens van become infinie, which is a drawback v) The Ahn and Gao model: 113) dr = αθ r )r d + σr 3/2 dw, where one could allow for deerminisic ime-dependen coefficiens also The firs aim of a shor rae model indeed, of any ineres rae model) is o price zero-coupon bonds A zero-coupon bond also called a discoun bond) is a bond which does no pay any coupons, bu which pays is nominal-, or face-, value a mauriy T The face value will usually be normalized o 1 of whaever currency we re working in: Serling, $, Euro, ec) We will le P,T, be he price a T of a -coupon wih mauriy T and face value P T,T = 1 We will suppose ha here exis -coupon bonds of all possible mauriies T This is of course no enirely realisic: firs of all, here are only a finiely many mauriies raded, and secondly, he longer-mauriy bonds will pay coupons; however, hese can be decomposed sripped ) ino a series of -coupon bonds wih mauriies corresponding o he coupon daes Noe, ha having -coupon bonds of all mauriies T means ha we are dealing wi a marke wih infiniely many in pracice, a very large number of) asses In a second sage, we would also like o price opions on bonds, like simple calls and pus on zero-coupon bonds of a given mauriy The fac ha ineres raes are sochasic, and closely correlaed wih he pay-offs will make he pricing of such opions more complicaed han for opions on simple sock The shor raes hemselves are no direcly raded, only he zero-) coupon bonds are This makes he shor rae models effecively incomplee: he basic risk-facor r is no a radable, and we can only rade r, which is basically a European derivaive on r, wih a pay-off of 1 a mauriy In paricular, we canno hedge P,T by buying or selling unis of r, we can only hedge a bond of a given mauriy by buying/selling bonds of anoher mauriy This will lead o he inroducion of a marke price of risk, which makes up for he difference in drif beween he sochasic evoluion of he ineres rae wih respec o he objecive or physical) probabiliy, and he risk-neural one Is origin is probably mos easily explained using he risk-less porfolio

4 PRICING II: MARTINGALE PRICING 57 approach from PDE-syle pricing, so we uncharacerisically, for hese lecures) do his approach firs 51 Derivaion of he Bond price equaion We suppose ha he price P,T is a funcion of ime and he shor-rae r a : P,T = pr, ; T ) This can be a poseriori jusified) An sandard applicaion of Io s lemma hen yields ha 114) dp,t = a,t P,T d + b,t P,T dw, < T, wih 115) b,t = σr, ) pr, ) and 116) a,t = all derivaives of p evaluaed in r, ) p r, 1 p pr, ) + αr, ) p r + 1 ) 2 σr, ) 2 2 p, r 2 Now consider wo zero-coupon bonds having mauriies T 1 < T 2 We ry o make a locally risk-free porfolio V = 1, P,T1 2, P,T2 Compuing he change in value beween and + d as dv = 1, dp,t1 2, dp,t2 = 1, a,t1 P,T1 2, a,t2 P,T2 ) d + 1, b,t1 P,T1 2, b,t2 P,T2 ) dw, we see ha he risk over [, + d] vanishes if: We can ake, for example, 1, 2, = b,t 2 P,T2 b,t1 P,T1 1, = b,t2 P,T2, 2,T = b,t1 P,T1 Since such a risk-free porfolio can only earn he risk-less rae, r, over [, + d], we mus have, for hese, ha or 1, a,t1 2, a,t2 ) d = r 1, P,T1 2, P,T2 ) d, a,t1 b,t2 a,t2 b,t1 ) P,T1 P,T2 = r b,t2 b,t1 ) P,T1 P,T2 Re-arranging, his gives: a,t1 r b,t1 = a,t 2 r b,t2

5 58 PRICING II: MARTINGALE PRICING In oher words, a,t r b,t = reurn on P,T r volailiy of P,T is independen of he mauriy T and can herefore only be a funcion of r and, say qr, ) This funcion q = qr, ) is called he marke price of risk: using he sandard deviaion as a measure of financial risk, we have ha qr, ) = exra reurn per uni of risk an invesor requires o hold he bond P,T a ime, if r = r for reurn on P,s = r + qr, ) volailiy of P,s ) ) Insering formulas 115) and 116) ino he relaion a,t r = qr, ), b,t we find: Proposiion 52 The bond pricing funcion pr, ; T ) saisfies he following PDE: p 117) σr, 2 p )2 + αr, ) qr, )σr, )) p r2 r = r p, < T, wih final value 118) pr, T ; T ) = 1 So we have derived a boundary value problem for he bond-price, and all ha is lef is o solve i for he various models proposed This can be done eiher by aacking he PDE direcly, or by using Feynman and Kac As an example we look a he exended Vasicek model of Hull and Whie 52 Solving he Vasicek equaion: analyic approach We ake ar, ) = αθ r) and σr, ) = σ, wih α θ and σ consans, so ha dr = αθ r )d + σdw For he marke price of risk we simply chooses a consan: qr, ) = q a consan Noe ha in ha case, αr, ) qr, )σr, ) = αθ r) qσ = αθ r), wih θ = θ qσ/α The bond price equaion becomes: P 119) + α θ r) P r + σ2 2 P = rp, on < T, 2 r2 which we have o solve wih boundary condiion 118) To do his analyically, we look for soluions having he special form: 12) e A;T ) rb;t ) To saisfy he boundary condiion, we have o have 121) AT ; T ) =, BT ; T ) =

6 Subsiuing 12) ino 119) gives: 122) PRICING II: MARTINGALE PRICING 59 A αθ B + σ2 2 B2 = r ) B αb + 1 Noe ha, for fixed mauriy T, A and B are funcions of only Deriving boh sides wih respec o r gives: 123) B αb + 1 =, and, upon subsiuing his again in 122): 124) General soluion of 123): A αθ B + σ2 2 B2 = Ce α + 1 α By he second equaion of 121), his has o be for = s I follows ha C = e αs /α, and herefore: 125) B, T ) = 1 α 1 e α T ) ) From his and 124) and 122), A can be found by inegraion: 126) A; T ) = αθ Bτ, T )dτ + σ2 2 Bτ, T ) 2 dτ Observe ha hese inegrals are auomaically if = T ; he exra minus-sign is explained by he fac ha differeniaing an inegral wr he lower bound of he inegraion domain gives minus he inegrand) Afer some calculaions one finds ha A; T ) can be wrien as: ) A; T ) = θ σ2 T )+ 1 ) 1 e α T ) θ ) σ2 σ2 ) 1 e α T ) 2 2α 2 α 2α 2 4α 3 Remembering 12) and 125), and inroducing he quaniy: we find ha: R = θ σ2 2α 2, 127) exp R s) + 1 α p Vasicek r, ; T ) = ) 1 e α T ) R r) ) ) σ2 4α 1 e α T ) 2 3

7 6 PRICING II: MARTINGALE PRICING 53 Solving he Vasicek equaion: probabilisic approach We know, by he Feynman-Kac heorem see Mah Mehods I), ha he soluion o 119) wih boundary value pr, T ; T ) = F r) is given by: 128) pr,, ; T ) = E exp ˆr τ dτ ) F ˆr T, T ) ˆr = r where he haed, risk-adjused process follows he SDE: d ˆr = αθ ˆr)d + σdw This is again an Ornsein-Ulenbeck SDE, whose soluion we know is for imes beween and T, wih iniial value r a, is given by: 129) ˆr τ = θ 1 e ατ )) + re ατ ) + σ τ ), e αs τ) dw s In paricular, we know ha r τ is Gaussian, whose mean and variance we compued in Mah Mehods I Applying 128) wih F r) = 1 gives ) ) pr, ; T ) = E exp ˆr τ dτ ˆr = r, and o be able o proceed, we have o analyze he inegraed process I T = ˆr τ dτ Insering 129), and changing order of inegraion, we find ha θ 1 e ατ )) dτ + I T = re ατ ) dτ + σ = θt ) θ α 1 e αt ) ) + r α 1 e αt ) ) + σ = θt ) θ α 1 e αt ) ) + r α 1 e αt ) ) + σ α τ e αs τ) dw s dτ dw s e αs τ) dτ s 1 e αs T ) )dw s I follows from he heory of Io inegrals wih deerminisic inegrands ha I T is again Gaussian, wih mean µ I = θt ) θ α 1 e αt ) ) + r α 1 e αt ) ), and variance ) 2 ) σi 2 = σ2 α E 1 e αs T ) )dw 2 s = σ2 α 2 1 2e αs T ) + e 2αs T ) ) ds = σ2 T ) 2σ2 α2 α 1 3 e αt ) + σ2 2α 1 3 e 2αT ) )

8 PRICING II: MARTINGALE PRICING 61 Now we know from Mah Mehods I and his is also easily verified direcly), ha if I Nµ I, σi 2 ), hen E e I) = e µ I+σ 2 I /2 Subsiuing he expressions we obained for µ I and σi 2, and re-arranging, we find once more formula 127) for p = p Vasicek = EexpI T )) The Vasicek model as hree parameers, α, θ and σ wih which o fi bond prices of all mauriies, which here are in principle an infinie number, and one doesn always ge a good fi This difficuly can be circumvened by leing he coefficiens depend deerminisically on : his will allow us o fi a leas any iniial yield curve: 54 Hull and Whie s exended Vasicek model We look a he simples varian, wih α and σ consan, and θ = θ ) a deerminisic funcion of Solving he new bond price equaion, p + α θ ) r) p r + σ2 2 p = rp, on < T, 2 r2 using for example he direc, analyical mehod, and looking for soluions of he same form 12), 121) as before, we find ha A; T ) and B; T ) have o saisfy he same equaions 123), 124), bu he laer of course wih θ ) B, T ) will be given by he same funcion 125) as before, bu 126) will have o be changed o: 13) A HW ; T ) = = θ τ)bτ, T )dτ+ σ2 2α 2 Hence, θ τ)bτ, T )dτ + σ2 Bτ; T ) 2 dτ 2 T ) 2 α eα T ) + 1 2α e2α T ) + 3 2α 131) p HW r, ; T ) = expa HW ; T ) rb; T )) where he suffix HW sands for Hull and Whie Making θ imedependen may look like inroducing an addiional complicaion, bu in fac gives an opporuniy o do: Exac Yield curve fiing: We suppose ha σ and α are already deermined from examining ime series daa for shor erm ineres raes a ypical economerical problem) Then we find θ by fiing a dae = oday ) he heoreical prices p HW r,, ; T ), where r is oday s observed shor rae, o he observed marke prices P,T = P,T Marke for all mauriies T > Taking logarihms, we find ha θ τ) has o saisfy he inegral equaion θ τ)bτ; T )dτ = log P,T + rb; T ) 1 2 σ2 T Bτ, T )2 dτ = 132) log P,T + r α 1 e αt ) σ2 2α T α e αt 1 2α e 2αT 2α) 3 )

9 62 PRICING II: MARTINGALE PRICING This equaion can be solved for θ T ) by differeniaing wice wih respec o he mauriy T : T whence 2 T 2 θ τ)bτ; T )dτ = θ T )BT ; T ) + I follows ha θ T ) = 2 T 2 = θ τ)e ατ T ) dτ = e αt θ τ)e ατ dτ, θ τ) Bτ; T ) T θ τ)bτ; T )dτ = θ T ) αe αt θ τ)e ατ dτ θ τ)bτ; T )dτ α T θ τ)bτ; T )dτ, and subsiuing he righ hand side of 132) one finds, afer some algebra, ha 133) θ T ) = 2 T log P 2,T α T log P,T + σ2 ) 1 e 2αT 2α To use formula 133) in pracice, we need o consruc a smooh ha is, wice-differeniable) curve P,T ou of he finiely many bonds which are in fac quoed corresponding o some finie sequence of mauriies T 1,, T N ) This is usually done by spline inerpolaion Finally we menion a problem wih yield curve fiing Suppose we calibrae wih oday s yield curve, and find a funcion θnow) If we calibrae again in one week s ime, say, we would in general find a differen funcion θhen ), whereas his funcion should be he same, a leas if he heory consisenly describes realiy A slighly non-realisic aspec of he Vasicek model, and oher one-facor models, is of course ha here is only one risk-facor: W, driving he sor rae, for he wole range of mauriies One could reasonably argue ha a he long end of he yield curve ha is, for big T ), oher risks will come ino play This has led o he inroducion of muli-facor models,: for a review of he lieraure, see Musiela and Rukowski, secion 124 The HJM-models which we will sudy in he nex lecure are ypically also muli-facor 55 Maringale approach o shor rae models Inroduce, as before, he savings accoun B 134) B = e rsds This is simply he amoun of money in a bank accoun o which an iniial deposi of 1, coninuously re-invesed a he shor rae r, as grown

10 PRICING II: MARTINGALE PRICING 63 a ime o see his, somply observe ha db = r B d, and inegrae) We are hen dealing wih a marke consising of he savings bond B, and infiniely many -coupons P,T, T > Exrapolaing from he muli-asse markes reaed in Lecure IV 1 risk facor, and infiniely many asses), we expec ha here will be absence of arbirage iff here is an Equivalen) Maringale) Measure), Q for he discouned asses P,T /B Tha is, if < u T, ) 135) B 1 P,T = E Q B 1 u P u,t F In paricular, aking u = T, and using P T,T = 1, and 134), we find he pricing equaion: 136) P,T = E Q e ) T r sds F Observe ha we obain a similar equaion bu wih r replaced by ˆr ) by applying he Feynman-Kac formula o he bond price equaion 117): changing from r o ˆr can also be undersood as replacing he SDE for r, dr = α d + σ dw where we have wrien α = αr, ), σ = σr, )) by dr = α q σ ) d + σ )dŵ, q = qr, ), wih dŵ = qr, )d + σr, )dw being a Brownian moion wih respec o some new probabiliy measure Q, obained from Girsanov s heorem Conversely, i can be proved ha on he Brownian F W is of he Girsanov form: dq = exp γ s dw s 1 ) γ 2 2 sds,, any EMM for some adaped process γ, wih Ŵ, defined by dŵ = γ d + dw, a Brownian moion wih respec o Q Wih respec o Q, r follows he SDE dr = α γ σ ) d + σdŵ, so ha γ is he price of risk In he case of for example he Vasicek model, formula 136), ogeher wih dr = αθ r )d + σdŵ, Ŵ a Q-Brownian moion, can be used o direcly compue he -coupon prices, by he same argumens as in 53 above

11 64 PRICING II: MARTINGALE PRICING 56 Exercises o Lecure VI Exercise 53 Wrie ou he deails of he compuaion leading o 114), 115), 116) Exercise 54 Verify equaion 133), and find a formula for P,T erms of oday s marke prices, P,T Exercise 55 A firs order ODE of he form y = px)y 2 + qx)y + rx), for an unknown funcion y = yx), wih given funcions px), qx) and rx), is called a Riccai equaion As usual, y = dy/dx Suppose we dispose of a paricular soluion y = y x) Then we can ransform he original equaion ino a firs order linear equaion, which being firs order) is explicily solvable Indeed, le u = ux) be defined by y = y + 1 u, or, equivalenly, 1 ux) = yx) y x) Le us no worry abou he possible singulariy in poins x for which yx) = y x): his would have o be sudied in any paricular case o which we would apply his mehod) a) Show ha u = ux) saisfies he ODE u = px) + 2px)y x)u + qx)u b) Consider from now he special case in which px), qx) and rx) are consans, p, q and r, respecively Show ha y x) = v ± are special soluions of 137) y = py 2 + qy + r, if v ± are he zeros of 138) pv 2 + qv + r =, ha is, v ± = q ± q 2 4pr 2p c) Take y = v as special soluion Show ha u = ux) saisfies u = p + du, where d := q 2 4pr is he discriminan of he quadraic equaion 138) Solve his equaion by he usual mehod general soluion = paricular soluion + general soluion homogeneous equaion ) o find ux) = p d + Cedx, wih C an arbirary consan d) Conclude ha he general soluion of in

12 137) is given by PRICING II: MARTINGALE PRICING ) yx) = pv + d) + C e dx p + C e dx, where C is some consan C = dv C if C is he consan from par c) ) e) Deermine C such ha y) =, and show ha he corresponding y is given by 14) yx) = v +1 e dx ) 1 v + e dx, where you will have o use ha verify his!) pv + d = pv + f) Show ha, if yx) is given by 14), x ) v ) ys)ds = x log1 v + e dx ) v + d Hin: Observe ha ) 1 e dx 1 v + e = 1+v e dx v+ 1 log1 + 1) = 1 dx 1 v + edx v+ e dx ) ) ) v + d Exercise 56 We can use he resuls of he previous exercise o find he price of a zero coupon bond in he CIR model: dr = αθ r )d + σ r dw a) Le P,T = p T r, ) be he price of a pure discoun bond of mauriy T Show ha, under he assumpion ha he marke price of risk is a consan imes r Correced!), p = p T solves he boundary value problem 142) p + α θ r) p r σ2 r 2 p = rp, < T, r2 pr, T ) = 1, where θ is he risk-adjused mean rae b) Making he Ansaz ) pr, ) = e Aτ) rbτ), 12 meaning rial soluion

13 66 PRICING II: MARTINGALE PRICING where τ := T, and Aτ) and Bτ) are funcions of τ only, show ha he laer ave o saisfy he following sysem of differenial equaions: da dτ = rb, db dτ = σ2 2 B2 αb + 1, wih iniial condiions A) = B) = 1 NB The reason for inroducing he new ime variable τ is o have iniial condiions a τ =, insead of final condiions a = T, simply because his is compuaionally more convenien) c) You should recognize he differenial equaion for Bτ) as a Riccai equaion Use he resuls of he previous exercise on such equaions o find Aτ) and Bτ), and o derive an explici formula for he bond price in he CIR model Ceck your answer agains formulas which can be found in he lieraure