where r() = r(s)e a( s) + α() α(s)e a( s) + σ e a( u) dw(u) s α() = f M (0, ) + σ a (1 e a ) Therefore, r() condiional on F s is normally disribued wi

Transcription

1 Hull-Whie Model Conens Hull-Whie Model Hull-Whie Tree Example: Hull-Whie Tree Calibraion Appendix: Ineres Rae Derivaive PDE Hull-Whie Model This secion is adaped from Brigo and Mercurio (006). As an exension of he Vasicek model, Hull- Whie model assumes ha he shor rae follows he mean-revering SDE: dr() = [θ() ar()]d + σdw() where a and σ are posiive consans; and θ() is ime-dependen funcion ha will be used o fi he curren zero curve. To solve his SDE, firs apply Io lemma o re a Then inegrae boh sides over [s,], d( re a ) = e a dr + are a d = (θd + σdw) e a r()e a r(s)e as = θ(u)e au du + σ e au dw(u) s r() = r(s)e a( s) + θ(u)e a( u) du + σ e a( u) dw(u) In order o fi he erm srucure of ineres raes, he ime-dependen θ(t) mus saisfy s θ(t) = fm (0, T) + af M (0, T) + σ T a (1 e at ) where f M (0, T) is he marke observed insananeous forward rae a ime 0 for he mauriy T. s s Then r() can be re-wrien as

2 where r() = r(s)e a( s) + α() α(s)e a( s) + σ e a( u) dw(u) s α() = f M (0, ) + σ a (1 e a ) Therefore, r() condiional on F s is normally disribued wih mean and variance given respecively by E{r() F s } = r(s)e a( s) + α() α(s)e a( s) Var{r() F s } = σ e a( u) du s = σ a [1 e a( s) ] HW model is an affine erm srucure model where he coninuously-compounded spo rae is an affine funcion in he shor rae, i.e., The zero coupon bond price is given by where R(, T) = α (, T) + β(, T)r() P(, T) = A(, T)e B(,T)r() A(, T) = PM (0, T) P M (0, ) exp {B(, T)fM (0, ) σ 4a (1 e a )B(, T) } B(, T) = 1 a [1 e a(t ) ] We can also find he close-form formulas for zero-coupon bond opions, caps/floors, and swapions. See Brigo and Mercurio (006) for deail. Hull-Whie Trinomial Tree To consruc HW ree, i is helpful o decompose he shor rae ino he following forma: Where r() = x() + α() α() = f M (0, ) + σ a (1 e a ) (1) dx() = ax()d + σdw x(0) = 0 () x() = x(s)e a( s) + σ e a( u) dw(u) s (3)

3 Wih his decomposiion in hand, he ree consrucion ask can be achieved in wo seps. In he firs sep one consrucs he rinomial ree for x(). Then in he second sep, one shifs he ree by α() o bring i in line wih he iniial erm srucure. Sep One: Consruc he Symmeric Trinomial Tree From equaions () and (3), i is known ha E{x() F s } = x(s)e a( s) Var{x() F s } = σ a [1 e a( s) ] Denoe he ree nodes by (i, j) where he ime index i ranges from 0 o N and he space index j ranges from some j i < 0 o some j i > 0. Using he resuls in A.F (Brigo and Mercurio 006), we have where i = i+1 i. E{x( i+1 ) x( i ) = x i,j } = x i,j e a i =: M ij Var{x( i+1 ) x( i ) = x i,j } = σ a [1 e a i] =: V i Now, given he node x i,j, we need o locae is subsequen nodes x i+1,k+1, x i+1,k and x i+1,k 1 wih he respecive ransiion probabiliies p u, p m, p d. This is done as follows. Firs find he space in he y direcion as Then locae he level k by x i+1 = V i 3 = σ 3 a [1 e a i] (5) k = round ( M i,j x i+1 ) (6) (4) where round(x) funcion indicaes he closes ineger o he real number x. We hen se x i+1,k+1 = (k + 1) x i+1, x i+1,k = k x i+1, x i+1,k 1 = (k 1) x i+1 (7) A las he ransiion probabiliies are chosen in a way o mach he condiional mean and variance,

4 where Sep Two: Displace he Tree p u = η j,k 6V + i p m = 3 η j,k 3V, i η j,k 3V i, η j,k p d = η j,k 6V i 3V i η j,k = M i,j x i+1,k = M i,j k x i+1 (9) The second sep consiss of displacing he ree nodes o obain he corresponding ree for r. An easy way o do so is hrough equaion (1), as in class HullWhie::FiingParameer, where he insananeous forward rae is approximaed by f M (0, ) F M (0;, + 0.5bps) This approach has o approximae coninuously-compounded rae R(0, ) wih shor rae r(0), herefore doesn fi exacly he zero curve. The oher way uses helps from Arrow-Debrew prices. Denoe α i be he displacemen a ime i, and Q i,j be he Arrow-Debrew price of node (i, j). A sae price securiy or an Arrow-Debreu securiy is defined as a conrac which pays off $1 in a paricular sae a a paricular ime and pays zero in all oher saes. Is price (ne presen value) is referred o as Arrow-Debreu price. The values of α i and Q i,j are calculaed recursively as follows. 1. Iniialize Q 0,0 = 1. Find α 0 = ln(p M (0, 1 )) 1 3. Wih α i, (i = 0 a his ime) in hand, calculae Q i+1,j = Q i,h q(h, j)exp{ (α i + h x i ) i } h where q(h, j) is he probabiliy of moving from node (i, h) o node (i + 1, j). (8)

5 4. Wih Q i,j (i = 1 a his ime) in hand, find α i by solving ha leads o j i P(0, i+1 ) = Q i,j exp{ (α i + j x i ) i } j=j i α i = 1 j=j Q i,j exp{ j x i } i i ln i P(0, i+1) 5. Loop sep 3 and 4 o discover α i and Q i,j for furher seps (i++). 6. Shor rae on each node Example: Hull-Whie Calibraion j i r i,j = x i,j + α i This secion illusraes he Hull-Whie model calibraion process wih a real example. I calibraes he Hull-Whie ree o he LIBOR marke on Monday, May 16, 011. The selemen dae is Wednesday, May 18, 011. Hull-Whie model has hree parameers: a, σ, and θ(). In his example, he firs wo will be calibraed o LIBOR swapions; and he hird one will be calibraed o LIBOR spo curve. To begin wih, a and σ are iniialized as Anoher inpu is LIBOR spo curve, which is given by a = 0.1, σ = 0.01 Table 1 -- LIBOR Curve Tenor Dae Time o Mauriy Year Fracion Rae 3m Thursday, Augus 18, days m Friday, November 18, days m Tuesday, February 1, days The raes are coninuously compounded. Convenions are Ac/360 and Modified Following. The calibraion process is carried ou in four seps: Sep 1. Sep. Sep 3. Sep 4. consruc rinomial ree for process x(); displace x() o obain he ree for r(); price swapions on his ree; calibrae he ree o swapions marke.

6 Sep One: Consruc Trinomial Tree for Process x() The oucome of his sep is shown in he following figure. This rinomial ree of x() is symmeric. Legend: rae pu (0,0) (Node) pm pd Figure 1 Trinomial Tree of x() (1,1) (1,0) (1,-1) Now le s walk hrough he seps o creae his figure. 1. From equaion (), x( 0 = 0) = 0, or node (0,0) is 0.. Consider node (0,0). From equaion (4) M 0,0 = x 0,0 e (0.1)(0.5556) = 0 V 0 = (0.01) ()(0.1) [1 e ()(0.1)(0.5556) ] = Go hrough equaions (3) (5) o locae is descendans: nodes (1,-1), (1,0) and (1,1). x 1 = V 0 3 = k = round ( M 0,0 x 1 ) = 0 x 1, 1 = x 1,0 = 0 (,) (,1) (,0) (,-1) (,-) Seven Nodes 0 = 0D 1 = 9D = 184D 3 = 79D

7 x 1,1 = Using (8) and (9) o ge ransiion probabiliies from x 0,0 o is descendans. η 0,0 = M 0,0 x 1,0 = 0 p u = η 0,0 6V + η 0,0 = V 0 p m = 3 η 0,0 3V = p d = η 0,0 6V η 0,0 = V 0 By now node (0,0) is finished. 5. Now consider node (1,-1). Follow sep o 4, and as well as M 1, 1 = x 1, 1 e (0.1)(0.5556) = V 1 = (0.01) ()(0.1) [1 e ()(0.1)(0.5556) ] = x = V 1 3 = k = round ( M 1, 1 ) = 1 x x, = x, 1 = x,0 = 0 η 1, 1 = M 1, 1 x, 1 = p u = η 1, 1 6V + η 1, 1 = V 1 p m = 3 η 1, 1 3V = p d = η 1, 1 6V + η 1, 1 = V 1 6. Process similarly node (1,0) and node (1,1), hen move on o he nex enor (6m). Then he figure will be creaed. When he volailiy is ime-dependen, his rinomial ree is recombining. Sep Two: Displace x() o Obain he Tree for r() We can ieraively find sae prices and α(), hen shif he symmeric ree from sep one by α() o obain he Hull-Whie ree. This has been explained in Secion. Le s follow he procedure inroduced in ha secion.

8 1. Iniialize Q 0,0 = 1. Find displacemen α 0 = ln(exp( )) = Move on o ime sep 1, compue he sae prices for he hree nodes: Q 1, 1 = Q 0,0 q(0, 1)exp{ (α 0 ) 0 } = Q 1,0 = Q 0,0 q(0,0)exp{ (α 0 ) 0 } = Q 1,1 = Q 0,0 q(0,1)exp{ (α 0 ) 0 } = The displacemen for ime sep 1 α 1 = 1 ln j= 1 Q 1,jexp{ x 1,j 1 } 1 exp ( ) = Move on o ime sep, compue he sae prices for he five nodes. For example, for he middle node (,0), i has hree incoming nodes: (1,-1), (1,0), and (1,1), hen Q,0 = Q 1, 1 q( 1,0)exp{ (α 1 x 1, 1 ) 1 } +Q 1,0 q(0,0)exp{ (α 1 + x 1,0 ) 1 } +Q 1,1 q(1,0)exp{ (α 1 + x 1,1 ) 1 } = Given he five sae prices on ime sep, he displacemen of his ime sep is The resuls are shown in he following able. α = 1 ln j= Q,jexp{ x,j } exp ( ) = Table Sae Prices and Displacemens Time Sep Q α By displacing he symmeric rinomial ree in sep one wih corresponding α i, he Hull-Whie ree is consruced as in he following figure.

9 Legend: rae pu (0,0) (Node) pm pd Figure Hull-Whie Trinomial Tree (1,1) (1,0) (1,-1) Noe ha he raes on node (1,-1) and node (,-) are negaive. Hull Whie model can produce negaive raes due o normal disribuion. Sep Three: Price Swapions on he Tree (,) (,1) (,0) (,-1) (,-) = 0D 1 = 9D = 184D 3 = 79D Before we can proceed o swapion pricing, i needs o poin ou he way in calculaing he discouning facor D(,T) on he shor rae ree. To calculae he discoun facor, for example D( 0, 1 ), we need o rerieve he rae a he beginning of he period, in his case on ime 0 = 0. This conrass wih he general case when he erm srucure curve is used, where we usually rerieve he rae from he end of he period, in his case on ime 1 = 3m. This rule applies o oher ime seps as well. Now we are ready o price on his ree a 3mx6m ATM European payer Swapion wih noional $1,000 and ATM rae The swapion can be exercised only on 1 = 3m, giving he owner he righ o ener ino he long posiion (pay fixed) of a 3mx6m forward saring swap. Therefore he value of his swapion on each of he hree nodes on 1 = 3m, or nodes (1,j), j= -1, 0, 1, is simply swapion value on node (1, j) = max {0, swap value on node (1, j)} Seven Nodes

10 So i comes down o price he underlying swap, which is priced by discouning is fixed leg cash flows and floaing leg cash flows respecively, via he following formula, NPV leg () = E { D(, T i ) CF i } = E { e R(,T i)τ(,t i ) CF i } Firs consider he fixed leg. I pays on 3 he amoun = where he year fracion (79 9) 360 according o 30/360 day coun. The resuls are shown in he following able. Table 3 -- Fixed Leg Pricing (Backward Inducion) Node Time sep 1 Time sep Time sep Now we explain how o discoun one sep backward, from ime sep 3 o from sep. Laer we will use he same logic o discoun backward he floaing leg. Consider he node (,-) for insance. I has hree descendans: node (3,-3), (3, -), and (3, -1). Denoe V 3,j he value of node (3, j). In his paricular case To roll one sep back, V 3,j = , for j = 3 o 3 V, = exp( r, ) {q(, 3) V 3, 3 +q(, ) V 3, +q(, 1) V 3, 1 } = exp( ) { } = We coninue o deal wih oher four nodes in sep and hen move on o sep 1. This leads o Table 3. For he floaing leg, we can do i similarly by firs idenifying he cash flows and hen discouning hem. An alernaive and quicker way is hrough equivalen cash flows.

11 Unlike fixed leg, floaing leg has enor of 3 monhs (see chaper LIBOR Raes). Therefore, i conains wo cash flows. The firs one reses a ime 1 and pays a ime ; he second one reses a ime and pays a ime 3. Look a he second cash flow. On node (,-), he rae is fixed a So he paymen made a ime 3 is which is equivalen o 1 L(, 3 ) = F( ;, 3 ) = τ(, 3 ) (P(, ) P(, 3 ) 1) N L(, 3 ) τ(, 3 ) N L(, 3 ) τ(, 3 ) P(, 3 ) = N{1 P(, 3 )} on ime afer discouning. Thus by saring wih via equivalen cash flows, i saves us one sep of backward inducion. The resuls are shown in he following able. Table 4 Two floaing leg cash flows (backward inducion) Node Second cash flow Firs cash flow Toal Value Time sep 1 Time sep Time sep 1 Time sep In Table 4, he wo floaing cash flows are reaed independenly, and hen added up ogeher. Cash flows sar wih equivalen cash flows. For example, second cash flow pays a ime sep 3. Is equivalen cash flow on node (, -) a ime sep is 1000 (1 exp( )) =.3463 Afer figuring ou he oher four nodes a ime sep, hey are discouned back o ime sep 1 via he same procedure as has been seen in he fixed leg par. Toal value a ime sep 1 is he sum of he firs and second cash flows. Now we have reaed boh floaing and fixed legs, i is ready o price he swap and he 3mx6m swapion. The pricing procedure is shown in he following able.

12 Table 5 3mx6m Swapion Pricing Node Fixed Leg Floaing Leg Swap Swapion Sae Prices NPV (1,1) (1,0) (1,-1) Sum In Table 5, he fixed leg column and floaing leg column are inheried from Table 3 and 4, respecively. Then a long swap posiion receives floaing leg while pays fixed leg. The Swapion is only exercised when i is in he money, or underlying swap has posiive value. The las column, NPV, is he produc of he Swapion value column and sae price column. Finally he NPV of swapion price is he sum of NPV column, or This example is done in Excel. In comparison, he accompanying C++ code gives Sep Four: Calibrae he Tree o Swapions Marke Sep hree calculaes he 3mx6m swapion on he HW ree. I is known as he model price, which depends on he model parameers. In his case, i depends on he (iniial) value of α = 0.1 and σ = Marke calibraes he Hull-Whie model o swapion volailiy cube by minimizing min f(α, σ) = ( PModel (α, σ) P Marke P Marke ) where model price P Model (α, σ) is calculaed by following sep hree; and P Marke is he marke price, obained by plugging he volailiy quoes ino Black model. Pay aenion o wheher he volailiies are quoed as log vol or normal vol (see Chaper LIBOR Volailiy). The opimizaion can be achieved by ieraions. In ha case, a new Hull-Whie ree will be consruced for each ieraion (when α and σ change). Bloomberg Commands: SWPM, VCUB. Appendix This appendix derives PDE for Ineres Rae Derivaives (IRDs) in shor rae model. Le he shor rae SDE be

13 dr = a(r, )d + b(r, )dw An ineres rae derivaive (IRD) has payoff V a ime T. Is value a ime is Using Io lemma by defining V = V(r, ; T) dv = V V d + r dr + 1 V d r, r r = ( V V + a r + 1 V V b r ) d + b r dw = μvd + σvdw μ = ( V V + a r + 1 V b r ) V σ = b V r V We consruc a hedging porfolio wih wo insrumens wih wo differen mauriies, T 1 and T In order o be risk-free and no-arbirage Π = 1 V 1 V dπ = ( 1 μ 1 V 1 μ V )d + ( 1 σ 1 V 1 σ V )dw 1 σ 1 V 1 = σ V dπ = rπd 1 μ 1 V 1 μ V = r( 1 V 1 V ) 1 V 1 (μ 1 r) = V (μ r) μ 1 r = μ r σ 1 which shows ha he marke price of risk is independen of mauriy T Then σ μ r σ = λ(r, ) μ = r + λσ = r + λb V r Subsiue i ino he drif equaion we obain he PDE

14 rv + λbv V r = V V + a r + 1 V b r A faser way o ge he PDE under risk-neural measure λ = 0 uses maringale propery. Le he bank accoun numeraire be B() = e 0 r(s)ds We know ha an insrumen wih payoff V(r, ; T), V(r, ; T)/B() is a maringale under risk-neural measure Q. Therefore, V(r, ) r(s)ds d ( ) = d (V(r, )e B() = ( V r dr + 1 V V d r, r + r whose drif erm should be 0. I leads o, Reference 0 ) r(s)ds d) e V V + a r + 1 V b r = rv 0 + Ve 0 r(s)ds ( r)d [1] Brigo, D. and Mercurio, F (006). Ineres rae models: heory and pracice: wih smile, inflaion, and credi. Springer Verlag. [] Daglish, T. Laice mehods for no-arbirage pricing of ineres rae securiies. The Journal of Derivaives. (18), pp. 7 19, 010.