An Analytical Implementation of the Hull and White Model

Transcription

1 Dwigh Gran * and Gauam Vora ** Revised: February 8, & November, Do no quoe. Commens welcome. * Douglas M. Brown Professor of Finance, Anderson School of Managemen, Universiy of New Mexico, Albuquerque, NM Tel: , fax: , dgran@swcp.com. ** Associae Professor of Finance, Anderson School of Managemen, Universiy of New Mexico, Albuquerque, NM Tel: , fax: , vora@unm.edu.

2 Absrac Ho and Lee inroduced he firs no-arbirage model of he evoluion of he spo ineres rae. Hull and Whie exended his work o include mean reversion of he spo ineres rae. When wriing abou he implemenaion of heir model in discree ime, hey have employed a search process a each dae and forward inducion o idenify he level of ineres raes in a rinomial laice. We derive an analyical soluion for he level of ineres raes. We apply analyical soluion o an example creaed by Hull and Whie.

3 An Analyical Implemenaion of he Hull and Whie Model Vasicek [977] iniiaed an imporan sream of research relaing o he evoluion of ineres raes. Cox, Ingersoll, and Ross (985) added an equilibrium model of he evoluion of ineres raes. While analyically racable hese models do no provide valuaions consisen wih he absence of arbirage. Ho and Lee [986] address his disadvanage wih a no-arbirage model ha incorporaes he marke erm and volailiy srucure of ineres raes. Heah, Jarrow and Moron [99] exend he basic Ho-Lee model o incorporae he enire srucure of forward raes; Ho and Lee assume ineres raes are normally disribued. Black, Derman, and Toy [99] and Black and Karasinski [99] conribue no-arbirage models ha assume ineres raes are log-normally disribued. This group of models relies on numerical mehods for heir implemenaion. Hull and Whie (HW) [99, 993, 994, and 996] exend he Ho and Lee approach by adding Vasicek s idea of a mean-revering ineres rae. In addiion o heir heoreical modeling, HW propose (996) a numerical implemenaion of heir model requiring a search process a each dae o idenify he level of ineres raes. This, in urn, requires implemenaion hrough forward inducion. We complemen HW s work by deriving an analyical soluion for he level of ineres raes and illusrae he soluion by applying i o an example used by HW (996). I. The Analyical Implemenaion The HW model expresses he coninuous ime evoluion of he insananeous spo rae as: { } dr() = µ () + α γ() r () d+ σ() dz(). () In equaion () he spo ineres rae a dae is r(). The drif in he spo rae is composed of wo erms, a pure drif erm () µ, plus a mean reversion erm, ( () r ()) α γ. The mean reversion erm causes he ineres rae o rever o a ime-varying normal value, γ (), a he insananeous

4 rae α. We wrie he insananeous volailiy of he spo ineres rae, σ (), in erms of a sandard Wiener process for which dz()~ N (,). The discree-ime analogue of equaion () for he change in he spo rae over he imeinerval, i.e., for he ime-period [, ] where r( ) and +, is ( µ γ ) σ r () = + k r + b z, () σ are respecively he spo rae and volailiy of he spo rae a ime for he ime-inerval from o + ; α is a posiive consan (< ); z is a uni normal random variable: k = e α, (-A) α e b= α, (-B) and he ime-varying normal rae of ineres a dae is ( ) ( ) ( ) ( ) ( ) ( j) γ = γ + µ = γ + µ + µ = γ + µ. (-C) Wihou loss of generaliy, we can se = and = and rewrie equaion of he evoluion of he spo rae as µ [ γ ] σ r r r = + k () r() + b z. (3) This yields, for example, for dae j= Hull and Whie (996, p.6) wrie he firs erm as a θ r d; herefore, a α θ = µ + αγ. We wrie he model as we have because he derivaion of = and µ is a cenral resul. See Hull and Whie (996, p. 9), Jamshidian (989) and Arnold (974) for he use and developmen of he componens k and b used in his discree ime expression.

5 = + µ + ( ) [ γ() ()] + σ = r + µ + γ r kγ + kr + bσ z = γ + µ ( ) + k[ r() γ() ] + bσ z(. ) r r k r b z (4) The mean reversion effec causes he deviaion of he spo ineres rae from he ime-varying normal value a dae, r γ, o decrease o k r γ a dae. Similarly, he spo rae a dae is = + µ + ( ) γ + σ = γ + µ + k r γ + bσ z(. ) r r k r b z A dae, he ime-varying normal rae is γ = γ + µ which gives r γ γ + σ = k r b z. Subsiuing his expression ino he above equaion, we obain { } r = γ + µ + µ + k k r γ + bσ z + bσ z k r kb z b z = γ + µ + µ + γ + σ + σ, (5) and in general, (6) j = γ + µ + γ + σ r j k r k b j z j. j= j= Equaion (6) indicaes ha he spo rae is he sum of a se of non-sochasic drif erms and a se of sochasic erms; he laer are all normally disribued. Consequenly, he spo ineres raes are normally disribued as follows: and in general, ( ) γ + µ + γ σ r ~ N k r, r. ( ) γ + µ + µ + γ σ + r ~ N k r, r r. r ~ N γ + µ ( j ) + k r γ ( ), σ r( j). j= j= (7) 3

6 The inpus for a HW no-arbirage ineres rae model in discree ime are ) a se of known { } prices of pure discoun bonds ha maure a daes,, 3,, n, (, ) (, ) ( 3,..., ) P P P P n, and ) he volailiy (sandard deviaion) of fuure one-period normally disribued spo ineres { } σ σ σ. 3 raes, (, ) (,..., ) ( n ) An evoluion of he spo ineres ha precludes arbirage mus saisfy he local expecaions condiion ha all bonds, regardless of mauriy, offer he same expeced rae of reurn in a given period under he equivalen maringale probabiliy (EMP) disribuion,. This is equivalen o he expecaion of he discouned value of each bond s erminal paymen being equal o is given marke (iniial) value. 4 Le he presen value, a dae =, of a bond s erminal paymen be given by n p( n) = exp r( j). Therefore, he no-arbirage condiions will be saed as j= and, in general, [ ] f r r( ) P = e E p() = E e = e, { f + f } { r + r } P = e E p E e =, n n P( n) = exp f ( j) E p( n) = E exp r( j). j= j= (8) E [ ] is he expecaion a dae under he EMP disribuion and f for he inerval from j o j +. j is he forward rae 3 For simpliciy, we assume ha r γ = for he reminder of his paper. 4

7 From saisics we know ha if x is normally disribued, (, ) N µ σ, hen: 5 x E e = e µ + σ. (9) Therefore, for dae = : { } { r E r } ( r( r + r + + σ ) ) P = E p( ) E e = = e. Upon simplificaion: lnp = r E r + σ ( r( ) ), or E r = lnp r + σ ( r( ) ). Because lnp f f r f = =, upon subsiuion in he above equaion, E r = f + σ ( r( ) ). () The expecaion of he spo rae a dae is he forward rae plus a erm deermined by he σ. variance of he spo rae, r Taking he expecaion of equaion (4), we have 6 E r = r + µ. () From equaions () and (), we derive: µ = f r + σ ( r( ) ). () 4 For example, we can illusrae he equivalence wih respec o he expeced rae of reurn on he wo-period bond from dae zero o dae. [ ] E is he expecaions operaor under he equivalen maringale probabiliy r r E e E e disribuion : ln r r r r( ) or = = e or P = E e. P P 5 See Mood, Graybill and Boes (974, p. 7) for a discussion of his resul. 6 Recall ha for simpliciy we have se ( ) r ( ) γ =. 5

8 Thus, he drif erm, µ, is equal o he sum of wo effecs: ) f() r() is he difference beween he forward rae and he spo rae, i.e., he spo ineres rae drifs up or down oward he forward rae, ) r o preclude arbirage. 7 σ is a posiive drif adjusmen erm (DAT) ha is required Le δ denoe he DAT for dae. Then: δ = ( ). (3) σ r Now, we can work ou he deails for dae = 3. Simplifying: { } r r r E r + r + r + σ ( r + r ) + + P 3 = E p 3 E e = = e. lnp( 3) = r E r E r + σ ( r + r ) = r f σ r E r + r + r E r = P r f + σ r + r r σ ( ) or σ ( ) ln 3. We know ha lnp( 3) f f f r f f equaion above: = =. Upon subsiuion in he E r = f + σ r + r r σ ( ). (4) The expecaion a dae = of he spo rae a dae is he forward rae plus a erm deermined by he variance, σ ( r r ) σ ( r ) +. Taking he expecaion of equaion (5), we have: E r = r + µ + µ. (5) 7 Boyle was he firs o poin ou his general resul. 6

9 From equaions (4) and (5) we derive: µ µ σ σ ( ) = f r + r + r r. Subsiue equaion () ino he above o ge: The drif erm, µ = f f + σ ( r + r ) σ ( r( ) ). (6) µ, is equal o he sum of wo effecs: ) f() f() is he difference beween ( r r ) σ ( r ) he forward rae a dae and he forward rae a dae, and ) σ he posiive DAT required o preclude arbirage. Le δ () denoe he DAT for dae. Then: If we add equaions for δ and = + is δ = σ ( r + r ) σ ( r( ) ). (7) δ (equaions (3) and (7)) we ge δ σ σ σ = σ ( r + r ) σ ( r( ) ). If we add equaions for µ and ( ) ( ) ( ) = r + r + r r µ (equaions () and (6)) we ge µ µ σ σ σ = f r + σ ( r + r ) σ ( r( ) ), + = f r + ( r ) + f f + ( r + r ) ( r ) which can be simplified o µ = f r( ) + δ. (8) = = The resuls of he firs wo daes can be generalized for he case of dae. 7

10 E r = f + σ r( j) σ r( j) < T. (9) j= j= n µ ( ) = f f ( ) + σ r( j) σ r( j) + σ r( j) 3. In addiion: () j= j= n= j= + δ ( n) = σ r( j) σ r( j). () n= j= j= µ ( n) = f ( + ) r( ) + δ ( n). () n= n= Equaions (9) () give he necessary recursive relaions o evolve he HW no-arbirage model of spo ineres rae. The inpus are he se of marke prices of (pure) discoun bonds, a srucure of volailiies for he spo raes, and oher parameric values. The above discussion is general in he sense ha i applies equally well o implemenaion based on ineres-rae rees and Mone Carlo simulaion. 8 II. Implemenaion Example We illusrae he implemenaion wih reference o an example developed by HW (996). They illusrae implemenaion of heir model wih he example of pricing a hree-year pu opion on a zero-coupon bond ha pays $ in 9 years. The exercise price is $63, he volailiy, s, is consan a % per annum for all daes, and he speed of reversion o he mean, α, is.. Exhibi displays he prices and yields of he zero-coupon bonds. The HW implemenaion uses a rinomial laice wih upper and lower bounds. Firs, HW idenify he sep sizes and he probabiliies necessary o achieve he desired volailiy, around zero, of he ineres raes. Second, hey find he expeced value of he ineres rae a each dae 8 Mone Carlo implemenaion of he HW model may be imporan for he valuaion of pah-dependen securiies. For example, a leas one derivaives firm values amorizing index swaps using Mone Carlo implemenaion of he HW model. 8

11 ha is consisen wih he iniial condiions. This requires he use of search process and forward inducion. We display HW s resuls in Exhibi. Our firs ask is o illusrae how o eliminae a numerical search procedure and forward inducion o idenify he mean value of he ineres raes. Specifically, given he iniial -period ineres rae 5.98%, how do we derive analyically he subsequen mean values of 6.56%, %, and 8.538%? Recall ha: E r = f + σ ( r( ) ), (3-A) ( σ ( ) ) E r = f + σ r + r r, (3-B) ( σ ( ) ) E r( 3) = f ( 3) + σ r + r + r( 3) r + r, (3-C) from equaions (), (5) and (9). Also: µ r = r + + bs z, (4-A) µ µ r = r bs k z + z, (4-B) µ µ µ r 3 = r bs k z + k z + z, (4-C) from equaions (4), (5) and (6), respecively. The expressions for k and b are given in equaions (-A) and (-B), respecively. Wih his informaion, we can calculae he variances of he sum of he spo raes. σ σ ( ) ( ) σ r = bs z = bs =.95 (.) =.963. ( { } ) ( ) σ { } r + r = bs z + k z + z = bs + k + = (.963)(4.684) =

12 ( { } ) ( r + r + r( 3) ) = bs + k + k z + z ( + k) + z = bs ( + k + k + ( + k) + ) σ σ = (.963)(.46) =.98. Wih his informaion, we can derive analyically he values ha HW derived hrough searches. E r = ln.896+ ln (.5)(.963) = 6.56%. E r = ln.877+ ln (.5)( ) = %. E r 3 = ln ln (.5)( ) = 8.538%. These resuls mach hose produced by HW s search mehod, as shown in Exhibi. III. Conclusion HW develop an aracive no-arbirage model of he evoluion of he spo ineres rae ha incorporaes mean reversion. Their implemenaion of he model requires boh he use of a search mehod o idenify he expeced value of he spo ineres rae a each dae and forward inducion. The analyical expression for he expeced value of he fuure spo raes derived in his paper eliminaes need for he search process. Our implemenaion described here applies equally well o ineres-rae binomial rees, rinomial laices and Mone Carlo simulaion implemenaion of he model and can be adaped o incorporae addiional complexiies such as ime-varying volailiy.

13 References Arnold, L. Sochasic Differenial Equaions. John Wiley and Sons, Inc. New York, N. Y Black, F., E. Derman and W. Toy. A One-Facor Model of Ineres Raes and Is Applicaion o Treasury Bond Opions. Financial Analyss Journal, 46 (99) (January/February), pp Boyle, P. P. Immunizaion Under Sochasic Models of he Ineres Rae Srucure. Journal of he Insiue of Acuaries, 5 (978) pp Cox, J. C. and S. A. Ross. The Valuaion of Opions for Alernaive Sochasic Processes. Journal of Financial Economics, 3 (976), pp Cox, J. C., J. E. Ingersoll and S.A. Ross. A Theory of he Term Srucure of Ineres Raes. Economerica, 53 (985), pp Harrison, J. M. and D. M. Kreps. Maringales and Arbirage in Muliperiod Securiy Markes. Journal of Economic Theory, (978), pp Heah, D., R. Jarrow and A. Moron. Bond Pricing and he Term Srucure of Ineres Raes: A Discree Time Approximaion. Journal of Financial and uaniaive Analysis, 5 (99), pp Heah, D., R. Jarrow and A. Moron. Coningen Claim Valuaion wih a Random Evoluion of Ineres Raes. Review of Fuures Markes, (99) pp Heah, D., R. Jarrow and A. Moron. Bond Pricing and he Term Srucure of he Ineres Raes: A New Mehodology, Economerica, 6 (99), pp Ho, T. S. Y. and S.-B. Lee. Term Srucure Movemens and Pricing Ineres Rae Coningen Claims. Journal of Finance, 4 (986), pp. 9. Hull, J. and A. Whie. Pricing Ineres Rae Derivaive Securiies. Review of Financial Sudies, 3 (99), pp Hull, J. and A. Whie. One-Facor Ineres Rae Models and he Valuaion of Ineres Rae Derivaive Securiies. Journal of Financial and uaniaive Analysis, (995), pp. 89. Hull, J. and A. Whie. Numerical Procedures for Implemening Term Srucure Models I: Single-Facor Models. Journal of Derivaives, (Fall, 994), pp Hull, J. and A. Whie. Numerical Procedures for Implemening Term Srucure Models II: Two- Facor Models. Journal of Derivaives, (Winer, 994), pp Hull, J. and A. Whie. Using Hull-Whie Ineres Rae Trees. Journal of Derivaives, 3 (Spring, 996), pp Hull, J. Opions, Fuures, and Oher Derivaives (Third Ediion). Prenice-Hall, Englewood Cliffs, NJ, 997. Jamshidian, F. An Exac Bond Opion Formula. Journal of Finance, 44 (989), pp Mood, A. M., F. A. Graybill and D. C. Boes. Inroducion o he Theory of Saisics (Third Ediion). McGraw-Hill, New York, NY, 974. Vasicek, O. A. An Equilibrium Characerizaion of he Term Srucure. Journal of Financial Economics, 5 (977), pp

14 Exhibi Iniial Prices and Yields in he Hull and Whie Example Mauriy Price $.953 $.896 $.877 $.7639 $.765 $.6536 $.6 $.5573 $.539 Yield 5.93% 5.795% 6.35% 6.733% 6.948% 7.87% 7.74% 7.38% 7.397%

15 Exhibi Inermediae Calculaions for he Four-Epoch Trinomial Laice for he Example of he Hull and Whie Shor Rae Calibraing he Variances in he Laice Seps Transiion Probabiliies Node Raes J p u p m p d i = i = i = i = % 3.979% %.649%.649% %.%.%.% %.6489%.6489% % 3.979% Calibraing he Prices in he Laice Seps Transiion Probabiliies Node Raes J p u p m p d i = i = i = i = %.357% % % 9.78% % 6.56% % 8.538% % 5.694% 6.449% % % 3