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1 Pricing Fixed-Income Derivaives wih he Forward-Risk Adjused Measure Jesper Lund Deparmen of Finance he Aarhus School of Business DK-8 Aarhus V, Denmark Homepage: Firs draf: April 998 I am graeful o Peer Honore for helpful commens.

2 he problem Consider a xed-income derivaive wih a single payo a ime which depends on he erm srucure. In paricular, we will look a opions on zero-coupon bonds and ineres-rae caps. For a call opion on a zero-coupon bond mauring a ime, he ime payo and hence value of he derivaive isgiven by V max P (; ), K; : () By he no-arbirage heorem, he price oday ( ) is V E e, R Q r sds V ; () where he expecaion is aken under he risk-neural disribuion (also called he Q-measure). hus, he price depends on he sochasic process for he shor rae and he conracual specicaion of he securiy (i.e., how he payo is linked o he erm srucure). he price V in equaion () is given by he expecaion of he produc of wo dependen random variables, and calculaing his expecaion is ofen quie dicul. he purpose of his noe is presening a change-of-measure echnique which considerably simplies he evaluaion of V. Specically, weare going o calculae V as V P (;)E Q (V ); (3) where Q is a new probabiliy measure (disribuion), he so-called forward-risk adjused measure. his echnique was inroduced in he xed-income lieraure by Jamshidian (99). Model seup and noaion Our erm-srucure model is a general one-facor HJM model, see Heah, Jarrow and Moron (99) or Lund (998) for an exposiion. Under he Q-measure, forward raes are governed by where df(; ),(; ) P (; )d + (; )dw Q ; (4) Z P (; ), (; u)du: (5) Bond prices evolve according o he SDE dp (; )r P(; )d + P (; )P (; )dw Q ; (6) so P (; )isheime volailiy of he zero mauring a ime.

3 3 he forward-risk adjused measure Under cerain regulariy condiions, he price of he derivaive securiy follows he SDE dv r V d + V ()V dw Q : (7) his means ha, under he risk-neural disribuion, he expeced rae of reurn equals he shor rae (jus like any oher securiy), and he reurn volailiy is V (). So far, neiher V nor V () are known, bu his is no essenial for he following argumens. In fac, he only hing ha maers is ha he price process has he form (7) since his faciliaes pricing by he forward-risk adjused measure. We begin by dening he deaed price process F V P(; ) (8) for [;]. We can inerpre F as he price of V in unis of he -mauriy bond price (i.e., as a relaive price). Using Io's lemma, i can be shown ha df P ( P, V )F d +( V, P )F dw Q (9) where P and V are shorhand noaion for P (; ) and V (), respecively. he proof of (9) is given in appendix A. Furhermore, we dene a new probabiliy measure, Q, such ha W Q W Q, Z P (u; )du; [;]; () is a Brownian moion under Q. and W Q is In dierenial form he relaionship beween W Q dw Q dw Q, P (; )d dw Q, P d: () he new probabiliy measure is known as he forward-risk adjused measure. I is very imporan o noe ha here is a dieren measure for each (payo dae). If we subsiue () ino (9), we obain he dynamics of F under he new probabiliy measure Q. Sraighforward calculaions give df, P ( V, P )F d +( V, P )F dw Q + P d ( V, P )F dw Q ; (3) We have, implicily, used a similar echnique when dening he risk-neural measure earlier. Specically, ifw is a Brownian moion under he original (rue) probabiliy measure, we dene Q such ha W Q W Q +Z (u)du () is a Brownian moion under Q. Noe ha (u) is he marke price.

4 as he wo erms wih d cancel ou. hus under Q, he drif is zero and F is a maringale. he new probabiliy measure was dened in order o obain his resul since he maringale propery implies ha F E Q (F ): (4) Moreover, by deniion P (;), so a mauriy we have F V, and using (4), he curren ( ) price of he derivaive securiy can now be calculaed as V P (;)F P(;)E Q (F ) P(;)E Q (V ); (5) which isp(;) imes he expeced payo under Q. Generally, he laer calculaion is a lo simpler han direc evaluaion of he expecaion under Q, as in equaion () above. Wih (5) a hand, he only remaining ask is deermining he disribuion of he payo under he forward-risk adjused measure. We conclude his secion by noing ha f(; ) is a maringale under Q. o see his, subsiue () ino he forward-rae SDE (4), df(; ),(; ) P (; )d + (; )dw Q + P (; )d (; )dw Q : (6) his propery urns ou be very useful when pricing a-he-money ineres-rae caps, cf. he second example in he nex secion. 4 wo examples For concreeness, we use he exended Vasicek model which is a special case of he one-facor HJM model wih and (; )e,(, ) ; (7) P (; ),Z (; u)du e,(, ), : (8) he exended Vasicek model is a Markovian HJM model, cf. Lund (998), bu he following pricing formulas for bond opions [equaion (3)] and ineres-rae caps [equaion (4)] do no depend on he Markov propery. Noe ha we are using he maringale propery in he \opposie" direcion (i.e., backwards) in equaions (4) and (5). Normally, we know F and use he maringale propery ocompue he expeced value a ime. his line of reasoning is implici in he weak form of marke eciency [Fama (97)] where we argue ha he bes forecas of he fuure sock price is he sock price oday. In he conex of pricing derivaives, we use our knowledge abou he Q -disribuion of he payo V, combined wih he maringale propery off (he relaive price), o compue he curren (relaive) price of he derivaive securiy. 3

5 4. Call opion on a zero-coupon bond In he rs example, he xed-income derivaive is a call opion on a zero-coupon bond mauriy a ime. he opion expires (maures) a ime < wih he following payo: C max ; (9) P (; ), K; where K is he srike (exercise) price of he opion. In order o price his securiy, we need he disribuion of C under he forwardrisk adjused measure. Since C only depends P (; ) and since P (;), we can calculae he expecaion of C from he disribuion of he relaive price, F (; ; )P(; )P (; ); () which is also he forward price of he -mauriy bond for delivery a ime. Using he resuls of secion 3, he SDE for F (; ; ) under Q is given by df (; ; ) f P (; ), P (; )g F (; ; )dw Q F (; ; )F (; ; )dw Q : () Since bond prices are always sricly posiive, he logarihm of F (; ; )iswelldened, and a simple applicaion of Io's lemma gives d log F (; ; ), F (; ; )d + F (; ; )dw Q : () Afer inegraing from o we have Z log F (;; ) log P (; ) log F (;; ), (; ; F )d +Z F (; ; )dw Q : (3) he rs equaliy in (3) follows because P (;). Moreover, if F (; ; ) is deerminisic, i.e. if he model is Gaussian, i follows from (3) ha log P (; ) is condiionally normally disribued wih variance! F (; ) Z and mean F (; ) log F (;; ), F (; ; )d; (4) Z F (; ; )d log F (;; ), (;! F ): (5) For he exended Vasicek model, F (; ; )isgiven by F (; ; ) e,(, ), ; (6) ), e,(, ) e,(, ) e,(, 4

6 cf. equaion (8), and he variance! F (; ) can be calculaed as! F (; ) e,(, ), e,(, ), A e,(, ) e, B (, )Var Q (r ); (7) since he las parenhesis in he second line can be recognized as he condiional variance of r, see equaion (4) in Jamshidian (989). he funcion B() is he \facor loading" (sochasic duraion) for he Vasicek model. 3 Finally, he price of he call opion, denoed C(;K), is given by: C(;K) P(;)E (C ) P(;)Z log K P (;)Z log K,P (;)KZ e x, K A p!f e,(x, F )! F dx e x p e,(x, F )! F dx!f log K p!f e,(x, F )! F dx P (; )N(d ),P(;)KN(d ); (3) where N() is he cumulaive normal disribuion funcion,! F is shorhand noaion for!(; ), and! d log P (; ) P(;),log K +! F! F (3) d d,! F : (3) he calculaion is compleely analogous o he Black-Scholes model for call opions on sock prices, so we skip he inermediae seps leading o he nal expression for C(;K) in equaion (3). 4 3 An alernaive derivaion for he exended Vasicek model can be based on he formula log P (; ) A(; )+B(,)r ; (8) cf. Lund (998), and since A(; ) is deerminisic, he variance of log P (; ) under Q is Var Q (log P (; )) B (, ) Var Q (r ): (9) Of course, wih his approach wewould sill need o deermine he variance of r. 4 Hin for your own derivaion: if x is N (; ), he runcaed mean of exp(x) is Z L e x p e,(x, ) dx 5

7 4. Ineres-rae caps Consider a derivaive wih he following payo a ime : C max(r, K; ): (34) his corresponds o a simple ineres-rae cap. 5 by: he price (oday) of he cap is given C(;K) P(;)E Q (C ) P(;)E Q [max(r, K; )] : (35) In order o calculae (35) we mus deermine he disribuion of r under Q (he forward-risk adjused measure). Firs, noe ha r f(;); (36) so we can obain he disribuion of r from f(;). Second, under Q he -mauriy forward rae is a maringale, as shown in equaion (6) in secion 3. his means ha r f(;) f(;)+z (; )dw Q : (37) If (; ) is deerminisic, r is condiionally normally disribued (a ime ) wih mean f(;) and variance Var Q (r ) Z (; )d v (;): (38) For he exended Vasicek model, his becomes Z v (;) e,(,), e, d : (39) Z L exp + p e +, (x,, ) dx,l +(+ ) N since he inegrand in he second line equals exp, + ; (33) imes he densiy funcion of a normal disribuion wih mean + and variance. 5 In he real world, caps are more complicaed. he underlying ineres rae is no he shor rae, bu (say) he hree-monh (LIBOR) ineres rae. Moreover, a cap conrac for years on he hree-monh rae is a porfolio of 4 so-called caples (single-paymen caps), and he paymen of he i'h caple is :5 max[r 3M ((i, )4), K; )], where R 3M () is he hree-monh ineres rae a ime. he paymens are made in arrear, which means ha he i'h paymen is made a ime i4 (hree monhs afer he xing dae). Of course, he \real world" cap can be priced by he same principles as he simple cap described in his secion, ha is by a suiable applicaion of he he forward-risk adjused measure for each caple. Needless o say, he algebra become more involved, bu ha is he only real dierence. 6

8 Finally, we compue he expeced payo under Q and hence he price of he cap. For reason of space, we concenrae on he a-he-money cap 6 where K f(;), and C(;f(;)) P (;)E Q [max(r, f(;);)] P (;) v(;) p ; (4) where v(;) is dened in (38) for any Gaussian one-facor HJM model, and in (39) for he exended Vasicek model. he second line in (4) follows by noing ha he payo can be wrien as max(r, f(;);) max Z (; )dw Q ;! (4) sz which has he same disribuion as (; )d max(x; ) v(;) max(x; ); (4) where x is N (; ), ha is normally disribued wih zero mean and uni variance. he expeced value of max(x; ) is given by E[max(x; )] Z Z x p, e x dx p e,u du p : (43) he second line follows by a change of variables from x o u x. his complees he proof of (4). 6 he price formula for a cap wih arbirary exercise price, K, (in he exended Vasicek model) can be found in Longsa (995). 7

9 Appendix A: proof of equaion (9) o simplify he noaion, we wrie he SDE for V subscrips, and P (; ) wihou he ime dv rv d + V VdW Q (44) dp rpd + P PdW Q : (45) Noe ha since V and P are driven by he same Brownian moion, changes in V and P are perfecly correlaed. he objecive is o deermine he SDE for he funcion F (V; P) VP. Firs, we compue he requisie parial derivaives of F wih respec o V and P P F (V; F (V; V P 3 F (V; P) P : (5) Second, an applicaion of Io's lemma gives us df P rv, V P rp + V P 3 P P, P F, V P F P V V, V P P P P V V P P d + dw Q (5) d +( V F, P F)dW Q (5) P ( P, V )Fd+( V, P )FdW Q (53) which is equaion (9) in secion 3. his complees he proof. 8

10 References Fama, E.F. (97), \Ecien Capial Markes: A Review of heory and Empirical ess," Journal of Finance, 5, 383{47. Heah, D., R. Jarrow and A. Moron (99), \Bond Pricing and he erm Srucure of Ineres Raes," Economerica, 6, 77{5. Jamshidian, F. (989), \An Exac Bond Opion Formula," Journal of Finance, 44, 5{9. Jamshidian, F. (99), \Bond and Opion Evaluaion in he Gaussian Ineres Rae Model," Research in Finance, 9, 3{7. Longsa, F.A. (995), \Hedging Ineres Rae Risk wih Opions on Average Ineres Raes," Journal of Fixed Income, March 995, 37{45. Lund, J. (998), \Review of Coninuous-ime erm-srucure Models Par II: Arbirage-Free Models," Lecure Noes, Deparmen of Finance, Aarhus School of Business, April

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