Tenamen i 5B1575 Finansiella Deriva. Måndag 27 augusi 2007 kl. 14.00 19.00. Answers and suggesions for soluions. 1. (a) For he maringale probabiliies we have q 1 + r d u d 0.5 Using hem we obain he following binomial ree where he value of he sock is wrien in he nodes, and he value of he opion is wrien in he adjacen boxes. 225 100 150 50 100 25 75 0 50 0 25 0 The price of he call opion is hus 25 kr. (b) A replicaing porfolio is a self-financing porfolio which has he same value as he claim a he exercise dae. I is easy o check
5B1575 Tenamen 2007-08-27 2 (c) ha he value of a porfolio consising of K bonds, a long posiion in a call opion wih srike price K and exercise dae T, and a shor posiion in he underlying sock iself will be equal o he value of he pu opion a ime T. The porfolio is also selffinancing since i is consan. i. A probabiliy measure Q is a risk neural maringale measure for he sandard Black-Scholes model if i is equivalen o P, and he process S/B is a Q-maringale. ii. Under he maringale measure Q he dynamics of S are given by ds rs d + σs dv, where V is a Q-Wiener process. Using Iô s formula on R ln(s /S 0 ) we obain dr 1 ds 1 1 S 2 S 2 (ds ) 2 (r 1 ) 2 σ2 d + σdv Since R 0 0 we obain R (r 1 ) 2 σ2 + σv, and hus R is normally disribued wih expecaion r σ 2 /2 and variance σ 2. 2. (a) The price of he claim a ime [0, T ] is given by [ ] Π [X] e r(t ) E Q S β T F The dynamics of S under Q are given by ds rs d + σs dv, where V is a Q-Wiener process. Le Z S β and use Iô s formula o find he differenial of Z dz βs β 1 ds + 1 2 β(β 1)Sβ 2 (ds) 2 (βr + 1 ) 2 β(β 1)σ2 Zd + βσzdv.
5B1575 Tenamen 2007-08-27 3 Le c βr + β(β 1)σ 2 /2. Inegraing we obain Z u Z + u cz s ds + u βσz s dv s. Now ake he condiional expecaion wih respec o F E[Z u F ] Z + u ce[z s F ]ds + 0. Le m u E[Z u F ] and ake derivaives wih respec o u { ṁ cm, m Z. Solving he ODE above we ge m u Z e c(u ). The price of he claim is given by e r(t ) m T, i.e. Π [X] S β e(c r)(t ), wih c βr + 1 2 β(β 1)σ2. (b) We know ha Π() F (, S ). Therefore using Iô s formula on F we obain (in he formulas below sub-indices denoe parial derivaives, for example F s F s ) df F d + F s ds + 1 2 F ss(ds) 2 (1) (F + 1 2 σ2 S 2 F ss ) d + F s ds. (2) In order for a porfolio h (h B, h S ) o be self-financing i mus saisfy dv h h B db + h S ds. Now comparing wih equaion (2) we see ha we mus have h S F s (, S ) βs β 1 e (c r)(t ), c βr 1 2 β(β 1)σ2. Since V h h B B + h S S we mus have h B F SF s e r (1 β)s β e(c r)(t ) e r c βr 1 2 β(β 1)σ2. You may recognize he coninuously rebalanced dela hedge.
5B1575 Tenamen 2007-08-27 4 3. (a) The price of he opion on he underlying sock which pays no dividends will be greaer han he price of he opion on he dividend paying sock, given ha he sock price is he same oday. This is because he price of he sock akes fuure dividend paymens ino accoun, bu he holder of he opion will no benefi from hem, and he dividend paymens will cause he sock price o fall making i less likely ha he opion will be exercised. (b) Jus as in he ordinary Black-Scholes model he price of he opion is given by Π e r(t ) E Q [max{s T K, 0} F ] However, in a model wih dividends i is he normalized gains process, G, which should be a maringale under he maringale measure Q. The normalized gains process for he sock is given by G Z () S δs u + du. B 0 B u This can be seen o be a maringale if he dynamics of S under Q are given by ds (r δ)s d + σs dv, where V is a Q-Wiener process. If we rewrie he price in he following way Π e δ(t ) e (r δ)(t ) E Q [max{s T K, 0} F ] }{{} BSformula we can as indicaed use he Black-Scholes formula on he second par wih r δ for he ineres rae and σ for he volailiy. We hus have where Π e δ(t ) [S Φ(d 1 ) e (r δ)(t ) KΦ(d 2 )] e δ(t ) S Φ(d 1 ) e r(t ) KΦ(d 2 ) d 1 { ( ) 1 S() σ ln T K d 2 d 1 σ T. + (r δ + 1 ) } 2 σ2 (T ),
5B1575 Tenamen 2007-08-27 5 4. (a) The Ho-Lee model possesses an affine erm srucure, i.e. he zero coupon bond prices are of he form p(, T ) e A(,T ) B(,T )r, where A and B are deerminisic funcions. To compleely specify he bond prices we need o find expressions for A and B. The Iô formula applied o p(, T ) ea(,t ) B(,T )r yields dp T (A B r)p T d Bp T dr + 1 2 B2 p T (dr) 2 [A B r θb + 1 ] 2 B2 ρ 2 p T d ρbp T du [ A θb + 1 ] 2 ρ2 B 2 B r p T d ρbp T du. Under Q we know ha p(, T )/B() is a maringale, which means ha p(, T ) has o have local reurn equal o he shor rae r. Thus, for 0, and r (, ) he following equaliy has o hold [ A θb + 1 2 ρ2 B 2 ] [B + 1]r 0 The only way his is possible is if boh square brackes equal zero, his will give you he ordinary differenial equaions solved by A, and B. The boundary condiions are obained from he condiion ha p(t, T ) 1. To sum up we have { B (, T ) 1, B(T, T ) 0, and why A (, T ) θb(, T ) 1 2 ρ2 B 2 (, T ), A(T, T ) 0. B(, T ) (T ), A(, T ) T θ(s)(s T )ds + ρ2 2 (T ) 3. 3 (b) Using he T -forward measure Q T we have he following pricing formula [ ] max{s(t ) K, 0} Π() p(, T )E T p(t, T ) F,
5B1575 Tenamen 2007-08-27 6 where he super index T indicaes ha he expecaion should be aken under he forward measure Q T. This can be wrien as Π() p(, T )E T [max{z(t ) K, 0} F ]. where Z() S()/p(, T ). In order o use he exended version of he Black-Scholes formula we need an ineres rae and a volailiy. Under Q T Z will be a maringale (i is a price process normalized by he numeraire p(, T )), which means ha he drif of Z is zero, and herefore also he ineres rae used should be zero. To obain he volailiy firs noe ha Iô applied o p(, T ) ea(,t ) B(,T )r yields dp(, T ) rp(, T )d ρb(, T )p(, T )du. Then use Iô once more o obain ( ) S dz (...)d + Z [σdw + ρb(, T )du ] p(, T ) σ 2 + ρ 2 B 2 (, T )dv From his we see ha he volailiy of Z is deerminisic. γ() σ 2 + ρ 2 B 2 (, T ). Then under Q T we have dz γ()z dv T Le for a Q T -Wiener process V T. Applying he exended version of he Black-Scholes formula now yields where Π() p(, T )E T [max{z(t ) K, 0} F ] d 1 ln ( p(, T ) [Z()Φ(d 1 ) KΦ(d 2 )] S()Φ(d 1 ) Kp(, T )Φ(d 2 ) S() Kp(,T ) T ) + 1 T 2 γ 2 (s)ds T, d 2 d 1 γ 2 (s)ds. γ 2 (s)ds
5B1575 Tenamen 2007-08-27 7 5. Recall ha f(, T ) Z + H(T ), (3) and ha dr µ(, r )d + σ(, r )dw. (4) By definiion, we have ha r f(, ), so (a) r f(, ) Z + H(0) Z. (b) Using (3), he relaion r Z, he dynamics of r in (4), and he Iô formula we obain df(, T ) [µ(, r ) h(t )]d + σ(, r )dw where h(x) H (x). The HJM drif condiion now reads µ(, r ) h(t ) σ(, r ) T σ(, r )ds i.e. we obain he following ideniy for all, T, and r µ(, r) h(t ) σ 2 (, r)(t ). Denoing T by x his reads µ(, r) h(x) σ 2 (, r)x. Seing x 0 yields µ(, r) c where c h(0). We hus have c h(x) σ 2 (, r)x which implies ha also σ has o be a consan as a funcion of (,r). This means ha we have he following degenerae form of he Ho-Lee model dr cd + σdw. For fun we noe ha h(x) c σ 2 x. Inegraing we obain H(x) σ2 2 x2 cx + d.