Tentamen i 5B1575 Finansiella Derivat. Tisdag 22 maj 2007 kl Answers and suggestions for solutions.

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1 Tenamen i 5B1575 Finansiella Deriva. Tisdag 22 maj 2007 kl Answers and suggesions for soluions. 1. (a) According o he Firs Fundamenal Theorem he model is free of arbirage if and only if here exiss a maringale measure. We hus need o prove ha here exis q 1, q 2, and q 3 all sricly beween zero and one, and such ha s r q 1 su + q 2 s + q 3 sd, 1 q 1 + q 2 + q 3. Leing q 1 ac as a parameer we obain for q 2 and q 3 q 2 2(0.6 q 1 ), q 3 q From his we see ha all values of q 1 such ha 0.2 < q 1 < 0.6 will resul in a maringale measure, and herefore he model is free of arbirage. (There are infiniely many maringale measures, bu o show ha he model is free of arbirage you only need o find one, so if you have found one soluion, say q 1 0.3, q and q 3 0.1, you need no worry abou he oher soluions.) (b) Since he ineres rae is zero he opion price is given by he following formula Since C Bach (0) E Q max{s T K, 0} E Q (S T K)I {ST K 0}. S T K S 0 + σs 0 V T K,

2 5B1575 Tenamen we see ha S T K N(S 0 K, σ 2 S0 2 T ). Now apply he saed resul wih µ S 0 K, σ 2 σ 2 S0 2 T, l 0, and h, and noe ha 1 Φ(x) Φ( x) and φ(x) φ( x). (c) By definiion we have ha call,bach C Bach, s once you have subsiued s for S in C Bach. Differeniaing you will find ( ) S K call,bach () Φ. σs 0 T Here we have subsiued back S o obain odays value of. If you have sold he opion ells you how many you have o buy of he underlying in order for your porfolio o become dela neural, i.e. insensiive o small changes in he sock price. 2. (a) We have he following equaion 0 Π(; X) e r(t ) E Q S T f(; T, S T ) F. Solving for he forward price we obain (use ha f(; T, S T ) F ) f(; T, S T ) E Q S T F. Since S/B is a Q-maringale and B e r we have ha f(; T, S T ) B T E Q ST B F S B T e r(t ) S. T B (b) The payoff of he range forward can be wrien as X max{min{s T, K 2 }, K 1 } f(0; T, S T ) K 1 + max{s T K 1, 0} max{s T K 2, 0} f(0; T, S T ). The price is herefore given by Π(; X) e r(t ) E Q K 1 + max{s T K 1, 0} max{s T K 2, 0} f(0; T, S T ) F C(, S, K 1, T, r, σ) C(, S, K 2, T, r, σ) +e r(t ) (K 1 S 0 e rt ), where C(, s, K, T, r, σ) denoes he sandard Black-Scholes price a ime of a European call opion wih exercise price K and

3 5B1575 Tenamen expiry dae T, when he curren price of he underlying is s, he ineres rae is r, and he volailiy of he underlying is σ. For fuure use we le P (, s, K, T, r, σ) denoe he price of he corresponding pu opion. (c) Using pu-call-pariy a ime T max{k S T, 0} K + max{s T K, 0} S T, we can decompose he payoff of he range forward in he following way insead X K 1 + max{s T K 1, 0} max{s T K 2, 0} f(0; T, S T ) max{k 1 S T } max{s T K 2, 0} + S T f(0; T, S T ). From his we see ha he price can also be wrien in he following way Π(; X) P (, S, K 1, T, r, σ) C(, S, K 2, T, r, σ) + S S 0 e r. Since S S 0 e r is he price of a forward, we see ha a range forward is equal o a porfolio composed of a long forward conrac, a long pu wih srike K 1, and a shor call wih srike K (a) We have p(, T ) E Q e T r(u)du F E Q e T X 1(u)+X 2 (u)du F E Q e T (e κ 1 (u ) u X 1 ()+κ 1 θ 1 e κ 1 (u s) u ds+σ 1 e T e T E Q e T (e κ 2 (u ) X 2 ()+κ 2 θ 2 u e κ 1 (u )du X 1 () e T e T (κ 2θ 2 u e κ 2 (u s) ds+σ 2 u e κ 2 (u )du X 2 () (κ u 1θ 1 e κ 1 (u s) u ds+σ 1 e κ 1 (u s) dw 1 (s))du e κ 2 (u s) u ds+σ 2 e κ 2 (u s) dw 2 (s))du which shows ha he zero coupon bond prices have he desired form. (Acually he expecaion can be compued direcly, since he sochasic variable in he exponenial is normally disribued.) e κ 1 (u s) dw 1 (s))du e κ 2 (u s) dw 2 (s))du F

4 5B1575 Tenamen (b) The Iô formula applied o p(, T ) e A(,T ) B(,T )X 1() C(,T )X 2 () yields dp T (A B X 1 C X 2 )p T d Bp T dx 1 Cp T dx B2 p T (dx 1 ) C2 p T (dx 2 ) 2 A B X 1 C X 2 Bκ 1 (θ 1 X 1 ) Cκ 2 (θ 2 X 2 ) B2 σ C2 σ2 2 p T d +... A Bκ 1 θ 1 Cκ 2 θ B2 σ C2 σ 2 2 (B Bκ 1 )X 1 (C Cκ 2 )X 2 p T d +... 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 X 1 + X 2. Thus, for 0, and x i (, ) i 1, 2 he following equaliy has o hold A Bκ 1 θ 1 Cκ 2 θ B2 σ C2 σ2 2 B Bκ 1 + 1x 1 C Cκ 2 + 1x 2 0 The only way his is possible is if all hree square brackes equal zero, his will give you he ordinary differenial equaions solved by A, B, and C. The boundary condiions are obained from he condiion ha p(t, T ) 1. So o sum up we have { B (, T ) B(, T )κ 1 1, B(T, T ) 0, { C (, T ) C(, T )κ 2 1, C(T, T ) 0, and A (, T ) κ 1 θ 1 B(, T ) + κ 2 θ 2 C(, T ) 1 2 σ2 1 B2 (, T ) 1 2 σ2 2 C2 (, T ), A(T, T ) (a) You can replicae he payoff 1/p d (T, U) a ime U by buying a domesic T -bond a ime and reinvesing he principal received

5 5B1575 Tenamen a ime T in 1/p d (T, U) domesic U-bonds. The price of a roll bond is herefore p d (, T ). (b) For T we have Π q roll E Q e U r d (s)ds 1 1 e p f (T, U) EQ U p d(, U) p f (T, U), p f (T, U) F r d (s)ds F where we have used ha 1/p f (T, U) F for T o obain he second equaliy. The price of a quano roll bond for T is hus p d (, U)/p f (T, U). (c) We have ha he value process of he porfolio is given by V h () h 1 ()p d (, U) + h 2 () p f (, U) + h 3 () p f (, T ) V (; T, U). A ime T he value process hen equals V h (T ) V (T ; T, U) p d(t, U)p f (T, T )G(T ; T, U) p f (T, U) p d(t, U) p f (T, U), where we have have used ha, as always, p f (T, T ) 1, and ha G(T ; T, U) 1. Thus he porfolio is equal o he desired claim a ime T. I remains o check ha he porfolio is self-financing, i.e. ha dv h () h 1 ()dp d (, U) + h 2 ()d p f (, U) + h 3 ()d p f (, T ) or wih he expressions for h 1, h 2 and h 3 insered ( dv h dpd (, U) () V (; T, U) p d (, U) d p f(, U) p f (, U) + d p ) f(, T ) (1) p f (, T ) Using he Iô formula and ha p f (, S) X()p f (, S) we obain d p S f Xd p S f + ps f dx + d ps f dx r d () p f (, S)d + σ() + ν f (, S) p f (, S)dW,

6 5B1575 Tenamen and insered in o (1) we ge ( ) dv h dpd (, U) () V (; T, U) p d (, U) ν f(, U)dW + ν f (, T )dw. Now use ha he dynamics of p f (, S) o see ha ( dv h dpd (, U) () V (; T, U) p d (, U) dp f(, U) p f (, U) + dp f(, T ) p f (, T ) ) + σ()ν f (, T ) ν f (, U) Since V h () V (; T, U) and he dynamics in he formula above are he same as hose of V (; T, U) given in he exercise, we conclude ha he porfolio is self-financing and replicaes he T -claim p d (T, U)/p f (T, U).. 5. If we use S as he new numeraire we know ha Π/S is a maringale under Q S. Wriing down he maringale propery we ge Π() S() ES Π(T ) S(T ) F, where he super scrip S indicaes ha he expecaion should be aken under Q S. Thus, Π() S() { E S max 1 T T 0 S udu S(T ), 0} S(T ) F E S max { 1 T T 0 S udu S(T ) 1, 0} E S max {Z(T ) 1, 0} F. In order o apply Feynman-Kač we need he Q S -dynamics of he process Z given by Z() 1 T 0 S(u)du. S() F

7 5B1575 Tenamen Recall ha he Girsanov kernel which akes you from Q o Q S is he volailiy of S. Using his we find ha he Q S -dynamics of S are given by ds (r + σ 2 )S d + σs du, where U is a Q S -Wiener process. Le Y () 1 T 0 S u du, i.e. dy 1 T S d. The dynamics of Z can now be found by an applicaion of he Iô formula dz 1 dy Y S S 2 ds + 1 2Y 2 S 3 (ds ) 2 ( ) 1 T rz d σz du. Now Feynman-Kač s heorem ells us ha Π() S() F (, Z()), where F solves he PDE saed in he exercise.

Chapter 7. Response of First-Order RL and RC Circuits

Chapter 7. Response of First-Order RL and RC Circuits Chaper 7. esponse of Firs-Order L and C Circuis 7.1. The Naural esponse of an L Circui 7.2. The Naural esponse of an C Circui 7.3. The ep esponse of L and C Circuis 7.4. A General oluion for ep and Naural