Tenamen i 5B1575 Finansiella Deriva. Torsdag 25 augusi 2005 kl. 14.00 19.00. Examinaor: Camilla Landén, el 790 8466. Tillåna hjälpmedel: Av insiuionen ulånad miniräknare. Allmänna anvisningar: Lösningarna skall vara läläsliga och välmoiverade. All införd noaion skall vara förklarad. Problem rörande inegrabilie behöver ej redas u. OBS! Personnummer skall anges på försäsblade. Numrera sidorna och skriv namn på varje blad! 24 poäng inklusive bonuspoäng ger säker godkän. Endas en uppgif på varje blad. 1. (a) In he binomial ree below he price of an American pu opion wih srike price K = 100 kr and exercise dae T = 2 years has been compued using he parameers s 0 = 100, u = 1.4, d = 0.8, r = 10%, and p = 0.75. (The value of he sock is wrien in he nodes, and he value of he opion is wrien in he adjacen boxes.) 196 0 140 0 100 10/1.1 112 0 80 20 64 36 Noe ha an early exercise of he opion is opimal a he node wih sock price 80 (his is why he las opion values have been pu in dashed boxes)! Your ask is o find he replicaing porfolio for his opion and o verify ha he porfolio is self-financing............................................ (3p) (b) Consider a sandard Black-Scholes marke, described in deail in Exercise 2. Suppose ha you presenly own a porfolio wih = 0 and Γ = 1. How many of he underlying sock, and derivaives wih payoff X = S 2 T, 1
5B1575 Tenamen 2005-08-25 2 (c) in T = 2 years, and pricing funcion ( F (, s) = s 2 exp r ) } 2 σ2 (T ), 8 should you buy or sell, in order o make your porfolio boh dela and gamma neural? Today s sock price is S 0 = 10. Use r=5% and σ=50% on a yearly basis.................................................................... (3p) i. Sae he Firs Fundamenal Theorem of arbirage pricing (a slighly sloppy version will do)..................................................... (2p) ii. Sae he Second Fundamenal Theorem of arbirage pricing........ (2p) 2. Consider a sandard Black-Scholes marke, i.e., a marke consising of a risk free asse, B, wih P -dynamics given by db = rb d, B 0 = 1, and a sock, S, wih P -dynamics given by ds = αs d + σs dw, S 0 = s 0. Here W denoes a P -Wiener process and r, α, and σ are assumed o be consans. A chooser opion is an agreemen which gives he owner he righ o choose a some prespecified fuure dae T 0, wheher he opion is o be a call or pu opion wih exercise price K and remaining ime o expiry T T 0. Noe ha K, T and T 0 < T are all prespecified by he agreemen. The only hing he owner can choose is wheher he opion should be a call or a pu opion, and his choice has o be made a ime T 0. Compue he arbirage price of he T 0 -claim called a chooser opion......... (10p) 3. Consider he Vasiček model for he shor rae, defined under he objecive measure P as ( ) dr() = κ θ P r() d + σdw (), where κ, θ P, and σ are posiive consans, and W denoes a P -Wiener process. If we assume he marke price of risk, λ, o be a consan, his will imply ha he dynamics of r under he risk neural maringale measure Q are given by ( ) dr() = κ θ Q r() d + σdv (), where θ Q = θ P σλ/κ, and V denoes a Q-Wiener process saisfying dv = λd+dw. Should you feel ha you need numbers, hen reasonable values are θ P = 0.04, θ Q = 0.04 ± 0.02 (you should choose he sign, see below), κ = 0.5, and σ = 0.01.
5B1575 Tenamen 2005-08-25 3 (a) Wha sign would you expec he marke price of risk λ o have, and why? (Noe ha σλ/κ will have he same sign as λ, and his is hus he sign you should use in he expression θ Q = 0.04 ± 0.02, should you need he number laer on). (2p) (b) Compue E Q [r(t )] and lim T EQ [r(t )].................................. (3p) Remark. I can be shown ha he saionary disribuion of r has lim T EQ [r(t )] as expeced value. (c) Recall ha he zero-coupon yield, y(, T ), solves p(, T ) = e y(,t ) (T ) 1. i. Suppose ha r 0 = θ Q. Wha does he iniial yield curve T y(0, T ) look like hen? (You need only give he big picure: where does he yield curve sar, wha happens as he mauriy ends o infiniy, and is he slope posiive, negaive, or none.) Hin: There are some resuls a he end of he exam which may be of use when looking a he yield. ii. Suppose ha r 0 = θ P. Wha does he iniial yield curve look like now (if needed, use he sign of λ you argued for in Exercise (a))? iii. The firs curve can be inerpreed as a Q-ypical shape of he yield curve, jus as he second curve can be called a P -ypical shape of he yield curve. Explain why! Hin: The remark afer Exercise (b) may help.......................................................................... (5p) 4. (a) Forward rae models have o be specified wih some care o avoid inroducing arbirage possibiliies. I is well known ha if he forward raes are modeled as df(, T ) = α(, T )d + σ(, T )dw, T 0, where W is an m-dimensional Wiener process, hen he he dynamics of he (zero-coupon) bond prices, p(, T ), are given by where dp(, T ) = r() + b(, T )}p(, T )d + a(, T )p(, T )dw (), T a(, T ) = σ(, u)du, b(, T ) = T α(, u)du + 1 2 a(, T ) 2. Your ask is o derive he Heah-Jarrow-Moron drif condiion (under Q). (2p) (b) Suppose you are o price a foreign bond derivaive, for example an opion. You hen need a erm srucure model describing he domesic and foreign bond markes, wih (zero-coupon) bond prices denoed by p d (, T ) (in unis of domesic currency) and p f (, T ) (in unis of foreign currency), respecively. i. Sar ou by assuming ha he marke consiss of a domesic and a foreign risk less asse, wih price processes B d and B f, respecively. B d is denominaed in unis of he domesic currency, and B f in unis of he foreign currency. The dynamics of hese price processes are given by db d () = r d ()B d ()d, and db f () = r f ()B f ()d,
5B1575 Tenamen 2005-08-25 4 where r d and r f are assumed o be adaped sochasic processes. Furhermore, i is assumed ha if you buy he foreign currency his is immediaely invesed in a foreign bank accoun (i.e. he foreign risk less asse). The exchange rae process X beween he domesic and foreign currency is assumed o be of he form dx() = α X ()X()d + σ X ()X()dV, where α X is a real-valued adaped process, σ X is an R m -valued adaped process, and V is an m-dimensional Wiener process under he domesic maringale measure Q d. Derive an expression for α X under he domesic maringale measure Q d. If you like you may assume ha V is one-dimensional, he resul will be he same............................................................... (3p) ii. Now, suppose we wan he domesic and foreign ineres raes o be given by forward rae processes f d and f f, respecively, as in he HJM seing. We hus model f d and f f under he domesic maringale measure in he following way df d (, T ) = α d (, T )d + σ d (, T )dv, df f (, T ) = α f (, T )d + σ f (, T )dv, where V is an m-dimensional Wiener process under he domesic maringale measure Q d. Again, in order o bar arbirage possibiliies he forward rae processes have o be modeled wih some care. The drif resricion which mus be placed on he domesic forward rae process under he domesic maringale measure is he well-known HJM drif resricion. Your ask is o derive he drif resricion on he foreign forward rae process f f under he domesic maringale measure Q d. (The expression for α X under Q d obained in previous exercise is sill valid in his new seing)..................................................................... (5p) 5. Consider a given marke consising of a risk free asse B and a risky asse wih price process S() and cumulaive dividend process (udelningsprocess) D(). Boh S() and D() are assumed o have sochasic differenials. The shor rae r is assumed o be consan and deerminisic. (a) Define he conceps porfolio, value process and self-financing porfolio......................................................................... (2p) (b) Define wha is mean by a maringale measure for his model........ (2p) (c) Now specialize o he case when he risky asse has a consan dividend yield δ, i.e. dd() = δs()d. Suppose ha you own he risky asse S and ha you inves all dividends in he risky asse in a self-financing way. How many risky asses will you hen own a ime T if you sared ou wih exacly one a ime = 0, and wha will he dynamics of your porfolio look like?........... (6p) Hin: The concep of a relaive porfolio migh be useful. Lycka ill!
5B1575 Tenamen 2005-08-25 5 Hins: You are free o use he following in any of he above exercises. The densiy funcion of a normally disribued random variable wih expecaion m and variance σ 2 is given by ϕ(x) = 1 σ 2π e (x m)2 /(2σ 2). Le Φ denoe he cumulaive disribuion funcion for he N(0, 1) disribuion. Then Φ( x) = 1 Φ(x). The sandard Black-Scholes formula for he price Π() of a European call opion wih srike price K and ime of mauriy T is Π() = F (, S()), where F (, s) = sφ[d 1 (, s)] e r(t ) KΦ[d 2 (, s)]. Here Φ is he cumulaive disribuion funcion for he N(0, 1) disribuion and d 1 (, s) = ( ) 1 s σ ln + (r + 12 ) } T K σ2 (T ), d 2 (, s) = d 1 (, s) σ T. Suppose ha here exis processes X(, T ) for every T 0 and suppose ha Y is a process defined by Y () = T0 X(, s)ds Then we have he following version of Iô s formula T0 dy = X(, )d + dx(, s)ds. Parameerize he Vasiček model in he following way under he risk neural maringale measure Q dr() = (b ar)d + σdv. (1) Proposiion 1 (The Vasiček erm srucure) In he Vasiček model, parameerized as in (1) under Q, he zero-coupon bond prices are given by where p(, T ) = e A(,T ) B(,T )r(), (2) B(, T ) = 1 a 1 e a(t )}, (3) A(, T ) = [B(, T ) T + ](ab 1 2 σ2 ) a 2 σ2 B 2 (, T ). (4) 4a