Remark2: There are one extra problem (the last one). You don t have to make it. But if you are close to a limit I will consider it. Good Luck!!

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1 MÄLARDALENS HÖGSKOLA Insiuiionen för Maemaik och fysik, IMa Examinaion in MMA707 & MT1410 Analyical Finance I Wednesday 24 of Ocober 2007, 14:30 18:30 Examiner: Jan Röman, phone You may use: Pencil, ruler, rubber gum and calculaor. General direcion: The soluion should be well moivaed and readable. All noaions mus be explained. Remark: Wrie your naional regisraion number (personnummer) and he number of pages on he firs page. Wrie only one soluion on per shee. Use page numbers and wrie your name on all pages. Remark2: There are one exra problem (he las one). You don have o make i. Bu if you are close o a limi I will consider i. Good Luck!! 1. Prove he pu-call relaionship for American opions (curren ime is = 0 mauriy a T): rt S K CA PA S Ke... (3p) 2. Assume a one-period financial marke model wih hree securiies on he probabiliy space (, F, P) wih = { 1, 2, 3 }, F = P() and P( i ) > 0 i = 1, 2, 3. The curren prices of he securiies are S(0) = (S 0 (0), S 1 (0), S 2 (0)) = (100, 150, ). A ime = 1 he prices are given by he following marix: S1(1, 1) S1(1, 2) S1(1, 3) S(1) S (1, ) S (1, ) S (1, ) S3(1, 1 ) S3(1, 2) S3(1, 3) (a) Name an equivalen characerizaion o freedom of arbirage in single period marke models. (b) Wha are he possible values for, so ha he marke remains arbirage-free? (c) Assume ha = 160. Calculae an equivalen maringale measure EMM wih he bond as numéraire. (d) Calculae he price of he asse wih payoff-vecor C(1) = (22, 66, 0)...(4p) 1

2 3. Assume a sandard 3-period CRR binomial model. The price of he sock is currenly $100. The risk-free ineres rae wih coninuous compounding is 6% per annum. Over he nex hree 4 monh periods, he sock is expeced o go up by 8% or go down by 7% in each period. (a) Wha is he value of a one-year European call wih srike price $103? (b) Wha is he value of a one-year European pu wih srike price $103? (c) Verify he Pu-Call pariy for he European call and he European pu...(3p) 4. Consider a financial marke in which he Black-Scholes formula for a European call opion holds. The risk-free ineres rae (con. compounding) is r. The underlying sock has value S wih volailiy. For a European call wih srike K and mauriy T, show ha he following relaions hold: Show ha he call saisfies he parial differenial equaion HINT! Firs, show ha (10p) 5. The price of he sock of ABC corporaion saisfies he SDE ds S d S dw where W is a Brownian moion. The corporaion eners ino a conrac wih is CEO, worh ST Aln K 2

3 a ime T. Noe ha if he sock price S T is greaer han K, he CEO receives a paymen, bu if S T < K hen she has o pay he corporaion. In oher words, his is an incenive for her o see ha he sock price goes up. In order o neuralize he conrac, she decides o hedge. Ignoring ransacion coss, how much does i cos her a ime = 0 o implemen a hedge ha will exacly balance his conrac a ime = T? You should obain your answer by (a) Expressing he hedging cos in erms of risk neural expecaions, (b) evaluaing hese expecaions. (c) Finally, work ou an acual cos, where T corresponds o 2 years, r = 3% per year, = 6% per year, = 30% per year, K = 10, he iniial price of he sock is S 0 = 12, and A = 100, (10p) 6. Consider a sandard Black-Scholes marke, i.e. a marke consising of a risk free asse, B, wih P-dynamics given by db( ) r B( ) d B(0) 1 and a sock, S, wih P-dynamics given by ds( ) S( ) d S( ) dw ( ) S(0) s Here W denoes a P-Wiener process and r, α and σ are assumed o be consans. (a) Suppose ha you for some reason are fairly cerain ha here will be a large move in he sock price unil ime T. However you are no cerain of wheher he price will increase or decrease. One way o make use of your informaion is o buy a srangle, which is a T-conrac wih a payoff srucure illusraed in he figure below. (For he applicaion described above oday's sock price should lie beween x 1 and x 2.) Compue he price of he srangle as explicily as possible. Hin: The easies way of doing his migh be o consruc a porfolio of derivaives wih known prices, which a ime T will pay exacly he same amoun as he opion above, i.e. o use he same mehod used o derive pu-call-pariy...(4p) 3

4 (b) Deermine he arbirage price of he coningen T-claim X = Φ(S T ) wih conrac funcion Φ given by s if s K ( s) 0 oherwise. Here K denoes a sricly posiive consan..... (6p) 7. Consider a model for wo counries. We hen have a domesic marke (Sweden) and a foreign marke (Japan). The domesic and foreign ineres raes, r d and r f, are assumed o be given real numbers. Consequenly, he domesic and foreign savings accouns saisfy d rd f B e B e rf where B d and B f are denominaed in unis of domesic and foreign currency, respecively. The exchange rae process X, which is used o conver foreign payoffs ino domesic currency (he "krona/yen"-rae), is modeled by he following sochasic differenial equaion under he objecive measure P dx Xd XdW X X where µ x and σ x are assumed o be consans and W is a P-Wiener process. A domesic maringale measure, Q d, is a measure which is equivalen o he objecive Measure P and which makes all a priori given price process, expressed in unis of domesic currency and discouned using he domesic risk-free rae, maringales. We assume ha if you buy he foreign currency his is immediaely invesed in a foreign bank accoun. All markes are assumed o be fricionless. (a) Deermine he Q d -dynamics of X. (b) Now ake he viewpoin of a foreign-based invesor, ha is an invesor who consisenly denominaes her profis and losses in unis of foreign currency. A foreign maringale measure, Q f, is a measure which is equivalen o he objecive measure P and which makes all a priori given price process, expressed in unis of foreign currency and discouned using he foreign riskfree rae, maringales. Find he Girsanov ransformaion beween Q d and Q f. (c) The domesic (foreign) marke is said o be risk neural if he domesic (foreign) maringale measure is equal o he objecive measure P. Under which condiions are boh markes risk neural?..(10p) 8. Exra problem(see page 1): Derive Black-Scholes Parial Differenial Equaion and solve i for a European Pu-opion. 4

5 Formulas: Suppose ha here exis processes X(., T) for every T 0 and suppose ha Y is a process defined by: T 0 Y ( ) X (, s) ds Then we have he following version of Iô's formula dy X (, ) d dx (, s) ds T 0 The sandard Black-Scholes formula for he price () of a European call opion wih srike price K and ime of mauriy T is () = F(, S()), where r( T ) F(, s) S N d (, S) e K N d (, S) 1 2 Here N is he cumulaive disribuion funcion for he N(0, 1) disribuion and 1 S 1 2 d1 (, S) ln r ( T ), T K 2 d (, S) d (, S) T 2 1 If N denoes he cumulaive disribuion funcion for he N(0; 1) disribuion, hen N(-x) = 1 - N(x). A linear SDE of he form dx ax b d dw X0 x0 where a is a consan and b and σ are deerminisic funcions, has he soluion a as as X e x e b ds e dw 0 s s s 0 0 5

6 6

4. International Parity Conditions

4. International Parity Conditions 4. Inernaional ariy ondiions 4.1 urchasing ower ariy he urchasing ower ariy ( heory is one of he early heories of exchange rae deerminaion. his heory is based on he concep ha he demand for a counry's currency