Advanced Tools for Risk Management and Asset Pricing

Transcription

1 MSc. Finance/CLEFIN 214/215 Ediion Advanced Tools for Risk Managemen and Asse Pricing May 215 Exam for Non-Aending Sudens Soluions Time Allowed: 13 minues Family Name (Surname) Firs Name Suden Number (Mar.) Please answer all quesions by choosing he mos appropriae alernaive and/or by wriing your answers in he spaces provided. You need o carefully jusify and show your work in he case of open quesions. There is only one correc answer(s) for each of he muliple choice quesions: each seleced alernaive ha is correc will be awarded one poin. Only answers explicily repored in he appropriae box will be considered. No oher answers or indicaions poining o poenial answers will be aken ino consideraion. In he case of open quesions, he maximum number of poins is indicaed. Quesion 1. Which of he following saemens abou Correlaion is FALSE? (A) Correlaion is well defined for heavy-ailed disribuions (B) Correlaion measures he linear dependence and akes values in he inerval [-1,1] (C) Correlaion is invarian under sricly increasing linear ransformaions (D) Correlaion is linked boh o he copula funcion and o he marginal disribuions Quesion 2. Which of he following funcions can NOT be a copula funcion? (A) C(u₁, u₂) = u₁u₂ (B) C(u₁, u₂) = min(u₁, u₂) (C) C(u₁, u₂) = max(u₁ + u₂, ) (D) None of he above 1

2 Quesion 3. Which of he following saemens abou Rank Correlaions is FALSE? (A) To compue rank correlaions one needs o know boh he numerical values of he variables and he ordering of he sample for each variable (B) Rank correlaions are invarian under sricly increasing ransformaions (C) Rank correlaions ake a value of -1 when he variables are counermonoonic and he value 1 when he variables are comonoonic (D) The populaion version of Kendall's au can be simply expressed as: ρ τ (X₁, X₂) = E(sign((X₁ X ₁)(X₂ X ₂))) Quesion 4. Which of he following saemens abou Copula Esimaion and Calibraion is FALSE? (A) A Gaussian copula is asympoically independen in boh ails (B) A Mehod-of-Momens approach involves calibraion of a copula using empirical esimaes for some rank correlaion measures (C) Boh Gumbel and Clayon copulas have upper ail dependence (D) Measures of exremal dependence beween a pair of random variables ₁ and X X₂ depend only on he copula of X₁ and X₂ Quesion 5. Which of he following saemens abou Sochasic Processes is TRUE? (A) Io's Inegral is a maringale (B) A Brownian moion is a saionary process wih independen Gaussian incremens. (C) The sandard chain rule does NOT work for Io's Inegral. (D) All he above saemens are TRUE Quesion 6. In Vasicek's ineres rae model, he dynamics of he shor rae process (r ) are given by he SDE dr = k(θ r )d + σdw, where k, θ and σ are sricly posiive consans. Show ha he soluion of SDE above is given by r = θ + (r θ)e k + σe k e ks dw s Hin: Consider he Io processes (X ) and (Y ) defined by X = e k and Y = r and use inegraion by pars. Answer. a) We consider he Io process (X ) and (Y ) defined by X = e k and Y = r and we noe ha 2

3 dx = ke k d, X₀ = 1, dy = k(θ r )d + σdw, Y₀ = r₀ Using he inegraion by pars formula, we calculae This implies ha e k r X Y = X₀Y₀ + Y s dx s + X s dy s = X₀Y₀ + [ke ks Y s + k(θ r )X s ]ds + (dy s )(dx s ) + σx s dw s = r₀ + [ke ks r s + k(θ r s )e ks ]ds + σe ks dw s = r₀ + θe k θ + σ e ks dw s. r = θ + (r θ)e k + σe k σe ks dw s. (b) Noing ha he sochasic inegral wih respec o he Brownian moion has expecaion, we can see ha he expecaion of he random variable r_{} is given by Using Io's isomery, we calculae E[r ] = θ + (r θ)e k var(r ) = E[(r E[r ])²] = σ 2 e 2k E e ks dw s = ((σ²)/(2k))(1 e 2k ) 2 Quesion 7. Assume a CDS quoed spread is 3 basis poins and he recovery rae is esimaed o be 4%. Under he assumpions ha i) he premium leg of he CDS pays coninuously and ii) he hazard rae is consan, wha is he value of he hazard rae? (A) 5% (B) 7.5% (C) 5 (D) 75 3

4 Quesion 8. The Gaussian Copula Approach allows o compue he join probabiliy of defaul of n names. Such probabiliy enails i) he calculaion of a muli-dimensional inegral and ii) he esimaion of he correlaion marix among names. Given ha dim I = dimension of he inegral and dim ρ = number of free correlaion parameers (enries of he correlaion marix), wha are he values of hese wo parameers under he Single-Facor Gaussian Copula Approach? (A) dim I = n, dim ρ = n(n 1) 2 (B) dim I = 1, dim ρ = n(n 1) 2 (C) dim I = n, dim ρ = n (D) dim I =1, dim ρ = n Quesion 9. Which of he following saemens abou implied correlaions is rue? (A) The compound correlaion is inconsisen a he level of a single ranche (B) The compound correlaion for a given ranche is always unique (C) Base correlaion can yield negaive expeced ranche losses (D)Base correlaion depends on pairs of aachmen poins Quesion 1. Which of he following saemens abou credi modeling in a muli-facor se up is false? (A) Secor concenraion risk is due o he correlaion srucure of obligors inside he porfolio (B) Secor concenraion risk affecs only he condiional variance erm (C) Name concenraion risk is a second order effec (D) Name concenraion risk can be diversified away Quesion 11. Which of he following saemens abou he Limiing Porfolio Loss Disribuion, inroduced by Vasicek, is false? (A) By he law of large numbers, for an infiniely granular porfolio, he porfolio loss disribuion converges o is expecaion (B) The limiing loss disribuion provides a good approximaion o he porfolio loss, also for a porfolio consising of uneven (bu no dominaing) weighs (C) The probabiliy densiy funcion of he porfolio loss can be U-shaped (D) The probabiliy densiy funcion of he porfolio loss can be unimodal 4

5 Quesion 12. Which of he following saemens abou differen approaches o price CDOs is rue? (A) The Implied Copula Approach by Hull and Whie solves he problems of inconsisencies across he capial srucure and mauriies (B) The Gaussian Copula Approach is able o capure he phenomenon of clusered (secor) defauls associaed o masses in he far righ ail of he loss disribuion (C) Dynamic(al) Loss Approaches model direcly aggregaed objecs (D) Dynamic(al) Loss Approaches allow o calculae single name sensiiviies Quesion 13. Which of he following saemens abou Analyically Tracable Firs Passage (AT1P) models is false? (A) AT1P models assume he exisence of a ime-dependen deerminisic barrier (B) AT1P models poses he problem of he correc deerminaion of shor erm credi spreads (C) AT1P models implicily assume ha i is possible o reach a considerable probabiliy of defaul in a very shor ime horizon smoohly (D)AT1P models implicily assume ha accouning daa are fully reliable and ransparen Quesion 14. Which of he following saemens abou Reduced Form (Inensiy) models is rue? (A) Sochasic inensiy models always allow o obain large levels of opion implied volailiies for CDS raes (B) In a muli-name seing, inroducing dependence in sochasic inensiies of differen names can lead o unrealisically low levels of dependence across defauls (C) Negaive values of he inensiy can never be aained in CDS calibraion (D) Calibraion o CDS spreads is more sable wih piece-wise linear inensiies compared o piece-wise consan ones Quesion 15. Which of he following saemens abou Reduced Form (Inensiy) models is false? (A) Cox processes allow o ake ino consideraion spread volailiy (B) Defaul is described by an endogenous jump process (C) In deerminisic inensiy models, survival probabiliies have he same srucure as discoun facors in shor rae models (D) Survival probabiliies obained hrough calibraion o CDS quoes can increase wih ime 5

6 Quesion 16. Consider a bespoke CDO ha conains he same credis as he index porfolio. The recovery rae for he index is 6% and for he bespoke i is. Which of he following saemens is false? (A) A 1% ranche on he bespoke will experience he same relaive losses as a 6% ranche on he index (B) A 1% ranche on he bespoke will experience he same relaive losses as a 4% ranche on he index (C) Losses on he bespoke are 2.5 imes he losses on he index (D) The 1% srike on he bespoke should be priced wih he same correlaion as he 4% srike on he index Quesion 17. Wihin he framework of srucural models of defaul, he firs model had been inroduced by Meron in Briefly describe he assumpions underlying he model, is limis and exensions (wrie a mos 2 lines of ex). Answer. See Lecure 9 Srucural Models : Assumpions: slide 8/65 Limis: slide 21/65 Exensions: slides 22-23/65 Quesion 18. In he framework of he Gaussian Copula approach, wo ypes of implied correlaion, i.e., compound correlaion and base correlaion have been inroduced. Briefly describe hem and heir limiaions (wrie a mos 2 lines of ex). Answer. See Lecure 1 CDO Basics Compound correlaion I is based on he assumpion ha each ranche [A,B] is characerized by a unique value of correlaion ρ AB. Typically, he compound correlaion srucure presens a smile. Limiaions: 1. I canno be easily inerpolaed/exrapolaed 2. I is unable o price bespoke ranches 3. I may no exis Base correlaion I is based on he assumpion ha each equiy ranche [,X] is characerized by a unique value of correlaion ρ X. I follows ha a ranche [A,B] depends on wo values of base correlaion. Typically, he compound correlaion srucure presens a skew. Limiaions: 1. I is inconsisen a he level of single ranche 2. I may yield negaive expeced ranche losses 6

7 Quesion 19. Considering he wo ermshees below, which of he following saemens is plausible: Bonus Cap A Bonus Cap B Underlying Fia Underlying Fia Mauriy 3 Years Mauriy 3 Years Barrier 7% Barrier 7% Barrier ype American Barrier ype European (A) Bonus Cap A has a Bonus equal o 115% and Bonus Cap B has a Bonus equal o 12%. (B) Bonus Cap A has a Bonus equal o 11% and Bonus Cap B has a Bonus equal o 18%. (C) Bonus Cap A has a Bonus equal o 18% and Bonus Cap B has a Bonus equal o 18%. (D) Bonus Cap A has a Bonus equal o 98% and Bonus Cap B has a Bonus equal o 95%. Quesion 2. A Shor Equiy Proecion which pays Eur 1 if a mauriy he underlying (wih price S()) is above he Srike and Eur 1 [1 + (1 S(T)/S() ) ] if he underlying is below he Srike is replicaed by: (A) Buying a Zero Coupon Bond and selling an ATM pu opion (B) Buying a Zero Coupon Bond and buying an ATM call opion (C) Buying a Zero Coupon Bond and selling an ATM call opion (D) Buying a Zero Coupon Bond and buying an ATM pu opion Quesion 21. You are srucuring an Equiy Proecion cerificae wih 1% capial proecion; he Zero Coupon Bond coss Eur 95. An ATM call opion on he FTSE MIB index coss 7 Euro. Because you would like your produc o offer 1% paricipaion o any poenial appreciaion of he underlying, which sraegy of selecion of an underlying differen from he FTSE MIB index would you consider? (A) I shall no need any alernaive selecion of he underlying asse because I can already offer 1% paricipaion o any poenial appreciaion of he underlying (B) I will be looking for an underlying asse wih lower volailiy and a higher dividend yield han he FTSE MIB so ha he opion will be cheaper o ry and aim a an opion cos of Eur 5 o make a 1% proecion possible (C) I will be looking for an underlying asse wih lower volailiy and dividend yield han he FTSE MIB so ha he opion will be cheaper o ry and aim a an opion cos of Eur 5 o make a 1% proecion possible (D) I will be looking for an underlying asse wih higher volailiy and dividend yield han he FTSE MIB so ha he opion will be cheaper o ry and aim a an opion cos of Eur 5 o make a 1% proecion possible 7

8 Quesion 22. An ouperformance cerificae is: (A) A cerificae wih capial proecion and one-o-one paricipaion o poenial appreciaion of he underlying (B) A cerificae wih capial proecion and magnified paricipaion o poenial appreciaion of he underlying (C) A cerificae wih condiional capial proecion and one-o-one paricipaion o poenial appreciaion of he underlying (D) None of he above Quesion 23. Which of he opions lised below will be he cheapes: (A) An ATM Asian opion on he FTSE MIB wih 3-year mauriy (B) An ATM European opion on he FTSE MIB wih 3-year mauriy (C) An ITM European opion on he FTSE MIB wih 3-year mauriy (D) I canno say Quesion 24. Please read he descripion of he srucured payoff below, draw he payoff graph (assume ha srike is 1% of he value of he underlying a he issuance dae) and explain which opion sraegy replicaes such a payoff. The cerificae guaranees ha a mauriy 1% of he capial invesed is paid back; in addiion, if he value of he underlying a mauriy is lower han 95% of he Srike, hen he cerificaes pays 1% (1 S() Srike ) and if he value of he underlying a mauriy is above 15% of he Srike hen i pays 1% ( S() Srike 1) Answer. This is special case of a double win cerificae similar o he ones analyzed in our lecures in which he payoff funcion is as follows. 2 Payoff

9 This srucure may be replicaed by buying a zero coupon bond wih a noional amoun of Eur 1, buying a European pu wih srike equal o 95% of he srike of he srucure, and buying a European call wih srike equal o 15% of he srike of he srucure. The idea is ha he capial saved wih respec o he 1 invesmen in he ZCB should allow o purchase boh he ou-of-he money European pus and calls ha he srucure requires. 9