Lectures on financial economics, antonio mele

Lectures on Financial Economics c ° by Antonio Mele University of Lugano and June 2012 c °by A Mele Front cover explanations Top: Illustration of the increased efficiency in maritime routing allowed by the Suez Canal (right panel) opened in 1869, and the Panama Canal (left panel) opened in 1913, two amongst the most enduring technological marvels with global economic and political implications Bottom: A 75 year 3% coupon bearing bond issued by the Panama Canal Company (“Compagnie Universelle du Canal Interoc´eanique de Panama”) in October 1884 The company defaulted in 1889 under the leadership of the Count Ferdinand de Lesseps, who during 1858 had also founded the Suez Canal Company (“Compagnie Universelle du Canal Maritime de Suez”) ii Preface These Lectures on Financial Economics are based on notes I wrote in support of advanced undergraduate and graduate lectures in financial economics, macroeconomic dynamics, financial econometrics and financial engineering Part I, “Foundations,” develops the fundamentals tools of analysis used in Part II and Part III These tools span such disparate topics as classical portfolio selection, dynamic consumptionand production- based asset pricing, in both discrete and continuous-time, the intricacies underlying incomplete markets and some other market imperfections and, finally, econometric tools comprising maximum likelihood, methods of moments, and the relatively more modern simulation-based inference methods Part II, “Applied asset pricing theory,” is about identifying the main empirical facts in finance and the challenges they pose to financial economists: from excess price volatility and countercyclical stock market volatility, to cross-sectional puzzles such as the value premium This second part reviews the main models aiming to take these puzzles on board Part III, “Asset pricing and reality,” aims just to this: to use the main tools in Part I and the lessons drawn from Part II, so as to cope with the main challenges occurring in actual capital markets, arising from option pricing and trading, interest rate modeling and credit risk and their associated derivatives In a sense, Part II is about the big puzzles we face in fundamental research, while Part III is about how to live within our current and certainly unsatisfactory paradigms, so as to cope with demand for intellectual expertise These notes are still underground The economic motivation and intuition are not always developed as deeply as they deserve, some derivations are inelegant, and sometimes, the English is a bit informal Moreover, I still have to include material on asset pricing with asymmetric information, monetary models of asset prices, theories about the nominal and the real term structure of interest rates, bubbles, asset prices implications of overlapping generations models, or financial frictions and their interconnections with business cycle developments Finally, I need to include more extensive surveys for each topic I cover, especially in Part II I plan to c °by A Mele revise these notes to fill these gaps Meanwhile, any comments on this version are more than welcome Antonio Mele June 2012 iv c °by A Mele “Antonio Mele does not accept any liability for any losses related to the use of the models, data, and methods described or developed in these lectures.” v Contents I Foundations The classic capital asset pricing model 1.1 Introduction 1.2 Portfolio selection 1.2.1 The wealth constraint 1.2.2 Portfolio choice 1.2.3 Without the safe asset 1.2.4 The market portfolio 1.3 The CAPM 1.4 The APT 1.4.1 A first derivation 1.4.2 The APT with idiosyncratic risk and a large number of assets 1.4.3 Empirical evidence 1.5 Appendix 1: Analytical details relating to portfolio choice 1.5.1 The primal program 1.5.2 The dual program 1.6 Appendix 2: The market portfolio 1.6.1 The tangent portfolio is the market portfolio 1.6.2 Tangency condition 1.7 Appendix 3: An alternative derivation of the SML 1.8 Appendix 4: Liquidity traps, portfolio selection and the demand for money 1.8.1 Dichotomy choices and aggregate money demand 1.8.2 Money demand in a theory of portfolio selection References 12 13 13 13 13 14 15 17 19 22 22 23 24 25 25 26 28 28 28 30 31 31 32 34 c °by A Mele Contents The CAPM in general equilibrium 2.1 Introduction 2.2 The static general equilibrium in a nutshell 2.2.1 Walras’ Law 2.2.2 Competitive equilibrium 2.2.3 Optimality 2.3 Time and uncertainty 2.4 Financial assets 2.5 Absence of arbitrage 2.5.1 How to price a financial asset? 2.5.2 The Land of Cockaigne 2.6 Equivalent martingales and equilibrium 2.6.1 The rational expectations assumption 2.6.2 Stochastic discount factors 2.6.3 Optimality and equilibrium 2.7 Consumption-CAPM 2.7.1 The risk premium 2.7.2 The beta relation 2.7.3 CCAPM & CAPM 2.8 Infinite horizon 2.9 Further topics on incomplete markets 2.9.1 Nominal assets and real indeterminacy of 2.9.2 Nonneutrality of money 2.10 Appendix 2.11 Appendix 2: Proofs of selected results 2.12 Appendix 3: The multicommodity case References Infinite horizon economies 3.1 Introduction 3.2 Consumption-based asset evaluation 3.2.1 Recursive plans: introduction 3.2.2 The marginalist argument 3.2.3 Intertemporal elasticity of substitution 3.2.4 Lucas’ model 3.3 Production: foundational issues 3.3.1 Decentralized economy 3.3.2 Centralized economy 3.3.3 Dynamics 3.3.4 Stochastic economies 3.4 Production-based asset pricing 3.4.1 Firms 3.4.2 Consumers the equilibrium 35 35 35 36 36 37 41 42 42 42 44 48 48 49 50 54 54 55 55 55 56 56 57 58 59 62 64 65 65 65 65 66 67 68 71 72 73 74 76 80 80 84 c °by A Mele Contents 3.4.3 Equilibrium 3.5 Money, production and asset prices in overlapping generations models 3.5.1 Introduction: endowment economies 3.5.2 Diamond’s model 3.5.3 Money 3.5.4 Money in a model with real shocks 3.6 Optimality 3.6.1 Models with productive capital 3.6.2 Models with money 3.7 Appendix 1: Finite difference equations, with economic applications 3.8 Appendix 2: Neoclassic growth in continuous-time 3.8.1 Convergence from discrete-time 3.8.2 The model 3.9 Appendix 3: Notes on optimization of continuous time systems References Continuous time models 4.1 Introduction 4.2 On lambdas and betas 4.2.1 Prices 4.2.2 Expected returns 4.2.3 Risk-adjusted discount rates 4.3 An introduction to methods or, the origins: Black & Scholes 4.3.1 Time 4.3.2 Asset prices as Feynman-Kac representations 4.3.3 Girsanov theorem 4.4 An introduction to no-arbitrage and equilibrium 4.4.1 Self-financed strategies 4.4.2 No-arbitrage in Lucas tree 4.4.3 Equilibrium with CRRA 4.4.4 Bubbles 4.4.5 Reflecting barriers and absence of arbitrage 4.5 Martingales and arbitrage 4.5.1 The information framework 4.5.2 Viability 4.5.3 Market completeness 4.6 Equilibrium with a representative agent 4.6.1 Consumption and portfolio choices: martingale approaches 4.6.2 The older, Merton’s approach: dynamic programming 4.6.3 Equilibrium 4.6.4 Continuous time Consumption-CAPM 4.7 Market imperfections and portfolio choice 4.8 Jumps 85 85 85 88 88 92 93 93 94 95 99 99 100 102 104 105 105 106 106 107 108 109 109 110 113 115 115 116 117 119 120 121 121 122 124 126 126 128 129 130 131 132 c °by A Mele Contents 4.8.1 Poisson jumps 4.8.2 Interpretation 4.8.3 Properties and related distributions 4.8.4 Asset pricing implications 4.8.5 An option pricing formula 4.9 Continuous time Markov chains 4.10 Appendix 1: Self-financed strategies 4.11 Appendix 2: An introduction to stochastic calculus for finance 4.11.1 Stochastic integrals 4.11.2 Stochastic differential equations 4.12 Appendix 3: Proof of selected results 4.12.1 Proof of Theorem 4.2 4.12.2 Proof of Eq (4.53) 4.12.3 Walras’s consistency tests 4.13 Appendix 4: The Green’s function 4.13.1 Setup 4.13.2 The PDE connection 4.14 Appendix 5: Portfolio constraints 4.15 Appendix 6: Models with final consumption only 4.16 Appendix 7: Topics on jumps 4.16.1 The Radon-Nikodym derivative 4.16.2 Arbitrage restrictions 4.16.3 State price density: introduction 4.16.4 State price density: general case References Taking models to data 5.1 Introduction 5.2 Data generating processes 5.2.1 Basics 5.2.2 Restrictions on the DGP 5.2.3 Parameter estimators 5.2.4 Basic properties of density functions 5.2.5 The Cramer-Rao lower bound 5.3 Maximum likelihood estimation 5.3.1 Basics 5.3.2 Factorizations 5.3.3 Asymptotic properties 5.4 M-estimators 5.5 Pseudo, or quasi, maximum likelihood 5.6 GMM 5.7 Simulation-based estimators 5.7.1 Three simulation-based estimators 132 133 134 135 136 136 137 138 138 148 153 153 153 154 155 155 156 157 159 161 161 162 162 163 165 166 166 166 166 167 168 168 169 169 169 169 170 172 173 174 177 178 c °by A Mele Contents 5.7.2 Asymptotic normality 5.7.3 A fourth simulation-based estimator: Simulated maximum likelihood 5.7.4 Advances 5.7.5 In practice? Latent factors and identification 5.8 Asset pricing, prediction functions, and statistical inference 5.9 Appendix 1: Proof of selected results 5.10 Appendix 2: Collected notions and results 5.11 Appendix 3: Theory for maximum likelihood estimation 5.12 Appendix 4: Dependent processes 5.12.1 Weak dependence 5.12.2 The central limit theorem for martingale differences 5.12.3 Applications to maximum likelihood 5.13 Appendix 5: Proof of Theorem 5.4 References II Applied asset pricing theory 180 183 184 184 185 189 190 193 194 194 194 194 196 197 200 Neo-classical kernels and puzzles 6.1 Introduction 6.2 The equity premium puzzle 6.2.1 A single-factor model 6.2.2 Extensions 6.2.3 The puzzles 6.3 Hansen-Jagannathan cup 6.4 Multifactor extensions 6.4.1 Exponential affine pricing kernels 6.4.2 Lognormal returns 6.5 Pricing kernels and Sharpe ratios 6.5.1 Market portfolios and pricing kernels 6.5.2 Pricing kernel bounds 6.6 Conditioning bounds 6.7 The cross section of stock returns and volatilities 6.7.1 Returns 6.7.2 Volatilities 6.8 Appendix References Aggregate fluctuations in equity markets 7.1 Introduction 7.2 The empirical evidence: bird’s eye view 7.3 Volatility: a business cycle perspective 7.3.1 Volatility cycles 201 201 202 202 205 205 207 209 209 211 213 213 214 216 217 217 218 219 222 224 224 225 231 231 c °by A Mele 13.6 A few hints on the risk-management practice It is easy to see that VaRCredit Risk () increases with  Basel II sets  = 0999 and, accordingly, it imposes a capital requirement equal to, Loss-given-default ∗ [VaRCredit Risk (0999) − PD] ∗ Maturity adjustment The reason Basel II requires the term VaRCredit Risk (0999)−PD, rather than just VaRCredit Risk , is that what is really needed here is the capital in excess of the 99.9% worst case loss over the expected idiosyncratic loss, PD Well functioning capital markets should already discount the idiosyncratic losses Finally, Basel II requires banks to compute  through a formula in which  is inversely related to PD The formula is based on empirical research (see Lopez, 2004): for a firm which becomes less creditworthy, the PD increases and its probability of default becomes less affected by market conditions Basel II requires banks to compute a maturity adjustment factor that takes into account that the longer the maturity the more likely it is a given name might eventually migrate towards a more risky asset class The previous model can be further elaborated We ask: (i) What is the unconditional probability of defaults, and (ii) what is the density function of the fraction of defaulting loans? First, note that conditionally upon the realization of the macroeconomic factor  , defaults are obviosly independent, being then driven by the idiosyncratic terms  in Eq (13.72) Given  loans, and the realization of the macroeconomic factor , these defaults are binomially distributed as: ả  Pr (No of defaults = |  ) =  ( ) (1 −  ( )) −   where  ( ) is as in Eq (13.73) Therefore, the unconditional probability of  defaults is: Z ∞ Pr (No of defaults = ) = Pr (No of defaults = |  )  ( )  −∞ where  denotes the standard normal density This formula provides a valuable tool analysis in risk-management It can be shown that VaR levels increase with the correlation  Next, let  denote the fraction of defaulting loans For a large portfolio of loans,  =  ( ), such that: Z ∞ Z ∞ Pr ( ≤ ) = Pr ( ≤ |  )  ( )  = I( )≤  ( )  = Φ ( ∗ )  (13.74) −∞ −∞ where the indicator function, and  ∗ satisfies, by Eq (13.73), − ∗ :  =  (− ∗ ) = ³ −1I denotes √ ∗´ Φ (PD)+  √ Solving for  ∗ leaves: Φ 1− √ − Φ−1 () − Φ−1 (PD)   = √  ∗ It is the threshold value taken by the macroeconomic factor that guarantees a frequency of defaults  less than  Replacing  ∗ into Eq (13.74) delivers the cumulative distribution function for  The density function  () for the frequency of defaults is then: r −  12 (Φ−1 ())2 − 21 (√1−Φ−1 ()−Φ−1 (PD))2  () =    649 13.7 Appendix 1: Present values contingent on future bankruptcy c °by A Mele 13.7 Appendix 1: Present values contingent on future bankruptcy The value of debt in Leland’s (1994) model can be written as: ả àZ Ê Ô −   + E − (1 − )    () = E (13A.1) where  is the time at which the firm is liquidated Eq (13A.1) simply says that the value of debt equals the expected coupon payments plus the expected liquidation value of the bond We have: Z ∞ ¡ ¢ ¡ − ¢  −  ;    ≡  ()  (13A.2) = E  ¢ ¡ where  ;   denotes the density of the first passage time from  to  It can be shown that  () is exactly as in Eq (13.11) of the main text Similarly, ả àZ ả àZ −   =  · E   E 0 ả Z àZ Â Ă    ;    =· 0 Z ∞ ¢ − − ¡ =·  ;      = · (1 −  ())  (13A.3)  Replacing Eq (13A.2)-(13A.3) into Eq (13A.1) yields Eq (13.10) 650 c °by A Mele 13.8 Appendix 2: Proof of selected results 13.8 Appendix 2: Proof of selected results Alternative derivation of Eq (13.21) Under the risk-neutral probability, the expected change of any bond price must equal zero when the safe short-term rate is zero,  () +  (Rec −  ()) =  = 0  with  ( ) =  where the first term, ()  , reflects the change in the bond price arising from the mere passage of time, and  (Rec −  ()) is the expected change in the bond price, arising from the event of default, i.e the probability of a sudden default arrival, , times the consequent jump in the bond price, Rec −  () The solution to the previous equation is, Z  − Rec ·   +  −   (0) = | {z } =Pr{Default at } which is Eq (13.21) Proof of Eq (13.22) The spread is given by: Ã ! ¡ ¢ Rec − − +  −  ( ) = − ln    With  = 1, and Rec =  · − , we have, ³ ´ ´ ³ ´ ´ ³ ³  ( ) = − ln − − − + − =  − ln −(−) − − +    or equivalently,  ( ) = − ´ ´ ´ ´ ³ − ³ ³ ³ ln  − − + − =  − ln  − − + −(−)    Therefore, if  ≥ , then, lim →∞  ( ) = , and if  ≤ , lim →∞  ( ) =  651 c °by A Mele 13.9 Appendix 3: Transition probability matrices and pricing 13.9 Appendix 3: Transition probability matrices and pricing Consider the matrix  ( − ) for  −  ≡ ∆,  (∆), and write, ½ +  ∆ =  (∆) ≡  6=   ∆ (13A.4) We are defining the constants  as they were the counterparts of the intensity of the Poisson process in Eq (13.20) Accordingly, these constants are simply interpreted as the instantaneous probabilities of migration from rating  to rating  over the time interval ∆ Naturally, for each , we have that P  =1  (∆) = 1, and using into Eq (13A.4), we obtain,  = −  X   (13A.5) =16= The matrix Λ containing the elements  defined in Eqs (13A.4) and (13A.5) is called the generating matrix Next, let us rewrite Eq (13A.4) in matrix form,  (∆) =  + Λ∆ Suppose we have a time interval [0  ], which we chop into  pieces, so to have = ả ( ) =  (∆) =  + Λ     We have, For large ,  ( ) = exp (Λ )  (13A.6) P∞ ( Λ) the matrix exponential, defined as, exp (Λ ) ≡ =0 ! To evaluate derivatives “written on states,” we proceed as follows Suppose  is the price of derivative in state  ∈ {1 · · ·  } Suppose the Markov chain is the only source of uncertainty relevant for the evaluation of this derivative Then,  =   + [˜ −  ]  ˜ ∈ {1 · · ·  }, with the usual conditional probabilities In words, the instantaneous change where   in the derivative value,  , is the sum of two components: one,   , related to the mere passage of time, and the other, [˜ −  ], related to the discrete change arising from a change in the rating Suppose that  = Then,   X  ( )  X  = = = + +  [ −  ] =  [ −  ]     =1 6= with the appropriate boundary conditions As an example, consider defaultable bonds In this case, we may be looking for pricing functions having the following form,  ( − ) =  ( − ) + −  ( − )  and then solve for  ( − ), for all  ∈ {1 · · ·  } Naturally, we have X  [ ( −  ) − ( −  )] = 0 − 0 + h X =  0 + 6= 6= i h X  ( −  ) − 0 + 6= 652 i  ( −  )  c °by A Mele 13.9 Appendix 3: Transition probability matrices and pricing which holds if and only if, hX X X X  ( −  ) = −   +   = − 0 = − 6= 6= 6= 6= i   +    That is, 0 = −Λ, which solved through the appropriate boundary conditions, yields precisely Eq (13A.6) 653 13.10 Appendix 4: Bond spreads in markets with stochastic default intensity c °by A Mele 13.10 Appendix 4: Bond spreads in markets with stochastic default intensity We derive Eq (13.34), by relying on the pricing formulae of Chapter 12 If the short-term is constant, the price of a defaultable bond derived in Section 13.4.7 of Chapter 12 can easily be extended to, with the notation of the present chapter, ∙ R ∙ ¸ Z  ¸ R − () − () − − E   E  ()  Rec ()  (13A.7)  (  ) =  + | {z } =Pr{Default∈(+)} The term indicated inside the integral of the second term, is indeed the density of default time at , because, ¸ ∙ R default by time  () = − E − ()  such that by differentiating with respect to , yields, under the appropriate regularity conditions, that Pr{Default∈ (  + )} is just the term indicated in Eq (13A.7) So Eq (13.34) follows Naturally, Pr{Default ∈ (  + )} = − Replacing this into Eq (13A.7), ¸ ∙ R Z  (  ) = − E − () + Rec  surv ( )   ∙ ¸  − − surv ( )   Z  ¢ ¡ − surv ( )  = − LGD − − surv (  ) − (1 − LGD)  where the second equality follows by integration by parts and the assumption of constant recovery rates Setting  = 0, produces Eq (13.35) 654 13.11 Appendix 5: Conditional probabilities of survival c °by A Mele 13.11 Appendix 5: Conditional probabilities of survival We prove Eqs (13.37)-(13.39) First, for (−1   ) small, the numerator in Eq (13.36) can be replaced by ¸ ∙ R  − ()  − surv ( ) ≡ E  ()   and rescaled by  Regularity conditions under which we can perform this differentiation can be found in a related context developed in Mele (2003) Eqs (13.37)-(13.38) follow As for Eq (13.39), the proof follows the same lines of reasoning as that in Appendix of Chapter 12 That is, we can define a density process, R ¯ ¸ ∙ R ¯ − () surv ( ( )     ) −  () ¯ ¸ ∙ R  surv ( ( )     ) ≡ E    ( ) = ¯ F  − () E  It is easy to show that the drift of surv is  ( )  , such that by Itˆo’s lemma,   ( ) = − [−Vol (surv ( ( )     ))]  ( )    ( ) where,  p surv ( ( )     ) p   ( ) =  ( −  )   ( ) −Vol (surv ( ( )     )) ≡ −  surv ( ( )     ) where the second line follows by the closed-form expression of surv in Eq (13.31) Therefore,  ( ) is a Brownian motion under  , where p  ( ) =  ( ) +  ( −  )   ( )  and Eq (13.39) follows 655 13.12 Appendix 6: Details regarding CDS index swaps and swaptions c °by A Mele 13.12 Appendix 6: Details regarding CDS index swaps and swaptions We prove Eq (13.51) Note that the expectation of the first term in Eq (13.48), conditional on the information set  , for  ≤ 0 is, now, for  = 1 · · ·  4 , ∙ R ¸   1X −  ( ) E  I{Surv at  } I{Def  ∈(−1  )} LGD  =1 ∙ R ¸ ∙ R ¸    1X −  ( ) −  ( ) I{Surv at  } E  I{Def  ∈(−1  )} = LGD ·  ( ) E  I{Def  ∈(−1  )}  = LGD  =1 (13A.8) where the last equality follows by the definition of the outstanding notional value in Eq (13.52), and the fact that the expectation in the first equality is the same for each name , due to the assumption that all names in the index have the same credit quality Summing over the reset dates,  = 1 · · ·  4 , delivers the first term in Eq (13.51) The second term in Eq (13.51) follows by elaborating the time  expectation of the second term in Eq (13.48), ∙ R ¸  −  ( ) · I{Surv at  } E  ∙ R ¸ ∙ R ¸   −  ( ) −  ( ) = E  · I{Surv at  } I{ Surv at  |Surv at  } = I{Surv at  } E  I{ Surv at  |Surv at  }  and summing over the reset dates and all names , using the definition of 1 in Eq (13.50), and noting, again, that the expectation in the second equality is the same for all the assets We now derive the value of the front-end protection in Eq (13.53) Note that the derivation of Eq (13A.8) relies on default events occurring after the swap origination, i.e over the reset dates, after  = 0 In evaluating the front-end protection, we we need to price securities that pay off over defaults possibly occurring over the life of the swaption, i.e before time  = 0 We have, ∙ R ¸ −  () F  = E   ⎤ ⎡ R  ´ ⎣ −  () X ³ I{Def  ∈( )} + I{Surv at  } I{Def  ∈(  )} ⎦ = LGD E    =1 ∙ R  ³ ´¸ 1X −  () = LGD D (  )  (   ) + LGD I{Surv at  } E  − I{ Surv at  |Surv at  }   =1 ả D ( ) (   ) +  ( ) ( (   ) − def (   ))  = LGD  where the third equality holds by the assumption that the names have the same credit quality, and   −  () −  (()+()) ] and def (   ) = E [ ] Note that the first term in the  (   ) = E [ brackets of the second equality is, obviously, always zero, when the timing of possible defaults does not overlap with the evaluation horizon, as for Eq (13A.8) ˆ sc defined in Eq (13.55) does integrate to Next, we show that the survival contingent measure  F 0 ˆ sc We shall need one, and that CDX ( ) = LGD 1 + ( )1 in Eq (13.54), is a martingale under  the equality summarized by the following lemma: R Lemma 13A.1 The following equality holds true: E [  ( )| F ( )] =  ( ) −  () 656 13.12 Appendix 6: Details regarding CDS index swaps and swaptions c °by A Mele Proof We have: ⎡  1X  ⎣ I{Surv E [  ( )| F ( )] = E  =  =  X =1 =1 I{Surv ¯ ⎤ ¯ ¯  ¯ ⎦ at  } ¯ F ( ) ¯ at  } E R −  ()  ( )  h k I{ Surv at  |Surv ¯ i ¯  at  } ¯ F ( ) ˆ sc , we have that under regularity conditions, As for the survival contingent measure R R ảá  −  () −  () ¯ E   ( ) 1 = E E   ( ) 1 ¯ F ( ) ∙ R ¸ −  (()+()) 1 =  ( ) E  =  ( ) 1  R ˆ sc = where the last equality follow by the definition of sc in Eq (13.46) Therefore,  ˆ sc [·] the time  conditional expectation ˆ sc , let E As for the martingale property of CDX ( ) under   ˆ sc in Eq ˆ operator under the the survival contingent measure sc We have, using the definition of  (13.55), ∙ ¸ ∙ ¸ F   0 sc sc sc  ˆ ˆ ˆ [CDX ( )] = LGD · E +E E    1  ( ) 1 ∙ R ¸ ∙ R ¸ F −  ()  ( ) 1 0 −  ()  ( ) 1 sc + E  = LGD · E   ( ) 1 1  ( ) 1  ( ) 1 ∙ R ¸ ∙ R ¸ 1 −  ()  ( ) −  () F 0 + · E  E   = LGD 1  ( )  ( ) 1 F 0 +  = LGD 1  ( ) 1 where the third equality follows by the Law of Iterated Expectations, Lemma 13A.1, and Eq (13.47), ảá R R  −  ()  ( ) −  ()  ( ) ¯ 0 = E E  0 ¯ F ( ) E   ( )  ( ) ∙ R ¸ = E −  (()+()) 0 = 0  657 13.13 Appendix 7: Modeling correlation with copulae functions c °by A Mele 13.13 Appendix 7: Modeling correlation with copulae functions A Statistical independence and correlation Two random variables are always uncorrelated, provided they are independently distributed Yet there might be situations where two random variables are not correlated and still exhibit statistical dependence As an example, suppose a random variable  relates to another, , through  = 3 , for  −1  0 1  · · ·  −1   }, some constant , and  can take on 2 + values,  ∈P{−  −1  · · ·P  3+ Then, we have that Cov ( ) ∝  (− )  and Pr { } = 2+1   =1 =1 ( )  = and yet,  and  are obviously dependent This example might be interpreted, economically, as one where  and  are two returns on two asset classes These two returns are not correlated, overall Yet the comove in the same direction in both very bad and in very good times This appendix is a succinct introduction to copulae, which are an important tool to cope with these issues Consider two random variables 1 and 2 We may relate 1 to another random variable 1 and we may relate 2 to a second random variable 2 , on a percentile-to-percentile basis, viz  ( ) =  ( )   = 1 2 (13A.8) where  are the cumulative marginal distributions of  , and  are the cumulative marginal distributions of  That is, for each  , we look for the value of  such that the percentiles arising through the mapping in Eq (13A.8) are the same Then, we may assume that 1 and 2 have a joint distribution and model the correlation between 1 and 2 through the correlation between 1 and 2 This indirect way to model the correlation between 1 and 2 is particularly helpful It might be used to model the correlation of default times, as in the main text of this chapter We now explain B Copulae functions We begin with the simple case of two random variables, This simple case shall be generalized to the multivariate one with a mere change in notation Given two uniform random variables 1 and 2 , consider the function  (1  2 ) = Pr (1 ≤ 1  2 ≤ 2 ), which is the joint cumulative distribution of the two uniforms A copula function is any such function , with the property of being capable to aggregate the marginals  into a summary of them, in the following natural way:  (1 (1 )  2 (2 )) =  (1  2 )  (13A.9) where  (1  2 ) is the joint distribution of (1  2 ) Thus, a copula function is simply a cumulative bivariate distribution function, as  (1 ) and  (2 ) are obviously uniformly distributed To prove Eq (13A.9), note that  (1 (1 )  2 (2 )) = Pr (1 ≤ 1 (1 )  2 ≤ 2 (2 )) ¡ ¢ = Pr 1−1 (1 ) ≤ 1  2−1 (2 ) ≤ 2 = Pr (1 ≤ 1  2 ≤ 2 ) =  (1  2 )  (13A.10) That is, a copula function evaluated at the marginals 1 (1 ) and 2 (2 ) returns the joint density  (1  2 ) In fact, Sklar (1959) proves that, conversely, any multivariate distribution function  can be represented through some copula function The most known copula function is the Gaussian copula, which has the following form: ¡ ¢ −1  (1  2 ) = Φ Φ−1 (13A.11) (1 )  Φ2 (2 )  where Φ denotes the joint cumulative Normal distribution, and Φ denotes marginal cumulative Normal distributions So we have, ¡ ¢ −1 (13A.12)  (1  2 ) =  (1 (1 )  2 (2 )) = Φ Φ−1 (2 (2 ))  Φ2 (2 (2 ))  658 13.13 Appendix 7: Modeling correlation with copulae functions c °by A Mele where the first equality follows by Eq (13A.10) and the second equality follows by Eq (13A.11) As an example, we may interpret 1 and 2 as the times by which two names default A simple assumption is to set: (13A.13)  ( ) = Φ ( )   = 1 2 for two random variables  that are “stretched” as explained in Part A of this appendix By replacing Eq (13A.13) into Eq (13A.12),  (1  2 ) = Φ (1  2 )  This reasoning can be easily generalized to the  -dimensional case, where:  (1  · · ·   ) =  (1 (1 )  · · ·   ( )) = Φ (1  · · ·   )  where  :  ( ) = Φ ( )  We use this approach to model default correlation among names, as explained in the main text, and in the next appendix 659 13.14 Appendix 8: Details on CDO pricing with imperfect correlation c °by A Mele 13.14 Appendix 8: Details on CDO pricing with imperfect correlation We follow the copula approach to price the stylized CDOs in the main text of this chapter For each name, create the following random variable, p √  =  + −    = 1 2 3 (13A.14) where  is a common factor among the three names,  is an idiosynchratic term, and  ∼  (0 1),  ∼  (0 1) Finally,  ≥ is meant to capture the default correlation among the names, as follows Assume that the risk-neutral probability each firm defaults, by  , is given by, Q ( ) = Φ ( 010 ) ≡ 10% where Φ is the cumulative distribution of a standard normal variable That is, by time  , each firm defaults any time that,    010 ≡ Φ−1 (10%)  Therefore,  is the default correlation among the assets in the CDO We can now simulate Eq (13A.14), build up payoffs for each simulation, and price the tranches by just averaging over the simulations, as explained below Naturally, the same simulation technique can be used to price tranches on CDOs with an arbitrary number of assets Precisely, simulate Eq (13A.14), and obtain values ˜ ,  = 1 · · ·  , where  is the number of simulations and  = 1 2 At simulation no , we have ˜1  ˜2  ˜3   ∈ {1 · · ·  }  We use the previously simulated values as follows: • For each simulation , count the number of defaults across the three names, defined as the number of times that ˜   010 , for  = 1 2 Denote the number of defaults as of simulation  with Def  • For each simulation , compute the total realized payoff of the asset pool, defined as,  ˜  = Def  · 40 + (3 − Def  ) · 100 • For each simulation , compute recursively the payoffs to each tranche,   , ắ ẵ ẵ ắ P        = max  =1 where  is the nominal value of each tranche (1 = 140, 2 = 90, 3 = 70) • Estimate the price of each tranche by averaging across the simulations, − Price Senior =     X X 1X − −  1  Price Mezzanine =   2  Price Junior =   3   =1  =1  =1 Note, the previous computations have to be performed under the risk-neutral probability  Using the probability  in the previous algorithm can only be lead to something useful for risk-management and VaR calculations at best Note, this model, can be generalized to a multifactor model where, p √ √  = 1 1 + · · · +   + − 1 − · · · −    with obvious notation 660 13.14 Appendix 8: Details on CDO pricing with imperfect correlation c °by A Mele References Amato, J D (2005): “Risk Aversion and Risk Premia in the CDS 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... These Lectures on Financial Economics are based on notes I wrote in support of advanced undergraduate and graduate lectures in financial economics, macroeconomic dynamics, financial econometrics... II I plan to c °by A Mele revise these notes to fill these gaps Meanwhile, any comments on this version are more than welcome Antonio Mele June 2012 iv c °by A Mele Antonio Mele does not accept... A Mele Contents 8.11 Appendix 1: Non-expected utility 8.11.1 Detailed derivation of optimality conditions and selected relations 8.11.2 Details concerning models of long-run