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  • 1. Arthur CHARPENTIER - cole dt EURIA. mesures de risques et dpendance Arthur Charpentier Universit de Rennes 1 & cole Polytechnique [email protected] http://blogperso.univ-rennes1.fr/arthur.charpentier/index.php/ 1
  • 2. Arthur CHARPENTIER - cole dt EURIA. Overview of the two sessionsConsider a set of risks, denoted by a random vector X = (X1 , . . . , Xd )The interest is an agregation function of those risks g(X), where g : Rd R, andwe wish to measure the risk of this quantity R(g(X)), for some risk measure R. Sance jeudi : Mesures de risques et allocation de capital Sance vendredi : Corrlations, copules et dpendance how to model X? what about diversication eects? what is the correlation of risks in X ? can we compare R(g(X)) and R(g(X )) (i.e. under independence)? what is the contribution of Xi in the overall risk? 2
  • 3. Arthur CHARPENTIER - cole dt EURIA. Some possible motivations... in nanceConsider a set of stock prices at time T denoted X = (X1 , . . . , Xd ) andY = (Y1 , . . . , Yd ) the ratio of the price at time T divided by the price at time 0,and and let g(X) denote the payo at time T of some nancial derivative, e.g. spread derivatives, g(x1 , x2 ) = (x1 x2 K)+ based on the spread between two assets, or more generally any extreme spread options, dual spread options, correlation options or ratio spread options, e.g. buttery derivatives, g(x) = (a x K)+ , i.e. call option on a portfolio of d assets, e.g. min-max derivatives or rainbow, g(x) = (min{x} K)+ , g(x) = (max{x} K)+ , i.e. call option on the minimum or maximum of d assets, i=i+ e.g. Atlas derivatives, g(x) = ( i=i Yi K)+ , where the sum is considered skipping the i lowest and the d i+ largest returns, or Himalaya i=d derivatives, g(x) = ( i=i+ Yi K)+ , 3
  • 4. Arthur CHARPENTIER - cole dt EURIA. Some possible motivations... in environmental risks 4
  • 5. Arthur CHARPENTIER - cole dt EURIA. Some possible motivations... in credit riskApplications with a high number of risks can also be considered, in credit risk forinstance. Let X = (X1 , ..., Xd ) denote the vector of indicator variables,indicating if the i-th contract defaulted during a given period of time. If a creditderivative is based on the occurrence of k defaults among d companies, and thus,the pricing is related to the distribution of the number of defaults, N , dened asN = X1 + ... + Xd . Under the assumption of possible contagious risks, thedistribution of N should integrate dependencies.CreditMetrics in 1995 suggested a Gaussian model for credit changes, based on a probit approach, Xi = 1(Xi < ui ), where Xi N (0, 1). 5
  • 6. Arthur CHARPENTIER - cole dt EURIA. 0.6 Probit model in dimension 1 Probit model in dimension 2 4 DEFAULT (1) DEFAULTS 0.5 2 Value of company (2) 0.4 0.3 0 0.2 !2 (2) DEFAULTS 0.1 0.0 !4 !6 !4 !2 0 2 4 6 !4 !2 0 2 4 Value of the company Value of company (1) Figure 1: Modeling defaults based on a probit approach. 6
  • 7. Arthur CHARPENTIER - cole dt EURIA. Some possible motivations... in risk managementConsider a set a risks X = (X1 , ..., Xd ) (returns in a portfolio, losses per line ofbusiness, positions of nancial desks)A classical risk measure (as in Markowitz (1959)) is the standard deviation, 2(Xi ) = V ar(Xi ) = E((Xi E(X)) ). The risk of the portfolioS = X1 + . . . + Xd is (S) = (X1 )2 + . . . + (Xd )2 + 2 r(Xi , Xj )(Xi )(Xj ). i
  • 8. Arthur CHARPENTIER - cole dt EURIA. The Dow Jones index, 18961935 0.15 q q 100 150 200 250 300 350 q 0.10 q Level of the Dow Jones index q q q qq q q q q q q q q q q qq q q q q q 0.05 q q qq q q q qq q qq q q q q qq q q q q qq q qq