The problem and its history Solution of the Problem Localisation and neighbourhood of worst case distributions Information Geometry in Mathematical Finance: Model Risk, Worst and Almost Worst Scenarios Imre Csisz´ ar Thomas Breuer PPE Research Centre, FH Vorarlberg, Austria MTA R´ enyi Institute of Mathematics, Hungary 12 June 2015, FEBS, Nantes
The problem and its history Solution of the Problem Localisation and neighbourhood of worst case distributions 1 The problem and its history 2 Solution of the Problem 3 Localisation and neighbourhood of worst case distributions
The problem and its history Solution of the Problem Localisation and neighbourhood of worst case distributions Outline 1 The problem and its history 2 Solution of the Problem 3 Localisation and neighbourhood of worst case distributions
The problem and its history Solution of the Problem Localisation and neighbourhood of worst case distributions The problem P ∈ Γ E P ( X ) inf (1) � V ( k ) � inf Xpd µ. (2) p : � pd µ =1 , H ( p ) ≤ k where • the real-valued function X depending on the risk factors r ∈ Ω describes the utility of a portfolio with a random payoff, • the distribution P of the risk factors r is uncertain, but known to be in Γ � { P : d P = pd µ, H ( p ) ≤ k } , a set of ‘plausible’ distributions defined in terms of the convex integral functional � H ( p ) = H β ( p ) � β ( r , p ( r )) µ ( dr ) Ω • determined by a function β ( r , s ) of r ∈ Ω , s ∈ R , measurable in r for each s ∈ R , strictly convex and differentiable in s .
The problem and its history Solution of the Problem Localisation and neighbourhood of worst case distributions Where does this problem arise? • Ambiguity averse decision makers rank alternatives X by (1): evaluation of multiple priors preferences (Gilboa and Schmeidler) • Every coherent risk measure can be represented by (1) for some Γ (Artzner et al. ) • Search for worst generalised scenarios (distributions) in the set Γ of plausible scenarios: systematic stress testing
The problem and its history Solution of the Problem Localisation and neighbourhood of worst case distributions Purpose of Stress Testing: Complement statistical risk measurement • Stress Tests: Which scenarios lead to big losses? Derive risk reducing action. (Statistical risk measurements: What are prob’s of big losses?) • Stress Tests: Address model risk. Consider alternative risk factor distribution. (Statistical risk measurement: Assume fixed model.)
The problem and its history Solution of the Problem Localisation and neighbourhood of worst case distributions Criticism of First Generation Stress Tests Are there any real stress tests whose results forced a bank to change strategy? Accidental or deliberate misrepresentation of risks: 1 Neglecting severe but plausible scenarios 2 Considering too implausible scenarios
The problem and its history Solution of the Problem Localisation and neighbourhood of worst case distributions Second Generation Stress Tests: Systematic Point Scenario Analysis • Set of plausible scenarios Ell h = { r : Maha ( r ) ≤ h } , where h is the plausibility threshold. • Systematic search of worst case scenario: min X ( r ) r ∈ Ell h
The problem and its history Solution of the Problem Localisation and neighbourhood of worst case distributions Advantages and Problems of Systematic Stress Testing with Point Scenarios All three requirements on stress testing are met: • Do not miss plausible but severe scenarios. • Do not consider scenarios which are too implausible. • Worst case scenario over Ell h gives information about portfolio structure and suggests risk reducing action. Disadvantages: • What if risk factor distributions ν is non-elliptical? • What if risk true factor distribution is not ν ? Model risk is not addressed. • Maha does not take into account fatness of tails. • min r ∈ Ell h X ( r ) depends on choice of coordinates.
The problem and its history Solution of the Problem Localisation and neighbourhood of worst case distributions Mixed Scenarios Mixed scenario: Probability distribution of point scenarios. • Interpretation 1: Risk factor distributions alternative to the prior ν . Model risk. • Interpretation 2: Generalisation of point scenarios, but support not concentrated on one point. Measure of plausibility: H ( p ) � � Ω β ( r , p ( r )) µ ( dr ) � Resulting problem: inf p : Xpd µ � pd µ =1 , H ( p ) ≤ k
The problem and its history Solution of the Problem Localisation and neighbourhood of worst case distributions Special cases • Take µ = P 0 , thus p 0 = 1, and let β ( r , s ) = f ( s ) be an autonomous convex integrand, with f ( s ) ≥ f (1) = 0. Then H ( p ) is the f -divergence D f ( P || P 0 ). • Let f be a strictly convex differentiable function on (0 , + ∞ ), and for s ≥ 0 let β ( r , s ) = ∆ f ( s , p 0 ( r )) where ∆ f ( s , t ) � f ( s ) − f ( t ) − f ′ ( t )( s − t ); (3) in case f ′ (0) = −∞ assume that p 0 > 0 µ -a.e. Then H ( p ) equals the Bregman distance. • In the special case f ( s ) = s log s − s + 1, both examples above give the I -divergence ball Γ = { P : D ( P || P 0 ) ≤ k } .
The problem and its history Solution of the Problem Localisation and neighbourhood of worst case distributions Outline 1 The problem and its history 2 Solution of the Problem 3 Localisation and neighbourhood of worst case distributions
The problem and its history Solution of the Problem Localisation and neighbourhood of worst case distributions Standing Assumptions • −∞ ≤ m < b 0 < M ≤ + ∞ where m � µ -ess inf X, b 0 � E P 0 ( X ) = � Ω X ( r ) p 0 ( r ) d µ ( r ), M � µ -ess sup X, • H ( p ) ≥ H ( p 0 ) = 0 � whenever pd µ = 1 . • 0 < k < k max � lim b ↓ m F ( b ).
The problem and its history Solution of the Problem Localisation and neighbourhood of worst case distributions Relation to the moment problem F ( b ) � inf Xpd µ = b H ( p ) . (4) p : � pd µ =1 , � io P µ, H ( p ) ≤ k } )) d P 0 ( || E P ( X ) = b In regular situations, the worst case distribution P over Γ = { p : H ( p ) ≤ k } equals the distribution with the smallest divergence H ( p ) among those satisfying the constraint E P ( X ) = b .
The problem and its history Solution of the Problem Localisation and neighbourhood of worst case distributions Relation to the moment problem Lemma If 0 < k < k max � lim b ↓ m F ( b ) , there exists a unique b satisfying F ( b ) = k , m < b < b 0 (5) and then V ( k ) = b. The minimum in (2) is attained if and only if that in (4) is attained (for the b in (5) ), and then the same p attains both minima.
The problem and its history Solution of the Problem Localisation and neighbourhood of worst case distributions Generalised exponential family • J ( a , b ) � inf p : Xpd µ = b H ( p ) , � � pd µ = a , is the value function for (4), F ( b ) = J (1 , b ). β ∗ ( r , θ 1 + θ 2 X ( r )) µ ( dr ) • K ( θ 1 , θ 2 ) � � = J ∗ ( θ 1 , θ 2 ) = sup a , b [ θ 1 a + θ 2 b − J ( a , b )] • p θ 1 ,θ 2 ( r ) � ( β ∗ ) ′ ( r , θ 1 + θ 2 X ( r )) , for ( θ 1 , θ 2 ) ∈ Θ � { ( θ 1 , θ 2 ) ∈ dom K : θ 1 + θ 2 X ( r ) < β ′ ( r , + ∞ ) µ -a.e. }
The problem and its history Solution of the Problem Localisation and neighbourhood of worst case distributions The first main result Theorem If for some ( θ 1 , θ 2 ) ∈ Θ � θ 2 < 0 , p θ 1 ,θ 2 d µ = 1 , (6) � θ 1 + θ 2 Xp θ 1 ,θ 2 d µ − K ( θ 1 , θ 2 ) = k (7) then the value of the inf in (2) is � V ( k ) = Xp θ 1 ,θ 2 d µ. (8) Essential smoothness of K is a sufficient condition for the existence of such ( θ 1 , θ 2 ) . Further, a necessary and sufficient condition for p to attain the minimum in (2) is p = p θ 1 ,θ 2 for some ( θ 1 , θ 2 ) ∈ Θ satisfying (6) and (7) .
The problem and its history Solution of the Problem Localisation and neighbourhood of worst case distributions The first main result Corollary If the equations ∂ K ( θ 1 , θ 2 ) = 1 , (9) ∂θ 1 ∂ θ 1 + θ 2 K ( θ 1 , θ 2 ) − K ( θ 1 , θ 2 ) = k (10) ∂θ 2 have a solution ( θ 1 , θ 2 ) ∈ int dom K with θ 2 < 0 then θ 1 , θ 2 satisfy (6) and (7) , and the solution to Problem (2) equals � V ( k ) = ∂ K ( θ 1 , θ 2 ) � . (11) � ∂θ 2 � ( θ 1 ,θ 2 )=( θ 1 ,θ 2 )
The problem and its history Solution of the Problem Localisation and neighbourhood of worst case distributions Outline 1 The problem and its history 2 Solution of the Problem 3 Localisation and neighbourhood of worst case distributions
The problem and its history Solution of the Problem Localisation and neighbourhood of worst case distributions Almost worst case scenarios Definition For a specified ǫ > 0 define an almost worst scenario to be a � density p satisfying H ( p ) ≤ k and achieving Xpd µ < V ( k ) + ǫ . Theorem (Clustering of almost worst case scenarios) Supposing 0 < k < k max , there exists ( θ 1 , θ 2 ) ∈ Θ with θ 2 < 0 � such that for each p with pd µ = 1 �� � B β,µ ( p , p θ 1 ,θ 2 ) ≤ H ( p ) − k − θ 2 Xpd µ − V ( k ) , (12) where B β,µ is generalised Bregman distance.
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