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Dynamic Convex Risk Measures: a Survey

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Dynamic Convex Risk Measures: a Survey 1 Dynamic convex risk measures: a survey Based on joint work with Beatrice Acciaio and Hans F¨ollmer Irina Penner Humboldt University of Berlin Email: [email protected] Workshop Risk Analysis in Economics and Finance, Guanajuato, February 2-4, 2011 Risk Analysis in Economics and Finance, Guanajuato, February 2-4, 2011 2 Outline • Static risk measures: a recap • Dynamic framework { Conditional risk measures { Time consistency { Asymptotics and bubbles { Risk measures for processes: capturing discounting ambiguity Risk Analysis in Economics and Finance, Guanajuato, February 2-4, 2011 Static risk measures 1 - 1 History • Starting point: Problem of quantifying the risk undertaken by a financial institution (bank, insurance company...) • From the point of view of a supervising agency a specific monetary purpose comes into play • Axiomatic analysis of capital requirements needed to cover the risk of some future liability was initiated by Artzner, Delbaen, Eber, Heath (1997, 1999) • Coherent risk measures: Artzner, Delbaen, Eber, Heath (1997, 1999), Delbaen (2000)... • Convex risk measures: F¨ollmerand Schied (2002), Frittelli and Rosazza Gianin (2002)... Risk Analysis in Economics and Finance, Guanajuato, February 2-4, 2011 Static risk measures 1 - 2 Monetary measures of risk The set of financial positions: X . Usually X = L1(Ω; F;P ). More generally: X is an ordered locally convex topological vector space. A mapping ρ : X! R is called a monetary convex risk measure if satisfies the following conditions for all X; Y 2 X : • Cash (or Translation) Invariance: If m 2 R, then ρ(X + m) = ρ(X) − m. • (Inverse) Monotonicity: If X ≤ Y , then ρ(X) ≥ ρ(Y ). • Convexity: ρ(λX + (1 − λ)Y ) ≤ λρ(X) + (1 − λ)ρ(Y ) for λ 2 [0; 1]. • (sometimes) Normalization: ρ(0) = 0. A convex risk measure with • Positive homogeneity: If λ ≥ 0, then ρ(λX) = λρ(X) is a coherent risk measure. Risk Analysis in Economics and Finance, Guanajuato, February 2-4, 2011 Static risk measures 1 - 3 Acceptance sets An important characterization of a monetary risk measure is the acceptance set A := X 2 X ρ(X) ≤ 0 : • By cash invariance, ρ is uniquely determined through its acceptance set: ρ(X) = inf m 2 R X + m 2 A (1) ! ρ(X) is the minimal capital requirement that has to be added to a financial position X in order to make it acceptable. • The acceptance set of a convex risk measure is convex, solid, and such that ess inf m 2 R m 2 A = 0 and 0 2 A. • On the other hand, every set A with the above properties defines a monetary convex risk measure via (1). Risk Analysis in Economics and Finance, Guanajuato, February 2-4, 2011 Static risk measures 1 - 4 Robust representation • A convex risk measure ρ : X! R has a robust representation if it can be written as ρ(X) = sup (Y (X) − α(Y )) ; Y 2X ∗ with some convex penalty function α : X ∗ ! [0; 1]. • Typically a penalty function is given by αmin(Y ) = sup (Y (X) − ρ(X)) = sup Y (X): X2X X2A • If ρ is coherent, then the robust representation is of the form ρ(X) = sup Y (X) Y 2Q for some convex set Q ⊆ X ∗. Risk Analysis in Economics and Finance, Guanajuato, February 2-4, 2011 Static risk measures 1 - 5 Robust representation (continued) For a convex risk measure ρ : L1 ! R the following are equivalent: • ρ has a robust representation in terms of probability measures, i.e. ρ(X) = sup (EQ[−X] − α(Q)) Q2M1(P ) with some penalty function α : M1(P ) ! [0; 1], where M1(P ) = Q Q probability measure on (Ω; F);Q P : • ρ is lower semicontinuous in the weak∗ topology σ(L1;L1). • The acceptance set A is weak∗ closed in L1. • ρ is continuous from above, i.e. ρ(Xn) % ρ(X) for any decreasing sequence (Xn) with Xn & X. • ρ has the Fatou property, i.e. ρ(X) ≤ lim inf ρ(Xn) for any bounded sequence (Xn) with Xn ! X. Risk Analysis in Economics and Finance, Guanajuato, February 2-4, 2011 Static risk measures 1 - 6 Important examples • Law-invariant risk measures • Shortfall risk • Measures of risk in a financial market Risk Analysis in Economics and Finance, Guanajuato, February 2-4, 2011 Static risk measures 1 - 7 Law-invariant risk measures Law-invariant risk measures depend only on the law of X. • A very common law-invariant monetary risk measure is Value at Risk at level λ 2 [0; 1]: V @Rλ(X) = inf m 2 R P [X + m < 0] ≤ λ -unfortunately not convex. • An alternative: Average Value at Risk (Tail Value at Risk, Conditional Value at Risk, Expected Shortfall) at level λ 2 [0; 1] 1 Z λ AV @Rλ(X) = V @Rγ (X)dγ = max EQ[−X]; λ 0 Q2Qλ n dQ 1 o where Qλ = Q P dP ≤ λ P -a.s. • AV @Rλ is the smallest law-invariant coherent risk measure that dominates V @Rλ. Risk Analysis in Economics and Finance, Guanajuato, February 2-4, 2011 Static risk measures 1 - 8 Shortfall risk Let l : R ! R be some convex and increasing loss function (or alter- natively u : R ! R some concave and increasing utility function). • The set 1 1 A := X 2 L E[l(−X)] ≤ l(0) = X 2 L E[u(X)] ≥ u(0) defines an acceptance set. • The corresponding convex risk measure ρ(X) is continuous from below. • The minimal penalty function in the robust representation of ρ is given by 1 dQ αmin(Q) = inf E l∗ λ λ>0 λ dP for all Q P . Risk Analysis in Economics and Finance, Guanajuato, February 2-4, 2011 Static risk measures 1 - 9 Shortfall risk: entropic risk measure • For the exponential loss function l(x) = eγx with the risk aversion γ > 0 we obtain the entropic risk measure 1 ρ(X) = log E[e−γX ]: γ • Entropic risk measure has the robust representation 1 ρ(X) = max EQ[−X] − H(QjP ) ; Q2M1(P ) γ i.e., the minimal penalty function of the entropic risk measure is given by 1 αmin(Q) = H(QjP ) = E [log(dQ=dP )]; γ Q where H(QjP ) denotes the relative entropy of Q w.r.t. P . Risk Analysis in Economics and Finance, Guanajuato, February 2-4, 2011 Static risk measures 1 - 10 Measures of risk in a financial market Consider a financial market model with a discounted price process given by a semi-martingale (St)t2[0;T ]. The set of admissible strategies is denoted by S. • The set of all financial positions that can be hedged with some admissible strategy at no cost n 1 R T o A := X 2 L 9ξ 2 S : 0 ξtdSt ≥ −X defines an acceptance set. • The corresponding risk measure ρ is the superhedging price. • ρ is a convex risk measure if S is convex and ρ is coherent if S is a cone. • The existence of a robust representation is related to the no-arbitrage condition. Risk Analysis in Economics and Finance, Guanajuato, February 2-4, 2011 Dynamic framework 2 - 1 Outline • Static risk measures: a recap • Dynamic framework: Assume there is an information flow described by a filtered probability space (Ω; F; (Ft)t=0;:::;T ;P ); F0 = f;; Ωg; F = FT : The time horizon T might be finite or infinite. How can this information be used for risk assessment? Risk Analysis in Economics and Finance, Guanajuato, February 2-4, 2011 Dynamic framework 2 - 2 Outline • Static risk measures: a recap • Dynamic framework: Assume there is an information flow described by a filtered probability space (Ω; F; (Ft)t=0;:::;T ;P ); F0 = f;; Ωg; F = FT : The time horizon T might be finite or infinite. How can this information be used for risk evaluation? { Conditional risk measures { Time consistency { Asymptotics and bubbles { Risk measures for processes: capturing discounting ambiguity Risk Analysis in Economics and Finance, Guanajuato, February 2-4, 2011 Dynamic framework: Conditional risk measures 3 - 1 Conditional convex risk measure 1 1 A conditional convex risk measure ρt is a map L (FT ) ! L (Ft) with the following properties for all X; Y 2 L1: 1 • Conditional Cash Invariance: 8 mt 2 L (Ft): ρt(X + mt) = ρt(X) − mt • (Inverse) Monotonicity: X ≤ Y ) ρt(X) ≥ ρt(Y ) 1 • Conditional Convexity: 8 λ 2 L (Ft); 0 ≤ λ ≤ 1: ρt(λX + (1 − λ)Y ) ≤ λρt(X) + (1 − λ)ρt(Y ) • Normalization: ρt(0) = 0. φt := −ρt is a conditional monetary utility function. Risk Analysis in Economics and Finance, Guanajuato, February 2-4, 2011 Dynamic framework: Conditional risk measures 3 - 2 Conditional coherent risk measure A conditional convex risk measure with 1 • Conditional positive homogeneity: 8 λ 2 L (Ft); λ ≥ 0: ρt(λX) = λρt(X) is called a conditional coherent risk measure. Risk Analysis in Economics and Finance, Guanajuato, February 2-4, 2011 Dynamic framework: Conditional risk measures 3 - 3 Acceptance sets The conditional version of acceptance set is given by 1 At := X 2 L ρt(X) ≤ 0 : Again, ρt is uniquely determined through its acceptance set: 1 ρt(X) = ess inf m 2 L (Ft) X + m 2 At ! ρt(X) is the minimal conditional capital requirement that has to be added to a financial position X in order to make it acceptable at time t. Risk Analysis in Economics and Finance, Guanajuato, February 2-4, 2011 Dynamic framework: Conditional risk measures 3 - 4 Robust representation (cf. Detlefsen and Scandolo (2005)) For a conditional convex risk measure ρt the following are equivalent: • ρt is continuous from above; • ρt has the robust representation min ρt(X) = ess sup EQ[−XjFt] − αt (Q) ; Q2Qt min where the penalty function αt is given by min αt (Q) = ess sup (EQ[−XjFt] − ρt(X)) = ess sup EQ[−XjFt] 1 X2L X2At for Q 2 Qt := Q P Q = P on Ft : Risk Analysis in Economics and Finance, Guanajuato, February 2-4, 2011 Dynamic framework: Conditional risk measures 3 - 5 Example: conditional Average Value at Risk The conditional version of Average Value at Risk (Tail Value at Risk, Conditional Value at Risk, Expected Shortfall) at level λt can be defined as AV @Rλt (X) = max EQ[−XjFt]; Q2Qλt n dQ 1 o 1 where Qλ = Q 2 Qt ≤ P -a.s.
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