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Mathematics and Financial Economics
Published in: Mathematics and Financial Economics 3/2020

24-02-2020

On the dynamic representation of some time-inconsistent risk measures in a Brownian filtration

Authors: Julio Backhoff-Veraguas, Ludovic Tangpi

Published in: Mathematics and Financial Economics | Issue 3/2020

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Abstract

It is well-known from the work of Kupper and Schachermayer that most law-invariant risk measures are not time-consistent, and thus do not admit dynamic representations as backward stochastic differential equations. In this work we show that in a Brownian filtration the “Optimized Certainty Equivalent” risk measures of Ben-Tal and Teboulle can be computed through PDE techniques, i.e. dynamically. This can be seen as a substitute of sorts whenever they lack time consistency, and covers the cases of conditional value-at-risk and monotone mean-variance. Our method consists of focusing on the convex dual representation, which suggests an expression of the risk measure as the value of a stochastic control problem on an extended the state space. With this we can obtain a dynamic programming principle and use stochastic control techniques, along with the theory of viscosity solutions, which we must adapt to cover the present singular situation.
previous article Consumption-investment optimization problem in a Lévy financial model with transaction costs and làdlàg strategies
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Appendix
Available only for authorised users
Footnotes
1
In fact, the conditional value-at-risk (also known as expected shortfall) has been praised, and its adoption recommended, by the Basel III Committee in the following terms [34, p. 18]:
“... the current framework’s reliance on VaR (value-at-risk) as a quantitative risk metric raises a number of issues, most notably the inability of the measure to capture the “tail risk” of the loss distribution. The Committee has therefore decided to use an expected shortfall (ES) measure for the internal models -based approach and will determine the risk weights for the revised standardised approach using an ES methodology...”
 
2
We denote by \(l^*\) the convex conjugate of l.
 
3
As pointed out by an anonymous referee, in some cases the upper lateral boundary condition may be expensive to obtain, since it involves an expected value. However, in the situation when \(\text {dom}(l^*)=[0,\infty )\), this boundary condition disappears and only the lower lateral boundary condition remains. The latter takes the form \(V(t,y,0)=-l^*(z)\) and is hence explicit.
 
4
\(\rho (X+ c) = \rho (X) +c\) for all \(X \in L^\infty ({{{\mathcal {F}}}})\) and \(c \in {\mathbb {R}} \). Translation invariance is a synonym for this.
 
5
In fact, it is \({\tilde{\rho }}(X):= \rho (-X)\) that satisfies the risk measures axioms as developed in Artzner et al. [1] and Föllmer and Schied [24], but we will work with the increasing functional \(\rho \) for ease of notation.
 
6
This condition is also known as the flow property. We refer to Cheridito et al. [11], Delbaen [15], Artzner et al. [2], Ruszczyński and Shapiro [40], Detlefsen and Scandolo [17] for discussions on the consequences of time-consistency.
 
7
This corresponds to the standard OCE risk measure up to a minus sign.
 
8
\(A'\) is the transpose of A.
 
9
To exemplify: Take \(m=d=1\) and \(\Gamma ^n_{1,2}=1,\Gamma ^n_{2,2}=-1/n\). Then \(H(t,y,1,\gamma ,\Gamma ^n)<\infty \) but in the limit (i.e. \(\Gamma _{1,2}=1,\Gamma _{2,2}=0\)) the Hamiltonian is infinite. Thus \(H<\infty \) is not closed and so there cannot be such continuous G.
 
10
See also Remark 3.5 for some comments on the growth properties of V.
 
11
The essential range of \(\eta Z\) is \(range(\eta Z):=[\mathop {\mathrm{ess}\,\mathrm{inf}}\eta Z,\mathop {\mathrm{ess}\,\mathrm{sup}} \eta Z]\).
 
12
This is definitely a technical point, and we see no reason why Theorem 1.1 would not hold without such assumption on the domain of \(l^*\). In fact, during revision of the present article, the authors and A. Max Reppen [3] derived existence and uniqueness for a related PDE in the case when the domain of \(l^*\) is bounded, but under a different notion of viscosity sub/super solution.
 
13
I.e. for all \(x_0=(s_0,y_0,z_0)\in [0,T]\times {\mathbb {R}}^m\times (0,+\infty )\) and \(\varphi \in C^2([0,T]\times {\mathbb {R}}^m\times (0,+\infty ))\) such that \(x_0\) is a local minimizer of \(w-\varphi \) and \(\varphi (x_0) = w(x_0)\), we have \(w(x_0)\ge \psi (y_0,z_0)\) if \(s_0=T\), and otherwise \( \partial _t\varphi (x_0) + H^n(x_0, D\varphi (x_0), D^2\varphi (x_0)) \le 0\).
 
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Metadata
Title
On the dynamic representation of some time-inconsistent risk measures in a Brownian filtration
Authors
Julio Backhoff-Veraguas
Ludovic Tangpi
Publication date
24-02-2020
Publisher
Springer Berlin Heidelberg
Published in
Mathematics and Financial Economics / Issue 3/2020
Print ISSN: 1862-9679
Electronic ISSN: 1862-9660
DOI
https://doi.org/10.1007/s11579-020-00261-2

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