TY - JOUR
T1 - A recursive algorithm for multivariate risk measures and a set-valued Bellman’s principle
AU - Feinstein, Zachary
AU - Rudloff, Birgit
N1 - Publisher Copyright:
© 2016, Springer Science+Business Media New York.
PY - 2017/5/1
Y1 - 2017/5/1
N2 - A method for calculating multi-portfolio time consistent multivariate risk measures in discrete time is presented. Market models for d assets with transaction costs or illiquidity and possible trading constraints are considered on a finite probability space. The set of capital requirements at each time and state is calculated recursively backwards in time along the event tree. We motivate why the proposed procedure can be seen as a set-valued Bellman’s principle, that might be of independent interest within the growing field of set optimization. We give conditions under which the backwards calculation of the sets reduces to solving a sequence of linear, respectively convex vector optimization problems. Numerical examples are given and include superhedging under illiquidity, the set-valued entropic risk measure, and the multi-portfolio time consistent version of the relaxed worst case risk measure and of the set-valued average value at risk.
AB - A method for calculating multi-portfolio time consistent multivariate risk measures in discrete time is presented. Market models for d assets with transaction costs or illiquidity and possible trading constraints are considered on a finite probability space. The set of capital requirements at each time and state is calculated recursively backwards in time along the event tree. We motivate why the proposed procedure can be seen as a set-valued Bellman’s principle, that might be of independent interest within the growing field of set optimization. We give conditions under which the backwards calculation of the sets reduces to solving a sequence of linear, respectively convex vector optimization problems. Numerical examples are given and include superhedging under illiquidity, the set-valued entropic risk measure, and the multi-portfolio time consistent version of the relaxed worst case risk measure and of the set-valued average value at risk.
KW - Bellman’s principle
KW - Dynamic programming
KW - Dynamic risk measures
KW - Set-valued risk measures
KW - Transaction costs
KW - Vector optimization
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U2 - 10.1007/s10898-016-0459-8
DO - 10.1007/s10898-016-0459-8
M3 - Article
AN - SCOPUS:84984804723
SN - 0925-5001
VL - 68
SP - 47
EP - 69
JO - Journal of Global Optimization
JF - Journal of Global Optimization
IS - 1
ER -