Minimal Entropy and Entropic Risk Measures: A Unified Framework via Relative Entropy
We introduce a new coherent risk measure, the minimal-entropy risk measure, which is built on the minimal-entropy <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>σ</mi></semantics></math>&l...
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MDPI AG
2025-04-01
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| Online Access: | https://www.mdpi.com/2227-9091/13/4/70 |
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| author | Moritz Sohns |
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| author_sort | Moritz Sohns |
| collection | DOAJ |
| description | We introduce a new coherent risk measure, the minimal-entropy risk measure, which is built on the minimal-entropy <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>σ</mi></semantics></math></inline-formula>-martingale measure—a concept inspired by the well-known minimal-entropy martingale measure used in option pricing. While the minimal-entropy martingale measure is commonly used for pricing and hedging, the minimal-entropy <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>σ</mi></semantics></math></inline-formula>-martingale measure has not previously been studied, nor has it been analyzed as a traditional risk measure. We address this gap by clearly defining this new risk measure and examining its fundamental properties. In addition, we revisit the entropic risk measure, typically expressed through an exponential formula. We provide an alternative definition using a supremum over Kullback–Leibler divergences, making its connection to entropy clearer. We verify important properties of both risk measures, such as convexity and coherence, and extend these concepts to dynamic situations. We also illustrate their behavior in scenarios involving optimal risk transfer. Our results link entropic concepts with incomplete-market pricing and demonstrate how both risk measures share a unified entropy-based foundation. |
| format | Article |
| id | doaj-art-555ab49400a24426bf3ff3bf2361e898 |
| institution | DOAJ |
| issn | 2227-9091 |
| language | English |
| publishDate | 2025-04-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Risks |
| spelling | doaj-art-555ab49400a24426bf3ff3bf2361e8982025-08-20T03:13:48ZengMDPI AGRisks2227-90912025-04-011347010.3390/risks13040070Minimal Entropy and Entropic Risk Measures: A Unified Framework via Relative EntropyMoritz Sohns0Faculty of Economic Studies, University of Finance and Administration, 10100 Prague, Czech RepublicWe introduce a new coherent risk measure, the minimal-entropy risk measure, which is built on the minimal-entropy <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>σ</mi></semantics></math></inline-formula>-martingale measure—a concept inspired by the well-known minimal-entropy martingale measure used in option pricing. While the minimal-entropy martingale measure is commonly used for pricing and hedging, the minimal-entropy <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>σ</mi></semantics></math></inline-formula>-martingale measure has not previously been studied, nor has it been analyzed as a traditional risk measure. We address this gap by clearly defining this new risk measure and examining its fundamental properties. In addition, we revisit the entropic risk measure, typically expressed through an exponential formula. We provide an alternative definition using a supremum over Kullback–Leibler divergences, making its connection to entropy clearer. We verify important properties of both risk measures, such as convexity and coherence, and extend these concepts to dynamic situations. We also illustrate their behavior in scenarios involving optimal risk transfer. Our results link entropic concepts with incomplete-market pricing and demonstrate how both risk measures share a unified entropy-based foundation.https://www.mdpi.com/2227-9091/13/4/70entropic risk measureminimal-entropy risk measurerelative entropyrisk-neutral measuresdynamic risk measures |
| spellingShingle | Moritz Sohns Minimal Entropy and Entropic Risk Measures: A Unified Framework via Relative Entropy Risks entropic risk measure minimal-entropy risk measure relative entropy risk-neutral measures dynamic risk measures |
| title | Minimal Entropy and Entropic Risk Measures: A Unified Framework via Relative Entropy |
| title_full | Minimal Entropy and Entropic Risk Measures: A Unified Framework via Relative Entropy |
| title_fullStr | Minimal Entropy and Entropic Risk Measures: A Unified Framework via Relative Entropy |
| title_full_unstemmed | Minimal Entropy and Entropic Risk Measures: A Unified Framework via Relative Entropy |
| title_short | Minimal Entropy and Entropic Risk Measures: A Unified Framework via Relative Entropy |
| title_sort | minimal entropy and entropic risk measures a unified framework via relative entropy |
| topic | entropic risk measure minimal-entropy risk measure relative entropy risk-neutral measures dynamic risk measures |
| url | https://www.mdpi.com/2227-9091/13/4/70 |
| work_keys_str_mv | AT moritzsohns minimalentropyandentropicriskmeasuresaunifiedframeworkviarelativeentropy |