COHERENT RISK MEASURES FOR DERIVATIVES UNDER BLACK–SCHOLES ECONOMY

2001 ◽  
Vol 04 (05) ◽  
pp. 819-835 ◽  
Author(s):  
H. YANG ◽  
T. K. SIU
Keyword(s):  
Risk Measures ◽  
Risk Measure ◽  
Fixed Time ◽  
Derivative Securities ◽  
Coherent Risk Measure ◽  
Subjective Probabilities ◽  
Coherent Risk ◽  
Subjective Beliefs ◽  
Black Scholes ◽  
Risk Managers

This paper proposes a risk measure for a portfolio of European-style derivative securities over a fixed time horizon under the Black–Scholes economy. The proposed risk measure is scenario-based along the same line as [3]. The risk measure is constructed by using the risk-neutral probability ([Formula: see text]-measure), the physical probability ([Formula: see text]-measure) and a family of subjective probability measures. The subjective probabilities are introduced by using Girsanov's theorem. In this way, we provide risk managers or regulators with the flexibility of adjusting the risk measure according to their risk preferences and subjective beliefs. The advantages of the proposed measure are that it is easy to implement and that it satisfies the four desirable properties introduced in [3], which make it a coherent risk measure. Finally, we incorporate the presence of transaction costs into our framework.

scholarly journals VECTOR-VALUED COHERENT RISK MEASURE PROCESSES

2014 ◽  
Vol 17 (02) ◽  
pp. 1450011 ◽  
Author(s):  
IMEN BEN TAHAR ◽  
EMMANUEL LÉPINETTE
Keyword(s):  
Continuous Time ◽  
Risk Measures ◽  
Risk Measure ◽  
Time Consistency ◽  
Axiomatic Characterization ◽  
Coherent Risk Measure ◽  
Coherent Risk ◽  
Dynamic Version ◽  
Vector Valued

Introduced by Artzner et al. (1998) the axiomatic characterization of a static coherent risk measure was extended by Jouini et al. (2004) in a multi-dimensional setting to the concept of vector-valued risk measures. In this paper, we propose a dynamic version of the vector-valued risk measures in a continuous-time framework. Particular attention is devoted to the choice of a convenient risk space. We provide dual characterization results, we study different notions of time consistency and we give examples of vector-valued risk measure processes.

IMPRECISE PREVISIONS FOR RISK MEASUREMENT

2003 ◽  
Vol 11 (04) ◽  
pp. 393-412 ◽  
Author(s):  
RENATO PELESSONI ◽  
PAOLO VICIG
Keyword(s):  
Value At Risk ◽  
Risk Measures ◽  
Risk Measure ◽  
Sufficient Conditions ◽  
Risk Measurement ◽  
Random Numbers ◽  
Coherent Risk Measure ◽  
Coherent Risk ◽  
Consistency Properties ◽  
Special Case

In this paper the theory of coherent imprecise previsions is applied to risk measurement. We introduce the notion of coherent risk measure defined on an arbitrary set of risks, showing that it can be considered a special case of coherent upper prevision. We also prove that our definition generalizes the notion of coherence for risk measures defined on a linear space of random numbers, given in literature. Consistency properties of Value-at-Risk (VaR), currently one of the most used risk measures, are investigated too, showing that it does not necessarily satisfy a weaker notion of consistency called 'avoiding sure loss'. We introduce sufficient conditions for VaR to avoid sure loss and to be coherent. Finally we discuss ways of modifying incoherent risk measures into coherent ones.

INSTABILITY OF PORTFOLIO OPTIMIZATION UNDER COHERENT RISK MEASURES

2010 ◽  
Vol 13 (03) ◽  
pp. 425-437 ◽  
Author(s):  
IMRE KONDOR ◽  
ISTVÁN VARGA-HASZONITS
Keyword(s):  
Portfolio Optimization ◽  
Risk Measures ◽  
Risk Measure ◽  
Expected Shortfall ◽  
Optimal Portfolio ◽  
The Other ◽  
Coherent Risk Measures ◽  
Coherent Risk ◽  
Large Samples

It is shown that the axioms for coherent risk measures imply that whenever there is a pair of portfolios such that one of them dominates the other in a given sample (which happens with finite probability even for large samples), then there is no optimal portfolio under any coherent measure on that sample, and the risk measure diverges to minus infinity. This instability was first discovered in the special example of Expected Shortfall which is used here both as an illustration and as a springboard for generalization.

scholarly journals Central Limit Theorems for Law-Invariant Coherent Risk Measures

2012 ◽  
Vol 49 (01) ◽  
pp. 1-21 ◽  
Author(s):  
Denis Belomestny ◽  
Volker Krätschmer
Keyword(s):  
Central Limit ◽  
Limit Theorems ◽  
Risk Measures ◽  
Risk Measure ◽  
Asymptotic Properties ◽  
Dependent Data ◽  
Distributed Data ◽  
Coherent Risk Measures ◽  
Coherent Risk ◽  
Weakly Dependent Data

In this paper we study the asymptotic properties of the canonical plugin estimates for law-invariant coherent risk measures. Under rather mild conditions not relying on the explicit representation of the risk measure under consideration, we first prove a central limit theorem for independent and identically distributed data, and then extend it to the case of weakly dependent data. Finally, a number of illustrating examples is presented.

scholarly journals Central Limit Theorems for Law-Invariant Coherent Risk Measures

2012 ◽  
Vol 49 (1) ◽  
pp. 1-21 ◽  
Author(s):  
Denis Belomestny ◽  
Volker Krätschmer
Keyword(s):  
Central Limit ◽  
Limit Theorems ◽  
Risk Measures ◽  
Risk Measure ◽  
Asymptotic Properties ◽  
Dependent Data ◽  
Distributed Data ◽  
Coherent Risk Measures ◽  
Coherent Risk ◽  
Weakly Dependent Data

In this paper we study the asymptotic properties of the canonical plugin estimates for law-invariant coherent risk measures. Under rather mild conditions not relying on the explicit representation of the risk measure under consideration, we first prove a central limit theorem for independent and identically distributed data, and then extend it to the case of weakly dependent data. Finally, a number of illustrating examples is presented.

scholarly journals Bigger Is Not Always Safer: A Critical Analysis of the Subadditivity Assumption for Coherent Risk Measures

Risks ◽  
2019 ◽  
Vol 7 (3) ◽  
pp. 91
Author(s):  
Hans Rau-Bredow
Keyword(s):  
Critical Analysis ◽  
Deposit Insurance ◽  
Risk Measures ◽  
Risk Measure ◽  
Coherent Risk Measures ◽  
Bank Mergers ◽  
Insurance Scheme ◽  
Bank Merger ◽  
Coherent Risk

This paper provides a critical analysis of the subadditivity axiom, which is the key condition for coherent risk measures. Contrary to the subadditivity assumption, bank mergers can create extra risk. We begin with an analysis how a merger affects depositors, junior or senior bank creditors, and bank owners. Next it is shown that bank mergers can result in higher payouts having to be made by the deposit insurance scheme. Finally, we demonstrate that if banks are interconnected via interbank loans, a bank merger could lead to additional contagion risks. We conclude that the subadditivity assumption should be rejected, since a subadditive risk measure, by definition, cannot account for such increased risks.

scholarly journals Continuous Time Portfolio Selection under Conditional Capital at Risk

2010 ◽  
Vol 2010 ◽  
pp. 1-26 ◽  
Author(s):  
Gordana Dmitrasinovic-Vidovic ◽  
Ali Lari-Lavassani ◽  
Xun Li ◽  
Antony Ware
Keyword(s):  
At Risk ◽  
Portfolio Selection ◽  
Continuous Time ◽  
Risk Measures ◽  
Risk Measure ◽  
Optimal Strategies ◽  
Time Dependent ◽  
Risky Assets ◽  
Black Scholes ◽  
Capital At Risk

Portfolio optimization with respect to different risk measures is of interest to both practitioners and academics. For there to be a well-defined optimal portfolio, it is important that the risk measure be coherent and quasiconvex with respect to the proportion invested in risky assets. In this paper we investigate one such measure—conditional capital at risk—and find the optimal strategies under this measure, in the Black-Scholes continuous time setting, with time dependent coefficients.

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