Abstract Major events like the COVID-19 crisis have impact both on the financial market and on claim arrival intensities and claim sizes of insurers. Thus, when optimal investment and reinsurance strategies have to be determined, it is important to consider models which reflect this dependence. In this paper, we make a proposal on how to generate dependence between the financial market and claim sizes in times of crisis and determine via a stochastic control approach an optimal investment and reinsurance strategy which maximizes the expected exponential utility of terminal wealth. Moreover, we also allow that the claim size distribution may be learned in the model. We give comparisons and bounds on the optimal strategy using simple models. What turns out to be very surprising is that numerical results indicate that even a minimal dependence which is created in this model has a huge impact on the optimal investment strategy.
{"title":"Bayesian optimal investment and reinsurance with dependent financial and insurance risks","authors":"N. Bäuerle, Gregor Leimcke","doi":"10.1515/strm-2021-0029","DOIUrl":"https://doi.org/10.1515/strm-2021-0029","url":null,"abstract":"Abstract Major events like the COVID-19 crisis have impact both on the financial market and on claim arrival intensities and claim sizes of insurers. Thus, when optimal investment and reinsurance strategies have to be determined, it is important to consider models which reflect this dependence. In this paper, we make a proposal on how to generate dependence between the financial market and claim sizes in times of crisis and determine via a stochastic control approach an optimal investment and reinsurance strategy which maximizes the expected exponential utility of terminal wealth. Moreover, we also allow that the claim size distribution may be learned in the model. We give comparisons and bounds on the optimal strategy using simple models. What turns out to be very surprising is that numerical results indicate that even a minimal dependence which is created in this model has a huge impact on the optimal investment strategy.","PeriodicalId":44159,"journal":{"name":"Statistics & Risk Modeling","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2021-02-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44313023","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract We consider multi-period cost-of-capital valuation of a liability cash flow subject to repeated capital requirements that are partly financed by capital injections from capital providers with limited liability. Limited liability means that, in any given period, the capital provider is not liable for further payment in the event that the capital provided at the beginning of the period turns out to be insufficient to cover both the current-period payments and the updated value of the remaining cash flow. The liability cash flow is modeled as a continuous-time stochastic process on [ 0 , T ] {[0,T]} . The multi-period structure is given by a partition of [ 0 , T ] {[0,T]} into subintervals, and on the corresponding finite set of times, a discrete-time cost-of-capital-margin process is defined. Our main objective is the analysis of existence and properties of continuous-time limits of discrete-time cost-of-capital-margin processes corresponding to a sequence of partitions whose meshes tend to zero. Moreover, we provide explicit expressions for the limit processes when cash flows are given by Itô diffusions and processes with independent increments.
{"title":"Continuous-time limits of multi-period cost-of-capital margins","authors":"Hampus Engsner, F. Lindskog","doi":"10.1515/strm-2019-0008","DOIUrl":"https://doi.org/10.1515/strm-2019-0008","url":null,"abstract":"Abstract We consider multi-period cost-of-capital valuation of a liability cash flow subject to repeated capital requirements that are partly financed by capital injections from capital providers with limited liability. Limited liability means that, in any given period, the capital provider is not liable for further payment in the event that the capital provided at the beginning of the period turns out to be insufficient to cover both the current-period payments and the updated value of the remaining cash flow. The liability cash flow is modeled as a continuous-time stochastic process on [ 0 , T ] {[0,T]} . The multi-period structure is given by a partition of [ 0 , T ] {[0,T]} into subintervals, and on the corresponding finite set of times, a discrete-time cost-of-capital-margin process is defined. Our main objective is the analysis of existence and properties of continuous-time limits of discrete-time cost-of-capital-margin processes corresponding to a sequence of partitions whose meshes tend to zero. Moreover, we provide explicit expressions for the limit processes when cash flows are given by Itô diffusions and processes with independent increments.","PeriodicalId":44159,"journal":{"name":"Statistics & Risk Modeling","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2020-04-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/strm-2019-0008","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47153238","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract Let 𝒳 be a subset of L 1 L^{1} that contains the space of simple random variables ℒ and ρ : X → ( - ∞ , ∞ ] rhocolonmathcal{X}to(-infty,infty] a dilatation monotone functional with the Fatou property. In this note, we show that 𝜌 extends uniquely to a σ ( L 1 , L ) sigma(L^{1},mathcal{L}) lower semicontinuous and dilatation monotone functional ρ ¯ : L 1 → ( - ∞ , ∞ ] overline{rho}colon L^{1}to(-infty,infty] . Moreover, ρ ¯ overline{rho} preserves monotonicity, (quasi)convexity and cash-additivity of 𝜌. We also study conditions under which ρ ¯ overline{rho} preserves finiteness on a larger domain. Our findings complement extension and continuity results for (quasi)convex law-invariant functionals. As an application of our results, we show that transformed norm risk measures on Orlicz hearts admit a natural extension to L 1 L^{1} that retains robust representations.
{"title":"On the extension property of dilatation monotone risk measures","authors":"Massoomeh Rahsepar, F. Xanthos","doi":"10.1515/STRM-2020-0006","DOIUrl":"https://doi.org/10.1515/STRM-2020-0006","url":null,"abstract":"Abstract Let 𝒳 be a subset of L 1 L^{1} that contains the space of simple random variables ℒ and ρ : X → ( - ∞ , ∞ ] rhocolonmathcal{X}to(-infty,infty] a dilatation monotone functional with the Fatou property. In this note, we show that 𝜌 extends uniquely to a σ ( L 1 , L ) sigma(L^{1},mathcal{L}) lower semicontinuous and dilatation monotone functional ρ ¯ : L 1 → ( - ∞ , ∞ ] overline{rho}colon L^{1}to(-infty,infty] . Moreover, ρ ¯ overline{rho} preserves monotonicity, (quasi)convexity and cash-additivity of 𝜌. We also study conditions under which ρ ¯ overline{rho} preserves finiteness on a larger domain. Our findings complement extension and continuity results for (quasi)convex law-invariant functionals. As an application of our results, we show that transformed norm risk measures on Orlicz hearts admit a natural extension to L 1 L^{1} that retains robust representations.","PeriodicalId":44159,"journal":{"name":"Statistics & Risk Modeling","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2020-02-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/STRM-2020-0006","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42969121","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract Based on an XVA analysis of centrally cleared derivative portfolios, we consider two capital and funding issues pertaining to the efficiency of the design of central counterparties (CCPs). First, we consider an organization of a clearing framework, whereby a CCP would also play the role of a centralized XVA calculator and management center. The default fund contributions would become pure capital at risk of the clearing members, remunerated as such at some hurdle rate, i.e. return-on-equity. Moreover, we challenge the current default fund Cover 2 EMIR sizing rule with a broader risk based approach, relying on a suitable notion of economic capital of a CCP. Second, we compare the margin valuation adjustments (MVAs) resulting from two different initial margin raising strategies. The first one is unsecured borrowing by the clearing member. As an alternative, the clearing member delegates the posting of its initial margin to a so-called specialist lender, which, in case of default of the clearing member, receives back from the CCP the portion of IM unused to cover losses. The alternative strategy results in a significant MVA compression. A numerical case study shows that the volatility swings of the IM funding expenses can even be the main contributor to an economic capital based default fund of a CCP. This is an illustration of the transfer of counterparty risk into liquidity risk triggered by extensive collateralization.
{"title":"XVA metrics for CCP optimization","authors":"C. Albanese, Yannick Armenti, S. Crépey","doi":"10.1515/strm-2017-0034","DOIUrl":"https://doi.org/10.1515/strm-2017-0034","url":null,"abstract":"Abstract Based on an XVA analysis of centrally cleared derivative portfolios, we consider two capital and funding issues pertaining to the efficiency of the design of central counterparties (CCPs). First, we consider an organization of a clearing framework, whereby a CCP would also play the role of a centralized XVA calculator and management center. The default fund contributions would become pure capital at risk of the clearing members, remunerated as such at some hurdle rate, i.e. return-on-equity. Moreover, we challenge the current default fund Cover 2 EMIR sizing rule with a broader risk based approach, relying on a suitable notion of economic capital of a CCP. Second, we compare the margin valuation adjustments (MVAs) resulting from two different initial margin raising strategies. The first one is unsecured borrowing by the clearing member. As an alternative, the clearing member delegates the posting of its initial margin to a so-called specialist lender, which, in case of default of the clearing member, receives back from the CCP the portion of IM unused to cover losses. The alternative strategy results in a significant MVA compression. A numerical case study shows that the volatility swings of the IM funding expenses can even be the main contributor to an economic capital based default fund of a CCP. This is an illustration of the transfer of counterparty risk into liquidity risk triggered by extensive collateralization.","PeriodicalId":44159,"journal":{"name":"Statistics & Risk Modeling","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/strm-2017-0034","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43561990","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract In this paper, we introduce a new interpolation method for call option prices and implied volatilities with respect to the strike, which first generates, for fixed maturity, an implied volatility curve that is smooth and free of static arbitrage. Our interpolation method is based on a distortion of the call price function of an arbitrage-free financial “reference” model of one’s choice. It reproduces the call prices of the reference model if the market data is compatible with the model. Given a set of call prices for different strikes and maturities, we can construct a call price surface by using this one-dimensional interpolation method on every input maturity and interpolating the generated curves in the maturity dimension. We obtain the algorithm of N. Kahalé [An arbitrage-free interpolation of volatilities, Risk 17 2004, 5, 102–106] as a special case, when applying the Black–Scholes model as reference model.
{"title":"Arbitrage-free interpolation of call option prices","authors":"Christian Bender, Matthias Thiel","doi":"10.1515/strm-2018-0026","DOIUrl":"https://doi.org/10.1515/strm-2018-0026","url":null,"abstract":"Abstract In this paper, we introduce a new interpolation method for call option prices and implied volatilities with respect to the strike, which first generates, for fixed maturity, an implied volatility curve that is smooth and free of static arbitrage. Our interpolation method is based on a distortion of the call price function of an arbitrage-free financial “reference” model of one’s choice. It reproduces the call prices of the reference model if the market data is compatible with the model. Given a set of call prices for different strikes and maturities, we can construct a call price surface by using this one-dimensional interpolation method on every input maturity and interpolating the generated curves in the maturity dimension. We obtain the algorithm of N. Kahalé [An arbitrage-free interpolation of volatilities, Risk 17 2004, 5, 102–106] as a special case, when applying the Black–Scholes model as reference model.","PeriodicalId":44159,"journal":{"name":"Statistics & Risk Modeling","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/strm-2018-0026","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48425053","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract We approximate Gerber–Shiu functions with heavy-tailed claims in a recently introduced risk model having both interclaim times and premiums depending on the claim sizes. We apply a technique known as “corrected phase-type approximations”. This results in adding a correction term to the Gerber–Shiu function with phase-type claim sizes. The correction term contains the heavy-tailed behavior at most once per convolution and captures the tail behavior of the true Gerber–Shiu function. We make the tail behavior specific in the classical case of one class of risk insured. After illustrating a use of such approximations, we study numerically the approximations’ relative errors for some specific penalty functions and claims distributions.
{"title":"On corrected phase-type approximations of the time value of ruin with heavy tails","authors":"D. Geiger, A. Adekpedjou","doi":"10.1515/strm-2019-0009","DOIUrl":"https://doi.org/10.1515/strm-2019-0009","url":null,"abstract":"Abstract We approximate Gerber–Shiu functions with heavy-tailed claims in a recently introduced risk model having both interclaim times and premiums depending on the claim sizes. We apply a technique known as “corrected phase-type approximations”. This results in adding a correction term to the Gerber–Shiu function with phase-type claim sizes. The correction term contains the heavy-tailed behavior at most once per convolution and captures the tail behavior of the true Gerber–Shiu function. We make the tail behavior specific in the classical case of one class of risk insured. After illustrating a use of such approximations, we study numerically the approximations’ relative errors for some specific penalty functions and claims distributions.","PeriodicalId":44159,"journal":{"name":"Statistics & Risk Modeling","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2019-11-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/strm-2019-0009","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41310496","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract Conditional excess risk measures like Marginal Expected Shortfall and Marginal Mean Excess are designed to aid in quantifying systemic risk or risk contagion in a multivariate setting. In the context of insurance, social networks, and telecommunication, risk factors often tend to be heavy-tailed and thus frequently studied under the paradigm of regular variation. We show that regular variation on different subspaces of the Euclidean space leads to these risk measures exhibiting distinct asymptotic behavior. Furthermore, we elicit connections between regular variation on these subspaces and the behavior of tail copula parameters extending previous work and providing a broad framework for studying such risk measures under multivariate regular variation. We use a variety of examples to exhibit where such computations are practically applicable.
{"title":"Conditional excess risk measures and multivariate regular variation","authors":"Bikramjit Das, Vicky Fasen-Hartmann","doi":"10.1515/strm-2018-0030","DOIUrl":"https://doi.org/10.1515/strm-2018-0030","url":null,"abstract":"Abstract Conditional excess risk measures like Marginal Expected Shortfall and Marginal Mean Excess are designed to aid in quantifying systemic risk or risk contagion in a multivariate setting. In the context of insurance, social networks, and telecommunication, risk factors often tend to be heavy-tailed and thus frequently studied under the paradigm of regular variation. We show that regular variation on different subspaces of the Euclidean space leads to these risk measures exhibiting distinct asymptotic behavior. Furthermore, we elicit connections between regular variation on these subspaces and the behavior of tail copula parameters extending previous work and providing a broad framework for studying such risk measures under multivariate regular variation. We use a variety of examples to exhibit where such computations are practically applicable.","PeriodicalId":44159,"journal":{"name":"Statistics & Risk Modeling","volume":null,"pages":null},"PeriodicalIF":1.5,"publicationDate":"2019-05-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/strm-2018-0030","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41487307","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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