${\scr E}$-martingales and their applications in mathematical finance

Choulli, Tahir; Krawczyk, Leszek; Stricker, Christophe

doi:10.1214/aop/1022855653

Cited by 58 publications

(44 citation statements)

References 24 publications

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“…In fact, under these two latter conditions, both models (S, P, F) and (S τ , P, G) are local martingales, and hence the Föllmer-Sondermann method will be applied simultaneously for both models. These two assumptions in (25) can be relaxed at the expense of considering the quadratic hedging method considered in [48,49], and the references therein. For the risk-minimization framework of these papers, the assumption sup 0≤t≤• |S t | 2 ∈ A + loc (F) will suffice together with some "no-arbitrage or viability" assumption on (S, τ), developed in [44].…”

Section: Impact's Quantification Of Mortality Risks: a General Formulamentioning

confidence: 99%

Mortality/Longevity Risk-Minimization with or without Securitization

2021

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This paper addresses the risk-minimization problem, with and without mortality securitization, à la Föllmer–Sondermann for a large class of equity-linked mortality contracts when no model for the death time is specified. This framework includes situations in which the correlation between the market model and the time of death is arbitrary general, and hence leads to the case of a market model where there are two levels of information—the public information, which is generated by the financial assets, and a larger flow of information that contains additional knowledge about the death time of an insured. By enlarging the filtration, the death uncertainty and its entailed risk are fully considered without any mathematical restriction. Our key tool lies in our optional martingale representation, which states that any martingale in the large filtration stopped at the death time can be decomposed into precise orthogonal local martingales. This allows us to derive the dynamics of the value processes of the mortality/longevity securities used for the securitization, and to decompose any mortality/longevity liability into the sum of orthogonal risks by means of a risk basis. The first main contribution of this paper resides in quantifying, as explicitly as possible, the effect of mortality on the risk-minimizing strategy by determining the optimal strategy in the enlarged filtration in terms of strategies in the smaller filtration. Our second main contribution consists of finding risk-minimizing strategies with insurance securitization by investing in stocks and one (or more) mortality/longevity derivatives such as longevity bonds. This generalizes the existing literature on risk-minimization using mortality securitization in many directions.

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Section: Impact's Quantification Of Mortality Risks: a General Formulamentioning

confidence: 99%

Mortality/Longevity Risk-Minimization with or without Securitization

2021

Self Cite

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“…In particular, the process xXy " pxXy t q tPI is the unique (up to indistinguishability) non-decreasing, predictable, càdlàg process with xXy 0 " 0 such that pX 2 t ´xXy t q tPI is a martingale. [11,13]. The BMO Φ p space was introduced and discussed in [20].…”

Section: Rzt0umentioning

confidence: 99%

On Riemann-Liouville type operators, BMO, gradient estimates in the Lévy-Itô space, and approximation

Geiss,

Nguyen

2020

Preprint

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In this article we discuss in a stochastic framework the interplay between Riemann-Liouville operators applied to càdlàg processes, real interpolation, weighted bounded mean oscillation, estimates for gradient processes on the Lévy-Itô space, and the connection to an approximation problem for stochastic integrals. We prove upper and lower bounds for gradient processes appearing in a Brownian setting within the Feynman-Kac theory for parabolic PDEs and in the setting of Lévy processes. The upper bounds are formulated by BMO-conditions on the fractional integrated gradient, the lower bounds are formulated in terms of oscillatory quantities. In the case of Lévy processes we are concerned with a gradient process with values in a Hilbert space where the regularity of this process depends on the direction within this Hilbert space. Moreover, it turns out that certain Hölder properties of terminal functions transfer into a singularity in time that can be compensated by Riemann-Liouville operators.

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“…In this subsection, we focus on the portfolio of l x pure endowment contracts with the discounted payoff given by (9). Let V P E (t, S t , λ t ) denote the expected present value of the total payoff of the portfolio with respect to the minimal martingale measure Q * at time t, that is,…”

mentioning

confidence: 99%

“…In fact, we can only need to prove the existence of the Föllmer-Schweizer decomposition since it leads to the equation ( 35) holds. We make an Assumption 1 in Section 3, it leads to Proposition 3.7 in [9]. Then the existence of the Föllmer-Schweizer decomposition can be obtained using Theorem 5.5 in [9].…”

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Hedging strategy for unit-linked life insurance contracts with self-exciting jump clustering

Wang¹,

Shen²,

Qian³

et al. 2022

JIMO

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This paper studies the hedging problem of unit-linked life insurance contracts in an incomplete market presence of self-exciting (clustering) effect, which is described by a Hawkes process. Applying the local riskminimization method, we manage to obtain closed-form expressions of the locally risk-minimizing hedging strategies for both pure endowment and term insurance contracts. Besides, we demonstrate the existence of the minimal martingale measure and perform numerical analyses. Our numerical results indicate that jump clustering has a significant impact on the optimal hedging strategies.

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${\scr E}$-martingales and their applications in mathematical finance

Cited by 58 publications

References 24 publications

Mortality/Longevity Risk-Minimization with or without Securitization

Mortality/Longevity Risk-Minimization with or without Securitization

On Riemann-Liouville type operators, BMO, gradient estimates in the Lévy-Itô space, and approximation

Hedging strategy for unit-linked life insurance contracts with self-exciting jump clustering

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