Extending dynamic convex risk measures from discrete time to continuous time: A convergence approach

Stadje, MA Mitja

doi:10.1016/j.insmatheco.2010.08.005

Cited by 44 publications

(42 citation statements)

References 45 publications

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“…However, concrete examples of dynamic, time-consistent convex risk measures are scarce, and they typically have to be defined abstractly, for example via the driver of a backward stochastic differential equation (BSDE) or as the limit of discrete time-consistent risk measures [Sta10]. As a result, intuition is lost, and there is at present little understanding what the choice of driver says about the measure of risk.…”

Section: Choice: What Is the "Right" Risk Measure?mentioning

confidence: 99%

From Smile Asymptotics to Market Risk Measures

Sturm

Sircar

2012

SSRN Journal

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The left tail of the implied volatility skew, coming from quotes on out-of-the-money put options, can be thought to reflect the market's assessment of the risk of a huge drop in stock prices. We analyze how this market information can be integrated into the theoretical framework of convex monetary measures of risk. In particular, we make use of indifference pricing by dynamic convex risk measures, which are given as solutions of backward stochastic differential equations (BSDEs), to establish a link between these two approaches to risk measurement. We derive a characterization of the implied volatility in terms of the solution of a nonlinear PDE and provide a small time-to-maturity expansion and numerical solutions. This procedure allows to choose convex risk measures in a conveniently parametrized class, distorted entropic dynamic risk measures, which we introduce here, such that the asymptotic volatility skew under indifference pricing can be matched with the market skew. We demonstrate this in a calibration exercise to market implied volatility data.

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Section: Choice: What Is the "Right" Risk Measure?mentioning

confidence: 99%

From Smile Asymptotics to Market Risk Measures

Sturm

Sircar

2012

SSRN Journal

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“…the convergence results of Briand et al [6], Stadje [26] characterized the driver of BSDE according to different risk measures.…”

Section: Introductionmentioning

confidence: 99%

A Risk-Averse Analog of the Hamilton–Jacobi–Bellman Equation

Ruszczyński

Yao

2015

2015 Proceedings of the Conference on Control and Its Applications

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In this paper, we study the risk-averse control problem for diffusion processes. We make use of a forward-backward system of stochastic differential equations to evaluate a fixed policy and to formulate the optimal control problem. Weak formulation is established to facilitate the derivation of the risk-averse dynamic programming equation. We prove that the value function of the risk-averse control problem is a viscosity solution of a risk-averse analog of the HamiltonJacobi-Bellman equation. On the other hand, a verification theorem is proved when the classical solution of the equation exists.

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“…See Riedel (2004), Cheridito et al (2006), Roorda et al (2005), Rosazza Gianin (2006), and Artzner et al (2007). As an example, Stadje (2010) showed how a large class of dynamic convex risk measures in continuous-time can be derived from the limit of their discrete time versions. Moreover, Jobert and Rogers (2008) showed how time-consistent valuations can be constructed through the backward induction of static one-period risk measures (or "valuations").…”

Section: Introductionmentioning

confidence: 99%

Time-Consistent Actuarial Valuations

Ghalehjooghi

Pelsser

2015

SSRN Journal

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Time-consistent valuations (i.e. pricing operators) can be created by backward iteration of one-period valuations. In this paper we investigate the continuous-time limits of well-known actuarial premium principles when such backward iteration procedures are applied. This method is applied to an insurance risk process in the form of a diffusion process and a jump process in order to capture the heavy tailed nature of insurance liabilities. We show that in the case of the diffusion process, the one-period time-consistent Variance premium principle converges to the non-linear exponential indifference price. Furthermore, we show that the Standard-Deviation and the Cost-of-Capital principle converge to the same price limit. Adding the jump risk gives a more realistic picture of the price. Furthermore, we no longer observe that the different premium principles converge to the same limit since each principle reflects the effect of the jump differently. In the Cost-of-Capital principle, in particular the VaR operator fails to capture the jump risk for small jump probabilities, and the time-consistent price depends on the distribution of the premium jump. * The research leading to these results has received funding from the European Union Seventh Framework Programme ([FP7

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Extending dynamic convex risk measures from discrete time to continuous time: A convergence approach

Cited by 44 publications

References 45 publications

From Smile Asymptotics to Market Risk Measures

From Smile Asymptotics to Market Risk Measures

A Risk-Averse Analog of the Hamilton–Jacobi–Bellman Equation

Time-Consistent Actuarial Valuations

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