In this paper we explore a novel way to combine the dynamic notion of time-consistency with the static notion of quantile-based coherent risk measure or spectral risk measure, of which Expected Shortfall is a prime example. We introduce a new class of dynamic risk measures given in terms of a certain family of g-expectations driven by Wiener and Poisson point processes, which we call dynamic spectral risk measures. We substantiate the link between dynamic spectral risk measures and their static counterparts by establishing a limit theorem for general path-functionals which shows that such dynamic risk measures arise as limits under vanishing time-step of iterated spectral risk measures driven by approximating lattice random walks. This involves a certain non-standard scaling of the corresponding spectral weight-measures that we identify explicitly. We also define a family of market-consistent dynamic spectral risk measures in the setting of a financial market. We identify the corresponding optimal hedging strategy and show that the negative of the value of such a risk measure is equal to the smallest price under a certain family of equivalent martingale measures. By varying the parameters of the risk measure we show that this price ranges between the subhedging price and the price under the minimal martingale measure .
In this paper we propose the notion of continuous-time dynamic spectral risk-measure (DSR). Adopting a Poisson random measure setting, we define this class of dynamic coherent risk-measures in terms of certain backward stochastic differential equations. By establishing a functional limit theorem, we show that DSRs may be considered to be (strongly) time-consistent continuous-time extensions of iterated spectral risk-measures, which are obtained by iterating a given spectral risk-measure (such as Expected Shortfall) along a given time-grid. Specifically, we demonstrate that any DSR arises in the limit of a sequence of such iterated spectral risk-measures driven by lattice-random walks, under suitable scaling and vanishing time- and spatial-mesh sizes. To illustrate its use in financial optimisation problems, we analyse dynamic portfolio optimisation problems under DSR, providing verification theorems for the optimal (equilibrium) strategies and the corresponding value functions in terms of associated (extended) HJB equations. In the case of a long-only investor we explicitly identify optimal portfolio allocation strategies .
R
explore a novel way to combine the dynamic notion of time-consistency with the static notion of quantile-based coherent risk measure or spectral risk measure, of which Expected Shortfall is a prime example. We introduce a new
propose the notion of continuous-time dynamic spectral risk-measure (DSR). Adopting a Poisson random measure setting, we define this
R
risk measures given
coherent risk-measures
R
a certain family of g-expectations driven by Wiener and Poisson point processes, which we call dynamic spectral risk measures. We substantiate the link between dynamic spectral risk measures and their static counterparts by establishing a limit theorem for general path-functionals which shows that such dynamic risk measures arise as limits under vanishing time-step
certain backward stochastic differential equations. By establishing a functional limit theorem, we show that DSRs may be considered to be (strongly) time-consistent continuous-time extensions
R
risk measures driven by approximating lattice random walks. This involves a certain non-standard scaling of the corresponding spectral weight-measures that we identify explicitly. We also define a family of market-consistent dynamic spectral risk measures in the setting of a financial market. We identify the corresponding optimal hedging strategy and show that the negative of the value of such a risk measure is equal to the smallest price under a certain family of equivalent martingale measures. By varying the parameters of the risk measure we show that this price ranges between the subhedging price and the price under the minimal martingale measure
risk-measures, which are obtained by iterating a given spectral risk-measure (such as Expected Shortfall) along a given time-grid. Specifically, we demonstrate that any DSR arises in the limit of a sequence of such iterated spectral risk-measures driven by lattice-random walks, under suitable scaling and vanishing time- and spatial-mesh sizes. To illustrate its use in financial optimisation problems, we analyse dynamic portfolio optimisation problems under DSR, providing verification theorems for the optimal (equilibrium) strategies and the corresponding value functions in terms of associated (extended) HJB equations. In the case of a long-only investor we explicitly identify optimal portfolio allocation strategies