Backward stochastic difference equations for dynamic convex risk measures on a binomial tree

Elliott, Robert J.; Siu, Tak Kuen; Cohen, Samuel N.

doi:10.1239/jap/1445543845

Cited by 12 publications

(7 citation statements)

References 32 publications

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“…The class of iterated spectral risk-measures defined as such contains in particular the Iterated Tail Conditional Expectation proposed in [29] and is closely related to the Dynamic Weighted V@R that is defined in [13] for adapted processes via its robust representation. As already noted in the proof of Proposition 3.5, in the static case such a representation was derived in [9] for bounded random variables; see also [27,Theorems 4.79 and 4.94] , and see [12] for the extension to the set of measurable random variables (we refer to [23] for families of dynamic risk measure defined via stochastic distortion probabilties in a binomial tree setting; see [14] for a general theory of finite state BSDEs). We show next that iterated spectral risk measures are discrete-time time-consistent dynamic coherent risk measures and identify the driver function of the associated BS∆E.…”

Section: Conditional and Iterated Choquet Integralsmentioning

confidence: 97%

On dynamic spectral risk measures, a limit theorem and optimal portfolio allocation

2017

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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 temporal and spatial mesh sizes. To illustrate its use in financial optimisation problems, we analyse a dynamic portfolio optimisation problem under a DSR.

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Section: Conditional and Iterated Choquet Integralsmentioning

confidence: 97%

On dynamic spectral risk measures, a limit theorem and optimal portfolio allocation

2017

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“…The ideas of the Malliavin calculus and BSDEs will now be described in the binomial framework. Definition and Lemma were presented in the work of Elliott et al…”

Section: Binomial Malliavin Calculusmentioning

confidence: 99%

Malliavin calculus in a binomial framework

Cohen

Elliott

Siu

2018

Appl Stoch Models Bus & Ind

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The binomial model is a standard framework used to introduce risk neutral pricing of financial assets. Martingale representation, backward stochastic differential equations, and the Malliavin calculus are difficult concepts in a continuous-time setting. This paper presents these ideas in the simple, discrete-time binomial model. KEYWORDS backward stochastic differential equation, binomial model, Malliavin calculus 774

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“…As it was shown in [RG06] there is a direct connection between convex risk measures and nonlinear expectations, and consequently there exists a direct link between convex risk measures and BSDEs. These connections were further studied in [CE11b], [ESC15], and [Sta09] for the case of discrete time setups, thus establishing a relationship between BS∆Es and DRM for terminal cashflows (random variables).…”

Section: Dynamic Convex Risk Measures and Dynamic Acceptability Indic...mentioning

confidence: 99%

Dynamic Conic Finance via Backward Stochastic Difference Equations

Bielecki¹,

Cialenco²,

Chen³

2015

SIAM J. Finan. Math.

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We present an arbitrage free theoretical framework for modeling bid and ask prices of dividend paying securities in a discrete time setup using theory of dynamic acceptability indices. In the first part of the paper we develop the theory of dynamic subscale invariant performance measures, on a general probability space, and discrete time setup. We prove a representation theorem of such measures in terms of a family of dynamic convex risk measures, and provide a representation of dynamic risk measures in terms of g-expectations, and solutions of BS∆Es with convex drivers. We study the existence and uniqueness of the solutions, and derive a comparison theorem for corresponding BS∆Es. In the second part of the paper we discuss a market model for dividend paying securities by introducing the pricing operators that are defined in terms of dynamic acceptability indices, and find various properties of these operators. Using these pricing operators, we define the bid and ask prices for the underlying securities and then for derivatives in this market. We show that the obtained market model is arbitrage free, and we also prove a series of properties of these prices.

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Backward stochastic difference equations for dynamic convex risk measures on a binomial tree

Cited by 12 publications

References 32 publications

On dynamic spectral risk measures, a limit theorem and optimal portfolio allocation

On dynamic spectral risk measures, a limit theorem and optimal portfolio allocation

Malliavin calculus in a binomial framework

Dynamic Conic Finance via Backward Stochastic Difference Equations

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