Mitja Stadje

Robust Portfolio Choice and Indifference Valuation

We solve, theoretically and numerically, the problems of optimal portfolio choice and indifference valuation in a general continuous-time setting. The setting features (i) ambiguity and time-consistent ambiguity-averse preferences, (ii) discontinuities in the asset price processes, with a general and possibly infinite activity jump part next to a continuous diffusion part, and (iii) general and possibly nonconvex trading constraints. We characterize our solutions as solutions to backward stochastic differential equations (BSDEs). Generalizing Kobylanskis result for quadratic BSDEs to an infinite activity jump setting, we prove existence and uniqueness of the solution to a general class of BSDEs, encompassing the solutions to our portfolio choice and valuation problems as special cases. We provide an explicit decomposition of the excess return on an asset into a risk premium and an ambiguity premium, and a further decomposition into a piece stemming from the diffusion part and a piece stemming from the jump part. We further compute our solutions in a few examples by numerically solving the corresponding BSDEs using regression techniques.

BS

We provide existence results and comparison principles for solutions of backward stochastic difference equations (BSΔEs) and then prove convergence of these to solutions of backward stochastic differential equations (BSDEs) when the mesh size of the time-discretizaton goes to zero. The BSΔEs and BSDEs are governed by drivers fN(t,ω,y,z) and f(t,ω,y,z), respectively. The new feature of this paper is that they may be non-Lipschitz in z. For the convergence results it is assumed that the BSΔEs are based on d-dimensional random walks WN approximating the d-dimensional Brownian motion W underlying the BSDE and that fN converges to f. Conditions are given under which for any bounded terminal condition ξ for the BSDE, there exist bounded terminal conditions ξN for the sequence of BSΔEs converging to ξ, such that the corresponding solutions converge to the solution of the limiting BSDE. An important special case is when fN and f are convex in z. We show that in this situation, the solutions of the BSΔEs converge to the solution of the BSDE for every uniformly bounded sequence ξN converging to ξ. As a consequence, one obtains that the BSDE is robust in the sense that if (WN,ξN) is close to (W,ξ) in distribution, then the solution of the Nth BSΔE is close to the solution of the BSDE in distribution too.

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We introduce two subclasses of convex measures of risk, referred to as entropy coherent and entropy convex measures of risk. Entropy coherent and entropy convex measures of risk are special cases of φ-coherent and φ-convex measures of risk. Contrary to the classical use of coherent and convex measures of risk, which for a given probabilistic model entails evaluating a financial position by considering its expected loss, φ-coherent and φ-convex measures of risk evaluate a financial position under a given probabilistic model by considering its normalized expected φ-loss. We prove that i entropy coherent and entropy convex measures of risk are obtained by requiring φ-coherent and φ-convex measures of risk to be translation invariant; ii convex, entropy convex, and entropy coherent measures of risk emerge as certainty equivalents under variational, homothetic, and multiple priors preferences upon requiring the certainty equivalents to be translation invariant; and iii φ-convex measures of risk are certainty equivalents under variational and homothetic preferences if and only if they are convex and entropy convex measures of risk. In addition, we study the properties of entropy coherent and entropy convex measures of risk, derive their dual conjugate function, and characterize entropy coherent and entropy convex measures of risk in terms of properties of the corresponding acceptance sets.

Es and BSDEs with non-Lipschitz drivers: Comparison, convergence and robustness

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 class of dynamic risk measures in terms of a certain family of g-expectations driven by Wiener and Poisson point processes. In analogy with the static case, we show that these risk measures, which we label dynamic spectral risk measures, are locally law-invariant and additive on the set of pathwise increasing random variables. 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.

Entropy Coherent and Entropy Convex Measures of Risk

This paper studies the optimal stopping problem in the presence of model uncertainty (ambiguity). We develop a numerically implementable method to solve this problem in a general setting, allowing for general time-consistent ambiguity-averse preferences and general payoff processes driven by jump diffusions. Our method consists of three steps. First, we construct a suitable Doob martingale associated with the solution to the optimal stopping problem using backward stochastic calculus. Second, we employ this martingale to construct an approximated upper bound to the solution using duality. Third, we introduce backward-forward simulation to obtain a genuine upper bound to the solution, which converges to the true solution asymptotically. We also provide asymptotically optimal exercise rules. We analyze the limiting behavior and convergence properties of our method. We illustrate the generality and applicability of our method and the potentially significant impact of ambiguity to optimal stopping in a few examples.

On Dynamic Spectral Risk Measures and a Limit Theorem

We study a utility maximization problem under multiple Value-at-Risk (VaR)-type constraints. The optimization framework is particularly important for financial institutions which have to follow short-time VaR-type regulations under some realistic regulatory frameworks like Solvency II, but need to serve long-term liabilities. Deriving closed-form solutions, we show that risk management using multiple VaR constraints is more useful for loss prevention at intertemporal time instances compared with the well-known result of the one-VaR problem in Basak and Shapiro (Rev Financ Stud 14:371–405, 2001), confirming the numerical analysis of Shi and Werker (J Bank Finance 36(12):3227–3238, 2012). In addition, the multiple-VaR solution at maturity on average dominates the one-VaR solution in a wide range of intermediate market scenarios, but performs worse in good and very bad market scenarios. The range of these very bad market scenarios is however rather limited. Finally, we show that it is preferable to reach a fixed terminal state through insured intertemporal states rather than through extreme up and down movements, showing that a multiple-VaR framework induces a preference for less volatility.

Optimal Stopping Under Uncertainty in Drift and Jump Intensity

We model a nonlinear price curve quoted in a market as the utility indifference curve of a representative liquidity supplier. As the utility function, we adopt a g

Risk management with multiple VaR constraints

g

Perfect hedging under endogenous permanent market impacts

-expectation. In contrast to the standard framework of financial engineering, a trader is no longer a price taker as any trade has a permanent market impact via an effect on the supplier’s inventory. The P&L of a trading strategy is written as a nonlinear stochastic integral. Under this market impact model, we introduce a completeness condition under which any derivative can be perfectly replicated by a dynamic trading strategy. In the special case of a Markovian setting, the corresponding pricing and hedging can be done by solving a semilinear PDE.

On dynamic deviation measures and continuous-time portfolio optimization

In this paper we propose the notion of dynamic deviation measure, as a dynamic time-consistent extension of the (static) notion of deviation measure. To achieve time-consistency we require that a dynamic deviation measures satisfies a generalised conditional variance formula. We show that, under a domination condition, dynamic deviation measures are characterised as the solutions to a certain class of backward SDEs. We establish for any dynamic deviation measure an integral representation, and derive a dual characterisation result in terms of additively