On behavioral Arrow Pratt risk process with applications to risk pricing, stochastic cash flows, and risk control

We introduce a closed form behavioural stochastic Arrow-Pratt risk process, decomposed into discrete asymmetric risk seeking and risk averse components that run on different local times in ϵ-disks centered at risk free states. Additionally, we embed Arrow-Pratt (“AP”) risk measure in a simple dynamic system of discounted cash flows with constant volatility, and time varying drift. Signal extraction of Arrow-Pratt risk measure shows that it is highly nonlinear in constant volatility for cash flows. Robust identifying restrictions on the system solution confirm that even for small time periods constant volatility is not a measure of AP risk. By contrast, time-varying volatility measures aspects of embedded AP risk. Whereupon maximal AP risk measure is obtained from a convolution of input volatility and idiosyncratic shocks to the system. We provide four applications for our theory. First, we find that Engle, Ng and Rothschild (1990) Factor-ARCH model for risk premia is misspecified because the factor price of risk is time varying and unstable. Our theory predicts that a hyper-ARCH correction factor is required to remove the Factor-ARCH specification. Second, when applied to analysts beliefs about interest rates and volatility, we find that AP risk measure is a feedback control over stochastic cash flows. Whereupon increased risk aversion to negative shocks to earnings increases volatility. Third, we use an oft cited example of Benes, Shepp and Witsenhausen (1980) to characterize a controlled AP diffusion for a conservative investor who wants to minimize the AP risk process for an asset. Fourth, we recover stochastic differential utility functional from the AP risk process and show how it is functionally equivalent to Duffie and Epstein’s (1992) parametrization.behavioural Arrow-Pratt risk process; asymmetric risk decomposition; asset pricing; Markov process; local martingale; local time change

Risk-Averse Receding Horizon Motion Planning

This paper studies the problem of risk-averse receding horizon motion planning for agents with uncertain dynamics, in the presence of stochastic, dynamic obstacles. We propose a model predictive control (MPC) scheme that formulates the obstacle avoidance constraint using coherent risk measures. To handle disturbances, or process noise, in the state dynamics, the state constraints are tightened in a risk-aware manner to provide a disturbance feedback policy. We also propose a waypoint following algorithm that uses the proposed MPC scheme for discrete distributions and prove its risk-sensitive recursive feasibility while guaranteeing finite-time task completion. We further investigate some commonly used coherent risk metrics, namely, conditional value-at-risk (CVaR), entropic value-at-risk (EVaR), and g-entropic risk measures, and propose a tractable incorporation within MPC. We illustrate our framework via simulation studies.Comment: Submitted to Artificial Intelligence Journal, Special Issue on Risk-aware Autonomous Systems: Theory and Practice. arXiv admin note: text overlap with arXiv:2011.1121

RAROC-Based contingent claim valuation

University of Technology Sydney. Faculty of Business.The present dissertation investigates the valuation of a contingent claim based on the criterion RAROC, an abbreviation of Risk-Adjusted Return on Capital. RAROC is defined as the ratio of expected return to risk, and may therefore be regarded as a performance measure. RAROC-based pricing theory can indeed be considered as a subclass of the broader `good-deal' pricing theory, developed by Bernardo and Ledoit (2000) and Cochrane and Saá-Requejo (2000). By fixing some specific target value of RAROC, a RAROC-based good-deal price for a contingent claim is determined as follows: upon charging the counterparty with this price and using available funds, we are able to construct a hedging portfolio such that the maximum achievable RAROC of our hedged position meets the target RAROC. As a first step, we consider the standard Black-Scholes model, but allow only static hedging strategies. Assuming that the contingent claim in question is a call option, we examine the behavior of maximum value of RAROC as a function of initial call price, as well as the corresponding optimal static hedging strategy. In this analysis we consider two specifications for the risk component of RAROC, namely Value-at-Risk and Expected Shortfall. Subsequently, we allow continuous-time trading strategies, while remaining in the Black-Scholes framework. In this case we suppose that the initial price of the call option is limited to be below the Black-Scholes price. Perfect hedging is thus impossible, and the position must contain some residual risk. For ease of analysis, we restrict our attention to a specific class of hedging strategies and examine the maximum RAROC for each strategy in this class. In the interest of tractability, the version of RAROC adopted risk is measured simply as expected loss. While the previous approach only permits us to examine the constrained maximum RAROC over a specific class of hedging strategies, we would like to employ a more general method in order to study the global maximum RAROC over all hedging strategies. To do so, we introduce the notion of dynamic RAROC-based good-deal prices. In particular, with reference to the dynamic good-deal pricing theory of Becherer (2009), such prices are required to satisfy certain dynamic conditions, so that inconsistent decision-making between different times can be avoided. This task is accomplished by constructing prices that behave like time-consistent dynamic coherent risk measures. As soon as the construction process is finished, we set up a discrete time incomplete market, and demonstrate how to determine the dynamic RAROC-based good-deal price for a call option. Furthermore, by following Becherer (2009), we derive the dynamics of RAROC-based good-deal prices as solutions for discrete-time backward stochastic difference equations. Finally, we introduce RAROC-based good-deal hedging strategies, and examine their representation in terms of discrete-time backward stochastic difference equations

A Dual Method For Backward Stochastic Differential Equations with Application to Risk Valuation

We propose a numerical recipe for risk evaluation defined by a backward stochastic differential equation. Using dual representation of the risk measure, we convert the risk valuation to a stochastic control problem where the control is a certain Radon-Nikodym derivative process. By exploring the maximum principle, we show that a piecewise-constant dual control provides a good approximation on a short interval. A dynamic programming algorithm extends the approximation to a finite time horizon. Finally, we illustrate the application of the procedure to financial risk management in conjunction with nested simulation and on an multidimensional portfolio valuation problem

Computational Dynamic Market Risk Measures in Discrete Time Setting

Different approaches to defining dynamic market risk measures are available in the literature. Most are focused or derived from probability theory, economic behavior or dynamic programming. Here, we propose an approach to define and implement dynamic market risk measures based on recursion and state economy representation. The proposed approach is to be implementable and to inherit properties from static market risk measures.Comment: 16 pages, 3 figure

Dynamic risk measures

This paper gives an overview of the theory of dynamic convex risk measures for random variables in discrete time setting. We summarize robust representation results of conditional convex risk measures, and we characterize various time consistency properties of dynamic risk measures in terms of acceptance sets, penalty functions, and by supermartingale properties of risk processes and penalty functions.Comment: 30 page