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.
Cadogan, Godfrey, "On behavioral Arrow Pratt risk process with applications to risk pricing, stochastic cash flows, and risk control" (2009). The Institute for Innovation and Technology Management. Paper 6.