Financial Markets under Knightian Uncertainty
Mathematics
Final Report Abstract
The project investigated the consequences for Knightian uncertainty on financial markets. In a first paper, we establish the negative finding that Knightian uncertainty about volatility in continuous time destroys a famous equivalence result from equilibrium theory; when asset prices are driven by Brownian motion and as many linearly independent assets as Brownian motions are traded in the financial market, then financial market equilibria are efficient and the first welfare theorem holds true. With Knightian uncertainty about volatility, this equivalence of static Arrow-Debreu and dynamic Radner equilibria breaks down. The second paper considers also general equilibrium economies with uncertainty about volatility. It highlights some additional important aspects that are different under Knightian uncertainty. For example, positive payoffs are for free on events outside the domain of the representing equilibrium pricing measure. Moreover, when aggregate risk is present and aggregate ambiguity is absent, the insurance properties of optimal allocations depend on the notion of ambiguity-free payoffs. The result is remarkable as it shows a new type of market incompleteness under volatility uncertainty. These two papers led us to the question whether the linear prices in classic Walrasian and Arrow-Debreu general equilibrium theory are appropriate for markets under uncertainty. We were thus led to introduce nonlinear notions of prices. In a third paper, we study economies with Knightian uncertainty about state prices.We introduce an equilibrium concept with sublinear prices and prove that equilibria exist under weak conditions. In general, such equilibria lead to inefficient allocations; they coincide with Arrow–Debreu equilibria if and only if the values of net trades are ambiguity-free in mean. In economies without aggregate uncertainty, inefficiencies are generic. Equilibrium allocations under price uncertainty are efficient in a constrained sense that we call uncertainty–neutral efficient. Arrow–Debreu equilibria turn out to be non-robust with respect to the introduction of Knightian uncertainty.
Publications
- (2020) Dynamically consistent alpha‐maxmin expected utility. Mathematical Finance 30 (3) 1073–1102
Beissner, Patrick; Lin, Qian; Riedel, Frank
(See online at https://doi.org/10.1111/mafi.12232) - Equilibrium prices and trade under ambiguous volatility, Economic Theory, 64 (2017), pp. 213-238
P. Beissner
(See online at https://doi.org/10.1007/s00199-016-0979-y) - Non-Implementability of Arrow-Debreu Equilibria by Continuous Trading under Knightian Uncertainty, Finance and Stochastics 22 (2018), pp. 603-620
P. Beissner, F. Riedel
(See online at https://doi.org/10.1007/s00780-018-0362-x) - Equilibria under Knightian Price Uncertainty, Econometrica 87 (2019), pp. 37–64
P. Beissner, F. Riedel
(See online at https://doi.org/10.3982/ECTA14934)
