Knightian Uncertainty, k-Ignorance, and Optimal Timing

We investigate within a continuous time setting how Knightian uncertainty characterized by k-ignorance affects the optimal timing policies of a risk-neutral and uncertainty averse investor in the case where the exercise payoff is monotonic. We prove that increased Knightian uncertainty unambiguously decreases the value of the optimal timing policy of an uncertainty averse investor. We also show that higher Knightian uncertainty accelerates timing by shrinking the continuation region whenever the termination payoff is independent of Knightian uncertainty. If this independence condition is not fulfilled, then our results indicate that higher Knightian uncertainty may decelerate optimal timing.Knightian uncertainty, k-ambiguity, optimal stopping, diffusions

"A Note on Learning under the Knightian Uncertainty"

In contrast to the traditional model of uncertainty, where the uncertainty is characterized by a single distribution function that a decision maker faces, the Knightian-uncertainty approach characterizes it as a set of distributions rather than a single one. Hence, learning in the context of Knightian uncertainty is characterized by an update process of the set of distributions after each of random sampling. This note presents two examples in which the Dempster-Shafer update rule, the one which attracts much attention since it seems intuitive, does not at all reduce the Knightian uncertainty (Example 1) and it actually increases the Knightian uncertainty (Example 2). Thus, what is a sensible update process is still an open question under the Knightian uncertainty.

"Irreversible Investment and Knightian Uncertainty"

When firms decide about irreversible investment, they may not have perfect confidence about their perceived probability measure describing future uncertainty. They may think other probability measures perturbed from the original one are also probable. Uncertainty characterized by not a single probability measure but a set of probability measures is called Knightian uncertainty. The effect of Knightian uncertainty on the value of irreversible investment opportunity is shown to be drastically different from that of the traditional uncertainty in the form of risk. Specifically, an increase in Knightian uncertainty decreases the value of investment opportunity while an increase in risk increases it.

Dark Clouds or Silver Linings? Knightian Uncertainty and Climate Change

This paper examines the impact of Knightian uncertainty upon optimal climate policy through the prism of a continuous-time real option modelling framework. We analytically determine optimal intertemporal climate policies under ambiguity. Additionally, numerical simulations are provided to illustrate the properties of the model. The results indicate that increasing Knightian uncertainty accelerates climate policy, i.e. policymakers become more reluctant to postpone the timing of climate policies into the future.climate change, Knightian uncertainty, ê ambiguity, real options

Investment Model Uncertainty and Fair Pricing

Modern investment theory takes it for granted that a Security Market Line (SML) is as certain as its "corresponding" Capital Market Line. (CML). However, it can be easily demonstrated that this is not the case. Knightian non-probabilistic, information gap uncertainty exists in the security markets, as the bivariate "Galton's Error" and its concomitant information gap proves (Journal of Banking & Finance, 23, 1999, 1793-1829). In fact, an SML graph needs (at least) two parallel horizontal beta axes, implying that a particular mean security return corresponds with a limited Knightian uncertainty range of betas, although it does correspond with only one market portfolio risk volatility. This implies that a security' risk premium is uncertain and that a Knightian uncertainty range of SMLs and of fair pricing exists. This paper both updates the empirical evidence and graphically traces the financial market consequences of this model uncertainty for modern investment theory. First, any investment knowledge about the securities risk remains uncertain. Investment valuations carry with them epistemological ("modeling") risk in addition to the Markowitz-Sharpe market risk. Second, since idiosyncratic, or firm-specific, risk is limited-uncertain, the real option value of a firm is also limited-uncertain This explains the simultaneous coexistence of different analyst valuations of investment projects, particular firms or industries, included a category "undecided." Third, we can now distinguish between "buy", "sell" and "hold" trading orders based on an empirically determined collection of SMLs, based this Knightian modeling risk. The coexistence of such simultaneous value signals for the same security is necessary for the existence of a market for that security! Without epistemological investment uncertainty, no ongoing markets for securities could exist. In the absence of transaction costs and other inefficiencies, Knightian uncertainty is the necessary energy for market trading, since it creates potential or perceived arbitrage (= trading) opportunities, but it is also necessary for investors to hold securities. Knightian uncertainty provides a possible reason why the SEC can't obtain consensus on what constitutes "fair pricing." The paper also shows that Malkiel's recommended CML-based investments are extremely conservative and non-robust.capital market line, security market line, beta, investments, decision-making, Knightian uncertainty, robustness, information-gap, Galton's Error, real option value

Equilibrium Asset Pricing with Time-Varying Pessimism

capital asset pricing;general equilibrium;uncertainty;financial risk;model misspecification;Knightian uncertainty;first order risk aversion

The Arbitrage Pricing Theorem with Incomplete Preferences

This paper proves existence of equilibrium and the arbitrage pricing theorem for an asset exchange economy, where the individual's preferences may be incomplete or intransitive. This extends existing results to a more general set of individual preferences. We also prove the arbitrage pricing theorem for a theory of choice under uncertainty by Bewley [1986]. These preferences model Knightian uncertainty by allowing for the possibility that preferences are incomplete.Incomplete Preferences, Equilibrium Existence, Arbitrage Pricing Theorem, Knightian Uncertainty

Risk and Uncertainty in Health Investment

Extending the Grossman [12] model of health capital into a stochastic one, we analyze how the presence of Knightian uncertainty about the efficacy of health care affects the optimal health investment behavior of individuals. Using Gilboa and Schmeidler's [11] model of maxmin expected utility (MMEU) with multiple priors, we show that an agent retains the initial level of health capital if the price of health care lies within a certain range. We also show that the no-investment range expands as the degree of Knightian uncertainty rises.Health Investment, Risk, Uncertainty