Smooth Ambiguity Preferences Research Articles

Equilibrium Portfolio Selection for Smooth Ambiguity Preferences

This paper investigates the equilibrium portfolio selection for smooth ambiguity preferences in a continuous-time market. The investor is uncertain about the risky asset’s drift term and updates the subjective belief according to the Bayesian rule. A verification theorem is established, and an equilibrium strategy can be decomposed into a myopic demand and two hedging demands. When the prior is Gaussian, we provide an equilibrium solution in closed form. Moreover, a puzzle in the numerical results is interpreted via an alternative representation of the smooth ambiguity preferences. Funding: This work was supported by the National Key R&D Program of China [Grant 2020YFA0712700], the National Natural Science Foundation of China [Grants 11871036, 11901574, 12071146, 12271290, and 12371477], and the MOE Project of Key Research Institute of Humanities and Social Sciences [Grant 22JJD910003].

Optimal nonlinear pricing by a monopoly with smooth ambiguity preferences

This paper examines the optimal nonlinear pricing by a monopolist who has smooth ambiguity preferences and sells a single good to two types of buyers with high and low valuation. Ambiguity arises from the monopolist’s uncertainty about which of the subjective beliefs govern the unobserved types of buyers. We show that the prevalence of ambiguity aversion distorts the rent extraction-efficiency tradeoff under asymmetric information in the standard model. Specifically, the ambiguity-averse monopolist acts in a pessimistic manner with the perception that buyers are less likely to have high valuation than the objective beliefs. Compared with the standard model, low valuation buyers consume a less downward distorted quantity and high valuation buyers enjoy a greater information rent. The unit prices paid by all buyers are reduced and the ambiguity-averse monopolist earns a smaller expected profit.

Smooth ambiguity preferences and asset prices with a jump-diffusion process

The present study extends the continuous-time smooth ambiguity preferences model by introducing a Poisson jump component into the agent's consumption process. Under this setting, we examine the effect of ambiguity with respect to both Brownian motion and a Poisson jump process on asset prices. Using reasonable values for preferences parameters, our model replicates historical moments of the U.S. asset returns, including the equity premium, equity volatility, and risk-free rate. Our model also generates substantial equity variance premium and a mildly downward sloping volatility curve of equity options that are consistent with the U.S. data. In addition, our model captures the predictive power of the variance premium with respect to equity returns.

Model and Predictive Uncertainty: A Foundation for Smooth Ambiguity Preferences

Smooth ambiguity preferences (Klibanoff, Marinacci, and Mukerji (2005)) describe a decision maker who evaluates each act f according to the twofold expectation V ( f ) = ∫ P ϕ ( ∫ Ω u ( f ) d p ) d μ ( p ) defined by a utility function u, an ambiguity index ϕ, and a belief μ over a set P of probabilities. We provide an axiomatic foundation for the representation, taking as a primitive a preference over Anscombe–Aumann acts. We study a special case where P is a subjective statistical model that is point identified, that is, the decision maker believes that the true law p ∈ P can be recovered empirically. Our main axiom is a joint weakening of Savage's sure‐thing principle and Anscombe–Aumann's mixture independence. In addition, we show that the parameters of the representation can be uniquely recovered from preferences, thereby making operational the separation between ambiguity attitude and perception, a hallmark feature of the smooth ambiguity representation.

If It Is Surely Better, Do It More? Implications for Preferences Under Ambiguity

Consider a canonical problem in choice under uncertainty: choosing from a convex feasible set consisting of all (Anscombe–Aumann) mixtures of two acts f and g, [Formula: see text]. We propose a preference condition, monotonicity in optimal mixtures, which says that surely improving the act f (in the sense of weak dominance) makes the optimal weight(s) on f weakly higher. We use a stylized model of a sales agent reacting to incentives to illustrate the tight connection between monotonicity in optimal mixtures and a monotone comparative static of interest in applications. We then explore more generally the relation between this condition and preferences exhibiting ambiguity-sensitive behavior as in the classic Ellsberg paradoxes. We find that monotonicity in optimal mixtures and ambiguity aversion (even only local to an event) are incompatible for a large and popular class of ambiguity-sensitive preferences (the c-linearly biseparable class. This implies, for example, that maxmin expected utility preferences are consistent with monotonicity in optimal mixtures if and only if they are subjective expected utility preferences. This incompatibility is not between monotonicity in optimal mixtures and ambiguity aversion per se. For example, we show that smooth ambiguity preferences can satisfy both properties as long as they are not too ambiguity averse. Our most general result, applying to an extremely broad universe of preferences, shows a sense in which monotonicity in optimal mixtures places upper bounds on the intensity of ambiguity-averse behavior. This paper was accepted by Manel Baucells, decision analysis.

Robust Decision Theory and Econometrics

This review uses the empirical analysis of portfolio choice to illustrate econometric issues that arise in decision problems. Subjective expected utility (SEU) can provide normative guidance to an investor making a portfolio choice. The investor, however, may have doubts on the specification of the distribution and may seek a decision theory that is less sensitive to the specification. I consider three such theories: maxmin expected utility, variational preferences (including multiplier and divergence preferences and the associated constraint preferences), and smooth ambiguity preferences. I use a simple two-period model to illustrate their application. Normative empirical work on portfolio choice is mainly in the SEU framework, and bringing in ideas from robust decision theory may be fruitful.

Ambiguity and the Home Currency Bias

This paper addresses the question of optimal currency exposure for a risk-and-ambiguity-avers international investor. A robust mean-variance model with smooth ambiguity preferences is used to derive the optimal currency exposure. In the theoretical part, we show that the sample-efficient currency demand can be calculated as the solution to a generalized ridge regression. Through the lens of these results, we demonstrate that our ambiguity-based model offers a new explanation of the home currency bias. The investor's dislike for model uncertainty induces a disproportionately high currency hedging demand. The empirical analysis of currency overlay strategies employs the foreign exchange, equity, and bond returns over the period from 1999 to 2018. Our out-of-sample back-tests illustrate that accounting for ambiguity enhances the stability of estimated optimal currency exposures and significantly improves the portfolio performance net of transaction costs.

Precautionary Saving with Generalized Two-Period Smooth Ambiguity Preferences

In this paper, we study two classical saving-insurance problems for the intertemporal version developed by Hayashi and Miao (2011) of the smooth ambiguity model of Klibanoff et al. (2005). These models put risk, ambiguity and time preferences together in a Kreps-Porteus aggregator, and disentangle the effects among risk, ambiguity and time preferences. We show that the concepts and techniques developed by Topkis (1998) and others can be used to obtain a set of simple and intuitive sufficient conditions such that risk, ambiguity and time preferences together always raise the demand for saving and self-insurance.

Ambiguity Aversion and the Variance Premium

This paper offers an ambiguity-based interpretation of the variance premium — the difference between risk-neutral and objective expectations of market return variance — as a compounding effect of both belief distortion and variance differential regarding the uncertain economic regimes. Our calibrated model can match the variance premium, the equity premium, and the risk-free rate in the data. We find that about 97% of the mean–variance premium can be attributed to ambiguity aversion. A three-way separation among ambiguity aversion, risk aversion, and intertemporal substitution, permitted by the smooth ambiguity preferences, plays a key role in our model’s quantitative performance.

Does Smooth Ambiguity Matter for Asset Pricing?

We use the Bayesian method introduced by Gallant and McCulloch (2009) to estimate consumption-based asset pricing models featuring smooth ambiguity preferences. We rely on semi-nonparametric estimation of a flexible auxiliary model in our structural estimation. Based on the market and aggregate consumption data, our estimation provides statistical support for asset pricing models with smooth ambiguity. Statistical model comparison shows that models with ambiguity, learning, and time-varying volatility are preferred to the long-run risk model. We also analyze asset pricing implications of the estimated models. Received April 12, 2016; editorial decision September 11, 2018 by Editor Stijn Van Nieuwerburgh. Authors have furnished an Internet Appendix, which is available on the Oxford University Press Web site next to the link to the final published paper online

Asset pricing with time varying pessimism and rare disasters

We incorporate time-varying consumption volatility in the representative-agent asset pricing model with generalized recursive smooth ambiguity preferences developed by Ju and Miao (2012). We calibrate the model to data on consumption and asset returns since the Great Depression period. Uncertainty aversion amplifies the perceived probability of the disastrous state coupled with high consumption volatility. We find that the model with time-varying volatility generates a high equity risk premium. When we impose the condition that no consumption disasters ever realized in simulated samples, the model with time-varying volatility can reproduce predictability of returns and non-predictability of consumption growth simultaneously, which are consistent with empirical findings.

Continuous-time smooth ambiguity preferences

This study extends the smooth ambiguity preferences model proposed by Klibanoff et al. (2005) to a continuous-time dynamic setting. It is known that the original smooth ambiguity preferences converge to the subjective expected utility as the time interval shortens so that decision makers do not exhibit any ambiguity-sensitive behavior in the continuous-time limit. Accordingly, this study proposes an alternative model of these preferences that interchanges the role of the second-order utility function with that of the second-order probability to prevent the smooth ambiguity attitude of decision maker from evaporating in the continuous-time limit. By utilizing the utility convergence results established by Kraft and Seifried (2014), our model is eventually represented by the stochastic differential utility with distorted beliefs so that most existing techniques in economics and financial studies can be made applicable together with these distorted beliefs. We give an asset-pricing example to demonstrate the applicability of our model.

Ambiguity preferences, risk taking and the banking firm

This paper examines the risk taking behavior of a banking firm facing ambiguity and possessing smooth ambiguity preferences. Ambiguity is modeled by a second-order probability distribution that captures the bank’s uncertainty about which of the subjective beliefs govern the return on its loans. Ambiguity aversion is modeled by a concave transformation of the (first-order) expected utility of profit conditional on each plausible subjective distribution of the random loan return. Within this framework, we show that the bank finds it less attractive to take risk in the presence than in the absence of ambiguity. This result extends to the case of greater ambiguity aversion. Given that the bank’s smooth ambiguity preferences exhibit non-increasing absolute ambiguity aversion, imposing a more stringent capital requirement to the bank has the desired effect that limits the bank’s incentive to take on excessive risk.

Does Smooth Ambiguity Matter for Asset Pricing?

We use the Bayesian method introduced by Gallant and McCulloch (2009) to estimate consumption-based asset pricing models featuring smooth ambiguity preferences. We rely on semi-nonparametric estimation of a flexible auxiliary model in our structural estimation. Based on the market and aggregate consumption data, our estimation provides statistical support for asset pricing models with smooth ambiguity. Statistical model comparison shows that models with ambiguity, learning and time-varying volatility are preferred to the long-run risk model. We analyze asset pricing implications of the estimated models.

The Identification of Attitudes towards Ambiguity and Risk from Asset Demand

Individuals behave differently when they know the objective probability of events and when they do not. The smooth ambiguity model accommodates both ambiguity (uncertainty) and risk. For an incomplete, competitive asset market, we develop a revealed preference test for asset demand to be consistent with the maximization of smooth ambiguity preferences; and we show that ambiguity preferences constructed from finite observations converge to underlying ambiguity preferences as observations become dense. Subsequently, we give sufficient conditions for the asset demand generated by smooth ambiguity preferences to identify the ambiguity and risk indices as well as the ambiguity probability measure. We do not require ambiguity beliefs to be observable: in a generalized specification, they may not even be defined. An ambiguity free asset plays an important role for identification.

Characterizations of Smooth Ambiguity Based on Continuous and Discrete Data

In the Anscombe-Aumann setup, we provide conditions for a collection of observations to be consistent with a well-known class of smooth ambiguity preferences (Klibanoff P, Marinacci M, Mukerji S (2005) A smooth model of decision making under ambiguity. Econometrica 73(6):1849–1892.). Each observation is assumed to take the form of an equivalence between an uncertain act and a certain outcome. We provide three results that describe these conditions for data sets of different cardinality. Our findings uncover surprising links between the smooth ambiguity model and classic mathematical results in complex and functional analysis.

Does Smooth Ambiguity Matter for Asset Pricing?

We use the Bayesian method introduced by Gallant and McCulloch (2009) to estimate consumption-based asset pricing models featuring smooth ambiguity preferences. We rely on semi-nonparametric estimation of a flexible auxiliary model in our structural estimation. Based on the market and aggregate consumption data, our estimation provides statistical support for asset pricing models with smooth ambiguity. Statistical model comparison shows that models with ambiguity, learning and time-varying volatility are preferred to the long-run risk model. We analyze asset pricing implications of the estimated models.

HEDGING AND THE COMPETITIVE FIRM UNDER AMBIGUOUS PRICE AND BACKGROUND RISK

ABSTRACTThis paper examines the optimal production and hedging decisions of the competitive firm that possesses smooth ambiguity preferences and faces ambiguous price and background risk. The separation theorem holds in that the firm's optimal output level depends neither on the firm's attitude towards ambiguity nor on the incident to the underlying ambiguity. We derive necessary and sufficient conditions under which the full‐hedging theorem holds and thus options are not used. When these conditions are violated, we show that the firm optimally uses options for hedging purposes if ambiguity is introduced to the price and background risk by means of mean‐preserving spreads. We as such show that options play a role as a hedging instrument over and above that of futures.

Cross‐Hedging Ambiguous Exchange Rate Risk

AbstractThis paper examines the behavior of an exporting firm that sells its output to two foreign countries, only one of which has futures and options available for its currency. The firm possesses smooth ambiguity preferences and faces multiple sources of ambiguous exchange rate risk. We show that the separation theorem fails to hold in that the firm's production and export decisions depend on the firm's attitude toward ambiguity and on the incident to the underlying ambiguity. Given that the random spot exchange rates are first‐order independent with respect to each plausible subjective distribution, we derive necessary and sufficient conditions under which the full‐hedging theorem applies to the firm's cross‐hedging decisions. When these conditions are violated, we show that the firm includes options in its optimal hedge position. This paper as such offers a rationale for the hedging role of options under smooth ambiguity preferences and cross‐hedging of ambiguous exchange rate risk. © 2016 Wiley Periodicals, Inc. Jrl Fut Mark 37:132–147, 2017

A Continuous-Time Asset Pricing Model with Smooth Ambiguity Preferences

This study extends the smooth ambiguity preferences model proposed by Klibanoff et al. (2005, 2009) to a continuous-time dynamic setting. It is known that these preferences converge to the subjective expected utility as the time interval shortens so that decision makers do not exhibit any ambiguity-sensitive behavior in the continuous-time limit. Accordingly, this study proposes an alternative model of decision making that applies Yaari’s (1987) dual theory to the original preferences and interchanges the role of the second-order utility function with that of the second-order probability. We then formulate a recursive utility of smooth ambiguity-sensitive decision makers in continuous-time. Our model is represented by the stochastic differential utility with distorted beliefs so that most existing techniques in financial studies can be made applicable together with these distorted beliefs. We give an asset-pricing example to demonstrate the applicability of our model.