Convergence of optimal expected utility for a sequence of discrete‐time markets

We examine Kreps' conjecture that optimal expected utility in the classic Black–Scholes–Merton (BSM) economy is the limit of optimal expected utility for a sequence of discrete‐time economies that “approach” the BSM economy in a natural sense: The nth discrete‐time economy is generated by a scaled n‐step random walk, based on an unscaled random variable ζ with mean 0, variance 1, and bounded support. We confirm Kreps' conjecture if the consumer's utility function U has asymptotic elasticity strictly less than one, and we provide a counterexample to the conjecture for a utility function U with asymptotic elasticity equal to 1, for ζ such that .     

Convex duality and Orlicz spaces in expected utility maximization

In this paper, we report further progress toward a complete theory of state‐independent expected utility maximization with semimartingale price processes for arbitrary utility function. Without any technical assumptions, we establish a surprising Fenchel duality result on conjugate Orlicz spaces, offering a new economic insight into the nature of primal optima and providing a fresh perspective on the classical papers of Kramkov and Schachermayer. The analysis points to an intriguing interplay between no‐arbitrage conditions and standard convex optimization and motivates the study of the fundamental theorem of asset pricing for Orlicz tame strategies.     

On the optimal portfolio for the exponential utility maximization: remarks to the six-author paper

This note contains ramifications of results of Delbaen et al. (2002). Assuming that the price process is locally bounded and admits an equivalent local martingale measure with finite entropy, we show, without further assumption, that in the case of exponential utility the optimal portfolio process is a martingale with respect to each local martingale measure with finite entropy. Moreover, the optimal value always can be attained on a sequence of uniformly bounded portfolios.     

Strict local martingales and optimal investment in a Black–Scholes model with a bubble

There are two major streams of literature on the modeling of financial bubbles: the strict local martingale framework and the Johansen–Ledoit–Sornette (JLS) financial bubble model. Based on a class of models that embeds the JLS model and can exhibit strict local martingale behavior, we clarify the connection between these previously disconnected approaches. While the original JLS model is never a strict local martingale, there are relaxations that can be strict local martingales and that preserve the key assumption of a log‐periodic power law for the hazard rate of the time of the crash. We then study the optimal investment problem for an investor with constant relative risk aversion in this model. We show that for positive instantaneous expected returns, investors with relative risk aversion above one always ride the bubble.     

Dynamically consistent alpha‐maxmin expected utility

The alpha‐maxmin model is a prominent example of preferences under Knightian uncertainty as it allows to distinguish ambiguity and ambiguity attitude. These preferences are dynamically inconsistent for nontrivial versions of alpha. In this paper, we derive a recursive, dynamically consistent version of the alpha‐maxmin model. In the continuous‐time limit, the resulting dynamic utility function can be represented as a convex mixture between worst and best case, but now at the local, infinitesimal level. We study the properties of the utility function and provide an Arrow–Pratt approximation of the static and dynamic certainty equivalent. We then derive a consumption‐based capital asset pricing formula and study the implications for derivative valuation under indifference pricing.     

Superreplication with proportional transaction cost under model uncertainty

We consider a discrete‐time financial market with proportional transaction cost under model uncertainty, and study a superreplication problem. We recover the duality results that are well known in the classical dominated context. Our key argument consists in using a randomization technique together with the minimax theorem to convert the initial problem to a frictionless problem on an enlarged space. This allows us to appeal to the techniques and results of Bouchard and Nutz to obtain the duality result.     

Optimal insurance under rank‐dependent utility and incentive compatibility

Bernard, He, Yan, and Zhou (Mathematical Finance, 25(1), 154–186) studied an optimal insurance design problem where an individual's preference is of the rank‐dependent utility (RDU) type, and show that in general an optimal contract covers both large and small losses. However, their results suffer from the unrealistic assumption that the random loss has no atom, as well as a problem of moral hazard that provides incentives for the insured to falsely report the actual loss. This paper addresses these setbacks by removing the nonatomic assumption, and by exogenously imposing the “incentive compatibility” constraint that both indemnity function and insured's retention function are increasing with respect to the loss. We characterize the optimal solutions via calculus of variations, and then apply the result to obtain explicitly expressed contracts for problems with Yaari's dual criterion and general RDU. Finally, we use numerical examples to compare the results between ours and Bernard et al.     

Convergence of utility indifference prices to the superreplication price in a multiple‐priors framework

This paper formulates a utility indifference pricing model for investors trading in a discrete time financial market under nondominated model uncertainty. Investor preferences are described by possibly random utility functions defined on the positive axis. We prove that when the investors's absolute risk aversion tends to infinity, the multiple‐priors utility indifference prices of a contingent claim converge to its multiple‐priors superreplication price. We also revisit the notion of certainty equivalent for multiple‐priors and establish its relation with risk aversion.     

Maximum utility portfolio construction in the forward freight agreement markets: Evidence from a multivariate skewed t copula

We form portfolios consisting of diverse quarterly forward freight agreement (FFA) contracts to maximize the market participant's expected utility. The empirical findings indicate that individual FFA returns display clear autocorrelation, seasonality, fat tail, and heteroscedasticity. The multivariate positively skewed t copula is suggested for constructing maximum utility FFA portfolios, implying that the constituent FFA returns exhibit higher correlations when they rise together. The out-of-sample trading strategy performance metrics and various robustness checks further indicate that the aforementioned copula performs best and robustly for all portfolios. These findings provide profound methodological and managerial implications for market participants to improve risk management.     

On controlling chaos in a discrete‐time Walrasian tâtonnement process

The simple pure exchange model with two individuals and two goods introduced by Day and Pianigiani in 1991, later extensively analyzed by Day and taken up again by Mukherji, is discussed and extended with the purpose of showing that chaos in a discrete tâtonnement process of this kind can be controlled if the auctioneer uses a smooth, non‐linear formulation of the price evolution process such that the price adjustment is a sigmoid‐shaped function of the excess demand, with given lower and upper limits. In particular, we show that, given the price adjustment speed and the excess demand function, the auctioneer can (a) stabilize the dynamics, (b) reduce the complexity of the attractor and (c) increase the economic significance of the adjustment process by simply acting on the lower and/or upper limits that constrain price dynamics.     

Optimal equilibria for time‐inconsistent stopping problems in continuous time

For an infinite‐horizon continuous‐time optimal stopping problem under nonexponential discounting, we look for an optimal equilibrium, which generates larger values than any other equilibrium does on the entire state space. When the discount function is log subadditive and the state process is one‐dimensional, an optimal equilibrium is constructed in a specific form, under appropriate regularity and integrability conditions. Although there may exist other optimal equilibria, we show that they can differ from the constructed one in very limited ways. This leads to a sufficient condition for the uniqueness of optimal equilibria, up to some closedness condition. To illustrate our theoretic results, a comprehensive analysis is carried out for three specific stopping problems, concerning asset liquidation and real options valuation. For each one of them, an optimal equilibrium is characterized through an explicit formula.     

DYNAMIC PORTFOLIO OPTIMIZATION WITH A DEFAULTABLE SECURITY AND REGIME‐SWITCHING

We consider a portfolio optimization problem in a defaultable market with finitely‐many economical regimes, where the investor can dynamically allocate her wealth among a defaultable bond, a stock, and a money market account. The market coefficients are assumed to depend on the market regime in place, which is modeled by a finite state continuous time Markov process. By separating the utility maximization problem into a predefault and postdefault component, we deduce two coupled Hamilton–Jacobi–Bellman equations for the post‐ and predefault optimal value functions, and show a novel verification theorem for their solutions. We obtain explicit constructions of value functions and investment strategies for investors with logarithmic and Constant Relative Risk Aversion utilities, and provide a precise characterization of the directionality of the bond investment strategies in terms of corporate returns, forward rates, and expected recovery at default. We illustrate the dependence of the optimal strategies on time, losses given default, and risk aversion level of the investor through a detailed economic and numerical analysis.     

Small‐time,large‐time,and asymptotics for the Rough Heston model

We characterize the behavior of the Rough Heston model introduced by Jaisson and Rosenbaum (2016, Ann. Appl. Probab., 26, 2860–2882) in the small‐time, large‐time, and (i.e., ) limits. We show that the short‐maturity smile scales in qualitatively the same way as a general rough stochastic volatility model , and the rate function is equal to the Fenchel–Legendre transform of a simple transformation of the solution to the same Volterra integral equation (VIE) that appears in El Euch and Rosenbaum (2019, Math. Financ., 29, 3–38), but with the drift and mean reversion terms removed. The solution to this VIE satisfies a space–time scaling property which means we only need to solve this equation for the moment values of and so the rate function can be efficiently computed using an Adams scheme or a power series, and we compute a power series in the log‐moneyness variable for the asymptotic implied volatility which yields tractable expressions for the implied vol skew and convexity which is useful for calibration purposes. We later derive a formal saddle point approximation for call options in the Forde and Zhang (2017) large deviations regime which goes to higher order than previous works for rough models. Our higher‐order expansion captures the effect of both drift terms, and at leading order is of qualitatively the same form as the higher‐order expansion for a general model which appears in Friz et al. (2018, Math. Financ., 28, 962–988). The limiting asymptotic smile in the large‐maturity regime is obtained via a stability analysis of the fixed points of the VIE, and is the same as for the standard Heston model in Forde and Jacquier (2011, Finance Stoch., 15, 755–780). Finally, using Lévy's convergence theorem, we show that the log stock price tends weakly to a nonsymmetric random variable as (i.e., ) whose moment generating function (MGF) is also the solution to the Rough Heston VIE with , and we show that tends weakly to a nonsymmetric random variable as , which leads to a nonflat nonsymmetric asymptotic smile in the Edgeworth regime, where the log‐moneyness as , and we compute this asymptotic smile numerically. We also show that the third moment of the log stock price tends to a finite constant as (in contrast to the Rough Bergomi model discussed in Forde et al. (2020, Preprint) where the skew flattens or blows up) and the process converges on pathspace to a random tempered distribution which has the same law as the hyper‐rough Heston model, discussed in Jusselin and Rosenbaum (2020, Math. Finance, 30, 1309–1336) and Abi Jaber (2019, Ann. Appl. Probab., 29, 3155–3200).     

Equilibrium concepts for time‐inconsistent stopping problems in continuous time

A new notion of equilibrium, which we call strong equilibrium, is introduced for time‐inconsistent stopping problems in continuous time. Compared to the existing notions introduced in Huang, Y.‐J., & Nguyen‐Huu, A. (2018, Jan 01). Time‐consistent stopping under decreasing impatience. Finance and Stochastics, 22(1), 69–95 and Christensen, S., & Lindensjö, K. (2018). On finding equilibrium stopping times for time‐inconsistent markovian problems. SIAM Journal on Control and Optimization, 56(6), 4228–4255, which in this paper are called mild equilibrium and weak equilibrium, respectively, a strong equilibrium captures the idea of subgame perfect Nash equilibrium more accurately. When the state process is a continuous‐time Markov chain and the discount function is log subadditive, we show that an optimal mild equilibrium is always a strong equilibrium. Moreover, we provide a new iteration method that can directly construct an optimal mild equilibrium and thus also prove its existence.     

The robust pricing–hedging duality for American options in discrete time financial markets

We investigate the pricing–hedging duality for American options in discrete time financial models where some assets are traded dynamically and others, for example, a family of European options, only statically. In the first part of the paper, we consider an abstract setting, which includes the classical case with a fixed reference probability measure as well as the robust framework with a nondominated family of probability measures. Our first insight is that, by considering an enlargement of the space, we can see American options as European options and recover the pricing–hedging duality, which may fail in the original formulation. This can be seen as a weak formulation of the original problem. Our second insight is that a duality gap arises from the lack of dynamic consistency, and hence that a different enlargement, which reintroduces dynamic consistency is sufficient to recover the pricing–hedging duality: It is enough to consider fictitious extensions of the market in which all the assets are traded dynamically. In the second part of the paper, we study two important examples of the robust framework: the setup of Bouchard and Nutz and the martingale optimal transport setup of Beiglböck, Henry‐Labordère, and Penkner, and show that our general results apply in both cases and enable us to obtain the pricing–hedging duality for American options.     

Robust consumption‐investment problem under CRRA and CARA utilities with time‐varying confidence sets

We consider a robust consumption‐investment problem under constant relative risk aversion and constant absolute risk aversion utilities. The time‐varying confidence sets are specified by Θ, a correspondence from [0, T] to the space of the Lévy triplets, and describe a priori drift, volatility, and jump information. For each possible measure, the log‐price processes of stocks are semimartingales, and the triplet of their differential characteristics is almost surely a measurable selector from the correspondence Θ. By proposing and investigating the global kernel, an optimal policy and a worst‐case measure are generated from a saddle point of the global kernel, and they constitute a saddle point of the objective function.     

Asymmetric effects of demand uncertainty on intrafirm trade in durable and non‐durable industries

This paper unveils a new empirical regularity regarding the asymmetric patterns of international sourcing modes in the durable and non‐durable industries under uncertainty, and explains the asymmetry based on the traditional lens of the transaction cost economics. Under demand uncertainty, firms face trade‐offs between outsourcing and vertical integration: while outsourcing requires less initial investment and allows easier market entry and exit, vertical integration offers better management and communication systems. In the durable industries, consistent with the transaction cost economics, the relationship between vertical integration and uncertainty is positive. In the nondurable industries, however, such relationship is weaker because inelastic demand and shorter gaps between production and sales mitigate the effect of uncertainty. I show the asymmetric effects of uncertainty based on a simple general equilibrium model and provide consistent empirical evidence using US intrafirm trade and microeconomic uncertainty data.     

Robust Markowitz mean‐variance portfolio selection under ambiguous covariance matrix

This paper studies a robust continuous‐time Markowitz portfolio selection problem where the model uncertainty affects the covariance matrix of multiple risky assets. This problem is formulated into a min–max mean‐variance problem over a set of nondominated probability measures that is solved by a McKean–Vlasov dynamic programming approach, which allows us to characterize the solution in terms of a Bellman–Isaacs equation in the Wasserstein space of probability measures. We provide explicit solutions for the optimal robust portfolio strategies and illustrate our results in the case of uncertain volatilities and ambiguous correlation between two risky assets. We then derive the robust efficient frontier in closed form, and obtain a lower bound for the Sharpe ratio of any robust efficient portfolio strategy. Finally, we compare the performance of Sharpe ratios for a robust investor and for an investor with a misspecified model.