We study combinations of risk measures under no restrictive assumption on the set of alternatives. The main result is the representation for resulting risk measures from the properties of both alternative functionals and combination functions. To that, we develop a representation for a mixture of convex risk measures. As an application, we address the context of probability-based risk measurements based on a functional on the set of distribution functions. We develop results related to this specific context. We also explore features of individual interest generated by our framework, such as the preservation of continuity properties, the representation of worst-case risk measures, stochastic dominance and elicitability.

Let $\Omega$ be a Polish space with Borel $\sigma$-field $\mathcal{F}$ and countably generated sub $\sigma$-field $\mathcal{G}\subset\mathcal{F}$. Denote by $\mathcal{L}(\mathcal{F})$ the set of all bounded $\mathcal{F}$-upper semianalytic functions from $\Omega$ to the reals and by $\mathcal{L}(\mathcal{G})$ the subset of $\mathcal{G}$-upper semianalytic functions. Let $\mathcal{E}(\cdot|\mathcal{G})\colon\mathcal{L}(\mathcal{F})\to\mathcal{L}(\mathcal{G})$ be a sublinear increasing functional which leaves $\mathcal{L}(\mathcal{G})$ invariant. It is shown that there exists a $\mathcal{G}$-analytic set-valued mapping $\mathcal{P}_{\mathcal{G}}$ from $\Omega$ to the set of probabilities which are concentrated on atoms of $\mathcal{G}$ with compact convex values such that $\mathcal{E}(X|\mathcal{G})(\omega)=$ $\sup_{P\in\mathcal{P}_{\mathcal{G}}(\omega)} E_P[X]$ if and only if $\mathcal{E}(\cdot |\mathcal{G})$ is pointwise continuous from below and continuous from above on the continuous functions. Further, given another sublinear increasing functional $\mathcal{E}(\cdot)\colon\mathcal{L}(\mathcal{F})\to\mathbb{R}$ which leaves the constants invariant, the tower property $\mathcal{E}(\cdot)=\mathcal{E}(\mathcal{E}(\cdot|\mathcal{G}))$ is characterized via a pasting property of the representing sets of probabilities, and the importance of analytic functions is explained. Finally, it is characterized when a nonlinear version of Fubini's theorem holds true and when the product of a set of probabilities and a set of kernels is compact.

In this paper, we present a family of a control-stopping games which arise naturally in equilibrium-based models of market microstructure, as well as in other models with strategic buyers and sellers. A distinctive feature of this family of games is the fact that the agents do not have any exogenously given fundamental value for the asset, and they deduce the value of their position from the bid and ask prices posted by other agents (i.e. they are pure speculators). As a result, in such a game, the reward function of each agent, at the time of stopping, depends directly on the controls of other players. The equilibrium problem leads naturally to a system of coupled control-stopping problems (or, equivalently, Reflected Backward Stochastic Differential Equations (RBSDEs)), in which the individual reward functions (or, reflecting barriers) depend on the value functions (or, solution components) of other agents. The resulting system, in general, presents multiple mathematical challenges due to the non-standard form of coupling (or, reflection). In the present case, this system is also complicated by the fact that the continuous controls of the agents, describing their posted bid and ask prices, are constrained to take values in a discrete grid. The latter feature reflects the presence of a positive tick size in the market, and it creates additional discontinuities in the agents reward functions (or, reflecting barriers). Herein, we prove the existence of a solution to the associated system in a special Markovian framework, provide numerical examples, and discuss the potential applications.

The market efficiency hypothesis has been proposed to explain the behavior of time series of stock markets. The Black-Scholes model (B-S) for example, is based on the assumption that markets are efficient. As a consequence, it is impossible, at least in principle, to "predict" how a market behaves, whatever the circumstances. Recently we have found evidence which shows that it is possible to find self-organized behavior in the prices of assets in financial markets during deep falls of those prices. Through a kurtosis analysis we have identified a critical point that separates time series from stock markets in two different regimes: the mesokurtic segment compatible with a random walk regime and the leptokurtic one that allegedly follows a power law behavior. In this paper we provide some evidence, showing that the Hurst exponent is a good estimator of the regime in which the market is operating. Finally, we propose that the Hurst exponent can be considered as a critical variable in just the same way as magnetization, for example, can be used to distinguish the phase of a magnetic system in physics.

This paper considers a time-inconsistent stopping problem in which the inconsistency arises from non-constant time preference rates. We show that the smooth pasting principle, the main approach that has been used to construct explicit solutions for conventional time-consistent optimal stopping problems, may fail under time-inconsistency. Specifically, we prove that the smooth pasting principle solves a time-inconsistent problem within the intra-personal game theoretic framework if and only if a certain inequality on the model primitives is satisfied. We show that the violation of this inequality can happen even for very simple non-exponential discount functions. Moreover, we demonstrate that the stopping problem does not admit any intra-personal equilibrium whenever the smooth pasting principle fails. The "negative" results in this paper caution blindly extending the classical approaches for time-consistent stopping problems to their time-inconsistent counterparts.

Real life hedging in the Black-Scholes model must be imperfect and if the stock's drift is higher than the risk free rate, leads to a profit on average. Hence the option price is examined as a fair game agreement between the parties, based on expected payoffs and a simple measure of risk. The resulting prices result in the volatility smile.

In their seminal work Carr and Lee (2008) show how to robustly price and replicate a variety of claims written on the quadratic variation of a risky asset under the assumption that the asset's volatility process is independent of the Brownian motion that drives the asset's price. Additionally, they propose a correlation immunization strategy that minimizes the pricing and hedging error that results when the correlation between the risky asset's price and volatility is nonzero. In this paper, we show that the correlation immunization strategy is the only strategy among the class of strategies discussed in Carr and Lee (2008) that results in real-valued hedging portfolios when the correlation between the asset's price and volatility is nonzero. Additionally, we perform a number of Monte Carlo experiments to test the effectiveness of Carr and Lee's immunization strategy. Our results indicate that the correlation immunization method is an effective means of reducing pricing and hedging errors that result from nonzero correlation.

We derive new formulas for the price of the European call and put options in the Black-Scholes model, under the form of uniformly convergent series generalizing previously known approximations. We also provide precise boundaries for the convergence speed and apply the results to the calculation of hedge parameters (Greeks).

We consider the optimal dividend problem under a habit formation constraint that prevents the dividend rate to fall below a certain proportion of its historical maximum, the so-called drawdown constraint. This is an extension of the optimal Duesenberry's ratcheting consumption problem, studied by Dybvig (1995) [Review of Economic Studies 62(2), 287-313], in which consumption is assumed to be nondecreasing. Our problem differs from Dybvig's also in that the time of ruin could be finite in our setting, whereas ruin was impossible in Dybvig's work. We formulate our problem as a stochastic control problem with the objective of maximizing the expected discounted utility of the dividend stream until bankruptcy, in which risk preferences are embodied by power utility. We semi-explicitly solve the corresponding Hamilton-Jacobi-Bellman variational inequality, which is a nonlinear free-boundary problem. The optimal (excess) dividend rate $c^*_t$ - as a function of the company's current surplus $X_t$ and its historical running maximum of the (excess) dividend rate $z_t$ - is as follows: There are constants $0 < w_{\alpha} < w_0 < w^*$ such that (1) for $0 < X_t \le w_{\alpha} z_t$, it is optimal to pay dividends at the lowest rate $\alpha z_t$, (2) for $w_{\alpha} z_t < X_t < w_0 z_t$, it is optimal to distribute dividends at an intermediate rate $c^*_t \in (\alpha z_t, z_t)$, (3) for $w_0 z_t < X_t < w^* z_t$, it is optimal to distribute dividends at the historical peak rate $z_t$, (4) for $X_t > w^* z_t$, it is optimal to increase the dividend rate above $z_t$, and (5) it is optimal to increase $z_t$ via singular control as needed to keep $X_t \le w^* z_t$. Because, the maximum (excess) dividend rate will eventually be proportional to the running maximum of the surplus, "mountains will have to move" before we increase the dividend rate beyond its historical maximum.

We introduce a new class of forward performance processes that are endogenous and predictable with regards to an underlying market information set and, furthermore, are updated at discrete times. We analyze in detail a binomial model whose parameters are random and updated dynamically as the market evolves. We show that the key step in the construction of the associated predictable forward performance process is to solve a single-period inverse investment problem, namely, to determine, period-by-period and conditionally on the current market information, the end-time utility function from a given initial-time value function. We reduce this inverse problem to solving a functional equation and establish conditions for the existence and uniqueness of its solutions in the class of inverse marginal functions.

Equity risk premium is a central component of every risk and return model in finance and a key input to estimate costs of equity and capital in both corporate finance and valuation. An article by Damodaran examines three broad approaches for estimating the equity risk premium. The first is survey based, it consists in asking common investors or big players like pension fund managers what they require as a premium to invest in equity. The second is to look at the premia earned historically by investing in stocks, as opposed to risk-free investments. The third method tries to extrapolate a market-consensus on equity risk premium (Implied Equity Risk Premium) by analysing equity prices on the market today. After having introduced some basic concepts and models, I'll briefly explain the pluses and minuses of the first two methods, and analyse more deeply the third. In the end I'll show the results of my estimation of ERP on real data, using variants of the Implied ERP (third) method.

The seminal idea of quantum money not forgeable due to laws of Quantum Mechanics proposed by Stephen Wiesner, has laid foundations for the Quantum Information Theory in early '70s. Recently, several other schemes for quantum currencies have been proposed, all however relying on the assumption that the mint does not cooperate with the counterfeiter. Drawing inspirations from the semi-device independent quantum key distribution protocol, we introduce the first scheme of quantum money with this assumption partially relaxed, along with the proof of its unforgeability. Significance of this protocol is supported by an impossibility result, which we prove, stating that there is no both fully device independent and secure money scheme. Finally, we formulate a quantum analogue of the Oresme-Copernicus-Gresham's law of economy.

How would the frontier have evolved in the absence of homestead policies? I apply a matrix completion method to predict the counterfactual time-series of frontier state capacity had there been no homesteading. In placebo tests, the matrix completion method outperforms synthetic controls and other regression-based estimators in terms of minimizing prediction error. Causal estimates signify that homestead policies had significant and long-lasting negative impacts on state government expenditure and revenue. These results are similar to difference-in-difference estimates that exploit variation in the timing and intensity of homestead entries aggregated from 1.46 million individual land patent records.

The international community was caught by surprise on 5 June 2017 when Saudi Arabia, the United Arab Emirates (UAE), Bahrain and Egypt severed diplomatic ties with Qatar, accusing it of destabilizing the region. More than one year after this diplomatic rift, several questions remain unaddressed. This study focuses on the regional business costs of the year-long blockade on Qatar. We split the sample to compare the stock market performances of Qatar and its Middle Eastern neighbors before and after the Saudi-led Qatar boycott. We focus our attention on the conditional volatility process of stock market returns and risks related to financial interconnectedness. We show that the Gulf crisis had the most adverse impact on Qatar together with Saudi Arabia and the UAE. Although not to the same degree as these three countries, Bahrain and Egypt were also harmfully affected. But shocks to the volatility process tend to have short-lasting effects. Moreover, the total volatility spillovers to and from others increase but moderately after the blockade. Overall, the quartet lobbying efforts did not achieve the intended result. Our findings underscore Qatar's economic vulnerability but also the successful resilience strategy of this tiny state. The coordinated diplomatic efforts of Qatar have been able to fight the economic and political embargo.

Planetary life support systems are collapsing due to climate change and the biodiversity crisis. The root cause is the existing consumer economy, coupled with profit maximisation based on ecological and social externalities. Trends can be reversed, civilisation may be saved by transforming the profit maximising consumer economy into an ecologically and socially just economy, which we call the prosumer economy. Prosumer economy is a macro scale circular economy with minimum negative or positive ecological and social impact, an ecosystem of producers and prosumers, who have synergistic and circular relationships with deepened circular supply chains, networks, where leakage of wealth out of the system is minimised. In a prosumer economy there is no waste, no lasting negative impacts on the ecology and no social exploitation. The prosumer economy is like a lake or a forest, an economic ecosystem that is productive and supportive of the planet. We are already planting this forest through Good4Trust.org, started in Turkey. Good4Trust is a community platform bringing together ecologically and socially just producers and prosumers. Prosumers come together around a basic ethical tenet the golden rule and share on the platform their good deeds. The relationship are already deepening and circularity is forming to create a prosumer economy. The platforms software to structure the economy is open source, and is available to be licenced to start Good4Trust anywhere on the planet. Complexity theory tells us that if enough agents in a given system adopt simple rules which they all follow, the system may shift. The shift from a consumer economy to a prosumer economy has already started, the future is either ecologically and socially just or bust.

This paper examines the effects of institutional quality on financing choice of individual using a large dataset of 137,160 people from 131 countries. We classify borrowing activities into three categories, including formal, constructive informal, and underground borrowing. Although the result shows that better institutions aids the uses of formal borrowing, the impact of institutions on constructive informal and underground borrowing among three country sub-groups differs. Higher institutional quality improves constructive informal borrowing in middle-income countries but reduces the use of underground borrowing in high- and low-income countries.

We propose a combinatorial model of economic development. An economy develops by acquiring new capabilities allowing for the production of an ever greater variety of products of increasingly complex products. Taking into account that economies abandon the least complex products as they develop over time, we show that variety first increases and then decreases in the course of economic development. This is consistent with the empirical pattern known as 'the hump'. Our results question the common association of variety with complexity. We further discuss the implications of our model for future research.