We consider a central bank strategy for maintaining a two-sided currency target zone, in which an exchange rate of two currencies is forced to stay between two thresholds. To keep the exchange rate from breaking the prescribed barriers, the central bank is generating permanent price impact and thereby accumulating inventory in the foreign currency. Historical examples of failed target zones illustrate that this inventory can become problematic, in particular when there is an adverse macroeconomic trend in the market. We model this situation through a continuous-time market impact model of Almgren--Chriss-type with drift, in which the exchange rate is a diffusion process controlled by the price impact of the central bank's intervention strategy. The objective of the central bank is to enforce the target zone through a strategy that minimizes the accumulated inventory. We formulate this objective as a stochastic control problem with random time horizon. It is solved by reduction to a singular boundary value problem that was solved by Lasry and Lions (1989). Finally, we provide numerical simulations of optimally controlled exchange rate processes and the corresponding evolution of the central bank inventory.

We propose three different data driven approaches for pricing European style call options using supervised machine-learning algorithms. The proposed approaches are tested on two stock market indices, NIFTY50 and BANKNIFTY from the Indian equity market. Although neither historical nor implied volatility is used as an input, the results show that the trained models have been able to capture the option pricing mechanism better than or similar to the Black Scholes formula for all the experiments. Our choice of scale free I/O allows us to train models using combined data of multiple different assets from a financial market. This not only allows the models to achieve far better generalization and predictive capability, but also solves the problem of paucity of data, the primary limitation of using machine learning techniques. We also illustrate the performance of the trained models in the period leading up to the 2020 Stock Market Crash, Jan 2019 to April 2020.

The current crisis, at the time of writing, has had a profound impact on the financial world, introducing the need for creative approaches to revitalising the economy at the micro level as well as the macro level. In this informal analysis and design proposal, we describe how infrastructure for digital assets can serve as a useful monetary and fiscal policy tool and an enabler of existing tools in the future, particularly during crises, while aligning the trajectory of financial technology innovation toward a brighter future. We propose an approach to digital currency that would allow people without banking relationships to transact electronically and privately, including both internet purchases and point-of-sale purchases that are required to be cashless. We also propose an approach to digital currency that would allow for more efficient and transparent clearing and settlement, implementation of monetary and fiscal policy, and management of systemic risk. The digital currency could be implemented as central bank digital currency (CBDC), or it could be issued by the government and collateralised by public funds or Treasury assets. Our proposed architecture allows both manifestations and would be operated by banks and other money services businesses, operating within a framework overseen by government regulators. We argue that now is the time for action to undertake development of such a system, not only because of the current crisis but also in anticipation of future crises resulting from geopolitical risks, the continued globalisation of the digital economy, and the changing value and risks that technology brings.

In this short note we develop a model for discounting.

A focus of the model is the discounting, when discount factors cannot be derived from market products. That is, a risk-neutralizing trading strategy cannot be performed. This is the case, when one is in need of a risk-free (default-free) discounting, but default protection on funding providers is not traded. For this case, we introduce a default compensation factor ($\exp(+\tilde{\lambda} T)$) that describes the present value of a strategy to compensate for default (like buying default protection would do).

In a second part, we introduce a model, where the survival probability depends on the required notional. This model is different from the classical modelling of a time-dependent survival probability ($\exp(-\lambda T)$). The model especially allows that large liquidity requirements are instantly more likely do default than small ones.

Combined the two approaches build a framework in which discounting (valuation) is non-linear.

The non-linear discounting presented here has several effects, which are relevant in various applications:

* If we consider the question of default-free valuation, i.e., factoring in the cost of default protection, the framework can will lead to over-proportional higher values (or cost) for large projects (or damages). The framework can lead to the effect that discount-factors for very large liquidity requirements or projects are an increasing function of time. It may even lead to discount factors larger than one. This may have relevance in the assessment of event like climate change.

* For the valuation of defaultable products, e.g., like a defaultable swap, the framework leads to the generation of a continuum of (defaultable) par rate curves (interest rate curve) and the valuation of a payer and a receiver swap differs by more than just a sign.

This paper studies a dynamic optimal reinsurance and dividend-payout problem for an insurer in a finite time horizon. The goal of the insurer is to maximize its expected cumulative discounted dividend payouts until bankruptcy or maturity which comes earlier. The insurer is allowed to dynamically choose reinsurance contracts over the whole time horizon. This is a singular control problem and the corresponding Hamilton-Jacobi-Bellman equation is a variational inequality with fully nonlinear operator and with gradient constraint. A comparison principle and $C^{2,1}$ smoothness for the solution are established by penalty approximation method. We find that the surplus-time space can be divided into three non-overlapping regions by a ceded risk and time dependent reinsurance barrier and a time dependent dividend-payout barrier. The insurer should be exposed to higher risk as surplus increases; exposed to all risk once surplus upward crosses the reinsurance barrier; and pay out all reserves in excess of the dividend-payout barrier. The localities of these regions are explicitly estimated.

We study how transaction cost affects to the equilibrium return and optimal stock holdings in equilibrium. To this end, we develop a continuous-time risk-sharing model where heterogenous agents trade toward terminal target holdings subject to a quadratic transaction cost. The equilibrium stock holdings and trading rate under transaction cost are characterized by a unique solution to a forward-backward stochastic differential equation (FBSDE). The equilibrium return is also characterized as the unique solution of a system of coupled but linear FBSDEs.

Inspired by Stein's lemma, we derive two expressions for the joint moments of elliptical distributions. We use two different methods to derive $E[X_{1}^{2}f(\mathbf{X})]$ for any measurable function $f$ satisfying some regularity conditions. Then, by applying this result, we obtain new formulae for expectations of product of normally distributed random variables, and also present simplified expressions of $E[X_{1}^{2}f(\mathbf{X})]$ for multivariate Student-$t$, logistic and Laplace distributions.

We have embedded the classical theory of stochastic finance into a differential geometric framework called Geometric Arbitrage Theory and show that it is possible to:

--Write arbitrage as curvature of a principal fibre bundle.

--Parameterize arbitrage strategies by its holonomy.

--Give the Fundamental Theorem of Asset Pricing a differential homotopic characterization.

--Characterize Geometric Arbitrage Theory by five principles and show they they are consistent with the classical theory of stochastic finance.

--Derive for a closed market the equilibrium solution for market portfolio and dynamics in the cases where:

-->Arbitrage is allowed but minimized.

-->Arbitrage is not allowed.

--Prove that the no-free-lunch-with-vanishing-risk condition implies the zero curvature condition. The converse is in general not true and additionally requires the Novikov condition for the instantaneous Sharpe Ratio Dynamics to be satisfied.

This paper is one of a collection of papers on prime-age male earnings volatility. Each paper produces a similar set of statistics for the same reference population using a different primary data source. Our primary data source is the Census Bureau's Longitudinal Employer-Household Dynamics (LEHD) infrastructure files. Using LEHD data from 1998 to 2016, we create a well-defined population frame to facilitate accurate estimation of temporal changes comparable to designed longitudinal samples of people. We show that earnings volatility, excluding increases during recessions, has generally declined over the analysis period, a finding robust to various sensitivity analyses. Although volatility is found to generally be declining, the effect is not homogeneous, particularly for workers with tenuous labor force attachment for whom volatility is increasing. These "not stable" workers have earnings volatility approximately 30 times larger than stable workers, but more important for earnings volatility trends we observe a large increase in the share of stable employment from 60% in 1998 to 67% in 2016, which we show to largely be responsible for the decline in overall earnings volatility. To further emphasize the importance of not stable and/or low earning workers we also conduct comparisons with the PSID and show how changes over time in the share of workers at the bottom tail of the cross-sectional earnings distributions can produce either declining or increasing earnings volatility trends.

Bernard et al. (2015) studied an insurance contract design problem under rank-dependent utility (RDU) theory. Their results, however, suffer from a moral hazard problem, namely, providing incentives for the insured to falsely report the actual loss. Xu et al. (2019) investigated the same problem, but took the {incentive compatibility} constraint into consideration to avoid that moral hazard. Mathematically speaking, the model reduces to a quantile optimisation problem with a compatibility constraint. They solved the problem by imposing assumptions on the loss and the probability weighting function. This paper solves the problem completely by a new quantile optimisation approach under general setting. The optimal solution is expressed by the solution of an obstacle problem for a semilinear second-order elliptic operator with mixed boundary conditions. Surprisingly, it is shown that every reasonable moral-hazard-free contract is optimal for infinitely many RDU maximisers with different utility functions and probability weighting functions.

We present multigrid iterative algorithms for solving a system of coupled free boundary problems for pricing American put options with regime-switching. The algorithms are based on our recent developed compact finite difference scheme coupled with Hermite interpolation for solving the m coupled partial differential equations consisting of the asset, delta, gamma, and speed options. In the algorithms, we first use the Gauss-Seidel as a smoother, and then implement V-cycle and modified multigrid strategies for solving our discretized equations. Hermite interpolation with Newton interpolatory divided difference (as the basis) is used in estimating the coupled asset, delta, gamma, and speed options in the set of equations. A numerical experiment is performed with the two-regimes example and compared with other existing methods to validate the optimal strategy. Results show that these algorithms provide fast and efficient tools for pricing American put options with regime-switching.

In this paper, we focus on a new generalization of multivariate general compound Hawkes process (MGCHP), which we referred to as the multivariate general compound point process (MGCPP). Namely, we applied a multivariate point process to model the order flow instead of the Hawkes process. Law of large numbers (LLN) and two functional central limit theorems (FCLTs) for the MGCPP were proved in this work. Applications of the MGCPP in the limit order market were also considered. We provided numerical simulations and comparisons for the MGCPP and MGCHP by applying Google, Apple, Microsoft, Amazon, and Intel trading data.

This paper studies an optimal investment and consumption problem with heterogeneous consumption of basic and luxury goods, together with the choice of time for retirement. The utility for luxury goods is not necessarily a concave function. The optimal heterogeneous consumption strategies for a class of non-homothetic utility maximizer are shown to consume only basic goods when the wealth is small, to consume basic goods and make savings when the wealth is intermediate, and to consume small portion in basic goods and large portion in luxury goods when the wealth is large. The optimal retirement policy is shown to be both universal, in the sense that all individuals should retire at the same level of marginal utility that is determined only by income, labor cost, discount factor as well as market parameters, and not universal, in the sense that all individuals can achieve the same marginal utility with different utility and wealth. It is also shown that individuals prefer to retire as time goes by if the marginal labor cost increases faster than that of income. The main tools used in analysing the problem are from PDE and stochastic control theory including viscosity solution, variational inequality and dual transformation.

We consider the optimal prediction problem of stopping a spectrally negative L\'evy process as close as possible to a given distance $b \geq 0$ from its ultimate supremum, under a squared error penalty function. Under some mild conditions, the solution is fully and explicitly characterised in terms of scale functions. We find that the solution has an interesting non-trivial structure: if $b$ is larger than a certain threshold then it is optimal to stop as soon as the difference between the running supremum and the position of the process exceeds a certain level (less than $b$), while if $b$ is smaller than this threshold then it is optimal to stop immediately (independent of the running supremum and position of the process). We also present some examples.

This paper proposes a theory of pricing premised upon the assumptions that customers dislike unfair prices---those marked up steeply over cost---and that firms take these concerns into account when setting prices. Since they do not observe firms' costs, customers must extract costs from prices. The theory assumes that customers infer less than rationally: when a price rises due to a cost increase, customers partially misattribute the higher price to a higher markup---which they find unfair. Firms anticipate this response and trim their price increases, which drives the passthrough of costs into prices below one: prices are somewhat rigid. Embedded in a New Keynesian model as a replacement for the usual pricing frictions, our theory produces monetary nonneutrality: when monetary policy loosens and inflation rises, customers misperceive markups as higher and feel unfairly treated; firms mitigate this perceived unfairness by reducing their markups; in general equilibrium, employment rises. The theory also features a hybrid short-run Phillips curve, realistic impulse responses of output and employment to monetary and technology shocks, and an upward-sloping long-run Phillips curve.

The first moment and second central moments of the portfolio return, a.k.a. mean and variance, have been widely employed to assess the expected profit and risk of the portfolio. Investors pursue higher mean and lower variance when designing the portfolios. The two moments can well describe the distribution of the portfolio return when it follows the Gaussian distribution. However, the real world distribution of assets return is usually asymmetric and heavy-tailed, which is far from being a Gaussian distribution. The asymmetry and the heavy-tailedness are characterized by the third and fourth central moments, i.e., skewness and kurtosis, respectively. Higher skewness and lower kurtosis are preferred to reduce the probability of extreme losses. However, incorporating high-order moments in the portfolio design is very difficult due to their non-convexity and rapidly increasing computational cost with the dimension. In this paper, we propose a very efficient and convergence-provable algorithm framework based on the successive convex approximation (SCA) algorithm to solve high-order portfolios. The efficiency of the proposed algorithm framework is demonstrated by the numerical experiments.

This paper examines Fama's (1970) semi-strong form efficient market hypothesis (EMH) in the prewar Japanese stock market using a new dataset. We particularly focus on the relationship between the prewar Japanese stock market and several government policy interventions to explore whether the semi-strong form stock market efficiency evolves over time. To capture the long-run impact of government policy interventions against stock markets, we measure the time-varying joint degree of market efficiency and the time-varying impulse responses based on Ito et al.'s (2014; 2017) generalized least squares-based time-varying vector autoregressive model. The empirical results reveal that (1) the joint degree of market efficiency in the prewar Japanese stock market fluctuated over time because of external events such as policy changes and wars, (2) the semi-strong form EMH is almost supported in the prewar Japanese stock market, and (3) the markets rapidly reflect the information of the external events through time. Therefore, we conclude that Lo's (2004) adaptive market hypothesis is supported in the prewar Japanese stock market even if we consider that the public information affects the stock market.

Since 1986, Government of Botswana has been running an Agricultural Credit Guarantee Scheme for dry-land arable farming. The scheme purports to assist dry-land crop farmers who have taken loans with participating banks or lending institutions to help them meet their debt obligations in case of crop failure due to drought, floods, frost or hailstorm. Nonetheless, to date, the scheme has focused solely on drought. The scheme has placed an unsustainable financial burden on Government because it is not based on sound actuarial principles. This paper argues that the level of Government subsidies should take into account the gains made by farmers during non-drought years. It is an attempt to circumvent the challenges of correlated climate risks and recommends a quasi self-financing mechanism, assuming that the major driver of crop yield failure is drought. Moreover, it provides a novel subsidy and premium rate setting method.