Optimal Dividend Policy when Cash Reserves Follow a Jump-Diffusion Process Under Markov-Regime Switching

Optimal Dividend Policy When Cash Reserves Follow a Jump-Diffusion Process Under Markov-Regime Switching

This study aims to understand the contagion effect of the major equity market sentiment events, defined as jumps in the VIX index, on cryptocurrency price jumps and the corresponding feedback effect on investors' sentiment. Using recent high frequency intraday data with multivariate Hawkes processes, we find that several noteworthy contagion effects exist between Bitcoin and the market sentiment proxied by the VIX index. First of all, we find that positive market sentiment jumps tend to trigger a moderate level of cross-contagion on both positive and negative jumps in the Bitcoin market, whereas negative market sentiment jumps have no contagion effect on the Bitcoin prices. We also find that the Bitcoin market shows stronger positive self-contagion and cross-contagion effects than the equity market, in general. We also observe a lasting fear of missing out (FOMO) phenomenon in Bitcoin. It is supported by the evidence that the effect of positive jumps in the Bitcoin market lasts three times as long as that in both the equity market and the negative Bitcoin price jumps, and these positive jumps in Bitcoin may serve as a harbinger of equity market movement. These findings provide a better understanding of risk control and policy guidance for the cryptocurrency market.

Compared with investing in individual stocks, ETF investment is capable of diversifying the non-systematic risk or exposure to broad market or industry sectors. The aim of this paper is to develop a jump contagion modeling framework to understand the contagion effect of market jump events of energy sector ETFs using multivariate Hawkes process modeling approach. Through analyzing intraday high-frequency market data, we find that negative index jumps lead index price discovery processes, and their influences disappear faster than the positive index jumps in both the S&P500 and the crude oil futures. And on average, the self contagion in negative jumps is stronger than the self contagion in the positive jumps across all ETF groups. However, the ETFs focused on the master limited partnership (MLP) segment show less negative self contagion and relatively stronger positive self contagion than the other energy ETFs. Overall, the influence of negative jumps on ETFs from both the equity index and the energy future index is stronger than that of the positive jumps. And the influence of the equity index (S&P500) jump on ETFs lasts longer than that of the crude oil futures index (CLC1).

This work deals with an optimal dividend payment problem for a piecewise-deterministic compound Poisson insurance risk model. The objective is to maximize the expected discounted dividend payout up to the time of ruin. When the dividend payment rate is restricted, the value function is shown to be a solution of the corresponding Hamilton-Jacobi-Bellman equation, which in turn leads to a tractable methodology to find an optimal threshold dividend payment policy. For the case of unrestricted payment rate, the value function and an optimal barrier strategy are determined explicitly with exponential claim size distributions. A comparison of two examples is provided to illustrate the main results.

Assets are assumed to follow a diffusion process subject to some conditions. The owners can pay dividends at their discretion, but whenever assets reach zero, they have to reinvest money so that assets never go negative. With each dividend payment there is a fixed and a proportional cost, and so with reinvestments. The goal is to maximize the expected value of discounted net cash flow, i.e., dividends paid minus reinvestments. It is shown that there can be two different solutions depending on the model parameters and the costs as follows. (1) Whenever assets reach a barrier $y^*$ they are reduced to $y^*-\delta^*$ through a dividend payment, and whenever they reach 0 they are increased to $\gamma^*\leq y^*-\delta^*$ by a reinvestment. (2) There is no optimal policy, but the value function is approximated by policies of the form described in (1) for increasing barriers. We provide criteria to decide whether an optimal solution exists, and when it does not, we show how to calculate the value function. We discuss how the problem can be solved numerically and give numerical examples.

This paper addresses the problem of finding an optimal dividend policy for a class of jump-diffusion processes. The jump component is a compound Poisson process with negative jumps, and the drift and diffusion components are assumed to satisfy some regularity and growth restrictions. Each dividend payment is changed by a fixed and a proportional cost, meaning that if ξ is paid out by the company, the shareholders receive kξ−K, where k and K are positive. The aim is to maximize expected discounted dividends until ruin. It is proved that when the jumps belong to a certain class of light-tailed distributions, the optimal policy is a simple lump sum policy, that is, when assets are equal to or larger than an upper barrier $\bar{u}^{*}$ , they are immediately reduced to a lower barrier $\underline{u}^{*}$ through a dividend payment. The case with K=0 is also investigated briefly, and the optimal policy is shown to be a reflecting barrier policy for the same light-tailed class. Methods to numerically verify whether a simple lump sum barrier strategy is optimal for any jump distribution are provided at the end of the paper, and some numerical examples are given.

We consider the optimal stopping problem, based on the compound binomial model with capital injection controlled by optimal dividend strategy. First, the value function V(x) is established. Second, via the Bellman equation satisfied by the value function V(x) which maximizes the discounted value of define between dividend payment and the penalized discounted capital injection, we find the optimal stopping time τ*. Furthermore an optimal stopping model with capital injection in order to maximize the shareholders’ interests is obtained.

We consider the one-sided and the two-sided first-exit problem for a compound Poisson process with linear deterministic decrease between positive and negative jumps. This process (X(t)) t≥0 occurs as the workload process of a single-server queueing system with random workload removal, which we denote by M/G u /G d /1, where G u (G d ) stands for the distribution of the upward (downward) jumps; other applications are to cash management, dams, and several related fields. Under various conditions on G u and G d (assuming e.g. that one of them is hyperexponential, Erlang or Coxian), we derive the joint distribution of τ y =inf{t≥0|X(t)∉(0,y)}, y>0, and X(τ y ) as well as that of T=inf{t≥0|X(t)≤0} and X(T). We also determine the distribution of sup{X(t)|0≤t≤T}.

A model in which a foraging animal can reproduce if its energy reserves reach a critical level is presented. The animal's reserves are modelled as a finite-state Markov chain. The animal has the choice between foraging options which may have the same mean net gain but differ in their variances. In addition to starvation, animals are subject to death because of predation, bad weather, and so on. We focus on the case where the rate of mortality due to these sources is the same under all options. We investigate policies that maximise the expected lifetime reproductive success. It is found that the optimal value function is concave at low reserves (low-variance action region) and convex at high reserves (high-variance action region). The value function under the low-variance action has also the same shape and the same inflexion point. This result allows us to compute optimal policies just by looking at the low-variance value function. The result is also used to show that increasing the mortality rate increases the high-variance region under the optimal policy. The pattern of risk-sensitive behaviour predicted by this model is in contrast to that predicted by a similar model in which no reproduction occurs and the optimality criterion is to minimise the probability of death (McNamara (1990)).

Optimal dividend policies for compound Poisson processes: The case of bounded dividend rates

We study a problem of optimal consumption and portfolio selection in a market where the logreturns of the uncertain assets are not necessarily normally distributed. The natural models then involve pure-jump Levy processes as driving noise instead of Brownian motion like in the Black and Scholes model. The state constrained optimization problem involves the notion of local substitution and is of singular type. The associated Hamilton-Jacobi-Bellman equation is a nonlinear first order integro-differential equation subject to gradient and state constraints. We characterize the value function of the singular stochastic control problem as the unique constrained viscosity solution of the associated Hamilton-Jacobi-Bellman equation. This characterization is obtained in two main steps. First, we prove that the value function is a constrained viscosity solution of an integro-differential variational inequality. Second, to ensure that the characterization of the value function is unique, we prove a new comparison (uniqueness) result for the state constraint problem for a class of integro-differential variational inequalities. In the case of HARA utility, it is possible to determine an explicit solution of our portfolio-consumption problem when the Levy process posseses only negative jumps. This is, however, the topic of a companion paper [7].

In this work, we study a dynamic portfolio optimization problem related to pairs trading, which is an investment strategy that matches a long position in one security with a short position in another security with similar characteristics. The relationship between pairs, called a spread, is modeled by a Gaussian mean-reverting process whose drift rate is modulated by an unobservable continuous-time, finite-state Markov chain. Using the classical stochastic filtering theory, we reduce this problem with partial information to an equivalent one with full information and solve it for the logarithmic utility function, where the terminal wealth is penalized by the riskiness of the portfolio according to the realized volatility of the wealth process. We characterize optimal dollar-neutral strategies as well as optimal value functions under full and partial information and show that the certainty equivalence principle holds for the optimal portfolio strategy. Finally, we provide a numerical analysis for a toy example with a two-state Markov chain.

In this paper, having been inspired by the work of Kunita and Seko, we study the pricing of δ‐penalty game call options on a stock with a dividend payment. For the perpetual case, our result reveals that the optimal stopping region for the option seller depends crucially on the dividend rate d. More precisely, we show that when the penalty δ is small, there are two critical dividends 0 < d1 < d2 < ∞ such that the optimal stopping region for the option seller takes one of the following forms: (1) an interval if d < d1; (2) a singleton if d∈ [d1, d2]; or (3) an empty set if d > d2. When d∈ [d1, d2], the value function is not continuously differentiable at the optimal stopping boundary for the option seller, therefore our result in the perpetual case cannot be established by the free boundary approach with smooth‐fit conditions imposed on both free boundaries. For the finite time horizon case, the dependence of the optimal stopping region for the option seller on the time to maturity is exhibited; more precisely, when both δ and d are small, we show that there are two critical times 0 < T1 < T2 < T, such that the optimal stopping region for the option seller takes one of the following forms: (1) an interval if t < T1; (2) a singleton if t∈ [T1, T2]; or (3) an empty set if t > T2. In summary, for both the perpetual and the finite horizon cases, we characterize in terms of model parameters how the optimal stopping region for the option seller shrinks when the dividend rate d increases and the time to maturity decreases; these results complete the original work of Emmerling for the perpetual case and Kunita and Seko for the finite maturity case. In addition, for the finite time horizon case, we also extend the probabilistic method for the establishment of existence and regularity results of the classical American option pricing problem to the game option setting. Finally, we characterize the pair of optimal stopping boundaries for both the seller and the buyer as the unique pair of solutions to a couple of integral equations and provide numerical illustrations.

We consider a model in which a firm faces two types of liquidity risks: a Brownian risk and a Poisson risk. The firm chooses a dividend policy to maximize shareholder value. We characterize the optimal firm value and we show that the optimal dividend policy is a barrier strategy: the firm keeps cash inside when the cash reserves level is less than a critical threshold and pays cash in excess of this threshold. We also analyze the problem of insurance against the Poisson risk. We find that it is optimal for the firm to buy full insurance when its cash reserves are above a critical threshold and not to insure otherwise.

In this note, we consider the optimal dividend payment strategy in regime-switching model under the bounded dividend rate restriction. The surplus of the insurance company is modelled as a regime switching Brownian motion, which has the drift and volatility from two-states Markov process control. The objective is to find the dividend policy which maximizes the expected total discounted dividend pay-outs until the time of bankruptcy. We derive the Hamilton-Jacobi-Bellman equation and closed form solution for the problem. Finally, a numerical simulation is presented to characterize the behavior of the optimal dividend payment strategy.