Optimal Dividend Payment Strategy under Stochastic Interest Force

This paper attempts to study the optimal dividend barrier strategy in risk analysis of an insurance company under stochastic discount interest. Based on stochastic perturbation methodology, we first describe the random of interest by Wiener Process and Poisson process and yield some theoretical results satisfied by optimal dividend barrier. In the case of an exponential individual claim distribution, a group of barrier values are obtained. Meanwhile we also discuss the effect of stochastic interest on the barrier by data analysis and direct interpretations about interest models. It is found that the barrier is more sensitive to constant interest force than other parameters in interest model and the effect of diffusion coefficient on barrier is less sensitive than that of Poisson coefficient. These all provide insights into the effect of stochastic interest on the optimal barrier, and show the importance of introducing stochastic interest. Finally, we propose several meaningful and follow-up problems, for example, changing the criterion of finding the optimal barrier and discussing under more extended risk models.     

Exit problems in regime-switching models

This paper provides a general framework for pricing of perpetual American and real options in regime-switching Lévy models. In each state of the Markov chain, which determines switches from one Lévy process to another, the payoff stream is a monotone function of the Lévy process labeled by the state. This allows for additional switching within each state of the Markov chain (payoffs can be different in different regions of the real line). The pricing procedure is efficient even if the number of states is large provided the transition rates are not very large w.r.t. the riskless rates. The payoffs and riskless rates may depend on a state. Special cases are stochastic volatility models and models with stochastic interest rate; both must be modeled as finite-state Markov chains. As an application, we solve exit problems for a price-taking firm, and study the dependence of the exit threshold on the interest rate uncertainty.     

The optimal consumption function in a Brownian model of accumulation Part A: The consumption function as solution of a boundary value problem

We consider a neo-classical model of optimal economic growth with c.r.r.a. utility in which the traditional deterministic trends representing population growth, technological progress, depreciation and impatience are replaced by Brownian motions with drift. When transformed to ‘intensive’ units, this is equivalent to a stochastic model of optimal saving with diminishing returns to capital. For the intensive model, we give sufficient conditions for optimality of a consumption plan (open-loop control) comprising a finite welfare condition, a martingale condition for shadow prices and a transversality condition as t→∞. We then replace these by conditions for optimality of a plan generated by a consumption function (closed-loop control), i.e. a function expressing log-consumption as a time-invariant, deterministic function of log-capital . Making use of the exponential martingale formula we replace the martingale condition by a non-linear, non-autonomous second-order o.d.e. which an optimal consumption function must satisfy; this has the form , where . Economic considerations suggest certain limiting values which and should satisfy as , thus defining a two-point boundary value problem (b.v.p.) — or rather, a family of problems, depending on the values of parameters. We prove two theorems showing that a consumption function which solves the appropriate b.v.p. generates an optimal plan. Proofs that a unique solution of each b.v.p. exists are given in a separate paper (Part B).     

Dealing with a liquidity trap when government debt matters: Optimal time-consistent monetary and fiscal policy

How does the need to preserve government debt sustainability affect the optimal monetary and fiscal policy response to a liquidity trap? To provide an answer, we employ a small stochastic New Keynesian model with a zero bound on nominal interest rates and characterize optimal time-consistent stabilization policies. We focus on two policy tools, the short-term nominal interest rate and debt-financed government spending. The optimal policy response to a liquidity trap critically depends on the prevailing debt burden. While the optimal amount of government spending is decreasing in the level of outstanding government debt, future monetary policy is becoming more accommodative, triggering a change in private sector expectations that helps to dampen the fall in output and inflation at the outset of the liquidity trap.     

Pricing executive stock options under employment shocks

We obtain explicit expressions for the subjective, objective and market value of perpetual executive stock options (ESOs) under exogenous employment shocks driven by an independent Poisson process. Previously, we obtain the executive's optimal exercise policy from the subjective valuation that is necessary for the objective one, or fair value. The perpetual ESO is compared with the true finite maturity ESO finding that the approximation is reasonably good. To illustrate the usefulness of the objective valuation for accounting purposes, we analyze the statistical distribution of the fair value when there is uncertainty about the employment shock intensity. Finally, the role of ESOs in the design of executives’ incentives is also discussed.     

Analysis of optimal dynamic withdrawal policies in withdrawal guarantee products

Guaranteed Minimum Withdrawal Benefits (GMWB) are popular riders in variable annuities with withdrawal guarantees. With withdrawals spread over the life of the annuities contract, the benefit promises to return the entire initial annuitization amount irrespective of the market performance of the underlying fund portfolio. Treating the dynamic withdrawal rate as the control variable, the earlier works on GMWB have considered the construction of a continuous singular stochastic control model and the numerical solution of the resulting pricing model. This paper presents a more detailed characterization of the pricing properties of the GMWB and performs a full mathematical analysis of the optimal dynamic withdrawal policies under the competing factors of time value of fund, optionality value provided by the guarantee and penalty charge on excessive withdrawal. When a proportional penalty charge is applied on any withdrawal amount, we can reduce the pricing formulation to an optimal stopping problem with lower and upper obstacles. We then derive the integral equations for the determination of a pair of optimal withdrawal boundaries. When a proportional penalty charge is applied on the amount that is above the contractual withdrawal rate, we manage to characterize the behavior of the optimal withdrawal boundaries that separate the domain of the pricing models into three regions: no withdrawal, continuous withdrawal at the contractual rate and an immediate withdrawal of a finite amount. Under certain limiting scenarios such as a high policy fund value, the time close to expiry, or a low value of guarantee account, we manage to obtain analytical approximate solution to the singular stochastic control model of dynamic withdrawals.     

Optimal fiscal and monetary policy with occasionally binding zero bound constraints

This paper studies optimal fiscal and monetary policy when the nominal interest rate is subject to the zero lower bound (ZLB) constraint in a stochastic New Keynesian economy. In the baseline model calibrated to match key features of the U.S. economy, it is optimal for the government to increase its spending when at the ZLB in the stochastic environment by about 60 percent more than it would in the deterministic environment. The presence of uncertainty creates a unique time-consistency problem if the steady state is inefficient. Although access to government spending policy increases welfare in the face of a large deflationary shock, it decreases welfare during normal times as the government reduces the nominal interest rate less aggressively before reaching the ZLB.     

Good timing: The economics of optimal stopping

This paper presents an economic interpretation of the optimal “stopping” of perpetual project opportunities under both certainty and uncertainty. Prior to stopping, the expected rate of return from delay exceeds the rate of interest. The expected rate of return from delay is the sum of the expected rate of change in project value and the expected rate of change in the option premium associated with waiting. At stopping the expected rate of return from delay has fallen to the rate of interest. Viewing stopping in this way unifies the theoretical and practical insights of the theory of stopping under certainty and uncertainty.     

Consumption optimization for recursive utility in a jump-diffusion model

In this paper, we consider a market model with prices and consumption following a jump-diffusion dynamics. In this setting, we first characterize the optimal consumption plan for an investor with recursive stochastic differential utility on the basis of his/her own beliefs, then we solve the inverse problem to find what beliefs make a given consumption plan optimal. The problem is viewed in general for a class of homogeneous recursive utility, and later we choose a logarithmic model for the utility aggregator as an explicitly computable example. When beliefs, represented via Girsanov’s theorem, get incorporated into the model, the change of measure gives rise, up to a transformation, to a backward stochastic differential equation whose generator exhibits a quadratic behavior in the Brownian component and a locally Lipschitz one in the jump component, which is solvable on the basis of some recent results.     

Mortgage Valuation with Optimal Intertemporal Refinancing Strategies

This article extends previous work on mortgage valuation in two ways. First, I identify the prepayment boundary by solving for the borrower's optimal prepayment strategy over the expected tenure in the house. Previous work has treated the prepayment decision as a one-time decision, not as an element of a multiperiod strategy. Second, the model incorporates borrower heterogeneity in terms of expected tenure in the house. The results show that the optimal refinancing strategy differs significantly from a sequence of one-time decisions. A borrower following the optimal strategy is less aggressive in refinancing and pays more interest and less transaction costs than does a borrower following a myopic strategy. Estimated mortgage values are higher and interest rate sensitivity is lower when compared to values calculated using the traditional approach.     

随机波动率LIBOR市场利率动态模型的理论估计与蒙特卡罗模拟

本文首先基于诸多Libor市场模型改进方法的基础之上,在标准市场模型中加入Heston随机波动率过程,建立随机波动率假设的新型Libor市场模型;其次,运用Black逆推参数校正方法和MCMC参数估计方法对该Libor利率市场模型中的局部波动率和随机波动率过程中的参数进行校正和估计;最后是实证模拟。研究结论认为,在构建Libor利率动态模型时,若在单因子Libor利率市场模型基础上引入随机波动率过程,可大大提高利率模型的解释力。     

Egalitarianism under population change: Age structure does matter

We study the compatibility of the optimal population size concepts produced by different social welfare functions and egalitarianism meant as “equal consumption for all individuals of all generations”. Social welfare functions are parameterized by an altruism parameter generating the Benthamite and Millian criteria as polar cases. The economy considered is in continuous time and is populated by homogeneous cohorts with a given life span. Production functions are linear in labor, (costly) procreation is the unique way to transfer resources forward in time. First, we show that egalitarianism is optimal whatever the degree of altruism in “perpetual youth” model, that is when lifetime span is finite but age structure does not matter: in this case egalitarianism does not discriminate between the social welfare functions considered. Then we show that, when life span is finite but age structure matters, egalitarianism does not arise systematically as an optimal outcome. In particular, in a growing economy, that is when population growth is optimal in the long-run, this egalitarian rule can only hold when the welfare function is Benthamite. When altruism is impure, egalitarianism is impossible in the context of a growing economy. Either in the Benthamite or impure altruism cases, procreation is never optimal for small enough life spans, leading to finite time extinction and maximal consumption for all existing individuals.     

On the asymptotic behavior of stochastic economic processes : Two examples from intertemporal allocation under uncertainty

Some of the recently developed models to deal with economic problems involving uncertainty are based on simplifying assumptions on the nature of the stochastic law of the environment influencing economic decisions. Relying on the theory of martingales, we derive some general results on the asymptotic behavior of two dynamic processes that are of interest in the theory of intertemporal resource allocation. The first example is related to the ‘turnpike’ theory of optimal allocation. The second is addressed to the question of allocation of a scarce resource by using prices when the supply of the resource is random.     

内部收益保证下养老基金的最优资产配置

研究内部收益保证下DC型养老基金的最优资产配置问题。利用鞅方法，在HJM利率期限结构下求得了最优资产配置的显性解。结论表明最优投资策略分为四部分：投机策略、利率套期保值策略、基准组合的复制策略及一揽予债券卖空策略。最后对最优策略的动态行为进行了数值分析。     

Optimal structure of joint inventory-pricing management with dual suppliers and different lead times

We consider a joint inventory-pricing control problem in a single-product, periodic-review, dual-supplier inventory system. The two suppliers have different lead times. One expedited supplier offers instantaneous replenishment, and one regular supplier requires an L-period lead time for delivery. The supply quantity is stochastic and the demand is price-dependent. For the expedited inventory replenishment, we characterize the optimal policy as a state-dependent almost-threshold policy by extending the stochastically linear in mid-point to a multidimensional setting. To investigate the optimal regular inventory replenishment and pricing policy, we propose the notions of partially stochastic translation (PST) and increasing partially stochastic translation (IPST), which help in obtaining the antimultimodularity preservation in dynamic programming problems. We provide properties, sufficient conditions, and examples for PST and IPST functions. By applying PST and IPST, we obtain the antimultimodularity of the profit functions. The antimultimodular profit functions ensure that the optimal regular ordering quantity and the optimal price are monotone in the current inventory level and outstanding order quantities. Moreover, we reveal that as the time interval increases, the effects of previous outstanding orders on the optimal regular ordering and pricing decisions are decreasing and increasing, respectively. PST and IPST also enable us to further characterize the optimal expedited ordering quantity as decreasing in the inventory level. However, the optimal expedited ordering quantity can be non-monotone with respect to the outstanding order quantities, as shown in the example.     

Can a stochastic cusp catastrophe model explain housing market crashes?

Like stock market prices, housing prices often exhibit temporary booms and busts. A possible explanation for the observed abrupt changes is offered by the stochastic catastrophe model. This paper addresses the question whether the catastrophe model can describe and predict the dynamics of housing markets. We fit a stochastic cusp catastrophe model to empirical housing market data for six OECD countries, US, JP, UK, NL, SE and BE. Two different estimation approaches are considered – Cobb׳s method and Euler discretization. The analysis shows that while Cobb׳s approach describes the long-run stationary density better, Euler discretization is more tailored for time series, as it provides better one-step-ahead predictions. Proceeding using the Euler discretization method we discuss the dynamics of housing markets in terms of the multiple equilibria cusp catastrophe model. By considering the long-term interest rate as an exogenous variable we obtain new insights into the policy implications of interest rate levels, in particular concerning the stability of housing markets.     

Usual stochastic ordering of the sample maxima from dependent distribution-free random variables

In this paper, we discuss stochastic comparison of the largest order statistics arising from two sets of dependent distribution-free random variables with respect to multivariate chain majorization, where the dependency structure can be defined by Archimedean copulas. When a distribution-free model with possibly two parameter vectors has its matrix of parameters changing to another matrix of parameters in a certain mathematical sense, we obtain the first sample maxima is larger than the second sample maxima with respect to the usual stochastic order, based on certain conditions. Applications of our results for scale proportional reverse hazards model, exponentiated gamma distribution, Gompertz–Makeham distribution, and location-scale model, are also given. Meanwhile, we provide two numerical examples to illustrate the results established here.     

A flexible lattice framework for valuing options on assets paying discrete dividends and variable annuities embedding GMWB riders

In a market where a stochastic interest rate component characterizes asset dynamics, we propose a flexible lattice framework to evaluate and manage options on equities paying discrete dividends and variable annuities presenting some provisions, like a guaranteed minimum withdrawal benefit. The framework is flexible in that it allows to combine financial and demographic risk, to embed in the contract early exercise features, and to choose the dynamics for interest rates and traded assets. A computational problem arises when each dividend (when valuing an option) or withdrawal (when valuing a variable annuity) is paid, because the lattice lacks its recombining structure. The proposed model overcomes this problem associating with each node of the lattice a set of representative values of the underlying asset (when valuing an option) or of the personal subaccount (when valuing a variable annuity) chosen among all the possible ones realized at that node. Extensive numerical experiments confirm the model accuracy and efficiency.

     

Approximate arbitrage-free option pricing under the SABR model

The stochastic-alpha-beta-rho (SABR) model introduced by Hagan et al. (2002) provides a popular vehicle to model the implied volatilities in the interest rate and foreign exchange markets. To exclude arbitrage opportunities, we need to specify an absorbing boundary at zero for this model, which the existing analytical approaches to pricing derivatives under the SABR model typically ignore. This paper develops closed-form approximations to the prices of vanilla options to incorporate the effect of such a boundary condition. Different from the traditional normal distribution-based approximations, our method stems from an expansion around a one-dimensional Bessel process. Extensive numerical experiments demonstrate its accuracy and efficiency. Furthermore, the explicit expression yielded from our method is appealing from the practical perspective because it can lead to fast calibration, pricing, and hedging.