An optimal stopping problem in risk theory

Abstract

In classical risk theory often stationary premium and claim processes are considered. In some cases it is more convenient to model non-stationary processes which describe a movement from environmental conditions, for which the premiums were calculated, to less favorable circumstances. This is done by a Markov-modulated Poisson claim process. Moreover the insurance company is allowed to stop the process at some random time, if the situation seems unfavorable, in order to calculate new premiums. This leads to an optimal stopping problem which is solved explicitly to some extent.     

An optimal execution problem with market impact

We study an optimal execution problem in a continuous-time market model that considers market impact. We formulate the problem as a stochastic control problem and investigate properties of the corresponding value function. We find that right-continuity at the time origin is associated with the strength of market impact for large sales; otherwise the value function is continuous. Moreover, we show the semigroup property (Bellman principle) and characterise the value function as a viscosity solution of the corresponding Hamilton–Jacobi–Bellman equation. We present some examples where the form of the optimal strategy changes completely, depending on the amount of the trader’s security holdings, and where optimal strategies in the Black–Scholes type market with nonlinear market impact are not block liquidation but gradual liquidation, even when the trader is risk-neutral.     

Free boundary and optimal stopping problems for American Asian options

We give a complete and self-contained proof of the existence of a strong solution to the free boundary and optimal stopping problems for pricing American path-dependent options. The framework is sufficiently general to include geometric Asian options with nonconstant volatility and recent path-dependent volatility models.      

An optimal investment model with Markov-driven volatilities

We consider a multi-stock market model. The processes of stock prices are governed by stochastic differential equations with stock return rates and volatilities driven by a finite-state Markov process. Each volatility is also disturbed by a Brownian motion; more exactly, it follows a Markov-driven Ornstein–Uhlenbeck process. Investors can observe the stock prices only. Both the underlying Brownian motion and the Markov process are unobservable. We study a discretized version, which is a discrete-time hidden Markov process. The objective is to control trading at each time step to maximize an expected utility function of terminal wealth. Exploiting dynamic programming techniques, we derive an approximate optimal trading strategy that results in an expected utility function close to the optimal value function. Necessary filtering and forecasting techniques are developed to compute the near-optimal trading strategy.     

A note on the optimal control of a cash balance problem

This paper provides an analytical solution to a cash management problem when cash income and demand are described by Compound Poisson processes. The paper generalizes past results in the cash management literature to arbitrary income and demand distribution functions. Further, our results can be applied as well in the area of banking. Throughout the paper we restrict attention to the family of control barrier policies. These consist in hedging cash up to a critical level and investing all incoming cash exceeding this level. We employ a long-run average cost criterion to determine an optimal control barrier. A diffusion approximation of the cash level process (income less demand) is used to obtain a simpler expression for the average cost and to yield a closed form solution to the optimal control barrier. For demonstration purposes, an example is resolved.     

Taxation and the optimal constraint on corporate debt finance: why a comprehensive business income tax is suboptimal

The tax bias in favour of debt finance under the corporate income tax means that corporate debt ratios exceed the socially optimal level. This creates a rationale for a general thin capitalization rule limiting the amount of debt that qualifies for interest deductibility. This paper sets up a model of corporate finance and investment in a small open economy to identify the optimal constraint on tax-favoured debt finance, assuming that a given amount of revenue has to be raised from the corporate income tax. For plausible parameter values, the socially optimal debt-asset ratio is 2–3% points below the average corporate debt level currently observed. Driving the actual debt ratio down to this level through limitations on interest deductibility would generate a total welfare gain of about 5% of corporate tax revenue. The welfare gain would arise mainly from a fall in the social risks associated with corporate investment, but also from the cut in the corporate tax rate made possible by a broader corporate tax base.     

Optimal design of equity-linked products with a probabilistic constraint

There is a rich variety of tailored investment products available to the retail investor. These products combine upside participation in bull markets with downside protection in bear markets. Examples include the equity-linked products sold by insurance companies and the structured products marketed by banks. This paper examines a particular contract design for products of this nature. The paper finds the optimal design from the investor's viewpoint. It is assumed that the investor wishes to maximize expected utility of the terminal wealth subject to certain constraints. These constraints include a guaranteed rate of return as well as the opportunity to outperform a benchmark portfolio with a given probability. We derive the explicit form of the optimal design assuming both constraints apply and we illustrate the nature of the solution using some specific examples.     

A risk-sensitive stochastic control approach to an optimal investment problem with partial information

We consider an infinite time horizon optimal investment problem where an investor tries to maximize the probability of beating a given index. From a mathematical viewpoint, this is a large deviation probability control problem. As shown by Pham (in Syst. Control Lett. 49: 295–309, 2003; Financ. Stoch. 7: 169–195, 2003), its dual problem can be regarded as an ergodic risk-sensitive stochastic control problem. We discuss the partial information counterpart of Pham (in Syst. Control Lett. 49: 295–309, 2003; Financ. Stoch. 7: 169–195, 2003). The optimal strategy and the value function for the dual problem are constructed by using the solution of an algebraic Riccati equation. This equation is the limit equation of a time inhomogeneous Riccati equation derived from a finite time horizon problem with partial information. As a result, we obtain explicit representations of the value function and the optimal strategy for the problem. Furthermore we compare the optimal strategies and the value functions in both full and partial information cases.

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The agency problem,investment decision,and optimal financial structure

This article constructs a real options model in which a firm has a privileged right to exercise an irreversible investment project with a stochastic payoff. Supposing that the investment costs are fully sunk, a firm that exercises the investment option after debt is in place will then choose a better state to exercise this option as it issues more bonds. This debt-overhang phenomenon, however, benefits the firm since waiting is itself valuable. Accordingly, the firm will both exercise the investment option later and issue more bonds as compared with a firm that issues bonds upon exercising the investment option.     

Optimal insurance contract under a value-at-risk constraint

This study develops an optimal insurance contract endogenously under a value-at-risk (VaR) constraint. Although Wang et al. [2005] had examined this problem, their assumption implied that the insured is risk neutral. Consequently, this study extends Wang et al. [2005] and further considers a more realistic situation where the insured is risk averse. The study derives the optimal insurance contract as a single deductible insurance when the VaR constraint is redundant or as a double deductible insurance when the VaR constraint is binding. Finally, this study discusses the optimal coverage level from common forms of insurances, including deductible insurance, upper-limit insurance, and proportional coinsurance. JEL Classification G22     

Optimal investment-reinsurance with dynamic risk constraint and regime switching

We study an optimal investment–reinsurance problem for an insurer who faces dynamic risk constraint in a Markovian regime-switching environment. The goal of the insurer is to maximize the expected utility of terminal wealth. Here the dynamic risk constraint is described by the maximal conditional Value at Risk over different economic states. The rationale is to provide a prudent investment–reinsurance strategy by taking into account the worst case scenario over different economic states. Using the dynamic programming approach, we obtain an analytical solution of the problem when the insurance business is modeled by either the classical Cramer–Lundberg model or its diffusion approximation. We document some important qualitative behaviors of the optimal investment–reinsurance strategies and investigate the impacts of switching regimes and risk constraint on the optimal strategies.     

Long-term optimal portfolios with floor

Long-term risk-sensitive portfolio optimization is studied with floor constraint. A simple rule to characterize its solution is mentioned under a general setting. Following this rule, optimal portfolios are constructed in several ways, using the optimal portfolio without floor constraint, combined with ideas of dynamic portfolio insurance, such as CPPI (constant proportion portfolio insurance), OBPI (option-based portfolio insurance), and DFP (dynamic fund protection). In addition, examples are presented with explicit computations of solutions.     

Optimal investment of an insurer with regime-switching and risk constraint

We investigate an optimal investment problem of an insurance company in the presence of risk constraint and regime-switching using a game theoretic approach. A dynamic risk constraint is considered where we constrain the uncertainty aversion to the ‘true’ model for financial risk at a given level. We describe the surplus of an insurance company using a general jump process, namely, a Markov-modulated random measure. The insurance company invests the surplus in a risky financial asset whose dynamics are modeled by a regime-switching geometric Brownian motion. To incorporate model uncertainty, we consider a robust approach, where a family of probability measures is cosidered and the insurance company maximizes the expected utility of terminal wealth in the ‘worst-case’ probability scenario. The optimal investment problem is then formulated as a constrained two-player, zero-sum, stochastic differential game between the insurance company and the market. Different from the other works in the literature, our technique is to transform the problem into a deterministic differential game first, in order to obtain the optimal strategy of the game problem explicitly.