No-arbitrage up to random horizon for quasi-left-continuous models

This paper studies the impact, on no-arbitrage conditions, of stopping the price process at an arbitrary random time. As price processes, we consider the class of quasi-left-continuous semimartingales, i.e., semimartingales that do not jump at predictable stopping times. We focus on the condition of no unbounded profit with bounded risk (called NUPBR), also known in the literature as no arbitrage of the first kind. The first principal result describes all the pairs of quasi-left-continuous market models and random times for which the resulting stopped model fulfils NUPBR. Furthermore, for a subclass of quasi-left-continuous local martingales, we construct explicitly martingale deflators, i.e., strictly positive local martingales whose product with the price process stopped at a random time is a local martingale. The second principal result characterises the random times that preserve NUPBR under stopping for any quasi-left-continuous model. The analysis carried out in the paper is based on new stochastic developments in the theory of progressive enlargements of filtrations.     

The valuation and behavior of black-scholes options subject to intertemporal default risk

This paper addresses the valuation and behavior of European options subject to intertemporal writer default risk. The framework allows the timing of default and recovery value to be uncertain. Default is said to occur if the writer's creditworthiness violates a specified critical level-both stochastic. Various recovery scenarios are considered including linking recovery to the moneyness of the option at the time of default. In an application of the model, it is estimated that current customer margin requirements for exchange-traded options are set far in excess of the fair market value.     

Fiscal Multipliers under an Interest Rate Peg of Deterministic versus Stochastic Duration

This paper revisits the size of the fiscal multiplier. The experiment is a fiscal expansion under the assumption of a pegged nominal rate of interest. We demonstrate that a quantitatively important issue is the articulation of the exit from the policy experiment. If the monetary‐fiscal expansion is stochastic with a mean duration of T periods, the fiscal multiplier can be unboundedly large. However, if the monetary‐fiscal expansion is for a fixed T periods, the multiplier is much smaller. Our explanation rests on a Jensen's inequality type argument: the deterministic multiplier is convex in duration, and the stochastic multiplier is a weighted average of the deterministic multipliers. The quantitative difference in the two multipliers also arises in a model with capital, and in the baseline nonlinear model. However, the differences between the two are less pronounced in the nonlinear models. The errors from a linear approximation are much larger for the stochastic exit model then for the deterministic exit model.     

Signaling with dividends and share repurchases: a choice between deterministic and stochastic cash disbursements

We study firms signaling with cash disbursements and show thatthe choice of a deterministic or a stochastic disbursement dependson a property of the firm's production function that is analogousto absolute risk aversion for a utility function. With decreasing(increasing) absolute risk aversion, the high-quality firm prefersto distinguish itself from the low-quality firm with a stochastic(deterministic) outlay. We then study in detail two common formsof corporate cash distributions: dividends, a deterministicdisbursement, and share repurchases, a stochastic disbursement.     

Reserves and cash flows under stochastic retirement

Uncertain time of retirement and uncertain structure of retirement benefits are risk factors for life insurance companies. Nevertheless, classical life insurance models assume these are deterministic. In this paper, we include the risk from stochastic time of retirement and stochastic benefit structure in a classical finite-state Markov model for a life insurance contract. We include discontinuities in the distribution of the retirement time. First, we derive formulas for appropriate scaling of the benefits according to the time of retirement and discuss the link between the scaling and the guarantees provided. Stochastic retirement creates a need to rethink the construction of disability products for high ages and ways to handle this are discussed. We show how to calculate market reserves and how to use modified transition probabilities to calculate expected cash flows without significantly more complexity than in the traditional model. At last, we demonstrate the impact of stochastic retirement on market reserves and expected cash flow in numerical examples.     

The Dynamics of Discrete Bid and Ask Quotes

This paper presents an empirical microstructure model of bid and ask quotes that features discreteness, random costs of market making, and ARCH volatility effects. Applied to intraday quotes at 15-minute intervals for Alcoa (a randomly chosen Dow stock), the results show that quote exposure costs contain stochastic components that are persistent and large relative to the deterministic intraday "U" components. Analysis of the filtered estimates of the system suggest that bid and ask costs contain common components, and that these costs reflect risk as proxied by ARCH variance forecasts.     

Compound trend renewal process with discounted claims: a unified approach

We derive recursive formulas for the moments of compound trend renewal sums with discounted claims. An integral expression for the moment generating function of this risk process is then obtained, from which particular distribution functions are found. We extend the compound (deterministic) trend renewal process by assuming a stochastic trend, a stochastic force of net interest and a stochastic dependence between the inter-occurrence times and the severities of the claims. Finally, stochastic dominance ordering is also observed between the compound trend renewal process and an associated non-homogeneous Poisson process.     

Does Insurance Help to Escape the Poverty Trap?—A Ruin Theoretic Approach

Poverty trapping refers to the fact that poor people in developing countries cannot escape their poverty without help from outside. This is worsened by extreme events, for example, floods or hurricanes, sending people to poverty who have not been poor before. Often, insurance is seen as a way out. This article studies poverty trapping in the context of catastrophic risk and introduces a ruin‐type model, combining deterministic growth with a stochastic loss model. We analyze the properties of the resulting piecewise deterministic Markov process, especially its trapping risk, and discuss for which groups of people insurance can reduce trapping probability.     

Volatility dynamics under duration-dependent mixing

This paper proposes a discrete-state stochastic volatility model with duration-dependent mixing. The latter is directed by a high-order Markov chain with a sparse transition matrix. As in the standard first-order Markov switching (MS) model, this structure can capture turning points and shifts in volatility, due for example, to policy changes or news events. However, the duration-dependent Markov switching model (DDMS) can also exploit the persistence associated with volatility clustering. To evaluate the contribution of duration dependence, we compare with a benchmark Markov switching-ARCH (MS-ARCH) model. The empirical distribution generated by our proposed structure is assessed using interval forecasts and density forecasts. Implications for areas of the distribution relevant to risk management are also assessed.     

Bounds for Right Tails of Deterministic and Stochastic Sums of Random Variables

We investigate lower and upper bounds for right tails (stop-loss premiums) of deterministic and stochastic sums of nonindependent random variables. The bounds are derived using the concepts of comonotonicity, convex order, and conditioning. The performance of the presented approximations is investigated numerically for individual life annuity contracts as well as for life annuity portfolios, where mortality is modeled by Makeham's law, whereas investment returns are modeled by a Brownian motion process.     

Relevant costs,congestion and stochasticity in production environments

Conventional management accounting principles used to evaluate relevant costs have been developed under the assumption of deterministic manufacturing settings. Manufacturing operations, however, are complex and stochastic. In this paper we examine the impact of stochasticity in the production process on relevant costs based on a dynamic assessment of capacity constraints. We develop a model to analyze the behavior of relevant costs with respect to changes in the expected duration and variability in set-ups and processing. An implication of this analysis is that for profit maximization capacity will exceed expected demand if production rates or demand are stochastic.     

Trends and random walks in macroeconmic time series: Some evidence and implications

This paper investigates whether macroeconomic time series are better characterized as stationary fluctuations around a deterministic trend or as non-stationary processes that have no tendency to return to a deterministic path. Using long historical time series for the U.S. we are unable to reject the hypothesis that these series are non-stationary stochastic processes with no tendency to return to a trend line. Based on these findings and an unobserved components model for output that decomposes fluctuations into a secular or growth component and a cyclical component we infer that shocks to the former, which we associate with real disturbances, contribute substantially to the variation in observed output. We conclude that macroeconomic models that focus on monetary disturbances as a source of purely transitory fluctuations may never be successful in explaining a large fraction of output variation and that stochastic variation due to real factors is an essential element of any model of macroeconomic fluctuations.     

Seasonal and stochastic effects in commodity forward curves

In this paper we develop a new model for the dynamics of forward curves of commodities exhibiting seasonalities, such as natural gas, electricity or agricultural commodities. In the existing literature on the subject, the first state variable in multi-factor models is the commodity price, which combines seasonal and stochastic features and may be unobservable. We propose to use instead the average forward price, which is devoid of seasonality and conveys a more robust representation of the current forward curve level. The second factor in the model is a quantity analogous to the stochastic convenience yield, which accounts for the random changes in the forward curve shape. The well-known cost-of-carry relationship is significantly improved by introducing a deterministic seasonal premium within the convenience yield. We develop model estimation procedures and apply them to a number of energy markets.     

Modeling of Contagious Credit Events and Risk Analysis of Credit Portfolios

We present a new model of the occurence of credit events such as rating changes and defaults for risk analyses of some portfolio credit derivatives. The framework of our model is based on a so-called top-down approach. Specifically, we first consider modeling the point process of each type of credit event in the whole economy using a self-exciting intensity process. Next, we characterize the point processes of credit events in the underlying sub-portfolio using random thinning processes specified by the distribution of credit ratings in the sub-portfolio. One of the main features of our model is that the model can capture credit risk contagion simultaneously among several credit portfolios. We present a credit event simulation algorithm based on our model and illustrate an application of the model to risk analyses of loan portfolios.     

A model for evaluating extreme risks with stochastic sustainability criteria: a case study of soil remediation on landfill sites

This article addresses the issue of risk management for extreme events. The term stochastic sustainability will be defined in relation to the risk management of extreme events. On the basis of stochastic sustainability, a risk evaluation function is defined as an efficient reserve to remedy contaminations just as a premium rating in insurance policy. A term of generalized safety loading is used as a key concept to evaluate the degree of extremeness of risk. A simple model of the environmental or infrastructure systems is used to illustrate the procedure of risk evaluation where stochastic sustainability is required in case of accidental

environmental contamination. A case study of soil contamination at landfill sites is presented as an example to test this risk evaluation function.     

High-speed rail transport valuation and conjecture shocks

In this paper, we derive the optimal investment policy in a high-speed rail transport (HSR) project. We assume that the source of uncertainty comes from the annual demand, and that it follows a geometric Brownian motion with jumps of random magnitude, occurring in random times, according to a Poisson process. We assess the impact of these shocks on the demand threshold, along with the investment opportunity value and option to differ. We consider several distributions for these jumps, and we compare with the no-jumps case. Numerical results are presented, showing the importance of assumptions about the underlying stochastic process.     

Are the Liquidity and Collateral Roles of Asset Bubbles Different?

Several papers explain why asset bubbles are observed when growth is large. These papers differ in the role of the bubble, used to provide liquidities or as collateral in a borrowing constraint. We compare the liquidity and collateral roles of bubbles in an overlapping generations model. When the bubble is deterministic, the equilibrium is identical under these two roles, implying that the same mechanism explains the crowding-in effect of the bubble on growth. With stochastic bubbles, growth is larger when bubbles play the liquidity role, because the burst of a bubble used for liquidity is less damaging to capital investors.     

Stochastic duration and fast coupon bond option pricing in multi-factor models

Generalizing Cox, Ingersoll, and Ross (1979), this paper defines the stochastic duration of a bond in a general multi-factor diffusion model as the time to maturity of the zero-coupon bond with the same relative volatility as the bond. Important general properties of the stochastic duration measure are derived analytically, and the stochastic duration is studied in detail in various well-known models. It is also demonstrated by analytical arguments and numerical examples that the price of a European option on a coupon bond (and, hence, of a European swaption) can be approximated very accurately by a multiple of the price of a European option on a zero-coupon bond with a time to maturity equal to the stochastic duration of the coupon bond. This revised version was published online in June 2006 with corrections to the Cover Date.     

First-Order Mortality Basis for Life Annuities

Mortality improvements pose a challenge for the life annuity business. For the management of such portfolios, it is important to forecast future mortality rates. Standard models for mortality forecasting assume that the force of mortality at age x in calendar year t is of the form exp, where the dynamics of the time index is described by a random walk with drift. Starting from such a best estimate of future mortality (called second-order mortality basis in actuarial science), the paper explains how to determine a conservative life table serving as first-order mortality basis. The idea is to replace the stochastic projected life table with a deterministic conservative one, and to assume mutual independence for the remaining life times. The paper then studies the distribution of the present value of the payments made to a closed group of annuitants. It turns out that De Pril–Panjer algorithm can be used for that purpose under first-order mortality basis. The connection with ruin probabilities is briefly discussed. An inequality between the distribution of the present value of future annuity payments under first-order and second-order mortality basis is provided, which allows to link value-at-risk computed under these two sets of assumptions. A numerical example performed on Belgian mortality statistics illustrates how the approach proposed in this paper can be implemented in practice.     

Optimal capital growth with convex shortfall penalties

The optimal capital growth strategy or Kelly strategy has many desirable properties such as maximizing the asymptotic long-run growth of capital. However, it has considerable short-run risk since the utility is logarithmic, with essentially zero Arrow–Pratt risk aversion. It is common to control risk with a Value-at-Risk (VaR) constraint defined on the end of horizon wealth. A more effective approach is to impose a VaR constraint at each time on the wealth path. In this paper, we provide a method to obtain the maximum growth while staying above an ex-ante discrete time wealth path with high probability, where shortfalls below the path are penalized with a convex function of the shortfall. The effect of the path VaR condition and shortfall penalties is a lower growth rate than the Kelly strategy, but the downside risk is under control. The asset price dynamics are defined by a model with Markov transitions between several market regimes and geometric Brownian motion for prices within a regime. The stochastic investment model is reformulated as a deterministic programme which allows the calculation of the optimal constrained growth wagers at discrete points in time.