A theory of Markovian time-inconsistent stochastic control in discrete time

We develop a theory for a general class of discrete-time stochastic control problems that, in various ways, are time-inconsistent in the sense that they do not admit a Bellman optimality principle. We attack these problems by viewing them within a game theoretic framework, and we look for subgame perfect Nash equilibrium points. For a general controlled Markov process and a fairly general objective functional, we derive an extension of the standard Bellman equation, in the form of a system of nonlinear equations, for the determination of the equilibrium strategy as well as the equilibrium value function. Most known examples of time-inconsistent stochastic control problems in the literature are easily seen to be special cases of the present theory. We also prove that for every time-inconsistent problem, there exists an associated time-consistent problem such that the optimal control and the optimal value function for the consistent problem coincide with the equilibrium control and value function, respectively for the time-inconsistent problem. To exemplify the theory, we study some concrete examples, such as hyperbolic discounting and mean–variance control.     

The Moments of the Time of Ruin in Markovian Risk Models

Abstract

We present an approach based on matrix-analytic methods to find moments of the time of ruin in Markovian risk models. The approach is applicable when claims occur according to a Markovian arrival process (MAP) and claim sizes are phase distributed with parameters that depend on the state of the MAP. The method involves the construction of a sample-path-equivalent Markov-modulated fluid flow for the risk model. We develop an algorithm for moments of the time of ruin and prove the algorithm is convergent. Examples show that the proposed approach is computationally stable.     

Risk measures based on behavioural economics theory

Coherent risk measures (Artzner et al. in Math. Finance 9:203–228, 1999) and convex risk measures (Föllmer and Schied in Finance Stoch. 6:429–447, 2002) are characterized by desired axioms for risk measures. However, concrete or practical risk measures could be proposed from different perspectives. In this paper, we propose new risk measures based on behavioural economics theory. We use rank-dependent expected utility (RDEU) theory to formulate an objective function and propose the smallest solution that minimizes the objective function as a risk measure. We also employ cumulative prospect theory (CPT) to introduce a set of acceptable regulatory capitals and define the infimum of the set as a risk measure. We show that the classes of risk measures derived from RDEU theory and CPT are equivalent, and they are all monetary risk measures. We present the properties of the proposed risk measures and give sufficient and necessary conditions for them to be coherent and convex, respectively. The risk measures based on these behavioural economics theories not only cover important risk measures such as distortion risk measures, expectiles and shortfall risk measures, but also produce new interesting coherent risk measures and convex, but not coherent risk measures.     

On the analysis of a multi-threshold Markovian risk model

We consider a class of Markovian risk models perturbed by a multiple threshold dividend strategy in which the insurer collects premiums at rate c _i whenever the surplus level resides in the i-th surplus layer, i=1, 2, …,n+1 where n<∞. We derive the Laplace-Stieltjes transform (LST) of the distribution of the time to ruin as well as the discounted joint density of the surplus prior to ruin and the deficit at ruin. By interpreting that the insurer, whose gross premium rate is c, pays dividends continuously at rate d _i =c?c _i whenever the surplus level resides in the i-th surplus layer, we also derive the expected discounted value of total dividend payments made prior to ruin. Our results are obtained via a recursive approach which makes use of an existing connection, linking an insurer's surplus process to an embedded fluid flow process.     

A Markovian cross-impact model

The author describes a cross-impact method which is based on Markov-chain theory. This allows scenarios to be derived which explicitly include the time dimension. An example of the use of the method is included which deals with the installation of nuclear reactors for energy production.     

An interest rate model with a Markovian mean reverting level

Abstract

A two‐factor Vasicek model, where the mean reversion level changes according to a continuous time finite state Markov chain, is considered. This model could capture the behaviour of monetary authorities who normally set a reference rate which changes from time to time. We derive the term structure via the analytic expression of the bond price that involves a fundamental matrix. The validity of the bond price closed form solution is verified via the forward rate dynamics.     

Risk aversion with state-dependent preferences in the rank-dependent expected utility theory

We extend the analysis of risk aversion with state-dependent preferences to the rank-dependent expected utility theory. We find that in this extended theory, for two preference relations to be comparable in risk aversion not only do their reference sets need to coincide (a condition first introduced by Karni [1983, 1985] in the original expected utility framework), but they must also rank the prospective state-dependent outcomes in the same manner. We formalize this additional condition by introducing the concept of certainty sets. Under our condition of comparability, various results and characterizations of interpersonal comparison of risk aversion are obtained. The implications for a specific insurance problem are also discussed.     

Pricing and managing risks of European-style options in a Markovian regime-switching binomial model

We discuss the pricing and risk management problems of standard European-style options in a Markovian regime-switching binomial model. Due to the presence of an additional source of uncertainty described by a Markov chain, the market is incomplete, so the no-arbitrage condition is not sufficient to fix a unique pricing kernel, hence, a unique option price. Using the minimal entropy martingale measure, we determine a pricing kernel. We examine numerically the performance of a simple hedging strategy by investigating the terminal distribution of hedging errors and the associated risk measures such as Value at Risk and Expected Shortfall. The impacts of the frequency of re-balancing the hedging portfolio and the transition probabilities of the modulating Markov chain on the quality of hedging are also discussed.     

Markovian Approaches to Joint-Life Mortality

Abstract

Many insurance products provide benefits that are contingent on the combined survival status of two lives. To value such benefits accurately, we require a statistical model for the impact of the survivorship of one life on another. In this paper we first set up two models, one Markov and one semi-Markov, to model the dependence between the lifetimes of a husband and wife. From the models we can measure the extent of three types of dependence: (1) the instantaneous dependence due to a catastrophic event that affect both lives, (2) the short-term impact of spousal death, and (3) the long-term association between lifetimes. Then we apply the models to a set of jointlife and last-survivor annuity data from a large Canadian insurance company. Given the fitted models, we study the impact of dependence on annuity values and examine the potential inaccuracy in pricing if we assume lifetimes are independent. Finally, we compare our Markovian models with two copula models considered in previous research on modeling joint-life mortality.     

A note on the existence of unique equivalent martingale measures in a Markovian setting

Simple sufficient conditions for the existence of a unique equivalent martingale measure are provided. Furthermore, these conditions give us a handle on situations where an equivalent martingale measure cannot exist. The existence of a unique equivalent martingale measure is of relevance to problems in mathematical finance. Two examples of models for which the question of existence was unresolved are studied. By means of our results existence of a unique equivalent measure up to an explosion time is proved.     

Risk can be good for self-esteem: beyond self-determination theory

Abstract

Despite many decades of research that has highlighted all risk-taking sport activities as a means to satisfy sensation seeking needs (e.g., Zuckerman 1979 Zuckerman, M. 1979. Sensation Seeking: Beyond the Optimal Level of Arousal. Cambridge, UK: Cambridge University Press. [Google Scholar]), recent research has challenged that view and has revealed that some high-risk activities provide opportunities for agentic emotion regulation during participation, and are not driven by sensation-seeking needs (e.g., Barlow, Woodman, and Hardy 2013 Barlow, M., T. Woodman, and L. Hardy. 2013. “Great Expectations: Different High-Risk Activities Satisfy Different Motives.” Journal of Personality & Social Psychology 105: 458–475. doi:10.1037/a0033542.[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]). Participation in high-risk sports is also associated with increased self-esteem (e.g., A?çi, Demirhan, and Dinç 2007 A?çi, F. H., G. Demirhan, and S. C. Dinç. 2007. “Psychological Profile of Turkish Rock Climbers: An Examination of Climbing Experience and Route Difficulty.” Perceptual and Motor Skills 104 (3): 892–900. doi:10.2466/pms.104.3.892-900.[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]). The aim of the present study was to investigate the link between the agentic and emotion regulation benefits of specific high-risk activities and any associated self-esteem benefits. We hypothesized that the emotion regulation and agency experiences in high-risk physical activities would mirror the elevated self-esteem derived from these activities. We examined high-risk activity (n?=?84), low-risk activity (n?=?65), and control (n?=?45) groups and found that the experience of agentic emotion regulation was greater during participation for high-risk sport participants. High-risk sport participants also had less post-activity difficulty with emotion regulation and higher self-esteem. This study provides the first support that activities that require greater agentic emotion regulation during participation also lead to elevated self-esteem. Basic psychological needs satisfaction did not account for the differences between groups, suggesting that people have other needs (e.g., the need to self-regulate) that are not incorporated into self-determination theory.     

Strategic foresight in a high-speed environment

This paper explores how a top management team developed strategic foresight and decided to launch an Internet bank in a context of uncertainty about the future take up of e-commerce. For this purpose, a single inductive case study is used. The settings are those of the UK financial services industry, characterised by rapid change, mainly driven by the new technology. The focus of analysis is Sunshine, a stand-alone Internet bank. The study, which is part of a broader project on the management of innovation in financial services, is based on qualitative data captured from semi-structured interviews undertaken with a number of Sunshine’s directors.The case study reveals that developing strategic foresight is a learning process, which takes place within a broad vision, and enacts the future by a mechanism of probing it through cheap multiple devices. At a more general level, the data suggest that in turbulent environments the retention of the unity of the whole organisational system is a challenging task, particularly when its physical dimensions grow too quickly. In this context, the data suggest that nimbleness, visible and structured processes, extensive communication glued together by a focused and eccentric management team form an important core capability that impacts on the firm’s ability to develop strategic foresight and innovate continuously without falling apart.     

Valuation of quanto options in a Markovian regime-switching market: A Markov-modulated Gaussian HJM model

We consider the valuation of European quanto call options in an incomplete market where the domestic and foreign forward interest rates are allowed to exhibit regime shifts under the Heath–Jarrow–Morton (HJM) framework, and the foreign price dynamics is exogenously driven by a regime switching jump-diffusion model with Markov-modulated Poisson processes. We derive closed-form solutions for four different types of quanto call options, which include: options struck in a foreign currency, a foreign equity call struck in domestic currency, a foreign equity call option with a guaranteed exchange rate, and an equity-linked foreign exchange-rate call.     

A Complete Markovian Stochastic Volatility Model in the HJM Framework

This paper considers a stochastic volatility version of the Heath, Jarrow and and Morton (1992) term structure model. Market completeness is obtained by adapting the Hobson and Rogers (1998) complete stochastic volatility stock market model to the interest rate setting. Numerical simulation for a special case is used to compare the stochastic volatility model against the traditional Vasicek (1977) model.     

Predicting earnings in a poor information environment

Financial intermediaries, such as analysts, play an important role in providing information to investors. However, a large segment of the market (about 39% of CRSP firms between 1992 and 2009) is not served by financial analysts, leaving investors in a poor information environment. In this paper, we examine whether other publicly available information signals, such as insider trades, institutional holdings, and firms’ stock repurchases, can be used to predict information about earnings for these firms. We find that CFOs’ trading decisions are associated with new information contained in the annual earnings reports for firms with no or scant analyst coverage. In contrast, for firms with multiple analyst coverage, insider trading decisions are not predictive of new information in earnings reports. Our results suggest that some public information signals, such as insider trades, can be used to alleviate the poor information environment faced by investors. However, the market may not have fully priced the information contained in these signals.     

Portfolio risk management in a data-rich environment

We study risk assessment using an optimal portfolio in which the weights are functions of latent factors and firm-specific characteristics (hereafter, diffusion index portfolio). The factors are used to summarize the information contained in a large set of economic data and thus reflect the state of the economy. First, we evaluate the performance of the diffusion index portfolio and compare it to both that of a portfolio in which the weights depend only on firm-specific characteristics and an equally weighted portfolio. We then use value-at-risk, expected shortfall, and downside probability to investigate whether the weights-modeling approach, which is based on factor analysis, helps reduce market risk. Our empirical results clearly indicate that using economic factors together with firm-specific characteristics helps protect investors against market?risk.     

A binomial approximation for two-state Markovian HJM models

This article develops a lattice algorithm for pricing interest rate derivatives under the Heath et al. (Econometrica 60:77–105, 1992) paradigm when the volatility structure of forward rates obeys the Ritchken and Sankarasubramanian (Math Financ 5:55–72) condition. In such a framework, the entire term structure of the interest rate may be represented using a two-dimensional Markov process, where one state variable is the spot rate and the other is an accrued variance statistic. Unlike in the usual approach based on the Nelson-Ramaswamy (Rev Financ Stud 3:393–430) transformation, we directly discretize the heteroskedastic spot rate process by a recombining binomial tree. Further, we reduce the computational cost of the pricing problem by associating with each node of the lattice a fixed number of accrued variance values computed on a subset of paths reaching that node. A backward induction scheme coupled with linear interpolation is used to evaluate interest rate contingent claims.     

Default and Recovery Risk Dependencies in a Simple Credit Risk Model

This paper provides evidence for the relationship between credit quality, recovery rate, and correlation. The paper finds that rating grade, rating shift, and macroeconomic factors provide a highly significant explanation for default risk and recovery risk of US bond issues. The empirical data suggest that default and recovery processes are highly correlated. Therefore, a joint approach is required for estimating time‐varying default probabilities and recovery rates that are conditional on default. This paper develops and applies such a model.