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We propose a dynamic programming (DP) approach for pricing options embedded in bonds, the focus being on call and put options with advance notice. An efficient procedure is developed for the cases where the interest-rate process follows the Vasicek, Cox-IngersollRoss (CIR), or generalized Vasicek models. Our DP methodology uses the exact joint distribution of the interest rate and integrated interest rate at a future date, conditional on the current value of the interest rate. We provide numerical illustrations, for the Vasicek and CIR models, comparing our DP method with finite-difference methods. Our procedure compares quite favorably in terms of both efficiency and accuracy. An important advantage of the our DP approach is that it can be applied to more general models calibrated to capture the term structure of interest rates (e.g., the generalized Vasicek model).

This article presents a novel framework integrating a dynamic states approach to firm growth and a portfolio perspective of board roles. In our framework we highlight four typical firm growth states aligned with primary board roles to guide firms effectively through the transitions associated with each state to reach a new equilibrium. The particular contribution of our framework is that it takes SME heterogeneity into account, while capturing the dynamic nature of these states which are reached along non-linear, non-sequential and non-deterministic transitional pathways, as well as incorporating reverse and counterintuitive trajectories between states. KeywordsTransitions, SMEs, board role portfolio, dynamic states 2 IntroductionA large and growing literature relates to the governance of small and medium firms (SMEs), with extensive debate over the value and contribution of boards of directors and their roles in this context (Boxer et al., 2016;Gnan et al., 2015; Wilson et al., 2014). Much of this debate considers board value from a static perspective. However, the question of board value and contribution is more complex when considered in circumstances of dynamic change. Recent studies have attempted to address this question by researching board roles in conjunction with small firm growth paths and the changes experienced over the course of their existence (Bonn and Pettigrew, 2009;Filatotchev et al., 2006;Perrault and McHugh, 2015).Two key issues arise from this debate. The first issue relates to an almost exclusive use of life cycle models to match board roles with SME growth patterns (Bonn and Pettigrew, 2009;Lynall et al., 2003) and transitions accompanying growth (Filatotchev et al., 2006;Perrault and McHugh, 2015). The focus on growth and life cycles implies unidirectional, deterministic patterns of development and ignores the complex realities that most firms encounter, such as erratic growth, turnaround situations and reversals due to decline or other negative impacts (Perrault and McHugh, 2015). The second issue concerns the highly heterogeneous and dynamic nature of SMEs (Delmar et al., 2003;Empson, 2010;Phelps et al., 2007), making it difficult to encapsulate theoretically their governance needs in a comprehensive model. 3These issues indicate the need for a broader conceptualisation that reflects more adequately the complexities associated with small firm heterogeneity and dynamics, in addressing the question of how SME boards can add strategic value. The novel framework we develop in this paper provides a more complete specification of both board and firm characteristics associated with transitional dynamics through growth and transformation.While recognizing the overlap of board roles, as well as the need for boards to perform more than a single role, our model highlights the most valuable board role for each transition we depict. To ground our framework, we review and build on two streams of literature relating to board roles in smaller firms, and SME growth and development models.Nume...

The extended Kalman filter, which linearizes the relationship between security prices and state variables, is widely used in fixed-income applications. We investigate whether the unscented Kalman filter should be used to capture nonlinearities and compare the performance of the Kalman filter with that of the particle filter. We analyze the cross section of swap rates, which are mildly nonlinear in the states, and cap prices, which are highly nonlinear. When caps are used to filter the states, the unscented Kalman filter significantly outperforms its extended counterpart. The unscented Kalman filter also performs well when compared with the much more computationally intensive particle filter. These findings suggest that the unscented Kalman filter may be a good approach for a variety of problems in fixed-income pricing. This paper was accepted by Wei Xiong, finance.

Purpose The purpose of this paper is to model SME board configurations and then to examine empirically their diversity. Polarity in corporate board research around two primary tasks (control and service/strategy), neither captures comprehensively the range of SME board types, based on what they actually do, nor elucidates how boards configure and why. SME heterogeneity is problematic for understanding how the triumvirate of power and control – owners, directors and executives – governs in such firms. Design/methodology/approach Survey research is used to examine 186 French private SMEs. Factorial and cluster analyses are used to classify board configurations according to board task performance. Findings Results reveal six different board types among small firms. The findings indicate that both organisational and board design need to be adjusted to align with the differentiation between the ownership and the management, and between the ownership and the directorship. The greater the differentiation between these power/control functions in response to increased internal and/or external contingencies, the more varied will be the board’s portfolio of tasks, with implications for the director capabilities and board competence. Research limitations/implications The research extends SME board governance theory and practice by bringing greater clarity to the field of board task performance in SMEs. It provides insights into explicit board task-related configurational behaviour through recognising the degree of differentiation between the triumvirate power/control functions at the apex of the small firm. SME boards in the sample show not just a single configuration but a combination from a portfolio of tasks with different emphases on each according to their circumstances. This finding implies that a particular type of board may select a task, or set of tasks, from the portfolio, depending on the nature of the SME in terms of its proximity – whether it is characterised more by specificity or by denaturation. Further research is needed to understand the variation in these configurations over time in response to internal and external contingencies and what board emphases and processes are involved in transitioning through these evolutions. Practical implications The findings are important because the extent of knowledge about what the configuration comprises will determine how effectively a board will execute its tasks. This knowledge is useful in helping boards place emphasis on how best to concentrate their efforts on creating value for the SME, by selecting an effective combination of tasks from a given board configuration depending on their circumstances. Originality/value The research extends SME board governance theory and practice by bringing greater clarity to the field of board task performance in SMEs. It provides insights into explicit board task-related configurational behaviour through recognising the degree of differentiation between the triumvirate power/control functions at the apex of the small firm.