Masaaki KIJIMA, Dean
School of Informatics and Data Science Hiroshima University, Japan
"Transition from mathematical reliability to financial engineering"
In the reliability literature, it is often assumed that, upon failure, the system is either replaced by a new one (perfect repair) or repaired minimally (minimal repair). However, the assumption that a repair (and preventive maintenance) can
be either perfect or minimal may be overly restrictive in practice given the wide variety of maintenance actions.
In this talk, I first show some reliability models that allow repairs (and preventive maintenance) to be more general and discuss the associated stochastic processes. Then, I explain the resemblance between reliability models and credit models in finance. In particular, a default of corporate bond can be seen as a failure of system. Repair of failed system is similar to debt rescheduling, maintenance activity resembles debt restructuring, etc.
In mathematical reliability theory, failure processes are modeled by, e.g., renewal processes, (inhomogeneous) Poisson processes, Cox processes, and Markov additive processes, whereas the modelling of default of corporate bonds is the same. Also, some environmental effects are introduced as covariates in both models. For example, in the reliability context, the covariate may be the total working time (or the virtual age as in Kijima’s model) of the system, whereas it is often the credit quality (or asset value) of the corporate firm in finance.
Despite the resemblance, however, there are several notable differences between them. The crucial and important difference, among others, is that the financial models aim to obtain the prices of financial products, while reliability models seek to calculate availability and/or reliability of systems in certain sense. Because the traded prices are observed in the market subject to financial risks, the prices must be calculated so as to reflect the risk premium, namely under the pricing (risk-neutral) measure, which is usually different from the physical measure. The two probability measures coincide with each other only when the risk premium is zero or the investors are risk neutral. On the other hand, calculation in reliability context is always performed under the physical probability measure.
In this talk, I use a finite, absorbing Markov chain to explain the resemblance and the difference between the reliability and financial models. Suppose that the state of the Markov chain represents the virtual age (credit quality, respectively) and the failure (default) state is appended as an absorbing state in the reliability (financial) context. Suppose further that maintenance activity (restructuring) takes place periodically and the maintenance (restructuring) is formulated as Kijima’s model I. Then, the Markov chain becomes inhomogeneous in time with absorbing state and we are interested in the time to failure (default), whose probability distribution can be calculated numerically with ease. Any performance measure in the reliability/availability context is evaluated from the distributional result.
Up until this point, the formulation is exactly the same for the two models. However, in the finance context, the absorbing Markov chain must be adjusted by the risk premium so that it is consistent with the market price of the corporate bond. The risk-premium adjustment is done through, e.g., Kijima-Komoribayashi scheme. Based on the risk-adjusted Markov chain, the value of options written on the bond can be evaluated.
Masaaki Kijima is currently the Dean of School of Informatics and Data Science, Hiroshima University. He graduated from Department of Information Sciences, Tokyo Institute of Technology in 1980, and received PhD from the Simon Business School, University of Rochester in 1986. Since then, he has held multiple professorships with the leading economic and mathematical departments in Japan, including Tokyo Institute of Technology and Kyoto University. He started his research in the area of Markov processes with applications to mathematical reliability and queuing models and then switched gradually to financial engineering when he joined the business school, Graduate School of Systems Management, University of Tsukuba. He has published numerous papers in international journals specializing applied probability, operations research and financial engineering. Also, he is the author of two books entitled "Markov Processes for Stochastic Modeling" in 1997 and "Stochastic Processes with Applications to Finance" in 2013. He had served as a council member of Bachelier Finance Society and Society for Computational Economics, and associate editors of SIAM Journal on Financial Mathematics and Journal of Economic Dynamics and Control.