This research project deals with pricing path-dependent or exotic options. A path-dependent option is an option whose payoff at exercise depends on the past history of the underlying asset price as well as its spot price at exercise. An American-style option is a weakly path-dependent option such that their historical paths affect the value but the option payoff does not contain the paths explicitly. Various options such as binaries, compounds, barriers, Asians, lookbacks and so on are included in the class of path-dependent options. In this research project, we develop mathematical models for generalized path-dependent options and analyze them by numerical and/or approximate methods.
■Related Work
Kimura, T., "An Approximate Barrier Option Model for Valuing Executive Stock Options," Journal of the Operations Research Society of Japan, Vol. 61, No. 1, pp. 110-131, 2018. DOI
Kimura, T., "A Refined Laplace-Carson Transform Approach to Valuing Convertible Bonds," Journal of the Operations Research Society of Japan, Vol. 60, No. 2, pp. 50-65, 2017. DOI
Kimura, T., "American Fractional Lookback Options: Valuation and Premium Decomposition," SIAM Journal on Applied Mathematics, Vol. 71, No. 2, pp.517-539, 2011. DOI
Kimura, T., "Valuing Executive Stock Options: A Quadratic Approximation," European Journal of Operational Research, Vol. 207, No. 3, pp. 1368-1379, 2010. DOI
Kimura, T., "Valuing Continuous-Installment Options," European Journal of Operational Research, Vol. 201, No. 1, pp. 222-230, 2010. DOI
Kimura, T., "American Continuous-Installment Options: Valuation and Premium Decomposition," SIAM Journal on Applied Mathematics, Vol. 70, No. 3, pp. 803-824, 2009. DOI
Kimura, T., "Valuing Finite-Lived Russian Options," European Journal of Operational Research, Vol. 189, No. 2, pp. 363-374, 2008. DOI
Kimura, T. and Shinohara, T., "Monte Carlo Analysis of Convertible Bonds with Reset Clauses," European Journal of Operational Research, Vol. 168, No. 2, pp. 301-310, 2006. DOI
Stochastic service systems in the fields of computer, communication and manufacturing often experience congestion due to irregular flows. Performance evaluation of these systems has been addressed through the analysis of queues therein. Exact solutions are available only under restrictive assumptions that scarcely fit in with the reality, making approximate solutions practically needed. Approximation methods for queues have thus been of practical interests.
This research project deals with so-called system approximations. A typical subclass of system approximations is a system-interpolation approximation that combines known analytical solutions of simpler systems in a closed form. Perhaps the most important contribution of this project is to point out that system approximations can be viewed in a new perspective that provides additional insight.
■Exact Data for the Waiting Time in M/PH/s Queues
Text files in mphs.zip (52 KB) contain exact data for the waiting time in some M/PH/s queues with the FCFS (first-come first-served) discipline. These data have been computed by using the aggregation/disaggregation algorithm. The data may be useful, e.g., for checking the accuracy of approximations for the M/G/s queue. See mphs.pdf (46 KB) for details.
■Related Work
Kimura, T., "The M/G/s Queue," Wiley Encyclopedia of Operations Research and Management Science, Volume 4, J.J. Cochran et al. (eds.), John Wiley & Sons, pp. 2913-2921, 2011. DOI
Kimura, T., "Equivalence Relations in the Approximations for the M/G/s/s+r Queue," Mathematical and Computer Modelling, Vol. 31, No. 10-12, pp. 215-224, 2000. DOI
Kimura, T., "A Transform-Free Approximation for the Finite Capacity M/G/s Queue," Operations Research, Vol. 44, No. 6, pp. 984-988, 1996. JSTOR
Kimura, T., "Optimal Buffer Design of an M/G/s Queue with Finite Capacity," Stochastic Models, Vol. 12, No. 1, pp. 165-180, 1996. DOI
Kimura, T., "Approximations for Multi-Server Queues: System Interpolations," Queueing Systems: Theory and Applications, Vol. 17, No. 3-4, pp. 347-382, 1994. DOI
Kimura, T., "Approximations for the Waiting Time in the GI/G/s Queue," Journal of the Operations Research Society of Japan, Vol. 34, No. 2, pp. 173-186, 1991. DOI
Kimura, T., "A Two-Moment Approximation for the Mean Waiting Time in the GI/G/s Queue," Management Science, Vol. 32, No. 6, pp. 751-763, 1986. JSTOR
This project deals with another widely used approximation called diffusion approximation, where a queueing characteristic process (e.g., queue length, waiting time, unfinished work load, etc.) is approximated by a diffusion (Brownian motion) process. It has been known that diffusion approximations for unstable queues can be often justified by heavy traffic limit theorems (HTLTs): When a queueing characteristic process is appropriately scaled and translated, HTLTs show that the translated process in an unstable queue converges weakly to a Brownian motion process. However, this does not necessarily imply that the Brownian motion process still gives an accurate approximation when the queue is stable, because HTLTs provide us no information on the process behavior in stable situations. Hence, we must clearly distinguish two kinds of "diffusion approximations", i.e., diffusion limits justified by HTLTs for unstable queues and diffusion models as continuous approximations for stable queues. In this research project, we consider a general framework of diffusion models for stochastic service systems such as computer, communication and manufacturing systems.
■Related Work
Kimura, T., "Diffusion Approximations for Queues with Markovian Bases," Annals of Operations Research, Vol. 113, No. 1-4, pp. 27-40, 2002. DOI
Kimura, T., "An M/M/s-Consistent Diffusion Model for the GI/G/s Queue," Queueing Systems: Theory and Applications, Vol. 19, No. 3-4, pp. 377-397, 1995. DOI
Kimura, T., "A Unifying Diffusion Model for State-Dependent Queues," Optimization, Vol. 18, No. 2, pp. 256-283, 1987. DOI
Kimura, T., "Refining Diffusion Approximations for GI/G/1 Queues: A Tight Discretization Method," Teletraffic Issues in an Advanced Information Society, ITC11, pp. 317-323, M. Akiyama (ed.), North-Holland, 1986. ITC
Kimura, T. and Ohsone, T., "A Diffusion Approximation for an M/G/m Queue with Group Arrivals," Management Science, Vol. 30, No. 3, pp. 381-388, 1984. JSTOR
Kimura, T., "Diffusion Approximation for an M/G/m Queue," Operations Research, Vol. 31, No. 2, pp. 304-321, 1983. JSTOR