A semi-martingale representation for a semi-Markov chain with application to finance
Authors:
R. Elliott, A. Swishchuk and I. Y. Zhang
Journal:
Theor. Probability and Math. Statist. 96 (2018), 45-57
MSC (2010):
Primary 60K15, 60K10, 60G42, 60G51, 91B28
DOI:
https://doi.org/10.1090/tpms/1033
Published electronically:
October 5, 2018
MathSciNet review:
3666871
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Additional Information
Abstract: In this paper we present the semi-martingale representation for a discrete time semi-Markov chain, and consider its application to a semi-Markov regime-switching binomial model in finance. We also introduce a semi-Markov switching Lévy process. Estimation results for Markov and semi-Markov chains are presented as well.
References
- Vladimir V. Anisimov, Switching processes in queueing models, Applied Stochastic Methods Series, ISTE, London; John Wiley & Sons, Inc., Hoboken, NJ, 2008. MR 2437051
- David Applebaum, Lévy processes and stochastic calculus, Cambridge Studies in Advanced Mathematics, vol. 93, Cambridge University Press, Cambridge, 2004. MR 2072890
- D. S. Bates, Jumps and stochastic volatility: exchange rate processes implicit in deutsche mark options, Rev. Financ. Stud. 9 (1996), 69–107.
- Pierre Brémaud, Point processes and queues, Springer-Verlag, New York-Berlin, 1981. Martingale dynamics; Springer Series in Statistics. MR 636252
- John Buffington and Robert J. Elliott, American options with regime switching, Int. J. Theor. Appl. Finance 5 (2002), no. 5, 497–514. MR 1916958, DOI https://doi.org/10.1142/S0219024902001523
- Samuel N. Cohen and Robert J. Elliott, Stochastic calculus and applications, 2nd ed., Probability and its Applications, Springer, Cham, 2015. MR 3443368
- Erhan Çinlar, Time dependence of queues with semi-Markovian services, J. Appl. Probability 4 (1967), 356–364. MR 214157, DOI https://doi.org/10.2307/3212029
- G. D’Amico, J. Janssen, and R. Manca, Valuing credit default swap in a non-homogeneous semi-Markovian rating based model, Comput. Econ. 29 (2007), no. 2, 119–138.
- Robert J. Elliott, Stochastic calculus and applications, Applications of Mathematics (New York), vol. 18, Springer-Verlag, New York, 1982. MR 678919
- Robert J. Elliott and Carlton-James U. Osakwe, Option pricing for pure jump processes with Markov switching compensators, Finance Stoch. 10 (2006), no. 2, 250–275. MR 2223098, DOI https://doi.org/10.1007/s00780-006-0004-6
- Pietro Fodra and Huyên Pham, Semi-Markov model for market microstructure, Appl. Math. Finance 22 (2015), no. 3, 261–295. MR 3367230, DOI https://doi.org/10.1080/1350486X.2015.1037963
- S. Goutte and B. Zou, Foreign exchange rates under Markov regime switching model, CREA Discussion paper series, University of Luxemburg, 2011.
- James D. Hamilton, A new approach to the economic analysis of nonstationary time series and the business cycle, Econometrica 57 (1989), no. 2, 357–384. MR 996941, DOI https://doi.org/10.2307/1912559
- S. L. Heston, A closed-form solution for options with stochastic volatility with applications to bond and currency options, Rev. Financ. Stud. 6 (1993), 327–343.
- S. L. Heston, A simple new formula for options with stochastic volatility, Washington University of St. Louis Working Paper, 1999.
- Vladimir S. Koroliuk and Nikolaos Limnios, Stochastic systems in merging phase space, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2005. MR 2205562
- Paul Levy, Processus semi-markoviens, Proceedings of the International Congress of Mathematicians, 1954, Amsterdam, vol. III, Erven P. Noordhoff N.V., Groningen; North-Holland Publishing Co., Amsterdam, 1956, pp. 416–426 (French). MR 0088105
- Nikolaos Limnios and Anatoliy Swishchuk, Discrete-time semi-Markov random evolutions and their applications, Adv. in Appl. Probab. 45 (2013), no. 1, 214–240. MR 3077547
- A. Melino and S. M. Turnbull, Pricing foreign currency options with stochastic volatility, J. Econometrics 45 (1990), 239–265.
- A. Melino and S. M. Turnbull, The pricing of foreign-currency options, Canad. J. Econ. 24 (1991), 251–181.
- R. C. Merton, Option pricing when underlying stock returns are discontinuous, J. Finance Econ. 3 (1976), 125–144.
- K. Miltersen, K. Sandmann, and D. Sondermann, Closed form solutions for term structure derivatives with log-normal interest rates, J. Finance 52 (1997), 409–430.
- A. Papapantoleon, An Introduction to Lévy Processes with Applications to Mathematical Finance. Lecture notes, arXiv:0804.0482, 2000.
- Ronald Pyke, Markov renewal processes: definitions and preliminary properties, Ann. Math. Statist. 32 (1961), 1231–1242. MR 133888, DOI https://doi.org/10.1214/aoms/1177704863
- Ronald Pyke, Markov renewal processes with finitely many states, Ann. Math. Statist. 32 (1961), 1243–1259. MR 154324, DOI https://doi.org/10.1214/aoms/1177704864
- Ronald Pyke and Ronald Schaufele, Limit theorems for Markov renewal processes, Ann. Math. Statist. 35 (1964), 1746–1764. MR 168026, DOI https://doi.org/10.1214/aoms/1177700397
- Ronald Pyke and Ronald Schaufele, The existence and uniqueness of stationary measures for Markov renewal processes, Ann. Math. Statist. 37 (1966), 1439–1462. MR 203811, DOI https://doi.org/10.1214/aoms/1177699138
- J. Rumsey, Pricing cross-currency options, J. Futures Markets 11 (1991), 89–93.
- W. Schoutens, Lévy Processes in Finance: Pricing Financial Derivatives, Wiley, 2003.
- D. S. Sīl′vestrov, Polumarkovskie protsessy s diskretnym mnozhestvom sostoyaniĭ, “Sovet. Radio”, Moscow, 1980 (Russian). Biblioteka Inzhenera po Nadezhnosti. [Library for the Engineer in Reliability]; Osnovy rascheta funktsional′nykh i nadezhnostnykh kharakteristik stokhasticheskikh sistem. [Principles of the calculation of functional and reliability characteristics of stochastic systems]. MR 612026
- W. L. Smith, Regenerative stochastic processes, Proc. Roy. Soc. London Ser. A 232 (1955), 6–31. MR 73877, DOI https://doi.org/10.1098/rspa.1955.0198
- Anatoly V. Swishchuk, Pricing of variance and volatility swaps with semi-Markov volatilities, Can. Appl. Math. Q. 18 (2010), no. 4, 415–438. MR 2858147, DOI https://doi.org/10.1155/2010/537571
- A. Swishchuk and R. Elliott, Pricing Options. Hidden Markov Models in Finance, Springer, 2007.
- Anatoliy Swishchuk and Nelson Vadori, A semi-Markovian modeling of limit order markets, SIAM J. Financial Math. 8 (2017), no. 1, 240–273. MR 3625801, DOI https://doi.org/10.1137/15M1015406
- Anatoly Swishchuk and Jianhong Wu, Evolution of biological systems in random media: limit theorems and stability, Mathematical Modelling: Theory and Applications, vol. 18, Kluwer Academic Publishers, Dordrecht, 2003. MR 2031169
- L. Takács, On secondary stochastic processes generated by recurrent processes, Acta Math. Acad. Sci. Hungar. 7 (1956), 17–29 (English, with Russian summary). MR 78584, DOI https://doi.org/10.1007/BF02022961
- N. Zhou and R. S. Mamon, An accessible implementation of interest rate models with regime switching, Expert Syst. Appl. 39 (2012), no. 5, 4679–4689.
References
- V. Anisimov, Switching Processes in Queueing Models, Wiley, 2013. MR 2437051
- D. Applebaum, Lévy Processes and Stochastic Calculus, Cambridge University Press, 2004. MR 2072890
- D. S. Bates, Jumps and stochastic volatility: exchange rate processes implicit in deutsche mark options, Rev. Financ. Stud. 9 (1996), 69–107.
- P. Bremaud, Point Processes and Queues: Martingale Dynamics, Springer, 1981. MR 636252
- J. Buffington and R. J. Elliott, American options with regime switching, Int. J. Theor. Appl. Finance 5 (2002), 497–514. MR 1916958
- S. Cohen and R. Elliott, Stochastic Calculus and Applications, 2nd edition, Birkhauser, 2015. MR 3443368
- E. Cinlar, Time dependence of queues with semi-Markov services, J. Appl. Probab. 4 (1967), 356–364. MR 0214157
- G. D’Amico, J. Janssen, and R. Manca, Valuing credit default swap in a non-homogeneous semi-Markovian rating based model, Comput. Econ. 29 (2007), no. 2, 119–138.
- R. J. Elliott, Stochastic Calculus and Applications, Springer, Berlin–Heidelberg–New York, 1982. MR 678919
- R. J. Elliott and C. J. Osakwe, Option pricing for pure jump processes with Markov switching compensators, Finance Stoch. 10 (2006), 250–275. MR 2223098
- P. Fodra and H. Pham, Semi-Markov model for market microstructure, Appl. Math. Finance 22 (2013), no. 3, 261–295. MR 3367230
- S. Goutte and B. Zou, Foreign exchange rates under Markov regime switching model, CREA Discussion paper series, University of Luxemburg, 2011.
- J. D. Hamilton, A new approach to the economic analysis of nonstationary time series and the business cycle, Econometrica 57 (1989), no. 2, 357–384. MR 996941
- S. L. Heston, A closed-form solution for options with stochastic volatility with applications to bond and currency options, Rev. Financ. Stud. 6 (1993), 327–343.
- S. L. Heston, A simple new formula for options with stochastic volatility, Washington University of St. Louis Working Paper, 1999.
- V. Koroliuk and N. Limnios, Stochastic Systems in Merging Phase Space, World Scientific, 2005. MR 2205562
- P. Levy, Processus Semi-Markoviens, Proceedings of International Congress of Mathematics 3 (1954), 416–426. MR 0088105
- N. Liminios and A. Swishchuk, Discrete-time semi-Markov random evolutions and their applications, Adv. Appl. Probab. 45 (2013), no. 1, 214–240. MR 3077547
- A. Melino and S. M. Turnbull, Pricing foreign currency options with stochastic volatility, J. Econometrics 45 (1990), 239–265.
- A. Melino and S. M. Turnbull, The pricing of foreign-currency options, Canad. J. Econ. 24 (1991), 251–181.
- R. C. Merton, Option pricing when underlying stock returns are discontinuous, J. Finance Econ. 3 (1976), 125–144.
- K. Miltersen, K. Sandmann, and D. Sondermann, Closed form solutions for term structure derivatives with log-normal interest rates, J. Finance 52 (1997), 409–430.
- A. Papapantoleon, An Introduction to Lévy Processes with Applications to Mathematical Finance. Lecture notes, arXiv:0804.0482, 2000.
- R. Pyke, Markov renewal processes: definitions and preliminary properties, Ann. Math. Statistics 32 (1961), 1231–1242. MR 0133888
- R. Pyke, Markov renewal processes with finitely many states, Ann. Math. Statistics 32 (1961), 1243–1259. MR 0154324
- R. Pyke, and R. Schaufele, Limit theorems for Markov renewal processes, Ann. Math. Statistics 35 (1964), 1746–1764. MR 0168026
- R. Pyke and R. Schaufele, The existence and uniqueness of stationary measures for Markov renewal processes, Ann. Math. Statistics 37 (1966), 1439–1462. MR 0203811
- J. Rumsey, Pricing cross-currency options, J. Futures Markets 11 (1991), 89–93.
- W. Schoutens, Lévy Processes in Finance: Pricing Financial Derivatives, Wiley, 2003.
- D. S. Silvestrov, Semi-Markov Processes with a Discrete State Space, Library for an Engineer in Reliability, Sovetskoe Radio, Moscow, 1980. MR 612026
- W. L. Smith, Regenerative stochastic processes, Proceedings of the Royal Society of London Archives 232 (1955), 6–31. MR 0073877
- A. Swishchuk, Pricing of variance and volatility swaps with semi-Markov volatilities, Can. Appl. Math. Q. 18 (2010). MR 2858147
- A. Swishchuk and R. Elliott, Pricing Options. Hidden Markov Models in Finance, Springer, 2007.
- A. Swishchuk and N. Vadori, A semi-Markovian modelling of limit order markets, SIAM J. Finan. Math. (2017). MR 3625801
- A. Swishchuk and J. Wu, Evolution of Biological Systems in Random Media: Limit Theorems and Stability, Springer, 2003. MR 2031169
- L. Takacs, On secondary processes generated by recurrent processes, Archiv der Mathematik 7 (1956), 17–29. MR 0078584
- N. Zhou and R. S. Mamon, An accessible implementation of interest rate models with regime switching, Expert Syst. Appl. 39 (2012), no. 5, 4679–4689.
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Additional Information
R. Elliott
Affiliation:
University of Calgary, Calgary, Canada and University of South Australia
Email:
relliott@ucalgary.ca
A. Swishchuk
Affiliation:
University of Calgary, Calgary, Canada
Email:
aswish@ucalgary.ca
I. Y. Zhang
Affiliation:
University of Calgary, Calgary, Canada
Email:
yi.zhang@ucalgary.ca
Keywords:
Discrete time finite state semi-Markov chain,
semi-Markov switching Lévy process,
semi-martingale representation,
financial derivatives,
regime-switching binomial model
Received by editor(s):
February 27, 2017
Published electronically:
October 5, 2018
Dedicated:
Dedicated to the 70th Anniversary of Professor Dmitrii Silvestrov
Article copyright:
© Copyright 2018
American Mathematical Society
