Convergence of option rewards for multivariate price processes

Silvestrov, D.; Lundgren, R.

doi:10.1090/S0094-9000-2013-00879-5

Convergence of option rewards for multivariate price processes

Authors: D. S. Silvestrov and R. Lundgren
Journal: Theor. Probability and Math. Statist. 85 (2012), 115-131
MSC (2000): Primary 60J05, 60H10; Secondary 91B28, 91B70
DOI: https://doi.org/10.1090/S0094-9000-2013-00879-5
Published electronically: January 14, 2013
MathSciNet review: 2933708
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: American type options with general payoff functions possessing polynomial rate of growth are considered for multivariate Markov price processes. Convergence results for optimal reward functionals of American type options for perturbed multivariate Markov processes are presented. These results are applied to approximation tree type algorithms for American type options for exponential multivariate Brownian price processes and mean-reverse price processes used to model stochastic dynamics of energy prices.

References

Kaushik Amin and Ajay Khanna, Convergence of American option values from discrete- to continuous-time financial models, Math. Finance 4 (1994), no. 4, 289–304. MR 1299240, DOI https://doi.org/10.1111/j.1467-9965.1994.tb00059.x
François Coquet and Sandrine Toldo, Convergence of values in optimal stopping and convergence of optimal stopping times, Electron. J. Probab. 12 (2007), no. 8, 207–228. MR 2299917, DOI https://doi.org/10.1214/EJP.v12-288

G. Cortazar, M. Gravet, and J. Urzua, The valuation of multidimensional American real options using the LSM simulation method, Comp. Oper. Res. 35 (2008), 113–129.

Savas Dayanik and Ioannis Karatzas, On the optimal stopping problem for one-dimensional diffusions, Stochastic Process. Appl. 107 (2003), no. 2, 173–212. MR 1999788, DOI https://doi.org/10.1016/S0304-4149%2803%2900076-0

Paul Dupuis and Hui Wang, On the convergence from discrete to continuous time in an optimal stopping problem, Ann. Appl. Probab. 15 (2005), no. 2, 1339–1366. MR 2134106, DOI https://doi.org/10.1214/105051605000000034

Erik Ekström, Carl Lindberg, Johan Tysk, and Henrik Wanntorp, Optimal liquidation of a call spread, J. Appl. Probab. 47 (2010), no. 2, 586–593. MR 2668508, DOI https://doi.org/10.1239/jap/1276784911

Bin Gao, Jing-zhi Huang, and Marti Subrahmanyam, The valuation of American barrier options using the decomposition technique, J. Econom. Dynam. Control 24 (2000), no. 11-12, 1783–1827. Computational aspects of complex securities. MR 1784574, DOI https://doi.org/10.1016/S0165-1889%2899%2900093-7

Vicky Henderson and David Hobson, An explicit solution for an optimal stopping/optimal control problem which models an asset sale, Ann. Appl. Probab. 18 (2008), no. 5, 1681–1705. MR 2462545, DOI https://doi.org/10.1214/07-AAP511

S. D. Jacka, Optimal stopping and the American put, Math. Finance 1 (1991), 1–14.

H. Jönsson, Monte Carlo studies of American type options with discrete time, Theory Stoch. Process. 7(23) (2001), no. 1–2, 163–188.

H. Jönsson, Optimal Stopping Domains and Reward Functions for Discrete Time American Type Options, Ph.D. Thesis, vol. 22, Mälardalen University, 2005.

H. Jönsson, A. G. Kukush, and D. S. Silvestrov, Threshold structure of optimal stopping strategies for American type option. I, Teor. Ĭmovīr. Mat. Stat. 71 (2004), 82–92; English transl., Theory Probab. Math. Statist. 71 (2005), 93–103. MR 2144323, DOI https://doi.org/10.1090/S0094-9000-06-00650-8

H. Jönsson, A. G. Kukush, and D. S. Silvestrov, Threshold structure of optimal stopping strategies for American type option. II, Teor. Ĭmovīr. Mat. Stat. 72 (2005), 42–53; English transl., Theory Probab. Math. Statist. 72 (2006), 47–58. MR 2168135, DOI https://doi.org/10.1090/S0094-9000-06-00663-6

I. Kim, The analytic valuation of American options, Rev. Financ. Studies 3 (1990), 547–572.

Alexander G. Kukush and Dmitrii S. Silvestrov, Structure of optimal stopping strategies for American type options, Probabilistic constrained optimization, Nonconvex Optim. Appl., vol. 49, Kluwer Acad. Publ., Dordrecht, 2000, pp. 173–185. MR 1819411, DOI https://doi.org/10.1007/978-1-4757-3150-7_9

A. G. Kukush and D. S. Silvestrov, Skeleton approximation of optimal stopping strategies for American type options with continuous time, Theory Stoch. Process. 7(23) (2001), no. 1–2, 215–230.

Alexander G. Kukush and Dmitrii S. Silvestrov, Optimal pricing for American type options with discrete time, Theory Stoch. Process. 10 (2004), no. 1-2, 72–96. MR 2327852

Robin Lundgren, Structure of optimal stopping domains for American options with knock out domains, Theory Stoch. Process. 13 (2007), no. 4, 98–129. MR 2482254

R. Lundgren, Convergence of Option Rewards, Ph.D. Thesis, vol. 89, Mälardalen University, 2010.

R. Lundgren and D. S. Silvestrov, Convergence of Option Rewards for Multivariate Price Processes, Research Report 2009:10, Department of Mathematics, Stockholm University.

Robin Lundgren and Dmitrii S. Silvestrov, Optimal stopping and reselling of European options, Mathematical and statistical models and methods in reliability, Stat. Ind. Technol., Birkhäuser/Springer, New York, 2010, pp. 371–389. MR 2797677, DOI https://doi.org/10.1007/978-0-8176-4971-5_29

R. Lundgren, D. S. Silvestrov, and A.G. Kukush, Reselling of options and convergence of option rewards, J. Numer. Appl. Math., 1(96), no. 3–4 (2008), 149–172.

Yuliya Mishura and Georgiy Shevchenko, The optimal time to exchange one asset for another on finite interval, Optimality and risk—modern trends in mathematical finance, Springer, Berlin, 2009, pp. 197–210. MR 2648604, DOI https://doi.org/10.1007/978-3-642-02608-9_10

Goran Peskir and Albert Shiryaev, Optimal stopping and free-boundary problems, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2006. MR 2256030

Jean-Luc Prigent, Weak convergence of financial markets, Springer Finance, Springer-Verlag, Berlin, 2003. MR 2036683

E. S. Schwartz, The stochastic behaviour of commodity prices: implications for valuation and hedgeing, J. Finance. 52 (1997), no. 3, 923–973.

D. Silvestrov, V. Galochkin, and A. Malyarenko, OPTAN—a pilot program system for analysis of options, Theory Stoch. Process. 7(23) (2001), no. 1-2, 291–300.

D. S. Silvestrov, V. G. Galochkin, and V. G. Sibirtsev, Algorithms and programs for optimal Monte Carlo pricing of American type options, Theory Stoch. Process. 5(21) (1999), no. 1–2, 175–187.

D. S. Silvestrov, H. Jönsson, and F. Stenberg, Convergence of option rewards for Markov type price processes controlled by stochastic indices. 1, Research Report 2006-1, Department of Mathematics and Physics, Mälardalen University.

D. Silvestrov, H. Jönsson, and F. Stenberg, Convergence of option rewards for Markov type price processes, Theory Stoch. Process. 13 (2007), no. 4, 189–200. MR 2482260

D. S. Sīl′vestrov, Kh. Ĭonson, and F. Stenberg, Convergence of option rewards for Markov-type price processes modulated by stochastic indices. I, Teor. Ĭmovīr. Mat. Stat. 79 (2008), 138–154 (Ukrainian, with Ukrainian summary); English transl., Theory Probab. Math. Statist. 79 (2009), 153–170. MR 2494545, DOI https://doi.org/10.1090/S0094-9000-09-00776-5

D. S. Sīl′vestrov, Kh. Ĭonson, and F. Stenberg, Convergence of option rewards for Markov-type price processes modulated by stochastic indices. II, Teor. Ĭmovīr. Mat. Stat. 80 (2009), 138–155 (Ukrainian, with English, Russian and Ukrainian summaries); English transl., Theory Probab. Math. Statist. 80 (2010), 153–172. MR 2541960, DOI https://doi.org/10.1090/S0094-9000-2010-00802-7

A. V. Skorokhod, Sluchaĭ nye protsessy s nezavisimymi prirashcheniyami, 2nd ed., Teoriya Veroyatnosteĭ i Matematicheskaya Statistika. [Probability Theory and Mathematical Statistics], “Nauka”, Moscow, 1986 (Russian). MR 860563

F. Stenberg, Semi-Markov Models for Insurance and Option Rewards, Ph.D. Thesis, vol. 38, Mälardalen University, 2006.

Z. Zhang and K-G. Lim, A non-lattice pricing model of American options under stochastic volatility, J. Futures Markets 26, (2006), no. 5, 417–448.