×

A time-series approach to non-self-financing hedging in a discrete-time incomplete market. (English) Zbl 1152.91730

Summary: We present an algorithm producing a dynamic non-self-financing hedging strategy in an incomplete market corresponding to investor-relevant risk criterion. The optimization is a two-stage process that first determines market calibrated model parameters that correspond to the market price of the option being hedged. In the second stage, an optimal set of model parameters is chosen from the market calibrated set. This choice is based on stock price simulations using a time-series model for stock price jump evolution. Results are presented for options traded on the New York Stock Exchange.

MSC:

91B84 Economic time series analysis
91B28 Finance etc. (MSC2000)

Software:

S-PLUS
PDFBibTeX XMLCite
Full Text: DOI EuDML

References:

[1] N. Josephy, L. Kimball, V. Steblovskaya, A. V. Nagaev, and M. Pasniewski, “An algorithmic approach to non-self-financing hedging in a discrete-time incomplete market,” Discrete Mathematics and Applications, vol. 17, no. 2, pp. 189-207, 2007. · Zbl 1282.91111 · doi:10.1515/dma.2007.016
[2] A. V. Nagaev and S. A. Nagaev, “Asymptotics of riskless profit under selling of discrete time call options,” Applicationes Mathematicae, vol. 30, no. 2, pp. 173-191, 2003. · Zbl 1055.91031 · doi:10.4064/am30-2-3
[3] S. A. Nagaev, A. V. Nagaev, and R. M. Kunst, “A diffusion approximation to the Markov chains model of the financial market and the expected riskless profit under selling of call and put options,” Economics Series 165, Institute for Advanced Studies, Vienna, Austria, 2005.
[4] S. A. Nagaev, A. V. Nagaev, and R. M. Kunst, “A diffusion approximation for the riskless profit under selling of discrete time call options: non-identically distributed jumps,” Economics Series 164, Institute for Advanced Studies, Vienna, Austria, 2005.
[5] A. V. Nagaev, “A diffusion approximation of the expected risky profit on an investor under selling of discrete time options” (Russian), working paper.
[6] G. Tessitore and J. Zabczyk, “Pricing options for multinomial models,” Bulletin of the Polish Academy of Sciences, Mathematics, vol. 44, no. 3, pp. 363-380, 1996. · Zbl 0868.90010
[7] G. Wolczynska, “An explicit formula for option pricing in discrete incomplete markets,” International Journal of Theoretical and Applied Finance, vol. 1, no. 2, pp. 283-288, 1998. · Zbl 0909.90034 · doi:10.1142/S0219024998000151
[8] O. Hammarlid, “On minimizing risk in incomplete markets option pricing models,” International Journal of Theoretical and Applied Finance, vol. 1, no. 2, pp. 227-233, 1998. · Zbl 0909.90024 · doi:10.1142/S0219024998000126
[9] J. Staum, “Incomplete markets,” in Handbooks in Operations Research and Management Science, vol. 15, pp. 511-564, Elsevier Science, Amsterdam, The Netherlands, 2006. · doi:10.1016/S0927-0507(07)15012-X
[10] E. Jouini, J. Cvitanić, and M. Musiela, Eds., Option Pricing, Interest Rates and Risk Management, Handbooks in Mathematical Finance, Cambridge University Press, Cambridge, UK, 2001. · Zbl 0967.91001
[11] A. N. Shiryaev, Essentials of Stochastic Finance: Facts, Models, Theory, vol. 3 of Advanced Series on Statistical Science & Applied Probability, World Scientific, River Edge, NJ, USA, 1999. · Zbl 0926.62100
[12] A. V. Mel’nikov, S. N. Volkov, and M. L. Nechaev, Mathematics of Financial Obligations, vol. 212 of Translations of Mathematical Monographs, American Mathematical Society, Providence, RI, USA, 2002. · Zbl 1050.91001
[13] V. Ryabchenko, S. Sarykalin, and S. Uryasev, “Pricing European options by numerical replication: quadratic programming with constraints,” Asia-Paciffc Financial Markets, vol. 11, no. 3, pp. 301-333, 2004. · Zbl 1147.91330 · doi:10.1007/s10690-005-9004-3
[14] W. B. Arthur, J. H. Holland, B. LeBaron, R. Palmer, and P. Tayler, “Asset pricing under endogenous expectations in an artificial stock market,” in The Economy as an Evolving, Complex System II, Addison-Wesley, Reading, Mass, USA, 1997.
[15] J. Benhabib, Cycles and Chaos in Economic Equilibrium, Princeton University Press, Princeton, NJ, USA, 1992.
[16] W. A. Brock and C. Sayers, “Is the business cycle characterized by deterministic chaos?” Journal of Monetary Economics, vol. 22, no. 1, pp. 71-90, 1988. · doi:10.1016/0304-3932(88)90170-5
[17] C. W. Granger, “Is chaotic economic theory relevant for economics? A review essay,” Journal of International and Comparative Economics, vol. 3, pp. 139-145, 1994.
[18] L. Rüschendorf, “On upper and lower prices in discrete-time models,” Proceedings of the Steklov Institute of Mathematics, vol. 237, pp. 134-139, 2002. · Zbl 1021.91033
[19] R. Carmona, Statistical Analysis of Financial Data in S-Plus, Springer Texts in Statistics, Springer, New York, NY, USA, 2004. · Zbl 1055.62112
[20] E. McKenzie, “An autoregressive process for beta random variables,” Management Science, vol. 31, no. 8, pp. 988-997, 1985. · Zbl 0607.62104 · doi:10.1287/mnsc.31.8.988
[21] M. Ristić and B. Popović, “A new uniform AR(1) time series model (NUAR(1)),” Publications de l’Institut Mathématique, vol. 68(82), pp. 145-152, 2000. · Zbl 1087.62102
[22] M. R. Chernick, “A limit theorem for the maximum of autoregressive processes with uniform marginal distributions,” The Annals of Probability, vol. 9, no. 1, pp. 145-149, 1981. · Zbl 0453.60026 · doi:10.1214/aop/1176994514
[23] A. J. Lawrance, “Uniformly distributed first-order autoregressive time series models and multiplicative congruential random number generators,” Journal of Applied Probability, vol. 29, no. 4, pp. 896-903, 1992. · Zbl 0761.62120 · doi:10.2307/3214722
[24] L. S. Dewald Sr., P. A. W. Lewis, and E. McKenzie, “Marginally specific alternatives to normal ARMA processes,” in Proceedings of the 19th Winter Simulation Conference (WSC ’87), A. Thesen, H. Grant, and W. D. Kelton, Eds., pp. 300-301, Atlanta, Ga, USA, December 1987. · doi:10.1145/318371.318435
[25] C. H. Sim, “Modelling non-normal first-order autoregressive time series,” Journal of Forcasting, vol. 13, no. 4, pp. 369-381, 1994. · Zbl 04520374 · doi:10.1002/for.3980130403
[26] C. C. Heyde and N. N. Leonenko, “Student processes,” Advances in Applied Probability, vol. 37, no. 2, pp. 342-365, 2005. · Zbl 1081.60035 · doi:10.1239/aap/1118858629
[27] G. K. Grunwald, R. J. Hyndman, L. Tedesco, and R. L. Tweedie, “Non-Gaussian conditional linear AR(1) models,” Australian & New Zealand Journal of Statistics, vol. 42, no. 4, pp. 479-495, 2000. · Zbl 1018.62065 · doi:10.1111/1467-842X.00143
[28] J. C. Hull, Options, Futures and Other Derivatives, Prentice Hall, Upper Saddle River, NJ, USA, 6th edition, 2005. · Zbl 1087.91025
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.