Calculating the American options in the default model. (English. Russian original) Zbl 1126.93436

Ivanov, R. V.

Autom. Remote Control 68, No. 3, 513-522 (2007); translation from Avtom. Telemekh. 68, No. 3, 154-164 (2007).

Summary: For the binomial model of the derivative securities market, consideration was given to calculation of prices and optimal instants of execution for the American instruments in the model with possible default (repudiation of a contract) by the contract holder. The results were obtained for the buyer and seller options with discount.

MSC:

93E20	Optimal stochastic control
91G80	Financial applications of other theories

Keywords:

binomial model; derivative security market; options

References:

[1]	Shiryaev, A.N., Osnovy stokhasticheskoi finansovoi matematiki (Fundamentals of the Stochastic Financial Mathematics), Moscow: Fazis, 1998.
[2]	Shiryaev, A.N., Kabanov, Yu.M., Kramkov, D.O., and Mel’nikov, A.V., On the Theory of Calculations of the European and American Options. I, Teor. Veroyatn. Primen., 1994, vol. 39, no. 1, pp. 21–79.
[3]	Myneni, R., The Pricing of the American Option, Ann. Appl. Probab., 1992, vol. 2, no. 1, pp. 1–23. · Zbl 0753.60040
[4]	Peskir, G., On the American Option Problem, Math. Finance, 2005, vol. 15, no. 1, pp. 169–181. · Zbl 1109.91028
[5]	Broadie, M. and Detemple, J., American Option Valuation: New Bounds, Approximations, and a Comparison of Existing Methods, Rev. Finan. Studies, 1995, vol. 9, pp. 1211–1250.
[6]	Lamberton, D., Error Estimates for the Binomial Approximation of American Put Options, Ann. Appl. Probab., 1998, vol. 8, pp. 206–233. · Zbl 0939.60022
[7]	Ivanov, R.V., Discrete Approximation of Finite-Horizon American-Style Options, Lithuan. Math. J., 2005, vol. 45, no. 4, pp. 525–536. · Zbl 1152.91517
[8]	Ivanov, R.V., On Discrete Approximation of the American Options, Usp. Mat. Nauk, 2006, vol. 61, no. 1, pp. 179–180.
[9]	Shiryaev, A.N., Kabanov, Yu.M., Kramkov, D.O., and Mel’nikov, A.V., On the Theory of Calculations of the European and American Options. II, Teor. Veroyatn. Primen., 1994, vol. 39, no. 1, pp. 80–129. · Zbl 0833.60065
[10]	Shepp, L.A. and Shiryaev, A.N., The Russian Option: Reduced Regret, Ann. Appl. Probab., 1993, vol. 3, no. 3, pp. 631–640. · Zbl 0783.90011
[11]	Kramkov, D.O. and Shiryaev, A.N., Calculating the Rational Cost of the ”Russian Optiion” in the Symmetrical Binomial Model of the (B,S)-market, Teor. Veroyatn. Primen., 1994, vol. 39, no. 1, pp. 191–200. · Zbl 0836.90013
[12]	Kramkov, D.O., Optional Decomposition of Supermartingales and Hedging Contingent Claims in Incomplete Security Markets, Probab. Theory Rel. Fields, 1996, vol. 105, no. 4, pp. 459–479. · Zbl 0853.60041
[13]	Kallsen, J. and Kühn, C., Pricing of Derivatives of American and Game Type in Incomplete Markets, Finance Stoch., 2004, vol. 8, no. 2 pp. 261–284. · Zbl 1052.91039
[14]	Evans, J.D., Kuske, R., and Keller, J.B., American Options on Assets with Dividends Near Expiry, Math. Finance, vol. 12, no. 3, pp. 219–239. · Zbl 1031.91047
[15]	Chalasani, P. and Jha, S., Randomized Stopping Times and American Option Pricing with Transaction Costs, Math. Finance, vol. 11, no. 1, pp. 33–79. · Zbl 0993.91021
[16]	Szimayer, A., Valuation of American Options in the Presence of Event Risk, Finance Stoch., 2005, vol. 9, no. 1, pp. 89–107. · Zbl 1078.91011
[17]	Linetsky, V., Pricing Equity Derivatives Subject to Bankruptcy, Math. Finance, 2006, vol. 16, no. 2, pp. 255–282. · Zbl 1145.91351