The Annals of Applied Probability

Error estimates for binomial approximations of game options

Yuri Kifer

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We justify and give error estimates for binomial approximations of game (Israeli) options in the Black–Scholes market with Lipschitz continuous path dependent payoffs which are new also for usual American style options. We show also that rational (optimal) exercise times and hedging self-financing portfolios of binomial approximations yield for game options in the Black–Scholes market “nearly” rational exercise times and “nearly” hedging self-financing portfolios with small average shortfalls and initial capitals close to fair prices of the options. The estimates rely on strong invariance principle type approximations via the Skorokhod embedding.

Article information

Ann. Appl. Probab., Volume 16, Number 2 (2006), 984-1033.

First available in Project Euclid: 29 June 2006

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 91B28
Secondary: 60F15: Strong theorems 91A05: 2-person games

Game options Dynkin games complete markets binomial approximation Skorokhod embedding


Kifer, Yuri. Error estimates for binomial approximations of game options. Ann. Appl. Probab. 16 (2006), no. 2, 984--1033. doi:10.1214/105051606000000088.

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  • Amin, K. and Khanna, A. (1994). Convergence of American option values from discrete-to continuous-time financial models. Math. Finance 4 289--304.
  • Baurdoux, E. J. and Kyprianou, A. E. (2004). Further calculations for Israeli options. Stochastics Stochastics Rep. 76 549--569.
  • Berkes, I. and Philipp, W. (1979). Approximation theorems for independent and weakly dependent random vectors. Ann. Probab. 7 29--54.
  • Billingsley, P. (1986). Probability and Measure, 2nd ed. Willey, New York.
  • Cvitanic, J. and Ma, J. (2001). Reflected forward--backward SDEs and obstacle problems with boundary conditions. J. Appl. Math. Stochastic Anal. 14 113--138.
  • Cox, J. C., Ross, R. A. and Rubinstein, M. (1976). Option pricing: A simplified approach. J. Financ. Econom. 7 229--263.
  • Ekström, E. (2006). Properties of game options. Math. Methods Oper. Res. 63. To appear.
  • Gapeev, P. V. and Kühn, C. (2005). Perpetual convertible bonds in jump--diffusion models. Statist. Decisions 23 15--31.
  • Hamadene, S. (2002). Mixed zero--sum differential game, Dynkin games, and American game options. Preprint.
  • Hubalek, F. and Schachermayer, W. (1998). When does convergence of asset price processes imply convergence of option prices. Math. Finance 8 385--403.
  • Ikeda, N. and Watanabe, S. (1989). Stochastic Differential Equations and Diffusion Processes, 2nd ed. North-Holland, Amsterdam.
  • Kallsen, J. and Kühn, C. (2004). Pricing derivatives of American and game type in incomplete markets. Finance Stoch. 8 261--284.
  • Karatzas, I. and Shreve, S. (1988). Brownian Motion and Stochastic Calculus. Springer, New York.
  • Kiefer, J. (1969). On the deviations in the Skorokhod--Strassen approximation scheme. Z. Wahrsch. Verw. Gebiete 13 321--332.
  • Kifer, Yu. (2000). Game options. Finance Stoch. 4 443--463.
  • Komlós, J., Major, P. and Tusnády, G. (1976). An approximation of partial sums of independent RV's and the sample DF. II. Z. Wahrsch. Verw. Gebiete 34 33--56.
  • Kühn, C. and Kyprianou, A. E. (2003). Collable puts as composite exotic options. Preprint.
  • Kühn, C. and Kyprianou, A. E. (2003). Pricing Israeli options: A pathwise approach. Preprint.
  • Kyprianou, A. E. (2004). Some calculations for Israeli options. Finance Stoch. 8 73--86.
  • Kunita, H. and Seko, S. (2004). Game call options and their exercise regions. Preprint.
  • Lamberton, D. (1998). Error estimates for the binomial approximation of American put options. Ann. Appl. Probab. 8 206--233.
  • Lamberton, D. (2002). Brownian optimal stopping and random walks. Appl. Math. Optim. 45 283--324.
  • Lepeltier, J. P. and Maingueneau, J. P. (1984). Le jeu de Dynkin en theorie generale sans l'hypothese de Mokobodski. Stochastics 13 24--44.
  • Lamberton, D. and Rogers, L. C. G. (2000). Optimal stopping and embedding. J. Appl. Probab. 37 1143--1148.
  • Mulinacci, S. (2003). American path-dependent options: Analysis and approximations. Rend. Studi Econ. Quant. 2002 93--120.
  • Neveu, J. (1975). Discrete-Parameter Martingales. North-Holland, Amsterdam.
  • Rogers, L. C. G. and Stapleton, E. J. (1998). Fast accurate binomial pricing. Finance Stoch. 2 3--17.
  • Shiryaev, A. N. (1999). Essentials of Stochastic Finance. World Scientific Publishing, River Edge, NJ.
  • Stroock, D. W. and Varadhan, S. R. S. (1979). Multidimensional Diffusion Processes. Springer, Berlin.
  • Walsh, J. B. (2003). The rate of convergence of the binomial tree scheme. Finance Stoch. 7 337--361.
  • Zaitsev, A. Yu. (2001). Multidimensional version of a result of Sakhanenko in the invariance principle for vectors with finite exponential moments. I--III. Theory Probab. Appl. 45 624--641; 46 (2002) 490--514, 676--698.