Abstract
In this article, we consider a general class of binomial models with an additional parameter λ. We show that in the case of a European call option the binomial price converges to the Black–Scholes price at the rate 1/n and, more importantly, give a formula for the coefficient of 1/n in the expansion of the error. This enables us, by making special choices for λ, to prove that convergence is smooth in Tian’s flexible binomial model and also in a new center binomial model which we propose.
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Amin K., Khanna A. (1994) Convergence of American option values from discrete- to continuous-time financial models. Math. Financ. 4, 289–304
Cox J., Ross S.A., Rubinstein M. (1979) Option pricing: a simplified approach. J. Financ. Econ. 7, 229–263
Diener F., Diener M.: Asymptotics of the binomial formula for option pricing. http://www-math.unice.fr(00000708-LG) (1999)
Diener F., Diener M. (2004) Asymptotics of the price oscillations of a European call option in a tree model. Math. Financ. 14, 271–293
Heston S., Zhou G. (2000) On the rate of convergence of discrete-time contingent claims. Math. Financ. 10, 53–75
Hsia C.-C. (1983) On binomial option pricing. J. Financ. Res. 6, 41–50
Jarrow R., Rudd A. (1983) Option Pricing. Irwin, Homewood
Jiang L., Dai M. (1999) Convergence of binomial tree method for American options. In: Chen H., Rodino L. (eds) Partial Differential Equations and Their Applications (Proceedings of the conference, Wuhan, China). World Scientific, Singapore, pp. 106–118
Lamberton D. (1998) Error estimates for the binomial approximation of American put options. Ann. Appl. Probab. 8, 206–233
Leisen D.P.J. (1998) Pricing the American put option: a detailed convergence analysis for binomial models. J. Econ. Dyn. Control 22: 1419–1444
Leisen D.P.J., Reimer M. (1996) Binomial models for option valuation—examining and improving convergence. Appl. Math. Financ. 3, 319–346
Pliska S.R. (1997) Introduction to Mathematical Finance: Discrete Time Models. Blackwell, Oxford
Qian X., Xu C., Jiang L., Bian B. (2005) Convergence of binomial tree method for American options in a jump-diffusion model. SIAM J. Numer. Anal. 42, 1899–1913
Rendleman R., Bartter B. (1979) Two state option pricing. J. Financ. 34, 1093–1110
Tian Y.S. (1993) A modified lattice approach to option pricing. J. Futures Markets 13, 563–577
Tian Y.S. (1999) A flexible binomial option pricing model. J. Futures Markets 19, 817–843
Uspensky J.V. (1937) Introduction to Mathematical Probability. McGraw-Hill, New York
Walsh J.B. (2003) The rate of convergence of the binomial tree scheme. Financ. Stochastics 7, 337–361
Walsh J.B., Walsh O.D. (2002) Embedding and the convergence of the binomial and trinomial tree schemes. In: Lyons T.J., Salisbury T.S. (eds) Numerical Methods and Stochastics (Fields Institute Communications, vol 34). American Mathematical Society, Providence, pp. 101–121
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Ken Palmer was supported by NSC grant 93-2118-M-002-002.
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Chang, LB., Palmer, K. Smooth convergence in the binomial model. Finance Stoch 11, 91–105 (2007). https://doi.org/10.1007/s00780-006-0020-6
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DOI: https://doi.org/10.1007/s00780-006-0020-6