Abstract
We introduce the optimal-drift model for the approximation of a lognormal stock price process by an accelerated binomial scheme. This model converges with order o(1/N), which is superior compared to today’s benchmark methods. Our approach is based on the observation that risk-neutral binomial schemes converge to the lognormal limit independently of the choice of the drift parameter. We verify the improved order of convergence by an asymptotic expansion of the binomial distribution function. Further, we show that the above result on drift invariance implies weak convergence of the binomial schemes suggested by Tian (in J. Futures Mark. 19, 817–843, 1999) and Chang and Palmer (in Finance Stoch. 11, 91–105, 2007).
Similar content being viewed by others
Notes
In the original article by Tian, the author illustrates smooth convergence with numerical examples, while a mathematical proof is given by Chang and Palmer.
In the literature, the term ‘CRR model’ appears both for the discretisation scheme with transition probabilities \(p(N)=1/2+1/2(\frac{r-1/2\sigma^{2}}{\sigma})\sqrt {T/N}\) and for the discretisation scheme with risk-neutral transition probabilities. In this paper, we use the term ‘CRR model’ for the latter variant.
Compare Diener and Diener [6] for an extended asymptotic calculus to bounded, but non-constant, coefficients.
We obtain the pricing error for cash-or-nothing options by scaling the discretisation error with Ge −rT. It is well known that in the general case (N=10,11,12,…,4000), we additionally face the even-odd effect. In fact, the even-odd effect is reflected in the defining equation (2.8) of b(N) because −1/2N does not contribute to the fractional part for N even, while it does for N odd.
Due to the presence of an even-odd effect, the analysis is again limited to even values of N.
The Leisen–Reimer model is based on the use of normal approximations to the binomial distribution. It thus cannot be directly included in our theoretical analysis of the order of convergence. In Tables 3 and 4, results from the Leisen–Reimer model are obtained with the Preizer–Pratt method 2 inversion (compare Leisen and Reimer [8]).
References
Amin, K., Khanna, A.: Convergence of American option values from discrete- to continuous-time financial models. Math. Finance 4, 289–304 (1994)
Billingsley, P.: Convergence of Probability Measures. Wiley, New York (1968)
Björk, T.: Arbitrage Theory in Continuous Time, 2nd edn. OUP, Oxford (2004)
Chang, L.-B., Palmer, K.: Smooth convergence in the binomial model. Finance Stoch. 11, 91–105 (2007)
Cox, J.C., Ross, S.A., Rubinstein, M.: Option pricing: A simplified approach. J. Financ. Econ. 7, 229–263 (1979)
Diener, F., Diener, M.: Asymptotics of the price oscillations of a European call option in a tree model. Math. Finance 14, 271–293 (2004)
Leisen, D.P.J.: Pricing the American put option: A detailed convergence analysis for binomial models. J. Econ. Dyn. Control 22, 1419–1444 (1998)
Leisen, D.P.J., Reimer, M.: Binomial models for option valuation: Examining and improving convergence. Appl. Math. Finance 3, 319–346 (1996)
Staunton, M.: From top dog to straw man. Wilmott Magazine, January/February 2004, 34–36 (2004)
Staunton, M.: Efficient estimates for valuing American options. In: Wilmott, P. (ed.) The Best of Wilmott 2, pp. 91–97. Wiley, Chichester (2005)
Tian, Y.S.: A flexible binomial option pricing model. J. Futures Mark. 19, 817–843 (1999)
Uspensky, J.V.: Introduction to Mathematical Probability. McGraw-Hill, New York (1937)
Acknowledgements
Both authors thank the Rheinland-Pfalz Cluster of Excellence DASMOD and the Research Centre (CM)2 for financial support. We thank Mike Staunton and Frank Seifried for valuable comments and software code. We also thank two anonymous referees for valuable hints and comments.
Author information
Authors and Affiliations
Corresponding author
Additional information
Stefanie Müller is currently working for KPMG AG Wirtschaftsprüfungsgesellschaft.
Rights and permissions
About this article
Cite this article
Korn, R., Müller, S. The optimal-drift model: an accelerated binomial scheme. Finance Stoch 17, 135–160 (2013). https://doi.org/10.1007/s00780-012-0179-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00780-012-0179-y