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Finance and Stochastics

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01-01-2013

The optimal-drift model: an accelerated binomial scheme

Authors: Ralf Korn, Stefanie Müller

Published in: Finance and Stochastics | Issue 1/2013

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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).

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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]).

Amin, K., Khanna, A.: Convergence of American option values from discrete- to continuous-time financial models. Math. Finance 4, 289–304 (1994) MathSciNetMATHCrossRef

Billingsley, P.: Convergence of Probability Measures. Wiley, New York (1968) MATH

Björk, T.: Arbitrage Theory in Continuous Time, 2nd edn. OUP, Oxford (2004) MATHCrossRef

Chang, L.-B., Palmer, K.: Smooth convergence in the binomial model. Finance Stoch. 11, 91–105 (2007) MathSciNetMATHCrossRef

Cox, J.C., Ross, S.A., Rubinstein, M.: Option pricing: A simplified approach. J. Financ. Econ. 7, 229–263 (1979) MATHCrossRef

Diener, F., Diener, M.: Asymptotics of the price oscillations of a European call option in a tree model. Math. Finance 14, 271–293 (2004) MathSciNetMATHCrossRef

Leisen, D.P.J.: Pricing the American put option: A detailed convergence analysis for binomial models. J. Econ. Dyn. Control 22, 1419–1444 (1998) MathSciNetMATHCrossRef

Leisen, D.P.J., Reimer, M.: Binomial models for option valuation: Examining and improving convergence. Appl. Math. Finance 3, 319–346 (1996) CrossRef

Staunton, M.: From top dog to straw man. Wilmott Magazine, January/February 2004, 34–36 (2004)

10.

Staunton, M.: Efficient estimates for valuing American options. In: Wilmott, P. (ed.) The Best of Wilmott 2, pp. 91–97. Wiley, Chichester (2005)

11.

Tian, Y.S.: A flexible binomial option pricing model. J. Futures Mark. 19, 817–843 (1999) CrossRef

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Uspensky, J.V.: Introduction to Mathematical Probability. McGraw-Hill, New York (1937) MATH

Title: The optimal-drift model: an accelerated binomial scheme
Authors: Ralf Korn
Stefanie Müller
Publication date: 01-01-2013
Publisher: Springer-Verlag
Published in: Finance and Stochastics / Issue 1/2013
Print ISSN: 0949-2984
Electronic ISSN: 1432-1122
DOI: https://doi.org/10.1007/s00780-012-0179-y

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