O.R. Applications On the no-arbitrage condition in option implied trees V. Moriggia a , S. Muzzioli b, * , C. Torricelli b a Department of Mathematics, Statistics, Computer Science and Applications, University of Bergamo, via dei Caniana 2, 24127 Bergamo, Italy b Department of Economics and CEFIN, University of Modena and Reggio Emilia, V.le Berengario 51, 41100 Modena, Italy Received 9 October 2006; accepted 11 October 2007 Available online 18 October 2007 Abstract The aim of this paper is to discuss the no-arbitrage condition in option implied trees based on forward induction and to propose a no- arbitrage test that rules out the negative probabilities problem and hence enhances the pricing performance. The no-arbitrage condition takes into account two main features: the position of the node in the tree and the relation between the dividend yield and the risk-free rate. The proposed methodology is tested in and out of sample with Italian index options data and findings support a good pricing performance. Ó 2007 Elsevier B.V. All rights reserved. Keywords: Finance; No-arbitrage condition; Binomial tree; Implied volatility; Calibration 1. Introduction After the October 1987 crash, option markets exhibited implied volatilities that varied across different strikes (smile effect) and different times to expiration (term structure of the volatility), in contrast with the Black and Scholes (1973) assumption of constant volatility. In order to cap- ture the implied volatility dependence on strike and time to maturity, different smile-consistent no-arbitrage models have been proposed in the literature, which can be classi- fied either as deterministic or stochastic volatility models. 1 Deterministic volatility models (see e.g. Derman and Kani, 1994; Barle and Cakici, 1995, 1998; Rubinstein, 1994; Jackwerth, 1997; Dupire, 1994) derive endogenously from European option prices the instantaneous volatility as a deterministic function of the asset price and time. Stochas- tic volatility models (see e.g. Derman and Kani, 1997; Brit- ten-Jones and Neuberger, 2000; Ledoit et al., 2002) allow for a no-arbitrage evolution of the implied volatility surface. Deterministic volatility models have both theoretical and practical advantages: they preserve the no-arbitrage pricing property of the Black and Scholes model and are easily implementable. With the exception of Dupire (1994), which is developed in continuous time, most models are developed in discrete time. Among the latter, some (Derman and Kani, 1994; Barle and Cakici, 1995, 1998; Li, 2001) use forward induction in the derivation of the implied trees, others (Rubinstein, 1994; Jackwerth, 1997), use backward induction. 2 The Rubinstein (1994) model is based on the assumption that different paths that lead to the same ending node have the same risk neutral probabil- ity, it captures only the smile effect and it is not useful for pricing path dependent options. The Jackwerth (1997) model, extend Rubinstein’s by allowing the implied tree to fit intermediate maturity options, thus capturing both the smile effect and the term structure of the volatility. The main advantages of deriving implied trees by forward 0377-2217/$ - see front matter Ó 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2007.10.017 * Corresponding author. Tel.: +39 0 59 2056771; fax: +39 0 59 2056947. E-mail address: silvia.muzzioli@unimore.it (S. Muzzioli). 1 See Bates (2003) for a survey on the approaches taken in option pricing and Skiadopoulos (2001) for a taxonomy and an extensive survey on smile-consistent no-arbitrage models. 2 This paper departs here from the terminology used by Skiadopoulos (2001) in that forward induction models are meant as those that use also forward induction and backward induction ones are those that use only backward induction. www.elsevier.com/locate/ejor Available online at www.sciencedirect.com European Journal of Operational Research 193 (2009) 212–221