Artur Rodrigues

Optimal subsidies and guarantees in public–private partnerships

In this paper, we analyse how certain subsidies and guarantees given to private firms in public–private partnerships should be optimally arranged to promote immediate investment in a real options framework. We show how an investment subsidy, a revenue subsidy, a minimum demand guarantee, and a rescue option could be optimally arranged to induce immediate investment, compensating for the value of the option to defer. These four types of incentives produce significantly different results when we compare the value of the project after the incentive structure is devised and also when we compare the timing of the resulting cash flows.

Fast Trees for Options with Discrete Dividends

The valuation of options using a binomial non-recombining tree with discrete dividends can be intricate. This paper proposes three different enhancements that can be used alone or combined to value American options with discrete dividends using a nonrecombining binomial tree. These methods are compared in terms of both speed and accuracy with a large sample of options with one to four discrete dividends. This comparison shows that the best results can be achieved by the simultaneous use of the three enhancements. These enhancements when used together result in significant speed/accuracy gains in the order of up to 200 times for call options and 50 times for put options. These techniques allow the use of a non-recombining binomial tree with very good accuracy for valuing options with up to four discrete dividends in a timely manner. ∗The support of the Portuguese Foundation for Science and Technology Project PTDC/GES/78033/2006 is acknowledged. We thank Ana Carvalho for her helpful comments. †School of Economics and Management. University of Minho. 4710-057 Braga (Portugal). E-mail: nareal@eeg.uminho.pt; Phone: +351 253 601 923; Fax:+351 253 601 380. ‡Corresponding author. School of Economics and Management. University of Minho. 4710-057 Braga (Portugal). E-mail: artur.rodrigues@eeg.uminho.pt; Phone: +351 253 601 923; Fax:+351 253 601 380. Fast trees for options with discrete dividends American options valuation using binomial lattices can be cumbersome. Several authors have suggested different approaches to speed the computation or to increase the speed of convergence. Among others, some authors suggested ways to reduce the number of the nodes in the tree, thus speeding up the computation time (e.g.: Baule and Wilkens [2004]). One example of the latter approach is a paper by Curran [1995] that has been largely ignored in the literature. One advantage of Curran’s [1995] method is that it is not an approximation to the value of a binomial tree, since it produces exactly the same result as Cox, Ross, and Rubinstein’s [1979] (CRR) tree with the same number of steps at a fraction of the computation time. The gains of speeding the computation of a binomial tree are more relevant for valuing options with underling assets that pay discrete dividends. In such cases the binomial trees do not recombine and the number of nodes rapidly explode even for a small number of time steps. There are several approximations that allow the use of recombining binomial trees, but all of them can in some occasions produce large valuation errors (usually they occur when a dividend is paid at the very beginning of the option life), or require a very large number of steps to avoid such valuation errors.1 The advantages of using a non-recombining binomial tree are twofold: first it considers the true stochastic process for the underlying asset which excludes arbitrage opportunities; and secondly it eliminates large valuation errors irrespective of when the dividends occur. This, in turn, results in a accurate valuation of such options. The problem of these trees is that the number of nodes in the tree grows exponentially with the number of dividends. This explains why usually a recombining approximation is used instead of the non-recombining tree. Unfortunately the method proposed by Curran [1995] is not directly applied to options on assets which pay discrete dividends. In light of the techniques proposed by Curran [1995], which in turn are based on the work of Kim and Byun [1994], this paper adjusts their acceleration techniques to the valuation of American put options and also proposes a different accelerated binomial method to the valuation of American call options. We also suggest two other improvements with very good results. One, called here Adapted Binomial, consists in making a time step to coincide with the ex-dividend dates. The other is to apply the Black and Scholes [1973] formula to obtain the continuation value in the last steps of the binomial tree. These adaptations along with the improvements here suggested result in significant speed/accuracy gains in the order of up to 200 times for call options and 50 times for put options. These techniques allow the use of a non-recombining binomial tree with very good Examples of such methods are: Schroder [1988], Hull and White [1988], Harvey and Whaley [1992], Wilmott, Dewynne, and Howison [1998], Vellekoop and Nieuwenhuis [2006].

Discrete dividends and the FTSE-100 index options valuation

This paper studies the effect of discrete dividends on the FTSE-100 index options valuation, following closely Harvey and Whaleys [ J. Fut. Mkts , 1992, 12 (2), 123-137] study on the S&P-100 index. To the best of our knowledge, no such study has ever been performed on FTSE-100 options, where the dividends have a discreteness pattern different from the S&P-100. Unlike the Harvey and Whaley study, both American and European options are considered, a more accurate benchmark is proposed, and a comprehensive comparison of the accuracy of a larger set of valuation methods is performed. It is shown that there are significant differences in accuracy and speed among different methods, and that, for both American and European options, a great deal of accuracy can be gained by using an approximation that takes into account the discrete nature of the FTSE-100 index option dividends.

Real options - introduction to the state of the art

This special issue on Real Options of the European Journal of Finance includes 11 papers presented at the recent Annual International Conferences on Real Options. Eight of them were presented at the 13th Annual Conference held at the University of Minho (Braga, Portugal) and at the University of Santiago de Compostela (Spain) in 17–20 June 2009. The conferences brought together academics and practitioners at the forefront of real options and investment under uncertainty to discuss recent developments and applications. The conference, organised by the Real Options Group and the hosting universities, featured academic and professional presentations of theoretical and applied work, workshops and case discussions, experiences from the field and panel discussions. Tourinho (1979) proposed the first known real options model. In this issue, Tourinho revisits his contribution 30 years later, on the occasion of his keynote address at the 12th Annual Conference held in Rio de Janeiro (Brazil) in 2008. The paper addresses how the use of a real option models can be reconciled with the equilibrium required in the market of the asset being valued. In the case of natural resources he suggest that this can be ensured by modelling the commodity price as a mean-reverting process to the price that triggers investment. In the same conference and also in this issue, Adkins and Paxson also revisit the Tourinho (1979) model, highlighting the neglected contribution of that seminal model. They claim that combining a convenience yield, the most common approach in the real options models since Brennan and Schwartz (1985), and an option-holding cost, as proposed by Tourinho (1979), produces a better representation of the extraction paradox identified by Tourinho. The holding cost can be manipulated in order to influence the extraction timing and the profit of the option holder. Jaimungal, Souza and Zubelli, along with arguments of Tourinho, suggest that in equilibrium, output and input prices tend to be mean reverting and, therefore, propose a real options model where the option to invest is valued assuming that the project value (V) and the investment cost (I) are both mean reverting, showing that the trigger is no longer a linear function of the ratio V /I as in McDonald and Siegel’s (1986) model, which assumes both factors as geometric Brownian motions. Dockendorf and Paxson develop rainbow option models that are present in the case where firms have the flexibility to choose between two outputs, deriving a quasi-analytical solution and a numerical lattice solution. Using a real case with two commodities, they show that even for

On the Dangers of a Simplistic American Option Simulation Valuation Method

Chen and Shen [Chen, A.-S., and P.-F. Shen. 2003. Computational complexity analysis of least-squares Monte Carlo (LSM) for pricing US derivatives. Applied Economics Letters 10: 223-9] argue that we can improve the least squares Monte Carlo method (LSMC) to value American options by removing the least squares regression module. This would make it not only faster but also more accurate. We demonstrate, using a large sample of 2500 put options, that the proposed algorithm - the perfect foresight method (PFM) - is, as argued by the authors, faster than the LSMC algorithm but, contrary to what they state, it is not more accurate than the LSMC. In fact, the PFM algorithm incorrectly prices American options. We therefore, do not recommend the use of the PFM.

The Optimal Timing for the Construction of an Airport

In this paper we study the option to invest in a new airport, considering that the benefits of the investment behave stochastically. In particular, the number of passengers, and the cash flow per passenger are both assumed to be random. Additionally, positive and negative shocks are also incorporated, which seems to be realistic for this type of projects. Accordingly, we propose a new real options model which combines two stochastic factors with positive and negative shocks.