Revisiting the Tourinho real options model: outstanding issues 30 years later

Abstract

This article presents and extends the first known model in real options, proposed in Tourinho (1979), and provides thoughts on addressing issues that are still outstanding 30 years later. It discusses the need to ensure the existence of market equilibrium when applying real options valuation to price assets, once all agents behave as suggested by the solution to the pricing equation. It argues that this can be achieved by using a stochastic process for the price that is sufficiently general to respond to supply and demand imbalances in the market for the resource. Once the individual decision rules are derived, the parameters of the process must be determined to ensure market equilibrium exists. For reserves of natural resources, this can be done by using a mean-reverting process for the price of the commodity and ensuring that the long-term price to which it reverts equals the trigger price for development of the marginal reserve.

Keywords:

• real options
• general equilibrium
• mean reversion

JEL Classification: :

• C62
• G12
• G31

Acknowledgements

This is a revised version of the keynote address presented at the 12th Annual International Conference on Real Options: Theory Meets Practice, held in Rio de Janeiro, Brazil, 9–12 July 2008, which was partly based on my 1979 PhD dissertation at the University of California at Berkeley. I thank Hayne Leland for suggestions, comments, incentive, and support throughout my dissertation, Lenos Trigeorgis for extensive comments and editorial remarks on the manuscript, and Dean Paxson and Roger Adkins for comments on a previous version of this article. I assume full responsibility for any remaining errors.

Notes

In the model, there are many simplifications, as indicated by the assumptions below. This is helpful in testing the consistency of the approach and highlights the equilibrium problem we wish to address. The body of literature that has developed in this area shows that the real options model is quite robust to the relaxation of several of the assumptions made herein, so they should not be regarded as being unduly restrictive.

The diffusion in Equation (1) has several attractive features. Since successive price ratios are independent and identically distributed, the resulting price distribution is log-normal at any point in time and is consistent with non-negative prices. As Samuelson (1965a) points out, there is no a priori reason for the spot price of a commodity to follow a specific probability model since correctly anticipated futures prices must have unbiased price changes and its rate of return must follow a random walk.

The assumption regarding the specification of the variance can be removed at the cost of added complexity. For example, Dias and Nunes (2008) assume a constant elasticity of variance diffusion, and Heston (1993) solves the option pricing problem for a process whose instantaneous variance follows a Cox-Ingersoll-Ross stochastic process.

This does not mean that negative resource holdings could exist in aggregate since for every short position there must be an offsetting long position.

The formal implications of doing this are not detailed here, as it involves including in the model the determination of futures prices, as done by Brennan and Schwartz (1985). As a first approximation, it is possible to disregard the details of the interaction between futures and spot prices, focusing instead on long-term prices. Schwartz (1998) develops a one-factor model of the stochastic behavior of long-term prices that retains most of the characteristics of the more complex two-factor stochastic convenience yield model, yielding practically the same result when valuing investment projects. To the extent that there is a strong correlation between them, the choice of the underlying asset on which the value of the real option is based is not crucial. In cases where the correlation between spot and futures prices is not high, two-factor and three-factor models must be used. Lautier (2003) surveys the literature on the term structure of commodity prices, discusses its relation to the convenience yield, and proposes different stochastic specifications for it.

Situations where time-to-build is important can be analyzed as proposed by Friedl (2002). He solves the real option valuation problem for that case in a framework similar to the one used here.

This assumption is not as restrictive as may seem. The price can be interpreted as being that of a long-term supply contract where extraction proceeds at a (planned) set rate once a given section of the mine is open. In this case, the future flow of the commodity can be sold at the long-term contract price, when the decision to exercise the real option is made, and the reserve owner would not run any further risk. Alternatively, the reserve owner may hold on to it and speculate on the difference between the long-term price and the spot price of the commodity. The decision on what to do with the extracted resource can be separated from that of developing the reserve. It is not addressed by the real option valuation model. It presumably depends on the determinants of the difference between spot and futures prices and hinges, among other things, on the relation between storage costs and the convenience yield of the resource.

Copeland and Antikarov (2001) provide some empirical evidence that the value of the option to shut down is small in the case of petroleum reserves.

It is also required for the development below that the reserve is a twice continuously differentiable function of the price of the underlying resource and time (C=C(S, t)). Merton (1977) shows that this condition can be derived as consequence, not an assumption, of the analysis.

I thank Roger Adkins and Dean Paxson for pointing this out to me. The precision of the approximation can be seen, for example, comparing the analytical and numeric solutions for S=20. For the former, the value of the reserve is 0.47610, which compares favorably with the value found in Table 3.1 of Tourinho (1979) equal to 0.47354.

McDonald and Siegel (1986) later used the same approach to solve the two-factor model with a convenience yield.

An intuitive reason for the absence of the drift parameter in the real option valuation equation is that, for the serially uncorrelated process of Equation (1), the commodity price S, which already appears in the equation, conveys all the information that would be contained in μ.

Brennan and Schwartz (1985) introduced convenience yields in the valuation of real options on natural resource assets in a two-factor model that includes the futures prices. Adkins and Paxson (2011) include it in the original single-factor model. This can be justified using the insight from Schwartz (1998) whereby the results obtained from the two-factor model with stochastic convenience yield can be effectively approximated by a one-factor model if the underlying commodity is interpreted as being a long-term supply contract. A large fraction of the international physical trade of iron ore, other mineral commodities, and petroleum is governed by medium-term contracts, rather than spot prices.

The introduction of a convenience yield (δ) also enables the instantaneous extraction and no suspension assumptions to be relaxed (with S being replaced by , where τ is the time of delay).

‘Marginal’ is used here to refer to the reserve whose development was decided last, in the immediate past, and not the reserve capable of supplying the marginal unit of the commodity.

While closure of the model is crucial to determine the general equilibrium path of prices, it may not be so important to determine the option value of reserves as it may not affect significantly the valuation function. This might be the case if the current price contains most of the relevant information for pricing the real option, even with mean reversion. As a consequence, in any specific application of the methodology, it is important to verify whether the bucket-shop assumption is justified by checking whether the trigger rules derived from the analysis are consistent with equilibrium in the underlying asset market.