Gonzalo Cortazar

THE VALUATION OF COMMODITY- CONTINGENT CLAIMS

!This article describes a new approach to the valuation of commodity-contingent claims. The approach uses all the information contained in the term structure ofcommodity futures prices in addition to the historical ,volatilities of futures returnsfor diferent maturities. It is based on the principle that no arbitrage opportunities shoullf exist when trading in futures contracts. The jiamework is applied to price copper-contingent claims. We analyze the daily returnsfor all copper futures

Implementing a stochastic model for oil futures prices

This paper develops a parsimonious three-factor model of the term structure of oil futures prices that can be easily estimated from available futures price data. In addition, it proposes a new simple spreadsheet implementation procedure. The procedure is flexible, may be used with market prices of any oil contingent claim with closed form pricing solution, and easily deals with missing data problems. The approach is implemented using daily prices of all futures contracts traded at the New York Mercantile Exchange between 1991 and 2001. In-sample and out-of-sample tests indicate that the model fits the data extremely well. Though the paper concentrates on oil, the approach can be used for any other commodity with well-developed futures markets.

Optimal Exploration Investments under Price and Geological-Technical Uncertainty: A Real Options Model

This article develops a real options model for valuing natural resource exploration investments (e.g. oil or copper) when there is joint price and geological-technical uncertainty. After a successful several-stage exploration phase, there is a development investment and an extraction phase. All phases are optimised contingent on price and geological-technical uncertainty. Several real options are considered. There are flexible investment schedules for all exploration stages and a timing option for the development investment. Once the mine is developed, there are closure, opening and abandonment options for the extraction phase. Our model maintains a relatively simple valuation structure by collapsing price and geological-technical uncertainty into a one-factor model. A computational implementation of the model applied to a copper exploration prospect shows that a significant fraction of total project value is due to the operative, the development and the exploration options available to project managers.

The valuation of multidimensional American real options using the LSM simulation method

In this paper we show how a multidimensional American real option may be solved using the LSM simulation method originally proposed by Longstaff and Schwartz [2001, The Review of the Financial Studies 14(1): 113-147] for valuing a financial option and how this method can be used in a complex setting. We extend a well-known natural resource real option model, initially solved using finite difference methods, to include a more realistic three-factor stochastic process for commodity prices, more in line with current research. Numerical results show that the procedure may be successfully used for multidimensional models, expanding the applicability of the real options approach. Even though there has been an increasing literature on the benefits of using the contingent claim approach to value real assets, limitations on solving procedures and computing power have often forced academics and practitioners to simplify these real option models to a level in which they loose relevance for real-world decision making. Real option models present a higher challenge than their financial option counterparts because of two main reasons: First, many real options have a longer maturity which makes risk modeling critical and may force considering many risk factors, as opposed to the classic Black and Scholes approach with only one risk factor. Second, real investments many times exhibit a more complex set of interacting American options, which make them more difficult to value. In recent years new approaches for solving American options have been proposed which, coupled with an increasing availability of computing power, have been successfully applied to solving long-term financial options. In this paper we explore the applicability of one the most promising of these new methods in a multidimensional real option setting.

A Compound Option Model of Production and Intermediate Inventories

This article extends the option approach to valuing real assets by modeling the firm as a two-stage process with bounded output rates, in which the output of the first stage may be held as work-in-process. In this setting, the real asset becomes a compound option, which, if exercised, gives the option to finish the work-in-process and sell the output as its final payoff. The existence of intermediate inventories may arise as an optimal investment strategy for exploiting possible future price increases. The framework allows us to analyze the effect of uncertainty on output rates and the effect of interest rates changes on inventory levels. Copyright 1993 by University of Chicago Press.

Monte Carlo evaluation model of an undeveloped oil field

Abstract In this article we develop and implement a model to value an undeveloped oil field and to determine the optimal timing of investment. We assume a two factor model for the stochastic behavior of oil prices for which a closed form solution for futures prices can be obtained. The advantage of this model is that is allows for the term structure of futures prices to be upward sloping (contango), downward sloping (backwardation) and also humped. We use Monte Carlo simulation methods for solving the problem. Since the decision to develop the oil field can be taken at any time until the expiration of the concession, the option to invest is of the American type. This type of options are solved by the numerical solution of the appropriate partial differential equation. If we assume, however, that the decision to invest (exercise the option) can be made at a finite number of points in time instead of continuously, the problem can be solved using simulation methods. Apart from being more intuitive, Monte Carlo simulation methods easily allow for the consideration of many additional random variables such as costs, amount of reserves, etc.

Evaluation of Installation Stock Based (R, Q)-Policies Fortwo-Level Inventory Systems with Poisson Demands

We consider a two-level inventory system with one warehouse and N retailers. Leadtimes (transportation times) are constant, and the retailers face different Poisson demand processes. All facilities apply continuous review installation stock (R, Q)-policies with different reorder points and batch quantities. We show how to evaluate holding and shortage costs exactly in case of two retailers. In case of more than two retailers we use the same methodology as an approximation. When evaluating costs at a certain retailer, the other retailers are then aggregated into a single retailer.

Optimal timing of a mine expansion: Implementing a real options model

We present the results of implementing a real options model for valuing an investment project that expands production capacity and/or modifies unit costs of a copper mine. The model and its implementation addresses the three requirements we find necessary to increase the use of the real option methodology by the practitioner community: a user-acceptable stochastic model for commodity prices with mean reversion, a customized real asset model which includes the main managerial flexibilities of opening-closing production or delaying investments, and a user-friendly computer implementation. A case study shows that a significant fraction of investment value may be due to the flexibility of delaying investment, value that decreases as copper prices increase. Critical investment prices are analyzed.

Evidence Propagation and Value of Evidence on Influence Diagrams

In this paper, we introduce evidence propagation operations on influence diagrams, a concept of the value of evidence to measure the impact/value of new observations/experimentation, and a concept of the value of revelation. Evidence propagation operations are critical for the computation of the value of evidence, general update and inference operations in normative expert systems that are based on the influence diagram (generalized Bayesian network) paradigm. The value of evidence allows us to compute the outcome sensitivity directly defined as the maximum difference among the values of evidence, and the value of perfect information, as the expected value of the values of evidence. We define the value of revelation as the optimal value of the values of evidence. We discuss the relationship between the value of revelation and the value of control. We also discuss implementation issues related to computation of the value of evidence and the value of perfect information.

Term Structure Estimation in Low-Frequency Transaction Markets: A Kalman Filter Approach with Incomplete Panel-Data

There are two issues that are of central importance in term structure analysis. One is the modeling and estimation of the current term structure of spot rates. The second is the modeling and estimation of the dynamics of the term structure. These two issues have been addressed independently in the literature. The methods that have been proposed assume a sufficiently complete price data set and are generally implemented separately. However, when the methods are applied to markets with sparse bond price, results are unsatisfactory. We develop a method for jointly estimating the current term structure and its dynamics for markets with low-frequency transactions. We propose solving both issues by using a dynamic term structure model estimated from incomplete panel data. To achieve this, we modify the standard Kalman filter approach to deal with the missing-observation problem. In this way, we can use historic price data in a dynamic model to estimate the current term structure. With this approach we are able to obtain an estimate of the current term structure even for days with an arbitrary low number of price observations. The proposed methodology can be applied to a broad class of continuous-time term-structure models with any number of stochastic factors. To show the implementation of the approach, we estimate a three-factor generalized-Vasicek model using Chilean government bond price data. The approach, however, may be used in any market with low-frequency transactions, a common characteristic of many emerging markets.