The two-factor price process in optimal sequential exploration

Abstract

In a group of exploration prospects with common geological features, drilling a well reveals information about chances of success in others. In addition, oil prices vary during the exploration campaign and with them so do the economics of wells and the optimal decision to drill. With these dependencies and price dynamics, where do we drill first and what comes next given success or failure in previous wells? The solution to this valuation problem should compare the value of learning (drilling wells that provide valuable information) with the uncertain value of earning (drilling wells that have large payoffs, yet uncertain). We calculate a joint distribution for geological outcomes by applying information-theoretic methods and construct a two-dimensional binomial sequence to represent a two-factor stochastic price process. We then propose a Markov decision process that solves the optimal exploration problem. An Excel® VBA software implementation of this algorithm accompanies this paper.

Keywords:

• Sequential Exploration
• capital budgeting
• Markov decision processes
• investments
• real options

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 This model is in fact equivalent to the stochastic convenience yield model of Gibson and Schwartz (1990). Mean-reversion as an appropriate assumption for commodities is discussed in e.g. Laughton and Jacoby (1993), Cortazar and Schwartz (1994), and Dixit et al. (1994). Schwartz (1997) discusses mean-reversion in stochastic price models and their ability to price existing future contracts, as well as financial and real assets.

2 Previous attempts to discretize two-factor price diffusions, notably the dual trinomial lattice approach in Hull and White (1994) or the improved version of Tseng and Lin (2007), worked only under a specific range of correlation values and had computational limitations.

3 In commodity markets, a forward curve is a function that defines prices for a set of forward contracts; all contracts are identical except for their varying maturities.

4 For our ten well application, there will be 1024 unknown joint probabilities ($2^n, n=10$) and 56 constraints ($1+n+n\left(n-1\right)/2, n=10$). Bickel and Smith (2006) explain the details of Kullback-Leibler procedure and its Lagrangian dual.

5 In the first decision epoch, we have 10 alternative wells each with 2 geologic outcomes and 4 price moves. The next epoch has 9 wells, each with 8 outcomes. Continuing this trend, we will have$10!\times 8^{10}$ total outcomes.

6 Although this application seems ideal for array programming languages, it is arguably not the ideal choice for users in academia and industry. For example, prohibitively high software license fees and scarce availability of programming skills hamper MATLAB® implementations of valuation algorithms (e.g. exploration waiting option in Jafarizadeh & Bratvold 2015). Yet Microsoft Excel is perhaps the platform of choice for small and medium scale analysis tasks in industry and dissemination of an open-source VBA application may be more beneficial.