Continuous-Time Option Games Review of Models and Extensions

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Continuous-Time Option Games Review of Models and Extensions

    Continuous-Time Option Games: Review of Models and Extensions

    Part 2: Oligopoly and War of Attrition under Uncertainty

    ththFirst Version: February 9, 2004. Current Version: June 12, 2004

    By: Marco Antonio Guimarães Dias (*) and José Paulo Teixeira (**)


    This sequel paper analyzes other selected methodologies and applications from the theory of continuous-time (real) option games the combination of real options and game theory. In the first

    paper (Dias & Teixeira, 2003), we analyzed preemption and collusion models of duopoly under uncertainty. In this second paper we focus on models of oligopoly under uncertainty, war of attrition under uncertainty, and the changing the war of attrition game toward a bargaining game. In the oligopoly model we follow Grenadier (2002), discussing two important methodological insights that simplify many option games applications: the Leahy’s principle of optimality of myopic behavior and

    the "artificial" perfectly competitive industry with a modified demand function. We discuss both the

    potential and the limitations of these insights. Next, we extend to the continuous-time framework the option game model presented in Dias (1997), a war of attrition under uncertainty applied to oil

    exploration prospects. In this model of positive externality the follower acts as free rider receiving additional information revealed by the leader’s drilling outcome. The way to model the information

    revelation in oil exploration is another extension of the original option game model. In addition, we analyzed the possibility of changing the game with the oil firms playing the bargaining game be

    perfect Nash equilibrium. Cooperation can increase the value of the firms thanks to additional private

    information revelation provided by a contract. We quantify the degree of information revelation with the convenient learning measure named expected variance reduction. The bargaining game strategy

    must be compared mainly with the follower strategy in asymmetric war of attrition. We set the game threshold window where the bargaining alternative dominates any war of attrition outcome. We also show that the option game premium can be much higher than the traditional real option premium in either war of attrition or bargaining game. This is generally the opposite of the oligopoly under uncertainty case, when the option game premium is lower than the traditional option premium, is zero in the oligopoly limit of infinite firms, and can be even negative in special preemption cases.

    Keywords: option games, option exercise games, real options, stochastic game theory, oligopoly under uncertainty, Leahy's optimality of myopic behavior, war of attrition, information revelation, changing the game, cooperative bargaining, option game premium.

(*) Senior Consultant by Petrobras and Doctoral Candidate at PUC-Rio. E-mail:

    Address: Petrobras/E&P-Corp/EngP/DPR. Av. Chile 65, sala 1702 Rio de Janeiro, RJ, Brazil, 20035-900

    (**) Professor of Finance, Dept. of Industrial Engineering at PUC-Rio. E-mail:

    Address: PUC-Rio, Dep. Engenharia Industrial, Rua Marques de São Vicente, 225 Rio de Janeiro, RJ, Brazil,



    1 - Introduction

     This paper is a sequel of our previous work (Dias & Teixeira, 2003) on continuous-time (real) option games. Option games models comprise the combination of two very important (Nobel laureate) and complementary theories, namely options pricing and game theory. Although discrete-time models are

    generally more intuitive, in most cases continuous-time models permit more general conclusions and more professional software. In our previous paper, after a short historic on option games literature, we focused on two alternative methodologies to solve preemption and collusion models of duopoly under uncertainty. In addition, we examined the role of mixed strategies in both symmetric and asymmetric duopolies.

    In this second paper we focus mainly in two models. First, the oligopoly under uncertainty with

    new artifices to simplify the solution of option games models. Second, the war of attrition under uncertainty which can be viewed as the opposite to the preemption models. We also analyze the interesting possibility of changing the game from war of attrition to bargaining. We continue to highlight concepts, tools, and methodologies to solve option games rather than theoretical details. In the oligopoly model we follow Grenadier (2002), which extends the classic paper of Leahy (1993) with his principle of optimality of myopic behavior. The applicability of this principle is well

    discussed in Dixit & Pindyck (1994, mainly chapter 9, section 1; but also chapters 8 and 11), but Grenadier extends this principle to oligopoly models. Perhaps his main contribution in this paper is the solution of oligopoly exercise strategies using an "artificial" perfectly competitive industry with a modified demand function. With these two artifices, we can solve many option games models using single agent's optimization procedures and the usual real options tools, without the necessity to use more complex techniques adopted in game theory like searching fixed-points from the players’ best-

    response correspondences.

    It is not by chance that most contributions in option-games come from real options researchers rather than game-theoreticians. Tools like stochastic processes and optimal control are more useful than fixed-point theorems and other related tools. But knowledge of game theory is always necessary to the financial engineer to develop option-game models. However, another promising way to solve option-game models comes from two game-theoreticians, Dutta & A. Rustichini (1995), who proved that the best response map satisfies a strong monotonicity condition, which is used to set the


    existence of Markov-Perfect Nash equilibriums. This solution can be applied for example in our war of attrition game under uncertainty analyzed in section 3.

    The third related school that can contribute to option games literature comes from researchers in optimal control, e.g., Basar & Olsder (1995) and Dockner & Jorgensen & Van Long & Sorger (2000), mainly the stochastic differential games branch. However, the bridge between option games and that branch in optimal control literature remains to be constructed.

    In the war of attrition under uncertainty model we extend the paper of Dias (1997), who worked with discrete-time option game model applied to oil exploration of two neighboring correlated prospects owned by two different oil companies. The option exercise is the drilling of one exploratory well

    the wildcat, and part of the information revealed by the drilling is public so that the option exercise generates a positive externality that benefices the follower, who can decide about the option exercise with better information. So, there is a second mover advantage in contrast with the first mover

    advantage from the preemption models.

    This paper is divided into 5 sections, including this introduction. The second section presents the oligopoly under uncertainty model based in Grenadier (2002), but with some simulations and charts not presented there. Section 3 discusses the war of attrition under uncertainty applied to oil exploration, with discussion on information revelation modeling issues and equilibrium possibilities. Section 4 analyzes the “changing the game” possibility with the oil companies abandoning the war of attrition in favor of a cooperative bargaining game, with transactions of rights and options. Section 5 presents some conclusions and suggestions for future research.

    2 Oligopoly under Uncertainty: The Grenadier’s Approach

    This section is based on Steven Grenadier's working paper "Option Exercise Games: An Application to the Equilibrium Investment Strategies of Firms", Stanford University, November 2000. An almost equal version was published (Grenadier, 2002). For sake of space we present only selected results, but it includes some equilibria simulations with charts not presented in the original work. This addition is because we judge important to highlight important concepts such as the comparison between monopoly, duopoly, and oligopoly outputs for the same stochastic shock in the demand; and the concept of upper reflecting barrier limiting the maximum prices in oligopoly due to the (even

    imperfect) competition effect.


    Grenadier (2002) has at least two very important contributions to the option-games literature:

    ? Extension of the Leahy's "Principle of Optimality of Myopic Behavior" to oligopoly; and

    ? The determination of oligopoly exercise strategies using an "artificial" perfectly competitive

    industry with a modified demand function.

    Both insights simplify the problems solution because "the exercise game can be solved as a single

    agent's optimization problem" and the usual real options tools in continuous-time. In order to solve the option-game problem it is not necessary more complex techniques to search fixed-points from the players’ best-response correspondences. We can even use Monte Carlo simulation of the stochastic demand to solve this model, as we will see later.

    In the first insight, the myopic firm (denoted by i) is a firm that, when considering the optimal entry in a market, assumes that all the other firms production (denoted by Q) will remain constant forever. i

    As Dixit & Pindyck (1994, p.291) mention, "each firm can make its entry decision ... as if it were the

    last firm that would enter this industry, and then making the standard option value calculation" and

    "it can be totally myopic in the matter of other firms' entry decisions". The remarkable property of

    the optimality of myopic behavior was discovered by Leahy (1993, "Investment in Competitive Equilibrium: The Optimality of Myopic Behavior") and has been used and extended in many ways. See also Baldursson & Karatzas (1997).

    The Grenadier's paper is closely related to Dixit & Pindyck (1994, mainly chapter 9, section 1; but also chapters 8 and 11). In Dixit & Pindyck (chapter 9) each firm produces only one unit so that the total industry output is the number of firms, whereas in Grenadier’s model the number of firms is fixed (n) but each firm can add more than one unit of production. Perhaps the Grenadier's way to model oligopoly is more useful and realistic (an improvement over Dixit & Pindyck), e.g., monopoly, duopoly, and perfect competition are particular cases respectively for n = 1, 2, and ?. However, for

    the asymmetric firms case, the unit production firm approach of Dixit & Pindyck has advantages over the Grenadier's way, because it is only an ordering problem (low-cost firms enter first). However, in both cases are necessary to assume that the investment is infinitely divisible (firm i can

    add an infinitesimal capacity dq by an infinitesimal investment dK), see Grenadier's footnote 13. Although it is more realistic the assumption of discrete-size (lump-sum) additions of capacity by the firms, the approach may be a reasonable approximation in many industries (e.g., new investment is a


    small fraction of current industry capacity), mainly if the aim is the industry equilibrium study. But

    the model is less realistic at firm-level decision. This necessary approximation allows the extension of

    the Leahy's principle of optimality of myopic behavior, which simplifies a lot the problem solution. However, this assumption is not necessary for the perfectly competitive case of Leahy (see also the

    wonderful explanation of Dixit & Pindyck, chapter 8, section 2), where the competitive firm analyzes myopically a lump-sum investment to enter in the competitive industry.

    The second Grenadier's insight permits the application of the important results obtained from the perfectly competitive framework into the apparently more complex case of imperfect competition of dynamic oligopoly under uncertainty. As example, Grenadier presents an extension of his previous paper on real-estate markets that considers the time-to-build feature for a perfectly competitive

    industry (Grenadier, 2000). He obtained simple closed-form solutions for the equilibrium investment strategies using this smart artifice. Other results obtained for perfectly competitive markets could also be easily extended to the oligopoly case. Examples are the results from Lucas & Prescott (1971) on rational expectations equilibrium, Dixit (1989) on hysteresis models, and Dixit (1991) for price ceilings models, among other known results.

    Let us describe the model. Assume that each firm from the n-firms oligopoly holds a sequence of investment opportunities that are like compound perpetual American call options over a production project of capacity addition. The first assumption is that all firms are equal, with technology to produce a specific product. The output is infinitely divisible, and the unity price of this product is P(t). This price changes with the time because the demand D[X(t), Q(t)] evolves as continuous-time

    stochastic process. Assume either that the firms are risk-neutral or that the stochastic process X(t) is risk-neutral (that is, the drift is a risk-neutral drift = real drift less the risk-premium).

    Initially, as in Grenadier's paper, let us consider a more general diffusion process and a more general inverse demand function, given respectively by:

    dX = (X) dt + ((X) dz (1)

    P(t) = D[X(t), Q(t)] (2) For the popular geometric Brownian motion (GBM), just make (X) = ;;. X; and ((X) = ( . X. As

    usual, is the (real) drift, ( is the volatility, and dz is the Wiener increment.


    In the Cournot-Nash perfect equilibrium, strategies are quantities and the market clears the price at each state of the demand along the time. Firms choose quantities q*(t), i = 1, 2, ... n, maximizing its i

    payoffs and considering the competitors best response Q *. i

    With the simplifying assumption of equal firms, the natural consequence is the choice of symmetric

    Nash equilibrium, that is, q*(t) = q*(t) for all i, j. We can write the optimal output for n-firms ij

    oligopoly symmetric Nash equilibrium as:

    q*(t) = Q*(t) / n i

    The exercise price of the option to add a capacity increment of dq is the investment I . dq, where I is

    the unitary investment cost, equal for all firms. The option to add capacity is exercised by firm i when the demand shock X(t) reaches a threshold level X *(q, Q ). iii

    Grenadier summarizes the equilibrium in his Proposition 1, with a partial differential equation (PDE) and three boundary conditions. The PDE is obtained using the standard option pricing approach (Itô’s Lemma, risk-free portfolio, etc.). The first and second boundary conditions are the value-matching and smooth-pasting conditions, as usual in continuous-time real options framework. However, the third condition is the strategic one, requiring that each firm i is maximizing its value V(X, q, Q) iii

    given the competitors' strategies (thresholds).

    The third condition is a value-matching at the competitors' threshold X(q, Q)*, which is equal to iii

    X(q, Q)* due to the symmetric equilibrium. The third condition is also like a fixed-point search iii

    over the best response maps. However, this condition will not be necessary with the Grenadier's

    Proposition 2, extending the myopic optimality concept to oligopolies. Proposition 2 assumes that investment is infinitely divisible (see discussion above) and tells that the myopic firm threshold is equal to the firm's strategic (Cournot-Nash perfect equilibrium) threshold. Proposition 3 will set the main equilibrium parameters with only two boundary conditions.

    iDenote the value of myopic firm by M(X, q, Q). Let us to work with the value of a myopic firm's ii

    imarginal output m(X, q, Q) defined by: ii

    iim(X, q, Q) = ?M(X, q, Q) / ?q (3) iiiii


    iGiven the symmetry, we can write X(q, Q)* = X*(Q) because q = Q/n and Q = (n 1) . Q / n. iiii

     firm will exercise its investment His proposition 3 establishes the symmetric Nash equilibrium: each

    option whenever X(t) rises to the trigger X*(Q). Let m(X, Q) denote the value of a myopic firm's

    marginal investment. The following PDE and two boundary conditions determine both X*(Q) and m(X, Q):

    20.5 σ(X) m + α(X) m r m + D(X, Q) + (Q / n) D(X, Q) = 0 (4) XXXQ

    Subject to:

    m[X*(Q), Q] = I (5)

    ?m[X*(Q), Q] / ?X = 0 (6) Where the subscripts in the PDE (eq. 4) denote partial derivatives, the equation (5) is the value-matching at X*(Q), and equation (6) is the smooth-pasting condition. The last two terms in the right side of equation (4) comprise the non-homogeneous part of the PDE, the called "cash-flow" terms. This non-homogeneous part will play a very special role in Grenadier's paper, because it is the modified demand function mentioned early. The first three terms of the PDE comprise the

    homogeneous part of the PDE. It is very known in real options literature (Dixit & Pindyck, 1994). The nice issue is that only two "real options" boundary conditions at the common threshold level X*(Q) are sufficient for the optimal strategic exercise of the option due to his Proposition 2, which says that the myopic firm threshold is equal to the firm's strategic threshold.

    Grenadier (section 5) shows that, besides the monopoly and perfectly competitive industry cases, it is also possible to solve the oligopoly case as a single agent optimization problem. The procedure is just

    to pretend that the industry is perfectly competitive, maximizing a "fictitious" objective function. This "fictitious" objective function uses an "artificial" demand function defined by:

    D'(X, Q) = D(X, Q) + (Q / n) D(X, Q) (7) Q

    As mentioned in the introduction, this is a very important result because permits the extension of known (or easier to obtain) results in perfectly competitive setting to the oligopoly case. In section 6, Grenadier shows the equilibrium with time-to-build as example of this extension. Here we focus in the example from his section 3 but with some further simulations not showed in the paper.


    Consider a specific diffusion process geometric Brownian motion, and also a specific inverse

    demand function a multiplicative shock constant-elasticity demand curve, given respectively by:

    dX = ;;X dt + ( X dz (8)

     1/?P(t) = X(t) . Q(t);;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;!?

    Where ? > 1/n to ensure that marginal profits are increasing in X. Assume also that the risk-free

    1discount rate is strictly higher than the drift . The optimal threshold X*(Q) is given by:

    1 / γX*(Q) = v . Q (10) n

    Where v is an upper reflecting barrier, that is, the maximum price that the product can reach in the n

    oligopolistic market. When the price reaches this level, firms add capacity in a quantity so that the price is reflected-down due to the additional supply.

    For this multiplicative demand shock, while X(t) follows the (unrestraint) GBM, the price P(t) follows a constrained GBM with upper reflecting barrier v, given by: n

    ??β??11?? (11) v (r α) I??n????β - 11 1 / n γ??1??

    2Where ~ > 1 is the known positive root of the quadratic equation: 0.5 ( ~ (~ 1) + ~ r = 0. 1

    Note that the threshold X*(Q) is decreasing with the number of firms in the oligopoly (n), which looks intuitive. It is the competitive effect with intensity n, reducing the entry threshold. In order to keep the prices at or below v, the addition of capacity dQ (= n dq) when X(t) > X*(Q), n

    with cost I dQ, is larger as larger is the difference X(t) X*(Q). In other words, if X(t) > X*(Q) then

    Q(t) = (X(t) / v)?. n

    What is the option premium when exercising this strategic option in the n-firms oligopoly? Grenadier defines this option premium as the NPV at X* per unit of investment I, OP(n) given by:

    OP(n) = 1 / [(n ?) 1] (12)

     1 For the risk-neutral drift, which for the GBM is equal to ’ = r – ?, just assume that the dividend yield ? > 0.


    Hence, when n tends to infinite the OP(n) tends to zero, a consistent result. See in Dixit & Pindyck (1994, chapter 8) that the NPV is zero for the perfectly competitive case. For n finite the NPV is positive but as small as large is the number of firms (n), i.e. as intense is the competition. In Dias & Teixeira (2003) we saw that in a special case, the NPV of exercising an expansion option could even be negative, in order to avoid the competitor entry that could be even worse for the current firm operations. In the next section we will see the opposite: for war of attrition the option premium from optimal exercise can be even higher than the traditional (monopolistic) real option premium. Let us perform some numerical calculations in order to see the power of the above concepts to understand the oligopoly equilibrium under uncertainty. We use the same numerical values adopted in Grenadier's paper, section 3 for his figure 1, except where indicated. The values are: = 0.02 p.a.;

    r = 0.05 p.a.; ( = 0.175 p.a.; ? = 1.5; n = 10 firms; I = 1 $; Q(0) = 100 units; and X(0) = 1.74 $/unit. An interesting and practical feature of the principle of optimality of myopic threshold is that we can

    use Monte Carlo simulation in order to solve the model. It is not necessary to work backwards

    because we know the ("myopic") threshold level X*(Q(t)) in advance. So, if this threshold is triggered by the simulated sample-path of the demand X(t), new capacity is added by the oligopolistic firms and it is easy to study many properties of the strategic exercise of options in an oligopoly and the aggregate behavior of the industry in the long-run, such as the industry output Q(t), the investment along the years, the prices evolution, etc. Figure 1 shows some of these features for

    10-firms oligopoly case presenting a certain demand sample-path X(t) over 10 years.


    Figure 1 Demand Sample-Path and Strategic Exercise in 10-Firms Oligopoly

    When the demand rises at the threshold level all firms exercise options to expand capacity, increasing the aggregated industry output. In this model, the firms’ addition of capacity is proportional to the difference between the demand shock X(t) and the threshold level X*(Q(t)), if positive. In case of the demand below the threshold X*, no investment is performed (and no exit as well). In the Grenadier's model the firms are equals, so that in 10-firms oligopoly case each firm adds 1/10 of the new capacity Q(t) Q(t dt) in case of positive shock at t, if X(t) > X*(Q(t dt)).

    Figure 1 also shows that, for this specific sample-path, after the year 8 the demand drops to levels well below the demand level at t = 0, but the total industry output remains (so that the prices drops). This model does not consider reduction of industry output due to low demand state. A possible improvement in the model is to consider other options like the option to temporary stopping (at cost) and the option to exit (or at least the option to contract).

    Figure 2 shows for one demand evolution sample-path, that the industry total output Q(t) is much higher for the 10-firms oligopoly (n = 10) case than for duopoly (n = 2), which presents a higher industry output than the monopoly case (n = 1).

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