Oil price, mean reversion and zone readjustments.

Hammoudeh, Shawkat

I. Introduction

Observing OPEC's short-term price-output ceiling behavior during the late 1980s and 1990s, one can conclude that it attempts to stabilize the market price within a range of its announced target price by controlling the output ceiling.(1) If the price moves within four to five dollars below the target price, it usually reduces the output ceiling and assigns new quotas to its member countries to keep the price close to the target price. In reality, OPEC establishes a band for the market price positioned around the target price by basically choosing suitable upper and lower limits for the output or, at least in soft markets, it places a tolerance zone below the target price in order to restrict the discrepancy between the market price and the target price [8; 9]. The lower limit is particularly needed because it sets a price floor and ensures that the market price stays above the significantly lower marginal cost of oil production. If the limits of these zones are backed by a perfectly credible intervention policy, they can generate an expectations process that should turn the market prices around even before any intervention takes place.

While OPEC in some sense observes the target zones for its prices, those zones are neither well defined nor vigorously defended. It can not always or may not be willing to maintain the price within the limits of the desired zone by cutting the output ceiling; it must sometimes readjust the target price and output ceiling, and thus create a new target zone to reflect the market's new fundamentals. This is particularly true now because OPEC is losing market share to the other oil producers and is contemplating to shift the current band.

Actual readjustments in the target price can be so large, as in 1980 and 1985, that the new market price must jump as well. They can occur when both the market price is near the limits of the band as well as when it is inside the band but still further away from those limits. Therefore, OPEC's policy of defending the limits of a target zone for a given target price may be imperfectly credible.

In this paper, I examine OPEC's oil market price behavior: First, when OPEC's policy is credible and the market price is limited within a given target zone; Second, when OPEC policy is imperfectly credible and the price reverts to the free market price due to speculative attacks triggered by "too large" output ceiling; and third, when OPEC has the chance of: either defending the current price or shifting the current target zone and declaring a new one.

II. The Basic Oil Model of Target Zones

In this model there are marginal interventions by OPEC which occur when the market price hits the limits of the current zone while the target price remains unchanged. There are no intramarginal interventions that aim at returning the market price to the specified target price within the band (mean reversion).

The equation that describes the oil price behavior inside the price's target zone is

[Mathematical Expression Omitted]

where [Mathematical Expression Omitted] is OPEC's output ceiling, [q.sub.2] is the cumulative inventory shock, E(dP)/dt is the expected rate of change in the price, [Gamma] is the speed of price adjustment [7; 10; 11]. Therefore, Equation (1) describes the price behavior inside the target zone as a function of OPEC's output ceiling, cumulative oil shock or inventory shock and market participants' expectations of changes in oil price. The ceiling [Mathematical Expression Omitted] is the intervention policy variable which is bounded by the announced upper and lower limits on the market price (i.e., the limits of the target zone). In this case [Mathematical Expression Omitted] will only change only if the market price reaches the maximum or the minimum defined by the band in order to bring it back to the interior of the target zone.

The cumulative shock [q.sub.2], which is a shift factor, is assumed to follow a random walk with a trend drift independent of the oil price:

d[q.sub.2] = [Eta]dt + [Sigma]dz (2)

where [Eta] and [Sigma] are constants, dz is the increment of a standard Wiener process. The shock d[q.sub.2] could be a demand shock, a supply shock or the difference between the two. The differentiating factor is the sign. If d[q.sub.2] is positive then it must be that OPEC's production exceeded its output ceiling and the surplus was added to the inventories, [q.sub.2]. The opposite is true if d[q.sub.2] is negative. The sum [Mathematical Expression Omitted] is called the fundamental.

The expectations term, E(dP)/dt, is a balancing item that matches the demand and supply of oil. If the demand falls short of supply then this term is negative in order to ensure that the demand matches the given supply. On the other hand, if demand exceeds supply the expectations term is positive.

In the presence of perfectly credible policy, which sufficiently influences expectations, the target zone price differs from the price dictated by the fundamental q. As the price reaches the limits of the zone, market participants' expectations of future interventions by OPEC causes an expected turnaround in the price, which the market turns into an immediate change [14]. This effect renders the perfectly credible target zone to be inherently stable in the sense that the zone stabilizes the aggregate fundamental q by basically setting an upper and lower limit on it. Inside the target zone, all the changes in q are due to the changes in [q.sub.2]. But when q reaches its lower and upper limits, OPEC changes the output ceiling [Mathematical Expression Omitted] to maintain the price within the limits of the band. Moreover, there is a negative (decreasing) relationship between the limits on the price and the limits on the fundamental q.

If the price is outside the band, intervention is considered passive or ineffective and the price will be determined by the fundamental forces of supply and demand.

In order to understand the dynamics of the market price we need to find an explicit expression for the expectations term in equation (1). Let the general form of the solution be represented by P = g(q). The term E(dp)/dt can be derived by applying Ito's lemma:

dP = g[prime](q)d[q.sub.2] + 1/2g[double prime](q)[(d[q.sub.2]).sup.2]. (3)

Substituting equation (2) into equation (3) and taking expectations conditioned on current information yields

E(dP)/at = g[prime](q)[Eta] + 1/2g[double prime](q)[[Sigma].sup.2]. (4)

Again substituting this term into equation (1) gives

P = g(q) = [Gamma]q + [Theta][g[prime](q)[Eta] + 1/2g[double prime](q)[[Sigma].sup.2]]. (5)

The general solution to equation (5) is

P = g(q) = [Gamma]q + [Theta][Gamma][Eta] + A exp[[[Lambda].sub.1q]] + B exp [[Lambda].sub.2q]] (6)

where

[[Lambda].sub.1] = [-[Eta] + ([[[Eta].sup.2] + 2[[Sigma].sup.2]/[Theta]).sup.1/2]]/[[Sigma].sup.2] [greater than] 0

and

[[Lambda].sub.2] = [-[Eta] + ([[[Eta].sup.2] + 2[[Sigma].sup.2]/[Theta]).sup.1/2]]/[[Sigma].sup.2] [less than] 0

Then

[Mathematical Expression Omitted]

Equation (6) describes a family of solutions for the oil price. Any selected solution should satisfy the boundary conditions appropriate to target zone models. The constants A and B are determined by those conditions.

If the intervention policy is passive and [Mathematical Expression Omitted] is expected to remain unchanged at its initial level the oil price may take on any value. However, it should not deviate arbitrarily far from the fundamental level as [q.sub.2] takes on large values or it may asymptotically approach this level as [q.sub.2] tends to infinity. Thus, in this case we may assume that A = B = 0 and the general solution of the model represented in equation (6) can be reduced to the free fundamental solution(2)

[Mathematical Expression Omitted] (8)

which is a combination of the output ceiling, the inventory shock, the time trend, the sensitivity to changes in expectations and the market speed of adjustment.

III. The Price solution within a Given Target Zone

Marginal Interventions

OPEC pursues marginal (infinitesimal) interventions in the output ceiling at the limits of the target zone in order to turn the market price around as it hits those limits, without changing the target price. Figure 1 plots the within-band solution form of the target zone model for this case (the curve labeled 1). If there is no intervention, [Mathematical Expression Omitted] will not change and an increase in [q.sub.2] will lead to a decline in P while a decrease will cause an increase. The movement of the short run price within the band depends on the steepness of the price function and the position of the fundamental, as in equation (3), since in this infinitesimal case there is a one-to-one-relationship between the price and the fundamental.

In this model, which explicitly includes expectations formation based on a perfectly credible quantity-determined intervention policy, OPEC would stand ready to change [Mathematical Expression Omitted] while keeping the target price [P.sup.T] the same. In this case of infinitesimal interventions, it would decrease [Mathematical Expression Omitted] as the market price approaches some minimum market price [P.sub.min] and increase it as it approaches a maximum price [P.sub.max] to offset changes in [q.sub.2], thereby keeping P within a band. This means that near the top of the band there would be an expected fall in the market price P because of the expected output intervention (i.e., [Mathematical Expression Omitted] increases) by OPEC, resulting in an immediate actual reduction in P. The path of the expected change in price within the band is determined by equation (4). That is, the market price path flattens out to a slope of zero at the upper and lower limits of the band.

The flattening out of the price is due to the property of the fundamental q which follows a Brownian motion in the short run.(3) Within the target zone, q's expected change is a constant. At the limits of the zone, the expected change of q is not constant, but rather increases at the upper limit and decreases at the lower limit. Thus, there is a jump in the expected change in q. But by Ito's Lemma:

E(dP) = g[prime](q)E(dq) + 1/2g[double prime](q)E[[(dq).sup.2]].

The jump in E(dq) would imply a jump in E(dP). This cannot be the case because it results in a safe arbitrage (a one sided bet): the price would move straight into the target zone. Therefore, we must have

g[prime]([q.sub.max]) = [Gamma] + [[Lambda].sub.1]A exp[[[Lambda].sub.1][q.sub.max]] + [[Lambda].sub.2]B exp [[Lambda].sub.2][q.sub.max]] = 0

g[prime]([q.sub.min]) = [Lambda] + [[Lambda].sub.1]A exp[[Lambda].sub.1][q.sub.min] + [[Lambda].sub.2]B exp [[[Lambda].sub.2][q.sub.min] = 0 (9)

in order to ensure that E(dP) is zero when E(dq) is non-zero. In other words, the price within the target zone is tangential to the limits of the bands. That is, if the policy intervention is infinitesimal and effective, g[prime]([q.sub.max]) = 0 and g[prime]([q.sub.min]) = 0 when [P.sub.min] = g([q.sub.max]) and [P.sub.max] = g([q.sub.min]), respectively. This is the "smooth pasting solution" condition known in option pricing theory [4]. These two sets of boundary conditions should be used in order to solve for A and B in equation (6) [5]. They together with equation (5) characterize an intervention policy that defend the declared band with a probability 1. It should be clear that A [greater than] 0 and B [less than] 0 for interventions to defend the band because q has a negative impact on the market price.

The expectations term E(dP) defined in equation (4) characterizes the curvature of the price path at the limits of the band. If this term is negative, the solution function g (q) is concave at the upper limit of P (i.e. g[double prime](q) [less than] 0) where g[prime](q) = 0. On the other hand, if it is positive, g(q) is convex at the lower limit of P (i.e. g[double prime](q) [greater than] 0). Therefore, given the target price the market price solution displays an inverted S-shaped behavior within the target zone, bending away from the free market solution represented by equation (8).

The market price solution within the band is less responsive to the fundamental q than the free market price. This can be seen by differentiating equations (6) and (8) with respect to q, and making use of the signs in equation (9) which require that [[Lambda].sub.1] [greater than] 0, A [greater than] 0, [[Lambda].sub.2] [less than] 0 and B [less than] 0:

g[prime](P) = [Gamma] + [[Lambda].sub.1]A exp[[[Lambda].sub.1]q] + [[Lambda].sub.2]B exp[[[Lambda].sub.2]q] [less than] [Gamma]

in absolute value. At the lower limit, the market price within the band can either rise or stay constant.

The solution function P = g(q) allows an inverse q(P) because in this case there is, as mentioned above, a one-to-one relation between the market price and the fundamental. Thus, the long run density of the market price within the band, [[Omega].sup.P](P), can be derived by a change of variable in the long run distribution of the fundamental and in the slope of the price solution

[[Omega].sup.P](P) = [[Omega].sup.q](q(P))/g[prime])q(P))

where [[Omega].sup.q](q(P)), the long run distribution of the fundamental, is known [2]. In such infinitesimal interventions at the limits, [[Omega].sup.q](q(P)) is a truncated exponential distribution leaning in the direction of drift [15]. Since at the limits of the band g[prime](q) = 0 and it is large in the middle of the band, and that [[Omega].sup.q](q(P)) is known, the long run distribution of the market price is U-shaped with spikes at limits of the band as can be seen in Figure 1. Most of the observations of the market price support this distribution. During the 1970s, most of the observations were near the left spike while in the 1980s and 1990s they were near the right spike.

Marginal Interventions: Credible Policy

Here we examine the speculative attacks in a one-sided model where the lower limit is defined by [P.sup.T] - Z and Z is the width of the price band below the target price, [P.sup.T] [7; 12; 13]. In this case there is no lower limit on [q.sub.2] so we have A = 0. Equation (6) can then be written

P = g(q) = [Gamma]q + [Theta][Gamma][Eta]+ B [[Lambda].sub.2]q] (10)

where B is determined by the boundary condition g[prime]([q.sub.u]) = 0 when P = [P.sup.T] - Z.

An arbitrarily chosen output ceiling [Mathematical Expression Omitted] is perceived to be defensible and the policy to be credible if it generates price expectations that balance total supply and demand, where the balancing item is the expectations term, and leads to a turnaround in the market price. In other words, the market participants perceive the chosen ceiling to not only match their estimate of expected demand but also to be compatible with the size of the shock in the market. On that basis, they form expectations that lead to a turnaround in the market price.

If equation (10) is evaluated at [Mathematical Expression Omitted], the constant B is appropriately chosen so that P = [P.sub.min] and that the output ceiling [Mathematical Expression Omitted] is defensible as in Figure 2, then [Mathematical Expression Omitted] will be the output shock associated with smooth pasting. The price path at that point will be tangential to the minimum price as in the locus TT. Therefore,

[Mathematical Expression Omitted]

Once again using equation (10) at [Mathematical Expression Omitted] we have

[Mathematical Expression Omitted]

where [P.sub.min] = [P.sup.T] - Z and [Mathematical Expression Omitted] is the defensible ceiling.

Furthermore, let us consider a situation where [q[prime].sub.2] is associated with an output ceiling which is considered to be non-defensible at the lower limit of the band, whether in comparison to the level of a previously defensible ceiling or on the basis of the market's estimates of the expected demand and the size of the shock. Let [Mathematical Expression Omitted] represent the nondefensible ceiling. As the market perceives this ceiling as an imbalancing factor, it would cause a speculative attack at [P.sub.min] and the price equation would follow the post attack equation

[Mathematical Expression Omitted]

which is the locus MM[prime]. Thus, at [q[prime].sub.2] we have,

[Mathematical Expression Omitted]

Combining equations (11), (12), (13) (and using the speculative attack assumption which posits no jump in the price level by choosing the appropriate parameters) yields

[Mathematical Expression Omitted]

Since [[Lambda].sub.2] should be negative for the system to be stable, the right hand side of equation (14) is positive. In other words, it is clear that

[Mathematical Expression Omitted]

Thus, the locus MM[prime] should hit the curve TT from below. That is, within the context of fixed limits of a given band, OPEC's intervention policy is considered non defensible if the output ceiling chosen by OPEC is relatively "too large". The minimum reduction in the output ceiling for the OPEC's policy to be credible is -1/[[Lambda].sub.2]. In particular, we have the following result:

RESULT 1. Intervention is credible if [Mathematical Expression Omitted] and intervention is non-credible if [Mathematical Expression Omitted], where [Mathematical Expression Omitted] is the arbitrarily chosen credible policy parameter.

That is, any output ceilings at least as large as [Mathematical Expression Omitted] are not credible and any ceilings lower than [Mathematical Expression Omitted] are credible. Thus, we can write

[Delta][q.sup.c*] = -1/[[Lambda].sub.2] (15)

where [Delta][q.sup.c*] is the minimum credible ceiling reduction. In other words, a credible ceiling reduction must be at least as large as (-1/[[Lambda].sub.2]).

We know that if [Delta][q.sup.c] was not a credible reduction in the ceiling for any arbitrarily chosen [Mathematical Expression Omitted], then any reduction smaller than [Delta][q.sup.c] will not be credible because it means a relatively "too large" ceiling.

The size of the minimum ceiling reduction is related to the structural parameters of the model. The higher the level of the drift is, the smaller the minimum credible reduction is. However, the higher sensitivity of market price to expectations of intervention or to market risk is, the greater the minimum credible reduction is [7].

Intramarginal Discrete Interventions

In the case of discrete quantity-determined interventions, the events of hitting the limits of the band and having an intervention by OPEC will not coincide. In this case, OPEC announces a discrete intervention rule which specifies both the upper and lower limits of the fundamental q ([q.sub.u] and [q.sub.1], respectively) at which intervention will occur and the size of the intervention at each limit. If q is the independent variable as in the marginal intervention case, then once the fundamental reaches [q.sub.u], which is greater than [q.sub.max] in Figure 1, it instantaneously moves back to an interior point such as [r.sub.1] where the market price remains the same. Similarly, when q reaches [q.sub.l] [less than] [q.sub.min], it moves forward to [r.sub.2]. The intervals [q.sub.u] - [r.sub.1] and [q.sub.1] - [r.sub.2] represent the sizes of discrete intervention. The boundary condition for the intramarginal intervention case can thus be written as

P([q.sub.u]) = P([r.sub.1])

P([q.sub.1] = P ([r.sub.2]).

Then for each intervention interval, two different fundamental levels correspond to each market price and, thus, the inverse of the P(q) function is not single-valued. Once again, the movement of the market price is proportional to the steepness of the solution function, by equation (3), and that the expected change in the price is still given by equation (4). The variables in these equations now depend on the position of the fundamental within the band as well as on that of the market price. However, the shape and the position of this market price is still uniquely determined by the price band. That is, whatever the size of intramarginal discrete interventions, the price solution must satisfy the "smooth pasting" condition, equation (9), at the limits of the price band. The solution is still fiat at the limits and P must rise when it is close to the upper limit.

The impact of the quantity-determined intervention can be better appreciated if we explicitly consider [q.sub.2] to be the independent variable as shown in Figure 3. For a given output ceiling [Mathematical Expression Omitted], the curve labeled 1 represents the market price as a function of the cumulative shock [q.sub.2], where [q.sub.2] is permitted to reach an upper limit [q.sub.u] just before an intervention by OPEC occurs.

As we saw earlier, as [q.sub.2] increases the market price falls then rises before the actual reduction in output ceiling occurs. Since intervention is expected, there is no jump in the market price. Moreover, since [q.sub.2] is exogenous (to OPEC) it does not change from the upper limit [q.sub.u] as a result of OPEC's intervention.

After the intervention, the curve labeled 1 shifts to the right and intersects the original one in a way that maintains the market price continuity. The size of the shift is determined by requiting that the new price solution evaluated at [q.sub.u] maintains the same price. The market price follows curve 2 and [q.sub.2] can move freely up or down from the starting point [q.sub.u] and future interventions will be determined accordingly. If [q.sub.u] moves up towards [q[prime].sub.u] the price will initially dip then rise in anticipation of another cut in the output ceiling at [q[prime].sub.u].(4) If [q.sub.u] moves down, the price will initially rise then dip in anticipation of another increase in the output ceiling near the left end of curve 2.

Intramarginal Interventions: Mean Reversion

These are interventions by OPEC within the target zone to return the market price to the specified target price. Although they occur, they are not as common as the marginal interventions. The expected rate of oil price change within the band is negatively related to the price within the band [ILLUSTRATION FOR FIGURE 4 OMITTED]. When the price is at the upper limit of the band, it is strong and cannot rise further: It either stays there or falls towards the interior of the band. That is, there is an expected price decline or the expected price change is negative. The market price for the given zone must be below the free market price. Equivalently, when the price is at the lower limit of the band, there is an expected price increase which means that the expected price change is positive. The zoned market price must be above the free market price. This negative relationship between the expected rate of the price change within the band and the market price within the band suggests that the market price exhibits mean reversion.

Based on that, it is more likely that the distributions of the oil market price within the band are U-shaped [ILLUSTRATION FOR FIGURE 1 OMITTED] rather than hump-shaped as observed in target zone exchange rates. This means that most of the observations of the market price is near the limits of the band, in contrast to the most of the observations of the exchange rates in the middle of the exchange rate band [3].

The simple way to specify the intramarginal interventions in this case is to assume that these interventions result in the expected rate of change of the fundamental towards central parity is proportional to the distance to the central parity

E(d[q.sub.t]) / dt = - [Mu][q.sub.t]

where [q.sub.t] can be interpreted as deviation from its central parity and [Mu] is the rate of the mean reversion and is a positive constant.

In order to analyze the effect of mean reversion on the stability of the market price within the given band, it is useful to compare the behavior of the market price with mean reversion within the target zone with those of the free market and the managed price with mean reversion but without a target zone.

The free market price is described by

P = [Gamma]q + [Theta][Gamma][Eta].

The managed market price with mean reversion but without a target zone is assumed to satisfy

E(d[P.sub.t]) / dt = -[Mu][P.sub.t]. (16)

where [P.sub.t] can also be interpreted as deviation of the market price from the target price. That is, the expected rate of change of the market price depends on its position relative to its central parity. Using equation (1) to substitute for the expectations term in equation (16) yields(5)

[P.sub.t] = [Gamma]q / (1 + [Theta][Mu]) (17)

which is the equation for the managed market price with mean reversion but without a target zone (line labeled 3 in Figure 1). It means the slope of the unzoned, managed market price with mean reversion is less than that of the free market price described above since [Mu] is a positive constant. That is, near the upper boundary of the fundamental q, this managed price may be higher than the free market price which should help OPEC members, while near the lower boundary it is lower, which should help the consumers.

In the case of interventions with a target zone (which also means the presence of a specified band and the occurrence of interventions at the limits of that band as well as within the band) the resulting market price is closer to the target price (but without mean reversion) except that it has a slightly inverted S-shape and smooth pasting at the limits of the band. Using equation (17) and equation (4) yields the general solution for this price

P = g(q) = [Gamma]q / (1 + [Mu]) + [Theta][Gamma][Eta] + [A.sub.1] exp[[[Lambda].sub.1][Gamma]q] + [A.sub.2] exp[[[Lambda].sub.2][Gamma]q] (18)

where [A.sub.1] and [A.sub.2] satisfy the smooth pasting condition similar to that of equation (9).

IV. Target Zone Readjustments

The imperfect credibility of OPEC's policy and the non defensibility of the market price within the target zone in the case of speculative attacks at the lower limit assumes a once-for-all-abandonment of all interventions. In reality, due to the abundance of oil reserves and their free storage in natural reservoirs, OPEC rarely allows the price to revert to the free market price.(6) It would readjust the bands and set new output ceilings and target prices. Thus, OPEC cannot always maintain the market price within the limits of the desired target zone by merely reducing the output ceiling; it must consider adjusting the target and, thus, create new target zones. This might happen whether the oil market is booming or slackening. In the slackening market, which is the more interesting case, OPEC may satisfy its members' revenue needs by one of two alternative ways. It may defend the current target zone and prevent a major price decline by moving towards the center of this zone, or by declaring a new target zone with a new center. This second option aims at satisfying members' revenue needs by allowing them to increase production and at the same time allow a decrease in the target price. This is the more preferred case when those members have excess capacity and are concerned about market share. These interventions are carried out at pre-known points of the fundamental.

In this section, following Bertola and Svensson [1], Bertola and Caballero [2], and Svensson [15], we postulate that OPEC's interventions amount to specifying a target zone around the fundamental q. The new adjoining zones are established and each centered at a value of [c.sub.t], with unchanged width. It is also assumed that the width of the zone be [Mathematical Expression Omitted] around any of these centers and that the adjustments be symmetric. In the first option above, OPEC defends the current target zone whose center is c with a known probability of (1 - [p.sub.d]) through interventions by reducing q by a size - [r.sub.d] towards the center, where d is for a decrease. Alternatively, it might declare a new center for the fundamental q which is

[c.sub.(+)] = [c.sub.(-)] + [d.sub.d]

and at the same time allow a positive jump of size

[Mathematical Expression Omitted]

in that fundamental. This is known to occur with a probability of [p.sub.d] where d stands for a decline in the price. The fundamental is now a bivariate process and the free market price is a function of the current level of the fundamental and the current central parity, P = g(q; c). In the interior of the target zone, the probability of jumps is zero and the solution described in equation (6) can be written as

g(q; c) = [Gamma]q + [Theta][Gamma][Eta] + A exp[[[Lambda].sub.1](q - c)] + B exp[[[Lambda].sub.2](q - c)] (19)

where [[Lambda].sub.1] and [[Lambda].sub.2] are defined as in above. In order to determine the constants A and B, the no-safe arbitrage boundary condition, the market price is not expected to change when intervention is forthcoming, is imposed (i.e., the market price should not jump when the fundamental does). Thus, the boundary conditions for the market price are(7)

[Mathematical Expression Omitted]

[Mathematical Expression Omitted]

where [p.sub.i] is the upward jump probability of increases in the price through readjustment decreases in the fundamental q while [p.sub.d] is the downward jump probability of decreases in the price through readjustment increases in q. Using equation (19) in equations (20) we obtain a linear system of equations in the constants A and B.

The conditions (20a) and (20b) mean that the expected price of the two possible market prices must equal the one prevailing at the limits of the existing band, [Mathematical Expression Omitted] and [Mathematical Expression Omitted], respectively. They are required in order to prevent market participants from reaping unbounded profit (no arbitrage profit). These conditions along with equation (19) can be solved for A(c) and B(c), which should have opposite signs for every c as shown in the previous sections.

In the interior of every target zone, the relationship between the market price and the fundamental takes the inverted S-shaped form like that in Figures 1 and 5 if the jump probability [p.sub.d] is small, i.e. the probability of declaring a new target zone at the edge [Mathematical Expression Omitted] is small. The constants A(c) and B(c) have the usual signs. Then

P(q; c) [less than] [Gamma]q + [Theta][Gamma][Eta].

This is the case that is more favorable for both the consumers and OPEC member states. The same relationship holds if the jump probability [p.sub.i] is small. However, if [p.sub.i] and [p.sub.d] are large i.e. there will be a readjustment, the signs for A (c) and B(c) are reversed and the relationship between the price and the fundamental becomes

P(q; c) [greater than] [Gamma]q + [Theta][Gamma][Eta].

That is, the relation takes the nonstandard inverted S-shaped in which price is lower than the free market price when the market is weak and higher when the market is strong. This case is, therefore, unfavorable to both OPEC and the consumer.

In reality it seems that both the decrease probability [p.sub.d] and the increase probability [p.sub.i] are small while 1 - [p.sub.d] and 1 - [p.sub.i] are large. This means that the relationship between the price and the fundamental follows the usual inverted S-shape. Moreover, it is more likely that [p.sub.d] [greater than] [p.sub.i].

Common sense can be used to come up with measures of the sizes of the parameters. Note that the parameter [d.sub.d] represents the size of readjustments and [r.sub.d] represents the size of intervention to defend the current zone. If [d.sub.d] is larger than [Mathematical Expression Omitted], readjustments do not give rise to overlapping target zones. Moreover, [k.sub.d] should be small because the market price is near the top of its new zone after a readjustment.

At the pre-known point, [Mathematical Expression Omitted], of the fundamental and given low jump probabilities of readjustments (high intervention probabilities), a higher positive trend drift reduces the convexity of the price/fundamental relationship and makes it steeper, making life more difficult for OPEC to defend the existing target zone. This can be seen upon differentiating equation (19) at the lower bound yielding:

[Delta]g[double prime] (q; c) / [Delta][Eta] = ([Delta][[Lambda].sub.2] / [Delta][Eta])[[Lambda].sub.2][Be.sup.[[Lambda].sub.2](q - c)] [2 + [[Lambda].sub.2](q - c)] [less than] 0,

where B [less than] 0, [[Lambda].sub.2] [less than] 0 and

[Delta][[Lambda].sub.2] / [Delta][Eta] = 1 / [[Sigma].sup.2][-1 - [Eta][([[Eta].sup.2] + 2[[Sigma].sup.2] / [Theta]).sup.-1/2]] [less than] 0.

However, a higher level of risk increases the convexity of the price relationship. That is,

[Delta]g[double prime](q; c) / [Delta][Sigma] [greater than] 0.

where B [less than] 0, [[Lambda].sub.2] [less than] 0, and [Delta][[Lambda].sub.2]/[Delta][Sigma] [greater than] 0. That is, a higher level of risk makes the market participants including OPEC more prudent and less aggressive when the price hits the limit.

Moreover, a higher sensitivity of the market price to changes in the participants' expectations increases the convexity of the market price and makes the intervention policy less aggressive.

[Delta]g[double prime] (q; c) / [Delta][Theta] [greater than] 0,

where B [less than] 0, [[Lambda].sub.2] [less than] 0 and [Delta][[Lambda].sub.2]/[Delta][Theta] [greater than] 0.

Finally, a higher probability of readjustments, [p.sub.d], at the pre-known points, changes the signs of A(c) and B(c). The above sensitivities will be reversed in this case.

V. Conclusions

This paper examines the oil price dynamics when OPEC faces different sets of options. The first set includes only one option: that OPEC should defend the current target zone and the prevailing limits. Under this option, the zoned market price is more stable than both the free market price and the managed price with mean reversion. The credibility of OPEC policy depends on the level of the trend drift, the sensitivity of the market price to changes in market participants' expectations and the magnitude of risk. The alternative to this option would be a speculative attack and a collapse of the price.

In the second set of policy options, which is more realistic, OPEC has two choices: To either defend the prevailing target zone or readjust the fundamentals by declaring a new price band. The shape of the relationship between the price and the fundamental for all the zones depends on the levels of the jump probabilities of defending the current zone or readjusting the fundamental at both ends of the fundamental band. The shape of the price and the aggressiveness of OPEC intervention policy also depends on the size of the drift, the sensitivity to expectations and the magnitude of risk, given those probabilities. For small jump probabilities, the last two factors increase the convexity of the price path and lead to less aggressive intervention policy.

The author would like to thank Lars E. Svensson for helpful information provided in a private correspondence.

1. The price is known as the price of the OPEC reference basket of crudes.

2. This is represented by the line labeled 2 in Figure 1.

3. The short run is defined as the period during which [Mathematical Expression Omitted] is fixed.

4. For a similar discussion on the exchange rates with decline in foreign reserves see Flood and Garber [6].

5. The market price in this section is stripped of the target price.

6. Oil prices collapsed in the late 1985 when OPEC declared a war against the other oil producers. Currently, it is contemplating an organized decline in the price in order to increase market share.

7. The boundary conditions for the intra marginal interventions are a special case of these conditions.

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