A Capacitated Commodity Tading Model with Maket Powe Victo Matínez-de-Albéniz Josep Maia Vendell Simón IESE Business School, Univesity of Navaa, Av. Peason 1, 08034 Bacelona, Spain VAlbeniz@iese.edu JMVendell@iese.edu Submitted: Decembe 0, 007. Revised: June 1, 009; Mach 8, 010. Abstact In commodity makets, physical tades can take advantage of geogaphical pice speads by buying the commodity at the location whee the pice is low and eselling it whee it is high. Geneally, these tading decisions may have an impact on futue pices, which eflect the tade s maket powe. We ae inteested in finding the optimal tading policy of a tade between two makets, which also faces physical opeational constaints, e.g., logistics. In the optimization poblem, we include the maket powe effect by alteing the undelying stochastic pice pocesses to account fo the tade s decisions, which is novel to the liteatue. We find that the optimal policy in this situation is defined by two thesholds that detemine thee egions whee it is optimal to: (1) move as much as possible fom one maket to the othe; () the same in the opposite diection; o (3) do nothing. We chaacteize these thesholds both in isolation and in a competitive equilibium. Futhemoe, we illustate in a eal industy setting the potential benefits of taking into account maket powe compaed to othe simple tading ules that ignoe a tade s impact on futue pices. 1 Intoduction Commodities ae the building blocks of a lage pat of ou economy. Examples include not only enegy, mineals, metals and agicultual poducts such as oil, natual gas, ion oe, aluminum, silve, gold, suga, coffee, ice, wheat, but also intemediate o manufactued poducts such as chemicals o geneic dugs. They ae nowadays elatively easily taded, can be physically deliveed anywhee in the wold and sometimes can be stoed fo a easonable peiod of time. The volume of commodity tading in the wold is colossal: fo instance, in 009, moe than 13 million 60-kg coffee bags wee taded and moe than 85 million baels of oil wee consumed daily. Commodities ae taded in vey active makets, such as the Chicago Boad of Tade (CBOT), the New Yok Mecantile Exchange (NYMEX), o the London Metal Exchange (LME), to cite a few. The pice of a commodity is diffeent in each location whee it is taded, because it is adjusted fo logistics costs and local maket conditions (local balance of supply and demand) among othes. Fo example, a coffee poduce with seveal factoies in Euope pays diffeent pices fo delivey in each one of the factoies. Inteestingly, the pice spead, i.e., the pice diffeence between two locations, is usually quite vaiable ove time, fo most commodities. It 1
can also significantly deviate fom the long-tem aveage in the shot-tem. As an example, conside the daily spot pice evolution of jet fuel (keosene) in two locations, New Yok and Los Angeles, fom Januay 1991 to Decembe 009, shown in Figue 1. The commodity taded in these two makets is exactly the same, since the consumes (ailines) equie a standad chemical composition eveywhee. As can be seen in the left-hand figue, the evolution of the pice in these two locations is almost the same, implying that in the long-un pice speads ae small. Howeve, in the ight-hand figue we obseve that the pice diffeences ove the peiod can be quite lage and speads of tens of cents occu a few times. In fact, the spead is on aveage -.49 cents and a standad deviation of 7.46 cents (with a minimum of -53.50 and a maximum of 76.50 cents). This suggests that although expected speads ae small, in the shot tem they can be vey lage. 450 400 Pice spot in NY Pice spot in LA 80 60 Pice spead P NY P LA USD cents pe gallon 350 300 50 00 150 100 USD cents pe gallon 40 0 0 0 50 40 0 1990 1995 000 005 010 Date 60 1990 1995 000 005 010 Date Figue 1: Plot of keosene jet-fuel daily spot pices on the left (in USD cents pe Gallon) in New Yok Habo (NY) and Los Angeles (LA), and the coesponding spead (pice diffeence) P NY t Pt LA. The souce of the data is the Enegy Infomation Administation of the U.S. Govenment, see http://www.eia.doe.gov. These shot tem geogaphical speads ceate inteesting oppotunities fo playes that can buy in one maket and sell in the othe. In pactice, commodity tades such as Bunge, Cagill, Louis Deyfus, etc. and commodity desks in investment banking ae quite active in this aea and often have dedicated esouces (such as shipping and stoage capacity) to be able to ente into these tades. it is. A tade typically faces many impotant opeational decisions that dictate how successful Pobably the most citical decision is to detemine the quantity to be taded at any moment, given the cuent spead levels. This task equies caeful consideation of the tade s opeational envionment, mainly its physical constaints, i.e., shipping capacity, and its tading cost. In paticula, one elevant element in this decision is to assess how the tade s action will
change the local pice dynamics, at both locations fom/to which it buys/sells: if the tade is lage enough, binging additional supply (demand) to one maket will deflate (inflate) pices. This is what we call maket powe. The objective of this pape is pecisely to integate maket powe consideations into the tade s opeational decision-making. In this context, finding the optimal tading policy becomes quite challenging because the evolution of pices ove time is stochastic and is influenced by the tade s actions. We establish the ational behavio of a tade between two makets when the pice spead dynamics ae based on the tading volumes. Specifically, they ae diven by an exogenous evolution (supply and demand shocks) togethe with an endogenous effect (tade s actions). In this espect, ou pape combines elements fom the two majo steams of eseach in commodity pice modeling. On the one hand, one can find models whee pices evolve exogenously, as ou supply and demand shocks. These models ae extemely pactical and easy to use because they only equie estimating a few paametes. On the othe hand, thee also exist models whee pice evolution is endogenously set by the actions of the diffeent playes in the industy; they typically assume pefect competition (i.e., no maket powe), and ae appealing because they model the basic economic elements fom which maket pices emege in equilibium. Similaly, ou pape consides that the pice evolution is detemined by the maket playes, but in contast it also takes into account maket powe in thei decisionmaking. In summay, we popose a hybid modeling appoach that etains the tactability of the exogenous models and some of the appeal of the endogenous equilibium models. In ode to make the analysis tactable, we focus on a simple setting in which the tading policies ae optimized. We assume that the tade is capable of puchasing in one maket to esell in the othe, within a fixed tading capacity. The capacity can be thought of as the maximum possible volume pe peiod that the tade can ship between the two locations with its logistics esouces. In addition, we assume that the inventoy stoed between the two makets is constant. This is a easonable assumption when the shipping is done though a pipeline (e.g., gas, oil) o in systems whee stoage capacity is vey limited. In this setting, we establish the stuctue of the optimal tading policy that maximizes the tade s expected discounted pofit. Given a ealization of the cuent pice spead between the two makets, one must detemine how much (if any) must be bought/sold fom/at the fist maket and sold/bought at/fom the second maket. We show that the optimal policy can be descibed by two thesholds: when the spead is below the lowe theshold o above the uppe theshold, the tade buys in the lowpice maket and sells in the high-pice maket, up to capacity; othewise, it does not tade. We povide a chaacteization of these optimal thesholds and study thei sensitivity with espect to the model paametes (capacity, cost, pice evolution dives). Futhemoe, when many 3
tades ae pesent in the maket, each tade s policy depends on what the competitos do. We analyze with game-theoetical tools the equilibium situation too, which we chaacteize in a symmetic setting. Moeove, we apply ou model to the keosene example descibed above, and we analyze how the numbe of playes with maket powe and the capacity they contol affect the optimal tading policies and the esulting pofits. In paticula, we find that the benefits deived fom optimization can be significant, and that they ae highe when the competitos ae myopic athe than when they also use thei maket powe. Finally, we discuss how the model can be extended with moe geneal pice evolution pocesses and with moe than two makets. The pape thus makes two main contibutions. Fist, it poposes a novel hybid appoach that incopoates the effect of lage tades on the pice spead dynamics. It also chaacteizes the esulting equilibium spead evolution pocess. Second, it povides elevant guidelines fo tades that influence pices in local makets. In paticula, it shows the potential benefit of optimization in the pesence of maket powe, compaed to simple myopic policies. The est of the pape is oganized as follows. In, we eview the elated liteatue. 3 descibes in detail the model with its assumptions and notation. We then chaacteize in 4 the stuctue of the optimal policy. We complement these findings with a desciption of the maket equilibia when thee is moe than one stategic tade. In 5 we compae ou policy with othe basic models available in liteatue and highlight the potential benefits. We then discuss in 6 seveal extensions of the basic model. Finally, 7 concludes the pape and discusses futue eseach diections. The poofs of ou theoetical esults ae included in an online appendix. Liteatue Review Thee is a significant amount of eseach on commodity pice modeling and commodity tading, with at least two diffeent liteatue steams: the fist in economics and finance; the second in opeations management. Reseaches in economics and finance have tied to explain the evolution of spot pices in commodity makets. As pointed out in the intoduction, thee ae typically two ways of modeling this evolution. The fist option is to descibe spot pices with non-equilibium models whee the evolution is detemined in an exogenous manne. Gibson and Schwatz (1990) model convenience yields (a measue of the benefits fom owning a commodity asset vesus holding a long futues contact on the asset) and spot pices as sepaate stochastic pocesses, possibly coelated. Schwatz (1997) develops thee vaiations of a mean-eveting stochastic model diven by one, two o 4
thee factos taking into account mean evesion of commodity pices, convenience yields and stochastic inteest ates. Schwatz and Smith (000) povide a two-facto stochastic spot pice model detemined by the combination of shot-tem and long-tem factos, that allow volatility in both tems. These models ae empiically validated fo coppe, gold and cude oil. Ou pape does not diectly model mean-eveting spot pice pocesses o convenience yields, as in these papes, but focuses instead on modeling pice speads. The second option is to develop industy equilibium models, which endogenously detemine the evolution of commodity pices by captuing the undelying evolution of supply and demand. They usually assume pefect competition, which esults in social welfae being maximized. One of the fist impotant papes in this aea is Deaton and Laoque (199). They develop a ational expectations model fo commodity pices that explains thei high autocoelation and vaiability. They base thei model on the existence of competitive speculatos that hold inventoy of the commodity: they build inventoy when the pice is low, and deplete it when it is high. As a esult, the speculatos act as egulatos of the commodity pices. Deaton and Laoque (1996) extend this wok and find that speculation alone is not sufficient to explain the high positive autocoelation obseved and that pat of it is caused by the undelying pocesses of supply and demand. Williams and Wight (1991), Chambes and Bailey (1996) and Routledge et al. (000) also use a competitive ational expectations model to explain the popeties of commodity pices. Some equilibium models have also been developed to explain geogaphic pice diffeences, but typically do not conside maket powe, as we do. Williams and Wight (1991) popose an equilibium model of shipping between two locations, with o without stoage at each of these two locations. Routledge et al. (1998) conside the coelation between pices of diffeent commodities within boad families (e.g., natual gas and electicity in enegy). They model the substitutability of these commodities and find existence of equilibium pice pocesses, whee ational agents convet one commodity into the othe. In paticula, they focus the pice diffeence between natual gas and electicity, known in the industy as the spak spead, which is used extensively by commodity tades. De Vany and Walls (1996) study the natual gas pice evolution in diffeent points of the U.S. pipeline netwok. They show that these pices move togethe with the speads being bounded by the tansmission costs within the netwok, except when the links only pemit small flows. They suggest that this commodity netwok is geneally competitive and that abitage oppotunities ae limited to shot-tem imbalances when flow is constained. In ou pape, we also focus on pice speads between diffeent geogaphical locations, although the wok could be extended to speads between diffeent commodities as discussed in 7. In contast with these equilibium models, ou objective is to optimize the 5
individual opeational decisions of ational tades with maket powe, in isolation and in equilibium. Also elated to ou pape ae Dixit and Pindyck (1994), who pesent a geneal model fo enty and exit of fims in a maket. The key diffeence with ou wok is that they focus on capacity choices at the industy level. They descibe the evolution of pofits exogenously and detemine whethe a fim should ente o withdaw fom a maket knowing the cost of enty and a fixed distibution of its possible gains. Enty o exit occus once the dynamic zeo excess condition is eached. Unlike them, ou tade is aleady in the maket (we do not conside any capital investment) and its pofits depend on a distibution which in tun depends on its actions (it is not known a pioi). Also, while Dixit and Pindyck find the pice evolution implicitly, we povide a chaacteization of the points whee the tade should buy/sell, which ae equivalent to the thesholds fo enteing o withdawing fom a maket in Dixit and Pindyck. Ou wok is hence an attempt at bidging these two main steams of wok as we incopoate equilibium aspects in a non-equilibium model by adding an endogenous pat in the pice pocess to account fo tades actions. Finally, thee also exists a liteatue in finance on tades with maket powe. The undelying idea in these papes is that lage investos have significant pice impact in thin makets and ty to mitigate these advese effects though thei tading stategies. Indeed, lage tades ae likely to move pices in the diection of the tade, theeby advesely affecting the tems of tade. Fo example Basak (1997) develops a geneal equilibium model to include an agent who acts as a pice leade in the secuity and good makets. Some othe papes specifically study the pice diffeences of diffeent types of shaes of the same company. These pice diffeences ae usually explained by shot-sale constaints and heteogeneous beliefs on stock pices and tading volume. Fo instance Mei et al. (005) find that tading caused by a few investos speculative motives can help explain a significant faction of the pice diffeence between the dual-class shaes in China. Foot and Daboa (1999) study how this pice diffeence fo twin shaes is affected by location of tade. These anomalies in pice diffeences ae eflected in Lamont and Thale (003) who conclude that if the same asset is selling fo two diffeent pices simultaneously, then abitageus will step in and coect the situation to make a pofit. In ou pape, we also conside a tade with maket powe. Howeve, as opposed to the papes above, we conside a diffeent context and focus on showing how pice diffeences fo the same commodity at diffeent locations can esult fom the stategic inteaction of multiple, lage agents by endogenously chaacteizing the impact of thei tading actions on the evolution of pice speads ove time. Ou wok is also elated to the opeations management liteatue since it consides ope- 6
ational constaints on the tading activity. Specifically, thee ae papes that focus on the management of inventoy of commodities, in the pesence of pice uncetainty, with buy/sell decisions in a single maket. Zipkin (000) povides a eview of the extensive liteatue on inventoy management models. We efe eades to Goel and Gutieez (004, 006, 007), and efeences theein fo specific applications to commodities. These papes ty to incopoate the infomation given by the convenience yield in the inventoy and buy/sell decisions, e.g., Caldentey et al. (007) fo coppe mining opeations in Chile, whee they use the stochastic pocess in Schwatz (1997) to model coppe spot pices. In this line, Golabi (1985) models the pice of a commodity in futue peiods as a andom vaiable with a known distibution function. Assuming constant demand, he poves that a sequence of citical pice levels at a given peiod detemines the optimal odeing stategy. Wang (001) poves that a myopic inventoy policy is optimal fo a multi-peiod model with stochastic demand and deceasing pices. Beling and Matínez-de-Albéniz (009) descibe the optimal odeing policy when pices follow a geometic Bownian motion o ae meaneveting. Secomandi (009a) consides optimal commodity tading and povides a much moe detailed view of the opeations involved in tading. He focuses on stoage assets, i.e., stoage facilities o contacts that ensue that one will have access to the inventoy at a pe-detemined time. His model is based on inventoy and well-behaved flow constaints. He shows that the optimal policy is, depending on the egion, to buy (and withdaw), to do nothing, o to sell (and inject). A simila stuctue applied to shipping policies is used fo contact valuation in Secomandi (009b) and Wang et al. (007). Ou wok also combines pice evolution models fom economics and finance, togethe with a moe detailed view of opeations. In paticula the cental diffeence is that we include equilibium aspects in the pipeline wok of Secomandi (009b). We ae specifically inteested in optimizing the tading (shipping) policy, given that the tading activity may influence the pice spead pocess. 3 The Model 3.1 Two Makets Conside the makets of a same commodity at two locations A and B. Each one of these locations has its own local dynamics and as a esult the pice at which the commodity tades in each place may be diffeent. Tading may occu between A and B. In paticula, tades may choose to buy in one place 7
and ship the commodity, at a cost, to the othe. In eality, shipping of physical commodities cannot be done immediately and is only feasible afte a cetain lead-time. Fo the sake of simplicity (and tactability) howeve, we assume that this lead-time is zeo. Implicitly we ae assuming that at any given time what is bought in one location is sold at the othe. Thus, ou basic model descibes a pipeline whee the intemediate stoage is fixed. In othe wods, a tade simply opens a faucet so that the commodity flows fom A to B o vice-vesa. We conside in addition that the pipeline capacity is given by the maximum achievable flow, in units pe time-unit. Note that the model could also handle a shipping pocess that does not involve a pipeline, as long as the inventoy level in each pat of the oute emains constant. Fo instance, conside a set of tucks o ships that on each day can move towads A o towads B in a coodinated manne. The quantity oiginating in A and aiving in B will be identical and will be constained by the maximum speed in the system. We denote by k AB the tading capacity fom A to B, pe peiod, and k BA the one fom B to A, which might be diffeent. We model time as a continuous vaiable, t 0, and we denote the pice of the commodity in makets A and B as P A t and P B t espectively. In addition we conside constant linea tading costs. We denote c AB the cost of moving one unit fom A to B, and c BA fom B to A. Using constant maginal costs is a stong assumption since it is independent of the flow acoss the tading facility, but it has been made befoe in the liteatue, e.g., Secomandi (009b). Fo instance, if the tading cost is a faction of the dolla volume of tade, the cost would be a constant times the spot pice at the buying maket times the tading quantity. While this is numeically solvable, the analysis becomes intactable, see 6.1. Let u AB t and u BA t time t. The capacity constaint implies that be the tading quantity fom A to B and fom B to A espectively, at 0 u AB t k AB and 0 u BA t and the tading cost incued is equal to c AB u AB t + c BA u BA t. k BA The tading policy, if decided ationally, should obviously depend on the pices quoted in A and B. Fo example, when P B t P A t > c AB it is pofitable to tade fom A to B. Clealy, if the cuent tading activity does not influence the futue pices the tade would ty to maximize the pesent pofit fom tading at t, i.e., 0 u AB t max k AB, 0 u BA t k BA (P B t P A t )(u AB u BA ) c AB u AB c BA u BA. 8
It is easy to see that the optimal policy is such that eithe u AB = 0 o u BA = 0 and k AB if Pt B Pt A c AB u := u AB u BA = 0 if c AB Pt B Pt A c BA k BA othewise. In othe wods, a tade that does not influence futue pices will tade at maximum capacity povided that the pice spead is bigge than the tading cost as shown in Secomandi (009b). Figue summaizes the main model featues. Of couse, if many tades follow this type of policy the pice at the cheape maket would ise and the pice at the othe maket would go down so that the spead would tend to close ove time. Hence, it is clea that the tading volume must impact the pices in the futue. If this was not the case one would see lage speads between locations. Moe impotantly, these speads may not close ove time. As a esult, if the tading quantities do influence futue pices it is no longe clea what type of policy is now optimal fo the tade. This equies modeling the elationship between global tading policies and the pice spead pocess. This is done in the next section. (1) Figue : Summay of model featues. The tade can ship u AB units of commodity fom A to B and u BA units fom B to A. 3. The Pice Pocesses In the pevious section, we have defined the net tading quantity at time t as u t := u AB t Using the same notation, v t := v AB t v BA t u BA t. denotes what the othe playes in the maket (competitos) tade fom A to B. Of couse, this is positive when the quantity flows fom A to B and negative othewise. We model the pice pocesses at A and B as coelated andom walks, i.e., ( dp A t dp B t ) = ( α(p A t, P B t ) + β A (u t + v t ) α(p A t, P B t ) β B (u t + v t ) 9 ) dt + ( σa1 σ A σ B1 σ B ) ( dw 1 t dw t ) ()
whee Wt 1 and Wt ae independent Wiene pocesses, α can be any exogenous function, and β A, β B, σ A1, σ A, σ B1, σ B 0. As one can obseve, the tade between A and B has a diect impact on the pices pocess which captues the playes maket powe. Specifically, the total net tade fom A to B at t, i.e., u t + v t, inceases the pice at A and deceases it at B when it is positive. This is qualitatively intuitive. Futhemoe, we model this dependency as linea. This is a stong assumption, which is elated to having linea pice-quantity demand cuves. Essentially, in a given maket (conside A, but the same agument applies to B), the pice is detemined by supply and demand. Assuming linea functions, the demand minus the supply can thus be witten as M t N t Pt A and the cleaing pice is thus equal to M t /N t. If the pice elasticity N t is constant, the pice vaiation dpt A will be given by the vaiation dm t. In ou model, the changes in M t (demand and supply changes) ae due to (i) deteministic changes fom an exogenous facto, (α + β A v t )dt, plus an endogenous effect caused by the tading volume, β A u t dt; and (ii) an exogenous shock σ A1 dwt 1 + σ A dwt. Notice also that the poposed model could exhibit mean-evesion popeties, as long as the mean-eveting adjustment is the same in both makets. Fo example, one could let α(pt A, Pt B ) = κ [ µ (θ A Pt A + θ B Pt B ) ], whee µ would be the long-tem mean pice defined as a weighted aveage of pices at A and B. When the mean-eveting adjustment is diffeent in A and B, again the analysis becomes intactable, but we can still use numeical optimization, see 6.1. Finally, in contast with models in finance, e.g., Schwatz (1997), we conside linea andom walks instead of geometic ones. Ou assumption guaantees tactability of the tading poblem. Howeve, when geometic Bownian motions need to be used, 6.1 exploes numeically this altenative. Since the pofit captued by the tade depends on the pice spead, we define G t := Pt B Pt A. Fom Equation () we have that dg t = (β A +β B )(u t +v t )dt+(σ B σ A )dwt (σ A1 σ B1 )dwt 1. Thus G t is a stochastic pocess that can be expessed as dg t = β(u t + v t )dt + σdw t, (3) whee β = β A + β B, σ = (σ B σ A ) + (σ A1 σ B1 ) and W t a Wiene pocess. Hence, when tades take advantage of the existing spead (u t, v t 0 if G t > 0 and u t, v t 0 if G t < 0), then the spead pocess is mean-eveting to zeo. This is in fact what tades do in thei best inteest. In the emainde of the pape, we assume that the taded quantity v t (elated to the est of the maket) only depends on G t, and that it is piecewise constant with a finite numbe of jumps. This assumption guaantees tactability. As a esult, both evenue and cost depend only on G t 10
and hence at optimality the tade s policy u t should only be a function of the spead G t. In fact, in 4.3 we analyze the game between tades in the maket and show that in equilibium thei tading quantity only depends on G t. 3.3 Relating the Spead Distibution and the Tade s Policy Consideing that u t and v t ae functions of G t only, i.e., u t = u(g t ) and v t = v(g t ), the andom vaiable G satisfies the stochastic diffeential equation dg t = A(G)dW t + B(G)dt, with A(G) = σ and B(G) = β(u(g) + v(g)). Poposition 1 If fo a cetain ɛ > 0, p.d.f., denoted f(g), which satisfies lim B ɛ and G lim B ɛ, G has a stationay G + 1 A (G) f f B(G) = 0. (4) G G Equation 4 is known as the Kolmogoov equation, and is a backwad paabolic patial diffeential equation. In ou case, it can be witten as 1 σ f f + β(u(g) + v(g)) G G = 0. (5) Thus, the contol u(g) has a diect influence on the p.d.f. of the spead. We denote it by f u (G) to make that dependency appaent. This dependency is the base of the optimization, since the tade can captue pofit now at the expense of educing futue speads and hence futue pofits. As an illustation, we show in Figue 3 the stationay p.d.f. fo two contols u with v 0. 3.4 Optimal Tading Policies Given a ealization of the pice spead at time t, G t, and given capacities k AB, k BA, we ae inteested in finding the optimal opeating policy u t that maximizes the tade s pofits. Because the evolution of the system is govened by the pice spead G t, the optimal policy is a ule that dictates u t, i.e., how much poduct to buy and sell depending on G t. In othe wods, the tade must decide on a function u(g t ) that maximizes its pofit. Unde a given policy, the pofit is calculated fom the evenue G t u(g t ) and the tading cost c AB u(g t ) + + c BA u(g t ), whee a + = max{a, 0} and a = max{ a, 0}. We focus hee on maximizing the tade s expected discounted pofit, with a continuous ate > 0, although simila esults can be deived fo the expected aveage pofit. 11
0.01 6 x 10 4 Pobability Density Function 0.009 0.008 0.007 0.006 0.005 0.004 0.003 0.00 0.001 Pobability Density Function 4 0 0 15 10 5 0 5 10 15 0 G 0 0 15 10 5 0 5 10 15 0 G Figue 3: Stationay p.d.f. of G, with v 0, β = 1 and σ = 1, whee, in the figue on the left, u = fo G < 0 and u = + fo G 0; and, in the figue on the ight, u = fo G < 10, u = 0 fo 10 G < 10 and u = + fo G 10. Note that we assume that the tade is isk-neutal. That is, the expectation used in the objective is diectly deived fom the stochastic pocess that govens pices. This assumption seems easonable because in pactice opeatos of shipping capacity often take decisions by evaluating simple discounted cash flows with a given discount ate, based on the histoical evolution of pices. Futhemoe, it is woth pointing out that the assumption of isk neutality is equivalent to assuming that the pice evolution equation is fomulated unde an equivalent matingale measue. Unde the discounted pofit citeion, the tade maximizes [ ( ) ] J(u, G 0 ) := E G t u(g t ) c AB u + (G t ) c BA u (G t ) e t dt. (6) 0 The discounted pofit depends on the cuent pice spead G 0, but the optimal policy u does not depend on G 0, as shown next. 4 Analysis 4.1 Chaacteization of the Optimal Policy We use optimal contol theoy to solve the poblem. Denoting as J (G 0 ) the pofit-to-go function evaluated at G 0 the initial pice spead, the maximization poblem can be witten as J (G 0 ) = sup {J(u, G 0 )}. (7) k BA u k AB 1
The optimality conditions can be captued though the Hamilton-Jacobi-Bellman (HJB) equation, see Betsekas (000) fo moe details. Poposition The value function satisfies σ d J dj βv dg dg J + max k BA u k AB {Gu c BA u c AB u + βu dj } = 0. (8) dg In addition, a maximize u in Equation (8) is an optimal policy and J = J(u, ). The optimal contol u (G) is given by (G β dj max k BA u k AB dg ) u c BA u c AB u +. (9) This implies that u (G) = k BA, 0, k AB. Hence, a bang-bang policy is optimal. In fact, the stuctue of the policy is to apply one of these thee contols in thee adjacent pice spead intevals. The next theoem summaizes this obsevation. Theoem 1 Assume that v(g) is non-deceasing, v(g) 0 fo G 0, v(g) 0 fo G 0, lim v(g) > 0 and lim v(g) < 0. Then thee exists G + G mba and m AB with m BA m AB so that the optimal policy of (7) is k BA if G m BA u (G) = 0 if m BA < G m AB k AB if G > m AB. The theoem shows that the optimal policy is detemined by thee egions: (, m BA ] whee it is optimal to move as much poduct as possible fom maket B to the maket A; (m AB, ), whee the same is tue in the opposite diection, fom A to B; and (m BA, m AB ] whee it is optimal to wait fo lage speads to occu. This is the same stuctue as in the case with no maket powe, see Equation (1). Indeed, with no maket powe the thesholds ae c BA and c AB. Hence, maket powe changes the optimal tading policy by delaying the point whee a tade should begin to ship poduct fom one maket to the othe. The esult equies cetain egulaity conditions on v. These conditions ae quite natual, since v epesents the dift of the spead pocess when the tade is absent, i.e., u = 0. Essentially, when the spead is away fom zeo (positive o negative), it tends to get close to zeo ove time. When the spead is equal to m BA, espectively m AB, then the optimal contol shifts fom k BA to 0, espectively fom 0 to k AB. Inteestingly, Equation (9) povides an implicit equation to find these optimal thesholds. 13
Poposition 3 The thesholds that define the optimal policy satisfy In addition, c BA βkba m BA + c BA = β dj dg (mba ) and m AB c AB = β dj m BA c BA and c AB m AB c AB + βkab. dg (mab ) (10) Although no closed-fom expession is available fo the thesholds, the poposition above povides an implicit condition to compute them numeically by simple seach. It also shows that it is neve optimal to ship below cost, i.e., that if u (G) > 0 then G > c AB and if u (G) < 0 then G < c BA, which is quite intuitive. Indeed, tading below cost would on the one hand yield immediate negative pofits, and on the othe hand educe the pobability of lage speads which would also be detimental to the tade. The following coollay identifies the asymptotic behavio of the thesholds when the tade is small compaed to the maket. It is a diect consequence of Poposition 3. Coollay 1 If k AB, k BA 0, then m BA c BA and m AB c AB. This esult shows that ou model is consistent and that in the limit (when k AB = k BA = 0), ou optimal policy is the same as that of a ational tade with no maket powe. In addition, one is able to obtain fist-ode appoximations of m BA and m AB when the capacities k AB, k BA ae small. Thus, in a pefect maket whee all tades ae small, each tade becomes active when the spead is beyond its tansaction cost. Some additional popeties of m BA and m AB can be deived using the implicit expessions in Equation (10). Poposition 4 If c AB = c BA, k AB = k BA and v( G) = v(g) fo all G, then m BA = m AB and J ( G) = J (G) fo all G. This esult implies that in a symmetic system, whee the tade has the same cost and capacity fom A to B as fom B to A, the optimal policy is also symmetic. This simplifies the seach fo the optimal policy in symmetic systems, since only one theshold must be found instead of two. 4. Sensitivity of the Optimal Policy In this section, we investigate the sensitivity of the optimal policy and pofit-to-go with espect to the model paametes (capacity, cost, volatility and pice-elasticity). Since studying the sensitivity analytically is difficult, we poceed numeically. 14
The esults ae obtained by computing fo each theshold policy u the coesponding value function J(u, G). The optimal policy is identified though Equation (10) and also coesponds to the policy that, fo a given G 0, maximizes J(u, G 0 ) ove u. Fo simplicity of exposition, we pesent the symmetic situation, whee c AB = c BA = c, k AB = k BA = k and v( G) = v(g) fo all G. Accoding to Poposition 4, in this case J ( G) = J (G) and only one theshold needs to be calculated. We fist show the impact of cost and capacities. Figue 4 shows on the left-hand side the impact of cost on the optimal thesholds and the discounted pofit; on the ight-hand side it illustates the impact of capacity. The discounted pofit J (G) is shown fo two values of G. G = 0 coesponds to a stating situation whee thee is no spead and the tade must wait until the spead appeas to stat making a pofit (and thus J (0) is low). G = 5 coesponds to a stating situation whee the spead is vey lage and thus J (5) is lage. m AB = m BA Discounted pofit 4 3 1 k=0.5 k=1.5 0 0 0.5 1 1.5.5 3 10 5 J * (0) with k=0.5 J * (0) with k=1.5 J * (5) with k=0.5 J * (5) with k=1.5 0 0 0.5 1 1.5.5 3 Cost c m AB = m BA 1.5 1 0.5 c=0.5 c=1.5 0 J * 0.5 (0) with c=0.5 1 1.5.5 3 J * (0) with c=1.5 8 J * (5) with c=0.5 J * (5) with c=1.5 6 Discounted pofit 4 0 0 0.5 1 1.5.5 3 Capacity k Figue 4: The top figues depict the changes of the optimal m AB = m BA with changes in c = c AB = c BA (left) and q = k AB = k BA (ight). The bottom figues show the coesponding changes in the discounted pofit J. In these figues, we conside that v(g) = 1 when G 1, v(g) = 1 when G 1 and v(g) = 0 othewise. We set β = 1, σ = 0.5 and = 10%. As one could expect, pofits ae deceasing and convex in c AB, and inceasing and concave in k AB. It is pehaps moe supising to see that the optimal theshold is at fist inceasing quite fast, suggesting that m AB c AB inceases with c AB aound zeo, and then inceasing moe slowly, implying that m AB c AB pogessively deceases to zeo as c AB keeps inceasing. Thus, at a high level of cost, the tade does ente the maket as soon as the spead is lage 15
than its cost, as when thee is no maket powe. Also, the theshold is inceasing in k AB, which implies that the lage the tade is, the moe it will wait to stat tading. In paticula, when k AB is close to zeo, it behaves as when thee is no maket powe, see Poposition 1. We next analyze the impact of the pice evolution paametes: the elasticity to pice β, i.e., the impact of a unit of tade into spead changes, and the volatility σ. Figue 5 shows that both thesholds and pofits ae deceasing in β and inceasing in σ. This is quite intuitive. Indeed, fo lage β, the tading activity tends to educe the speads faste and hence the pobability of seeing lage speads gets smalle. As a esult, a tade will ente the maket ealie. Similaly, fo smalle σ, the pobability of seeing lage speads is educed (in the exteme case of σ = 0, speads lage than the cost neve occu) and hence the tade will set a lowe theshold. Inteestingly, similaly to eal option valuation (see Tigeogis 1999), volatility of the pice spead inceases the expected discounted pofit of the tade. m AB = m BA 1.4 1.3 1. 1.1 σ=0.5 σ=1 m AB = m BA 1.4 1.3 1. 1.1 β=0.5 β=1.5 Discounted pofit 1 0. 0.4 0.6 0.8 1 1. 1.4 1.6 0 15 10 5 J * (0) with σ=0.5 J * (0) with σ=1 J * (5) with σ=0.5 J * (5) with σ=1 0 0. 0.4 0.6 0.8 1 1. 1.4 1.6 Elasticity to volume β Discounted pofit 1 0. 0.4 0.6 0.8 1 1. 1.4 1.6 15 10 5 J * (0) with β=0.5 J * (0) with β=1.5 J * (5) with β=0.5 J * (5) with β=1.5 0 0. 0.4 0.6 0.8 1 1. 1.4 1.6 Pice volatility σ Figue 5: The top figues depict the changes of the optimal m AB = m BA with changes in β (left) and σ (ight). The bottom figues show the coesponding changes in the discounted pofit J. In these figues, we conside that v(g) = 1 when G 1, v(g) = 1 when G 1 and v(g) = 0 othewise. We set c AB = 1, k AB = 1 and = 10%. 4.3 Maket Equilibium So fa we have consideed the optimization poblem of a tade given that all the othe tades follow a fixed tading policy, denoted as v(g). Howeve, in pactice these tades should be ational playes as well and hence optimize thei actions themselves. Hence, the optimal 16
policy u (G) deived in 4.1 was in fact the best-esponse of one tade to the competitos tading stategies. In this section, we study the game-theoetic dynamics between n playes in the maket, each one tading u i t = u i (G t ) when the spead is G t, i = 1,..., n. Each playe can have a diffeent cost and capacity constaint and we assume that it is fully ational. Thus, the ule u i is the best-esponse of tade i to the competitos actions, captued though v i t = v i (G t ) = j i u j (G t ). We analyze the equilibia of the game in pue stategies. We focus hee on the symmetic case, whee all tades have the same cost c BA, c AB, the same capacity k BA, k AB and thei objective is the same, namely the expected pofit discounted at the same ate. In addition, fo simplicity we assume that c BA = c AB and k BA = k AB, although the esults could be extended fo geneal costs and capacities. When all tades ae identical, it tuns out that thee is a unique symmetic equilibium. Indeed, in such an equilibium all playes use the same tading stategy u i = u, and hence v i (G) = (n 1)u(G). Since v satisfies the assumption of Theoem 1, thee exists two common thesholds m BA and m AB beyond which all playes stat tading. Hence, the equilibium value function of a tade J satisfies AB dj βnk σ d J dg = J dg + J kab (G c AB ) BA dj βnk dg + J kba ( G c BA ) when m AB G when m BA G m AB when G m BA In addition, m BA + c BA = β dj dg (mba ) and m AB c AB = β dj dg (mab ). These equations can be solved in closed-fom to deive the equilibium thesholds. Theoem If c AB = c BA = c and k AB = k BA = k, thee is a unique symmetic equilibium such that m BA = m AB = m eq whee whee ν = m eq c = ( (nβk) + σ nβk σ βθ ( ) ( e θmeq e θmeq νβnk ) ( e θmeq θ + ν βθνk and θ = /σ. + k θ ν βθνk ) ) e θmeq The theoem povides the value at which all tades will in equilibium stat using thei tading capacity. This can be used to investigate the sensitivity of the equilibium theshold m eq with espect to the model paametes. It is paticulaly inteesting to see the effect of the numbe of playes n given that the total capacity is fixed to k tot = nk. In accodance with the esults fom 4., one can check that m eq is deceasing in n. Thus, as competition becomes 17
moe intense, each playe sets its theshold close and close to its cost. In the limit n, the playes behave as if thee was no maket powe. 5 Numeical Study 5.1 A Case Study: Paamete Estimates Thee ae many paametes that need to be detemined in the model. Fist, one needs to detemine the pice pocess paametes, i.e., β and σ. Second, one needs to know who the diffeent tades between A and B ae, how much capacity and what tading cost each one has. We use the keosene data mentioned in the intoduction to geneate a easonable estimate of these paametes fo this paticula case. The data contains the daily spot pices of jet fuel in New Yok and Los Angeles, which ae the pices paid fo delivey in these two locations. Jet fuel is used almost exclusively by the ailine industy, is poduced in oil efineies (1 bael of oil typically povides 4.1 gallons of jet fuel, among othe efined poducts) and is typically tanspoted by land (tuck o pipeline) fom the efineies to the final place of consumption. Oil supply and efiney tansfomation capacities ae quite constant so basically the main dives of keosene pices ae feight ates and local demand. As a esult, when feight ates ae fixed, the cost of tading should be constant. The basic statistics of the spead G t = P NY t P LA t ae summaized in Table 1. Table 1: Basic statistics of G t = P NY t P LA t Numbe of obsevations (maket days) 4687 Minimum -53.50 Maximum 76.5 Mean -.49 Standad deviation 7.51 Notice that the maximum and minimum value of the spead ae quite lage, as well as the standad deviation. These facts ae inteesting fo a playe tading in these makets. Thee ae seveal actos that we can conside tades, and that might want to take advantage of the NY/LA spead. Thee ae mainly small physical tades that own tanspotation capacity and can buy and sell in the two makets. Thee ae also lage oil companies that contol the supply of jet fuel fom the efineies, which may have some maket powe. Given that it is impossible to know exactly when these actos buy and sell in the maket, we assume, fo the 18
sake of obtaining appoximate estimates, that most of the tading volume is moved by small playes; in othe wods, we assume that the maket is faily competitive. As a esult, one can conside that each one will stat tading as soon as the pice diffeential is lage enough. This assumption allows us to make a few estimates fom the obseved pices. Indeed, if all the tades ae small compaed to the maket, then ou model povides a way to estimate how much total tading capacity k(g) is available at each cost level G. Note that since the effect of capacity and elasticity (β) ove the spead appea jointly in Equation (3), we can conside without loss of geneality that β = 1. Detailed estimates can be obtained by diectly expessing all the pice vaiations, dg t = G t+1 G t, when G t [G δ, G + δ], as the sum of a constant plus a white noise: dg t = βk(g t )dt + σ(g t )dw t. Note that the time peiod we use is equal to dt = 1 day. This appoach has the advantage of estimating diectly βk(g) (the constant) and σ (the standad deviation of the noise). Howeve, as shown in Figue 6 it tuns out that these estimates ae not eliable, and only look consistent when we aggegate them significantly (lage δ). Notice that the volatility σ is lage compaed with the dift βk(g). Fom this pocedue, we obtain an estimate of the volatility ˆσ =.03. 8 6 4 β k(g) 0 4 0 15 10 5 0 5 10 15 Spead G Figue 6: Fo each spead level G t and consideing δ = 1 cent, we show the estimated dift βk(g) (cente of the bas) and the coesponding volatility (half of the length of the bas). We can also calibate the paamete ˆk(G) in a way that povides moe eliable estimates. We depict the spead distibution of the seies. Figue 7 shows both the p.d.f. of the speads and its logaithm. In ou model, fom 3.3, the slope of the logaithm of the p.d.f., f /f, is equal to βk(g)/σ. This can be calculated fom the figue. In othe wods, we can estimate βk(g)/σ fom the data. In a sense, this povides an aggegate measue fo the intensity 19
with which the maket bings an off-aveage spead back to the aveage. Inteestingly, this measue can be well appoximated though a piecewise constant function, as follows. 0.9 fo G 5.3 ( f ˆ f ) (G) = βˆk(g) ˆσ = 0.00 fo 5.3 < G 0.3 0.4 fo G > 0.3 Among all piecewise-linea functions with thee pieces (constant in the middle piece), this appoximation is the one that minimizes the sum of squaes between the actual f /f and the appoximation. It fits the data well, as shown in Figue 7. Pobability density function 0.1 0.1 0.08 0.06 0.04 0.0 0 0 15 10 5 0 5 10 15 Spead G Log(p.d.f.).5 3 3.5 4 4.5 5 5.5 6 6.5 0 15 10 5 0 5 10 15 Spead G Figue 7: On the left, the shape of the p.d.f. fo the data of jet keosene is simila to what we could have expected when applying ou tading policies. On the ight, we show that the linea appoximation (dashed line) fits the logaithm of the p.d.f. well. The vetical line eflects the mean -.49 USD cents of the data. On the x-axis, we use a ange of mean plus/minus two standad deviations. Table summaizes the estimates of the model paametes fo the keosene data. Note that c AB is slightly negative, which is easonable because hee the aveage spead is negative (in ou model, the cost paamete neve appeas independent of the spead, and hence both G t, c AB, c BA can be shifted by a constant without changing any of the esults). 5. Equilibium Pedictions While the estimates ae geneated assuming that no playe has maket powe, ou model can now detemine the effect of maket powe in the jet fuel example. Fo this pupose, using the estimates fom above, we study the tades equilibium actions when thee ae n < identical playes (with costs c BA, c AB and capacities k BA = k BA tot /n, k AB = k AB tot /n). Theoem povides the value m eq at which all tades stat using thei tading capacity in equilibium, 0
Table : Paamate estimates β 1 σ.0 c BA 5.3 c AB -0.3 ktot BA 0.58 ktot AB 0.48 when k AB = k BA and c AB = c BA. In contast, hee the symmety conditions ae not satisfied. We thus detemine the equilibium values m BA, m AB numeically. Figue 8 shows the effect of the numbe of playes n given that the total capacity is fixed to k BA tot, k AB tot. As one can see, m BA and m AB ae deceasing in n. Thus, as competition becomes moe intense, each playe sets its theshold close and close to its cost. In the limit n, the playes behave as if thee was no maket powe. A side obsevation fom the figue is that, as the numbe of playes inceases, the standad deviation of the equilibium spead pocess deceases. Indeed, with fewe playes, the ange ove which thee ae no flows between A and B is wide, and hence the standad deviation highe. 1 0 1 3 4 m AB c AB c BA m BA 5 6 7 3 4 5 6 7 8 9 10 Numbe of tades n Figue 8: Sensitivity of the equilibium m BA, m AB to the numbe of playes n, fo the paametes identified in Table and = 0.1/365 (10% pe yea). Futhemoe, we can also illustate how m AB, m BA is set by a lage tade when it contols 1
a pecentage α of the total capacity: k BA = αk BA tot, k AB = αk AB tot. We conside that the est of the capacity is contolled by small tades (which use it as soon as G c BA o G c AB ). The esults ae shown in Figue 9. We obseve that the lage the shae of capacity contolled by a tade, the late the tade entes the maket, i.e., the egion whee no tade occus is lage. In paticula, as α becomes close to one, the thesholds m BA, m AB deviate fom c BA, c AB vey significantly. 10 5 m AB c AB c BA m BA 0 5 10 15 0.1 0. 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Pecentage of capacity contolled by the lage tade, α Figue 9: Values of the equilibium m BA, m AB as a function of the capacity contolled by one lage tade facing small tades, fo the paametes identified in Table and = 0.1/365 (10% pe yea). 5.3 Potential Impovements due to Policy Optimization: Compaison with Myopic Policies Ou analysis eveals that a tade with maket powe should use an optimal tading policy that deviates fom the myopic one descibed in Equation (1). The objective of this section is to shed some light on the potential pofit impovement due to optimization. We compae the pefomance of the optimal policy compaed to a myopic policy in diffeent scenaios. This compaison eveals the loss of optimality due to ignoing one s effect on futue pices. Fo this pupose, we conside the jet fuel case with two playes, whee each one has costs c BA, c AB and capacities k BA = k BA tot /, k AB = k AB tot /, with the paametes of Table and = 0.1/365 (10% pe yea). As seen in 4.1, the discounted pofits achieved by a tade that uses policy u ae denoted by J(u, G) whee G is the cuent spead level. Since these pofits depend on the actions of the
competito, we denote them hee as Π(u, v, G), to highlight the dependency on v. We denote Π (v, G) = sup u Π(u, v, G), as in Equation (7). Fo a given G 0, we can hence compae the pofits due to optimization, when a tade faces a myopic o a stategic competito. Π MM (G 0 ) = Π(u myopic, u myopic, G 0 ), whee u myopic is defined in Equation (1), is the pofit achieved by two competing myopic tades. Π SM (G 0 ) = Π (u myopic, G 0 ) = Π(u SM, u myopic, G 0 ) is the pofit achieved by a stategic tade competing against a myopic one, which esults in an optimal policy u SM. Π MS (G 0 ) = Π(u myopic, u SM, G 0 ) denotes the pofit achieved by a myopic tade against a stategic one (which uses policy u SM ). Finally, Π SS (G 0 ) = Π (u SS, G 0 ) = Π(u SS, u SS, G 0 ) is the pofit discussed in 5., when both tades ae stategic. Table 3 illustates the pofits fo G 0 set to the aveage spead found in the data. It is woth noting that although pofits depend on the value of G 0, since is small, they ae elatively insensitive to G 0 and ae appoximately equal to the long-un aveage pofit achieved with the optimal policy. Table 3: Pofits Π ij achieved fo diffeent tade types, G 0 =.49 cents (the elative values compaed to Π MM (G 0 ) ae shown in backets). Competito Tade M S M 89 36 (100%) (103%) S 556 57 (110%) (11%) Fom the table we obseve that a tade is always bette off using optimization. To genealize this finding, and noting that the pofit incease depends on the cuent value of G 0, we depict in Figue 10 the atios ΠMS ΠSM 1, 1 and ΠSS 1, which show pecentage pofit ΠMM ΠMM ΠMM inceases compaed to the scenaio whee both tades ae myopic (whee the lowest pofit is achieved). Fist, obseve that Π MS > Π MM, which implies that a myopic tade pefes a stategic competito, since in that case competition is less intense (as the speed at which the speads go back to the aveage is smalle). Second, Π SM > Π SS : a tade that uses optimization pefes facing a myopic competito. This seems supising. On the one hand, a myopic competito stats tading as soon as speads ae lage than costs and hence the pobability of lage speads is educed. On the othe hand, the egion whee the tade is inactive is wide, which implies that the pobability of lage speads is inceased. Oveall, the 3
pobability of tading is educed, but the spead distibution has fatte tails. Compaed to the case whee the competito is stategic, pofits ae lage when the competito is myopic. Finally, note that the atios become smalle when the spead is well above/beyond the aveage. This is tue because, when the initial spead G 0 is lage, the initial pofits captued fom tading ae identical egadless of whethe the tade o its competitos ae stategic o myopic. This occus until spead eaches a level whee the policy used by the tade and its competito depends on whethe they ae stategic o myopic, at which point the pofits become diffeent. Hence, the elative diffeence in pofits is educed. 15% Π SM /Π MM 1 Π SS /Π MM 1 Π MS /Π MM 1 10% 5% 0 50 0 50 Spead G 0 Figue 10: Π MS ΠSM 1, 1 and ΠSS ΠMM ΠMM Π MM 1 fo diffeent values of G 0. Futhemoe, fom the figue we also obseve that ΠSM ΠSS > > 1. Thus, the impovement in pofits due to optimization is highe when the competito is myopic. Note that the ΠMM ΠMS last atio ΠSS takes into account that the competito sets its policy depending on whethe the ΠMS tade is myopic o not (if it is, the competito sets v = u SM, othewise it uses v = u SS ). Hence, this suggests that the impact of optimization is highe the less sophisticated the competitos. This obsevation would also imply that when a lage tade faces many vey small competitos (in which case these would use at optimality the myopic policy), the potential of optimization is highest. 4
6 Extensions 6.1 Altenative Pice Pocesses and Cost Stuctues One of the main advantages of ou model is that we can ecast a poblem with two stochastic pocesses, Pt A and Pt B, into a poblem with a single stochastic pocess G t = Pt B Pt A. This can be done unde two main assumptions: that dg t can be expessed only as a function of G t, fo instance, that the actions of the competitos v t only depend on G t, not on (Pt A, Pt B ); and that the cost function of the tade is the poduct of a constant times of the quantity taded, which implies that the cost pe unit is independent of (Pt A, Pt B ). Ou assumptions ae thus invalidated when one consides geneal cost functions found in the liteatue, fo example those including shipping losses, e.g., Secomandi (009b). Unfotunately, when any of these two assumptions is not satisfied, then the appoach developed in the pape becomes significantly moe complicated. We can nevetheless povide guidelines on how to deal with these moe geneal models. The geneic fomulation fo the pice pocesses, eplacing Equation (), is: [ ( )] dpt A = α A (Pt A, Pt B ) + β A (Pt A, Pt B ) u t + v t (Pt A, Pt B ) dt + σ A (Pt A, Pt B )dwt A [ ( )] dpt B = α B (Pt A, Pt B ) β B (Pt A, Pt B ) u t + v t (Pt A, Pt B ) dt + σ B (Pt A, Pt B )dwt B (11) The objective function thus becomes [ { )} ] J(u, G 0 ) := E (Pt B Pt A )u t c (Pt A, Pt B, u t e t dt, 0 whee Pt A, Pt B evolve following Equation (11) and u t = u(pt A, Pt B ). In this case, the analysis becomes two-dimensional and the solution cannot be easily computed. One paticula case of this extended fomulation is the geometic Bownian motion, whee α i = 0, β i = β i P i t, and σ i = σ i P i t, with β i, σ i constants. Figue 11 shows the optimal policy fo this case with a linea cost. We obseve that the bang-bang natue of the policy is peseved, but the theshold does not depend on G t = P B t P A t but moe geneally on (P A t, P B t ). Anothe inteesting example is when pices in each maket ae mean-eveting. Ou model aleady exhibits mean evesion ceated by the tading inteaction between the makets, but moe geneally thee may be extenal easons why the pice in each maket sepaately evets to a local aveage. To illustate this effect, we conside α i = κ i (ᾱ i ln ( P i t ) ) P i t, β i = β i P i t, and σ i = σ i P i t. Figue 1 shows how the conclusions fom ou oiginal model ae alteed. As befoe, 5
4 Slope < 1: the theshold does not depend only on P t B Pt A 3.5 u(p t A,Pt B )=q P t B 3.5 P t B Pt A =c P t B Pt A = c u(p t A,Pt B )=0 1.5 u(p t A,Pt B )= q 1 1 1.5.5 3 3.5 4 A P t Figue 11: Optimal policy fo the geometic Bownian motion, with β i = 1, σ i = 0.. Costs ae assumed to ( ) be linea, i.e., c Pt A, Pt B, u = c u with c = 1, and the capacity equal to k AB = k BA = 1. The competito is assumed to tade v t (Pt A, Pt B ) = 1 if Pt B Pt A, v t (Pt A, Pt B ) = 1 if Pt B Pt A and zeo othewise. The inteest ate is set to = 10%. the bang-bang natue of the policy is peseved, but again the theshold does not depend on G t = Pt B Pt A but moe geneally on (Pt A, Pt B ). Compaed to the geometic Bownian motion scenaio fom Figue 11, with mean evesion, even without competitos, a tade tends to stat tading ealie, i.e., the egion whee it is optimal not to tade at all is educed. This is tue because mean evesion in each maket educes the pobability of lage speads (in the figue, the median spead is e 1 e 0.8 = 0.49). 6. Tading in a Netwok The model developed in this pape consides tading between two pe-detemined locations. Howeve, in eality tading oppotunities ae possible between many potential oigins and destinations. Fo this pupose, in a netwok of n points (a connected gaph), one can define the tading quantities between i and j as u ij and v ij, the existing tading capacity as k ij and the tading unit cost c ij. We must also edefine the pice pocess evolution fom Equation () as 6
4 3.5 3 u(p t A,Pt B )=q P t B Pt A =c P t B.5 u(p t A,Pt B )=0 P t B Pt A = c 1.5 u(p t A,Pt B )= q 1 1 1.5.5 3 3.5 4 A P t Figue 1: Optimal policy fo a scenaio with sepaate mean evesion in each maket, with κ i = 0.5, ᾱ A = 0.8, ᾱ B = 1 and β ( ) i = 1, σ i = 0.. Costs ae assumed to be linea, i.e., c Pt A, Pt B, u = c u with c = 1, and the capacity equal to k AB = k BA = 1. The competito does not tade, i.e., v = 0. The inteest ate is set to = 10%. dp i t = α(p t ) + j i β ij (u ij t + v ij t ) β ji (u ji t + v ji j i t ) dt + σ i dw i t fo all i {1,..., n} (1) whee P t = (Pt 1,..., Pt n ), and Wt i ae (possibly coelated) Wiene pocesses. With this new fomulation, simila to 6.1, it is impossible to tansfom the poblem into a single-dimension state space (the pice spead). The analysis consequently becomes intactable, and numeical methods must be used. When costs ae linea, the bang-bang natue of the tading policy seems to be peseved (numeically). Howeve, the egions in which u ij = 0 o k ij ae now defined as n-dimensional sets that ae no longe simple to chaacteize. 7 Conclusions The model pesented in this pape analyzes the optimal tading stategy fo a capacitated playe between makets. We have focused on two makets only, A and B, whee pices evolve following a coelated andom walk and ae influenced by the tade s actions. Unde this 7
assumption, we show that it is possible to tansfom the tade s decision, based on both pices (Pt A, Pt B ), into one that only depends on the pice spead G t = Pt B Pt A. We chaacteize the optimal tading policy which is descibed by two thesholds m BA m AB such that it is optimal to tade as much as possible (at capacity) fom B to A when G t m BA, do nothing when m BA < G t m AB, and again tade as much as possible, fom A to B this time, when G t > m AB. We know that this stuctue is optimal when the tade has no impact on futue pices. We show hee that the stuctue emains the same even when the tade moves the maket. Inteestingly, the thesholds ae vey close to the tading costs c BA, c AB when the tading capacity is small, confiming that when the tade is small in the maket, then it can safely assume that it has no maket powe (although its tading volume has an impact on the long-un distibution of pices). On the othe hand, when the tading capacity is lage, the thesholds can significantly deviate fom the tading costs. We also examine the equilibium situation whee a few tades detemine thei tading policies in esponse to thei competitos stategies, and chaacteize the equilibium thesholds in a symmetic setting. The model is used in a case study on keosene speads between Los Angeles and New Yok. We detemine how the optimal policy depends on how many playes with maket powe ae pesent and how much capacity they contol. We find that, with the paametes estimated fom the data, the impovement due to optimization is lage when the competitos ae myopic athe than when they also use thei maket powe. This suggests that a tade that uses ou model to optimize its tading policies can incease its pofits significantly (aound 10% with the paametes used in ou study). Of couse, this assessment is based solely on ou model, and an expeimental validation would be necessay, which goes beyond the scope of the cuent pape. Moe geneally, it would be quite inteesting to test empiically the validity of the model pesented hee, in the line of De Vany and Walls (1996). Such a study would establish the elationship between the shipping capacity and the speed at which speads etun to zeo, and would equie complete data on pice evolution in diffeent makets, the cost stuctue and the capacity available to all the tades. This constitutes a pomising line of futue eseach. Thee ae also othe extensions of this pape that can be consideed. We discuss some of them in 6, when the pice pocesses follow geneal evolution dynamics o thee ae moe than two makets and the tade opeates in a netwok. One othe possible extension is to integate the inventoy aspect of the tade. We have consideed a shipping system whee the quantity bought at A is equal to the one sold at B. This is a fai assumption if we focus on a pipeline. Howeve, if the shipping system is a seial netwok of waehouses, then one could conside the possibility of stoing inventoy somewhee in that netwok. The analysis becomes significantly moe complex. 8
Finally, we have intepeted the makets A and B as locations whee an identical commodity is taded. The tading capacity epesents the shipping capacity pe peiod available, such as a pipeline, a system of tucks, etc. Inteestingly, thee is an altenative intepetation of ou esults: A and B could epesent two diffeent commodities piced in the same location. Thee, the tading capacity would epesent the tansfomation capacity pe peiod. Conside fo example Pt A being the pice of natual gas, and Pt B the pice of electicity, in the same maket. A powe geneation company could conside its geneation capacity as tading capacity fom A to B. Ou model would dictate the geneation policy fo this company, and the distibution of the pice spead (called the spak spead) could be found. Thus, ou model is able to descibe analytically what the tade/geneato should do. In that espect, ou esults complement Routledge et al. (1998). Acknowledgments The authos would like to thank the depatment edito, an associate edito and an anonymous efeee fo thei comments and suggestions which helped us impove significantly this manuscipt. Refeences [1] Basak S. 1997. Consumption Choice and Asset Picing with a non-pice-taking Agent. Economic Theoy, 10, pp. 437-46. [] Betsekas D. P. 000. Dynamic Pogamming and Optimal Contol. Athena Scientific, Belmont, Massachusetts. [3] Beling P. and V. Matínez-de-Albéniz 009. Optimal Inventoy Policies when Puchase Pice and Demand ae Stochastic. Fothcoming in Opeations Reseach. [4] Beling P. and K. Rosling 005. The Effects of Financial Risks on Inventoy Policy. Management Science, 51(1), pp. 1804-1815. [5] Boyle P. P 1989. The Quality Option and Timing Option in Futues Contacts.The Jounal of Finance, XLIV(1), pp. 101-113. [6] Caldentey R., R. Epstein and D. Saué 007. Optimal Explotation of Nonenewable Resouce. Woking pape, Sten School, New Yok Univesity. [7] Chance D.M and M.L. Hemle 1993. The Impact of Delivey Options on Futues Pices: A Suvey.The Jounal of Futues Makets, 13(), pp. 101-113. 9
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Online Appendix: Poofs Poposition 1 Poof. In steady-state, it is not necessay to conside the tansients solutions of the Kolmogoov equation, that involve f. We conside the backwad poblem: the pobability that t the spead is G at time t + dt, denoted as f(g, t + dt), depends only on whee the spead was at time t, as we deal with a Makovian pocess. So we have [ f(g, t + dt) = E f(g B(G)dt A(G) ] dtɛ, t) whee ɛ = 1 with pobability 1/ and 1 othewise. When the p.d.f. is stationay, a second ode Taylo appoximation yields f(g) = f(g, t + dt) = f(g) f G B(G)dt + 1 f G A(G) dt. This yields Equation (4). See Wilmott (1997) fo details on the methodology. Poposition Poof. The poposition is an adaptation of Popositions 3..1 and 3.3.1 in Betsekas (000). A heuistic deivation is the following. Unde the optimal policy, letting u = u (G 0 ) and v = v(g 0 ), consideing a small time incement, we must have J (G 0 ) = ( G 0 u c BA u c AB u +) + (1 )EJ ( G 0 β(u + v) + σ ) ɛ whee ɛ = 1 with pobability 1/ and 1 othewise. A Taylo expansion of the equality above yields J (G 0 ) = J (G 0 )+ [G 0 u c BA u c AB u + J (G 0 ) β(u + v) dj dg (G 0) + σ whee o( ) 0 as = 0. Hence G 0 u c BA u c AB u + J (G 0 ) β(u + v) dj dg (G 0) + σ d J dg (G 0) = 0. d J ] dg (G 0) +o( ) Theoem 1 Poof. We stat the poof by descibing the value of the objective J (G) fo G, denoted hee J to simplify notation. Conside G lage enough ove which v(g) = v > 0 constant 33
(this is tue if v is piecewise constant with a finite numbe of pieces). Fo G = G 0 >> G the spead tends to evet to smalle values. Thus the tade is bette off shipping maximum volume fom A to B to avoid losing a potential pofit that will be gone anyway. As a esult, the value of J can be computed as the net pesent value of the pofits: J(G 0 ) = E 0 [ ( )] e t G t u(g t ) c u(g t ) dt 0 e t k AB ( EG t c AB) dt J AB (G 0 ) whee J AB (G 0 ) := kab G 0 βkab (k AB + v ) because EG t = G 0 β(k AB + v )t. Thus, J is asymptotically linea at +. The same holds tue at. Conside now the solution to Equation (8). This is a second-ode homogeneous diffeential equation. Its solution is continuously diffeentiable eveywhee. In addition, J is infinitely diffeentiable at points G whee u and v emain fixed. The solution to the HJB equation has two degees of feedom, which will detemined by the bode conditions, i.e., the fact that J is asymptotically linea at G = +,. We also know that J(G) 0 fo all G. ) We define fo all G, φ AB (G) = J(G) kab (G c AB ) βk (k AB AB + v(g), piecewise linea with jumps up (at points whee v jumps up). Similaly φ BA (G) = J(G) ) kba (G + c BA ) βk (k BA BA v(g), piecewise linea with jumps down. Finally φ 0 (G) = J(G). Fom Equation (8) we know that if u(g) = 0, if u(g) = k AB, and if u(g) = k BA, σ σ σ d φ AB dg d φ BA dg d φ 0 dg dφ0 = βv dg + φ0 ; = β(k AB + v) dφab dg + φab ; = β( k BA + v) dφba dg + φba. The asymptotic condition descibed above can be expessed as lim G + φab (G) = 0 and lim G φba (G) = 0. We focus on G 0 (when G 0 the analysis is analogous). We pove that thee exists m AB such that once u(m AB ) = k AB, any G m AB also dictates u(g) = k AB. We fist show that when u(g) = 0 then d J dg > 0. Then we show that if we each a point whee u(g) = kab then u stays equal to k AB fo lage values of G. 34
Take G 0 such that at optimality u(g 0 ) = 0 (do nothing), i.e., J(G) = φ 0 (G). Suppose that d J dg (G 0) 0. This implies that dj dg (G 0) 0 since v(g) 0 fo G 0. Then, σ d 3 J dg 3 = βv d J dg + dj dg 0. Thus when J becomes concave at G 0, it stays deceasing and concave fo G G 0 while u(g) = 0, because the jumps up v(g) can only make d J dg moe negative. Eventually, since G β dj dg inceases, thee is G 1 such that u(g 1 ) = k AB. At this point J(G) = φ AB (G) + ) kab G βk (k AB AB + v(g) and σ d φ AB dg By continuity of dj dg and d J dg at G 1 (because max u is deceasing and concave at G 1. Since = β(k AB + v) dφab dg + φab ; { Gu c(u) βu dj dg } is continuous), φ AB σ d 3 φ AB dg 3 = β(k AB + v) d φ AB dg + dφab dg 0, φ AB stays deceasing and concave and u(g) = k AB fo G > G 1. Eventually J(G) < 0 which is a contadiction. Hence, when u(g 0 ) = 0 then d J dg (G 0) > 0. Let G 1 be the lowest G G 0 such that u(g 1 ) = k AB. At this point and fo G > G 1, povided that u(g) = k AB, G 1 β dj dg (G 1) = c AB and 1 β d J dg (G 1) > 0. σ d φ AB dg ( ) dφ = β k AB AB + v dg + φab. At G = G 1 by continuity d φ AB dg (G 1) 0. Suppose that σ d 3 φ AB ( dg 3 (G 1) = β k AB + v ) d φ AB dg (G 1) + dφab dg (G 1) 0. Then φ AB would become moe and moe convex, and hence at G + φ AB (G) cannot convege to zeo. This esults in a contadiction. Thus d3 φ AB dg 3 (G 1) < 0 and as a esult dφ AB dg (G 1) < 0 since d φ AB dg (G 1) 0. This also implies that φ AB (G 1 ) > 0. 35
We thus know that at G = G 1, φ AB is positive, deceasing, convex but less and less convex. The same agument applies as G inceases, even when v(g) jumps up. Afte a jump, φ AB has a jump up, dφab dg is continuous and as a esult d φ AB dg has a jump down. The second deivative cannot go negative since in that case J(G) < 0 eventually. Hence G β dj is inceasing at a dg ate 1 β d J dg, which was lage than zeo at G = G 1 and thus emains lage than zeo fo G > G 1. This shows that fo G G 1, u(g) = k AB, and thus the existence of m AB > 0. Fo G 0, the same agument applies: stating fom zeo, we can find the fist G < 0 such that u(g ) = k BA, below which the policy u emains the same. Hence, thee is m BA < 0 below which u(g) = k BA. Clealy, since at + u(g) = k AB and at u(g) = k BA, since G β dj is continuous in G, in between these thesholds u(g) = 0. dg Poposition 3 Poof. In the pevious poof, we established that the thesholds ae pecisely detemined by Equation (10). In addition, we also know that fo G m AB, φ AB is deceasing and convex. This implies that dj dg = dφab dg + kab is non-deceasing. Futhemoe dj dg kab when G. Hence, Similaly, m AB c AB = β dj m BA + c BA = β dj dg (mab ) βkab dg (mba ) βkba In addition, m AB c AB and m BA c BA. Indeed, the pofit is inceasing when the spead is lage than c AB since highe shot-tem pofit can be obtained with highe speads; similaly, the pofit is deceasing when the spead is below c BA, i.e. pofit is highe when the negative spead is lage... Coollay 1 Poof. Letting k AB, k BA 0 in Poposition 3 yields the esult. Poposition 4 Poof. When the cost and capacities ae symmetic, it is immediate to obseve that J ( G) = J (G), since if a function J(G) satisfies the optimality HJB equation, then J( G) also does. 36
Poof of Theoem Poof. When costs and capacities in both diections ae the same, J( G) = J(G) so m = m AB = m BA. Thus dj (0) = 0. Fo G m, dg J(G) = q ( q ) (G c) βn + A e νg (nβq) fo some paamete A 0, whee ν = + σ nβq σ. Fo 0 G m, ( J(G) = A m e θg e θg) fo some paametes A m, whee θ = /σ. Note that θ ν. To guaantee that J is continuously diffeentiable at m, we have that Thus and hence q ( q ) (m c) βn + A e νm = A m (e θm + e θm) q ( νa e νm = θa m e θm e θm) ( ν q (m c) βn ( q ) ) + q A m = (θ + ν)e θm (θ ν)e θm βθ ( e θm e θm) [ ( ν q (m c) βn ( q ) ) ] + q m c = (θ + ν)e θm (θ ν)e θm. Letting = m c, this expession can be solved into [ (θ + ν)e θm (θ ν)e θm] ( = βθ e θm e θm) ( νq νβnq + q ) o in othe wods [( θ + ν βθνq ) ( e θm θ ν βθνq ) ] e θm = βθ (e θm e θm) ( νβnq + q ) o m c = ( which has a unique solution m. βθ ( e θm e θm) ( νβnq ) ( e θm θ + ν βθνq + q θ ν βθνq ) ) e θm 37