No-Arbitrage Commodity Option Pricing with Market Manipulation
NNo–Arbitrage Commodity Option Pricingwith Market Manipulation
René Aïd ∗ Giorgia Callegaro † Luciano Campi ‡ March 4, 2020
Abstract
We design three continuous–time models in finite horizon of a commodity price,whose dynamics can be affected by the actions of a representative risk–neutralproducer and a representative risk–neutral trader. Depending on the model, theproducer can control the drift and/or the volatility of the price whereas the tradercan at most affect the volatility. The producer can affect the volatility in two ways:either by randomizing her production rate or, as the trader, using other meanssuch as spreading false information. Moreover, the producer contracts at time zeroa fixed position in a European convex derivative with the trader. The trader canbe price-taker, as in the first two models, or she can also affect the volatility of thecommodity price, as in the third model.We solve all three models semi–explicitly and give closed–form expressions ofthe derivative price over a small time horizon, preventing arbitrage opportunitiesto arise. We find that when the trader is price-taker, the producer can alwayscompensate the loss in expected production profit generated by an increase ofvolatility by a gain in the derivative position by driving the price at maturity toa suitable level. Finally, in case the trader is active, the model takes the form ofa nonzero-sum linear-quadratic stochastic differential game and we find that whenthe production rate is already at its optimal stationary level, there is an amount ofderivative position that makes both players better off when entering the game.
Keywords: price manipulation, fair game option pricing, martingale optimalityprinciple, linear-quadratic stochastic differential games. ∗ Université Paris-Dauphine, PSL Research University, LEDa. Email: [email protected]. † Università degli Studi di Padova, Dipartimento di Matematica, Via Trieste 63, I-35121 Padova,Italy. Email: [email protected]. ‡ London School of Economics, Department of Statistics, Columbia House, 10 Houghton Street,WC2A 2AE, London, United Kingdom. Email: [email protected]. a r X i v : . [ q -f i n . M F ] M a r Introduction
The methods and techniques of manipulation are limited only by the ingenuity of man ,in Cargill vs Hardin, US Court of Appeal, 8 th circuit, Dec 7, 1971.Price manipulation in financial markets is not the rare event as one may think. Intheir paper, Aggarwal and Wu [1] provide data on more than 140 cases of market stockprice manipulation in the sole period of ten years from 1990 to 2001 released by theSecurity Exchange Commission. As the authors quote, those cases only correspond tothose who were caught. On commodity markets, illegal practices of market manipu-lations are abundantly documented and can compete with stock markets (see Pirrong[20] for a survey of those practices). More recently, the LIBOR itself was the object ofa coordinated manipulation by a cartel of banks. The LIBOR, created more than fiftyyears ago and controlled by the British Bankers Association, serves as a benchmark forloans and as an index in hundreds of trillions of nominal in derivatives. It is enoughto read Duffie and Stein [7] to measure the extent of social welfare loss induced bythe manipulators actions. In their report for the Federal Reserve Bank of New Yorkon the LIBOR scandal, Hou and Skeie [14] explain that if the first motivation for thismanipulation in the aftermath of the 2008 financial crisis was to maintain a signal ofcredit worthiness, the second motivation was the express intent of benefiting the bank’sderivatives positions .Indeed, if the first generation of market price manipulation concentrated on usingsome market power to increase the market price and then resell the good at a higherprice (unravelling strategy), it seems that the second generation of market manipulationwill use the leverage effect provided by derivatives. Worrying enough to support thisprognosis, the recent paper of Griffin and Shams [12] asserts the possibility of an on–going VIX manipulation using large position in the out–of–the money options used tocompute the VIX. If true, it would mean that some traders are already engaged in whatwas thirty years ago a theoretical problem in derivative pricing when academics wouldrelax the hypothesis of no–market impact in the Black & Scholes pricing framework (seeJarrow [15] for a seminal work on this subject).In this paper, we take market price manipulation models one step further in consid-ering the possibility of the joint control of the average (the drift) and of the volatilityof a commodity price by the actions of a producer and a trader who exchange a deriva-tive. To analyse the behaviours of both players and the distortion of the prices of thecommodity and of the derivative, we design three continuous–time models of increasingcomplexity. In each model, the commodity price is impacted by the actions of a rep-resentative risk–neutral producer and a representative risk–neutral trader. Both agentswant to maximise their respective expected profits. The representative producer hasmarket power and can increase or decrease the price by reducing or increasing her pro-duction rate. Actions on the volatility can be performed either by randomizing theproduction rate or by spreading false information. Production randomization is justmaking a strategic use of outages and the question answered in this paper is when thisdevice has an interest for the producer. Further, regarding the use of information on2he volatility, we suppose that the trader or the producer has identified some channelsallowing her to act on the nominal volatility of the underlying by an appropriate rate of information. For both agents, manipulation of the commodity price comes at somecosts, which are included in their profit functions.We consider first the case of production–based manipulation: the producer actsalone and can impact both the average price and its volatility by changing her averageproduction rate and the volatility of her production rate. Second, we consider the case ofproduction and information–based manipulation by a producer: the producer acts alone,she can affect the average price by changing her production rate and the volatility asmentioned above, e.g. by spreading appropriate information. Finally, we consider thecase of a competition between a producer who can exert market power on the drift ofthe price and a trader who has an impact on the volatility of the price. In each case, wesuppose that the producer contracts at time zero a constant (long or short) position of aEuropean convex derivative and delivers (or receives) its payoff at maturity. The traderhas the opposite position in the derivative. Since they have an asymmetric impact onthe dynamics of the commodity price, we are able to assess which instrument is moreefficient, manipulation of the drift or manipulation of the volatility. We aim at studyingto which extent the producer and the trader can profit from their market power andhow the prices of the commodity and its derivatives can be distorted by their actions.In the classification of market manipulation provided by Allen and Gorton [3], thefirst model corresponds to an action–based manipulation (using physical means suchas production); the second model is a mixture of action–based and information–based (spreading false rumours on commodity scarcity or accounting and earnings’ manipu-lation); the last one is a mixture of action–based, information–based and trading–based (buying to increase the price and then selling back).To our knowledge, this is the first paper to study the joint manipulation of a com-modity price and a derivative. We give now some reasons why our analysis might geteven more relevant in a near future.First, the commodity business has gone through a concentration trend in the lastdecades with the emergence of major players like Glencore/Xstrata, Rio Tinto or BHPBillinton. A small amount of international firms concentrate in their hands a significantvolume of minerals production. For instance, Glencore concentrate 60% of the zinc,50% of the copper, 45 % of the lead and 38% of the aluminium. At the same time, theyhave to take significant position in the financial markets to hedge their big physicalpositions. For instance, Glencore [10, note 28, p. 201] shows a position of $3.2 billionof commodity related contracts including futures, options, swaps and physical forwardscompared to an adjusted EBITDA of $15.8 billion or a total asset value of $128 billion.Rio Tinto [21, notes p. 193] presents an exposition in nominal value of derivatives inaluminum of $1.786 billion for an EBITDA for aluminum of $3.1 billion and operatingasset value of $16.5 billion. Players of this size cannot ignore that the impact they mayhave on the price of a commodity will affect the value of their portfolio derivatives too.Second, large commodity firms are not the only big players in financial markets tohold significant positions in commodity derivatives. With the financialization of com-3odity markets, large hedge funds, banks and institutional players have increased theirposition in commodity derivatives (see Cheng and Xiong [5] for an overview). Thus,when trying to move the price at her own advantage, a producer may find some oppo-sition from financial actors harmed by her action. This problem is already documentedin the case of large position of derivatives exchanged between financial institution (seethe case of Merrill Lynch selling $500 million of knock–in put options to Leiter’s Inter-national in Gallmeyer and Seppi [11]).Third, the activities of trading in commodity firms are in general isolated in a sub-sidiary because they fall within the scope of financial regulation. Thus, the tradingactivity might end up in conflict with the production activity regarding the use of mar-ket power.Our main results can be summarized as follows. For each model we provide closed–form solutions in terms of a coupled Riccati systems of ordinary differential equations.In each model, the price of the derivative is a fair market price in the sense that it isconsistent with no–arbitrage condition. First, we find that in all models, the optimalproduction rate of the producer follows the same pattern: during a transitory phase,it reaches the production rate that maximises the profit rate, then it stays there andat maturity, the production rate increases (resp. decreases) in case of short position(resp. long position). In the case of production–based manipulation, it is optimal forthe producer to increase the volatility of the production rate to induce an increase in thevolatility of the derivative if and only if he has a short position in the derivative exceedinga given threshold. Without derivative position, the value function of the producer isa non–increasing function of the volatility. Thus the producer prefers to reduce thevolatility. But, when she holds a sufficiently large short position in the derivative, theincrease in volatility that pushes the price of the derivative up can compensate theinduced indirect cost of volatility. Since her impact on the average price is significant,the producer can compensate the loss in expected profit from production due to anincrease of volatility by an increased profit from the derivative position. When theproducer action on the volatility is information–based, the previous observations stillhold, except that now her value function is increasing in the volatility, providing strongincentive to raise the volatility even without derivative position. Thus, it results that ifthe producer can impact the price to increase her profit in her derivative position, shedoes it.What happens if the producer manipulation can be challenged by a trader takingan opposite position in the derivative? We find that the actions of the trader on thevolatility only reduces the potential profit made by the producer on the derivative.Further, despite the asymmetry of powers of the two players, we find that when theproduction rate is already at its optimal stationary level, there is an amount of derivativeposition that makes both players better off entering the game.There is a considerable financial economics literature about market manipulation,which follows in particular a game theoretic approach. A short review of the portion ofsuch a literature related to stock markets has to start with the seminal work of Kyle [16]and the work of Allen and Gale [2] who provide a simple information condition under4hich an uninformed trader can make a profitable unravelling strategy (buying thestock, making the price rise and sell the stocks at an average higher price). Chatterjeaand Jarrow [4] provide a game theoretic model of US Treasury Securities manipulation.Cooper and Donaldson [6] design a dynamic game theoretic model of corner strategy.Regarding commodity price manipulation strategy, thorough analysis are available inthe works of Pirrong [18, 19, 20].It is also worth mentioning that our modelling and contribution are different fromthose in the rich literature on market impact, for which we refer the reader to, e.g., therecent book by Guéant [13] and the references therein. Indeed, while in market impactmodels the drift of the market price is affected by the traders as a consequence of anoptimal execution of a market order, in our setting both drift and volatility are affectedand the impact comes directly from market manipulation. Moreover, the modelledfinancial phenomena are different and so are the problems solved (optimal execution vsprofit maximization).The closest work to ours is the paper by Nyström and Parviainen [17]. The authorsprovide a zero–sum game between two players who can control the drifts and the volatil-ities of a multidimensional stock market and show under mild conditions that the gamehas a value and that it is given by the unique viscosity solution of degenerate parabolicPDE. Considering a more specific model of actors and impact functions, we are able toprovide more insights in the gains of the producer and the trader and the distortion ofprices.The paper is organised in the following way. Sections 2, 3 and 4 present respectivelythe model of manipulation through production, manipulation through information andcompetition of manipulation. Section 5 provides numerical illustration of the threemodels. This section contains all our results on the first model of a producer of a commodity,who can manipulate the price of the commodity through production.More in detail, we consider a producer whose objective is to maximise her profitfrom production and from investment in a financial market over the time period [0 , T ] for some T > . The producer can increase her production rate q t with the instantaneouscontrol rate u t at the expense of a cost κ u with κ > . We suppose that the productionis entirely sold at a market price ˜ S , which can be affected by the producer: the morethe production rate the less the market price. This effect leads to observed market price ˜ S t := s − a q t where a > is some fixed parameter and s > is the market price beforeaction of the producer (which in this case is constant). We will relax the hypothesis ofa constant market price before impact in the second model (see Section 3). The formerrelation can also be seen as an inverse demand function of the good, where a is itselasticity. Thus, the instantaneous profit rate is P t := q t ˜ S t .We suppose that the production rate q t is affected by a random factor that gathersall the randomness that usually affects production processes (outages, strikes and so5n). We suppose that, without intervention of the producer, uncertainty is normallydistributed with standard deviation σ > . Further, we suppose that the producerhas an effect on the uncertainty of his production rate. These hypotheses lead to thefollowing dynamics for the production rate: dq t = u t dt + σ √ z t dW t , q ∈ R , (2.1)where W is a standard Brownian motion, defined on some probability space (Ω , F , P ) ,and z t is the effect (in percentage) on the variance of the production rate. The in-formation available to the producer is modelled by the natural filtration, ( F t ) t ∈ [0 ,T ] =( F Wt ) t ∈ [0 ,T ] , generated by the Brownian motion W and completed with all P -null sets.Hence, anywhere in this section adaptedness will always be referred to this filtration.We suppose that controlling z t requires some financial cost g z with g > . Theproducer can choose either to decrease or to increase the volatility of the production rate q t . Although the costs incurred to increase the volatility are less easy to grasp than thecost involved to decrease it, they can be interpreted as the costs of the actions neededto hide them.At this stage of the model description, we notice that an increase of the volatility of q t has a negative impact on the expected instantaneous profit E [ P t ] = s E [ q t ] − a E [ q t ] .In other terms, the producer is Gamma negative. Thus, he has no incentive to increasethe volatility of his production facilities.Most large commodity producers make an important use of financial market forhedging purposes. Hence, we suppose that the producer intervenes in the financial mar-ket for his production good by selling derivatives at the initial time. Since the produceris Gamma negative, a natural hedge would be to sell a Gamma positive derivative suchas a call option. Here, we suppose that the producer has a net derivative position λ which can be positive (short, sale) or negative (long, purchase) with maturity T andpayoff h T := ˜ S T . Such a quadratic payoff can be seen as a position over a portfolio ofcall options with the same maturity T and different strike prices. We denote by h theprice at time of that option, its precise definition will be given when specifying theset of admissible controls. Indeed, h is not given from the outset, as it depends on theunderlying which is in turn controlled.The aim of the producer is to maximise the following objective functional J λ ( u, z, h ) := E (cid:20)(cid:90) T (cid:16) P t − κ u t − g z t (cid:17) dt + λ (cid:0) h − h T (cid:1)(cid:21) . (2.2)Now, within the producer firm, there are two distinct departments, a production de-partment and an investment department. The former takes any production–relateddecisions, i.e. it controls u and z , while the latter is responsible for selling/buying thederivatives at a fair price and pursuing the corresponding hedging strategy. It is naturalto assume that the investment department is using the nowadays classical no-arbitragemachinery to propose a derivative’s price. The two departments are aware of the factthat their decision affects each other. In particular, the fact that the investment depart-ment uses the no-arbitrage approach to price derivatives implies the following natural6onstraints for the production side: the chosen production plan should not lead toarbitrage opportunities. We are going to incorporate this idea into the definition ofadmissible policies. After that, we will give the precise formulation of the produceroptimization problem. Definition 2.1.
We say that any pair ( u, z ) is admissible if the following properties aresatisfied:(i) ( u t , z t ) t ∈ [0 ,T ] are progressively measurable processes with values in R × ( − , ∞ ) such that E (cid:20)(cid:90) T ( u t + z t ) dt (cid:21) < ∞ , E (cid:20)(cid:90) T q t (1 + z t ) dt (cid:21) < ∞ ; (ii) there exists a unique equivalent martingale measure Q u,z for the price process ˜ S ,equivalently for the production process ( q t ) t ∈ [0 ,T ] , with h T ∈ L ( P ) ∩ L ( Q u,z ) ;(iii) there exists a real-valued progressively measurable process (∆ u,zt ) t ∈ [0 ,T ] satisfying E (cid:20)(cid:90) T (cid:0) | ∆ u,zt u t | + | ∆ u,zt | (1 + z t ) (cid:1) dt (cid:21) < ∞ , and such that the following holds Q u,z -a.s. h u,zt := E Q u,z [ h T |F t ] = E Q u,z [ h T ] + (cid:90) t ∆ u,zs dq s , for all t ∈ [0 , T ] .The set of all admissible pairs will be denoted by A . Hence h u,z = E Q u,z [ h T ] can be viewed as the price of the option h T under theproduction control ( u, z ) . Notice that such a price is clearly affected by the controlsvia the risk neutral measure in (ii). It can be interpreted as “commitment price”: afterselling the option, the producer could deviate from the implementation of the hedgingstrategy that leads to the measure Q u,z . Here we make the assumption that the producerimplements the production controls leading to precisely that measure and thus, thatprice.Notice that from q ’s dynamics we have that the measure Q u,z , whose existence ispostulated in (ii) above, is necessarily given by the following Radon-Nikodym derivative d Q u,z d P = exp (cid:26) − (cid:90) T δ t dW t − (cid:90) T δ t dt (cid:27) , δ t = u t σ √ z t . Before giving the final formulation of the producer optimization problem, we can exploitthe admissibility properties above to rewrite the objective functional (2.2) as follows J λ ( u, z, h ) = E (cid:20)(cid:90) T (cid:16) P t − κ u t − g z t − λ ∆ u,zt u t (cid:17) dt (cid:21) =: ˜ J λ ( u, z ) . h T − h = (cid:90) T ∆ u,zt dq t = (cid:90) T ∆ u,zt ( u t dt + σ √ z t dW t ) . Condition (iii) implies that the dW -part above has zero expectation under P . Moreover,we observe that the integrability assumptions in (i) and (iii) ensure that ˜ J λ ( u, z ) isfinite.Finally, after all these preliminaries, we can formulate the producer’s optimizationproblem sup ( u,z ) ∈A ˜ J λ ( u, z ) . (2.3) In this part we develop the heuristics needed to obtain a candidate for the solution ofproblem (2.2). In the next sub-section, we will verify that the candidate is indeed theoptimal solution according to the definition above.First, notice that, since the market is complete, there exists only one possible no-arbitrage price for the derivative h T , which also gives the initial wealth needed to fundthe hedging strategy. The derivative can be perfectly replicated by trading in a self-financing way in the underlying ˜ S t = s − aq t or, equivalently, in q t . Therefore h T = E Q [ h T ] + (cid:90) T ∆ t dq t , where Q is the unique equivalent martingale measure for ˜ S (or, equivalently, for q ), and ∆ is the delta hedging. More precisely, using Girsanov’s theorem we get the dynamicsof q under Q , which is dq t = σ √ z t dW Q t , dW Q t = dW t − δ t dt, where δ t = u t σ √ z t for t ∈ [0 , T ] . Hence, defining the price at time t of the derivative as h t = E Q [( s − aq T ) | q t ] := ϕ ( t, q t ) , we obtain the usual PDE for the price ϕ t + 12 σ (1 + z ( t, q )) ϕ qq = 0 , ϕ ( T, q ) = ( s − aq ) . (2.4)Finally, we have the usual relationship ∆ t = ϕ q ( t, q t ) . Remark 2.1.
Notice from Equation (2.4) that ϕ depends on z in a functional way.However we expect the optimal control to be Markovian, which justifies replacing z t (which could in principle depend on the whole path of the state variable ( q t ) ) with thefunction z ( t, q ) of time and of the value q of state variable at time t . The PDE aboveneeds to be solved together with the HJB equation for the value function, since thecoefficient of the second derivative ϕ qq depends on the control z .8he perfect replicability of the derivative allows us to rewrite the objective functionin a more suitable way as at the end of the previous sub-section. Indeed, observe firstthat h − h T = − (cid:90) T ∆ t dq t = − (cid:90) T ϕ q ( t, q t )( u t dt + σ √ z t dW t ) , where recall that W is a Brownian motion under P . Hence, given the hedging strategy ϕ q ( t, q t ) , the maximization problem on the production side can be expressed as follows v λ (0 , q ) := sup u,z E (cid:20)(cid:90) T (cid:16) q t ( s − aq t ) − g z t − κ u t − λϕ q ( t, q t ) u t (cid:17) dt (cid:21) . (2.5)In other terms, the gain coming from selling the derivative has been absorbed by therunning profit term. Therefore, we can get the HJB equation − v t = sup u,z (cid:26) q ( s − aq ) − g z − κ u − λϕ q ( t, q ) u + uv q + σ z ) v qq (cid:27) , (2.6)with terminal condition v ( T, q ) = 0 . Notice that the PDE for the price (2.4) and theHJB equation for the value function are clearly coupled as the optimal z appears inthe pricing PDE, while the derivative of the price, ϕ q , appears in the HJB equation(compare to Remark 2.1). The first order conditions give the two (candidate) optimalcontrols (cid:98) u = 1 κ ( v q − λϕ q ) , (cid:98) z = σ g v qq . (2.7)Notice that we have dropped the dependence upon λ in the value function for sake ofreadability. In order to get the full solution, it is natural to make the following Ansatz 2.1.
Both solutions ϕ and v λ are quadratic in q , i.e. ϕ ( t, q ) = A ( t ) q + B ( t ) q + C ( t ) , v λ ( t, q ) = D ( t ) q + E ( t ) q + F ( t ) , where A, B, C, D, E, F are deterministic functions of time, to be determined.
Solving for ϕ . To ease the notation, we drop the dependence of time from
A, B andso on. First of all, applying the Ansatz 2.1 to the candidate optimal controls gives (cid:98) u = 1 κ (2 q ( D − λA ) + E − λB ) , (cid:98) z = σ g D. Next, we substitute the expression above for (cid:98) u and (cid:98) z in the pricing PDE (2.4) and weobtain σ (cid:18) σ g D (cid:19) A + A (cid:48) q + B (cid:48) q + C (cid:48) = 0 , A ( T ) q + B ( T ) q + C ( T ) = s − as q + a q . In particular, the terminal condition for ϕ gives the corresponding terminal conditionsfor A, B, C as A ( T ) = a , B ( T ) = − as , C ( T ) = s .
9y identification of the terms in q , we get the following ODEs for A, B and CA (cid:48) = 0 , B (cid:48) = 0 , C (cid:48) = − σ a (cid:18) σ g D (cid:19) , which can be easily solved using the terminal conditions above. Indeed, we obtain A ( t ) = a , B ( t ) = − as , C ( t ) = s + σ a (cid:90) Tt (cid:18) σ g D ( r ) (cid:19) dr, (2.8)for all t ∈ [0 , T ] . Notice that the function D ( t ) will be obtained when solving the HJBequation (2.6). Solving for v λ . Substituting the Ansatz 2.1 for v λ (together with the optimal controls)in the HJB equation (2.6) and identifying the terms in q , we obtain the following ODEsfor D, E and F − D (cid:48) = − a + 2 κ ( D − λA ) , D ( T ) = 0 , − E (cid:48) = s + 2 κ ( D − λA )( E − λB ) , E ( T ) = 0 , − F (cid:48) = σ g D + σ D + 12 κ ( E − λB ) , F ( T ) = 0 . Now, using (2.8) implies − D (cid:48) = − a + 2 κ ( D − λa ) , (2.9) − E (cid:48) = s + 2 κ ( D − λa )( E + 2 λas ) , (2.10) − F (cid:48) = σ g D + σ D + 12 κ ( E + 2 aλs ) , (2.11)with null terminal conditions D ( T ) = E ( T ) = F ( T ) = 0 . Remark 2.2.
We observe that, while the equation for D is a one-dimensional RiccatiODE, the second one is linear and the third one can be solved just by integration. TheRiccati equation (2.9) can be easily proved to have a unique solution over the wholetime interval [0 , T ] . Indeed, this is a direct consequence of, e.g., Lemma 10.12 in [9].Moreover, that lemma also implies that D ( t ) ≤ for all t ∈ [0 , T ] ⇔ D (cid:48) ( T ) = a − λ a κ ≥ , which will be important later for the interpretation of our results. The value a − λ a κ corresponds to the slope of D close to T . 10et θ = (cid:112) a/κ . Solving the equations above gives the following expressions D ( t ) = − a − λ a κ )( e θ ( T − t ) − θ ( e θ ( T − t ) + 1) + λa κ ( e θ ( T − t ) − , (2.12) E ( t ) = s (cid:90) Tt e (cid:82) ut κ ( D ( r ) − λa ) dr (cid:20) aλκ ( D ( u ) − λa ) (cid:21) du, (2.13) F ( t ) = (cid:90) Tt (cid:18) σ g D ( u ) + σ D ( u ) + 12 κ ( E ( u ) + 2 aλs ) du (cid:19) , (2.14)for all t ∈ [0 , T ] . We conclude the section with the verification that the candidate described above isindeed a solution to problem (2.3).
Theorem 2.1.
Let
D, E be deterministic functions of time as in, respectively, (2.12) and (2.13) . Whenever H := 1 − λaκ ( λa − gσ ) > , we assume that the maturity T issmall enough, more precisely T < T max := 2 θ coth − (cid:18) σ aθg H (cid:19) . (2.15) Then there exists an optimal policy ( (cid:98) u, (cid:98) z ) ∈ A for problem (2.3) , where the productionpolicies are (cid:98) u t = 1 κ (cid:0) (cid:98) q t ( D ( t ) − λa ) + 2 as λ + E ( t ) (cid:1) , (cid:98) z t = σ g D ( t ) , t ∈ [0 , T ] . (2.16) The no-arbitrage price process for the derivative h T is given by (cid:98) h t := h (cid:98) u, (cid:98) zt = ( s − a (cid:98) q t ) + σ a (cid:90) Tt (cid:18) σ g D ( u ) (cid:19) du, t ∈ [0 , T ] , (2.17) and the hedging process is (cid:98) ∆ t := ∆ (cid:98) u, (cid:98) zt = 2 a ( a (cid:98) q t − s ) , t ∈ [0 , T ] , (2.18) where the production rate is (cid:98) q t = e R ( t ) (cid:26) q + (cid:90) t e − R ( s ) κ ( − λa + 2 as λ + E ( s )) ds + (cid:90) t e − R ( s ) σ (cid:115) σ g D ( s ) dW s (cid:41) , (2.19) with R ( t ) := (cid:82) t κ D ( s ) ds . roof. The proof is structured in two steps.1.
Admissibility . Let us verify that the pair ( (cid:98) u, (cid:98) z ) given in the statement above belongsto A . We start from condition (i) in Definition 2.1. First, (cid:98) u, (cid:98) z are trivially progressivelymeasurable and real valued. We need to check (cid:98) z t > − , which is equivalent to σ g D ( t ) > − . We distinguish two cases: if a ≤ λ a κ we have D ( t ) ≥ (this is a consequenceof Remark 2.2 or, alternatively, the explicit formula (2.12)), hence (cid:98) z t > − for all t ∈ [0 , T ] . On the other hand, if a > λ a κ , it follows from expression (2.12) that D ( t ) isnondecreasing with D ( T ) = 0 . Therefore, it suffices to check that D (0) > − g/σ , where D (0) = − a − λ a κ ) θ coth( θT ) + λa κ . After some computation, we obtain that D (0) > − g/σ if and only if coth (cid:18) θT (cid:19) > σ aθg H, where H is the constant defined in the statement. Now, if H < the inequality aboveis always satisfied as the LHS above is nonnegative. If H > , the inequality aboveis guaranteed by the condition T < T max . We can conclude that even in this secondcase, provided
T < T max , we have (cid:98) z t > − for all t ∈ [0 , T ] . Regarding the integrabilityproperties, we verify now that E (cid:20)(cid:90) T ( (cid:98) u t + (cid:98) z t ) dt (cid:21) < ∞ , E (cid:20)(cid:90) T (cid:98) q t (1 + (cid:98) z t ) dt (cid:21) < ∞ , (2.20)where (cid:98) q = q (cid:98) u, (cid:98) z . Since (cid:98) u is affine in (cid:98) q with continuous time-dependent coefficients and (cid:98) z is deterministic and continuous in t , checking the properties above boils down to show E (cid:20)(cid:90) T (cid:98) q t dt (cid:21) < ∞ . First, we use Fubini’s theorem to get E [ (cid:82) T (cid:98) q t dt ] = (cid:82) T E [ (cid:98) q t ] dt . Moreover, since (cid:98) q t is aGaussian random variable for any fixed t (see Remark 2.3 below), the function t (cid:55)→ E [ (cid:98) q t ] is continuous over [0 , T ] , so its integral is finite.Regarding condition (ii), we need to show that there exists a unique EMM (cid:98) Q = Q (cid:98) u, (cid:98) z for the production process (cid:98) q . Let us recall that d (cid:98) Q d P = exp (cid:26) − (cid:90) T (cid:98) δ t dW t − (cid:90) T (cid:98) δ t dt (cid:27) , (cid:98) δ t = (cid:98) u t σ √ (cid:98) z t . We use [23, Theorem 2.1] to prove that under our assumptions the probability (cid:98) Q iswell-defined (see also [22] for more general results of the same type). According to thatresults, we need to check Assumption 2.2 in [23], which in our case is satisfied as long12s σ (1 + (cid:98) z t ) > for all t ∈ [0 , T ] . By the same arguments used for condition (i), we getthe result. A standard application of Girsanov theorem, together with the integrabilityproperties in (2.20), yields immediately that (cid:98) q is a martingale under (cid:98) Q .To end checking condition (ii), we have to show h T ∈ L ( P ) ∩ L ( (cid:98) Q ) . Now, h T =( s − a (cid:98) q T ) , hence quadratic in (cid:98) q T . Since under both probability measures (cid:98) q T is aGaussian random variable, we have q T ∈ L ( P ) ∩ L ( (cid:98) Q ) , which gives the desired property.We pass to condition (iii) in Definition 2.1. First, (cid:98) ∆ is trivially a progressively measur-able process with real values. For the integrability property, since both (cid:98) u and (cid:98) ∆ arelinear in (cid:98) q t , we are again reduced to the square integrability E [ (cid:82) T (cid:98) q t dt ] < ∞ , which hasbeen proved just before.To conclude this part of the proof, it remains to check that, given ( (cid:98) u, (cid:98) z ) as above, (cid:98) h t := h (cid:98) u, (cid:98) zt = E (cid:98) Q [ h T |F t ] = E (cid:98) Q [ h T ] + (cid:82) t (cid:98) ∆ s dq s a.s. under (cid:98) Q , for all t ∈ [0 , T ] . This canbe done by direct computation as follows: applying Itô’s formula to (cid:98) h t in (2.17) we get d (cid:98) h t = − a ( s − a (cid:98) q t ) d (cid:98) q t whence, in integral form, (cid:98) h t = (cid:98) h − a (cid:90) t ( s − a (cid:98) q s ) d (cid:98) q s = (cid:98) h + (cid:90) t (cid:98) ∆ s d (cid:98) q s . Moreover, one easily find (cid:98) E [ h T ] = (cid:98) E [( s − a (cid:98) q T ) ] = s − as q + a (cid:98) E [ (cid:98) q T ] , where (cid:98) E denotes the expectation with respect to the measure (cid:98) Q . Using Itô’s isometry,we also have (cid:98) E [ (cid:98) q T ] = q + (cid:90) T σ (cid:18) σ g D ( t ) (cid:19) dt, which leads to the remaining property in (iii).2. Optimality . To check the optimality condition, we are going to use the martingaleoptimality principle (see [8]). Let us define the process Y u,zt := (cid:90) t (cid:16) q r ( s − aq r ) − g z r − κ u r − λ (cid:98) ∆ r u r (cid:17) dr + V ( t, q t ) , (2.21)with V ( t, q ) = D ( t ) q + E ( t ) q + F ( t ) . Thanks to the martingale optimality principle, proving that Y u,z is a supermartingalefor all ( u, z ) ∈ A and a martingale for ( u, z ) = ( (cid:98) u, (cid:98) z ) , will give us the result. Itô’sformula yields dY u,zt = (cid:20) q t ( s − aq t ) − g z t − κ u t − λ (cid:98) ∆ t u t + V t + V q u t + 12 V qq σ (1 + z t ) (cid:21) dt + V q σ √ z t dW t , (2.22)13here V t , V q , V qq denote partial derivative of the function V ( t, q ) . We have omitted thedependence on ( t, q ) for sake of simplicity. First, notice that due to the integrabilityproperties in Definition 2.1(i) the process (cid:82) t V q σ √ z r dW r is a true P -martingale.Therefore, it remains to show that the dt -part in (2.22) above is a nonincreasing process,that is it is lower or equal to zero almost everywhere. By construction (see heuristics),we know that the function V ( t, q ) satisfies the HJB equation (2.6), with ϕ q ( t, q ) = 2 a q − as , hence ϕ q ( t, q t ) = (cid:98) ∆ t for all t . We can conclude that the drift in (2.22) is lower or equal tozero a.e., yielding that Y u,z is a supermartingale for all ( u, z ) ∈ A . To show that Y (cid:98) u, (cid:98) z isa martingale, one proceeds in the same way getting equalities instead of inequalities. Inparticular one gets that the drift in (2.22) is equal to zero a.e., implying the martingaleproperty.Finally, we can solve for (cid:98) q t in explicit form, via standard resolution methods, since it isthe unique solution of the following linear SDE d (cid:98) q t = 1 κ (cid:0) (cid:98) q t ( D ( t ) − λa ) + 2 as λ + E ( t ) (cid:1) dt + σ (cid:115) σ g D ( t ) dW t , (cid:98) q = q . Remark 2.3.
Notice, from Equation (2.19), that (cid:98) q t is a Gaussian random variable withtime-dependent mean and variance. In this section we describe and solve explicitly a variant of the model presented before.We still have a producer of a commodity, who is maximizing her profit coming fromboth production and a short/long position in some derivative. The main differences arethat the market price of the commodity is no longer a constant as it is driven by theBrownian motion W , that in turn does not affect the production rate anymore and,finally, the producer can directly control the volatility of the market price (by spreadingfalse information on the state of his production, for instance). Thus, the dynamics ofthe state variables is now given by (cid:26) dS t = µ dt + σ √ z t dW t ,dq t = u t dt. (3.1)Moreover in this model the market price ˜ S is given by ˜ S t = S t − aq t , t ∈ [0 , T ] . Theobjective of the producer is the same as before, i.e. J λ ( u, z, h ) := E (cid:20)(cid:90) T (cid:16) P t − κ u t − g z t (cid:17) dt + λ (cid:0) h − h T (cid:1)(cid:21) . (3.2)Analogously to the previous model, we will be working with the following definition ofadmissible policies, which admits the same interpretation as before.14 efinition 3.1. We say that any pair ( u, z ) is admissible if the following properties aresatisfied:(i) ( u t , z t ) t ∈ [0 ,T ] are progressively measurable processes with values in R × ( − , ∞ ) such that E (cid:20)(cid:90) T ( u t + z t ) dt (cid:21) < ∞ , E (cid:20)(cid:90) T ˜ S t (1 + z t ) dt (cid:21) < ∞ ; (ii) there exists a unique equivalent martingale measure Q u,z for the price process ˜ S ,with h T ∈ L ( P ) ∩ L ( Q u,z ) ;(iii) there exists a real-valued progressively measurable process (∆ u,zt ) t ∈ [0 ,T ] satisfying E (cid:20)(cid:90) T (cid:0) | ∆ u,zt u t | + | ∆ u,zt | (1 + z t ) (cid:1) dt (cid:21) < ∞ , and such that the following holds Q u,z -a.s. h u,zt := E Q u,z [ h T |F t ] = E Q u,z [ h T ] + (cid:90) t ∆ u,zs d (cid:101) S s , for all t ∈ [0 , T ] .The set of all admissible pairs will be denoted by A . Analogously as in the previous model, from ˜ S ’s dynamics we have that the measure Q u,z , whose existence is postulated in (ii) above, is necessarily given by the followingRadon-Nikodym derivative d Q u,z d P = exp (cid:26) − (cid:90) T γ t dW t − (cid:90) T γ t dt (cid:27) , γ t = µ − au t σ √ z t . Before giving the final formulation of the producer’s optimization problem in this modelas well, we can exploit the admissibility properties above to rewrite the objective func-tional (3.2) as follows J λ ( u, z, h ) = E (cid:20)(cid:90) T (cid:16) P t − κ u t − g z t − λ ∆ u,zt ( µ − au t ) (cid:17) dt (cid:21) =: ˜ J λ ( u, z ) . Finally, the producer’s optimization problem, that we are going to solve in the nextsub-section, is given by sup ( u,z ) ∈A ˜ J λ ( u, z ) . (3.3)15 .1 Heuristics In this sub-section we describe the heuristics that led us to propose some candidatesolution. It follows the same lines as in the first model. Being the market complete,there exists a unique martingale measure Q under which (cid:101) S is a martingale. Indeed wehave d (cid:101) S t = ( µ − au t ) dt + σ √ z t dW t = ( µ − au t − γ t σ √ z t ) dt + σ √ z t dW Q t , where W Q is a Q -Brownian motion and by choosing γ t = µ − au t σ √ z t , (cid:101) S is a Q -martingale.The price at time t = 0 of the European claim h T equals E Q [ (cid:101) S T ] , so that we expect that h T = h + (cid:82) T ∆ t d (cid:101) S t , where ∆ is the corresponding delta hedging strategy. Hence, thedifference h − h T also reads h − h T = − (cid:90) T ∆ t d (cid:101) S t . (3.4)The last quantity to be introduced is the financial claim’s price, that we denote by ϕ : ϕ ( t, q, s ) := E Q (cid:104) (cid:101) S T | q t = q, S t = s (cid:105) . Remark 3.1.
Notice that the filtration generated by q and (cid:101) S is the same as the oneassociated with q and S , namely F (cid:101) St ∨F qt = F St ∨F qt for every t ∈ [0 , T ] . We convenientlychoose to consider q and S as state variables.We clearly expect ϕ to solve the following PDE (cid:40) ϕ t + σ (1 + z ( t, q, s )) ϕ ss = 0 ϕ ( T, q, s ) = ( s − aq ) . (3.5)Since we formally have ∆ := ∂ϕ∂ (cid:101) s = ∂ϕ∂s ∂s∂ (cid:101) s = ∂ϕ∂s , using the dynamics (cid:101) S under P togetherwith (3.4) we find that the value function v λ satisfies v λ (0 , q , s ) = sup u,z E q ,s (cid:20)(cid:90) T (cid:16) q t ( S t − aq t ) − g z t − κ u t − λϕ s ( t, q t , S t )( µ − au t ) (cid:17) dt (cid:21) . The value function v λ is solution to the following HJB equation, which depends on ϕ (satisfying (3.5)) − v t = sup u,z (cid:8) q ( s − aq ) − g z − κ u − λϕ s ( µ − au ) + v q u + v s µ + σ (1 + z ) v ss (cid:9) v ( T, q, s ) = 0 (3.6)
Ansatz 3.2.
We guess the value function v and the price ϕ have the following form v ( t, q, s ) = A ( t ) q + B ( t ) s + C ( t ) qs + D ( t ) q + E ( t ) s + F ( t ) ,ϕ ( t, q, s ) = ¯ A ( t ) q + ¯ B ( t ) s + ¯ C ( t ) qs + ¯ D ( t ) q + ¯ E ( t ) s + ¯ F ( t ) . (cid:98) z = σ g v ss , (cid:98) u = 1 κ ( aλϕ s + v q ) , and thus, using the ansatz above, (cid:98) z t = σ g B ( t ) (cid:98) u t = 1 k (cid:8) λa (cid:2) B ( t ) s + ¯ C ( t ) q + ¯ E ( t ) (cid:3) + 2 A ( t ) q + C ( t ) s + D ( t ) (cid:9) , (3.7)for all t ∈ [0 , T ] . Solving for ϕ . We can now explicitly find ϕ by exploiting the Ansatz 3.2 and replacingthe control pair ( (cid:98) u, (cid:98) z ) into (3.5). We find: ¯ A ( t ) ≡ a ¯ B ( t ) ≡ C ( t ) ≡ − a ¯ D ( t ) ≡ E ( t ) ≡ F ( t ) = (cid:90) Tt σ (cid:20) σ g B ( u ) (cid:21) du Solving for v λ . We proceed in the same way as above to find the value function v λ .We find the following systems of ODEs A (cid:48) ( t ) = a − κ ( A ( t ) − λa ) (3.8) B (cid:48) ( t ) = − κ (2 λa + C ( t )) (3.9) C (cid:48) ( t ) = − − κ (2 λa + C ( t ))( A ( t ) − λa ) (3.10) D (cid:48) ( t ) = − k D ( t ) (cid:0) A ( t ) − a λ (cid:1) − µ (cid:0) C ( t ) + 2 aλ (cid:1) (3.11) E (cid:48) ( t ) = − k D ( t ) (cid:0) C ( t ) + 2 aλ (cid:1) − µ ( B ( t ) − λ ) (3.12) F (cid:48) ( t ) = − σ B ( t ) g − D ( t ) κ − µE ( t ) − σ B ( t ) (3.13)with null terminal conditions A ( T ) = · · · = F ( T ) = 0 . Remark 3.3.
Notice, in particular, that B is a positive decreasing function of time,which implies (recall Equation (3.7)) that the control (cid:98) z is always positive. This meansthat there is always interest in increasing the market price volatility, even if the producerbuys the derivative. 17 .2 Verification Theorem 3.1.
Let
A, B, C, D be solutions to the system (3.8) - (3.13) . There exists anoptimal policy ( (cid:98) u, (cid:98) z ) ∈ A for problem (3.3) , where (cid:98) u t = 1 κ (cid:104) (2 λa + C ( t )) (cid:98) S t + 2( A ( t ) − λa ) (cid:98) q t + D ( t ) (cid:105) , (cid:98) z t = σ g B ( t ) , t ∈ [0 , T ] . (3.14) The no-arbitrage price process for the derivative h T is given by (cid:98) h t := h (cid:98) u, (cid:98) zt = ( (cid:98) S t − a (cid:98) q t ) + σ (cid:90) Tt (cid:18) σ g B ( u ) (cid:19) du, t ∈ [0 , T ] , (3.15) and the hedging process is (cid:98) ∆ t := ∆ (cid:98) u, (cid:98) zt = 2( (cid:98) S t − a (cid:98) q t ) , t ∈ [0 , T ] . (3.16) Proof.
The proof is very similar to the previous model (cf. Theorem 2.1), hence we givedetails only for those steps which are slightly different.1.
Admissibility.
First, we observe that checking the admissibility property (i), namely (cid:98) z t > − for all t ∈ [0 , T ] , is actually easier as no conditions on small T are needed.Indeed, (cid:98) z t > − if and only if B ( t ) > − g/σ , where B solves the corresponding equationin the system (3.8)-(3.13). The latter inequality is satisfied since, being B (cid:48) ( t ) ≤ and B ( T ) = 0 , we have B ( t ) ≥ for all t ∈ [0 , T ] .Regarding the integrability properties in (i), checking them is equivalent to showthat E (cid:20)(cid:90) T (cid:98) q t dt (cid:21) < ∞ , E (cid:20)(cid:90) T (cid:98) S t dt (cid:21) < ∞ . As for the condition on (cid:98) S , observe that under P the process (cid:98) S satisfies d (cid:98) S t = µdt + σ (cid:113) σ g B ( t ) dW t , namely it is a Gaussian process. So, as previously noticed in theproof of Theorem 2.1, we apply first of all Fubini’s theorem and then we remark thatthe function t (cid:55)→ E [ (cid:98) S t ] is continuous over [0 , T ] and its integral is finite. We now workon E [ (cid:98) q t ] , which is more delicate, since d (cid:98) q t = (cid:98) u t dt and (cid:98) u depends also on (cid:98) S (see (3.14)).We have (cid:98) q t = q + 1 κ (cid:90) t (2 λa + C ( s )) (cid:98) S s ds + 2 κ (cid:90) t ( A ( s ) − λa ) (cid:98) q s ds + 1 κ (cid:90) t D ( s ) ds, E [ (cid:98) q t ] ≤ (cid:18) q + 1 κ (cid:90) t D ( s ) ds (cid:19) + 3 κ E (cid:18)(cid:90) t (2 λa + C ( s )) (cid:98) S s ds (cid:19) + 12 κ E (cid:18)(cid:90) t ( A ( s ) − λa ) (cid:98) q s ds (cid:19) ≤ (cid:18) q + 1 κ (cid:90) t D ( s ) ds (cid:19) + 3 tκ (cid:90) t [2 λa + C ( s )] E [ (cid:98) S s ] ds (cid:124) (cid:123)(cid:122) (cid:125) := α ( t ) + 12 tκ (cid:90) t [ A ( s ) − λa ] E [ (cid:98) q s ] ds. Now, E [ (cid:98) S s ] is positive and finite and so we can safely introduce the positive continuousfunction α as above and write E [ (cid:98) q t ] ≤ α ( t ) + 12 tκ (cid:90) t [ A ( s ) − λa ] E [ (cid:98) q s ] ds. An application of Gronwall’s lemma leads to E [ (cid:98) q t ] ≤ α ( t ) + (cid:90) t α ( s ) K ( s ) e (cid:82) ts K ( u ) du ds, with K ( u ) := tκ [ A ( u ) − λa ] > . So, t → E [ (cid:98) q t ] is bounded by a continuous functionand its integral over [0 , T ] is finite.Regarding condition (ii), we proceed again as in the proof of Theorem 2.1 by checkingAssumption 2.2 in [23], which in our case is satisfied as long as σ (1 + (cid:98) z t ) > for all t ∈ [0 , T ] . This is automatically true, since here B ( t ) ≥ for all t ∈ [0 , T ] and so σ (1 + (cid:98) z t ) = σ (1 + σ g B ( t )) ≥ σ > . To end checking condition (ii), we have to show (cid:98) h T ∈ L ( P ) ∩ L ( (cid:98) Q ) . Now, (cid:98) h T = ( (cid:98) S T − a (cid:98) q T ) , hence quadratic in both (cid:98) S t and in (cid:98) q T .For what we have seen up to now q T and (cid:98) S T belong to L ( P ) and they also belong to L ( (cid:98) Q ) , which gives the desired property.It remains to check (iii). We proceed again as in the previous theorem, except thatnow (cid:98) h t in (3.15) is a function of both (cid:98) S and (cid:98) q . First, notice that (cid:98) h t = (cid:98) h + 2 (cid:90) t ( (cid:98) S s − a (cid:98) q s )( d (cid:98) S s − ad (cid:98) q s ) = (cid:98) h + (cid:90) t (cid:98) ∆ s d (cid:101) S s . Now, since (cid:98) h T = ( s − aq ) + 2 (cid:82) T ( (cid:98) S s − a (cid:98) q s ) σ √ (cid:98) z s d (cid:99) W s we have (cid:98) E [ h T |F t ] = ( s − aq ) + 2 (cid:90) t ( (cid:98) S s − a (cid:98) q s ) σ (cid:112) (cid:98) z s d (cid:99) W s +2 (cid:98) E (cid:20)(cid:90) Tt ( (cid:98) S s − a (cid:98) q s ) σ (cid:112) (cid:98) z s d (cid:99) W s (cid:21) = (cid:98) h t . (cid:98) ∆ in condition (iii) in Definition3.1 can be treated exactly as in the proof of Theorem 2.1, using (i).2. Optimality
This can be proved by proceeding exactly as in the proof of Theorem 2.1,by taking into account that now the state variable is two-dimensional.
In this section, we finally consider a game between a producer who can manipulate thecommodity price through the drift and a trader who can manipulate the volatility ofthe price. Hence, the trader is no longer price-taker or passive as in the previous twomodels. Here, she can affect the price of commodity by paying some (quadratic) cost.The dynamics for S and q are as in (3.1). The controls are still given by ( u, z ) withthe big difference that now only u is controlled by the producer, while z is controlledby the trader. Clearly, the corresponding costs are allocated accordingly. When thestrategy profile for both players is ( u, z ) and the derivative price is h , the producerpayoff is J pr ( u, z, h ) = E (cid:20)(cid:90) T (cid:16) q t ( S t − aq t ) − κ u t (cid:17) dt + λ ( h − h T ) (cid:21) . (4.1)On the other hand, the trader who has the opposite position in the derivatives will getthe payoff J tr ( u, z, h ) = E (cid:20) − g (cid:90) T z t dt − λ ( h − h T ) (cid:21) . (4.2)Now, we can give the definition of admissible policies, including the strategy profiles ofboth players together with the pricing and hedging strategy of the investment depart-ment in the production firm. Notice that it is the same as for the model in Section 3.We recall that ˜ S t = S t − aq t , for t ∈ [0 , T ] . Definition 4.1.
We say that any pair ( u, z ) is admissible if the following properties aresatisfied:(i) ( u t , z t ) t ∈ [0 ,T ] are progressively measurable processes with values in R × ( − , ∞ ) such that E (cid:20)(cid:90) T ( u t + z t ) dt (cid:21) < ∞ , E (cid:20)(cid:90) T ˜ S t (1 + z t ) dt (cid:21) < ∞ ; (ii) there exists a unique equivalent martingale measure Q u,z for the price process ˜ S ,with h T ∈ L ( P ) ∩ L ( Q u,z ) ;(iii) there exists a real-valued progressively measurable process (∆ u,zt ) t ∈ [0 ,T ] satisfying E (cid:20)(cid:90) T (cid:0) | ∆ u,zt u t | + | ∆ u,zt | (1 + z t ) (cid:1) dt (cid:21) < ∞ , nd such that the following holds Q u,z -a.s. h u,zt := E Q u,z [ h T |F t ] = E Q u,z [ h T ] + (cid:90) t ∆ u,zs dq s , for all t ∈ [0 , T ] .The set of all admissible pairs will be denoted by A . On the other hand the definition of solution is slightly different, due to the factthat the trader can also play strategically in this model. Before proceeding, we exploitthe definition of admissibility, conditions (ii) and (iii) in particular, to rewrite as in theprevious two models the payoffs of both the trader and the producer as follows J pr ( u, z, h ) = E (cid:20)(cid:90) T (cid:16) q t ( S t − aq t ) − κ u t − λ ( µ − au t )∆ u,zt (cid:17) dt (cid:21) =: ˜ J pr ( u, z ) ,J tr ( u, z, h ) = E (cid:20)(cid:90) T (cid:16) − g z t + λ ( µ − au t )∆ u,zt (cid:17) dt (cid:21) =: ˜ J tr ( u, z ) , for any admissible pair ( u, z ) ∈ A . Definition 4.2.
We say that the pair ( (cid:98) u, (cid:98) z ) ∈ A is a Nash equilibrium if it is a Nashequilibrium between the producer and the trader, i.e., ˜ J pr ( (cid:98) u, (cid:98) z ) ≥ ˜ J pr ( u, (cid:98) z ) , ˜ J tr ( (cid:98) u, (cid:98) z ) ≥ ˜ J tr ( (cid:98) u, z ) , (4.3) for all deviations u, z such that ( u, (cid:98) z ) and ( (cid:98) u, z ) belong to A . We want to compute explicitly a Nash equilibrium for the game between producer andtrader, described just above. We start from some heuristics that would lead to somecandidate equilibrium, while the rigorous verification is postponed to the next sub-section as usual.First, assuming h t = ϕ ( t, q t , S t ) and consequently ∆ t = ϕ s ( t, q t , S t ) , while exploitingthe market completeness as for the previous two models, we can rewrite the runningbest-response functions of the two players as follows v ( t, q, s ; z ) = sup u E (cid:20)(cid:90) Tt (cid:16) q r ( S r − aq r ) − κ u r − λ ( µ − au r ) ϕ s (cid:17) dr | q t = q, S t = s (cid:21) ,w ( t, q, s ; u ) = sup z E (cid:20)(cid:90) Tt (cid:16) − g z r + λ ( µ − au r ) ϕ s (cid:17) dr | q t = q, S t = s (cid:21) , where ϕ s is the delta hedging of the derivative, that will have to be determined at theequilibrium. It is reasonable to expect that the derivative’s price ϕ ( t, q, s ) is the solutionto the following PDE: ϕ t + σ z ) ϕ ss = 0 , ϕ ( T, s, q ) = ( s − aq ) , (4.4)21or some function z = z ( t, q, s ) coming from the trader’s best-response. The PDEabove is coupled with the following two HJB equations, arising from the best-responsefunctions of the two players: − v t = sup u (cid:26) q ( s − aq ) − κ u − λϕ s ( µ − au ) + v q u + v s µ + σ z ) v ss (cid:27) , (4.5) − w t = sup z (cid:26) − g z + λϕ s ( µ − au ) + w q u + w s µ + σ z ) w ss (cid:27) , (4.6)with terminal conditions v ( T ) = w ( T ) = 0 . Solving the optimization problems withinthe HJB equations above, we get the (candidate) equilibrium strategies for the producerand the trader in terms of the corresponding payoff functions: (cid:98) u = 1 κ (cid:0) v q + λaϕ s (cid:1) , (cid:98) z = σ g w ss . The HJB equations for the producer and the trader become respectively as − v t = q ( s − aq ) + 12 κ (cid:0) v q + λaϕ s (cid:1) − µλϕ s + µv s + σ (cid:18) σ g w ss (cid:19) v ss and − w t = λµϕ s − κ (cid:0) v q + λaϕ s (cid:1)(cid:0) λaϕ s − w q (cid:1) + µw s + σ w ss + σ g w ss . Furthermore, the PDE giving the option equilibrium price ϕ is given: ϕ t + σ (cid:18) σ g w ss (cid:19) ϕ ss = 0 , with ϕ ( T, q, s ) = ( s − aq ) . Analogously as in the previous two models, we use the following ansatz for w : w ( t, q, s ) = A w ( t ) q + B w ( t ) s + C w ( t ) qs + D w ( t ) q + E w ( t ) s + F w ( t ) , and similarly for v and ϕ with self-explanatory notation for their coefficients. Therefore,using the ansatz and proceeding in the usual way, we easily get ϕ ( t, q, s ) = ( s − aq ) + F ϕ ( t ) , where F ϕ ( t ) := σ (cid:90) Tt (cid:18) σ g B w ( r ) (cid:19) dr. − A (cid:48) v = − a + 2 κ (cid:0) A v − a λ (cid:1) (4.7) − B (cid:48) v = 12 κ ( C v + 2 aλ ) (4.8) − C (cid:48) v = 1 + 2 κ (cid:0) A v − a λ (cid:1) ( C v + 2 aλ ) (4.9) − D (cid:48) v = µ ( C v + 2 aλ ) + 2 κ D v (cid:0) A v − a λ (cid:1) (4.10) − E (cid:48) v = 2 µ ( B v − λ ) + 1 κ D v ( C v + 2 aλ ) (4.11) − F (cid:48) v = µE v + 12 κ D v + σ (cid:18) σ g B w (cid:19) B v (4.12) − A (cid:48) w = 4 κ (cid:0) A v − a λ (cid:1) (cid:0) A w + a λ (cid:1) (4.13) − B (cid:48) w = 1 κ ( C w − aλ ) ( C v + 2 λa ) (4.14) − C (cid:48) w = 2 κ (cid:2) ( A v − λa )( C w − aλ ) + ( A w + a λ )( C v + 2 λa ) (cid:3) (4.15) − D (cid:48) w = µ ( C w − aλ ) + 2 κ D v (cid:0) A w + a λ (cid:1) + 2 κ D w (cid:0) A v − a λ (cid:1) (4.16) − E (cid:48) w = 2 µ ( B w + λ ) + 1 κ D w ( C v + 2 aλ ) + 1 κ D v ( C w − aλ ) (4.17) − F (cid:48) w = µE w + 1 κ D v D w + σ B w + σ g B w (4.18)with zero terminal condition for all ODEs above. The first equation, which is a RiccatiODE, can be solved explicitly giving the same expression as for D in the first model: A v ( t ) = − a − λ a κ )( e θ ( T − t ) − θ ( e θ ( T − t ) + 1) + λa κ ( e θ ( T − t ) − , θ := (cid:114) aκ . The other equations are linear, hence they can be solved in integral form. For themoment, we give only the expressions for the coefficients that we need in order tocompute the equilibrium strategy (cid:98) z of the trader. They are given by: A w ( t ) = 4 a λκ (cid:90) Tt e κ (cid:82) ut ( A v ( r ) − a λ ) dr ( A v ( u ) − a λ ) du,B w ( t ) = 1 κ (cid:90) Tt ( C w ( r ) − aλ )( C v ( r ) + 2 aλ ) dr,C w ( t ) = 2 κ (cid:90) Tt e κ (cid:82) ut ( A v ( r ) − a λ ) dr (cid:2) ( A w ( u ) + a λ )( C v ( u ) + 2 aλ ) − aλ ( A v ( u ) − a λ ) (cid:3) du,C v ( t ) = (cid:90) Tt e κ (cid:82) ut ( A v ( r ) − a λ ) dr (cid:18) aλκ ( A v ( u ) − a λ ) (cid:19) du. .2 Verification Theorem 4.1.
Let A v , C v , D v , B w be solutions to the system (4.7) - (4.18) such that B w ( t ) > − gσ , t ∈ [0 , T ] . (4.19) Then there exists a Nash equilibrium ( (cid:98) u, (cid:98) z ) ∈ A as in Definition 4.2, where (cid:98) u t = 1 κ (cid:104) C v ( t ) (cid:98) S t + 2 A v ( t ) (cid:98) q t + D v ( t ) + 2 λa ( (cid:98) S t − a (cid:98) q t ) (cid:105) , (cid:98) z t = σ g B w ( t ) , t ∈ [0 , T ] . (4.20) The no-arbitrage equilibrium price process for the derivative h T is given by (cid:98) h t := h (cid:98) u, (cid:98) zt = ( (cid:98) S t − a (cid:98) q t ) + σ (cid:90) Tt (cid:18) σ g B w ( u ) (cid:19) du, t ∈ [0 , T ] , (4.21) and the hedging process at equilibrium is (cid:98) ∆ t := ∆ (cid:98) u, (cid:98) zt = 2( (cid:98) S t − a (cid:98) q t ) , t ∈ [0 , T ] . (4.22) Proof.
Analogously as for the previous two models, the proof is structured in two steps.1.
Admissibility . We start from the verification that the pair ( (cid:98) u, (cid:98) z ) belongs to A . Thetwo processes (cid:98) u, (cid:98) z are clearly progressively measurable by definition, where (cid:98) u takes realvalues. Moreover, assumption (4.19) implies that (cid:98) z t > − for all t ∈ [0 , T ] . Concerningthe integrability properties in Definition 4.1(i), since both (cid:98) u and (cid:98) z are affine in the statevariables (cid:98) q t , (cid:98) S t with time continuous (hence bounded) coefficients, they boil down tochecking E (cid:20)(cid:90) T ( (cid:98) q t + (cid:98) S t ) dt (cid:21) < ∞ . (4.23)For the square integrability of (cid:98) S , observe that since (cid:98) z is a deterministic continuousfunction of time, each (cid:98) S t is normally distributed, hence it has every moment and theyare continuous in time. Therefore E [ (cid:82) T (cid:98) S t dt ] < ∞ . To verify the square integrabilityof (cid:98) q , notice that at equilibrium we have d (cid:98) q t = [ α ( t ) (cid:98) q t + β ( t ) (cid:98) S t + γ ( t )] dt, (cid:98) q = q , for some deterministic continuous functions of time α, β and γ . Such a linear ODE canbe solved pathwise, giving (cid:98) q t = e (cid:82) t α ( r ) dr (cid:18) q + (cid:90) t e − (cid:82) r α ( u ) du (cid:16) β ( t ) (cid:98) S r + γ ( r ) (cid:17) dr (cid:19) , t ∈ [0 , T ] . This implies that showing E [ (cid:82) T (cid:98) q t dt ] < ∞ reduces to E [ (cid:82) T ( (cid:82) t (cid:98) S r dr ) dt ] < ∞ , whichfollows since Σ t := (cid:82) t (cid:98) S r dr is a Gaussian process with time continuous second moment.24egarding condition (ii), we need to show that there exists a unique EMM (cid:98) Q = Q (cid:98) u, (cid:98) z for the production process (cid:98) q . Let (cid:98) γ t := µ − a (cid:98) u t σ √ (cid:98) z t and let us consider L (cid:98) u, (cid:98) zt := exp (cid:26) − (cid:90) t (cid:98) γ t dW t − (cid:90) t (cid:98) γ t dt (cid:27) , t ∈ [0 , T ] . We use once more [23, Theorem 2.1] to prove that under our assumptions the probability d (cid:98) Q := L (cid:98) u, (cid:98) zT d P is well-defined (see also [22]). For this model, Assumption 2.2 in [23] issatisfied as long as σ (1 + (cid:98) z t ) > for all t ∈ [0 , T ] , which is immediately given by ourassumption that B w ( t ) > − g/σ ensuring, as we already saw above, that (cid:98) z t > − forall t ∈ [0 , T ] .To end this part, we need to check property (iii) in Definition 4.1. The first part, relativeto (cid:98) h , is done as in the proof of Theorem 2.1, hence the details are omitted. Concerning (cid:98) ∆ , first notice that it is clearly progressively measurable and takes real values. Due tothe fact that (cid:98) ∆ t is linear in both (cid:98) q t and (cid:98) S t , its integrability property is equivalent to(4.23), which has already been checked before.2. Equilibrium . We are going to use the martingale optimality principle here as wellto verify that at the proposed equilibrium ( (cid:98) u, (cid:98) z ) both players are implementing anoptimal response to each other strategy. Let us consider the producer first and definethe following process Y u, (cid:98) zt := (cid:90) t (cid:16) q r ( (cid:98) S r − aq r ) − κ u r − λ ∆ u, (cid:98) zr ( µ − au r ) (cid:17) dr + W ( t, q t , (cid:98) S t ) , (4.24)with W ( t, q, s ) = A w ( t ) q + B w ( t ) s + C w ( t ) qs + D w ( t ) q + E w ( t ) s + F w ( t ) . Similarly as in the proof of Theorem 2.1, one can verify by applying Itô’s formula and theHJB equation (4.5) satisfied by the function W ( t, q, s ) by construction in the heuristicspart, that Y u, (cid:98) zt is a supermartingale for all u such that ( u, (cid:98) z ) ∈ A , and a martingale for u = (cid:98) u . For the trader, we consider the process Z (cid:98) u,zt := (cid:90) t (cid:16) − g z r + λ ∆ (cid:98) u,zr ( µ − a (cid:98) u r ) (cid:17) dr + U ( t, (cid:98) q t , S t ) , (4.25)with U ( t, q, s ) = A v ( t ) q + B v ( t ) s + C v ( t ) qs + D v ( t ) q + E v ( t ) s + F v ( t ) . Applying the same arguments as for the producer, one can easily check that Z (cid:98) u,z is asupermartingale for all z such that ( (cid:98) u, z ) ∈ A and a martingale for z = (cid:98) z .Finally, an application of the martingale optimality principle combined with the ad-missibility of ( (cid:98) u, (cid:98) z ) , gives that the latter is a Nash equilibrium as in Definition 4.2.Therefore, the proof is complete. 25 emark 4.1. Observe that in the theorem above we assumed that B w ( t ) > − g/σ forall t ∈ [0 , T ] . This is satisfied when the maturity T is small enough. Indeed, one canreason heuristically in the following way: when T ≈ , using the equation (4.14) for B w we have B (cid:48) w ( T ) ≈ κ (2 λa ) > and we also have B w ( T ) = 0 . Therefore, it is naturalto expect B w ( t ) > − g/σ for all t ∈ [0 , T ] when T is small enough, which would alsoguarantee that the Radon-Nikodym derivative d Q (cid:98) u, (cid:98) z /d P = L (cid:98) u, (cid:98) zT is well-defined (see thesecond part of the proof above). Unfortunately, the study of the function B w is muchmore difficult in this case than in the model of Section 2, where we were able to quantifyprecisely how small T must be. In this section, we use numerical simulations to illustrate and explain the behaviours ofthe producer in the three models and of the trader in the third one. We set the drift ofthe commodity price to zero, µ = 0 , in Models 2 and 3, to simplify the analysis. Model 1: production-based manipulation.
The understanding of the first modelis quite straightforward and it is illustrated by the first column of Figure 1. Startingfrom a zero production rate q , the optimal strategy of the producer, whether or not sheholds a derivative position, is to reach as fast as possible the optimal production ratethat maximises the running profit q (cid:63) := s a . We have seen that when the producer hasno position in the derivative market, she has no interest in increasing the volatility usingsome randomisation of her production rate q . Indeed, her expected profit is proportionalto E (cid:2) − q t ] and thus, increasing the volatility decreases her expected profit. On thecontrary, since controlling the volatility has a cost, she makes costly efforts to reduce it.When the producer holds a derivative position, we first note that she uses her marketpower to drive the price of the commodity at maturity to a level that suits her profit.If she has bought (resp. sold) the derivative, she drives the commodity price up (resp.down). For instance, in case of a sale, the derivative is sold at, say, 100, but at maturityits payoff is close to zero, ensuring a profit of nearly 100. Figure 2 (left) gives the valuefunction of the producer at time zero as a function of the derivative position. We seethat selling derivatives ( λ > ) can only make her better off while buying derivativesrequires a certain amount of sales before it is worth the cost.Regarding the volatility, since the price h of the derivative is an increasing functionof the realised volatility, the producer may have an interest in increasing the volatilityto push the value of the derivative up. But, since increasing the volatility has a negativeeffect on the expected profit, the producer has to assess this trade-off. Using the Remark2.2 together with the expression for (cid:98) z in (2.1) and noting that it makes sense to increasethe volatility only in case the producer has sold the derivative ( λ > ) we have that (cid:98) z ≥ ⇔ λ ≥ (cid:114) κ a . (5.1)26f the net position exceeds the threshold above, the benefit of increasing the volatilityoutweighs the cost. The higher the market power, the lower the threshold. Besides, it isworth noting that this threshold does not depend on the cost of intervention g to reducethe volatility. It only depends on the parameters affecting the drift of the commodityprice process. In Figure 1, we choose a large short position of λ = 1 which makes theprofit on the derivative as important as the profit from production. In that case, theproducer increases more than by half the volatility. Figure 3 illustrates the variationof the price of the derivative h , of the expected payoff E P [ h T ] but also of the price ofthe derivative if no volatility manipulation was undertaken, noted h z =00 , as a functionof the holding position λ . In these simulations, we started the initial production rateat its optimal stationary level q (cid:63) to get rid of transitory effects. For the first model, weobserve that h is an increasing function of the position, but it varies much less than theexpected terminal payoff. It means that much of the benefit from holding a derivativeposition comes from the manipulation of the price at maturity. Model 2: production and information based manipulation.
The story for thesecond model is illustrated by the second column of Figure 1 and it has many points incommon with the first model: the producer optimal production strategy is to reach thestationary optimal level q (cid:63) and to drive the price at maturity up in case of a purchaseand down in case of a sale. But, now, contrary to the former case, as pointed outin Remark 3.3, the producer always increases the volatility whatever her net position,long or short, because her profit rate is an increasing function of the volatility. As aconsequence, we observe in the second column of Figure 3 that h is always greaterthan h z =00 . Further, the variation of E P [ h T ] − h is much larger now, when the producercan separate the manipulation of the drift and of the volatility, than in the first model.Figure 2 (middle) provides the value function of the producer at time zero as a functionof the derivative position. The situation here is very similar to what we observed in thefirst model, namely selling derivatives ( λ > ) results in a profit, while buying derivativesis worth the cost as soon as the amount of sales exceeds a threshold depending on themodel parameters.In both models 1 and 2, the capacity of driving the price of the commodity atmaturity at a desired level reveals itself an efficient tool to take advantage of a derivativeposition. If the producer has sold the derivative at, say, 100, she increases her productionrate so that at terminal date, the price of the commodity decreases, making the price ofthe derivative decrease below the initial price and thus ensuring a profit on the derivative. Model 3: producer-trader competition.
What happens when the producer isfacing an opponent who can control the level of volatility? The third column of Figure 1illustrates the interaction between the producer and the trader. The fact that theproducer now faces an opponent does not change her overall production strategy: shestill reaches the stationary optimal level of production rate q (cid:63) and she manipulates thecommodity price at maturity at her own advantage. But, the actions of the trader onthe volatility reduces the potential profit made by the producer on the derivative. When27he producer has sold (resp. bought) the derivative to the trader, the trader reduces(resp. increases) the volatility to push the price h of the derivative down. The thirdcolumn of Figure 3 shows a much lower variation of the derivative profit h − E P [ h T ] than in the first two models. Besides, we observe that for λ > , we have h − E P [ h T ] > and for λ < , we have h − E P [ h T ] < , meaning that in each case, the trader is makinga loss.We have seen that the producer can drive the price at maturity at a level thatwould make her derivative position a profitable trade for her, providing her with anefficient tool in this asymmetric game of price manipulation. In this situation, consideredthe potential strong asymmetry of power in this game, a natural question on whetheran exchange level λ that would make both players better off exchanging might arise.Figure 2 (right) gives the value functions of the producer and of the trader at time zeroas a function of the derivative position. We observe that even if the value function ofthe producer exhibits the same pattern as in the first two models, her expected profitis now considerably reduced due to the counteraction of the trader. Besides, the valuefunction of the trader is concave and admits an optimum at a position that makes theproducer worse off trading. Further, we observe that in this situation, with a zero initialrate of production, neither the producer nor the trader are better off trading.But, if we consider that the production rate starts at its optimal stationary level q (cid:63) ,we find that whatever the market power of the producer, there is an exchange positionmaking both the producer and the trader better off than not making a trade. Figure 4presents the value functions of the producer and the trader at initial time for differentvalues of the market power parameter a and different cost of intervention for the trader g . In each case, we chose as an initial production rate q = q (cid:63) = s / (2 a ) , the stationarylevel of production, avoiding in this way the transitory phase to optimal productionrate. When the producer and the trader do not trade ( λ = 0 ), their respective value v (0 , q , s ) and w (0 , q , s ) stand at the intersection of the black axis. As the traderstarts to sell the derivative, λ becomes negative and both values are greater than when λ = 0 , showing that both are better off making the trade. Further, we observe thatboth are worse off in the case where the trader buys the derivative from the producer.28odel 1 Model 2 Model 3 t ^ q t ^ q t ^ q t ^ ' t ^ ' t ^ ' t < p
1+ ^ z t < p
1+ ^ z t < p
1+ ^ z t ~ S t ~ S t ~ S Figure 1:
Optimal production rate (cid:98) q , derivative price (cid:98) ϕ , volatility (cid:98) σ and commodity price (cid:98) S when the producer has no derivative position λ = 0 (blue), bought the derivative λ < (black)and sold the derivative, λ > (red). Parameter values: s = 10 , a = 0 . , g = 0 . , κ = 0 . , σ = 1 , T = 1 , µ = 0 . , q = 0 , λ ∈ {− . , } for Model 1 & 2, λ ∈ {− . , . } for Model 3. v (0 ;q ;S ) v (0 ;q ;S ) -0.1 -0.05 0 0.05 0.1 v (0 ;q ;S ) w (0 ;q ;S ) Figure 2:
Producer’s and trader’s value function at initial time as a function of derivativeposition λ . Parameter values: s = 10 , a = 0 . , g = 0 . , κ = 0 . , σ = 1 , T = 1 , µ = 0 . , q = 0 . Model 1 Model 2 Model 3 -0.2 0 0.2 0.4 0.6 0.8 1050100150200250300350400 -0.2 0 0.2 0.4 0.6 0.8 105001000150020002500 -0.1 -0.05 0 0.05 0.1102030405060708090100-0.2 0 0.2 0.4 0.6 0.8 12525.125.225.325.425.525.625.725.8 -0.2 0 0.2 0.4 0.6 0.8 12426283032343638 -0.1 -0.05 0 0.05 0.12525.225.425.625.82626.226.4
Figure 3:
Values of the derivative h , the expected value of the payoff E P [ h T ] and the valueof the derivative in case of no volatility manipulation h z =00 as a function of the net position λ .Parameter values: s = 10 , a = 0 . , g = 0 . , κ = 0 . , σ = 1 , T = 1 , µ = 0 . , q = q (cid:63) . = 0 . a = 0 . a = 0 . -0.08 -0.06 -0.04 -0.02 0250.6250.8251251.2251.4251.6 -20 -15 -10 -5 0 510 -3 -4 -3 -4 -4 Figure 4:
Values of v (0 , q , s ) and w (0 , q , s ) as a function of the trading position λ , λ > in red, λ < in blue, for q = s / a , and for different values of a . eferences [1] R. K. Aggarwal, G. Wu. Stock price manipulations. Journal of Business , 79(4):1915-1953, 2006.[2] F. Allen, D. Gale. Stock–price manipulation.
Review of Financial Studies , 5(3):503-529, 1992.[3] F. Allen, G. Gorton. Stock price manipulation, market microstructure and asym-metric information.
European Economic Review , 36:624-630, 1992.[4] A. Chatterjea, R. J. Jarrow. Market manipulation, price bubbles, and a model ofthe U.S. treasury securities auction market.
Journal of Financial and QuantitativeAnalysis , 33(2):255-289, 1998.[5] I.-H. Cheng, W. Xiong. The Financialization of Commodity Markets.
Annual Reviewof Financial Economics , 6:419–441, 2014.[6] D. J. Cooper, R. G. Donaldson. A strategic analysis of corners and squeezes.
Journalof Financial and Quantitative Analysis , 33(1):117-137, 1998.[7] D. Duffie, J. C. Stein. Reforming LIBOR and Other Financial Market Benchmarks.
Journal of Economic Perspectives , 29(2):191–212, 2015.[8] N. El Karoui. Les aspects probabilistes du contrôle stochastique. , Springer, 73-238, 1981.[9] D. Filipovic. Term-Structure Models. A Graduate Course. Springer, 2009.[10] Glencore. Annual Report, 2018.[11] M. Gallmeyer and D. Seppi. Derivative Security Induced PriceManipulation.
Work-ing Paper 2000-E41 , Tepper School of Business, Carnegie Mellon University, 2000.[12] J. M. Griffin, A. Shams. Manipulation in the VIX?
Review of Financial Studies ,31(4):1377–1417, 2017.[13] O. Guéant. The Financial Mathematics of Market Liquidity: From optimal execu-tion to market making. Chapman and Hall/CRC, 2016.[14] D. Hou, D. R. Skeie. LIBOR: Origins, Economics, Crisis, Scandal, and Reform.
FRB of New York Staff Report No. 667 , April 2014.[15] R. J. Jarrow. Derivative security markets, market manipulation, and option pricingtheory.
Journal of Financial and Quantitative Analysis , 29(2):241-261, 1994.[16] A. S. Kyle. Continuous auctions and insider trading.
Econometrica , 53(6):1315-1336, 1985. 3217] K. Nyström, M. Parviainen. Tug-of-war, market manipulation, and option pricing.
Mathematical Finance , 27(2):279-312, 2017.[18] G. Pirrong. Manipulation of the commodity futures market delivery process.
Jour-nal of Business , 66(3):335-369, 1993.[19] G. Pirrong. Mixed manipulation strategies in commodity futures markets.
TheJournal of Futures Markets , 15(1):13-38, 1995.[20] G. Pirrong. The economics of commodity markets manipulation: a survey.
TheJournal of Commodity Markets , 5:1–17, 2017.[21] RioTinto. Annual Report, 2018.[22] J. Ruf. The martingale property in the context of stochastic differential equations.
Electronic Communications in Probability , vol. 20, paper 34, 2015.[23] T. H. Rydberg. A note on the existence of unique equivalent martingale measuresin a Markovian setting.