Asymptotic Exponential Arbitrage in the Schwartz Commodity Futures Model

In this paper, we consider the Schwartz’s one-factor model for a storable commodity and a futures contract on that commodity. We introduce the analysis of asymptotic arbitrage in storable commodity models by proving that the futures prices process allows asymptotic exponential arbitrage with geometric decaying failure probability. Next, we find by comparison that, under some similar conditions, our result is a corresponding commodity assets (stronger) version of Föllmer and Schachermayer’s result stated in the modeling setting of geometric Ornstein-Uhlenbeck process for financial security assets.


Introduction
Classical Arbitrage Theory in financial market models (including both security and commodity models) has been basically concerned with the study of investors' trading opportunities that generate risk-less profit in any finite time horizon  > 0. Prohibiting such trading opportunities in any such financial model is characterized in the classical literature by the existence of an equivalent martingale measure (), a result celebrated as the "First Fundamental Theorem of Asset Pricing" stated for instance in [1,2].It has been proved that most of these market models (financial securities models especially) share the following feature: for any finite time horizon  > 0, there is a possibility of excluding arbitrage opportunities from the model by verifying the so-called Novikov condition on the associated "market price of risk" (see for instance [3]).But, when the trading time horizon  is large enough (i.e., when  → ∞), one may generate risk-less profits.The name "asymptotic arbitrage" was first given to such long-term risk-less profits by Kabanov and Kramkov in their article "Asymptotic arbitrage in large financial markets" (see [4]).Next, Föllmer and Schachermayer ( [5]) in a single financial market (of security assets) discussed this concept of asymptotic arbitrage in a slightly different form first in a typical case of Ornstein-Uhlenbeck process and next in a general continuous-time diffusion setting in which they conjectured the possibility of generating exponential growth profit on investors wealth in long-term (i.e., when  → ∞).Mbele Bidima and Rásonyi proved this conjecture in a corresponding discretetime setting under two different sets of conditions in [6,7], proposing the name "asymptotic exponential arbitrage" to the literature.Moreover, the authors of [6,7] clarified the concept by improving the later definition to "asymptotic exponential arbitrage with geometrically/exponentially decaying failure probability" by requiring the control at an exponential/geometrical decaying rate with the probability of failing to achieve such an exponential growth profit in long-term.Next, Du and Neufeld used this later definition in [8] and proved Föllmer and Schachermayer's conjecture in continuous-time setting under stronger conditions using general security models which are not necessarily diffusion processes.And recently, Haba and Jacquier discussed also in [9] the later form of asymptotic arbitrage in the typical case of Heston stochastic volatility model.
We have noticed that those studies on asymptotic arbitrage are carried out since the past decade only for standard security models (equity, bonds, interest rates, etc.) and not yet already for commodity models (oil, gas, coffee, energy, etc.).In this article, given the particularities of commodity modeling (especially for storable commodities), we intend to introduce the analysis of asymptotic (exponential) arbitrage in those models, in the typical case of Schwartz one-factor model of commodity futures, just as Haba and Jacquier discussed it in the special case of Heston model of equities.And we compare our result to a result from our motivating paper [5] of Föllmer and Schachermayer where they treated the special case of geometric Ornstein-Uhlenbeck process for security assets.But first, in Section 2 below, we start with a comparative analysis of (storable) commodity markets with standard security markets, recalling basic definitions and concepts that we use.Next, in Section 3, the key part of this paper, we present the Schwartz's model for storable commodity futures.After verifying absence of arbitrage and model completeness on each finite time horizon, we state and prove our main result.And in Section 4, we end our work with a technical comparative discussion and concluding remarks.

Storable Commodities versus Financial Securities
Unlike standard financial securities (such as stock equities, bonds, etc.) and their derivatives which are traded in a single market counter, storable commodities (such as oil, natural gas, coffee, etc.) are related to two interconnected markets: the cash-storage (inventory) market where one (or more) storable commodity is traded at a spot price   ,  > 0, and the futures market in which a futures contract on this underlying commodity, initiated at any time  ≥ 0 and with delivery date (time horizon)  > , is traded at price   fl (, ).
Under the standard assumption of no-arbitrage, the relationship between the spot price and the futures price is given, as stated in [10] or [11,Section 5.6.3], by where  is a constant interest rate on an accompanying risk-free bond paying 1 at time .This relation is also true in standard security markets.But it is known in the literature that the relationship (1) does not hold in practice for most commodity markets, especially in storable commodity markets, because of the inability of investors and speculators to short the underlying asset,   .Hence as indicated in [12], there is an adjustment by the cost of carry in the nonarbitrage pricing formula above which is given by the so-called "(net) convenience yield", a sort of rate of the cost of carrying the physical commodity denoted   , letting the relationship above to become By comparison to financial security markets, as established in [13, p. 37], note that the no-arbitrage price of a forward contract on an underlying security asset (such as stock) with prices process   which pays a continuous dividend at (stochastic) rate, say   , with the same risk-free interest rate and the same maturity, is mathematically the same as formula (2) above by replacing   with   .The only difference resides on the two types of market and the corresponding type of gain on holding or carrying the underlying asset: in financial security markets, an asset may be assumed paying a continuous dividend, and in storable commodity markets, carrying a storable commodity necessarily requires a cost; i.e., the presence of convenience yield is consistent with the theory of storage as explained above.
Next, another distinction of storable commodity markets with security markets is highlighted by Liu and Tang in [14, Lemma 1, Theorem 1] where they proved that the following.

Theorem 1.
(1) If short-selling is allowed in any physical commodity market under no-arbitrage condition, then the convenience yield process   is zero almost surely.
(2) Under short-selling prohibition, a storable commodity market is arbitrage-free if and only if the convenience yield is always nonnegative i.e.,   ≥ 0.
The interpretation of these results is as follows: in regulating any storable commodity market with futures contracts, while one should not allow short-selling of physical goods, absence of arbitrage opportunity on any finite trading time period [0, ],  > 0, is guaranteed by insuring that the convenience yield is always nonnegative.
This key initial comparative study will lead our choice of realistic assumptions on the study of finite time horizon arbitrage and the investigation of long-term arbitrage in the Stchwartz one-factor model of storable commodity futures below.

The Schwartz Commodity Futures Model
For a fixed time horizon  > 0, consider a cash-storage market where a single factor, i.e., the only underlying asset, is a single physical commodity with prices process   .Besides there is a futures exchange where a single futures contract on this commodity with prices process   := (, ) for all times  ∈ [0, ].There is also an accompanying bank account or a risk-free bond which serves as numéraire in the no-arbitrage relation (2).It pays 1 at time  with a constant continuously compounded interest rate  > 0; hence, its price is   =  − at any time  ≤ .
According to Schwartz in [15], we assume that the spot prices process   is governed by the dynamics, or equivalently, the log spot prices process   fl ln   follows a mean-reverting process of Ornstein-Uhlenbeck (OU) type as follows: where  0 fl ln  0 is an initial real constant, ,  (with  > 0) are real constants representing respectively the long-term mean and the volatility of the spot prices process   ,  > 0 and ] fl  −  2 /2 are respectively the speed and the level of mean-reversion for the OU log prices process   , and   is a standard Brownian motion on a modeling probability space (Ω, F, P) with respect to which the underlying process   (with any other derivative and claim) verifies the usual integrability properties (such as the square-integrability) required in the sequel.We equip this space with the natural filtration F fl (F  )  of the spot prices process   , which is the same as the natural filtration of   .E and V will denote respectively the expectation and the variance with respect to the measure P, and E Q will denote the expectation relative to another appropriate measure Q on the same measurable space (Ω, F).
We formulate the following set of assumptions in line with the general conditions discussed in the preceding section.
Assumption 2. (1) Short-selling the physical commodity is not allowed, but (2) one may short-sell any real quantity in the futures contract.
(3) The convenience yield is a constant  and is strictly positive; i.e.,  > 0.
Itô's Lemma on ( 2) and (3) implies that the futures prices process   , still under the probability measure P, follows the dynamics, where stands for the "market price of risk" for the futures prices contract.As in financial security markets, its interpretation is straightforward: since the sign of   at any time  depends on the sign of the log-price   = ln   of the commodity, this gives an insight to investors for when to hold negative or positive quantities of the futures contract when building their portfolio in this spot-futures market.
Definition 3. (i) We call the dynamics (5) the Schwartz's model for the commodity futures.
(ii) According to [14] of Liu and Tang, a trading portfolio in this commodity futures model is a pair of R-valued and Fadapted processes   fl (  ,   ) representing respectively the investor's position in the risk-free bond of the money market and in the futures contract in the futures exchange.
(iii) The value process of such a trading portfolio (or wealth of the investor) denoted    is defined by (iv) Any such trading opportunity is said to be selffinancing if it is a pair of predictable process such that Assumption 4. For computational simplicity, we also assume that the risk-free interest rate  on the bond is zero, i.e., the bond price is   fl 1 for all times  ∈ [0, ].

Remark 5. (1)
This assumption implies that the discounted futures prices process F fl   /  is just   and the discounted value Ṽ  :=    /  of any portfolio is again    .
(2) Similar to the restriction assumed in [12, Section 2.2], the assumption above hints that we may restrict the definition of self-financing portfolios only to the part   of investment in the futures contract.Denoting the value process now    ≡  (10)

No Finite Horizon Arbitrage and Model Completeness.
Recall the following definitions from [16,17].
(ii) The model is said to be arbitrage-free for any finite time horizon  > 0 if there is no arbitrage opportunity in the sense above.

Definition 8 (model completeness).
A market model is said to be complete if every contingent claim is attainable; i.e., there is a self-financing strategy whose final value equals the payoff of the claim.Theorem 1 (2) (recalled from [14]) applied on our Assumption 2 (3) insures there is no arbitrage opportunity in the Schwartz one-factor model of storable commodity (3).
We check in the result below, using the "First Fundamental Theorem of Asset Pricing", that the Schwartz commodity futures model ( 5) is arbitrage-free on any finite time horizon.

Proposition 9. The Schwartz's model of commodity futures
(5) admits an equivalent martingale measure; hence, it is arbitrage-free, for any finite time horizon  > 0.
Proof.Solving the stochastic differential equation ( 4) we get for all  ∈ [0, ], This (as argued for instance in [13, pp.65-66]) implies that the random variable   is normally distributed as From this we get E(  ) < ∞ and E( 2  ) = V(  ) + [E(  )] 2 < ∞ for all times  ∈ [0, ], and next we obtain that Hence E[exp((1/2) ∫  0  2  )] < ∞, showing that the market price of risk   = / − (/)  for the futures prices satisfies the Novikov condition.It follows by Radon-Nikodym Theorem that the strictly positive and P-integrable random variable   : Ω → R + defined as defines an equivalent probability measure Q by setting Q/P fl   .Applying Girsanov's Theorem (see [17, Theorem 4.5.1]),this measure is an equivalent martingale measure, i.e., under which the (discounted) futures prices process   is a martingale and is written as where   fl   + ∫  0   ,  ∈ [0, ], is a Brownian motion under Q, showing the result, as required.
Next, we state and prove the following result.

Proposition 10. The Schwartz commodity futures model (5) is complete.
Proof.For any fixed time horizon  > 0, we have to show that for any contingent claim with payoff a (square integrable) random variable   , there is a self-financing portfolio   such that    =   .The self-financing condition (9) means, we seek a predictable process   such that under the equivalent martingale measure Q.
Noting that the filtration (F  )  is also generated by the Brownian motion   , the (discounted) portfolio value process    is a Q-martingale if and only if for all times  ∈ [0, ] , and the process Hence it is enough in (17) to take   fl   /  for all times  ∈ [0, ], ending the proof.
Remark 11.Denote M   () the set of equivalent localmartingale measures (which are equivalent martingale measures since   is a diffusion) in the model up to time  > 0. By the "Second Fundamental Theorem of Asset Pricing" (see [11,Theorem 5.4.9]), the completeness of this model entitles that M   () = {Q} where Q is the above equivalent martingale probability measure.We will need this fact as one key argument in proving the main result of this paper in the subsection below.2) is   ≡ (, ) fl    −(−) for all times  ∈ [0, ], where  > 0 is a fixed time horizon, and that for any admissible self-financing portfolio   in H, the investors' wealth in the futures contract which we considered in (9) is

Existence of Asymptotic Exponential Arbitrage. Recall under Assumptions 2 and 4 that the futures prices of the underlying commodity in (
We see that, unlike in financial security models, since the futures price depends on two time parameters  and , to discuss the asymptotic behavior of the wealth process    , i.e., from long-term trading in the futures contract when the delivery date  becomes larger and larger, it is enough to do it when the running time  is getting larger and larger.
Next, Föllmer and Schachermayer introduced the following form of asymptotic arbitrage, which we adapt here in our present modeling setting of commodity futures.
Definition 12 (Definition 1.1 in [5]).The futures price process   is said to allow (strong) asymptotic arbitrage if, for any threshold  > 0, there are  < ∞ and And they stated the following general result to prove existence of (strong) asymptotic arbitrage in a general semimartingale setting.
Proposition 13 (Proposition 2.1 in [5]).For any locally bounded semimartingale   such that the set M   () of equivalent local martingales, for any time  > 0, is not empty, then for any constants 0 <  1 ,  2 < 1, for any time  > 0, the following are equivalent: (i) There is In our present paper we use this result in proving that the futures prices process   in (5) allows asymptotic exponential arbitrage in the sense below, defined by Mbele Bidima and Rásonyi, and we compare our result to their result in a special case of standard geometric Ornstein-Uhlenbeck process for security assets.Definition 14 (from Definition 1.1 in [6]).We say that the futures prices process   generates asymptotic exponential arbitrage (AEA) with geometrically decaying failure probability (GDP-F), if there are constants ,  1 ,  2 > 0 and a trading strategy   ∈ H such that the value process    satisfies the following conditions: (i)    ≥ − − 1  , P − a.., (ii) and P[ This definition presents a better economic interpretation than Föllmer and Schachermayer's definition.Indeed, Definition 12 above means that while the maximum loss an investor may achieve is bounded by , its wealth up to time  is at least  −1 with probability larger than 1 − .But there is no clear relationship between the threshold  and the time  from which one realizes a potential profit.This drawback is solved in the preceding definition which says that the maximal loss of the investor's wealth at any time  is exponentially bounded in (i) by  − 1  and (ii) means that, even from zero initial capital, an investor may generate a profit that grows exponentially fast in time with probability converging to 1 exponentially fast.And (ii)' states that failing to achieve such a growth profit can be controlled in time by a probability converging to 0 exponentially fast.Now we state below the main theorem of this paper.
Theorem 15.Consider the Schwartz commodity futures model (5).Under Assumptions 2 and 4, there exists a trading strategy   ∈ H such that its corresponding value process  The proof of this result relies on the application of Proposition 13 above, an appropriate use of Large Deviation Principle in the following preliminary result, which we prove analogously to that derived in [5].
Indeed, consider, for any time  > 0 and for any constant  > 0, the event, where   is the market price of risk process for the futures prices process   ,   fl exp(− ) is the Radon-Nikodym density process similarly defined as in equation ( 15) and   fl (1/)  is the log spot prices process weighted by the volatility .Then, we have the following.Lemma 16.Assume for computational simplicity that the level of mean-reversion of log spot prices in ( 4) is 0; i.e., ] :=  −  2 /2 = 0.Then, for any  ∈ (0, /2 +  2 /2 2 ), we have where  is defined as above.
Proof.Let us mention first that our proof below is analogue (but not exactly) to part of that of Proposition 4.1 in [5] from which we avoid a complicating used argument on ergodic properties of the process   by defining our   above differently.
Next, since we assumed ] fl − 2 /2 = 0; i.e.,  =  2 /2, then (4) becomes For all time  > 0 we have ln(   (  − 0 )/ ) = ln   + (  −  0 )/, and International Journal of Mathematics and Mathematical Sciences So for all times  > 0, where Applying Theorem 2.2 of [18], the stochastic process Γ  / satisfies a large deviations principle with good rate function ; i.e., (see for instance [19]) for any open set  and any closed set  in the natural topology on R, with the function  defined by this function is strictly increasing on (−1/2, ∞), the large deviations upper bound above implies for  fl [, ∞] that lim →∞ (1/) ln P[Γ  / ≥ ] ≤ −(), for any  > −1/2.Next, for all times  > 0, we obtain from (26) that Remark 17.Note that we have proved this result after proving Lemma 16 under the condition that the mean-reversion level ] of the log spot prices process   is zero.This assumption was made to simplify computations.Indeed, if we assume that ] fl  −  2 /2 ̸ = 0, then Theorem 15 is still stated and proved similarly.We only modify the proof of Lemma 16 by replacing the term ( 2 /2 2 ) in (25) with the term (]/ 2 −  2 ]/ −

𝜑 9 )
, the self-financing relation above becomes    =     ..,    for all times  ∈ [0, ] .(Definition 6 (admissible strategy).A self-financing trading portfolio   is called an admissible strategy, if there exists a constant  ≥ 0 such that its corresponding value process    verifies the constraint    ≥ −, P − ..for all  ≤ .Let H be the set of admissible self-financing strategies in this model.For any time  > 0 (in particular for fixed time horizon  > 0), denote the set of value processes for all admissible self-financing strategies   up to time  with   0 = 0 as K  ; i.e., K  fl {∫  0     :   ∈ H with   0 = 0} .