A Stochastic Control Approach to Managed Futures Portfolios
aa r X i v : . [ q -f i n . M F ] N ov A Stochastic Control Approach toManaged Futures Portfolios
Tim Leung ∗ Raphael Yan † November 6, 2018
Abstract
We study a stochastic control approach to managed futures portfolios. Buildingon the Schwartz (1997) stochastic convenience yield model for commodity prices, weformulate a utility maximization problem for dynamically trading a single-maturityfutures or multiple futures contracts over a finite horizon. By analyzing the associatedHamilton-Jacobi-Bellman (HJB) equation, we solve the investor’s utility maximizationproblem explicitly and derive the optimal dynamic trading strategies in closed form.We provide numerical examples and illustrate the optimal trading strategies using WTIcrude oil futures data.
Keywords: commodity futures, dynamic portfolios, trading strategies, utility maximization
JEL Classification:
C61, D53, G11, G13
Mathematics Subject Classification (2010): ∗ Department of Applied Mathematics, University of Washington, Seattle WA 98195. Email:[email protected]. Corresponding author. † Department of Mathematics and Statistics, McMaster University, 1280 Main Street West, Hamilton,Ontario L8S 4K1, Canada. Email: [email protected] Introduction
Managed futures funds constitute a significant segment in the universe of alternative assets.These investments are managed by professional investment individuals or management com-panies known as Commodity Trading Advisors (CTAs), and typically involve trading futureson commodities, currencies, interest rates, and other assets. Regulated and monitored byboth government agencies such as the U.S. Commodity Futures Trading Commission andthe National Futures Association, this class of assets has grown to over US$350 billion in2017. One appeal of managed-futures strategies is their potential to produce uncorrelatedand superior returns, as well as different risk-return profiles, compared to the equity market(Gregoriou et al., 2010; Elaut et al., 2016). While the types of securities traded and strate-gies are conceivably diverse among managed futures funds, details of the employed strategiesare often unknown. Hurst et al. (2013) suggest that momentum-based strategies can helpexplain the returns of these funds.In this paper, we analyze a stochastic dynamic control approach for portfolio optimizationin which the commodity price dynamics and investor’s risk preference are incorporated. Thecommodity futures used in our model have the same spot asset but different maturities.Futures with the same spot asset share the same sources of risk. We apply a no-arbitrageapproach to construct futures prices from a stochastic spot model. Specifically, we adoptthe well-known two-factor model by Schwartz (1997), which also takes into account thestochastic convenience yield in commodity prices. We determine the optimal futures tradingstrategies by solving the associated Hamilton-Jacobi-Bellman (HJB) equations in closedform. The explicit formulae of our strategies allow for financial interpretations and instantimplementation. Moreover, our optimal strategies are explicit functions of the prices ofthe futures included in the portfolio, but do not require the continuous monitoring of thespot price or stochastic convenience yield. Related to the strategies, we also discuss thecorresponding wealth process and certainty equivalent from futures trading. We providesome numerical examples and illustrate the optimal trading strategies using WTI crude oilfutures data.There is a host of research on the pricing of futures, but relatively few studies apply dy-namic stochastic control methods to optimize futures portfolios. Among them, Bichuch and Shreve(2013) consider trading a pair of futures but use the arithmetic Brownian motion. In a re-cent study, Angoshtari and Leung (2018) study the problem of dynamically trading the pricespread between a futures contract and its spot asset under a stochastic basis model. Theymodel the basis process by a scaled Brownian bridge, and solve a utility maximization prob-lem to derive the optimal trading strategies. These two related studies do not account forthe well-observed no-arbitrage price relationships and term-structure in the futures mar-ket. They motivate us to consider a stochastic spot model that can generate no-arbitragefutures prices and effectively capture their joint price evolutions. In our companion paperLeung and Yan (2018), we focus on dynamic pairs trading of VIX futures under a Cen-tral Tendency Ornstein-Uhlenbeck no-arbitrage pricing model. All these studies propose astochastic control approach to futures trading. In contrast, Leung et al. (2016) introducean optimal stopping approach to determine the optimal timing to open or close a futures Source: BarclayHedge ( ). Let us denote the commodity spot price process by ( S t ) t ≥ . Under the Schwartz (1997)model, the spot price is driven by a stochastic instantaneous convenience yield, denoted by( δ t ) t ≥ here. This convenience yield, which was originally used in the context of commodityfutures, reflects the value of direct access minus the cost of carry and can be interpreted asthe“dividend yield” for holding the physical asset. It is the “flow of services accruing to theholder of the spot commodity but not to the owner of the futures contract” as explained inSchwartz (1997).For the spot asset, we consider its log price, denoted by X t . Under the Schwartz (1997)model, it satisfies the system of stochastic differential equations (SDEs) under the physicalprobability measure P : X t = log( S t ) , (1) dX t = (cid:18) µ − η − δ t (cid:19) dt + ηdZ st , (2) dδ t = κ ( α − δ t ) dt + ¯ ηdZ δt . (3)Here, Z st and Z δt are two standard Brownian motions under P with instantaneous correlation ρ ∈ ( − , α , volatility ¯ η , and speed of mean-reversionequal to κ . We require that κ, ¯ η, η > µ, α ∈ R .The investor’s portfolio optimization problem will be formulated under the physical mea-sure P , but in order to price the commodity futures we need to work with the risk-neutralpricing measure Q . To this end, we assume a constant interest rate r ≥
0, and apply a changeof measure from P to Q . The Q -dynamics of the correlated Brownian motions ( Z st , Z δt ) aregiven by d ˜ Z st = µ − rη dt + dZ st , (4) d ˜ Z δt = λ ¯ η dt + dZ δt . (5)3onsequently, the risk-neutral log spot price evolves according to dX t = (cid:18) r − δ t − η (cid:19) dt + ηd ˜ Z st ,dδ t = κ ( ˜ α − δ t ) dt + ¯ ηd ˜ Z δt , where we have defined the risk-neutral equilibrium level for the convenience yield by˜ α ≡ α − λκ . It is adjusted by the ratio of the market price of risk λ associated with Z δt and the speedof mean reversion κ . With a constant λ , the convenience yield again follows the Ornstein-Uhlenbeck model under measure Q but with a different equilibrium level compared to thatunder measure P .We consider a commodity market that consists of n traded futures contracts with matu-rities T i , i = 1 , . . . , n . Let F ( i ) t ≡ F ( i ) ( t, X t , δ t ) = E [ e X T | X t , δ t ]be the price of the T i -futures at time t , which is a function of time t , current log spot price X t , and convenience yield δ t . For any i = 1 , . . . , n , the price function F ( i ) ( t, X, δ ) satisfiesthe PDE η ∂ F ( i ) ∂X + ρη ¯ η ∂ F ( i ) ∂X∂δ + ¯ η ∂ F ( i ) ∂δ + (cid:18) r − δ − η (cid:19) ∂F ( i ) ∂X + κ ( ˜ α − δ ) ∂F ( i ) ∂δ = − ∂F ( i ) ∂t , (6)for ( t, x, δ ) ∈ [0 , T i ) × ( −∞ , ∞ ) × ( −∞ , ∞ ), where we have compressed the dependence of F ( i ) on ( t, X, δ ). The terminal condition is F ( i ) ( T i , X, δ ) = exp( X ) for x ∈ R . As is wellknown (see Schwartz (1997); Cortazar and Naranjo (2006)), the futures price admits theexponential affine form: F ( i ) t = exp ( X t + A i ( t ) + B i ( t ) δ t ) (7)for some functions A i ( t ) and B i ( t ) that depend only on time t and not the state variables.The functions A i ( t ) and B i ( t ) are found from the ODEs r + ¯ η B i ( t ) + B i ( t )( ακ + ρη ¯ η ) + A ′ i ( t ) = 0 , (8) B ′ i ( t ) − κB i ( t ) − , (9)for t ∈ [0 , T i ), with terminal conditions A i ( T i ) = 0 and B i ( T i ) = 0. The ODEs (8) and (9)admit the following explicit solutions: A i ( t ) = (cid:18) r − ˜ α + ¯ η κ − η ¯ ηρκ (cid:19) ( T i − t )+ ¯ η − e − κ ( T i − t ) κ + (cid:18) ˜ ακ + η ¯ ηρ − ¯ η κ (cid:19) − e − κ ( T i − t ) κ , (10) B i ( t ) = − − e − κ ( T i − t ) κ . (11)4pplying Ito’s formula to (7), the T i -futures price evolves according to the SDE dF ( i ) t F ( i ) t = µ i ( t ) dt + ηdZ st + ¯ ηB i ( t ) dZ δt , (12)under the physical measure P , where the drift is given by µ i ( t ) = ( λ + ˜ ακ + ρ ¯ ηη ) B i ( t ) + ¯ η B i ( t ) + µ + A ′ i ( t ) + δ ( B ′ i ( t ) − κB i ( t ) −
1) (13)= µ − r − λ (1 − e − κ ( T i − t ) ) κ . (14)The last equality follows from (8) and (11). As a consequence, the drift of F ( i ) t is independentof X t and δ t , meaning that the investor’s value function (see (22) or (32)) will also beindependent of X t and δ t . This turns out to be a crucial feature that greatly simplifies theinvestor’s portfolio optimization problem and ultimately leads to an explicit solution.To facilitate presentation, let us rewrite the linear combination of dZ st and dZ δt in (12) as σ i ( t ) dZ ( i ) t ≡ ηdZ st + ¯ ηB i ( t ) dZ δt , where Z ( i ) t is a standard Brownian motion and σ i ( t ) = η + 2 ρ ¯ ηηB i ( t ) + ¯ η B i ( t ) (15)is the instantaneous volatility coefficient.Under this model, futures prices are not independent and admit a specific correlationstructure. For example, consider the T and T contracts. The SDE for the respectivefutures price is dF ( i ) t F ( i ) t = µ i ( t ) dt + σ i ( t ) dZ ( i ) t , i ∈ { , } , (16)The two Brownian motions, Z (1) t and Z (2) t , are correlated with dZ (1) t dZ (2) t = ρ ( t ) dt. where ρ ( t ) = ¯ η B ( t ) B ( t ) + ( B ( t ) + B ( t )) ρη ¯ η + η σ ( t ) σ ( t ) (17)is the instantaneous correlation that depends not only on the spot model parameters ( ρ, η, ¯ η )but also the two futures price functions through B ( t ) and B ( t ).5 Utility Maximization Problem
We now present the mathematical formulation for the futures portfolio optimization problem.To begin, we discuss the case where the investor trades only futures with the same maturityin Section 3.1. Then, we extend the analysis to optimize a portfolio with two different futuresin Section 3.2. We will also investigate in Section 3.3 the value of trading using the notionof certainty equivalent.
Suppose that the investor trades only futures of a single maturity T i for some chosen i ∈{ , , . . . , n } . The trading horizon, denoted by T , must be equal to or shorter than thechosen maturity T i , so we require T ≤ T i .We will let ˜ π i ( t, F i ) denote the number of T i -futures contracts held in the portfolio. Theinvestor can choose the size of the position in the T i -futures, and the position can be long orshort at anytime. For brevity, we may write ˜ π i ≡ ˜ π i ( t, F i ).Without loss of generality, we arbitrarily set i = 1 in our presentation of the optimizationproblem and solution. The investor is assumed to trade only the futures contract and notother risky or risk-free assets. The dynamic portfolio consists of ˜ π ( t, F ) units of T -futuresat time t . The self-financing condition means that the wealth process satisfies d ˜ W t = ˜ π ( t, F (1) t ) dF (1) t . (18)Applying the futures price equations (7) and (12), we can express the system of SDEs forthe wealth process and futures price as (cid:20) d ˜ W t dF (1) t (cid:21) = " ˜ π µ ( t ) F (1) t µ ( t ) F (1) t dt + " ˜ π ηF (1) t ˜ π ¯ ηB ( t ) F (1) t ηF (1) t ¯ ηB ( t ) F (1) t dZ st dZ δt (cid:21) , (19)= " ˜ π µ ( t ) F (1) t µ ( t ) F (1) t dt + " ˜ π σ ( t ) F (1) t σ ( t ) F (1) t dZ (1) t . (20)A control ˜ π is said to be admissible if ˜ π is real-valued progressively measurable, andis such that the system of SDE (19) admits a unique solution ( ˜ W t , F (1) t ) and the integra-bility condition E (cid:16)R Tt ˜ π ( s, F (1) s ) ( F (1) s ) ds (cid:17) < ∞ is satisfied. We denote by ˜ A t the set ofadmissible strategies in this case given an initial investment time t .The investor’s risk preference is described by the exponential utility function U ( w ) = − e − γw , for w ∈ R , (21)where γ > , T ], theinvestor seeks an admissible strategy that maximizes the expected utility of terminal wealthat time T by solving the optimization problem˜ u ( t, w, F ) = sup ˜ π ∈ ˜ A t E (cid:16) U ( ˜ W T ) | ˜ W t = w, F (1) t = F (cid:17) . (22)6e note that the value function is only a function of time t , current wealth w , and currentfutures price F , and does not depend on the current spot price or convenience yield.To facilitate presentation, we define the following partial derivatives˜ u t = ∂ ˜ u∂t , ˜ u w = ∂ ˜ u∂w , ˜ u ww = ∂ ˜ u∂w , ˜ u = ∂ ˜ u∂F , ˜ u = ∂ ˜ u∂F , ˜ u w = ∂ ˜ u∂w∂F . We expect the value function ˜ u ( t, w, F ) to solve the HJB equation˜ u t + sup ˜ π { ˜ π µ ( t ) F ˜ u w + ˜ π σ ( t ) F ˜ u w + 12 ˜ π σ ( t ) F ˜ u ww } (23)+ σ ( t ) F ˜ u + µ ( t ) F ˜ u = 0 , for ( t, w, F ) ∈ [0 , T ) × R × R + , with terminal condition ˜ u ( T, w, F ) = e − γw for ( w, F ) ∈ R × R + . Performing the optimization in (25), we can express the optimal control ˜ π ∗ as˜ π ∗ ( t, F ) = ˜ u w µ ( t ) + F ˜ u w σ ( t ) F ˜ u ww σ ( t ) . (24)Substituting this into (25), we obtain the nonlinear PDE˜ u t − ˜ u w µ ( t ) u ww σ ( t ) − F ˜ u w ˜ u w µ ( t )˜ u ww + F (2˜ u ˜ u ww µ ( t ) − F (˜ u w − ˜ u ˜ u ww ) σ ( t ) )2˜ u ww = 0 . (25)Next, we conjecture that ˜ u depends on t and w only, and apply the transformation˜ u ( t, w ) = − e − γw − ˜Φ( t ) , (26)for some function ˜Φ( t ) to be determined. By direct substitution and computation, we obtainthe ODE d ˜Φ dt = − µ ( t ) σ ( t ) = −
12 ( λ (1 − e − κ ( T − t ) ) − κ ( µ − r )) (1 − e − κ ( T − t ) ) ¯ η − − e − κ ( T − t ) ) κρη ¯ η + κ η , (27)subject to ˜Φ( T )=0. In turn, we obtain ˜Φ( t ) by integration˜Φ( t ) = Z Tt µ ( t ′ ) σ ( t ′ ) dt ′ , ≤ t ≤ T. Applying (26) to (24), we obtain the optimal strategy˜ π ∗ ( t, F ) = µ ( t ) − σ ( t ) ˜Φ γF σ ( t ) = µ ( t ) γF σ ( t ) . (28)7sing (11), (14), and (15), the optimal strategy ˜ π ∗ in the single-contract case is explicitlygiven by ˜ π ∗ ( t, F ) = 1 γF κ ( λ (1 − e − κ ( T − t ) ) − κ ( µ − r ))(1 − e − κ ( T − t ) ) ¯ η − − e − κ ( T − t ) ) κρη ¯ η + κ η . (29)We observe from (29) that ˜ π ∗ is inversely proportional to γ and F . This means that a higherrisk aversion will reduce the size of the investor’s position. A higher futures price will alsohave the same effect. However, the total cash amount invested in the futures, i.e. ˜ π ∗ ( t, F ) F ,does not vary with the futures price, and is in fact a deterministic function of time. Notethat the investor’s position is independent of the equilibrium level of the convenience yield α or ˜ α , but it depends on the speed of mean reversion κ , volatility ¯ η , and market price ofrisk λ of the convenience yield. We now consider the utility maximization problem involving a pair of futures with differentmaturities. Without loss of generality, let T and T be the two maturities of the futures inthe portfolio. The trading horizon T satisfies T ≤ min { T , T } . The investor continuouslytrades only the two futures over time. The trading wealth satisfies the self-financing condition dW t = π ( t, F (1) t , F (2) t ) dF (1) t + π ( t, F (1) t , F (2) t ) dF (2) t , (30)where π i ( t, F (1) t , F (2) t ), i = 1 ,
2, denote the number of T i -futures held. If it is negative,the corresponding futures position is short. For notational simplicity, we may write π i ≡ π i ( t, F (1) t , F (2) t ) . Writing the trading wealth and two futures prices together in terms of twofundamental sources of randomness ( Z (1) t , Z (2) t ), we get dW t dF (1) t dF (2) t = π µ ( t ) F (1) t + π µ ( t ) F (2) t µ ( t ) F (1) t µ ( t ) F (2) t dt + π σ ( t ) F (1) t π σ ( t ) F (2) t σ ( t ) F (1) t σ ( t ) F (2) t " dZ (1) t dZ (2) t . (31)A pair of controls ( π , π ) is said to be admissible if it is real-valued progressively mea-surable, and such that the system of SDE (31) admits a unique solution ( W t , F (1) t , F (2) t )and the integrability condition E (cid:0) R Tt [ π i ( s, F (1) s , F (2) s ) F (1) s ] ds (cid:1) < ∞ , for i = 1 ,
2, is satisfied.We denote by A t the set of admissible controls with an initial time of investment t . Next,we define the value function u ( t, w, F , F ) of the investor’s portfolio optimization problem.The investor seeks an admissible strategy ( π , π ) that maximizes the expected utility fromwealth at time T , that is, u ( t, w, F , F ) = sup ( π ,π ) ∈A t E (cid:16) U ( W T ) | W t = w, F (1) t = F , F (2) t = F (cid:17) . (32) To facilitate presentation, we define the following partial derivatives u t = ∂u∂t , u w = ∂u∂w , u ww = ∂ u∂w , = ∂u∂F , u = ∂ u∂F , u = ∂u∂F , u = ∂ u∂F ,u w = ∂ u∂w∂F , u w = ∂ u∂w∂F , u = ∂ u∂F ∂F . We determine the value function u ( t, w, F , F ) by solving the HJB equation u t + sup π ,π (cid:2) ( π µ ( t ) F + π µ ( t ) F ) u w + ( π σ ( t ) F + π ρ ( t ) σ ( t ) σ ( t ) F F ) u w + ( π σ ( t ) F + π ρ ( t ) σ ( t ) σ ( t ) F F ) u w + 12 ( π σ ( t ) F + π σ ( t ) F + ρ ( t ) π π σ ( t ) σ ( t ) F F ) u ww (cid:3) + µ ( t ) F u + µ ( t ) F u + σ ( t ) F u + σ ( t ) F u + ρ ( t ) σ ( t ) σ ( t ) F F u = 0 , (33)for ( t, w, F , F ) ∈ [0 , T ) × R × R + × R + , along with the terminal condition u ( T, w, F , F ) = − e − γw , for ( w, F , F ) ∈ R × R + × R + . Next, we apply the transformation u ( t, w, F , F ) = − e − γw − Φ( t,f ,f ) , (34)with f = log F and f = log F . Substituting (34) into (33), we obtain the linear PDE forΦ: 0 = Φ t + (cid:18) µ (1 − ρ ) σ + 12 µ (1 − ρ ) σ − ρ µ µ (1 − ρ ) σ σ (cid:19) + σ − Φ ) + σ − Φ ) + ρ σ σ Φ , (35)with Φ( T, f , f )=0. We have defined the partial derivativesΦ t = ∂ Φ ∂t , Φ = ∂ Φ ∂f , Φ = ∂ Φ ∂f , Φ = ∂ Φ ∂f , Φ = ∂ Φ ∂f , Φ = ∂ Φ ∂f ∂f , and suppressed the dependence on t , in µ i , σ i , and ρ to simplify the notation.We can solve this linear PDE of Φ by using the ansatzΦ( t, f , f ) = a ( t ) f + a ( t ) f + a ( t ) f + a ( t ) f + a ( t ) f f + a ( t )to deduce that a ′ ( t ) = a ′ ( t ) = a ′ ( t ) = 0 , a ( t ) = a ( t ) = a ( t ) = 0 ,a ′ ( t ) = a ′ ( t ) = 0 , a ( t ) = a ( t ) = 0 . t only, independent of f and f , andsatisfies the first-order differential equation d Φ dt = − µ ( t ) σ ( t ) + µ ( t ) σ ( t ) − ρ ( t ) µ ( t ) µ ( t ) σ ( t ) σ ( t )2(1 − ρ ( t ) ) σ ( t ) σ ( t ) . Solving this and applying (14), (15), and (17), we obtain a closed-form expression for Φ.Precisely, Φ( t ) = ( T − t ) (cid:0) ( r − µ ) ¯ η + 2 λ ( r − µ ) ρ ¯ ηη + λ η (cid:1) − ρ ) ¯ η η . (36)Applying (36) to (34), the value function is given by u ( t, w ) = − e − γw − Φ( t ) . (37)Interestingly, as in the single-futures case, the value function is independent of the speedof mean reversion κ and equilibrium level α of the convenience yield process. Intuitively, itsuggests that the optimal strategy effectively removes the stochasticity of the convenienceyield in the investor’s maximum expected utility. This feature is evident again later in thecharacterization of the optimal wealth process. Moreover, the value function does not dependon the current futures prices ( F , F ). The simplicity of the value function is unexpected,especially since there are two stochastic factors and two futures in the trading problem.Nevertheless, it does not mean that the corresponding trading strategies are trivial. In fact,the strategies depend not only on other model parameters but also the futures prices, as wewill discuss next.By applying (34) and (36) to (33), we obtain the optimal trading strategies π ∗ ( t, F , F ) = 1 γ (1 − ρ ( t ) ) σ ( t ) F (cid:18) µ ( t ) σ ( t ) − ρ ( t ) µ ( t ) σ ( t ) (cid:19) , (38) π ∗ ( t, F , F ) = 1 γ (1 − ρ ( t ) ) σ ( t ) F (cid:18) µ ( t ) σ ( t ) − ρ ( t ) µ ( t ) σ ( t ) (cid:19) . (39)In this case with two futures, for either i = 1 ,
2, the corresponding optimal strategy π ∗ i is afunction of F i , but does not depend on the price of the other futures F j , for i = j . Also notethat if ρ ( t ) is zero, then the two-futures strategy reduces to the single-futures strategy, asin (28), which is given explicitly by (29).We recall (14), (15), and (17), and express the optimal strategies explicitly in terms ofmodel parameters. Precisely, π ∗ = − e κ ( T − t ) (cid:0)(cid:0) e tκ − e κT (cid:1) ( r − µ ) ¯ η + (cid:0) e tκ λ + e κT ( rκ − λ − κµ ) (cid:1) ρ ¯ ηη + e κT κλη (cid:1) F ( e κT − e κT ) γ (1 − ρ ) ¯ η η , (40) π ∗ = e κ ( T − t ) (cid:0)(cid:0) e tκ − e κT (cid:1) ( r − µ ) ¯ η + (cid:0) e tκ λ + e κT ( rκ − λ − κµ ) (cid:1) ρ ¯ ηη + e κT κλη (cid:1) F ( e κT − e κT ) γ (1 − ρ ) ¯ η η . (41)Thus we see that the optimal controls π ∗ and π ∗ do not depend on the current spot price S t or convenience yield δ t , and is also independent on the equilibrium of the convenience10ield α . For practical applications, this independence removes the burden to estimate orcontinuously monitor the spot price or convenience yield. Nevertheless, the optimal controlsdo depend on all the other parameters, namely µ, r, κ, η, ¯ η, ρ, and λ . Lastly, we notice from(38) that, when ρ ( t ) (see (17)) equals zero, π ∗ in this two-futures case is identical to ˜ π ∗ from the single-futures case (see (28)). Remark 1
Naturally, one can consider trading futures with more than two maturities. How-ever, in such case under the Schwartz two-factor model, there is an infinite number ofsolutions to the corresponding utility maximization problem and the additional futures areredundant, as we show in Appendix A.
To derive the optimal wealth process, we substitute the optimal futures positions, π ∗ and π ∗ , into the wealth equation (30) and get dW t = π ∗ dF (1) t + π ∗ dF (2) t = µ W dt + ( π ∗ F (1) t + π ∗ F (2) t ) ηdZ st + ( π ∗ F (1) t B ( t ) + π ∗ F (2) t B ( t ))¯ ηdZ δt ≡ µ W dt + σ W dZ Wt , where we have defined µ W = π ∗ F (1) t µ ( t ) + π ∗ F (2) t µ ( t )= ( r − µ ) ¯ η + 2 λ ( r − µ ) ρ ¯ ηη + λ η γ (1 − ρ ) ¯ η η (42)and σ W = ( π ∗ F (1) t + π ∗ F (2) t ) η + ( π ∗ F (1) t B ( t ) + π ∗ F (2) t B ( t )) ¯ η +2 ρη ¯ η ( π ∗ F (1) t + π ∗ F (2) t )( π ∗ F (1) t B ( t ) + π ∗ F (2) t B ( t ))= ( r − µ ) ¯ η + 2 λ ( r − µ ) ρ ¯ ηη + λ η γ (1 − ρ ) ¯ η η = µ W γ . (43)In (42) and (43) we have used (40) and (41).Note that both µ W and σ W are constant. This implies that the wealth process, underthe optimal trading strategy, is an arithmetic Brownian motion with constant drift andvolatility. Moreover, these two constants do not depend on the speed of mean reversion κ and equilibrium level α of the convenience yield process. This is why the value function isalso independent of these two parameters. The financial intuition is that the optimal strategysuggests trading in a way that removes the randomness stemmed from the convenience yieldprocess. As a special case, when µ = r and λ = 0, the P measure is identical to Q . This willlead to π ∗ i = 0 , i = 1 ,
2, and in turn a constant wealth, with µ W = σ W = 0.11 .3 Certainty Equivalent Next, we consider the certainty equivalent associated with the trading opportunity in thefutures. The certainty equivalent is the cash amount that derives the same utility as thevalue function. First, we consider the single-futures case. Recall from (21) and (26) that theinvestor’s utility and value functions are both of exponential form. Therefore, the certaintyequivalent is given by ˜ C ( i ) ( t, w ) ≡ U − (˜ u ( t, w )) = w + ˜Φ ( i ) ( t ) γ . (44)Here, the superscript ( i ) refers to the futures with maturity T i in the portfolio. From (44),we observe that the certainty equivalent is the sum of the investor’s wealth w and thetime-deterministic component ˜Φ ( i ) ( t ) /γ , which is positive and inversely proportional to therisk aversion parameter γ . All else being equal, a more risk averse investor has a lowercertainty equivalent, valuing the futures trading opportunity less. Interestingly, the certaintyequivalent does not depend on the current futures prices F but it does depend on the modelparameters that appear in the futures price dynamics.Similarly, the certainty equivalent from dynamically trading two futures with differentmaturities is given by C ( t, w ) ≡ U − ( u ( t, w )) = w + Φ( t ) γ , (45)where u ( t, w ) is the value function in (37) and Φ is given by (36).Since the certainty equivalents in both the single-futures and two-futures cases have thesame linear dependence on wealth w , we will for simplicity set w = 0 in our numericalexamples to compare across these cases. To this end, we denote ˜ C ( i )0 ( t ) ≡ ˜ C ( i ) ( t,
0) and C ( t ) ≡ C ( t, We now examine our model through a number of numerical examples using simulated andempirical data. For our examples, we will use the estimated parameters values found inEwald et al. (2018). They are displayed here in Table 1. The drift parameter µ of thespot price was not given in Ewald et al. (2018), so we set µ = 1% for our examples. Weuse federal funds rate as a proxy for the instantaneous interest rate r which, during thecalibration period, hovered around 0 . The default value for the risk aversion coefficient γ is 1% unless noted otherwise. µ κ η ¯ η ρ λ r .
010 0 .
800 0 .
450 0 .
500 0 .
750 0 .
050 0 . Table 1: The Schwartz (1997) model parameters estimated by Ewald et al. (2018). .25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 η π ∗ π ∗ at γ = 0 . π ∗ at γ = 0 . π ∗ at γ = 0 . η -25-20-15-10-50 π ∗ π ∗ at γ = 0 . π ∗ at γ = 0 . π ∗ at γ = 0 . Figure 1: Optimal positions, π ∗ and π ∗ , respectively in the T -futures and T -futures in the two-futures case plotted for ¯ η ∈ [0 . , . γ . Common parameters are displayed in Table 1, with F = 100 and F = 100. In Figure 1, we show the dependence of the optimal positions, π ∗ and π ∗ , respectivelyin the T -futures and T -futures in the two-futures case on the volatility parameter ¯ η of theconvenience yield process, for three different risk aversion levels. Observe that π ∗ at all threelevels of γ is positive and decreasing in ¯ η while π ∗ is negative and increasing in ¯ η . Withthe parameters given in Table 1, we are long the T -futures F (1) and short the T -futures F (2) . When we rearrange the formulae (40) and (41) for π ∗ and π ∗ , respectively, and collectterms involving ¯ η , we see that for both i = 1 ,
2, the optimal strategies are of the form A i + B i / ¯ η + C i / ¯ η , which means that the absolute value of the each strategy π ∗ i decreasesas ¯ η increases, with other variables held constant. The practical consequence is that thenumber of contracts held, on both the long and short sides, are decreasing as the volatilityof the stochastic convenience yield process δ t increases. This is in line with a risk-aversetrader’s intuition that less exposures on both legs of the traded pair should be preferred, ifthe volatility of the stochastic convenience yield is high. Furthermore, the positions increasein size (more positive for π ∗ and more negative for π ∗ ) as risk aversion decreases. This isobvious given the inverse relationship between γ and π ∗ i as seen in Eq (38) and (39).Figure 2 illustrates how the optimal futures positions, π ∗ and π ∗ , vary with respect tomaturity. First of all, the two positions are of different signs and their sizes are very close.As maturity T or T lengthens, the size of the corresponding futures position increases, with π ∗ becoming more positive and π ∗ more negative. However, the change is very small as thescale on the y-axis shows, so one can interpret this as the positions are not very sensitive tothe futures maturities. 13 .08 0.09 0.1 0.11 0.12 0.13 0.14 0.15 0.16 0.17 T π ∗ T -4.96-4.94-4.92-4.9 π ∗ Figure 2: Optimal positions, π ∗ in the T -futures and π ∗ in the T -futures inthe two-futures portfolio, plotted as a function of T and T respectively, withparameters as displayed in Table 1, and F = 100 and F = 100. η -20246 π ∗ ˜ π ∗ η -6-4-202 π ∗ ˜ π ∗ Figure 3: Optimal futures position π ∗ i (dashed) in the 2-contract portfolio and˜ π ∗ i (solid) in the single-contract portfolio (with the T i -futures) plotted over η ∈ [0 . , . F = 100 and F = 100.
14n Figure 3 we compare the optimal trading strategies, π ∗ and π ∗ for two futures to theoptimal strategy ˜ π ∗ i for trading a single futures. We plot the strategies as functions of η ,the volatility of the spot price, using same set of parameters as in Table 1. When tradinga single contract, the corresponding optimal strategy, ˜ π ∗ and ˜ π ∗ , are both very small nearzero. However, it can be seen that they do increase slightly in size when η becomes small,as volatility decreases.This is in contrast to the two-contract case where the optimal strategies are π ∗ and π ∗ .Both increase, in opposite directions, as η increases. This shows that despite the increase inrisk as η increases, paired positions in π ∗ and π ∗ , of opposite signs, will increase as volatilityof the spot process increases.It is also interesting to note the size of the positions in the single contract cases ascompared to the pair-trading case. When we are constrained to trade only single contracts,that is when the admissible set is ˜ A t as opposed to A t , the position is much smaller. Underthe current model, the presence of multiple contracts of different maturities significantlyincreases trade volume and allows the trader to take much bigger hedged trades.In Figure 4 we plot the optimal strategies as functions of γ , the risk aversion coefficient.Obviously, given the inverse relationship between γ and π ∗ i as seen in Eq (38) and (39), aswell as between γ and ˜ π ∗ i as seen in Eq (28), the optimal positions are expected to decreasein magnitude. What is interesting to note is the insensitivity of ˜ π ∗ i with respect to γ , incomparison to π ∗ i . This means that in the single futures case, the position will be smallregardless of the level of risk aversion. γ -20246 π ∗ ˜ π ∗ γ -6-4-202 π ∗ ˜ π ∗ Figure 4: Optimal futures position π ∗ i (dashed) in the 2-contract portfolio and˜ π ∗ i (solid) in the single-contract portfolio (with the T i -futures) plotted over γ ∈ [0 . , . F = 100 and F = 100. Having analyzed the parameter dependence of the optimal strategies in details, now weturn to their path behavior based on historical data. We consider the June 2014 and July15 π ∗ π ∗ π ∗ + π ∗ Figure 5: Optimal strategies π ∗ , π ∗ and π ∗ + π ∗ based on historical WTI crude oilfutures data over the period Mar 2014 - Jun 2014 using parameters as displayedin Table 1. π ∗ , π ∗ ,and π ∗ + π ∗ based on the daily settlement prices of these contracts as well as the parametersin Table 1. As shown in Figure 5, the optimal strategy π ∗ is positive throughout this period,corresponding to a long position in the front-month contract, and the opposite holds for π ∗ .Taken together, the sum of both positions is negligibly small, corresponding to a net neutralposition. Overall, the positions changed little when the parameters η and ¯ η are kept fixed.The only variables that change are F i and T i − t , of which we have already seen the relativeinsensitivity in Figure 2.We now turn our attention to the certainty equivalents. With reference to Section 3.3,we plot in Figure 6 the following certainty equivalents: ˜ C (1) in the single-futures case with T -futures traded, ˜ C (2) in the single-futures case with T -futures traded, and C in the two-futures case with T -futures and T futures traded. Their numerical values are given inTable 2. C (0) ˜ C (1)0 (0) ˜ C (2)0 (0)0 . . . Table 2: Values of certainty equivalent: ˜ C (1) in the single-futures case with T -futures traded, ˜ C (2) in the single-futures case with T -futures traded, and C in thetwo-futures case with T -futures and T futures traded. The certainty equivalentsare evaluated at t = 0 and w = 0. t C ˜ C ˜ C Figure 6: The certainty equivalents C for the two-futures portfolio, as well as˜ C (1)0 and ˜ C (2)0 for the single-futures portfolios, respectively with T -futures and T -futures (see (44)). The certainty equivalents are evaluated at time t = 0 withinitial wealth w = 0. The trading horizon is T = 1, maturity of F is T = 13 / T = 14 /
12. Other common parameters are from Table 1, alongwith F = 100 and F = 100. We observe from Figure 6 that the certainty equivalent for trading two contracts si-multaneously is significantly greater than that derived from trading only a single contractregardless of the choice of maturity. In fact, the certainty equivalent C is much larger thanthe sum of the two certainty equivalents ˜ C (1) and ˜ C (2) . This makes sense since the single-contract case can be viewed as two-contracts case but with one strategy constrained at zero.Effectively, the single-contract case is restricting the admissible set from A t to ˜ A t , thus re-ducing the maximum expected utility as well as the certainty equivalent. Our result confirmsthe intuition that more choices of trading instruments are preferable to fewer.Lastly, we examine the behavior of C at different risk aversion levels with focus on itssensitivity with respect to the market price of risk λ . In Figure 7, we see that the certaintyequivalent at time 0, C , is increasing and quadratic in λ , and tends to infinity as λ increases.This holds for all three values of γ shown, but a lower risk aversion suggests that the certaintyequivalent is higher and faster growing in λ . 17 .01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11 λ C C with γ =10% C with γ =5% C with γ =1% Figure 7: Certainty equivalent C , at time t = 0 with zero initial wealth W = 0,as a function of the market price of risk λ , with parameters as displayed in Table 1. We have analyzed the problem of dynamically trading two futures contracts with the sameunderlying. Under a two-factor mean-reverting model for the spot price, we derive the futuresprice dynamics and solve the portfolio optimization problem in closed form and give explicitoptimal trading strategies. By studying the associated Hamilton-Jacobi-Bellman equation,we solve the utility maximization explicitly and provide the optimal trading strategies inclosed form. In addition to the analytic properties of our solutions, we also apply ourresults to commodity futures trading and present numerical examples to illustrate the optimalholdings.There are several natural directions for future research on managed futures. First, addi-tional factors and sources of risks can be incorporated in the spot model, including randomjumps, stochastic volatility, and stochastic interest rate. Nevertheless, more complex mod-els typically mean that the value function and optimal trading strategies are not availablein closed form and thus require numerical approximations. In reality, futures are typicallytraded with leverage, and margin requirement is a core issue. Incorporating this feature tofutures portfolio optimization may not be straightforward, but will certainly have practicalimplications. 18
Appendix
A.1 Portfolio with Three Futures Contracts
Let us consider a dynamic portfolio of three futures contracts with different maturities T , T and T . In this case, the wealth and futures prices follow the system of SDEs dW t dF (1) t dF (2) t dF (3) t = π µ ( t ) F (1) t + π µ ( t ) F (2) t + π µ ( t ) F (3) t µ ( t ) F (1) t µ ( t ) F (2) t µ ( t ) F (3) t dt + π ηF (1) t + π ηF (2) t + π ηF (3) t π ¯ ηB ( t ) F (1) t + π ¯ ηB ( t ) F (2) t + π ¯ ηB ( t ) F (3) t ηF (1) t ¯ ηB ( t ) F (1) t ηF (2) t ¯ ηB ( t ) F (2) t ηF (3) t ¯ ηB ( t ) F (3) t (cid:20) dZ st dZ δt (cid:21) . (46)The HJB equation associated with the value function u ( t, w, F , F , F ) is u t + sup π ,π ,π [ π µ F u w + π µ F u w + π µ F u w + F ( π σ F + π ( η + ¯ η B B + ρη ¯ η ( B + B )) F + π ( η + ¯ η B B + ρη ¯ η ( B + B )) F ) u w + F ( π ( η + ¯ η B B + ρη ¯ η ( B + B )) F + π σ F + π ( η + ¯ η B B + ρη ¯ η ( B + B )) F ) u w + F ( π ( η + ¯ η B B + ρη ¯ η ( B + B )) F + π ( η + ¯ η B B + ρη ¯ η ( B + B )) F + π σ F ) u w + 12 ( F π σ + F π σ + F π σ + 2( η + ¯ η B B + ρη ¯ η ( B + B )) F F π π + 2( η + ¯ η B B + ρη ¯ η ( B + B )) F F π π + 2( η + ¯ η B B + ρη ¯ η ( B + B )) F F π π ) u ww ]+ σ F u + σ F u + σ F u + µ F u + µ F u + µ F u + ( η + ¯ η B B + ρη ¯ η ( B + B )) F F u + ( η + ¯ η B B + ρη ¯ η ( B + B )) F F u + ( η + ¯ η B B + ρη ¯ η ( B + B )) F F u = 0 , where we suppress the dependence on t in µ i ( t ) , σ i ( t ) and B i ( t ), for i = 1 , , π , π , π ), we impose the first-order conditions. Tofacilitate the presentation, we define the constants a ij ≡ η + ¯ η B i B j + ρη ¯ η ( B i + B j ) , i, j = 1 , , . This leads to the following system of equations u ww F σ F F a F F a F F a F σ F F a F F a F F a F σ π π π = − F u w µ + F u w σ + F F u w a + F F u w a F u w µ + F F u w a + F u w σ + F F u w a F u w µ + F F u w a + F F u w a + F u w σ , (47)which is singular as verified by computation.19 eferences Angoshtari, B. and Leung, T. (2018). Optimal dynamic basis trading. working paper.Bichuch, M. and Shreve, S. (2013). Utility maximization trading two futures with transaction costs.
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