Optimal Dynamic Futures Portfolios Under a Multiscale Central Tendency Ornstein-Uhlenbeck Model
OOptimal Dynamic Futures Portfolios Undera Multiscale Central Tendency Ornstein-Uhlenbeck Model
Tim Leung and Yang Zhou Abstract — We study the problem of dynamically trading mul-tiple futures whose underlying asset price follows a multiscalecentral tendency Ornstein-Uhlenbeck (MCTOU) model. Underthis model, we derive the closed-form no-arbitrage prices for thefutures contracts. Applying a utility maximization approach, wesolve for the optimal trading strategies under different portfolioconfigurations by examining the associated system of Hamilton-Jacobi-Bellman (HJB) equations. The optimal strategies dependon not only the parameters of the underlying asset price process,but also the risk premia embedded in the futures prices.Numerical examples are provided to illustrate the investor’soptimal positions and optimal wealth over time.
I. I
NTRODUCTION
Futures are standardized exchange-traded bilateralcontracts of agreement to buy or sell an asset at a pre-determined price at a pre-specified time in the future. Theunderlying asset can be a physical commodity, like goldand silver, or oil and gas, but it can also be a market indexlike the S&P 500 index or the CBOE volatility index.Futures are an integral part of the derivatives market. TheChicago Mercantile Exchange (CME), which is the world’slargest futures exchange, averages over 15 million futurescontracts traded per day. Within the universe of hedgefunds and alternative investments, futures funds constitute amajor part with hundreds of billions under management. Thismotivates us to investigate the problem of trading futuresportfolio dynamically over time.In this paper, we introduce a multiscale central tendencyOrnstein-Uhlenbeck (MCTOU) model to describe the pricedynamics of the underlying asset. This is a three-factor modelthat is driven by a fast mean-reverting OU process and a slowmean-reverting OU process. Similar multiscale frameworkhas been widely used for modeling stochastic volatility ofstock prices [1]. The flexibility of multifactor models permitsgood fit to empirical term structure as displayed in themarket. Especially in deep and liquid futures markets, such ascrude oil or gold, with over ten contracts of various maturitiesactively traded at any given time, multifactor models areparticular useful. In the literature, we refer to [2] for amultifactor Gaussian model for pricing oil futures, and [3]for a multifactor stochastic volatility model for commodityprices to enhance calibration against observed option prices. Department of Applied Mathematics, University of Washington, SeattleWA 98195. E-mail: [email protected]. Corresponding author. Department of Applied Mathematics, University of Washington, SeattleWA 98195. E-mail: [email protected]. Source: CME Group daily exchange volume and open interest re-port, available at
Under the MCTOU model, we first derive the no-arbitrageprice formulae for the futures contracts. In turn, we solve autility maximization problem to derive the optimal tradingstrategies over a finite trading horizon. This stochastic controlapproach leads to the analysis of the associated Hamilton-Jacobi-Bellman (HJB) partial differential equation satisfiedby the investor’s value function. We derive both the investor’svalue function and optimal strategy explicitly.Our solution also yields the formula for the investor’scertainty equivalent, which quantifies the value of the futurestrading opportunity to the investor. Surprisingly the valuefunction, optimal strategy and certainty equivalent dependnot on the current spot and futures prices, but on theassociated risk premia. In addition, we provide the numericalexamples to illustrate the investor’s optimal futures positionsand optimal wealth over time.In the literature, stochastic control approach has beenwidely applied to continuous-time dynamic optimization ofstock portfolios dating back to [4], but much less has beendone for portfolios of futures and other derivatives. For fu-tures portfolios, one must account for the risk-neutral pricingbefore solving for the optimal trading strategies. To that end,our model falls within the multi-factor Gassian model forfutures pricing, as used for oil futures in [2]. The utilitymaximization approach is used to derive dynamic futurestrading strategies under two-factor models in [5] and [6].A general regime-switching framework for dynamic futurestrading can be found in [7]. As an alternative approach forcapturing futures and spot price dynamics, the stochasticbasis model [8], [9] directly models the difference betweenthe futures and underlying asset prices, and solve for theoptimal trading strategies through utility maximization.In comparison to these studies, we have extended theinvestigation of optimal trading in commodity futures marketunder two-factor models to a three-factor model. Closed-form expressions for the optimal controls and for the valuefunction are obtained. Using these formulae, we illustrate theoptimal strategies. Intuitively, it should be more beneficialto be able to access a larger set of securities, and thisintuition is confirmed quantitatively. Therefore, we considerall available contracts of different maturities in that market.From our numerical example, the highest certainty equivalentis achieved from trading every contract that is available.There are a few alternative approaches and applicationsof dynamic futures portfolios. In [10] and [11], an optimalstopping approach for futures trading is studied. In practice,dynamic futures portfolios are also commonly used to tracka commodity index [12]. a r X i v : . [ q -f i n . M F ] F e b I. T HE M ULTISCALE C ENTRAL T ENDENCY O RNSTEIN -U HLENBECK M ODEL
We now present the multiscale central tendency Ornstein-Uhlenbeck (MCTOU) model that describes the price dynam-ics of the underlying asset. This leads to the no-arbitragepricing of the associated futures contracts. Hence, the dy-namcis under both the physical measure P and risk-neutralpricing measure Q are discussed. A. Model Formulation
We fix a probability space (Ω , F , P ) . The log-price of theunderlying asset S t is denoted by X (1) t . Its evolution underthe physical measure P is given by the system of stochasticdifferential equations: dX (1) t = κ (cid:16) X (2) t + X (3) t − X (1) t (cid:17) dt + σ dZ P , t , (1) dX (2) t = 1 (cid:15) (cid:16) α − X (2) t (cid:17) dt + 1 √ (cid:15) σ (cid:18) ρ dZ P , t + (cid:113) − ρ dZ P , t (cid:19) , (2) dX (3) t = δ (cid:16) α − X (3) t (cid:17) dt + √ δσ (cid:18) ρ dZ P , t + ρ dZ P , t + (cid:113) − ρ − ρ dZ P , t (cid:19) , (3)where Z P , , Z P , and Z P , are independent Brownianmotions under the physical measure P .Under this model, the mean process of log-price X (1) isthe sum of two stochastic factors, X (2) t and X (3) t , modeledby two different OU processes. The first factor X (2) t is fastmean-reverting. The rate of mean reversion is represented by /(cid:15) , with (cid:15) > being a small parameter corresponding to thetime scale of this process. X (2) t is an ergodic process and itsinvariant distribution is independent of (cid:15) . This distributionis Gaussian with mean α and variance σ / . In contrast,the second factor X (3) t is a slowly mean-reverting OUprocess. The rate of mean reversion is represented by a smallparameter δ > .The three processes X (1) t , X (2) t , and X (3) t can be cor-related. The correlation coefficients ρ , ρ , and ρ areconstants, which satisfy | ρ | < and ρ + ρ < .We specify the market prices of risk as ζ i , for i = 1 , , ,which satisfy dZ Q ,it = dZ P ,it + ζ i dt, (4)where Z Q , , Z Q , and Z Q , are independent Brownianmotions under risk-neutral pricing measure Q . We introducethe combined market prices of risk λ i , for i = 1 , , , definedby λ = ζ σ , (5) λ = 1 √ (cid:15) σ (cid:18) ζ ρ + ζ (cid:113) − ρ (cid:19) , (6) λ = √ δσ (cid:18) ζ ρ + ζ ρ + ζ (cid:113) − ρ − ρ (cid:19) . (7) Then, we write the evolution under the risk-neutral measure Q as: dX (1) t = κ (cid:16) X (2) t + X (3) t − X (1) t − λ /κ (cid:17) dt + σ dZ Q , t , (8) dX (2) t = 1 (cid:15) (cid:16) α − X (2) t − (cid:15)λ (cid:17) dt + 1 √ (cid:15) σ (cid:18) ρ dZ Q , t + (cid:113) − ρ dZ Q , t (cid:19) , (9) dX (3) t = δ (cid:16) α − X (3) t − λ /δ (cid:17) dt + √ δσ (cid:18) ρ dZ Q , t + ρ dZ Q , t + (cid:113) − ρ − ρ dZ Q , t (cid:19) . (10)For convenience, we define X t = ( X (1) t , X (2) t , X (3) t ) (cid:48) , (11) Z P t = ( Z ( P , t , Z ( P , t , Z ( P , t ) (cid:48) , (12) Z Q t = ( Z ( Q , t , Z ( Q , t , Z ( Q , t ) (cid:48) , (13) µ = (0 , α /(cid:15), δα ) (cid:48) , (14) λ = ( λ , λ , λ ) (cid:48) , (15) K = κ − κ − κ /(cid:15)
00 0 δ , (16) Σ = σ σ / √ (cid:15)
00 0 √ δσ , (17)and C = ρ (cid:112) − ρ ρ ρ (cid:112) − ρ − ρ . (18)Then, the evolution for X t under measures P and Q can bewritten concisely as d X t = ( µ − KX t ) dt + Σ C d Z P t , (19)and d X t = (cid:18) µ − λ − KX t (cid:19) dt + Σ C d Z Q t . (20) Remark 1:
If the stochastic mean of log price X (1) is onlymodeled by X (2) or X (3) , instead of their sum, it will reduceto the CTOU model, which is used in [13] for pricing VIXfutures. Under this model, the futures portfolio optimizationproblem has been studied in [5].III. F UTURES P RICING AND F UTURES T RADING
A. Futures Pricing
Let us consider three futures contracts F (1) , F (2) and F (3) , written on the same underlying asset S with threearbitrarily chosen maturities T , T and T respectively.Recall that the asset price is given by S t = exp( X (1) t ) , t ≥ . hen, the futures price at time t ∈ [0 , T ] is given by F ( k ) ( t, x ) := IE Q (cid:2) exp( X (1) T k ) | X t = x (cid:3) , (21)for k = 1 , , . Define the linear differential operator L Q · = (cid:18) µ − λ − Kx (cid:19) (cid:48) ∇ x · + 12 Tr (cid:18) ΣΩΣ ∇ xx · (cid:19) , (22)where ∇ x · = ( ∂ x · , · · · , ∂ x N · ) (cid:48) is the nabla operator andHessian operator ∇ xx · satisfies ∇ xx · = ∂ x · ∂ x x · . . . ∂ x x N · ∂ x x · ∂ x · . . . ∂ x x N · ... ... . . . ... ∂ x x N · ∂ x x N · . . . ∂ x N · . (23)Then, for k = 1 , , , the futures price function F ( k ) ( t, x ) solves the following PDE ( ∂ t + L Q ) F ( k ) ( t, x ) = 0 , (24)for ( t, x ) ∈ [0 , T ) × R N , with the terminal condition F ( k ) ( T, x ) = exp( e (cid:48) x ) for x ∈ R N , where e = (1 , , (cid:48) . Proposition 2:
The futures price F ( k ) ( t, x ) is given by F ( k ) ( t, x ) = exp (cid:18) a ( k ) ( t ) (cid:48) x + β ( k ) ( t ) (cid:19) , (25)where a ( k ) ( t ) and β ( k ) ( t ) satisfy a ( k ) ( t ) = exp (cid:18) − ( T − t ) K (cid:48) (cid:19) e (26) β ( k ) ( t ) = (cid:90) Tt ( µ − λ ) (cid:48) a ( k ) ( s )+ 12 Tr (cid:18) ΣΩΣ a ( k ) ( s ) a ( k ) ( s ) (cid:48) (cid:19) ds. (27) Proof:
We substitute the ansatz solution (25) into PDE(24). The t -derivative is given by ∂ t F ( t, x ) = (cid:32)(cid:18) d a ( k ) ( t ) dt (cid:19) (cid:48) x + dβ ( k ) ( t ) dt (cid:33) × exp (cid:18) a ( k ) ( t ) (cid:48) x + β ( k ) ( t ) (cid:19) . (28)Then, the first and second derivatives satisfy ∇ x F ( t, x ) = a ( k ) ( t ) F ( t, x ) , (29) ∇ xx F ( t, x ) = a ( k ) ( t ) a ( k ) ( t ) (cid:48) F ( t, x ) . (30)By substituting (28), (29), and (30) into PDE (24), we obtain d a ( k ) ( t ) dt − K (cid:48) a ( k ) ( t ) = 0 , (31)and dβ ( k ) ( t ) dt + ( µ − λ ) (cid:48) a ( k ) ( t )+ 12 Tr (cid:18) ΣΩΣ a ( k ) ( t ) a ( k ) ( t ) (cid:48) (cid:19) = 0 . (32) The terminal conditions of a ( k ) ( t ) and β ( k ) ( t ) are given by a ( k ) ( T ) = e , β ( k ) ( T ) = 0 . (33)By direct substitution, the solutions to ODEs (31) and (32)are given by (26) and (27). B. Dynamic Futures Portfolio
Now consider a collection of M contracts of differentmaturities available to trade, where M = 1 ,
2, 3. We notethat there are only three sources of randomness, so tradingthree contracts is sufficient. Any additional contract wouldbe redundant in this model. By Proposition 2, we have dF ( k ) t F ( k ) t = a ( k ) ( t ) (cid:48) λ dt + a ( k ) ( t ) (cid:48) Σ C d Z P t (34) ≡ µ ( k ) F ( t ) dt + σ ( k ) F ( t ) (cid:48) d Z P t , (35)where we have defined µ ( k ) F ( t ) ≡ a ( k ) ( t ) (cid:48) λ , σ ( k ) F ( t ) ≡ C (cid:48) Σ (cid:48) a ( k ) ( t ) . (36)Define d F t = (cid:32) dF (1) t F (1) t , · · · , dF ( M ) t F ( M ) t (cid:33) (cid:48) . (37)Then, in matrix form, the system of futures dynamics is givenby the set of SDE: d F t = µ F ( t ) dt + Σ F ( t ) d Z P t , (38)where µ F ( t ) = (cid:16) µ (1) F ( t ) , · · · , µ ( M ) F ( t ) (cid:17) (cid:48) (39) Σ F ( t ) = (cid:16) σ (1) F ( t ) , · · · , σ ( M ) F ( t ) (cid:17) (cid:48) . (40)Here, we assume there be no redundant futures contract,which means any futures contract could not be replicated byother M − futures contracts, indicating that rank ( Σ F ) = M .Next, we consider the trading problem for the investor.Let strategy π t = (cid:16) π (1) t , · · · , π ( M ) t (cid:17) (cid:48) , where the element π ( k ) t denotes the amount of money invested in k -th futurescontract. In addition, we assume the interest rate be zero forsimplicity. Then, for any admissible strategy π , the wealthprocess is dW πt = M (cid:88) k =1 π ( k ) t dF ( k ) t F ( k ) t = π (cid:48) t µ F ( t ) dt + π (cid:48) t Σ F ( t ) d Z P t . (41)We note that the wealth process is only determined by thestrategy π t and it is not affected by factors variable X andfutures prices F .The investor’s risk preference is described by the expo-nential utility: U ( w ) = − exp( − γw ) , (42)where γ > denotes the coefficient of risk aversion. Atrategy π is said to be admissible if π is real-valuedprogressively measurable and satisfies the Novikov condition[14]: IE P (cid:20) exp (cid:32)(cid:90) ˜ Tt γ π (cid:48) s Σ F ( s ) Σ (cid:48) F ( s ) π s ds (cid:33) (cid:21) < ∞ . (43)The investor fixes a finite optimization horizon < ˜ T ≤ T , which means that ˜ T has to be less than or equal tothe maturity of the earliest expiring contract, and seeks anadmissible strategy π that maximizes the expected utility ofwealth at ˜ T : u ( t, w ) = sup π ∈A t IE [ U ( W π ˜ T ) | W t = w ] , (44)where A t denotes the set of admissible controls at the initialtime t . Since the wealth SDE (41) does not depend on thefactors variable X and futures prices F , the value functiondoes not depend on them either.To facilitate presentation, we define L π · = π (cid:48) t µ F ( t ) ∂ w · + 12 π (cid:48) t Σ F ( t ) Σ (cid:48) F ( t ) π t ∂ ww · . (45)Then, following the standard verification approach to dy-namic programming [15], the candidate value function u ( t, w ) and optimal trading strategy π ∗ is found from theHamilton-Jacobi-Bellman (HJB) equation ∂ t u + sup π L π u = 0 , (46)for ( t, w ) ∈ [0 , ˜ T ) × R , along with the terminal condition u ( T, w ) = − e − γw , for w ∈ R . Theorem 3:
The unique solution to the HJB equation (46)is given by u ( t, w ) = − exp (cid:32) − γw − (cid:90) ˜ Tt Λ ( s ) ds (cid:33) , (47)where Λ ( t ) = µ F ( t ) (cid:48) ( Σ F ( t ) Σ F ( t ) (cid:48) ) − µ F ( t ) . (48)The optimal futures trading strategy is explicitly given by π ∗ ( t ) = 1 γ ( Σ F ( t ) Σ (cid:48) F ( t )) − µ F ( t ) . (49) Proof:
We will first use the ansatz u ( t, w ) = − e − γw h ( t ) . (50)Then, using the relations ∂ t u = − e − γw ∂ t h ( t ) , ∂ w u = γe − γw h ( t ) , (51)and ∂ ww u = − γ e − γw h ( t ) , (52)the PDE (46) becomes − ddt h ( t )+ sup π t (cid:20) γ π (cid:48) t µ F ( t ) h − γ π (cid:48) t Σ F ( t ) Σ (cid:48) F ( t ) π t h (cid:21) = 0 , (53) with terminal condition h ( ˜ T ) = 1 . From the first-ordercondition, which is obtained from differentiating the termsinside the supremum with respect to π t and setting theequation to zero, we have γ µ F ( t ) − γ Σ F ( t ) Σ (cid:48) F ( t ) π t = 0 . (54)Recall that rank( Σ F ( t )) = M . Then, Σ F ( t ) Σ (cid:48) F ( t ) is an M × M invertible matrix. Accordingly, we have the optimalstrategy (49). Given the fact that A (cid:48) A is the semi-positivedefinite matrix for any matrix A , the time-dependent com-ponent Λ ( t ) = µ F ( t ) (cid:48) ( Σ F ( t ) Σ F ( t ) (cid:48) ) − µ F ( t ) is alwaysnon-negative.Substituting π ∗ back, the equation (53) becomes − ddt h ( t ) + 12 Λ ( t ) h ( t ) = 0 . (55)Accordingly, we have h ( t ) = exp (cid:18) − (cid:90) ˜ Tt Λ ( s ) ds (cid:19) . (56) Example 4:
If there is only one futures contract F (1) available in the market. Then by (39), we have µ F = µ (1) F , Σ F = σ (1) F ( t ) (cid:48) . (57)Then, the optimal strategy (49) becomes π (1) ∗ ( t ) = 1 γ µ F ( t ) Σ F ( t ) Σ (cid:48) F ( t ) = 1 γ µ (1) F σ (1) F ( t ) (cid:48) σ (1) F ( t ) , (58)where µ (1) F and σ (1) F ( t ) are shown as (35).In order to quantify the value of trading futures to theinvestor, we define the investor’s certainty equivalent asso-ciated with the utility maximization problem. The certaintyequivalent is the guaranteed cash amount that would yieldthe same utility as that from dynamically trading futuresaccording to (44). This amounts to applying the inverse ofthe utility function to the value function in (47). Precisely,we define C ( t, w ) := U − ( u ( t, w )) (59) = w + 12 γ (cid:90) ˜ Tt Λ ( s ) ds. (60)Therefore, the certainty equivalent is the sum of theinvestor’s wealth w and a non-negative time-dependentcomponent γ (cid:82) ˜ Tt Λ ( s ) ds . The certainty equivalent is alsoinversely proportional to the risk aversion parameter γ ,which means that a more risk averse investor has a lowercertainty equivalent, valuing the futures trading opportunityless. From (26), (36), (39) and (48), we see that the certaintyequivalent depends on the constant matrix K , volatilitymatrix Σ , correlation matrix C and market prices of risk λ . Nevertheless, the certainty equivalent does not depend onthe current values of factors X t . (1)0 X (2)0 X (3)0 α α (cid:15) δ σ σ σ ρ ρ ρ λ λ λ T T T ˜ T γ κ /
12 2 /
12 3 /
12 1 /
12 1 TABLE IP
ARAMETERS FOR THE
MCTOU
MODEL . Asset's Log-price X (1) Fast-Varying Factor X (2) Slow-Varying Factor X (3) Spot Price T -Futures Price T -Futures Price T -Futures Price Trading Day P r i c e Fig. 1. Top: simulation paths for asset’s log-price X (1) . Dashed curvesrepresent confidence interval. Middle: simulation path for fast varyingfactor X (2) and slow-varying factor X (3) . Dashed and dotted curvesrepresent confidence interval for X (2) and X (3) , respectively. Bottom:Sample price paths for the underlying asset and associated futures. IV. N
UMERICAL I LLUSTRATION
In this section, we simulate the MCTOU process andillustrate the outputs from our trading model. With theclosed-form expressions obtained in the Section III, we nowgenerate the futures prices, optimal strategies and wealthprocesses numerically, using the parameters in Table I.Primarily, we let (cid:15) and δ be small parameters and weconsider trading three futures with maturities T = 1 / year, T = 2 / year and T = 3 / year. Then, our tradinghorizon will be ˜ T = 1 / year, no greater than the futuresmaturities. We assume 252 trading days in a year and 21trading days in a month (or 1/12 year). In our figures, weshow the corresponding trading days on the x axis.In Figure 1, we plot the simulation paths and confi-dence intervals for three factors in the top figure and middlefigure. As shown in the middle panel, the confidence O p t i m a l S t r a t e g y π ( * ) Fig. 2. Optimal strategies π ( ∗ ) for different futures combinations. Solid,dashed and dotted lines represent the optimal position (in $) on T -futures, T -futures and T -futures, respectively. interval of the slow-varying factor X (3) is much narrowerthan the one for fast-varying factor X (2) . At the bottom, weplot the spot price and futures prices. The three paths for thefutures prices are highly correlated and T -futures price isequal to the asset’s spot price at its maturity date T , whichis the 21st trading day.In Figure 2, we plot the optimal strategies as functionsof time for different portfolios and different correlationparameters. In each sub-figure, from top to bottom, we showthe optimal strategies for one-contract portfolio, two-contractportfolio and three-contract portfolio respectively. The opti-mal investments on T -futures, dashed lines represent theoptimal investment on T -futures, T -futures, and T -futuresare represented by solid, dashed, and dotted lines respec-tively. The optimal cash amount invested are deterministicfunctions for time, but the optimal units of futures held dovary continuously with the prevailing futures price.Moreover, the investor takes large long/short positionsin three-contract portfolio since all sources of risk can behedged. We provide sample path for wealth process for three-contract portfolios in the Figure 3.In Figure 4, we see that the certainty equivalent increasesas a function of trading horizon ˜ T , which means that themore time the investor has, the more valuable is the tradingopportunity. As the trading horizon reduces to zero, thecertainty equivalent converges to the initial wealth w , whichis set to be 0 in this example, as expected from (60). Also,with a lower risk aversion parameter γ , the investor has a arameters Futures Combinations (Maturity) T T T { T , T } { T , T } { T , T } { T , T , T } ρ = 0 ρ = − . ρ = 0 ρ = 0 . ρ = 0 . ρ = − . ρ = 0 ρ = 0 . ρ = − . ρ = − . ρ = 0 ρ = 0 . TABLE IIC
ERTAINTY EQUIVALENTS ( × − ) FOR ALL POSSIBLE FUTURES COMBINATIONS UNDER DIFFERENT CORRELATIONS . higher certainty equivalent for any given trading horizon.Table II shows the certainty equivalents for all possible fu-tures combinations under various correlation configurations.The certainty equivalent is much higher when more contractsare traded. In addition, if there is only one futures contract totrade, the certainty equivalent is increasing with respect to itsmaturity, see first three columns. The certainly equivalentstend to be higher when ρ and ρ are negative. Trading Day W e a l t h Three-Contract Portfolio
Fig. 3. Sample path for wealth process for the three-futures portfolio.
V. C
ONCLUSION
We have studied the optimal trading of futures under amultiscale multifactor model. Closed-form expressions forthe optimal controls and value function are derived throughthe analysis of the associated HJB equation. Using these,we have illustrated the path behaviors of the futures pricesand optimal positions. We also quantify the values of thetrading different combinations of futures under differentmodel parameters. R
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