Optimal Dynamic Futures Portfolios Under a Multiscale Central Tendency Ornstein-Uhlenbeck Model
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Optimal Dynamic Futures Portfolios Under
a Multiscale Central Tendency Ornstein-Uhlenbeck Model
Tim Leung1 and Yang Zhou2
Abstract— We study the problem of dynamically trading mul- Under the MCTOU model, we first derive the no-arbitrage
tiple futures whose underlying asset price follows a multiscale price formulae for the futures contracts. In turn, we solve a
central tendency Ornstein-Uhlenbeck (MCTOU) model. Under utility maximization problem to derive the optimal trading
this model, we derive the closed-form no-arbitrage prices for the
strategies over a finite trading horizon. This stochastic control
arXiv:2102.12601v1 [q-fin.MF] 24 Feb 2021
futures contracts. Applying a utility maximization approach, we
solve for the optimal trading strategies under different portfolio approach leads to the analysis of the associated Hamilton-
configurations by examining the associated system of Hamilton- Jacobi-Bellman (HJB) partial differential equation satisfied
Jacobi-Bellman (HJB) equations. The optimal strategies depend by the investor’s value function. We derive both the investor’s
on not only the parameters of the underlying asset price process, value function and optimal strategy explicitly.
but also the risk premia embedded in the futures prices.
Numerical examples are provided to illustrate the investor’s Our solution also yields the formula for the investor’s
optimal positions and optimal wealth over time. certainty equivalent, which quantifies the value of the futures
trading opportunity to the investor. Surprisingly the value
I. I NTRODUCTION function, optimal strategy and certainty equivalent depend
not on the current spot and futures prices, but on the
Futures are standardized exchange-traded bilateral
associated risk premia. In addition, we provide the numerical
contracts of agreement to buy or sell an asset at a pre-
examples to illustrate the investor’s optimal futures positions
determined price at a pre-specified time in the future. The
and optimal wealth over time.
underlying asset can be a physical commodity, like gold
In the literature, stochastic control approach has been
and silver, or oil and gas, but it can also be a market index
widely applied to continuous-time dynamic optimization of
like the S&P 500 index or the CBOE volatility index.
stock portfolios dating back to [4], but much less has been
Futures are an integral part of the derivatives market. The
done for portfolios of futures and other derivatives. For fu-
Chicago Mercantile Exchange (CME), which is the world’s
tures portfolios, one must account for the risk-neutral pricing
largest futures exchange, averages over 15 million futures
before solving for the optimal trading strategies. To that end,
contracts traded per day.3 Within the universe of hedge
our model falls within the multi-factor Gassian model for
funds and alternative investments, futures funds constitute a
futures pricing, as used for oil futures in [2]. The utility
major part with hundreds of billions under management. This
maximization approach is used to derive dynamic futures
motivates us to investigate the problem of trading futures
trading strategies under two-factor models in [5] and [6].
portfolio dynamically over time.
A general regime-switching framework for dynamic futures
In this paper, we introduce a multiscale central tendency
trading can be found in [7]. As an alternative approach for
Ornstein-Uhlenbeck (MCTOU) model to describe the price
capturing futures and spot price dynamics, the stochastic
dynamics of the underlying asset. This is a three-factor model
basis model [8], [9] directly models the difference between
that is driven by a fast mean-reverting OU process and a slow
the futures and underlying asset prices, and solve for the
mean-reverting OU process. Similar multiscale framework
optimal trading strategies through utility maximization.
has been widely used for modeling stochastic volatility of
In comparison to these studies, we have extended the
stock prices [1]. The flexibility of multifactor models permits
investigation of optimal trading in commodity futures market
good fit to empirical term structure as displayed in the
under two-factor models to a three-factor model. Closed-
market. Especially in deep and liquid futures markets, such as
form expressions for the optimal controls and for the value
crude oil or gold, with over ten contracts of various maturities
function are obtained. Using these formulae, we illustrate the
actively traded at any given time, multifactor models are
optimal strategies. Intuitively, it should be more beneficial
particular useful. In the literature, we refer to [2] for a
to be able to access a larger set of securities, and this
multifactor Gaussian model for pricing oil futures, and [3]
intuition is confirmed quantitatively. Therefore, we consider
for a multifactor stochastic volatility model for commodity
all available contracts of different maturities in that market.
prices to enhance calibration against observed option prices.
From our numerical example, the highest certainty equivalent
1 Department of Applied Mathematics, University of Washington, Seattle is achieved from trading every contract that is available.
WA 98195. E-mail: timleung@uw.edu. Corresponding author. There are a few alternative approaches and applications
2 Department of Applied Mathematics, University of Washington, Seattle
of dynamic futures portfolios. In [10] and [11], an optimal
WA 98195. E-mail: yzhou7@uw.edu. stopping approach for futures trading is studied. In practice,
3 Source: CME Group daily exchange volume and open interest re-
port, available at https://www.cmegroup.com/market-data/ dynamic futures portfolios are also commonly used to track
volume-open-interest/exchange-volume.html a commodity index [12].II. T HE M ULTISCALE C ENTRAL T ENDENCY Then, we write the evolution under the risk-neutral measure
O RNSTEIN -U HLENBECK M ODEL Q as:
(1) (2) (3) (1)
We now present the multiscale central tendency Ornstein- dXt = κ Xt + Xt − Xt − λ1 /κ dt
Uhlenbeck (MCTOU) model that describes the price dynam-
ics of the underlying asset. This leads to the no-arbitrage + σ1 dZtQ,1 , (8)
pricing of the associated futures contracts. Hence, the dy-
1
(2) (2)
namcis under both the physical measure P and risk-neutral dXt α2 − Xt − λ2 dt
=
pricing measure Q are discussed.
1
q
+ √ σ2 ρ12 dZtQ,1 + 1 − ρ212 dZtQ,2 , (9)
A. Model Formulation
√
(3) (3)
We fix a probability space (Ω, F, P). The log-price of the dXt = δ α3 − Xt − λ3 /δ dt + δσ3 ρ13 dZtQ,1
(1)
underlying asset St is denoted by Xt . Its evolution under q
Q,2 2 2
+ ρ23 dZt + 1 − ρ13 − ρ23 dZt Q,3
. (10)
the physical measure P is given by the system of stochastic
differential equations:
(1)
(2) (3) (1)
For convenience, we define
dXt = κ Xt + Xt − Xt dt + σ1 dZtP,1 , (1) (1) (2) (3)
Xt = (Xt , Xt , Xt )0 , (11)
(2) 1
(2)
dXt = α2 − Xt dt ZtP = (Zt
(P,1) (P,2)
, Zt , Zt
(P,3) 0
), (12)
1
q (Q,1) (Q,2) (Q,3) 0
P,1
+ √ σ2 ρ12 dZt + 1 − ρ12 dZt 2 P,2
, (2) ZtQ = (Zt , Zt , Zt ), (13)
µ= (0, α2 /, δα3 ) , 0
(14)
√
(3) (3)
dXt = δ α3 − Xt dt + δσ3 ρ13 dZtP,1 λ= (λ1 , λ2 , λ3 )0 , (15)
q κ −κ −κ
+ ρ23 dZtP,2 + 1 − ρ213 − ρ223 dZtP,3 , (3) K = 0 1/ 0 , (16)
0 0 δ
where Z P,1 , Z P,2 and Z P,3 are independent Brownian
σ1 0√ 0
motions under the physical measure P. Σ = 0 σ2 / √ 0 , (17)
Under this model, the mean process of log-price X (1) is 0 0 δσ3
(2) (3)
the sum of two stochastic factors, Xt and Xt , modeled
(2) and
by two different OU processes. The first factor Xt is fast
mean-reverting. The rate of mean reversion is represented by 1 p 0 0
1/, with > 0 being a small parameter corresponding to the C = ρ12 1 − ρ212 p 0 . (18)
(2) ρ13 ρ23 2 2
1 − ρ13 − ρ23
time scale of this process. Xt is an ergodic process and its
invariant distribution is independent of . This distribution Then, the evolution for Xt under measures P and Q can be
is Gaussian with mean α2 and variance σ22 /2. In contrast, written concisely as
(3)
the second factor Xt is a slowly mean-reverting OU
process. The rate of mean reversion is represented by a small dXt = (µ − KXt )dt + ΣCdZtP , (19)
parameter δ > 0. and
(1) (2) (3)
The three processes Xt , Xt , and Xt can be cor- dXt = µ − λ − KXt dt + ΣCdZtQ . (20)
related. The correlation coefficients ρ12 , ρ13 , and ρ23 are
constants, which satisfy |ρ12 | < 1 and ρ213 + ρ223 < 1.
Remark 1: If the stochastic mean of log price X (1) is only
We specify the market prices of risk as ζi , for i = 1, 2, 3,
modeled by X (2) or X (3) , instead of their sum, it will reduce
which satisfy
to the CTOU model, which is used in [13] for pricing VIX
dZtQ,i = dZtP,i + ζi dt, (4)
futures. Under this model, the futures portfolio optimization
where Z Q,1 , Z Q,2 and Z Q,3 are independent Brownian problem has been studied in [5].
motions under risk-neutral pricing measure Q. We introduce
the combined market prices of risk λi , for i = 1, 2, 3, defined III. F UTURES P RICING AND F UTURES T RADING
by A. Futures Pricing
λ1 = ζ1 σ1 , (5) Let us consider three futures contracts F (1) , F (2) and
(3)
1 F , written on the same underlying asset S with three
q
λ2 = √ σ2 ζ1 ρ12 + ζ2 1 − ρ212 , (6)
arbitrarily chosen maturities T1 , T2 and T3 respectively.
√
q Recall that the asset price is given by
λ3 = δσ3 ζ1 ρ13 + ζ2 ρ23 + ζ3 1 − ρ213 − ρ223 . (7) (1)
St = exp(Xt ), t ≥ 0.Then, the futures price at time t ∈ [0, T ] is given by The terminal conditions of a(k) (t) and β (k) (t) are given by
(1)
F (k) (t, x) := IE Q exp(XTk ) | Xt = x , a(k) (T ) = e1 , β (k) (T ) = 0.
(21) (33)
for k = 1, 2, 3. Define the linear differential operator By direct substitution, the solutions to ODEs (31) and (32)
0 are given by (26) and (27).
Q 1
L · = µ − λ − Kx ∇x · + Tr ΣΩΣ∇xx · , (22)
2 B. Dynamic Futures Portfolio
0 Now consider a collection of M contracts of different
where ∇x · = (∂x1 ·, · · · , ∂xN ·) is the nabla operator and
Hessian operator ∇xx · satisfies maturities available to trade, where M = 1, 2, 3. We note
2 that there are only three sources of randomness, so trading
∂x1 · ∂x1 x2 ·
. . . ∂ x1 xN ·
three contracts is sufficient. Any additional contract would
∂x1 x2 · ∂x22 ·. . . ∂x2 xN ·
∇xx · = .
.. .. .
(23) be redundant in this model. By Proposition 2, we have
.. ..
. . . (k)
2 dFt
∂x1 xN · ∂x2 xN · . . . ∂xN ·
(k)
= a(k) (t)0 λdt + a(k) (t)0 ΣCdZtP (34)
(k)
Ft
Then, for k = 1, 2, 3, the futures price function F (t, x) (k) (k)
solves the following PDE ≡ µF (t)dt + σF (t)0 dZtP , (35)
(∂t + LQ )F (k) (t, x) = 0, (24) where we have defined
(k) (k)
for (t, x) ∈ [0, T ) × RN , with the terminal condition µF (t) ≡ a(k) (t)0 λ, σF (t) ≡ C 0 Σ0 a(k) (t). (36)
F (k) (T, x) = exp(e01 x) for x ∈ RN , where e1 = (1, 0, 0)0 . Define !0
(1) (M )
dFt dFt
dFt = ,··· , . (37)
Proposition 2: The futures price F (k) (t, x) is given by (1)
Ft
(M )
Ft
F (k) (t, x) = exp a(k) (t)0 x + β (k) (t) , (25) Then, in matrix form, the system of futures dynamics is given
by the set of SDE:
where a(k) (t) and β (k) (t) satisfy dFt = µF (t)dt + ΣF (t)dZtP , (38)
(k)
a (t) = exp − (T − t)K e1 0
(26) where
0
(1) (M )
Z T µF (t) = µF (t), · · · , µF (t) (39)
(k)
β (t) = (µ − λ)0 a(k) (s) 0
(1) (M )
t ΣF (t) = σF (t), · · · , σF (t) . (40)
1
+ Tr ΣΩΣa(k) (s)a(k) (s)0 ds. (27) Here, we assume there be no redundant futures contract,
2
which means any futures contract could not be replicated by
Proof: We substitute the ansatz solution (25) into PDE other M − 1 futures contracts, indicating that rank (ΣF ) =
(24). The t-derivative is given by M.
(k) 0 ! Next, we consider the trading problem for the investor.
0
da (t) dβ (k) (t)
(1) (M )
∂t F (t, x) = x+ Let strategy πt = πt , · · · , πt , where the element
dt dt (k)
πt denotes the amount of money invested in k-th futures
× exp a(k) (t)0 x + β (k) (t) . (28) contract. In addition, we assume the interest rate be zero for
simplicity. Then, for any admissible strategy π, the wealth
Then, the first and second derivatives satisfy process is
M (k)
∇x F (t, x) = a(k) (t)F (t, x), (29) X (k) dFt
dWtπ = πt (k)
∇xx F (t, x) = a(k) (t)a(k) (t)0 F (t, x). (30) k=1 Ft
By substituting (28), (29), and (30) into PDE (24), we obtain = πt0 µF (t)dt + πt0 ΣF (t)dZtP . (41)
da(k) (t) We note that the wealth process is only determined by the
− K 0 a(k) (t) = 0, (31) strategy πt and it is not affected by factors variable X and
dt
futures prices F .
and
The investor’s risk preference is described by the expo-
dβ (k) (t) nential utility:
+ (µ − λ)0 a(k) (t)
dt
1
U (w) = − exp(−γw), (42)
+ Tr ΣΩΣa(k) (t)a(k) (t)0 = 0. (32)
2 where γ > 0 denotes the coefficient of risk aversion. Astrategy π is said to be admissible if π is real-valued with terminal condition h(T̃ ) = 1. From the first-order
progressively measurable and satisfies the Novikov condition condition, which is obtained from differentiating the terms
[14]: inside the supremum with respect to πt and setting the
Z T̃ 2 ! equation to zero, we have
P γ 0 0
IE exp π ΣF (s)ΣF (s)πs ds < ∞. (43) γµF (t) − γ 2 ΣF (t)Σ0F (t)πt = 0.
t 2 s (54)
Recall that rank(ΣF (t)) = M . Then, ΣF (t)Σ0F (t)
is an
The investor fixes a finite optimization horizon 0 < T̃ ≤
M × M invertible matrix. Accordingly, we have the optimal
T1 , which means that T̃ has to be less than or equal to
strategy (49). Given the fact that A0 A is the semi-positive
the maturity of the earliest expiring contract, and seeks an
definite matrix for any matrix A, the time-dependent com-
admissible strategy π that maximizes the expected utility of
ponent Λ2 (t) = µF (t)0 (ΣF (t)ΣF (t)0 )−1 µF (t) is always
wealth at T̃ :
non-negative.
u(t, w) = sup IE[U (WT̃π )|Wt = w], (44)
π∈At Substituting π ∗ back, the equation (53) becomes
where At denotes the set of admissible controls at the initial d 1
− h(t) + Λ2 (t)h(t) = 0. (55)
time t. Since the wealth SDE (41) does not depend on the dt 2
factors variable X and futures prices F , the value function Accordingly, we have
does not depend on them either.
1 T̃ 2
Z
To facilitate presentation, we define h(t) = exp − Λ (s)ds . (56)
1 2 t
π
L ·= · + πt0 ΣF (t)Σ0F (t)πt ∂ww · . (45)
πt0 µF (t)∂w
2
Then, following the standard verification approach to dy- Example 4: If there is only one futures contract F (1)
namic programming [15], the candidate value function available in the market. Then by (39), we have
u(t, w) and optimal trading strategy π ∗ is found from the
(1) (1)
Hamilton-Jacobi-Bellman (HJB) equation µF = µF , ΣF = σF (t)0 . (57)
∂t u + sup Lπ u = 0, (46) Then, the optimal strategy (49) becomes
π
(1)
1 µF (t) 1 µF
for (t, w) ∈ [0, T̃ ) × R, along with the terminal condition π (1)∗ (t) = = , (58)
u(T, w) = −e−γw , for w ∈ R. γ ΣF (t)Σ0F (t) γ σ (1) (t)0 σ (1) (t)
F F
Theorem 3: The unique solution to the HJB equation (46) (1) (1)
where µF and σF (t) are shown as (35).
is given by
! In order to quantify the value of trading futures to the
1 T̃ 2
Z
u(t, w) = − exp −γw − Λ (s)ds , (47) investor, we define the investor’s certainty equivalent asso-
2 t ciated with the utility maximization problem. The certainty
where equivalent is the guaranteed cash amount that would yield
−1
the same utility as that from dynamically trading futures
Λ2 (t) = µF (t)0 (ΣF (t)ΣF (t)0 ) µF (t) . (48) according to (44). This amounts to applying the inverse of
The optimal futures trading strategy is explicitly given by the utility function to the value function in (47). Precisely,
we define
1 −1
π ∗ (t) = (ΣF (t)Σ0F (t)) µF (t). (49) C(t, w) := U −1 (u(t, w)) (59)
γ
Proof: We will first use the ansatz Z T̃
1
=w+ Λ2 (s)ds. (60)
u(t, w) = −e−γw h(t). (50) 2γ t
Then, using the relations
Therefore, the certainty equivalent is the sum of the
∂t u = −e−γw ∂t h(t), ∂w u = γe−γw h(t), (51) investor’s wealth w and a non-negative time-dependent
1
R T̃ 2
component 2γ Λ (s)ds. The certainty equivalent is also
and t
inversely proportional to the risk aversion parameter γ,
∂ww u = −γ 2 e−γw h(t), (52)
which means that a more risk averse investor has a lower
the PDE (46) becomes certainty equivalent, valuing the futures trading opportunity
d
less. From (26), (36), (39) and (48), we see that the certainty
− h(t)+ sup γπt0 µF (t)h equivalent depends on the constant matrix K, volatility
dt πt
matrix Σ, correlation matrix C and market prices of risk
1 λ. Nevertheless, the certainty equivalent does not depend on
− γ 2 πt0 ΣF (t)Σ0F (t)πt h = 0, (53)
2 the current values of factors Xt .(1) (2) (3)
X0 X0 X0 α2 α3 One Futures Contract
1 0.5 0.5 0.5 0.5 0.08
δ σ1 σ2 σ3
0.05 0.01 0.8 0.02 0.3
0.06
ρ12 ρ13 ρ23 λ1 λ2 λ3
0 0 0 0.02 0.02 0.02
T1 T2 T3 T̃ γ κ 0.04
1/12 2/12 3/12 1/12 1 5
0 5 10 15 20
TABLE I Two Futures Contracts
1.0
Optimal Strategy π (*)
PARAMETERS FOR THE MCTOU MODEL .
0.5
0.0
1.4
Asset's Log-price X (1)
−0.5
1.2
0 5 10 15 20
Three Futures Contracts
1.0
0.8 2000
0.6
0 5 10 15 20 0
0.52 −2000
0 5 10 15 20
Price
0.50 Trading Day
0.48 Fast-Varying Factor X (2)
Slow-Varying Factor X (3) Fig. 2. Optimal strategies π (∗) for different futures combinations. Solid,
0 5 10 15 20 dashed and dotted lines represent the optimal position (in $) on T1 -futures,
T2 -futures and T3 -futures, respectively.
2.8
2.7
2.6
interval of the slow-varying factor X (3) is much narrower
2.5 Spot Price
T1-Futures Price
than the one for fast-varying factor X (2) . At the bottom, we
2.4 T2-Futures Price
T3-Futures Price plot the spot price and futures prices. The three paths for the
2.3
0 5 10 15 20 futures prices are highly correlated and T1 -futures price is
Trading Day equal to the asset’s spot price at its maturity date T1 , which
is the 21st trading day.
Fig. 1. Top: simulation paths for asset’s log-price X (1) . Dashed curves In Figure 2, we plot the optimal strategies as functions
represent 95% confidence interval. Middle: simulation path for fast varying of time for different portfolios and different correlation
factor X (2) and slow-varying factor X (3) . Dashed and dotted curves
represent 95% confidence interval for X (2) and X (3) , respectively. Bottom: parameters. In each sub-figure, from top to bottom, we show
Sample price paths for the underlying asset and associated futures. the optimal strategies for one-contract portfolio, two-contract
portfolio and three-contract portfolio respectively. The opti-
mal investments on T1 -futures, dashed lines represent the
IV. N UMERICAL I LLUSTRATION optimal investment on T1 -futures, T2 -futures, and T3 -futures
In this section, we simulate the MCTOU process and are represented by solid, dashed, and dotted lines respec-
illustrate the outputs from our trading model. With the tively. The optimal cash amount invested are deterministic
closed-form expressions obtained in the Section III, we now functions for time, but the optimal units of futures held do
generate the futures prices, optimal strategies and wealth vary continuously with the prevailing futures price.
processes numerically, using the parameters in Table I. Moreover, the investor takes large long/short positions
Primarily, we let and δ be small parameters and we in three-contract portfolio since all sources of risk can be
consider trading three futures with maturities T1 = 1/12 hedged. We provide sample path for wealth process for three-
year, T2 = 2/12 year and T3 = 3/12 year. Then, our trading contract portfolios in the Figure 3.
horizon will be T̃ = 1/12 year, no greater than the futures In Figure 4, we see that the certainty equivalent increases
maturities. We assume 252 trading days in a year and 21 as a function of trading horizon T̃ , which means that the
trading days in a month (or 1/12 year). In our figures, we more time the investor has, the more valuable is the trading
show the corresponding trading days on the x axis. opportunity. As the trading horizon reduces to zero, the
In Figure 1, we plot the simulation paths and 95% confi- certainty equivalent converges to the initial wealth w, which
dence intervals for three factors in the top figure and middle is set to be 0 in this example, as expected from (60). Also,
figure. As shown in the middle panel, the 95% confidence with a lower risk aversion parameter γ, the investor has aFutures Combinations (Maturity)
Parameters
T1 T2 T3 {T1 , T2 } {T1 , T3 } {T2 , T3 } {T1 , T2 , T3 }
ρ13 = −0.5 0.563 1.58 3.25 5.36 4.65 4.41 419
ρ12 = 0 ρ13 = 0 0.502 1.09 1.74 3.09 2.62 2.41 417
ρ13 = 0.5 0.456 0.837 1.19 2.88 2.35 2.01 417
ρ13 = −0.5 0.561 1.56 3.23 5.34 4.64 4.40 543
ρ12 = 0.5 ρ13 = 0 0.500 1.08 1.73 3.08 2.62 2.40 542
ρ13 = 0.5 0.454 0.833 1.18 2.87 2.34 2.01 541
ρ13 = −0.5 0.565 1.59 3.27 5.39 4.66 4.42 571
ρ12 = −0.5 ρ13 = 0 0.504 1.10 1.75 3.11 2.63 2.41 569
ρ13 = 0.5 0.457 0.842 1.20 2.90 2.36 2.02 569
TABLE II
C ERTAINTY EQUIVALENTS (×10−4 ) FOR ALL POSSIBLE FUTURES COMBINATIONS UNDER DIFFERENT CORRELATIONS .
higher certainty equivalent for any given trading horizon. 7 γ=1
Table II shows the certainty equivalents for all possible fu- γ = 0.8
γ = 0.6
tures combinations under various correlation configurations. 6
The certainty equivalent is much higher when more contracts
Certainty Equivalent (10−2)
are traded. In addition, if there is only one futures contract to 5
trade, the certainty equivalent is increasing with respect to its
4
maturity, see first three columns. The certainly equivalents
tend to be higher when ρ12 and ρ13 are negative. 3
2
1.8
1
1.6
0
0 5 10 15 20
1.4 Trading Horizon (Days)
Wealth
1.2 Fig. 4. Certainty equivalents for the three-futures portfolio as the trading
horizon T̃ and risk aversion parameter γ vary.
1.0
0.8 Three-Contract Portfolio
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