optimal portfolios in commodity futures markets

Finance StochDOI 10.1007/s00780-013-0224-5

Optimal portfolios in commodity futures markets

Fred Espen Benth · Jukka Lempa

Received: 18 October 2012 / Accepted: 13 September 2013© Springer-Verlag Berlin Heidelberg 2014

Abstract We develop a general approach to portfolio optimization in futures mar-kets. Following the Heath–Jarrow–Morton (HJM) approach, we model the entire fu-tures price curve at once as a solution of a stochastic partial differential equation.We also develop a general formalism to handle portfolios of futures contracts. In theportfolio optimization problem, the agent invests in futures contracts and a risk-freeasset, and her objective is to maximize the utility from final wealth. In order to captureself-consistent futures price dynamics, we study a class of futures price curve modelswhich admit a finite-dimensional realization. More precisely, we establish conditionsunder which the futures price dynamics can be realized in finite dimensions. Usingthe finite-dimensional realization, we derive a finite-dimensional form of the portfoliooptimization problem and study its solution. We also give an economic interpretationof the coordinate process driving the finite-dimensional realization.

Keywords Futures contract · Commodity markets · Optimal portfolios · Stochasticpartial differential equations · Finite-dimensional realization · Invariant foliation

Mathematics Subject Classification 91G10 · 60H15

JEL Classification G11 · G13 · C61

F.E. BenthCentre of Mathematics for Applications, University of Oslo, PO Box 1053, Blindern, 0316 Oslo,Norwaye-mail: [email protected]

J. Lempa (B)School of Business, Faculty of Social Sciences, Oslo and Akershus University College, P.O. Box 4,St. Olavs Plass, 0130 Oslo, Norwaye-mail: [email protected]

mailto:[email protected]

mailto:[email protected]

F.E. Benth, J. Lempa

1 Introduction

Futures contracts [16] form an important class of financial instruments exchanged oncommodity markets. These contracts convey the right to purchase or sell a specifiedquantity of the commodity at a fixed price at a fixed future date. At maturity, theholder must settle the contract. The price fixed in the contract is called the futuresprice and it is set such that no money is paid upfront, i.e., the initial value of a futurescontract is zero. In spite of this, a futures contract yields a cash flow during its lifetime generated by the changes in futures prices over time. Indeed, the party in whosefavor the futures price change occurs must immediately be paid the full amount ofthe change by the losing party. Typically, a margin account is set up for this purpose.

The purpose of this paper is to develop and analyze a general approach to portfoliooptimization in futures markets. We study utility maximization on a financial marketwhere the agent invests in futures contracts and a risk-free asset. The futures pricesare assumed to be settled continuously in time and the futures contracts are traded liq-uidly for every time-to-delivery y > 0. Following the Heath–Jarrow–Morton (HJM)approach (see [28]), we model the evolution of the futures price curve under a pric-ing measure using a stochastic partial differential equation. The price curve evolvesin an appropriate Hilbert space, whereas the futures portfolios are elements of itsdual. As a result, we face a portfolio optimization problem in infinite dimensions,since we can take positions in futures contracts for a continuum of times to deliv-ery. To tackle this issue, we restrict our attention to the class of futures price modelswhich admit a finite-dimensional realization in the following sense: for any given(sufficiently nice) initial curve, there exists a foliation, i.e., a parameterized familyof finite-dimensional linear submanifolds, such that the futures price curve will stayon this foliation over time. Consequently, the randomness in the price evolution willbe captured by a finite-dimensional Markov process, which keeps track of the coor-dinates of the price on the foliation. From the modeling point of view, the restrictionto models with finite-dimensional realizations is natural, since it guarantees that fu-tures prices are self-consistent. Having then the finite-dimensional structure at ourdisposal, we derive a finite-dimensional form of the portfolio optimization problemand study its solution.

Various classes of models admitting finite-dimensional realizations have beenstudied in recent years, mostly in connection with term structure models of interestrates; see also [13] for a study on self-consistent variance curve models. In a series ofpapers including [9–12], the authors study geometric aspects and finite-dimensionalrealizations of interest rate models using differential geometry and systems and con-trol theoretic methods. In another series of papers including [20, 21], see also [19], theauthors analyze the properties of finite-dimensional realizations using the so-calledconvenient analysis on Fréchet space. In these papers, they provide a general theoryon invariance of manifolds for infinite-dimensional stochastic equations and studyterm structure models of interest rates as an application. While mathematically verygeneral, their analysis is quite demanding for the reader. Recently, a more direct ap-proach to affine realizations of term structure models was introduced in [34]—wefollow this approach when studying our futures price model. We also refer to [18]and [32], where portfolio optimization with an infinite-dimensional state variable is


studied. These studies are concerned with optimal bond portfolios where the under-lying interest rate term structure follows a solution of a stochastic partial differentialequation. In addition to addressing a different application, these studies operate onthe infinite-dimensional level, whereas we study models with a finite-dimensionalrealization.

Our paper makes the following main contributions. First, we provide a generalmathematical framework for portfolio optimization on futures markets based on theHJM approach. Furthermore, we analyze this optimization problem in detail in thecase when the underlying price dynamics (described on an appropriate Hilbert space)admit a finite-dimensional realization. We provide conditions under which a giveninfinite-dimensional portfolio optimization problem can be solved in terms of a finite-dimensional control problem. In doing so, we contribute also to the existing theoryof realizations for our chosen class of stochastic partial differential equations servingas models of futures price evolution. In addition, we analyze economic interpreta-tions of the coordinate process and how a solution of the finite-dimensional controlproblem can be connected to the coordinate process and, consequently, back to theinfinite-dimensional portfolio problem. Indeed, we identify explicitly an affine rela-tion between the coordinate process and observable quantities on the futures pricecurve. This affine relation depends on the form of the family of affine manifolds onwhich the futures price evolves. Finally, we derive the HJB equation for the finite-dimensional portfolio optimization problem and establish a verification theorem.

The remainder of the paper is organized as follows. First we make a brief surveyof futures markets and the specifics of commodities. In Sect. 3, we set up a generalframework for modeling the futures curve and the portfolio optimization. In Sect. 4,we analyze the case of a finite-dimensional realization. The portfolio problem is re-cast as a finite-dimensional control problem in Sect. 5.

2 Commodity and energy futures markets

The classical commodity markets include trade in agricultural products, metals andenergy, but in recent decades, other more exotic markets for weather and freight haveemerged. Futures contracts constitute an important part of these markets, being toolsfor hedging and speculation.

If we consider for example a farmer, selling the crop in the futures market willlock in the price and therefore provide an insurance against potential losses in spotprices. On the other hand, in some markets like for example oil, there is no spot mar-ket, and the natural trading takes place in futures contracts delivering the commodityat future times. Interestingly, in most commodity markets the open interest in futurescontracts is far larger than the overall physical availability of the commodity, whichis a clear sign of large speculative positions taken in the market. As the players cancancel their contracts by taking an opposite position before maturity, one can do pureprice speculation without having to deliver or receive any commodity. This makescommodity futures markets attractive for investment banks and insurance companies,offering new asset classes for portfolio diversification different from more traditionalfinancial markets. Indeed, over the last decades great attention has been paid to in-vestments in various commodity markets based on speculative purposes rather than


physical needs. We refer to [25] and [29] for more on trading in commodity futuresmarkets. Our aim in the present paper is to investigate optimal portfolio managementin such markets.

An interesting example of a relatively new commodity market is that of electricity.This market has been a motivating example for the present study for several reasons.Electricity markets have emerged world-wide over the last decades, with a typicalsplit between a spot and a futures market. On the German power market EEX, forexample, an auction-based day ahead spot market is the underlying for the futurescontracts. Whereas the spot market is physical in the sense that a buyer must have thecapacity to receive power and a seller must be able to generate, the futures contractsare financial in the sense that the settlement is in terms of the money equivalent tothe power spot. Moreover, the contracts are settled against the average power price ina pre-defined period rather than at a fixed delivery time. There exist other markets,like the one for temperature futures at the Chicago Mercantile Exchange (CME), thatalso settle the contracts over periods. The periods in question are typically months,quarters or years. In a quantitative analysis of such futures, it is convenient to modelcontracts with fixed maturity times, although these do not exist for trade. We refer to[7] and [25] for more on power markets and their specific nature.

Trading in futures contracts is to some extent similar to more classical financial as-sets like stocks or bonds, as one can buy and sell these in an open market to speculateon prices going up or down. There are some major differences, though, that make theanalysis of portfolio optimization problems in these markets challenging. One majordifference is that one has many different contracts parameterized by time to maturity.On power, for example, one can enter futures contracts which settle next month, or intwo months from now, or in the next quarter, or next year. One can flexibly constructportfolios consisting of some or all of these available contracts. But as time moveson, some futures contracts run into delivery and therefore cannot be traded anymore.On the other hand, contracts with new delivery times enter the market and can beintegrated in the portfolio. Since one can also do over-the-counter deals, all deliverytimes are in principle available for trade, and the universe of assets to invest is notfinite.

The futures price is defined as the price to be paid for receiving the commodity.The price will be paid at delivery time, but is agreed upon when entering the contract.Typically, in a complete market where one can liquidly trade in the underlying spot,the futures price will be the cost of carrying the spot forward from contract entryuntil delivery. This spot-futures relationship is a simple consequence of the so-calledbuy-and-hold argument to hedge the futures (see e.g. [29, Chap. 3]). In commoditymarkets for agriculture, say, there are costs of storage involved, as well as the so-called convenience yield that gives a different relationship between the futures and thespot (see [25, Sect. 2.2] for a discussion of convenience yield and storage in variouscommodity markets). However, in some markets, like for example those of weatherand power, the underlying spot is not tradable in the sense that it cannot be integratedinto a portfolio like an asset. Temperature cannot be purchased and held like an asset,obviously, and the same goes for power as it is not a storable asset. Trading in spotelectricity means immediate consumption. Hence, for these commodities, there is nohedging argument in the underlying spot that naturally provides us with an arbitrage-free futures price dynamics. A reasonable way to mend this is to adopt the HJM


modeling approach from fixed-income theory (see [28]), and model the futures curvedynamics directly.

In [15, Chap. 4], an HJM approach is proposed for commodity futures markets, inparticular, for the case of energies like oil and gas. As in the case of interest rate the-ory, one models directly under the risk-neutral probability, which essentially meansto model the volatility term structure. Typical models propose a finite-dimensionalWiener process driving the stochastic dynamics, and under structural conditions onthe volatility term structure, one can recover a finite-dimensional realization of the fu-tures curve dynamics (see an extensive analysis of this based on the theory of Tappe[34] in the following sections). In [3], different HJM models for the power futuresmarket are analyzed.

As futures are tradable assets, we can use market prices for estimating the parame-ters in the HJM model. However, this requires a specification of the market dynamicsrather than a risk-neutral model. This means in practice that we must model the riskpremium in the market, defined as the difference between the forward price and thepredicted spot at delivery. In classical commodity markets, the risk premium is neg-ative, coming from the hedging needs of producers. In power markets, say, the riskpremium structure is more involved, and one can experience market situations wherethe premium is stochastically varying. We may see a positive premium in the shortend of the market as a result of consumers hedging their price risk (see [26]). Evenmore, there is empirical and theoretical evidence of a stochastic risk premium varyingwith the delivery times of the futures (see [2] and [5]).

In power markets, empirical studies show a high degree of idiosyncratic risk inthe logarithmic returns (log-returns) of futures prices. Koekebakker and Ollmar [31]found, using a principal component analysis, that a high number of factors is requiredto model the futures curve dynamics accurately. This is in sharp contrast to interestrate modeling, where typically the forward rates require 3 to 4 factors only (see [14,Sect. 1.7] for an extensive discussion on this). Later studies for power markets in[1] and [24] have shown that log-returns of futures prices with different times tomaturity (or different delivery times) are strongly correlated. The correlation is afunction of time to maturity and length of delivery of the futures contracts. Thisempirical evidence points strongly towards infinite-dimensional stochastic models,as each maturity will have its own risk factor. Moreover, this will also impact the riskpremium structure, as one may assign a risk loading to each maturity time.

The above-mentioned empirical studies also show that power futures log-returnsare leptokurtic, calling for non-Gaussian (Lévy) models. It is known that power spotprices are very volatile, with distinct spikes in the prices (see [7, Chaps. 1 and 3] forexamples). These extreme price variations are to some extent inherited in the futuresdynamics, but are considerably smoothed out by averaging over the delivery period.In other commodity markets like gas and oil, the spot fluctuates significantly lesscompared to power, and Gaussian models are reasonable. In our analysis, we focuson Gaussian models exclusively, although futures dynamics based on Lévy randomfields are called for in some commodity markets. We leave such an extension to futurestudies.

The Samuelson effect [33] is a crucial property of the volatility term structure incommodity markets. It is observed as an increasing volatility of the futures price as


time to maturity tends to zero. Eventually, the volatility becomes equal to the spotvolatility. The Samuelson effect is a consequence of stationarity properties of thespot price. It is notable that the Samuelson effect is violated in some futures marketswhere the contracts deliver over a period; see for example [7, Chap. 4] for the case ofpower markets and [6, Chap. 5] for weather markets.

3 The portfolio problem

3.1 Generation of wealth: a heuristic argument

As we mentioned already in the introduction, a futures contract is a derivative se-curity written on the futures price, which is denoted by ft . The initial market valueof this contract is zero, and it yields during its lifetime a cash flow generated by thefluctuations of the futures price. The profits (losses) caused by the fluctuations areput in (drawn from) a margin account such that the broker’s collateral remains on therequired level over time—we make later specific assumptions on the margin account.In order to describe the time evolution of the wealth generated by trading futurescontracts, consider first a single futures contract in discrete time. Denote the lengthof the time step as �t . The parties enter a single futures contract at time t with deliv-ery at time τ > t + �t . This position is canceled at time t + �t and a new contractis entered with the same time of delivery—this is the marking to market procedure.The resulting profit/loss is the associated fluctuation of the futures price, i.e.,

ft+�t (y − �t) − ft (y) = ft+�t (y) − ft (y) − (ft+�t (y) − ft+�t (y − �t)

),

where y = τ − t is the time-to-delivery at time t . When passing to the limit �t → 0,this gives the expression

dft (y) − ∂ft (y)

∂ydt.

For the sake of argument, we assume here that the futures price ft is a continuouslydifferentiable function with respect to time to maturity y for all t ≥ 0. The wealth dy-namics generated by a single futures contract investment in continuous time consistsnow of two parts: It is the sum of the interest generated by the current wealth on themoney account and the fluctuation in the futures price. Formally, it can be expressedas

dXt =(

rXt − ∂ft (y)

∂y

)dt + dft (y). (3.1)

Here, r > 0 denotes the risk-free rate of return.Next, we set up our model for the continuous-time futures price evolution and,

based on the intuition given by the previous heuristic argument, give a rigorous defi-nition of the portfolio optimization problem.


3.2 Futures price dynamics

We start by defining the function space in which the futures price curve t �→ ft

evolves. We use the Musiela parameterization and write the futures curve as a func-tion of time to maturity y ∈ R+. Following [19], see also [34], we fix a parameterα > 0 and let Hα be the space of all absolutely continuous functions h : R+ → Rsuch that

‖h‖α :=(

|h(0)|2 +∫ ∞

0eαy

∣∣∣∣∂h

∂y(y)

∣∣∣∣

2

dy

) 12

< ∞.

Here, the derivative ∂∂y

is understood in the weak sense. The space (Hα,‖ · ‖α) isa separable Hilbert space for which the point evaluation h �→ δy(h) : Hα → R isa continuous linear functional for each y ∈ R+—see e.g. [34, Th. 4.1]. We denoteby (St )t≥0 the semigroup of right shifts defined as Stf = f (· + t) for f ∈ Hα andt ≥ 0. Furthermore, denote the differential operator ∂

∂yby A and its domain by D(A).

Then we know that the semigroup (St )t≥0 is strongly continuous with infinitesimalgenerator A; see [34, Th. 4.1].

The Hilbert space Hα possesses many desirable properties for commodity futuresprice modeling. As the point evaluation map is continuous, we can naturally associatestochastic processes with values in Hα with the price dynamics of futures contractswith a fixed time to maturity. Empirically, one sees that futures price curves are rela-tively flat in the long end of the market, arguing for a space of curves where the deriva-tive tends asymptotically to zero. In the given Hilbert space, the (weak) derivative isasymptotically bounded by an exponential decay rate. One can dispense with this byintroducing other weight functions than exp(αy) in the definition of the Hilbert space(see [19, Sect. 5.1]) if one expects a different behavior of the curve in the long end ofthe market. Typical futures curves in commodity markets are in contango or backwar-dation, that is, either increasing or decreasing from current spot price towards somelong-run mean level. The curves are rather smooth as a function of time-to-delivery;see for example [25, Sect. 9.2] for different empirical futures curves in the oil andgas markets. In power markets, smooth curves have been applied to model the powercontracts with delivery period; see [4, 24].

Let U be a real separable Hilbert space. We assume that W is a Wiener processtaking values in U and defined on a complete filtered probability space (Ω,F ,F,Q),where the filtration F = {Ft }t≥0 satisfies the usual conditions. Denote the covarianceoperator of W as Q and, for an appropriate index set I , the associated eigenvectorsand values as {ei}i∈I and {λi}i∈I , respectively. We assume that Q has finite trace,i.e., that TrQ = ∑

λi < ∞. The family {ei}i∈I is an orthonormal basis for the spaceU , and we can represent W as

Wt =∑

i∈I

√λiW i

t ei ,

where W i are scalar Wiener processes. Finally, we define UQ = Q12 (U), where Q

12

is the pseudo-square root of the covariance operator Q.


To fix notation, denote the space of Hilbert–Schmidt operators from UQ to Hα

by LHS(UQ,Hα). Let Σ : Hα → LHS(UQ,Hα). Following the HJM approach, weassume that Q is a (local) martingale measure and model the futures price dynamicsas a solution of the stochastic partial differential equation

dft (·) = Aft(·) dt + Σ(ft (·)

)dWt = Aft(·) dt +

∑

i∈I

√λiσi

(ft (·)

)dW i

t (3.2)

with f0 ∈ Hα , where, for brevity, σi(h(·)) := Σi(h(·))(ei) ∈ Hα . Here, we assumethat the volatility Σ is Lipschitz-continuous. Then it is easy to check that the condi-tions of Theorem 7.4 in [17] are satisfied and, consequently, that (3.2) has a uniquemild solution. This solution can be expressed as

ft (·) = Stf0(·) +∫ t

0St−u

(Σ(fu(·)

)dWu

)

= Stf0(·) +∑

i∈I

√λi

∫ t

0St−u

(σi

(fu(·)

))dW i

u.

We point out that under our assumptions, the mild solution is also a weak solution;see e.g. Corollary 10.9 in [22].

We also need a description of the futures dynamics under the market measure P. Tothis end, we apply Girsanov’s theorem; see e.g. [17, Th. 10.14]. For a given ψ ∈ UQ,define the process W as

dWt = dWt − ψdt.

Then W is a U -valued Wiener process under an equivalent measure P defined via theRadon–Nikodým derivative dP

dQ = E(−ψ · W)T . Here, E denotes the Doléans–Dade

exponential; see [19, Chap. II]. Furthermore, the covariance operator of W is Q. Thefutures price dynamics can be written under the measure P as

dft (·) =(Aft(·) + Σ

(ft (·)

)(ψ)

)dt + Σ

(ft (·)

)dWt

=: ν(ft (·)

)dt + Σ

(ft (·)

)dWt . (3.3)

In financial terms, we can interpret the element ψ as the market price of risk. Be-ing an element of a Hilbert space, the market price of risk ψ can have essentiallyricher structure than its finite-dimensional counterpart, a constant vector. Indeed, ifthe space UQ is a space of functions of time to maturity y, the market price of riskwill also depend on y. Furthermore, we can interpret the term Σ(ft (·))(ψ) as therisk premium. We observe immediately that the risk premium is proportional to thevolatility of the futures price curve. Our stochastic dynamics allow a very flexiblemodeling of the risk premium, taking into account possible idiosyncrasies betweenfutures contracts with different maturities (recall the discussion in Sect. 2). Since theP-dynamics of the futures price curve is completely determined by the volatility Σ

and the market price of risk ψ , we call the pair (Σ,ψ) a futures price model.


3.3 Portfolios of futures contracts

Having set up the futures price dynamics, we proceed by defining admissible futuresportfolios. To this end, recall the expression (3.1),

dXt =(

rXt − ∂ft (y)

∂y

)dt + dft (y),

for a given y ≥ 0. We use this as a formal starting point. Using the expression (3.3),we rewrite (3.1) as

dXt =(

rXt + δy

(Σ(ft (·)

)(ψ)

))dt + δy

(Σ(ft (·)

)dWt

),

for a given y ≥ 0. This expression has a natural interpretation. Indeed, we ob-serve that the wealth dynamics are driven by the continuously compounding inter-est, the risk premium, and the random fluctuations in the futures price. Let Γ be anF-progressively measurable process taking values in the dual H ∗

α . Similarly to [18],we call this process a futures portfolio and define the wealth generated by this port-folio as

XΓt =

∫ t

0

(rXΓ

s + ⟨Γ (s),Σ

(fs(·)

)(ψ)

⟩)ds +

∫ t

0

⟨Γ (s),Σ

(fs(·)

)dWs

⟩, (3.4)

where XΓ0 ≥ 0, for all t ∈ [0, T ]. Here, 〈·, ·〉 is the bilinear pairing on (Hα,H ∗

α ). Forthe right-hand side of (3.4) to make sense, we assume that the portfolio process Γ

satisfies the L2-condition

E[(∫ T

0

⟨Γ (t),Σ

(ft (·)

)(ψ)

⟩dt

)2

+∫ T

0

∑

i∈Iλi

⟨Γ (t), σi

(ft (·)

)⟩2dt

]< ∞. (3.5)

Portfolio processes Γ satisfying the conditions (3.5) and

XΓt ≥ 0, 0 ≤ t ≤ T , P-a.s. (3.6)

will be called admissible. Given the time t and the value XΓt = x of the wealth pro-

cess, we denote the set of admissible portfolios by A(t, x).

Remark 3.1 We impose the constraint (3.6) on the wealth process as a simple condi-tion to rule out, in particular, possible doubling strategies. The credit line is assumedto be zero for reasons of convenience. A way of dealing with a negative (but finite)credit line can be described for instance as follows. Assume that the investor can bor-row money with the interest rate r . If the credit line is −a for 0 < a < ∞, we caninstead of (3.6) impose the admissibility condition

XΓt ≥ −e−r(T −t)a, 0 ≤ t ≤ T , P-a.s.

Indeed, if XΓ0 > −e−rT a and τa = inf{t > 0 : XΓ

t = −e−r(T −t)a} < T , the debt willbe equal to a at the time T and, consequently, no further credit will be given afterthe first hitting time τa . In this case, we have A(t, x) = {0} for all τa ≤ t ≤ T . For arecent study on the notion of admissibility with infinite credit line, we refer to [8].


Remark 3.2 The admissible portfolio strategies Γ describe the investor’s allocationof wealth over the whole futures curve. This will include so-called roll-over strate-gies, where one invests in a single futures contract with a given time to maturity y.By picking more times to maturity y1, . . . , yn, one can create roll-over portfolios offutures contracts.

3.4 The optimization problem

The objective of the investor is to determine an admissible investment policy whichmaximizes the expected utility at a terminal time T . To study this problem, we firstpropose an appropriate set of utility functions. We call a function u : R → [−∞,∞)

a utility function if

(1) it is concave, nondecreasing and upper-semicontinuous,(2) the half line dom(u) := {x ∈ R | u(x) > −∞} is a nonempty subset of [0,∞),(3) u′ is continuous, positive and strictly decreasing on the interior of dom(u), and

limx→∞ u′(x) = 0;

see [30, Sect. 3.4]. Given a utility function u, the portfolio optimization problemreads

V (t, x) = supΓ ∈A(t,x)

E[u(XΓ

T )∣∣XΓ

t = x], V (T , x) = u(x), (3.7)

with 0 ≤ t ≤ T and x ≥ 0.

4 Finite-dimensional realizations

4.1 Invariant foliation: a characterization

In this section, we assume that the real separable Hilbert space U is truncated to befinite-, say, n-dimensional. We can identify the truncated U with the Euclidean spaceRn. From a practical point of view, this corresponds for example to the case where theeigenvalues λi are very small for all i > n. Thus the associated scalar processes W i ,i > n, contribute very little to the overall fluctuations of the process W , and conse-quently the processes Wi , i ≤ n, can be identified as the principal components drivingthe futures curve ft ; see e.g. [14, Sect. 1.7]. We denote the truncated (i.e., Rn-valued)Wiener process as W . We remark that the market price of risk ψ degenerates nowinto a constant vector (ψ1, . . . ,ψn) ∈ Rn. For instance, in [27] a two-factor modelfor the oil futures price dynamics is proposed, which corresponds in our context toassuming that the futures curve is driven by a two-dimensional Wiener process. Onthe other hand, in electricity there is empirical and theoretical evidence for a highdegree of idiosyncratic risk, which means that a high-dimensional Brownian motionis required for the futures curve dynamics (see [2, 31]).

The first objective of this section is to pin down conditions on the volatility struc-ture Σ under which the price dynamics given as the solution of the SPDE (3.3) admitsa finite-dimensional realization. From the modeling point of view, this is importantsince it will guarantee that the futures prices are self-consistent. The existence of a


finite-dimensional realization means, roughly speaking, that for a given initial curveh0, there exists a nice family of manifolds in Hα such that the initial curve is on theinitial manifold and that the evolution of the curve in confined to the family of mani-folds. This problem setting has been studied extensively during the last decade or so;references include [9–12, 19, 20, 34]. In our analysis, we follow the approach of [34].We proceed by making the following definitions.

Definition 4.1 Let V be a linear d-dimensional subspace of Hα . Then:

(A) A family M := (Mt )t≥0 of linear submanifolds of Hα is called a foliation gen-erated by V if there exists a continuously differentiable θ : R+ → Hα such thatMt = θ(t) + V for all t ≥ 0. Here, θ is called the parameterization of M. Anal-ogously, the tangent space is defined as TMt := θ ′(t) + V for all t ≥ 0.

(B) A foliation M is called invariant for the SPDE (3.3) if for all t0 ∈ R+ andh0 ∈ Mt0 , we have P(ft ∈ Mt0+t ) = 1, t ≥ 0, where f is the (weak) solutionfor (3.3) with f0 = h0.

Definition 4.2 Let V be a linear d-dimensional subspace of Hα . Then Equation (3.3)is said to have a finite-dimensional realization generated by V if for all h ∈ D(A),there exists a foliation Mh = (Mh

t )t≥0 generated by V with h ∈ Mh0 which is invari-

ant for (3.3).

As we see from these definitions, the invariant foliation is the basic building blockof a finite-dimensional realization. The next proposition gives necessary and suffi-cient conditions for invariance of a given foliation for the SPDE (3.3). We remarkthat this proposition is analogous to Theorem 5.3 in [34], where a similar characteri-zation is proved when the underlying price dynamics are driven by a one-dimensionalWiener process. In our result, we consider the case where the driver is a multi-dimensional Wiener process. In addition, the drift term in the futures price dynamics(3.3) cannot be handled immediately using existing results. Indeed, Theorem 5.3 in[34] is concerned with the HJM approach of term structure modeling of forwardrates which leads to the well-known HJM drift condition for the underlying dynam-ics, whereas we consider a different drift condition stemming from the fact that weanalyze futures prices. For convenience, we recall the definition of ν as

ν(h) = Ah + Σ(h)(ψ),

where h ∈ D(A). Furthermore, we make the following assumption on the volatilitystructure. These assumptions allow us in particular to use Theorem 2.11 in [34]. (Wepoint out that Theorem 2.11 in [34] is proved for a one-dimensional driving Brownianmotion, but we observe that the result holds also for a multi-dimensional Browniandriver by simply plugging it into the proof.)

Assumption 4.3 We assume that the components σj are continuously differentiableand Lipschitz-continuous.

Proposition 4.4 Let M be a foliation generated by the d-dimensional linear spaceV ⊂ Hα spanned by elements {vi}di=1. The following statements are equivalent:


(A) The foliation M is invariant for Equation (3.3).(B) We have

{θ(t) ∈ D(A) for all t ≥ 0,

vi ∈D(A) for all i = 1, . . . , d,(4.1)

and there exist functions β ∈ C0,1(R+ × Rd;Rd), κ ∈ C0,1(R+ × Rd;Rd × Rn)

such that⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

ν(θ(t)

) = θ ′(t) +d∑

i=1

βi(t,0)vi,

σj

(θ(t) +

d∑

i=1

zivi

)=

d∑

i=1

viκij (t, z), j = 1, . . . , n,

(4.2)

for all (t, z) ∈ R+ × Rd , and the elements vk satisfy the ordinary differentialequations

d

dyvk −

d∑

i=1

vi

∂

∂zk

(βi(t, z) −

n∑

j=1

κij (t, z)ψj

)= 0, (4.3)

for all k = 1, . . . , d .

Proof (A) ⇒ (B) : Fix t ≥ 0 and assume that the foliation M is invariant for (3.3).Then we know from Theorem 2.11 in [34] that Mt and consequently θ(t) and vi arein D(A) for all t ≥ 0 and i = 1, . . . , d . Let h ∈ Mt and write h = θ(t) +∑d

i=1 zhi vi ,

where zh ∈ Rd . Again, we know from Theorem 2.11 in [34] that ν(h) ∈ TMt andσj (h) ∈ V for all j = 1, . . . , n. This yields

σj (h) =d∑

i=1

zσj (h)

i vi,

where zσj (h) ∈ Rd for each j = 1, . . . , n. Define the linear isomorphism I : Rd → Vas I(z) = ∑d

i=1 zivi . Then we can write

κ·j (t, zh) := zσj (h) = I−1

(

σj

(θ(t) +

d∑

i=1

zhi vi

))

(4.4)

for all j = 1, . . . , n. Since I is linear and σj is continuously differentiable, it is easyto check that κ·j is continuously differentiable in z.

To prove the claim on ν, we find similarly that

ν(h) = d

dyθ(t) +

d∑

i=1

zhi

d

dyvi +

d∑

i=1

n∑

j=1

vizσj (h)

i ψj = θ ′(t) +d∑

i=1

zν(h)i vi, (4.5)


where zν(h) ∈ Rd . Then we find using (4.4) that

β(t, zh) := zν(h)

= I−1(

d

dyθ(t) +

d∑

i=1

(zhi

d

dyvi +

n∑

j=1

viκij (t, zh)ψj

)− θ ′(t)

). (4.6)

The linearity of I and differentiability of σj ensure that β is continuously differ-entiable in z. Finally, the differential equations (4.3) follow by differentiating theexpression (4.5) with respect to zh

k for all k = 1, . . . , d .(B) ⇒ (A): Assume now that the conditions (4.1)–(4.3) hold and let h ∈ Mt .

Using Theorem 2.11 in [34], it suffices to show that

ν

(θ(t) +

d∑

i=1

zhi vi

)= θ ′(t) +

d∑

i=1

βi(t, zh)vi (4.7)

to prove the claim. To this end, we observe first that σj (θ(t)) = ∑di=1 viκij (t,0) for

all j = 1, . . . , n. Thus we can write

ν

(θ(t) +

d∑

i=1

zhi vi

)

= ν(θ(t)

)+d∑

i=1

zhi

d

dyvi +

n∑

j=1

d∑

i=1

vi

(κij (t, z

h) − κij (t,0))ψj

= θ ′(t) +d∑

i=1

(vi

(βi(t,0) +

n∑

j=1

(κij (t, z


)+ zh

i

d

dyvi

).

By using the differential equations (4.3) and changing the order of summation, wefind that

d∑

i=1

(vi

(βi(t,0) +

n∑

j=1

(κij (t, z


)+ zh

i

d

dyvi

)

=d∑

i=1

(vi

(βi(t,0) +

n∑

j=1

(κij (t, z


)

+ zhi

d∑

k=1

vk

∂

∂zi

(βk(t, z

h) −n∑

j=1

κkj (t, zh)ψj

))

=d∑

k=1

((βk(t,0) −

n∑

j=1

κkj (t,0)ψj

)

+d∑

i=1

zhi

∂

∂zi

(βk(t, z

h) −n∑

j=1

κkj (t, zh)ψj

)+

n∑

j=1

κkj (t, zh)ψj

)vk.


To proceed, we readily verify that

β(t, z) −n∑

j=1

κ·j (t, z)ψj = β(t, z) − (I−1 ◦ I)

( n∑

j=1

κ·j (t, z)ψj

)

= I−1(

d

dyθ(t) −

d∑

i=1

zhi

d

dyvi − θ ′(t)

). (4.8)

On the other hand, since I is a linear isomorphism, the inverse I−1 : V → Rd isalso linear. Thus we observe from the expression (4.8) that the partial derivatives∂

dzi(β(t, z) −∑n

j=1 κ·j (t, z)ψj ) are independent of z. Consequently, we find that

β(t,0) −n∑

j=1

κ·j (t,0)ψj +d∑

i=1

zi

∂

∂zi

(β(t, z) −

n∑

j=1

κ·j (t, z)ψj

)

is the Maclaurin series of the function β − ∑nj=1 κ·jψj . This proves the iden-

tity (4.7). �

Proposition 4.4 gives a convenient characterization of invariance for a given fo-liation. First, it gives a set of ordinary differential equations (4.3) which have thespanning functions vi as their solutions. The coefficients of these ODEs are charac-terized in terms of the original volatility structure Σ and the market price of risk ψ .Furthermore, we observe that the volatility Σ must be of a very specific type in orderto be associable to an finite-dimensional realization. Indeed, each of the componentsσj must map the manifold M into the linear space V for all t . This means that therandomness in the price dynamics is confined to the finite-dimensional linear struc-ture V . Put differently, the price dynamics admitting a finite-dimensional realizationcan diffuse only in finitely many directions in the infinite-dimensional state space Hα .

We can identify the coordinate process Z driving the finite-dimensional realizationusing Proposition 4.4. To this end, fix t0 ∈ R+ and h ∈ Mt0 . Then we have a uniquez ∈ Rd such that

h = θ(t0) +d∑

i=1

zivi .

Now, let Z be the strong solution of the Itô equation

dZt = β(t0 + t,Zt )dt + κ(t0 + t,Zt )dWt , Z0 = z,

where β and κ are given by Proposition 4.4. Then it is straightforward to verify usingItô’s formula that the Hα-valued process f defined as

ft := θ(t0 + t) +d∑

i=1

Zit vi, f0 = h, (4.9)

is the strong solution of the SPDE (3.3).


Remark 4.5 By coupling the expression (4.9) with the condition (4.1), we observethat ft ∈ D(A) for all t ≥ 0. In particular, the curve ft is continuously differentiablefor all t ≥ 0. Hence, the heuristic derivation of the wealth process X in Sect. 3.1 isrigorous in the finite-dimensional setup of this section. The wealth process XΓ cannow be expressed as

XΓt =

∫ t

0

(rXΓ

s −⟨Γ (s),

∂fs(·)∂y

⟩)ds +

∫ t

0〈Γ (s), dfs(·)〉 , (4.10)

for all t ≥ 0.

4.2 Interpretation of the coordinate process

The coordinate process Z has a priori no intrinsic economical interpretation. How-ever, we can equip it with one. To this end, we observe that the futures curve can bedecomposed as

ft = πVft + πV⊥ft = πVft + πV⊥Mt+t0 .

Here, the latter equality (which holds almost surely) follows from the invariance ofthe foliation M and the fact that the projection πV⊥Mt+t0 is a singleton set. We pointout that the projection πV⊥Mt+t0 can be used as a parameterization of the foliationand that it is the unique parameterization which is in V⊥ for all t ≥ 0. According to(4.9), we can rewrite the futures price as

ft = πV⊥Mt+t0 +d∑

i=1

Zit vi .

Denote a basis of the subspace V⊥ as {wj }∞j=1 and let Λ : Hα → Rd be a linear

continuous operator such that Λ(V) = Rd . Then we can write

Λ(ft ) = Λ(πV⊥Mt+t0

)+ Λ

( d∑

i=1

Zit vi

)

=∞∑

j=1

cjt Λ(wj ) +

d∑

i=1

Zit Λ(vi)

=∞∑

j=1

cjt

d∑

i=1

bjiΛ(vi) +d∑

i=1

Zit Λ(vi)

=d∑

i=1

(Zi

t +∞∑

j=1

cjt bji

)Λ(vi).


Define the matrix Θ as Θij = Λi(vj ), where i, j = 1, . . . , d . Then we can write

Zt = Λ(ft )Θ−1 −

∞∑

j=1

cjt bj ·.

In other words, we observe that the state process Z can be associated with the imageof f of an arbitrary linear operator Λ modulo an affine transformation depending onΛ and the invariant foliation M. This means in particular that the coordinate processZ becomes an observable quantity as a functional transformation of quantities whichare observed from the futures price curve.

Example 4.6 Consider the operator Λ given by benchmark contracts for times tomaturity yi , i = 1, . . . , d , defined as

Λi(h) = δyi(h), h ∈ Hα.

Here δ is the evaluation functional on Hα , δy(h) = h(y). Then we can write

Zit =

d∑

j=1

ft (yj )Θ−1ji −

∞∑

j=1

cjt bji ,

for all i = 1, . . . , d . The dimension d gives the number of benchmark contractsneeded to reconstruct the whole futures curve t �→ ft via the identity (4.9). Indeed,we need d benchmark contracts in order to have a connection between the coordi-nate process Z and the benchmark contracts f·(yi). This connection is given by theinvertible d × d-matrix Θ with coefficients Λi(vj ) = vj (yi), i, j = 1, . . . , d .

4.3 Existence of a finite-dimensional realization: sufficient conditions

In the previous subsection, we have proved a characterization of the invariance of agiven foliation M for the SPDE (3.3). In this subsection, we use this result to givesufficient conditions for the existence of a finite-dimensional realization. To this end,consider the futures price dynamics given by (3.3) when the volatility is of the form

σj (h) =p∑

i=1

viΦij (h), h ∈ Hα, (4.11)

for all j = 1, . . . , n. Here, the functions vi , i = 1, . . . , p, are linearly independentand each Φij maps Hα into R. We assume that the functionals Φij are continuouslydifferentiable and Lipschitz-continuous for all i = 1, . . . , d and j = 1, . . . , n. Thisguarantees that we can use Proposition 4.4. Equation (3.3) can now be written as

dft (·) =(

Aft(·) +n∑

j=1

p∑

i=1

viΦij

(ft (·)

)ψj

)dt +

n∑

j=1

p∑

i=1

viΦij

(ft (·)

)dW

jt , (4.12)


with f0 = f , where W j are scalar P-Brownian motions. We remark that the ex-pression (4.11) gives actually a necessary condition for the existence of a finite-dimensional realization; see [34, Lemma 3.2]. Thus this form of volatility is the mostgeneral we can consider in this framework.

Define inductively the domains D(An) := {h ∈ D(An−1) | An−1h ∈ Hα} for alln ≥ 2, and the intersection D(A∞) = ⋂∞

n=1 D(An). An element h ∈ D(A∞) iscalled quasi-exponential if the linear space spanned by the family {Anh}n≥1 is finite-dimensional. The following proposition, which is analogous to Proposition 6.2 in[34], gives sufficient conditions for the existence of a finite-dimensional realization.

Proposition 4.7 Assume that the volatility structure of (3.3) is given by (4.11) andthat the functions vi , i = 1, . . . , p, are quasi-exponential. Then Equation (3.3) admitsa finite-dimensional realization.

Proof Since the elements vi , i = 1, . . . , p, are quasi-exponential, the linear space

Y :=p⊕

i=1

span

{dn

dynvi : n ≥ 0

}⊂ D(A)

is by definition finite-dimensional. Furthermore, we observe that

d

dyv ∈ Y, v ∈ Y. (4.13)

Set d := dimY and choose vp+1, . . . , vd ∈ Y such that {v1, . . . , vd} is a basis of Y .Fix an arbitrary h0 ∈D(A). First, define the function θ : R+ → Hα as

θ(t) := Sth0 ∈ D(A),

and the function κ ∈ C0,1(R+ × Rd;Rd × Rn) as

κij (t, z) ={

Φij (θ(t) +∑dk=1 zkvk), i = 1, . . . , p,

0, i = p + 1, . . . , d,(4.14)

for all j = 1, . . . , n. We observe from (4.14) that the latter condition in (4.2) is satis-fied. Furthermore, define the function β ∈ C0,1(R+ × Rd;Rd) as

βi(t, z) =n∑

j=1

κij (t, z)ψj +d∑

k=1

aikzk, (4.15)

where aij are chosen, due to (4.13), such that

d

dyvi =

d∑

j=1

vjaji, (4.16)

for all i = 1, . . . , d . With this specification, we verify by differentiating (4.15) withrespect to zk that the elements vi satisfy the differential equations (4.3). Finally, thedefinition of β coupled with the fact that d

dyθ(t) = θ ′(t) = h0

′(y + t) implies that theformer condition in (4.1) is also satisfied. �


Remark 4.8 Proposition 4.7 points out the important role of quasi-exponential func-tions in our theory. As we have seen, we can start by specifying the volatility in theunderlying dynamics as a (potentially state-dependent) linear combination of quasi-exponential functions, and as a result obtain a futures price model admitting a finite-dimensional realization.

Proposition 4.7 and its proof give us not only sufficient conditions for the existenceof a finite-dimensional realization, but also a recipe for its construction. To illustratethis, we consider the example from Sect. 6 in [9].

Example 4.9 Assume that the driving Brownian motion in (4.12) is one-dimensional,the volatility functional Φ ≡ 1 and the quasi-exponential function v1(x) = xe−ax

for a > 0, i.e., σ = σ1 = v1. In addition, let for simplicity ψ = 0. We observe thatV = span{v1, v2}, where v2(y) = e−ay , so the dimension of the affine realization is 2.It is a matter of differentiation to show that (see (4.16)):

(v′1 v′

2) = (v1 v2)

(−a 01 −a

)=: vA.

Thus the state variable Z is given, due to (4.14) and (4.15), as the strong solution ofthe two-dimensional Itô equation

dZt = AZt dt + BdWt ,

where B = (1,0)� and W is a Q-Brownian motion. Now, given the initial h0 ∈ D(A),the affine realization of futures price reads as

ft (·) = h0(· + t0 + t) + Z1t v1(·) + Z2

t v2(·).

It is worth pointing out that the process Z becomes an Ornstein–Uhlenbeck processwith solution given by

Zt = exp(At)Z0 +∫ t

0exp

(A(t − s)

)BdWs.

As the eigenvalue of the matrix A is −a, that is, negative, Z becomes stationary.Many popular spot and futures price dynamics of commodities and energy have sta-tionarity as a crucial property; see [7, Chap. 3] for examples in energy. The model inthe current example gives rise to a hump-shaped futures price curve, which is relevantin, say, oil markets (see [25, Chap. 9.2]).

5 Portfolio optimization revisited: finite-dimensional state variable

In this section, we recast the optimization problem (3.7) into a finite-dimensionalsetting using the identity (4.9). Assume that the initial futures price curve can be


expressed as f0 = St0h0 +∑di=1 zivi , where h0 ∈ D(A) and t0 ∈ R+. Then the affine

futures price curve dynamics is given by

ft = StSt0h0 +d∑

i=1

Zit vi .

Furthermore,

dft = StSt0

∂h0

∂ydt +

d∑

i=1

dZit vi .

By coupling this with the representation (4.10), we find that

XΓt =

∫ t

0

(rXΓ

s −⟨Γ (s),

∂fs

∂y

⟩)ds +

∫ t

0

⟨Γ (s),

d∑

i=1

dZisvi

⟩

+∫ t

0

⟨Γ (s), SsSt0

∂h0

∂y

⟩ds. (5.1)

It is a matter of differentiation to show that

SsSt0

∂h0

∂y− ∂fs

∂y= SsSt0

∂h0

∂y− SsSt0

∂h0

∂y−

d∑

i=1

Zis

∂vi

∂y= −

d∑

i=1

Zis

∂vi

∂y.

By coupling this with the expression (4.16), we observe that (5.1) can be written as

XΓt =

∫ t

0rXΓ

s ds +∫ t

0

⟨Γ (s),

d∑

i=1

dZisvi −

d∑

i=1

Zis

∂vi

∂yds

⟩

=∫ t

0rXΓ

s ds +∫ t

0

⟨Γ (s),

d∑

i=1

dZisvi −

d∑

i=1

Zis

d∑

j=1

vjaji ds

⟩

=∫ t

0rXΓ

s ds +∫ t

0

⟨Γ (s),

d∑

i=1

vi

(dZi

s −d∑

j=1

aijZjs ds

)⟩

=∫ t

0rXΓ

s ds +∫ t

0

d∑

i=1

γ is

(dZi

s −d∑

j=1

aijZjs ds

),

where the real-valued processes γ i := 〈Γ,vi〉 satisfy the condition (3.5). Finally, byusing (4.15), we find

XΓt = X

γt =

∫ t

0rX

γs ds +

∫ t

0

d∑

i=1

γ is

n∑

j=1

κij (s,Zs)(dWjs + ψj ds)

=∫ t

0rX

γs ds +

∫ t

0γ �s (dZs − AZs ds), (5.2)


where W is a P-Wiener process, the superscript � denotes the transpose, and thematrix A is given by (4.16). By plugging this into the utility maximization problem(3.7), we reduce the initial infinite-dimensional control problem to a classical finite-dimensional control problem. We write this in the Markovian form

V (t, x, z) = supγ

E[u(X

γ

T )∣∣Xγ

t = x,Zt = z]. (5.3)

We can use classical control-theoretic techniques to study this problem; see e.g. [23,Chap. IV]. To illustrate this, we derive the HJB equation for this problem. Assumingthat the function V is sufficiently smooth, Itô’s formula yields

dV (t,Xγt ,Zt ) =

[Vt (t,X

γt ,Zt ) + Vx(t,X

γt ,Zt )

(rx +

d∑

i=1

γi

n∑

k=1

κik(t,Zt )ψk

)

+d∑

i=1

Vzi(t,X

γt ,Zt )βi(t,Zt )

+ 1

2Vxx(t,X

γt ,Zt )

d∑

i=1

n∑

k=1

γiκij (t,Zt )κji(t,Zt )γi

+ 1

2

d∑

i,j=1

Vzizj(t,X

γt ,Zt )

n∑

k=1

κjk(t,Zt )κki(t,Zt )

+d∑

i=1

Vxzi(t,X

γt ,Zt )

n∑

k=1

κik(t,Zt )κki(t,Zt )γi

]dt

+ Vx(t,Xγt ,Zt )

d∑

i=1

γi

n∑

k=1

κik(t,Zt )dW kt

+d∑

i=1

Vzi(t,X

γt ,Zt )

n∑

k=1

κik(t,Zt )dW kt . (5.4)

By a standard martingale argument from stochastic control, see e.g. [23, Sects. IV.2and IV.3], this yields the HJB equation

supγ

{Vt (t, x, z) + Vx(t, x, z)

(rx +

d∑

i=1

γi

n∑

k=1

κik(t, z)ψk

)

+d∑

i=1

Vzi(t, x, z)βi(t, z) + 1

2Vxx(t, x, z)

d∑

i=1

n∑

k=1

γiκij (t, z)κji(t, z)γi

+1

2

d∑

i,j=1

Vzizj(t, x, z)

n∑

k=1

κjk(t, z)κki(t, z)

+d∑

i=1

Vxzi(t, x, z)

n∑

k=1

κik(t, z)κki(t, z)γi

}= 0,


where V (T , x, z) = u(x) for all x, z ∈ R. As a consequence, we can formulate thefollowing verification result.

Proposition 5.1 Assume that we have:

(1) a smooth function V : [0, T ] × Rd × R → R with V (T , z, x) = u(x) such that

Vt (t, x, z) + Vx(t, x, z)

(rx +

d∑

i=1

γi

n∑

k=1

κik(t, z)ψk

)+

d∑

i=1

Vzi(t, x, z)βi(t, z)

+ 1

2Vxx(t, x, z)

d∑

i=1

n∑

k=1

γiκij (t, z)κji(t, z)γi

+ 1

2

d∑

i,j=1

Vzizj(t, x, z)

n∑

k=1


+d∑

i=1

Vxzi(t, x, z)

n∑

k=1

κik(t, z)κki(t, z)γi ≤ 0

for all t, γ, x, z;(2) a function γ ∗ : [0, T ] × R × Rd → Rd such that

Vt (t, x, z) + Vx(t, x, z)

(rx +

d∑

i=1

γ ∗i (t, x, z)

n∑

k=1

κik(t, z)ψk

)

+d∑

i=1

Vzi(t, x, z)βi(t, z)

+ 1

2Vxx(t, x, z)

d∑

i=1

n∑

k=1

γ ∗i (t, x, z)κij (t, z)κji(t, z)γ

∗i (t, x, z)

+ 1

2

d∑

i,j=1

Vzizj(t, x, z)

n∑

k=1


+d∑

i=1

Vxzi(t, x, z)

n∑

k=1

κik(t, z)κki(t, z)γ∗i (t, x, z) = 0

for all t, z, x;(3) a wealth process X∗ corresponding to an admissible control γ such that we have

γt = γ ∗(t,X∗t , z) for all t ≥ 0 and z ∈ Rd .

Then the function V solves the control problem (5.3) (i.e., V = V ), and an optimalcontrol is given by γ .


Proof Let γ be an admissible control and Xγ the associated wealth process. ThenItô’s formula yields (5.4) with V replaced by V . Since the term on the right-handside inside the square brackets is nonpositive by (1), we obtain

u(Xγ

T ) = V (T ,Xγ

T ,ZT )

≤ V (t,Xγt ,Zt ) +

∫ T

t

(Vx(s,X

γs ,Zs)

d∑

i=1

γi(s)

n∑

k=1

κik(s,Zs) dW ks

+d∑

i=1

Vzi(s,X

γs ,Zs)

n∑

k=1

κik(s,Zs) dW ks

).

Since γ is admissible, the stochastic integral is a true martingale, and consequentlyE[u(X

γ

T ) |Xγt = x,Zt = z] ≤ V (t,X

γt ,Zt ). This shows that V dominates the solu-

tion V . To show that V ≤ V and that γ gives an optimal control, we use the sameargument coupled with the conditions (2) and (3). �

Proposition 5.1 gives a set of sufficient conditions for a given smooth functionto coincide with the value function of the control problem (5.3). Moreover, it givesan identification of an optimal control. Even though the control γ does not ap-pear to have a direct economic interpretation, it can be traced back for example tobenchmark futures prices following Example 4.6. In this example, the d-dimensionalcoordinate process Z was identified with an affine transform of d benchmark fu-tures prices. On the other hand, as we observe from Equation (5.2), the processγ can be understood as “portfolio weights” on d-dimensional diffusion dynamicst �→ ∫ t

0 κ(s,Zs)(dWs + ψ ds). These dynamics can be extracted from the evolutionof the coordinate process Z.

We end our paper with discussing a futures price dynamics typical for energymarkets. Suppose that the dynamics is given as

dft (y) = (Aft(y) + σ1ψ1 + σ2e

−ayψ2)

dt + σ1 dW 1t + σ2e

−ay dW 2t (5.5)

for a two-dimensional Brownian motion W and positive constants σ1, σ2, a. For sim-plicity, we also assume the market prices of risk ψi , i = 1,2, to be constants. In thismodel, we identify Φ12 = Φ21 = 0, Φ11 = σ1 and Φ22 = σ2. Furthermore, v1(y) = 1and v2(y) = exp(−ay), which obviously are quasi-exponential functions spanning alinear space of dimension 2. As v′

1(y) = 0 and v′2(y) = −av2(y), we find the dynam-

ics of the two-dimensional process Z to be

dZt = (Ψ + AZt) dt + BdWt

with Ψ = (σ1ψ1, σ2ψ2)� and

A =(

0 00 −a

), B =

(σ1 00 σ2

).

We conclude from the analysis above on finite-dimensional realizations that

ft (y) = h0(y + t0 + t) + Z1t + e−ayZ2

t , (5.6)


for some t0 > 0. By splitting into the two components of Z, we find that Z1 is adrifting Brownian motion,

dZ1t = σ1ψ1 dt + σ1 dW 1

t ,

and Z2 is a mean-reverting process,

dZ2t = (σ2ψ2 − aZ2

t ) dt + σ2 dW 2t .

Optimizing a portfolio invested in the futures price curve will with this model beequivalent to optimizing a portfolio investment in two “assets” with dynamics Z. Asexp(−ay) tends to zero when y tends to infinity, we find that

ft (y) ∼ h0(y + t0 + t) + Z1t

for large values of y. Hence, an investment in Z1 can be viewed as holding a portfolioposition in a futures with long time to maturity, that is, a position in a contract in thefar end of the futures curve. The “asset” Z2 can then be interpreted as the differencebetween a futures far out on the curve (being Z1) and one with short time to maturity.Our investment problem will therefore be to select optimally a portfolio of contractsin the short and long end of the curve.

The model in (5.5) or (5.6) can be viewed as the implied futures price dynamicsfrom a two-factor spot model. In fact, following Gibson and Schwartz [27], we canassume that the spot price of some commodity is given by

St = Λ(t) + Z1t + Z2

t ,

where Λ(t) is some deterministic seasonality function. This spot price model will bean arithmetic analog of the dynamics proposed for oil in [27]. Here, Z2 is interpretedas the short-term variations of the oil spot price, while Z1, the non-stationary part, isthe long-term trends in oil prices including inflation and extinction of reserves. Thiscorresponds to the view of investing in the long and short end of the futures curve.

Acknowledgements Two anonymous referees and an Associate Editor are acknowledged for construc-tive comments. Financial support from the project “Energy markets: modeling, optimization and simula-tion (EMMOS)”, funded by the Norwegian Research Council under grant 205328, is gratefully acknowl-edged.

References

1. Andresen, A., Koekebakker, S., Westgaard, S.: Modeling electricity forward prices using the multi-variate normal inverse Gaussian distribution. J. Energy Mark. 3, 3–25 (2010)

2. Benth, F.E., Cartea, A., Kiesel, R.: Pricing forward contracts in power markets by the certainty equiv-alence principle: Explaining the sign of the market risk premium. J. Bank. Finance 32, 2006–2021(2008)

3. Benth, F.E., Koekebakker, S.: Stochastic modelling of financial electricity contracts. Energy Econ. 30,1116–1157 (2008)

4. Benth, F.E., Koekebakker, S., Ollmar, F.: Extracting and applying smooth forward curves fromaverage-based commodity contracts with seasonal variation. J. Deriv. 15, 52–66 (2007)


5. Benth, F.E., Meyer-Brandis, T.: The information premium for non-storable commodities. J. EnergyMark. 2, 111–140 (2009)

6. Benth, F.E., Šaltyte Benth, J.: Modeling and Pricing in Financial Markets for Weather Derivatives.World Scientific, Singapore (2013)

7. Benth, F.E., Šaltyte Benth, J., Koekebakker, S.: Stochastic Modelling of Electricity and Related Mar-kets. World Scientific, Singapore (2008)

8. Biagini, S., Sîrbu, M.: A note on admissibility when the credit line is infinite. Stochastics 84, 157–169(2012)

9. Björk, T., Gombani, A.: Minimal realizations of interest rate models. Finance Stoch. 3, 413–432(1999)

10. Björk, T., Svensson, L.: On the existence of finite-dimensional realizations for nonlinear forward ratemodels. Math. Finance 11, 205–243 (2001)

11. Björk, T., Landén, C.: On the construction of finite dimensional realizations for nonlinear forward ratemodels. Finance Stoch. 6, 303–331 (2002)

12. Björk, T., Blix, M., Landén, C.: On finite dimensional realizations for the term structure of futuresprices. Int. J. Theor. Appl. Finance 9, 281–314 (2006)

13. Buehler, H.: Consistent variance curve models. Finance Stoch. 10, 178–203 (2006)14. Carmona, R., Tehranchi, M.: Interest Rate Models: An Infinite Dimensional Stochastic Analysis Per-

spective. Springer Finance. Springer, Berlin (2006)15. Clewlow, L., Strickland, C.: Energy Derivatives: Pricing and Risk Management. Lacima (2000)16. Cox, J.C., Ingersoll, J.E., Ross, S.A.: The relation between forward prices and futures prices. J. Financ.

Econ. 9, 321–346 (1981)17. Da Prato, G., Zabczyk, J.: Stochastic Equations in Infinite Dimensions. Cambridge University Press,

Cambridge (1992)18. Ekeland, I., Taflin, E.: A theory of bond portfolios. Ann. Appl. Probab. 15, 1260–1305 (2005)19. Filipovic, D.: Consistency Problems for Heath–Jarrow–Morton Interest Rate Models. Springer, Berlin

(2001)20. Filipovic, D., Teichmann, J.: Existence of invariant manifolds for stochastic equations in infinite di-

mensions. J. Funct. Anal. 197, 398–432 (2003)21. Filipovic, D., Teichmann, J.: Regularity of finite-dimensional realizations for evolution equations.

J. Funct. Anal. 197, 433–446 (2003)22. Filipovic, D., Tappe, S., Teichmann, J.: Jump-diffusions in Hilbert spaces: existence, stability and

numerics. Stochastics 82, 475–520 (2010)23. Fleming, W.H., Soner, M.: Controlled Markov Processes and Viscosity Solutions, 2nd edn. Springer,

Berlin (2006)24. Frestad, D., Benth, F.E., Koekebakker, S.: Modeling term structure dynamics in the Nordic electricity

swap market. Energy J. 31, 53–86 (2010)25. Geman, H.: Commodities and Commodity Derivatives. Wiley-Finance, New York (2005)26. Geman, H., Vašícek, O.: Forwards and futures on non-storable commodities: the case of electricity.

RISK, 93–97 (2001)27. Gibson, R., Schwartz, E.S.: Stochastic convenience yield and the pricing of oil contingent claims.

J. Finance 45, 959–976 (1990)28. Heath, D., Jarrow, R., Morton, A.: Bond pricing and the term structure of interest rates: a new method-

ology of contingent claims valuation. Econometrica 60, 77–105 (1992)29. Hull, J.C.: Options, Futures and Other Derivatives. Prentice-Hall, New York (2000)30. Karatzas, I., Shreve, S.: Methods of Mathematical Finance. Springer, Berlin (1998)31. Koekebakker, S., Ollmar, F.: Forward curve dynamics in the Nordic electricity market. Manag. Fi-

nance 31, 74–95 (2005)32. Ringer, N., Tehranchi, M.: Optimal portfolio choice in the bond market. Finance Stoch. 10, 553–573

(2006)33. Samuelson, P.: Proof that properly anticipated prices fluctuate randomly. Ind. Manage. Rev. 6, 13–32

(1965)34. Tappe, S.: An alternative approach on the existence of affine realizations for HJM term structure

models. Proc. R. Soc., Ser. A 466, 3033–3060 (2010)

optimal portfolios in commodity futures markets

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