Abstract
We propose an extension of the model introduced by Barndorff-Nielsen and Shephard, based on stochastic processes of Ornstein–Uhlenbeck type taking values in Hilbert spaces and including the leverage effect. We compute explicitly the characteristic function of the log-return and the volatility processes. By introducing a measure change of Esscher type, we provide a relation between the dynamics described with respect to the historical and the risk-neutral measures. We discuss in detail the application of the proposed model to describe the commodity forward curve dynamics in a Heath–Jarrow–Morton framework, including the modelling of forwards with delivery period occurring in energy markets and the pricing of options. For the latter, we show that a Fourier approach can be applied in this infinite-dimensional setting, relying on the attractive property of conditional Gaussianity of our stochastic volatility model. In our analysis, we study both arithmetic and geometric models of forward prices and provide appropriate martingale conditions in order to ensure arbitrage-free dynamics.
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1 Introduction
In this paper, we propose and analyse a Heath–Jarrow–Morton forward price dynamics in the Musiela parametrisation with stochastic volatility and leverage. This infinite-dimensional Ornstein–Uhlenbeck-type of dynamics, taking values in a separable Hilbert space, is assumed to be modulated by an operator-valued extension of the Barndorff-Nielsen and Shephard [3] stochastic volatility model. Moreover, we propose to include a leverage effect in the term structure of forward prices which, to the best of our knowledge, is new in the literature. We analyse both arithmetic and geometric models, where no-arbitrage conditions are derived and methods for option pricing by Fourier techniques are presented. Commodity markets typically organise trading in spot, forwards and options on these. In regulated markets for commodities as the CME (NYMEX), options are plain vanilla calls and puts on forwards. Energy markets like power, gas and weather are trading in various forwards where the contracts deliver the underlying commodity over a contracted delivery period, different from classical commodity markets where delivery happens at agreed calendar dates. Depending on the market, forward contracts may have financial settlement instead of physical delivery of the underlying. We refer to Geman [37] for a gentle introduction to commodity markets. In the present paper, the focus is on forward price modelling in commodity markets and the application to option pricing. In our exposition, we do not distinguish between forward and futures contracts in the analysis, but use the terminology forwards for all such contracts. Although we take a general perspective on commodity markets, we have a particular view towards energy markets.
Roughly speaking, one may divide the modelling of forward prices into two approaches: a spot-based approach and a direct modelling approach. Classically, a stochastic model for the spot market dynamics is introduced and the forward price is derived from this using no-arbitrage arguments. Frictions like storage and the notion of convenience yield can be added to obtain more realistic price dynamics; see e.g. [37, Chap. 2] and Eydeland and Wolyniec [34, Chap. 4, pp. 118–124] for discussions and analysis. One- and two-factor stochastic models for the spot based on Gaussian Ornstein–Uhlenbeck processes are proposed in Schwartz [61] and Lucia and Schwartz [51], while general multi-factor models of Ornstein–Uhlenbeck type including jumps are introduced in Benth et al. [20, Chap. 3]. Such models typically allow analytical forward prices. For instance, Nomikos and Soldatos [55] propose a two-factor affine jump-diffusion model for power spot prices in the Nordic power markets, while Kyriakou et al. [47] study an exponential jump-diffusion model with Heston stochastic volatility which is calibrated to refined oil futures prices in Europe and US markets (see also Kyriakou et al. [48] for a similar model applied to oil option pricing). Alternatively to spot-forward modelling, Clewlow and Strickland [28, Chap. 8] proposed to adopt the Heath–Jarrow–Morton paradigm (see Heath et al. [40]) from fixed-income theory and model directly the forward price dynamics. Following Musiela [52], when expressing the price in terms of time-to-maturity, one obtains a stochastic partial differential equation for the forward price dynamics. Recent papers using this approach are Barndorff-Nielsen et al. [2], Benth and Krühner [14], Benth et al. [18], Latini et al. [50] and Callegaro et al. [25]. In the present paper, we consider such models where the noise term is allowed to incorporate stochastic volatility, leverage and a dependence structure across maturities. Our analysis is based on the theory for Hilbert-valued stochastic processes as presented in Da Prato and Zabczyk [31] and Peszat and Zabczyk [56]. More precise references to [31] and [56] will be provided later when we use results from these books.
In contrast with interest rate modelling, a principal component analysis suggests that a huge number of factors are necessary in order to describe the forward dynamics in power markets, say. Koekebakker and Ollmar [45] performed this kind of investigation and showed that more than 12 factors are necessary to cover \(95\%\) of the data variation in the Nordic power market. Power is a rather peculiar commodity which is to a large extent non-storable. However, storage is scarce in other commodity markets as well, for example gas. Additionally, in the market for weather, one cannot talk about any “storage” of the underlying. Forward-looking information like maintenance, political decisions or weather forecasts may give economic rationale to high-dimensionality of the noise in the forward price dynamics, as argued and analysed in Benth and Meyer-Brandis [15]. On the other hand, one can also argue for the convenience of using infinite-dimensional noise modelling in forward markets as this provides a possibility to model the stochasticity across maturities by operators without having to resort to factor models. Operators may be conveniently expressed by integral operators. Benth and Paraschiv [17] propose a random field approach to the German power forward market in this vein, based on Gaussian random fields by adopting the Musiela parametrisation. In commodity markets, the Samuelson effect is fundamental for the term structure of volatility. As Samuelson [60] argues, the forward price volatility decays with time to maturity. On the other hand, it converges to the spot volatility when time to maturity goes to zero. One can trace this effect back to the mean-reverting behaviour of the spot price dynamics; see e.g. Benth et al. [20, Chap. 4] for an analysis of this. In [17], a Samuelson effect is empirically observed in the term structure of volatility of German power forwards. We propose a simple method to allow a Samuelson effect in our infinite-dimensional models.
Barndorff-Nielsen and Shephard [3] proposed a class of stochastic volatility models based on positive-valued Ornstein–Uhlenbeck processes driven by a subordinator. This class, in the following named BNS-SV model, has been applied to model financial time series like exchange rates and stock prices, see [3], and further analysed in derivatives pricing in Nicolato and Venardos [53] and Hubalek and Sgarra [42]. Benth [9] suggested the BNS-SV model with a mean-reverting dynamics to model gas prices in the UK market, while Issaka and SenGupta [44] propose to model variance swaps based on the BNS-SV model with leverage. Recently, Roberts and SenGupta [59] proposed the BNS-SV model for crude oil prices. Barndorff-Nielsen and Stelzer [4] extended the BNS-SV model to a multivariate setting. In Benth et al. [19], an infinite-dimensional version of the BNS-SV model is introduced with the purpose of describing the stochastic volatility of forward prices in energy markets by means of a Heath–Jarrow–Morton model. However, the leverage effect is not included in their description, which restricts the applicability of the model. The literature on infinite-dimensional stochastic volatility models is scarce. However, we should like to mention the papers by Cox et al. [30, 29], Benth and Simonsen [23] and Benth et al. [11].
In equity markets, leverage is commonly thought of as the negative dependence between returns and volatility. According to Black [24], asset price returns are changing opposite with the volatility. In commodity and energy markets, one often encounters what is called inverse leverage, i.e., a positive dependence between prices and volatility. As argued for instance in Geman [37, Chap. 2], this may be explained by inventory level and shows up as an upward sloping implied volatility skew (see [37, Sect. 5.5], where different oil and natural gas smiles from NYMEX are presented, as well as Eydeland and Wolyniec [34, Chap. 7, pp. 304–305]). Furthermore, Piccirilli et al. [57] detect a volatility smile in the German power forward market. It is to be noted that volatility smiles are signs of stochastic volatility in the underlying dynamics (in addition to leverage), which can be read from the shape of the smile. In a recent study by Kristoufek [46], the leverage effect in front month futures price series is analysed by sophisticated statistical estimation techniques to show that the picture is rather mixed: natural gas has a negative leverage, while Brent and WTI crude oils exhibit leverage as in equity markets. Heating oil, on the other hand, shows a more indeterminate picture. Wolyniec [62] explains the direct and inverse leverage of crude oil and natural gas by the relationship between the equity volatility, the slope of the forward curve and the theory of storage, which have different impacts in different markets. We should also like to mention the study of Nomikos and Andriosopoulos [54] that reveals direct and inverse leverage effects in eight different energy commodity forwards at NYMEX. In conclusion, leverage is an important aspect of commodity and energy futures price modelling.
As already mentioned, we propose and study the BNS-SV model with leverage in infinite dimensions. A motivation for the model is provided by a forward price dynamics derived from a one-factor Ornstein–Uhlenbeck spot model with BNS-SV and leverage. We formulate the Heath–Jarrow–Morton–Musiela dynamics of the forward price as a stochastic process taking values in a separable Hilbert space, where the noise term consists of a Wiener integral modulated by an operator-valued BNS-SV and a jump term which is the same as the driver in the BNS-SV. The jump process is an operator-valued process, and we transform this into the state space of forward prices by a linear operator modelling the impact of volatility on prices. Also, the Samuelson effect is introduced into the volatility process as a linear operator in the Wiener integral term of the dynamics. A typical state space is the Filipović space (see Filipović [36, Chap. 5]), which has been used for forward rates in fixed-income theory. This provides us with a very general term structure dynamics, and we study both arithmetic and geometric models, the latter requiring additional algebraic properties of the Hilbert space. We derive martingale conditions for both the arithmetic and geometric model classes, based on a Girsanov transform of the Gaussian noise and an Esscher transform of the Lévy noise. We present rather explicit expressions for the application of Fourier methods to option pricing in both model classes. Volatility spillover between maturities is also studied. A powerful and remarkable property of the BNS-SV is that the forward dynamics becomes Gaussian when we condition on the volatility. This is a key property showing an affine structure of the model class that enables us to find rather explicit formulas for characteristics of the model and option prices, say. A major challenge in the infinite-dimensional setting we operate in is the general non-commutativity of operators. This leads to difficulties when performing Fubini-type arguments, as we shall see. A key assumption in many of our results is a certain “commutativity” relation between the stochastic volatility and the covariance operator of the Wiener process driving the forward dynamics. In our exposition, we provide many concrete specifications and examples to connect the modelling framework with practice. Using the Filipović space for realising the states of the term structure, we can further detail our results. For example, the no-arbitrage condition for the exponential model leads to the question of whether an explicit function defined via the norm and Lévy integral of certain operators belong to this space. For the Filipović space, this turns out to be true.
The plan of the paper is the following. After a motivating example, we introduce and derive the basic characteristics of a general \(H\)-valued BNS-SV model with leverage in Sect. 2. Section 3 introduces the Esscher and Girsanov transforms suitable for our infinite-dimensional setting and provides the relation between the dynamics described with respect to the historical and the risk-neutral probability measures, respectively. Forward price models with the Samuelson effect are detailed in Sect. 4, where we demonstrate the volatility spillover between maturities in the proposed model. In Sect. 5, we apply the Fourier method to price options on forwards. We include in our discussion also forwards with a delivery period, which can be expressed as linear operators acting on the instantaneous forward curves, and can be easily included in the proposed pricing framework. Geometric forward models are introduced in Sect. 6, where particular emphasis is put on showing the martingale drift condition ensuring an arbitrage-free dynamics. We also discuss pricing of options under this model definition.
1.1 A motivating example
We consider a motivating example taken from commodity spot price modelling. In Benth [9], a spot dynamics of Ornstein–Uhlenbeck type with the Barndorff-Nielsen and Shephard stochastic volatility (BNS-SV) was proposed and fitted to UK gas price data. The model did not account for leverage, but an explicit dynamics for forward and futures prices was derived. Here we extend this model to allow leverage in the spot, and we derive the forward price which will provide us with a motivation for a general Heath–Jarrow–Morton (HJM) dynamics for the term structure of forward prices where leverage is accounted for. Throughout, we take as given a filtered probability space \((\Omega ,\mathcal {F},(\mathcal {F}_{t})_{t\geq 0},\mathbb{P})\) satisfying the usual conditions. Stochastic processes \((X(t))_{t\geq 0}\) are frequently denoted simply by \(X\). Introduce a Brownian motion \(B\) and a subordinator \(L\), i.e., a driftless pure jump Lévy process with positive jumps. Here \(B\) and \(L\) are assumed to be independent.
The simple BNS-SV model \(\sigma \) is defined as \(V(t)=\sigma ^{2}(t)\), where
for a constant \(\lambda >0\). The spot price dynamics is assumed to be
for constants \(\alpha >0,\mu \) and \(\gamma \). This adaption of the leverage model in Barndorff-Nielsen and Shephard [3] is an extension of the model considered in [9], where leverage is modelled by the factor \(\gamma dL(t)\) in the spot dynamics. The forward price \(F(t,T)\) at time \(t\geq 0\) for a contract delivering the spot at time \(T\geq t\) is defined as
where \(\widehat{\mathbb{E}}\) is the expectation operator under a pricing measure \(\widehat{\mathbb{P}}\approx \mathbb{P}\) with \(S(T)\) being \(\widehat{\mathbb{P}}\)-integrable for all \(T\geq 0\) (see Benth et al. [20, Chap. 4] for details). In this exposition, we choose \(\widehat{\mathbb{P}}\) as a measure which is an Esscher transform on \(L\) and a Girsanov transform on \(B\) so that \(L\) is still a subordinator under \(\widehat{\mathbb{P}}\) and
is a Brownian motion under \(\widehat{\mathbb{P}}\) for some constant \(\theta \). Here we suppose implicitly that this is a well-defined measure change. Hence the explicit dynamics of \(S\) under \(\widehat{\mathbb{P}}\) is
The dynamics of the forward price is stated in the next result.
Proposition 1.1
Suppose that \(\sigma \) is Itô-integrable for \(\widehat{B}\) on \([0,T]\). For \(0\leq t\leq T\) and \(\alpha \neq \lambda \), it holds that
with
Proof
First notice that the Itô-integrability of \(\sigma \) implies that the stochastic process \((\int _{0}^{t} \sigma (s) e^{-\alpha s} d\widehat{B}(s))_{t \geq 0}\) is a \(\widehat{\mathbb{P}}\)-martingale. Furthermore, by the Itô isometry, we see that \(\int _{0}^{t}\widehat{\mathbb{E}}[\sigma ^{2}(s)]ds<\infty \), and therefore \(L\) is also \(\widehat{\mathbb{P}}\)-integrable. This together implies the \(\widehat{\mathbb{P}}\)-integrability of \(S\). Let \(\widehat{L}(t):=L(t)-t\widehat{\mathbb{E}}[L(1)]\), \(t \geq 0\), which is a \(\widehat{\mathbb{P}}\)-martingale and a Lévy process. From (1.1) and (1.2), we have
where we appealed to the \(\mathcal {F}_{t}\)-measurability of \(S(t)\) and \(V(t)\) and the martingale property of \(\widehat{B}\) and \(\widehat{L}\). Since
for any \(u\geq t\), it follows that
after again appealing to the \(\mathcal {F}_{t}\)-measurability of \(V(t)\) and the \(\widehat{\mathbb{P}}\)-martingale property of \(\widehat{L}\). Inserting into the expression for \(F(t,T)\) above yields the result. □
Our concern in this paper is an HJM dynamics with stochastic volatility and leverage. HJM models are appropriately stated in the Musiela parametrisation, that is, in terms of time to maturity \(x:=T-t\) instead of time of maturity \(T\). Define
for \(x\geq 0\). The next result presents the dynamics of \(f\).
Proposition 1.2
Suppose that \(\sigma \) is Itô-integrable for \(\widehat{B}\) on \([0,T]\) and \(\alpha \neq \lambda \). Then the ℙ-dynamics of \(f\) defined in (1.3) is
where \(R(t,x)=R_{0}(x)-\theta e^{-\alpha x}V(t)\),
and
for \(\xi \) defined in Proposition 1.1.
Proof
Proposition 1.1 yields that
Hence
Computing \(\partial _{x}f(t,x)\) and rearranging completes the proof. □
The discounting term \(\exp (-\alpha x)\) in front of the Brownian motion and volatility \(\sigma \) is the so-called Samuelson effect (see Samuelson [60]). This well-known effect exhibited by forward prices says that the volatility of forward prices is decreasing with time to maturity. The leverage effect and the stochastic volatility give rise to a mixed Samuelson effect in the \(dL\)-term, as is evident from \(\rho (x)\) being the sum of two exponentially decaying functions. As discussed in Benth [9], the Samuelson effect appearing in both the \(dB\)- and \(dL\)-terms is an implication of the mean-reverting behaviour of the spot price and variance dynamics. We may view the term structure of forward prices \(x \mapsto f(t,x) \), \(f(t, \,\cdot \,): \mathbb{R}_{+} \rightarrow \mathbb{R}\) as a stochastic process taking values in a space of real-valued functions on \(\mathbb{R}_{+}\). For example, the so-called Filipović space (see Filipović [36, Chap. 5] for a definition and Benth and Krühner [13] for an application to energy forward markets), which is a separable Hilbert space, provides a convenient state space to model such a forward price dynamics. To this end, suppose \(w:\mathbb{R}_{+}\rightarrow [1,\infty )\) is a nondecreasing measurable function with \(w(0)=1\) and \(w^{-1}\in L^{1}(\mathbb{R}_{+})\) and define \(H:=H_{w}\) to be the space of absolutely continuous functions \(g:\mathbb{R}_{+}\rightarrow \mathbb{R}\) such that \(g^{2}(0)<\infty \) and \(\int _{0}^{\infty}w(x)g'(x)^{2}dx<\infty \), where \(g'\) is the weak derivative of \(g\). This space is equipped with the inner product
and induced norm \(\vert g\vert _{w}^{2}:=(g,g)_{w}\). The Filipović space is a separable Hilbert space for which the unbounded derivative operator \(\partial _{x}\) is densely defined and the generator of the shift semigroup (see e.g. [36, Chap. 5]). The space possesses many desirable properties for modelling forward rates in fixed-income theory and forward prices in commodity markets (see [13] for the latter). It is simple to see that the functions \(x\mapsto \exp (-\alpha x)\), \(\rho \) and \(R_{0}\) in Proposition 1.2 are elements in \(H_{w}\) for appropriately chosen exponential weight functions \(w\). Moreover, from [13, Proposition 4.18], \(H_{w}\) is a Banach algebra. Indeed, for two functions \(g,h\in H_{w}\), it holds that
for an explicitly given constant \(c>0\) (see [13, Proposition 4.18] for details).
2 A volatility-modulated Ornstein–Uhlenbeck process with leverage
The purpose of the present section is to illustrate a Hilbert-valued stochastic volatility model with jumps including the leverage effect and to prove that it is an affine model, by providing with Proposition 2.3 an explicit computation of the conditional characteristic function.
Let \(H\) denote a separable (real) Hilbert space with norm \(\vert \cdot \vert _{H}\) induced by an inner product \((\,\cdot \,,\,\cdot \,)_{H}\). Furthermore, we denote by ℋ the space of all Hilbert–Schmidt operators on \(H\). Recall that an operator \(A\) on a separable Hilbert space is a Hilbert–Schmidt operator if it is compact and satisfies \(\Vert A \Vert _{\mathcal{H}}^{2} = \sum _{i=1}^{ \infty} \vert A e_{i} \vert ^{2}_{H}<\infty \), where \((e_{i})_{i \in \mathbb{N}}\) is an orthonormal basis in \(H\). We denote the norm and inner product in ℋ by \(\Vert \cdot \Vert _{\mathcal {H}}\) and \(\langle \,\cdot \,,\,\cdot \,\rangle _{\mathcal {H}}\), respectively. By \(L(H_{1},H_{2})\), we mean the space of bounded linear operators between two Hilbert spaces \(H_{1}\) and \(H_{2}\). Moreover, for any \(\mathcal {T}\in L(H_{1},H_{2})\), we denote by \(\mathcal {T}^{*}\) its adjoint. We extend the Ornstein–Uhlenbeck dynamics with BNS stochastic volatility considered in Benth et al. [19] to allow leverage as follows.
Suppose that our probability space supports a Brownian motion and an independent Lévy process taking values in the Hilbert spaces \(H\) and ℋ, respectively. Let \(X\) and \(\mathcal{Y}\) be stochastic processes with values in the Hilbert spaces \(H\) and ℋ, respectively, satisfying the stochastic differential equations
Here, \(B\) is an \(H\)-valued Wiener process with covariance operator \(\mathcal{Q}\), which is a self-adjoint and nonnegative definite trace class operator on \(H\). We assume that \(X_{0} \in H\), \(R\) is an adapted stochastic process taking values in \(H\) and with locally Bochner-integrable paths, ℒ is an ℋ-valued Lévy process independent of \(B\) with nondecreasing paths (which means that \(\mathcal{L} (t)\) is self-adjoint and the map \(t \mapsto (\mathcal{L} (t) f, f)_{H} \) is a.s. nondecreasing for every \(f \in H\)), \(\mathcal {M}\in L(H)\) and \(\rho \in L(\mathcal {H},H)\). In our modelling, \(\rho \) accounts for the leverage effect while ℳ is the Samuelson effect (see Sect. 4). Moreover, \(\mathcal{A}\) is a linear operator on \(H\), possibly unbounded, densely defined and generating a \(C_{0}\)-semigroup \(\mathcal{S}\). Furthermore, we suppose that \(\mathcal{Y} _{0}\in \mathcal {H}\) is self-adjoint and nonnegative definite, and ℭ is a bounded linear operator on ℋ. We assume ℭ is such that the unique mild solution of (2.2) given by
is self-adjoint and nonnegative definite. Here, \(\mathfrak{S}(t)=\exp (\mathfrak{C}t)\) is the operator exponential (that is, the \(C_{0}\)-semigroup) of ℭ. In [19], a detailed investigation of the operator ℭ and on the conditions ensuring the positivity of the process \(\mathcal{Y}\) are provided. In (2.1), \(\mathcal{Y}^{1/2} (t)\) denotes the self-adjoint square root of \(\mathcal{Y} (t)\).
We have the following useful result concerning the leverage term.
Lemma 2.1
Assume \(\rho \in L(\mathcal {H},H)\) and ℒ is a Lévy process taking values in ℋ. Then \((\rho \mathcal {L}(t))_{t \geq 0}\) is an \(H\)-valued Lévy process with Lévy–Khintchine representation
for all \(h\in H\), where \(\Psi _{\mathcal {L}}(\mathcal {T}), \mathcal {T}\in \mathcal {H}\), is the characteristic exponent of ℒ defined by \(\exp{(\Psi _{\mathcal {L}}(\mathcal {T}))} = \mathbb{E} [\exp (\mathrm{i} \langle \mathcal {L}(1), \mathcal {T} \rangle _{\mathcal {H}}) ]\).
Proof
From Peszat and Zabczyk [56, Corollary 8.17], \(\rho \mathcal {L}\) is a Lévy process in \(H\). Moreover, for any \(h\in H\), we find that \((\rho \mathcal {L}(1),h)_{H}=\langle \mathcal {L}(1),\rho ^{*}h\rangle _{ \mathcal {H}}\), and the Lévy–Khintchine representation follows. □
Let us discuss a possible choice of the operator \(\rho :\mathcal {H}\rightarrow H\).
Example 2.2
Assume \(\Gamma \in \mathcal {H}\) and \(\widehat{\rho}\in H\) are given. For \(\mathcal {T}\in \mathcal {H}\), define
It is simple to see that \(\rho \) is a linear operator from ℋ into \(H\). By the Cauchy–Schwarz inequality, we also find
which shows that \(\rho \) is indeed a bounded linear operator. For any \(g\in H\), we readily see that
where ⊗ denotes the tensor product on the Hilbert space (i.e., for \(a,b\in H\), \(a\otimes b\) denotes the linear operator \((a\otimes b)(h)=(a,h)_{H} \,b\)). We can for example define an increasing, positive definite Lévy process in ℋ by \(\mathcal {L}(t):=\sum _{n=1}^{N(t)}Y_{n}^{\otimes 2}\) for i.i.d. random variables \((Y_{n})_{n\in \mathbb{N}}\) in \(H\) and a Poisson process \(N\), where \(Y_{n}^{\otimes 2}\) denotes the tensor power of order 2 on Hilbert spaces (i.e., \(Y_{n}^{\otimes 2} = Y_{n} \otimes Y_{n}\)). Then we find
If \(\Gamma \) is a self-adjoint and positive definite operator, then \(L(t):=\sum _{n=1}^{N(t)}(\Gamma Y_{n},Y_{n})_{H}\), \(t \geq 0 \), is a compound Poisson process with positive jumps (i.e., a subordinator). Recalling Proposition 1.2, we can choose \(H=H_{w}\), the Filipović space, and let \(\widehat{\rho}\) be some function \(\widehat{\rho}\in H_{w}\) incorporating a Samuelson effect.
We have for (2.1) the mild solution
Notice that by the assumption on \(R\) and because \(\mathcal {S}\) is a \(C_{0}\)-semigroup, the first integral in (2.3) is well defined. The last integral is also well defined as \(\rho \mathcal {L}\) is a Lévy process in \(H\). From [19, Proposition 3.1], we find that the Wiener integral is well defined.
Proposition 2.3
Suppose that \(R\) is non-stochastic and that there exists \(\mathcal {D}\) such that
Then for \(0 \leq s \leq t \), \(f \in H\) and \(\mathcal{T} \in \mathcal{H} \), we have
where
Proof
By the semigroup property, we can write for \(t\geq s\) that
By the \(\mathcal {F}_{s}\)-measurability of \(\mathcal{S} (t-s) X (s)\) and \(\mathfrak{S} (t-s) \mathcal{Y} (s)\) and the assumption that \(R\) is deterministic, we find
Consider the conditional expectation on the right-hand side. Recall that ℒ and thus also \(\mathcal {Y}\) is independent of \(B\) by assumption. Let \(\mathcal {G}_{t,s}\) be the \(\sigma \)-algebra generated by \(\mathcal {F}_{s}\) and ℒ up to time \(t\). By \(\mathcal {G}_{t,s}\)-measurability of the two \(d\mathcal {L}(u)\)-integrals, we find from the tower property of conditional expectation that
By focusing on the first term under the expectation above, we can write
where we used the covariance expression for a Gaussian stochastic integral by following the arguments in [19, proof of Proposition 3.2]. From the commutativity condition (2.4) on \(\mathcal {Y}\), we find that
From (2.6), we have for \(t\geq u\geq s\) that
and from the \(\mathcal {F}_{s}\)-measurability of \(\mathcal {Y}(s)\), it follows that
For the expression outside the conditional expectation in (2.7), we calculate for its exponent that
Let us next focus on the conditional expectation in (2.7). In the first term in the exponent, we appeal to a stochastic Fubini theorem as in Peszat and Zabczyk [56, Theorem 8.14]. We get
For partitions \(\mathcal {P}\) of \([s,t]\) with \(|\mathcal {P}| \rightarrow 0 \), we consider the discretisations of the three integrals in the exponent as
Here we denote by \(\Delta \mathcal {L}(u_{k}) := \mathcal {L}(u_{k}) - \mathcal {L}(u_{k-1}) \) the increments of the Lévy process. From the continuity of the integrands, we find that (2.8)–(2.10) converge, respectively, to
First, by appealing to the independent increment property of Lévy processes, we can replace the conditional expectation given \(\mathcal{F}_{s}\) in (2.7) by the expectation
Next, by appealing to the dominated convergence theorem, we can replace (2.11)–(2.13) by their discretisations (2.8)–(2.10) to obtain
Now we manipulate the terms in the infinite products to make them more easily interpretable. We first apply duality arguments in the first three equalities and in the last step use the definition of the characteristic exponent of ℒ to obtain, with the notation \(\Delta v_{k} = v_{k+1} - v_{k} \),
Finally, appealing once again to the dominated convergence theorem gives
Wrapping up the terms yields the result. □
If we suppose that \(R\) is independent of time, then \(( X(t), \mathcal{Y} (t))_{t \geq 0} \) is an affine time-homogeneous Markov process in \(H \times \mathcal{H} \). Moreover, in this case, the functions \(A(\,\cdot \,;f) , {\mathcal {B}}(\,\cdot \, ; f , \mathcal{T} ), C(\,\cdot \,; f, \mathcal{T} )\) are mild solutions of the system (with \(t>0\))
Here, we slightly abused the notation for \(C\), writing \(C(t;f,\mathcal {T}):=C(0,t; f,\mathcal {T})\).
Remark 2.4
The commutativity property (2.4) required by Proposition 2.3 is just a technical assumption. A sufficient condition granting \(\mathcal{Y}\) commuting with \(\mathcal{Q}\) is that \(\mathcal{Q}\) commutes with \(\mathcal{Y} _{0}\) and \(\mathcal{L} (t)\) for all \(t \geq 0\), and that \(\mathfrak{C} ( \mathcal{T} ) \mathcal{Q} = \mathfrak{C} ( \mathcal{T} \mathcal{Q} )\) and \(\mathcal{Q} \mathfrak{C} ( \mathcal{T} ) = \mathfrak{C} ( \mathcal{Q} \mathcal{T} )\) for every \(\mathcal{T} \in \mathcal{H}\). Some examples in which the commutativity property holds can be found in [19, Example 2.7 and Proposition 3.2]. One more example is provided by a compound Poisson process \(\sum _{k}^{N_{t} } J_{k} \), where \(J_{k} = Z_{k} e_{j}^{\otimes 2}\), with \(( Z_{k} )\) a sequence of real-valued positive random variables and \(e_{j}\) the \(j\)th eigenvector of the covariance operator \(Q\). See also Cox et al. [30, Remark 2.12].
3 An Esscher-type measure change
The purpose of this section is to introduce a class of parametric measure changes preserving the structure of the dynamics for the model introduced in the previous section. Proposition 3.2 provides conditions for existence of such a measure change and its construction, while Proposition 3.4 illustrates under which conditions the stochastic process describing the dynamics of \(X\) under the risk-neutral measure turns out to be a (local) martingale so that the new measure can be considered risk-neutral.
In order to deal with derivative pricing, a measure change must be introduced, providing a relation between the dynamics described with respect to the historical probability measure and with respect to the risk-neutral measure. Such a measure change would ideally keep the dynamics unchanged in its general form, but possibly with different parameter values, i.e., be a structure-preserving measure change. Moreover, we want the measure change itself to be specified by a parameter. By following a well-established tradition in the Lévy setting, see Gerber and Shiu [38] and Cox et al. [20, Chap. 3], we introduce an Esscher-type measure change. We recall that the Esscher transform turning the stochastic exponential of a Lévy process into a local martingale has the remarkable property of minimising the distance, expressed as the relative entropy, between the risk-neutral measure and the historical measure. For this reason, it can be characterised as the minimal entropy martingale measure; see Hubalek and Sgarra [42], Esche and Schweizer [33]. Hubalek and Sgarra [43] introduced the Esscher transform for the BNS model in a finite-dimensional setting, while Benth and Ortiz-Latorre [16] proposed an equivalent change of measure preserving the structure for a BNS model version for commodity markets. Benth and Sgarra [22] proved that the Esscher transform with a suitable choice of its parameters can explain the sign change in time of the risk premium for power markets. The previous list of references shows that a strong motivation stands behind the choice of our measure change in the present setting.
Let us consider our filtered probability space \(( \Omega , \mathcal{F} , (\mathcal{F} _{t})_{t\geq 0}, \mathbb{P})\) for \(t \in [0, T]\), where \(T<\infty \) is a fixed time horizon. The following results are based on the paper by Quiao and Wu [58].
Definition 3.1
We define the exponential tilting with parameter \(\Theta \in \mathcal{H}\) of a Lévy measure \(\nu (d\mathcal{Z})\) on the Borel \(\sigma \)-algebra of ℋ as the measure
provided we have the integrability condition
To a Lévy process \(\mathcal {U}\) valued in ℋ, we can associate the characteristic triplet \((\mathcal {C},\mathfrak{Q},\nu )\), where \(\nu \) is the Lévy measure, \(\mathfrak{Q}\) is the covariance operator of the continuous martingale part and \(\mathcal {C}\in \mathcal {H}\) is the drift (see e.g. Peszat and Zabczyk [56, Theorem 4.27]). As the next result shows, the exponential tilting gives rise to a measure change which preserves the Lévy property of \(\mathcal {U}\).
Proposition 3.2
Let \(\mathcal {U}\) be an ℋ-valued Lévy process which has the characteristic triplet \((\mathcal {C},\mathfrak{Q},\nu )\). Assume \(\Theta \in \mathcal {H}\) is such that \(\widehat{\nu}\) is an exponential tilting of \(\nu \). Then there exists a probability measure \(\widehat{\mathbb{P}}\approx \mathbb{P}\) on \({\mathcal {F}}_{T}\) such that \(\mathcal {U}\) has the characteristic triplet \((\widehat{\mathcal {C}} , \widehat{\mathfrak{Q}} , \widehat{\nu})\), where
Moreover, the Radon–Nikodým derivative of \(\widehat{\mathbb{P}}\) with respect to ℙ on \(\mathcal {F}_{t}\) is given by
for \(t\leq T\), where \(\Psi _{\mathcal{U}}\) is the characteristic exponent of \(\mathcal{U}\).
Proof
(sketch) We can apply [58, Theorem 2.1] by setting, in their notation,
where \(\nu (ds)du\) is the predictable compensator of \(N\). By choosing \(\gamma (s,x) \equiv 0\) and \(\lambda (t,\mathcal{Z}) = \exp{ ( \langle \Theta , \mathcal{Z} \rangle _{ \mathcal{H}} ) } \) for all \(t\), we get the result. □
The measure change from ℙ to \(\widehat{\mathbb{P}}\) is called the Esscher transform. For the Lévy process ℒ, we henceforth assume no continuous martingale part, which means \(\mathfrak{Q}=0\). From now on, we use the notation \(\widehat{\mathbb{E}}\) for the expectation operator with respect to the probability \(\widehat{\mathbb{P}}\). We discuss the measure change separately for the diffusion component and for the pure jump component, since they have been assumed independent from the beginning. So there is no restriction in this assumption on ℒ for the purpose of the present section. We remark that the integration domain of the Lévy measures \(\nu , \hat{\nu}\) for our specific choice of the Lévy process ℒ, which is assumed to be positive definite, is the subset \(\mathcal {H}_{+}\) of positive operators in ℋ.
Since the Wiener component \(B\) and the Lévy process ℒ in our model for \(X\) in (2.1) are assumed to be independent, it is possible to change measure for both by introducing a likelihood process as a mixture of the Esscher transform above and a Girsanov transform, see Da Prato and Zabczyk [31, Theorem 10.2.1]), via
where \(G\) is an \(H\)-valued stochastic process which is progressive measurable with respect to the filtration generated by the Wiener process \(B\) and with paths which are locally Bochner-integrable, and such that
Then the \(H\)-valued stochastic process defined by
will be a \(\widehat{\mathbb{P}}\)-Wiener process. It follows from [31, Lemma 10.15] that the covariance operator of \(\widehat{B}\) is the same as for \(B\), i.e., \(\widehat{\mathcal {Q}}=\mathcal {Q}\).
The dynamics of the process \(X\) under \(\widehat{\mathbb{P}}\) can be written as
where \(\widehat{B}\) is the \(\widehat{\mathbb{P}}\)-Wiener process in (3.7) and \(\widehat{\mathcal{L}}(t):=\mathcal {L}(t)-t\widehat{\mathbb{E}}[ \mathcal {L}(1)]\), \(0\leq t\leq T\), is a martingale Lévy process satisfying the integrability condition (3.1). Here, we have \(\widehat{\mathbb{E}}[\mathcal {L}(1)]\in \mathcal {H}\), with the expectation interpreted as the Bochner integral over the probability ℙ in the space ℋ of Hilbert–Schmidt operators. Furthermore, \(\widehat{R}\) is given by
The \(\mathcal{Y} \)-dynamics under \(\widehat{\mathbb{P}}\) is given by (2.2), but with the driving Lévy process ℒ having the \(\widehat{\mathbb{P}}\)-characteristics given by (3.2)–(3.4).
Remark 3.3
To apply the results of Sect. 2 on the computation of the characteristic function of \(X(t)\), we need \(\widehat{R}(t)\) to be deterministic. A sufficient condition for this requirement is that \(G\) is such that \(\mathcal {M}\mathcal {Y}^{1/2}(t)G(t)=R(t)\). A possible example where this condition is satisfied is the case of a risk premium \(R\) which is proportional to the variance, i.e., \(R(t)=\mathcal {M}\mathcal {Y}(t)\mathcal {M}^{*}\theta (t)\). By choosing then \(G(t)=\mathcal {Y}^{1/2}(t)\mathcal {M}^{*}\theta (t)\), where \(\theta \) is some \(H\)-valued process such that \(G\) is a progressively measurable \(H\)-valued process with locally Bochner-integrable paths, we obtain a deterministic \(\widehat{R}\).
When \(X\) is the model for forward prices in commodity markets, the so-called risk-neutral dynamics is fundamental for derivative pricing. We discuss this in the next section. The following proposition gives the necessary result.
Proposition 3.4
Suppose there exist an operator \(\Theta \in \mathcal {H}\) such that \(\widehat{\nu}\) is an exponential tilting of \(\nu \) and a progressively measurable \(H\)-valued stochastic process \(G\) with locally Bochner-integrable paths such that (3.6) holds. If \(\Theta \) and \(\mathcal{G}\) are such that \(\widehat{R}(t)=0\) a.s. for a.e. \(t\in [0,T]\), then
where \((\widehat{M}(t))_{t\in [0,T]}\) is a \(\widehat{\mathbb{P}}\)-martingale valued in \(H\) and defined by
Proof
By the assumptions, we have a probability \(\widehat{\mathbb{P}}\approx \mathbb{P}\) on \(\mathcal {F}_{T}\) with likelihood process (3.5). Furthermore, from (3.8), we find the claimed dynamics of \(\widehat{X}\) by using the definition of \(\widehat{M}\). By definition, \(\widehat{\mathcal {L}}\) is an ℋ-valued \(\widehat{\mathbb{P}}\)-martingale, and thus \(\rho \widehat{\mathcal {L}}\) becomes an \(H\)-valued \(\widehat{\mathbb{P}}\)-martingale. Finally, \(\int _{0}^{t} \mathcal {M} \mathcal {Y}^{1/2}(s)d\widehat{B}(s)\) is a well-defined square-integrable stochastic integral with respect to the Wiener process \(\widehat{B}\) valued in \(H\), and thus defines an \(H\)-valued \(\widehat{\mathbb{P}}\)-martingale; see Peszat and Zabczyk [56, Theorem 8.7]. The result follows. □
Remark 3.5
The previous construction to remove the drift \(\widehat{R}\) in the dynamics of \(X\) can be described by the following steps. First, one selects an operator \(\Theta \) providing the Esscher transform of the driving Lévy process which removes the “constant” drift. Then one searches a \(G\) such that \(\widehat{R}(t) \equiv 0\), i.e., an \(H\)-valued process \(G\) satisfying
which defines a Girsanov transform. Notice that as \(\mathcal {Y}\) is a Hilbert–Schmidt operator, it is not invertible. Hence \(\mathcal {Y}^{1/2}\) is not invertible either (see Akhiezer and Glazman [1, Chap. 5]). Suppose the risk premium \(R\) has an affine form of the type
for \(R_{0}\in H\) and some stochastic process \(R_{1}\) which is progressively measurable with respect to the filtration generated by \(B\) and ℒ. Then we first search for a \(\Theta \) with
If one exists, we define \(G(t):=\mathcal {Y}^{1/2}(t) \mathcal {M}^{*} R_{1}(t)\) (which depends on \(\Theta \) via \(R_{0}\) and the condition in the preceding line). If this gives rise to a Girsanov transform, we are done. We observe from this procedure that there will be rather restrictive requirements on the structure of the risk premium \(R\) in order to ensure the possibility to remove it by the Esscher and Girsanov transforms. We remark that the class of equivalent probabilities is of course wider than those spanned by the Esscher transforms, but then one might lose the analytical tractability. Also notice the resemblance between the above choice of \(R(t)\) and the univariate case presented in Proposition 1.2.
Remark 3.6
The operator \(\Theta \) can be interpreted in the present context as the market price of risk associated with the Lévy process, while \(G\) can be interpreted as the market price of risk associated with the Brownian component in the forward dynamics. It is interesting to further note that in strict analogy with the finite-dimensional case (see Benth and Sgarra [22]), for mean-reverting processes, the Esscher transform affects the long-term mean, but does not affect the mean-reversion speed (which is modelled by the semigroup in the present context).
4 Forward prices, the Samuelson effect and volatility spillover
As our interest is to model forward prices, it is natural to consider a Hilbert space \(H\) of real-valued functions on \(\mathbb{R}_{+}\). Indeed, the Filipović space \(H_{w}\) we defined at the end of Sect. 1.1 is a convenient choice that we apply here. In \(H_{w}\), \(\partial _{x}\) is the weak derivative operator with respect to \(x\in \mathbb{R}_{+}\), which is a densely defined unbounded operator on \(H_{w}\) with the associated \(C_{0}\)-semigroup \(\mathcal{S}\) being the shift-semigroup, i.e., \(\mathcal {S}(t)g(x)=g(x+t)\) for \(g\in H_{w}\) and \(t\geq 0\) (see Filipović [36, Chap. 4, Sect. 2] for details).
Let \(F(t,\tau )\) be the forward price at time \(t\geq 0\) of a contract delivering at time \(\tau \geq t\). Setting \(f(t,x):=F(t,t+x)\) for \(x=\tau -t\geq 0\), we can model the forward price dynamics with respect to the historical probability ℙ as \(f(t,x):=\delta _{x}(X(t))\), where \(\delta _{x}\) is the evaluation functional on \(H_{w}\), i.e., \(\delta _{x} (g) = g(x) \). In fact, \(\delta _{x}\) is a linear functional on \(H_{w}\) (see Benth and Krühner [13]), that is, a continuous linear operator from \(H_{w}\) to ℝ. As \(H_{w}\) is separable, there exists for every \(x\in \mathbb{R}_{+}\) an explicitly given function \(h_{x}\in H_{w}\) which is such that for all \(g\in H_{w}\),
If the conditions of Proposition 3.4 hold for the dynamics of \(X\), it follows that \(t\mapsto F(t,\tau )\), \(0\leq t\leq \tau \), is a \(\widehat{\mathbb{P}}\)-martingale, which means that we have an arbitrage-free model specification of the forward price with delivery time \(\tau \).
As already indicated, the operator ℳ in the dynamics of \(X\) in (2.1) plays the role of the Samuelson effect. Referring to the discussion after Proposition 1.2, this is an exponentially decaying function entering as a multiplication of the volatility term. Motivated by this, we introduce \(n\) operators \(\mathcal {M}_{k}\in L(H_{w})\) as well as \(n\) functions \(\xi _{k}\in H_{w}\). Define, for any \(g\in H_{w}\),
First of all, \(\mathcal {M}_{k}g\in H_{w}\), and if \(\xi _{k}\in H_{w}\), we find by the algebra property of \(H_{w}\) that \(\xi _{k}\mathcal {M}_{k}g\in H_{w}\). Also, the operator \(g\mapsto \xi _{k}\mathcal {M}_{k} g\) is linear and bounded, and thus an element of \(L(H_{w})\). In conclusion, \(\mathcal {M}\in L(H_{w})\). Moreover, its operator norm is bounded by a constant times \(\sum _{k=1}^{n}\vert \xi _{k}\vert _{w}\Vert \mathcal {M}_{k}\Vert _{{ \mathrm{{op}}}}\).
A simple Samuelson effect can now be achieved by choosing \(n=1\), \(\mathcal {M}_{1}=I\), the identity operator, and \(\xi _{1}(x):=\exp (-\alpha x)\). Here, \(\alpha >0\). We must require that \(\xi _{1}\in H_{w}\), which is indeed the case whenever
This is true if \(w(x)=\exp (\beta x)\) for some \(0<\beta <2\alpha \), for which we also have \(\int _{0}^{\infty}w^{-1}(x)dx<\infty \). We refer to [13, Theorem. 4.17] for details on these considerations. Many volatility term structures have a fixed volatility level at the far end of the term structure towards which there is an exponential decay. We can incorporate this by simply taking \(\xi _{1}(x):=c+\exp (-\alpha x)\), which is still an element of \(H_{w}\) under the condition (4.3). However, a more natural “two-factor” specification lifting a Gibson–Schwartz [39] model to an infinite-dimensional stochastic volatility framework is to choose \(n=2\), \(\mathcal {M}_{k}=\mathcal {P}_{k}\), where \(\mathcal {P}_{k}\) is the orthogonal projection onto the subspace spanned by \(\phi _{k}\in H_{w}\), with \(\phi _{1},\phi _{2}\) being two linearly independent functions in \(H_{w}\) (e.g. the first and second basis vector in the ONB of \(H_{w}\)). Letting \(\xi _{1}=1\) and \(\xi _{2}\) the function \(x\mapsto \exp (-\alpha x)\), we get back the Gibson–Schwartz two-factor volatility term structure with Samuelson effect and fixed long-term level of volatility. Moreover, we also have the stochastic volatility \(\mathcal {Y}^{1/2}\), of course.
In practice, there may also be seasonality in the volatility term structure. Fackler and Tian [35] observed this in soybean futures and Benth and Koekebakker [12] for Nordic electricity price swaps. [35] (see also [12]) suggest a seasonal volatility term structure of the form
where \(a(t)\) is some deterministic seasonality function. For example, we can choose \(a\) to be a truncated Fourier series of sine and cosine functions as in [12]. This suggests the specification of (4.2), in the context of a simple Samuelson effect, as \(n=1\), \(\xi _{1}(t,x)=a(t)\exp (-\alpha x)\) and \(\mathcal {M}_{1}=I\). We see that \(x\mapsto \xi _{1}(t,x)\) is in \(H_{w}\) so that we have a time-parametrised family of functions \((\xi _{1}(t,\,\cdot \,))_{t\geq 0}\subseteq H_{w}\). We may of course extend this to include seasonality also for the more general case of ℳ considered in (4.2). In that case, we should use the notation \(\mathcal {M}(t)\), indicating a (deterministic) family of bounded linear operators. To have a rigorous theory for such a case, we must impose additional integrability conditions. We do not study such conditions in more detail here, but remark that if \(a\) above is a measurable and bounded function, the mild solution in (2.3) is well defined. For an empirical analysis of a seasonality function specification \(a\), we refer the interested reader to [12].
With our stochastic volatility modulated forward price model, we can discuss the aspect of volatility spillover, that is, how much of the volatility of \(f(t,x)\) can be explained by the volatility of \(f(t,y)\), where volatility now refers to the square root of the total variance of the price process. We discuss this issue next.
We can measure the spillover effect by looking at the covariance between the forward prices at different maturities. Moreover, by considering the price increments, we can study the effect of stochastic volatility on maturities. Hence we introduce the so-called adjusted returns (see Benth [9])
By analysing the adjusted returns, we focus on the return for a given time of maturity, which is the time variable relevant in trading forwards on the market. We find the following result.
Proposition 4.1
Suppose that \(R\) is deterministic and ℒ is square-integrable. For any \(x,y\in \mathbb{R}_{+}\) and \(t\geq 0\), we have
Here \(\mathfrak {Q}\) is the covariance operator of \(\mathcal {L}(1)\).
Proof
From (2.3) and the semigroup property of \(\mathcal {S}\), one finds
Since ℒ is integrable, we find \(\mathbb{E}[\mathcal {L}(1)]\in \mathcal {H}\), where the expectation is interpreted as the Bochner integral over the probability ℙ in the space of Hilbert–Schmidt operators on \(H_{w}\). We define the martingale Lévy process \(\bar{\mathcal {L}}(t):=\mathcal {L}(t)-t\mathbb{E}[\mathcal {L}(1)]\), \(t \geq 0\). By the square-integrability of ℒ, that is, \(\mathbb{E}[\Vert \mathcal {L}(t)\Vert _{\mathcal {H}}^{2}]<\infty \), \(t \leq T\), the covariance of \(\int _{t}^{t+\Delta}\mathcal {S}(t+\Delta -s)\rho d\mathcal {L}(s)\) is well defined.
Recall the representation of the evaluation functional \(\delta _{x}\) in (4.1). Conditioning on the independent volatility process \(\mathcal {Y}\), we find
and
by the zero-mean property of Wiener integrals. Since \(R\) is deterministic, we find
In the second equality, we used the quadratic variation formula in [56, Theorem 4.47]. The result follows. □
In order to clarify the cumbersome notation appearing in the last two expressions, we recall that \(\delta _{x}\) is a linear functional from \(H_{w}\) to ℝ. Its adjoint \(\delta _{x}^{*}\) therefore maps real numbers to elements in \(H_{w}\). For the sake of argument, let the inner product in ℝ for the moment be denoted by \((a,b)=ab\) with \(a,b \in \mathbb{R}\). If \(f \in H_{w}\), then \(( \delta _{x} f, a)=f(x) a = (f,\delta _{x}^{*} a)_{w}\). But then \(\delta _{x} f =f(x) 1=( \delta _{x} f , 1)=(f,\delta _{x}^{*} (1))_{w}\). So the representative of \(\delta _{x}\) is \(\delta _{x}^{*} (1)\) when realising the linear functional \(\delta _{x}\) as an inner-product map.
In the expression for the covariance of the adjusted returns for maturities \(x\) and \(y\) in Proposition 4.1, we see that the stochastic volatility enters via a covariance operator \(\mathbb{E}[\mathcal {Y}^{1/2}(t)\mathcal {Q}\mathcal {Y}^{1/2}(t)]\) and \(\mathfrak {Q}\). Thus in general, the volatility for the maturity \(x\) is connected with that of maturity \(y\) and vice versa, yielding a spillover effect. In particular, if the variance of \(\tilde{\Delta}f(t,x)\) is large, so is the variance of \(\tilde{\Delta}f(t,y)\). One may suspect that the appearance of adjoint operators could make the calculation of the covariance operator intractable. However, this is not necessarily the case. Consider for example the \(H_{w}\)-element \(\mathcal {M}^{*}\mathcal {S}^{*}(t+\Delta -s)\delta _{x}^{*}(1)\). Recalling \(\delta _{x}^{*}(1)=h_{x}\) with \(h_{x}\) in (4.1), we can obtain an expression for this function as
Above, we have presented different relevant specifications of ℳ to model the Samuelson effect, which we can readily insert to obtain an explicit form for the desired function \(\mathcal {M}^{*}\mathcal {S}^{*}(t+\Delta -s)\delta _{x}^{*}(1)\). Similar considerations can be made for \(\rho ^{*}\mathcal {S}^{*}(t+\Delta -s)\delta _{x}^{*}(1)\).
If \(\Delta \) is small, we have that \(\mathcal {S}(\Delta )\approx I\), the identity operator, and
Furthermore, if there exists an operator \(\mathcal {D}\) with \(\mathcal {Y}^{1/2}(t)\mathcal {Q}\mathcal {Y}^{1/2}(t)=\mathcal {D}^{1/2} \mathcal {Y}(t)\mathcal {D}^{1/2}\) as in Proposition 2.3, we find
when assuming \(\mathcal {Y}_{0}=0\). Thus the volatility spillover is controlled by the specification of the volatility process in terms of the mean and covariance of ℒ together with the semigroup \(\mathfrak {S}\) in the dynamics of \(\mathcal {Y}\).
Let us consider \(\tilde{\Delta}f(s,x)\) and \(\tilde{\Delta}f(t,y)\) for times \(s,t\) such that \(\vert t-s\vert \geq \Delta \). Following the arguments in the proof of Proposition 4.1, where we additionally appeal to the independent increment properties of \(B\) and ℒ, yields
for all \(x,y\in \mathbb{R}_{+}\). The adjusted returns are uncorrelated over discrete times, but not independent as they are not Gaussian; see Benth [9, Eq. (2.10) and discussion below] for a similar analysis for spot markets, corresponding to the choice \(x=y=0\) in our context.
5 Applications to pricing
In this section, we analyse the arbitrage-free pricing of options written on forward contracts traded in commodity markets. Additionally, we look at some extensions of our forward models to allow a delivery period, which is relevant in many energy markets. Our pricing approach is based on the Fourier transform of the forward price. Since the literature on pricing methods based on the Fourier transform is huge, we briefly recall here the pioneer article by Carr and Madan [26], the survey by Eberlein [32] and the textbook by Cherubini et al. [27, Chap. 7, Sect. 1]. We recall that the forward price at time \(t\geq 0\) with time to maturity \(x\) is defined as \(f(t,x)=\delta _{x} X(t)\), following the setup of Sect. 4. Moreover, we focus on the risk-neutral dynamics of \(X\) as given in Proposition 3.4.
In commodity markets, options are written on forwards with a fixed delivery or delivering over a period of time. The former class of options usually occur in classical commodity markets like for oil, metals or agriculture, while options on forwards with delivery period are found in markets for electricity and gas as well as freight and weather (see Benth et al. [20, Chap. 1]). In Benth and Krühner [14], it is shown that the dynamics of a delivery-period forward can be represented as an integral operator on \(X\). Hence we can represent the payoff of a generic option written on a forward (whether fixed delivery or with delivery period) as \(g(\Lambda _{T} X(T))\), where \(T>0\) is the option’s exercise time and \(g\) is the payoff. We assume \(\Lambda _{T}\in H_{w}^{*}\) is a linear functional on \(H_{w}\). This linear functional typically depends on \(T\), but also on the delivery period of the forward which we ignore in our notation. Typical examples of payoff functions \(g\) are given by \(g(x)=\max (x-K,0)\) or \(g(x)=\max (K-x,0)\), corresponding to call or put options, respectively. If we consider an option on a fixed-delivery forward, say, then it has the payoff \(g(F(T,\tau ))\) with \(T\leq \tau \), and we see that \(F(T, \tau )=f(T,\tau -T)=\delta _{\tau -T}X(T)\), i.e., \(\Lambda _{T} =\delta _{\tau -T}\). The arbitrage-free price under \(\widehat{\mathbb{P}}\) at time \(t\leq T\) is given as
where we use the notation \(\widehat{\mathbb{E}}\) to denote the expectation operator with respect to the probability \(\widehat{\mathbb{P}}\) given in Proposition 3.4. Furthermore, we assume for simplicity that the risk-free rate of return is zero.
Now consider a payoff function \(g\in L^{1}(\mathbb{R})\) with the Fourier transform \(\widehat{g}\in L^{1}(\mathbb{R})\). (We admit that this restriction of \(g\) does not include call and put options; but as we shall see, these can be treated by an exponential damping at the expense of a moment condition on \(X\).) For such \(g\), we have the Fourier representation
and by appealing to the Fubini theorem, we find
In the next result, we state an explicit expression for the conditional characteristic function appearing in the above Fourier integral, a result analogous to Proposition 2.3.
Proposition 5.1
Assume the model is described by (2.1) and (2.2), there exists an operator \(\mathcal {D}\) such that \(\mathcal {Y}^{1/2}(t)\mathcal {Q}\mathcal {Y}^{1/2}(t)=\mathcal {D}^{1/2} \mathcal {Y}(t)\mathcal {D}^{1/2}\) for all \(0\leq t\leq T\) and the assumptions of Proposition 3.4hold. Then we have
where \(\widehat{\Psi}_{\mathcal {L}}\) is the characteristic exponent of ℒ with respect to the probability \(\widehat{\mathbb{P}}\).
Proof
We have from Proposition 3.4 that
where \(\widehat{B}\) is a \(\widehat{\mathbb{P}}\)-Wiener process with covariance operator \(\mathcal {Q}\) and the process \(\widehat{\mathcal {L}}\) given by \(\widehat{\mathcal {L}}(t):=\mathcal {L}(t)-t\widehat{\mathbb{E}}[ \mathcal {L}(1)]\) is a zero-mean \(\widehat{\mathbb{P}}\)-Lévy process. Since \(\Lambda _{T}\in H_{w}^{*}\) (note also that \(\Lambda _{T}=(\Lambda _{T}^{*}1,\,\cdot \,)_{H}\), where \(\Lambda _{T}^{*}\) is the adjoint of \(\Lambda _{T}\) and \(\Lambda _{T}^{*}1\in H_{w}\)), we find
After using the tower property of conditional expectations for \(\mathcal {F}_{T}^{\mathcal {Y}}\) which is generated by \(\mathcal {F}_{t}\) and the paths of \(\mathcal {Y}\) from zero to \(T\), we find from \(\mathcal {F}_{t}\)-measurability of \(X(t)\) and \(\mathcal {F}_{T}^{\mathcal {Y}}\)-measurability of \(\widehat{\mathcal {L}}\) that
where we appealed in the second equality to the conditional Gaussianity of the Wiener integral. From the assumptions and the dynamics of \(\mathcal {Y}\), we find
We notice that the first term is \(\mathcal {F}_{t}\)-measurable, while the second is independent of \(\mathcal {F}_{t}\) under both ℙ and \(\widehat{\mathbb{P}}\) due to the independent increments of ℒ. To complete the proof, it remains to calculate \(\widehat{\mathbb{E}}[\exp (Z)]\), where
This can be done by following the same steps as in the proof of Proposition 2.3, except that we compute under \(\widehat{\mathbb{P}}\) rather than ℙ and recall the Lévy property of ℒ under \(\widehat{\mathbb{P}}\). This completes the proof. □
In the expression of the characteristic exponent, we come across many adjoint operators. In the Filipović space, recall the representation of the evaluation operator \(\delta _{x}\) in (4.1). Thus for any linear operator \(\mathcal {V}\in L(H_{w})\),
This means that the adjoint operators can be expressed as integral operators via the inner product. For example, if ℳ is the multiplication operator by \(\exp (-\alpha \,\cdot \,)\), a simple model for the Samuelson effect, its adjoint has the explicit form
We also recall that \(\delta _{x}^{*}1=h_{x}\). Therefore, if we consider options on a fixed-delivery forward, we find \(\Lambda _{T}^{*}1=\delta _{\tau -T}^{*}1=h_{\tau -T}\).
As noted above, plain-vanilla call and put options do not have a payoff function \(g\) satisfying the integrability required to apply Fourier methods directly for pricing. Following Carr and Madan [26], the trick to overcome this is to introduce an exponential damping of the payoff function. For example, for the call option payoff, let \(\gamma >0\) be fixed and consider
for some strike \(K\in \mathbb{R}\). We obtain a bounded continuous function which is supported on \((K,\infty )\), where \(g_{\gamma}(x)=(x-K)\exp (-\gamma x)\) for \(x>K\). This is an integrable function on ℝ with an integrable Fourier transform. Hence we get
If we use this representation, we need to compute in the Fourier approach the expected value of \(\exp ((iy+\gamma )\Lambda _{T}X(T))\). This expectation is well defined whenever \(\Lambda _{T}X(T)\) has a finite exponential moment of some positive order, as we then can tune \(\gamma \) to obtain a well-defined moment. In the next result, we provide a sufficient condition for a finite exponential moment in terms of an integrability condition on the characteristic exponent of ℒ.
Lemma 5.2
Assume \(\Lambda \in H_{w}^{*}\) and there exists a positive definite operator \(\mathcal {D}\) such that \(\mathcal {D}^{1/2}\mathcal {Y}(t)\mathcal {D}^{1/2}=\mathcal {Y}^{1/2}(t) \mathcal {Q}\mathcal {Y}^{1/2}(t)\). Let \(\widehat{\Psi}_{\mathcal {L}}\) be the characteristic exponent of ℒ under \(\widehat{\mathbb{P}}\). If
then \(\widehat{\mathbb{E}}[\exp (\Lambda _{T} X(T))]<\infty \).
Proof
Without loss of generality, we suppose for simplicity that \(\mathcal {Y}(0)=0\). Appealing to the conditional Gaussianity of the stochastic integral with respect to \(\widehat{B}\) in \(X\), we find after conditioning on \(\mathcal {Y}\) that
Under the imposed integrability condition on \(\widehat{\Psi}_{\mathcal {L}}\), we can now proceed as in the proof of Proposition 2.3. The result follows. □
The above condition on \(\widehat{\Psi}_{\mathcal {L}}\) for finite exponential moments can be further worked out to obtain moment conditions on the Lévy measure \(\widehat{\nu}\). We refrain from elaborating further on these rather technical derivations here. We observe that due to the imaginary unit \(i\) inside the argument of \(\widehat{\Psi}_{\mathcal {L}}\), the integrability condition is on the so-called moment generating function of ℒ.
We can also treat calendar spread options within our framework. A calendar spread on two fixed-delivery forwards pays the difference \(F(T,\tau _{1})-F(T,\tau _{2})\), where we have \(T\leq \min (\tau _{1},\tau _{2})\). We choose \(\Lambda _{T}=\delta _{\tau _{1}-T}-\delta _{\tau _{2}-T}\) in this case, which is obviously a bounded linear functional on \(H_{w}\). It is worth observing here that in our infinite-dimensional framework, we cannot split the computation of the price into a Fourier transform of the payoff function and a characteristic exponent of the dynamics as in Carr and Madan [26]. The operator \(\Lambda _{T}\), which is a part of the option payoff, enters into the characteristic exponent. As is evident from the characteristic exponent in Proposition 5.1, both leverage and stochastic volatility impact the option price.
Let us now turn our attention to forwards delivering over a period, also known as flow forwards. Such products are traded in gas and power markets, but also in markets for freight and temperature (see Benth et al. [20, Chap. 1] for a discussion on these different markets and contracts). If we let the delivery period be the interval \([\tau _{1},\tau _{2}]\) with \(\tau _{1}<\tau _{2}\), a flow forward in power has a price at time \(t\leq \tau _{1}\) which can be expressed as
i.e., the average fixed-delivery forward price \(F(t,\tau )\) over the delivery period (see [20, Chap. 1] for more details). The reason for averaging is that prices are denominated per MWh, and so they refer to the total aggregated energy delivery per time unit. Using the definitions above, a simple derivation yields
Hence the flow forward price can be expressed as an integral operator acting on \(X\). Such linear integral operators and the pricing of options on forwards without stochastic volatility and leverage have been studied in detail in Benth and Krühner [14], where it is shown that the operator is a linear functional on \(H_{w}\). We observe that by choosing
we can price options on flow forwards using our Fourier approach. In fact, we can also consider options on calendar spreads, say, by an appropriate definition of \(\Lambda _{T}\) which includes two delivery periods.
Let us end this section with a remark on how to include seasonality in forward prices when pricing options. Indeed, in energy markets like electricity or gas, one may expect seasonal variations in the prices (see e.g. [20, Chaps. 3 and 5]). Going back to the example in Sect. 1.1, adding a deterministic seasonality function \(\varpi (t)\) to the arithmetic spot model would entail a forward price \(\widetilde{F}\) of the form
with \(F\) as before (see Proposition 1.1). Notice that \(t\mapsto \widetilde{F}(t,T)\) is a martingale if and only if \(t\mapsto F(t,T)\) is. In the Musiela parametrisation, we should get
Trigonometric functions are typical choices to define seasonality functions \(\varpi \), and as their derivatives become again trigonometric functions, \(x\mapsto \varpi (t+x)\) fails to be an element of the Filipović space. This is, however, not any obstacle when pricing derivatives. For example, assume we want to price a call option on a fixed-delivery forward. With a seasonal component, the payoff (with exercise time \(\tau \geq T\)) becomes
and we are back to the situation treated above after redefining the strike \(K\) to be \(K-\varpi (\tau )\). In general, we can express \(g(\widetilde{F}(T,\tau ))\) as
which shows that we recover the above considerations on pricing by substituting the function \(g\) by \(\widetilde{g}(\,\cdot \,):=g(\varpi (\tau )+\,\cdot \,)\). We can do analogous modifications for options written on delivery-period forwards.
6 A geometric forward model
Classical financial models are of geometric type, for example the geometric Brownian motion model in the Black [24] formula for call options on forwards. Geometric models ensure positive prices, and this is a crucial feature in many markets. In power markets like the German EEX, the spot price can often become negative if the production is influenced by intermittent sources of power like wind and photovoltaic (PV), but forward prices have not been below zero. On the other hand, in April 2020, we had negative oil futures prices in the US (see the news article on BBC.com [8]).
We recall that the Filipović space is an algebra under pointwise multiplication, and thus it makes sense to define a geometric model for the forward price as
where \(X\) is the dynamics in (2.1) which includes the operator ℳ modelling the Samuelson effect. We observe that the logarithmic forward prices are conditionally Gaussian when the risk premium \(R\) is deterministic (or a Gaussian process itself). We remark in passing that swap prices, i.e., forward prices for contracts with a delivery period, are represented by an integral operator, as studied above, acting on \(\exp (X(t))\).
In general, a Hilbert space (of functions on \(\mathbb{R}_{+}\)) may fail to be an algebra and thus \(\exp (X(t))\) need not make sense. On the other hand, if \(\delta _{x}\) is a bounded linear functional on \(H\), then \(\exp (\delta _{x}X(t))\) does make sense. Therefore we can define geometric forward price models also for general Hilbert spaces \(H\) of functions on \(\mathbb{R}_{+}\) if \(\delta _{x}\in H^{*}\) for all \(x\geq 0\). Such a model is given by
Of course, we need not have that \(x\mapsto f(t,x)\) is in \(H\). Let us consider this general case and derive a condition for when the forward price dynamics is arbitrage-free.
To have an arbitrage-free forward price dynamics, we want to ensure the existence of a probability \(\widehat{\mathbb{P}}\approx \mathbb{P}\) such that \(F(t,\tau ):=f(t,\tau -t), t\leq \tau \), is a \(\widehat{\mathbb{P}}\)-martingale. Suppose that \(\widehat{\mathbb{P}}\) is a probability equivalent to ℙ as introduced in Sect. 3, more precisely, having Radon–Nikodým density as in (3.5). We recall \(G\) as an adapted process shifting the ℙ-Brownian motion \(B\) into a \(\widehat{\mathbb{P}}\)-Brownian motion \(\widehat{B}\), while \(\Theta \) is the exponential tilting of the Lévy measure \(\nu \) of ℒ into a new Lévy measure \(\widehat{\nu}\) on ℋ. We find the \(\widehat{\mathbb{P}}\)-dynamics of \(X\) as in (3.8) to be
where \(\widehat{\mathcal {L}}(t):=\mathcal {L}(t)-t\widehat{\mathbb{E}}[ \mathcal {L}(1)]\) and
We notice that \(\widehat{\mathcal {L}}\) and \(\widehat{B}\) are \(\widehat{\mathbb{P}}\)-martingales. Additionally, as \(\rho \) is a bounded linear operator, \(\rho \widehat{\mathcal {L}}\) is a Lévy process valued in \(H\) which is a \(\widehat{\mathbb{P}}\)-martingale. The following result gives a drift condition ensuring the martingale property of \(t\mapsto F(t,\tau )\) for \(t\leq \tau \leq T\).
Proposition 6.1
Suppose \(\widehat{\mathbb{P}}\) is such that for any \(t\leq T\) and \(x\geq 0\),
Then \(t\mapsto F(t,\tau ), 0\leq t\leq \tau \leq T\), is a local \(\widehat{\mathbb{P}}\)-martingale.
Proof
Since \(\delta _{x}\mathcal {S}(t)=\delta _{0}\mathcal {S}(t+x)\), we find from (6.1) that
for \(0\leq t\leq \tau \). The first term \(\delta _{\tau}X_{0}\) is constant in time \(t\), and without loss of generality, we assume \(X_{0}=0\) in the rest of the proof.
Define for \(t\leq \tau \) the \(H\)-valued stochastic process
and observe that \(Y(t):=F(t,\tau )=\exp (\delta _{0}Z(t))=\psi (Z(t))\). Here, \(\psi :H\rightarrow \mathbb{R}_{+}\) is the smooth function \(\psi (g)=\exp (\delta _{0}g)\) for \(g\in H\). Its Fréchet derivatives are \(D\psi (g)=\psi (g)\delta _{0}\in H^{*}\) and \(D^{2}\psi (g)=\psi (g)(\delta _{0}\otimes \delta _{0})\), where \((\delta _{0}\otimes \delta _{0})(h)=\delta _{0}h \delta _{0}\). Hence \(\delta _{0}\otimes \delta _{0}\) defines a bounded linear operator \(H\rightarrow H^{*}\). Also, for \(u,v\in H\), we have \((\delta _{0}\otimes \delta _{0})(u\otimes v)=(\delta _{0}u)( \delta _{0}v)\), where \(u\otimes v\in \mathcal {H}\) is the tensor product in \(H\). Noticing that
for the martingale \(M\) (which is a true martingale by [56, Corollary 8.7 (iii)]) as both \(\mathcal {S}(\tau -s)\mathcal {M}\mathcal {Y}^{1/2}(s)\) and \(\mathcal {S}(\tau -s)\rho \) are bounded linear functionals on \(H\)) given by
we apply Itô’s formula in [56, Theorem D.2] to obtain
We analyse the terms in (6.3) and (6.4) further:
Concerning (6.3), we know that the continuous martingale part of \(M\) is given by \(\int \mathcal {S}(T-s)\mathcal {M}\mathcal {Y}^{1/2}(s)d \widehat{B}(s)\). According to the notation in [56, Theorem D.1 and the included comments],
where \((e_{i})_{i\in \mathbb{N}}\) is an orthonormal basis in \(H\), \(\langle \! \langle \,\cdot \,,\,\cdot \, \rangle \! \rangle \) is the angle bracket and
Here, \(\mathcal {P}_{k}:=(\,\cdot \,,e_{k})_{H}\) is the projection operator on the \(k\)th coordinate. Now [56, Corollary 8.17] together with Parseval’s identity yields
Applying the identities \(\delta _{0}^{*}1=h_{0}\in H, (\delta _{0}\otimes \delta _{0})(e_{k} \otimes e_{j})=e_{k}(0)e_{j}(0)\) and
it follows for (6.3) that
where in the first and second equalities dominated convergence allows interchanging the integrals and the infinite sums. This concludes the analysis of the term in (6.3).
For the term in (6.4), we observe that \(Z\) jumps only when \(\widehat{\mathcal {L}}\) jumps. Thus we get
where \(\widetilde{N}\) is the compensated Poisson random measure of ℒ on ℋ. Because we have \(\delta _{0} \mathcal {S}(\tau -s)\rho \mathcal {Z}=\langle \mathcal {Z},\rho ^{*} \mathcal {S}(\tau -s)^{*}h_{0}\rangle _{\mathcal {H}}\), the result follows after collecting the \(ds\)-integrals and noting that they add up to zero. □
The drift conditions yields a direct relationship between the drift process \(R\) and the parameters \(G\) and \(\Theta \) in the measure change. Equation (6.2) can thus be considered as a variant in commodity forward markets of the no-arbitrage HJM drift condition known from fixed-income markets. Moreover, as \(R\) and \(G\) are \(H\)-valued and \(\Theta \in H\), we must have that the right-hand side of (6.2) defines a stochastic process with values in \(H\). That is, we must have that
Whether this is true or not depends on the structure of the Hilbert space \(H\).
If we model directly under \(\widehat{\mathbb{P}}\), then the drift condition becomes a specification of the drift \(R\) as
If the purpose of modelling the dynamics of forward prices is option pricing, one frequently states the forward price dynamics directly under the pricing measure \(\widehat{\mathbb{P}}\). We analyse this situation further in the case of the Filipović space \(H=H_{w}\). We need to verify that \(R\) as defined in (6.5) is an \(H_{w}\)-valued adapted stochastic process. Adaptedness of \(t\mapsto R(t,x)\) for fixed \(x\geq 0\) follows from the definition of \(\mathcal {Y}\). We have the following results.
Lemma 6.2
The function \(x\mapsto \mathcal {Q}^{1/2}\mathcal {Y}^{1/2}(t)\mathcal {M}^{*}\mathcal {S}(x)^{*}h_{0} \) takes values in \(H_{w}\).
Proof
Notice that \(\mathcal {S}(x)^{*}h_{0}=(\delta _{0}\mathcal {S}(x))^{*}1=\delta _{x}^{*}1=h_{x}\) for \(h_{x}\in H_{w}\) defined in (4.1). For fixed \(t\geq 0\) and \(\omega \in \Omega \), define
To have a valid drift condition, we must check if \(x\mapsto g(x)\in H_{w}\). To simplify the exposition (and slightly abusing notation), set \(\mathcal {K}:=\mathcal {Q}^{1/2}\mathcal {Y}^{1/2}(t) \mathcal {M}^{*}\in L(H_{w})\). Obviously, \(\mathcal {K}^{*}\mathcal {K}\) is a symmetric and positive definite operator on \(H_{w}\), and since
it is trace class as well. Hence we find
for the symmetric, positive definite trace class operator \(\mathcal {K}^{*}\mathcal {K}\). From Benth and Krühner [13, Corollary 4.12], one can deduce that there exist a constant \(c\in \mathbb{R}\) and a symmetric function \(\kappa \in L^{2}(\mathbb{R}_{+}^{2})\) such that
We note that as \(\mathcal {K}\) is random and time-dependent, \(\kappa \) is so as well. However, our analysis here goes for fixed \(t\) and \(\omega \in \Omega \). Appealing to the symmetry of \(\kappa \), we get
and by the Cauchy–Schwarz inequality, we estimate
In deducing finiteness, we used the conditions \(\kappa \in L^{2}(\mathbb{R}_{+}^{2})\) and \(w^{-1}\in L^{1}(\mathbb{R}_{+})\). Moreover,
and we conclude that \(g\in H_{w}\) which proves the result. □
For the last term in (6.5), we have the following result.
Lemma 6.3
Assume
where \(c>0\) is the constant appearing in the norm estimate in (1.4). Then the function \(x\mapsto \int _{\mathcal {H}} (e^{\langle \mathcal {Z},\rho ^{*} \mathcal {S}(x)^{*}h_{0}\rangle _{\mathcal {H}}}-1-\langle \mathcal {Z}, \rho ^{*}\mathcal {S}(x)^{*}h_{0}\rangle _{\mathcal {H}} )\widehat{\nu}(d \mathcal {Z})\) belongs to \(H_{w}\).
Proof
It holds that
Hence we obtain
Now \(\rho \mathcal {Z}\in H_{w}\) for every \(\mathcal {Z}\in \mathcal {H}\), and by the algebra property of \(H_{w}\), it follows that \(\exp (\rho \mathcal {Z})-1 -\rho \mathcal {Z}\in H_{w}\). Moreover, by a Taylor expansion, we find that
and thus
for \(\Vert \mathcal {Z}\Vert _{\mathcal {H}}\leq 1\), and
otherwise. Since \(\widehat{\nu}\) is a Lévy measure, it integrates \(\Vert \mathcal {Z}\Vert ^{2}_{\mathcal {H}}\) on the set \(\{\Vert \mathcal {Z}\Vert _{\mathcal {H}}\leq 1\}\). From the assumption, it follows that \(\exp (\rho \mathcal {Z})-1-\rho \mathcal {Z}\) is Bochner-integrable on ℋ with respect to the Lévy measure \(\widehat{\nu}\), and thus the result follows. □
Hence under a mild exponential integrability hypothesis on \(\widehat{\nu}\), that is, an exponential moment condition on \(\mathcal {L}(1)\), we can define a martingale dynamics in \(H_{w}\) for the exponential model for forward prices.
We end this section with a discussion on option pricing for the exponential model. To this end, we suppose that we have a model for \(X\) in the Filipović space directly under \(\widehat{\mathbb{P}}\). Hence (6.5) holds for the drift \(R\). As in Sect. 5, we consider an option paying \(\xi (\Lambda _{T} \exp (X(T)))\) at the exercise time \(T\), where \(\xi :\mathbb{R}\rightarrow \mathbb{R}\) is some payoff function and \(\Lambda _{T}\in H_{w}^{*}\). From the algebra property of \(H_{w}\), we have
with \(g(x)=\xi (\exp (x))\). Therefore, we are back to the situation in Sect. 5, where the price of the option is expressible in terms of a Fourier integral of \(\widehat{g}\), the Fourier transform of \(g\), and the conditional characteristic function of \(\Lambda _{T}X(T)\). To obtain an expression for the conditional characteristic function, we can follow similar arguments as in the proof of Proposition 5.1, noticing that the drift \(R\) depends on \(\mathcal {Y}\).
Furthermore, allowing seasonality in forward prices would naturally lead to a model (in the Musiela parametrisation) of the form
Hence when pricing a fixed-delivery forward, we can use instead of \(g\) the function \(\widetilde{g}(x):=g(\exp (\varpi (\tau ))x)\). For delivery-period forwards, we can use the linear functional
where \([\tau _{1},\tau _{2}], \tau _{1}<\tau _{2}\), is the delivery period of the underlying forward.
7 Concluding remarks
In this paper, we have proposed a forward price dynamics based on the Musiela parameterisation of a Heath–Jarrow–Morton dynamics which accounts for stochastic volatility, leverage and statistical dependence across maturities. The BNS-SV model is set in infinite dimensions, and leverage comes in directly in the forward dynamics by the term \(\rho \mathcal {L}\). The operator \(\rho \) maps the positive noise ℒ from the operator space to the state space of the term structure. The BNS-SV model is an operator-valued Ornstein–Uhlenbeck process driven by ℒ. Hence leverage is controlled by \(\rho \), and we can achieve inverse and direct leverage across maturities by tuning this operator. By allowing an operator-valued stochastic volatility, we open the door for volatility spillover between maturities, an effect we can determine from the model analytically. Finally, we explicitly account for the Samuelson effect as well as possible correlation between maturities by a Wiener process driving the prices. The forward prices become conditionally Gaussian, a highly attractive property for analysis but also for modelling.
Our proposed model is very general, taking into account many of the stylised features one observes in the market. However, estimation or calibration of the model to market price data, i.e., observed forward prices or option data, is a challenging problem. Recently, Benth et al. [21] developed a law of large numbers result for the realised integrated variance of the type of Ornstein–Uhlenbeck processes we analyse in this paper. This is a promising direction for recovering the volatility in the forward market based on functional data analysis. On the other hand, Benth et al. [10] apply deep neural networks to calibrate Hilbert-space-valued forward dynamics to option prices, extending the framework of Bayer et al. [7] and Horvath et al. [41] to infinite dimensions.
Another promising stream of further development is the construction of suitable reliable and efficient numerical schemes in order to get explicit prices for all the considered contracts. Some of these methods can involve multilevel Monte Carlo methods as introduced by Barth and Lang [6] (and applied in Barth and Benth [5] to a class of infinite-dimensional forward models), or SUPG methods like those presented in Larsson and Thomée [49, Chap. 10]. These developments and applications will be the subject of future studies.
References
Akhiezer, N.I., Glazman, I.M.: Theory of Linear Operators in Hilbert Space. Dover Publications Inc., New York (1993)
Barndorff-Nielsen, O.E., Benth, F.E., Veraart, A.: Modelling electricity forward markets by ambit fields. Adv. Appl. Probab. 46, 719–745 (2014)
Barndorff-Nielsen, O.E., Shephard, N.: Non-Gaussian Ornstein–Uhlenbeck-based models and some of their uses in economics. J. R. Stat. Soc. B 63, 167–241 (2001)
Barndorff-Nielsen, O.E., Stelzer, R.: Multivariate supOU processes. Ann. Appl. Probab. 21, 140–182 (2011)
Barth, A., Benth, F.E.: The forward dynamics in energy markets – infinite-dimensional modelling and simulation. Stochastics 86, 932–966 (2014)
Barth, A., Lang, A.: Multilevel Monte Carlo method with applications to stochastic partial differential equations. Int. J. Comput. Math. 89, 2479–2498 (2012)
Bayer, C., Horvath, B., Muguruza, A., Stemper, B., Tomas, M.: On deep calibration of (rough) stochastic volatility models. Preprint (2019). Available online at https://arxiv.org/abs/1908.08806
BBC.com: US oil prices turn negative as demand dries up. Press article (2020). Available online at https://www.bbc.com/news/business-52350082
Benth, F.E.: The stochastic volatility model of Barndorff-Nielsen and Shephard in commodity markets. Math. Finance 21, 595–625 (2011)
Benth, F.E., Detering, N., Lavagnini, S.: Accuracy of deep learning in calibrating HJM forward curves. Digit. Finance 3, 209–248 (2021)
Benth, F.E., Di Nunno, G., Simonsen, I.: Sensitivity analysis in the infinite dimensional Heston model. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 24, 2150014 (2021)
Benth, F.E., Koekebakker, S.: Stochastic modeling of financial electricity contracts. Energy Econ. 30, 1116–1157 (2008)
Benth, F.E., Krühner, P.: Representation of infinite-dimensional forward price models in commodity markets. Commun. Math. Stat. 2, 47–106 (2014)
Benth, F.E., Krühner, P.: Derivatives pricing in energy markets: an infinite-dimensional approach. SIAM J. Financ. Math. 6, 825–869 (2015)
Benth, F.E., Meyer-Brandis, T.: The information premium for non-storable commodities. J. Energy Mark. 2(3), 111–140 (2009)
Benth, F.E., Ortiz-Latorre, S.: A change of measure preserving the affine structure in the Barndorff-Nielsen and Shephard model for commodity markets. Int. J. Theor. Appl. Finance 18, 1550038 (2015)
Benth, F.E., Paraschiv, F.: A space-time random field model for electricity forward prices. J. Bank. Finance 95, 203–216 (2018)
Benth, F.E., Piccirilli, M., Vargiolu, T.: Mean-reverting additive energy forward curves in a Heath–Jarrow–Morton framework. Math. Financ. Econ. 13, 543–577 (2019)
Benth, F.E., Rüdiger, B., Süss, A.: Ornstein–Uhlenbeck processes in Hilbert space with non-Gaussian stochastic volatility. Stoch. Process. Appl. 13, 543–577 (2018)
Benth, F.E., Šaltytė-Benth, J., Koekebakker, S.: Stochastic Modelling of Electricity and Related Markets. World Scientific, Singapore (2008)
Benth, F.E., Schroers, D., Veraart, A.: A weak law of large numbers for realised covariation in a Hilbert space setting. Stoch. Process. Appl. 145, 241–268 (2022)
Benth, F.E., Sgarra, C.: The risk premium and the Esscher transform in power markets. Stoch. Anal. Appl. 30, 1–20 (2012)
Benth, F.E., Simonsen, I.: The Heston stochastic volatility model in Hilbert space. Stoch. Anal. Appl. 36, 733–750 (2018)
Black, F.: Studies of stock price volatility changes. In: Proceedings of the 1976 Meetings of the American Statistical Association. Business and Economical Statistics Section, vol. 34, pp. 177–181 (1976)
Callegaro, G., Mazzoran, A., Sgarra, C.: A self-exciting modelling framework for forward prices in power markets. Appl. Stoch. Models Bus. Ind. 38, 27–48 (2022)
Carr, P., Madan, D.B.: Option valuation using the fast Fourier transform. J. Comput. Finance 2, 61–73 (1998)
Cherubini, U., Della Lunga, G., Mulinacci, S., Rossi, P.: Fourier Transform Methods in Finance. Wiley, London (2010)
Clewlow, L., Strickland, C.: Energy Derivatives: Pricing and Risk Management. Lacima Publications, London (2000)
Cox, S., Karbach, S., Khedher, A.: An infinite-dimensional affine stochastic volatility model. Math. Finance 32, 878–906 (2020)
Cox, S., Karbach, S., Khedher, A.: Affine pure-jump processes on positive Hilbert–Schmidt operators. Stoch. Process. Appl. 151, 191–229 (2022)
Da Prato, G., Zabczyk, J.: Stochastic Equations in Infinite Dimensions. Cambridge University Press, Cambridge (1996)
Eberlein, E.: Fourier-based valuation methods in mathematical finance. In: Benth, F.E., et al. (eds.) Quantitative Energy Finance: Modeling, Pricing, and Hedging in Energy and Commodity Markets, pp. 85–114. Springer, New York (2014)
Esche, F., Schweizer, M.: Minimal entropy preserves the Lévy property: how and why. Stoch. Process. Appl. 115, 299–327 (2005)
Eydeland, A., Wolyniec, K.: Energy and Power Risk Management. Wiley, Hoboken (2003)
Fackler, P.L., Tian, Y.: Volatility models for commodity markets. In: Proceedings of the NCR-134 Conference on Applied Commodity Price Analysis, Forecasting, and Market Risk Management, pp. 247–256 Department of Agricultural Economics of the Oklahoma State University, Chicago (1999)
Filipović, D.: Consistency Problems for Heath–Jarrow–Morton Interest Rate Models. Lecture Notes in Mathematics, vol. 1760. Springer, Berlin (2001)
Geman, H.: Commodities and Commodity Derivatives. Wiley, Chichester (2005)
Gerber, H., Shiu, S.W.: Option pricing by Esscher transforms. Trans. Soc. Actuar. 46, 99–191 (1994)
Gibson, R., Schwartz, E.S.: Stochastic convenience yield and the pricing of oil contingent claims. J. Finance 45, 959–976 (1990)
Heath, D., Jarrow, R., Morton, A.: Bond pricing and the term structure of interest rates: a new methodology for contingent claim valuation. Econometrica 60, 77–105 (1992)
Horvath, B., Muguruza, A., Tomas, M.: Deep learning volatility: a deep neural network perspective on pricing and calibration in (rough) volatility models. Quant. Finance 21, 11–27 (2021)
Hubalek, F., Sgarra, C.: Esscher transforms and the minimal entropy martingale measures for exponential Lévy models. Quant. Finance 6, 125–145 (2006)
Hubalek, F., Sgarra, C.: On the Esscher transforms and other equivalent martingale measures for Barndorff-Nielsen and Shephard stochastic volatility models with jumps. Stoch. Process. Appl. 119, 2137–2157 (2009)
Issaka, A., SenGupta, I.: Analysis of variance based instruments for Ornstein–Uhlenbeck type models: swap and price index. Ann. Finance 13, 401–434 (2017)
Koekebakker, S., Ollmar, F.: The forward curve dynamics in Nordic electricity market. Manag. Finance 31, 74–95 (2005)
Kristoufek, L.: Leverage effect in energy futures. Energy Econ. 45, 1–9 (2014)
Kyriakou, I., Nomikos, N.K., Papapostolou, N.C., Pouliasis, P.K.: Affine-structure models and the pricing of energy commodity derivatives. Eur. Financ. Manag. 22, 853–881 (2016)
Kyriakou, I., Pouliasis, P.K., Papapostolou, N.C.: Jumps and stochastic volatility in crude oil prices and advances in average option pricing. Quant. Finance 16, 1859–1873 (2016)
Larsson, S., Thomée, V.: Partial Differential Equations with Numerical Methods. Springer, Berlin (2003)
Latini, L., Piccirilli, M., Vargiolu, T.: Mean-reverting no-arbitrage additive models for forward curves in energy markets. Energy Econ. 79, 157–170 (2019)
Lucia, J., Schwartz, E.S.: Electricity prices and power derivatives: evidence from the Nordic power exchange. Rev. Deriv. Res. 5, 5–50 (2002)
Musiela, M.: Stochastic PDEs and Term Structure Models. Stochastic PDEs and term structure models. Journées Internationales de Finance, IGRAFFI, La Baule, Unpublished (1993)
Nicolato, E., Venardos, E.: Option pricing in stochastic volatility models of the Ornstein–Uhlenbeck type. Math. Finance 13, 445–466 (2003)
Nomikos, N.K., Andriosopoulos, K.: Modelling energy spot prices: empirical evidence from NYMEX. Energy Econ. 34, 1153–1169 (2012)
Nomikos, N.K., Soldatos, O.: Using affine jump diffusion models for modelling and pricing electricity derivatives. Appl. Math. Finance 15, 41–71 (2008)
Peszat, S., Zabczyk, J.: Stochastic Partial Differential Equations with Lévy Noise. Cambridge University Press, Cambridge (2007)
Piccirilli, M., Schmeck, M., Vargiolu, T.: Capturing the power options smile by an additive two-factor model for overlapping futures prices. Energy Econ. 95, 105006 (2021)
Quiao, H., Wu, J.-L.: On the path-independence of the Girsanov transformation for stochastic evolution equations with jumps in Hilbert spaces. Discrete Contin. Dyn. Syst., Ser. B 24, 1449–1467 (2019)
Roberts, M., SenGupta, I.: Sequential hypothesis testing in machine learning, and crude oil price jump size detection. Appl. Math. Finance 27, 374–395 (2020)
Samuelson, P.: Proof that properly anticipated prices fluctuate randomly. Ind. Manage. Rev. 6, 41–44 (1965)
Schwartz, E.S.: The stochastic behaviour of commodity prices: implications for valuation and hedging. J. Finance 52, 923–973 (1997)
Wolyniec, K.: Commodity volatility, skew and inverse leverage effect. Risk.net, November issue pp. 30–34 (2016)
Acknowledgements
This paper is dedicated to the memory of Ole Barndorff-Nielsen. FEB acknowledges financial support from SPATUS, a thematic research group funded by UiO:Energy, University of Oslo. The authors thank two anonymous referees and the Associate Editor for their valuable comments and hints.
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Benth, F.E., Sgarra, C. A Barndorff-Nielsen and Shephard model with leverage in Hilbert space for commodity forward markets. Finance Stoch 28, 1035–1076 (2024). https://doi.org/10.1007/s00780-024-00546-0
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DOI: https://doi.org/10.1007/s00780-024-00546-0
Keywords
- Forward prices
- Stochastic volatility
- Leverage
- Ornstein–Uhlenbeck
- Processes in Hilbert space
- Positive definite operators
- Fourier transform
- Lévy processes