Evaluating Energy Forward Dynamics Modeled as a Subordinated Hilbert-Space Linear Functional

In this study, we evaluate energy forward dynamics modeled as time-change Hilbert-space of linear functional. The energy forward is represented as an element of Hilbert-space of function. Representing energy forward and futures contracts as a time-changing stochastic process in a Hilbert-space of functions shows clearly, that an arbitrage-free forward price can be derived from the buy-and hold strategy in the energy market thereby enabling investors in the market willing to be salvage from the market uncertainties as well as Arrow-Debreu situations to execute a spot or forward contracts depending on the time and place the market becomes favorable. With a clock measuring speed of evolution or data frequency for the energy stock market, the distribution of the increments of the Lévy process with the subordinator is subordinated to the distribution of increments of the Lévy process and the results are utilized to price forward contracts of a sample electricity commodity.


Introduction
The forward pricing dynamics of an incomplete Arrow Debreu world reveals interesting challenges to speculators and one such challenge is the stochastic nature of the return process for every investment in an underlying commodity stock. In this study, we define such return process as ( ) model of the return process is assumed to be impacted by the fluctuation in the price caused either by contago or normal backwardation. Hence, the evaluation of the forward dynamics of the underlying commodity while incorporating the time-change component as an incomplete energy market is very significant for this study.
Interestingly, we adopt the completeness properties of a Banach space (a special type of Hilbert space) such that the return process ( ) X t is defined in some normed space (i.e. complete, without hole) to enable us capture all discrete moving forward rates in the corresponding forward curves of the pricing dynamics, as such adopting similar approach by [1] and [2]. This approach is considered such that the distance between two nodes defined by the daily change price on the curve is defined in norm spaces with no gap in the sequence 1 2 , , , , t t t t n X X X X + + +  , where 1, 2, 3, n =  with each representing a node in a forward curve of the energy forward contracts.
Furthermore, for our time-change evaluation process and according to [3], the usual subordination procedure can be used to generate a Banach space valued Lévy processes and with the dynamics from other commodities markets like power and gas we pick a motivation for this study due to the presence of strong seasonality patterns, high degree of idiosyncratic risk over different market segments and leading to spikes in the forwards contract curve. Similarly, in other to capture all forward driven prices and capture the corresponding rates, the return process will be moved from general Lévy process to a subordinated Lévy process in discrete time.
Meanwhile, the arbitrage-free forward price is derived from the buy-and-hold strategy in the underlying spot commodity and the forward price dynamics is thus implied from a given stochastic model of the spot commodity based upon similar method used by [4] and [5]. Therefore, the representation of the forward price as the conditional expected value of the spot at time of delivery is represented. As such the expectation is estimated with respect to an equivalent martingale measure Q which is only possible if the price of the spot commodity is specified by a semimartingale dynamics.
In literature ( [4] [6] [7] and [8]), it is clear that the fundamental relationship between the spot and forward is highly delicate in energy markets and it is only fair to model the forward price dynamics directly.  ( ) X t , the subordinated process is such

Characteristics Function of a Subordinated Lévy Process
where the subordinator ( ) t Θ and ( ) L t are increasing Lévy processes with independent and stationary increments.
According to [9] Hence, every semi-martingale can also be written as a time-changed Lévy process such as The distribution of increments Θ is a stochastic process evaluated at a stochastic time, its characteristics function involves expectation over two sources of randomness where the inside expectation is taken on t L Θ , conditional on a fixed value of t Θ =  and the outside expectation is on all possible value of t Θ . If the random time t Θ is independent of t L , the randomness due to the Lévy process can be Under independence, the characteristics function of is also a subordinated Lévy process with characteristics function Therefore, in defining a parametric Lévy process is to obtain an Lévy process by subordinating a Brownian motion with an independent increasing Lévy process. However, for a continuous time change process, we it is necessary to evaluate the stochastic time integral of the subordinated Lévy process linear in each argument.

Stochastic Integrals of a Subordinated Lévy Process in Hilbert Space
Since the time-changed return processes t X and t Y are independent subordinated Lévy processes evaluated at a stochastic time, and using the approach in [10], we derive some general results on the stochastic integral represented as linear functional in a subordinated Hilbert space. Suppose we define a subordinated stochastic integral Y in the form for a Lévy process ( ) L t with values in a separable Hilbert space U and an integral stochastic process : , and considering the stochastic partial differential equation is an operator-valued process.
Suppose for convenience, we are only interested in one-dimensional martingales, i.e., we are interested in where ϒ is a continuous linear functional on the state space H of Y. Therefore, by replacing ( ) Y t with ( ) X t assuming the characteristic of the subordinated process we have; Proof: Let Ψ be an elementary random variable such that there exist n ∈  , 0 here, N is an n-dimensional Lévy process with the desired properties and the martingale covariance L Q of L and W Q of W coincide based on the above theorem ( [10]).
Based on results in [3] we conclude the construction of the stochastic integral based on Equation (10); However, in order to express the subordination in terms of standard Brownian motion, we make the following definition: Definition 2. A subordinated Brownian moion L with values in some Hilbert space U is a Lévy process such that there is a U-valued Brownian motion B and a subordinator Θ which is independent of B such that Subordinated Brownian motion L, N are of same type if there are Brownian be the time-changed filtration given by Then L is a U-valued square integrable Lévy process and there is an isometric embedding where the left stochastic integral is with respect to the filtration ( ) Therefore, we can see from the above construction that Equation (14) holds for elementary integrands, and thus for all integrands by a density argument in the Hilbert space. We can use results from Lemma (1) to derive a link between functionals of the infinite-dimensional stochastic integral and finite dimensional versions of it.

Evaluating Energy Forward Dynamics as a Subordinated Hilbert Space
In this section, we use the properties of Hilbert space to represent the forward and futures prices in energy markets as an element of Hilbert space of functions.
Motivated by results in [10], it is observed that the various relevant forwards and futures contracts traded in energy markets, which deliver the underlying over a period rather than at a fixed time in the future, can be understood as a bounded operator on a suitable Hilbert space.
We begin by stating the following relevant assumption from the Filipovic space [11] which supports the Hilbert space appropriate for our considerations.
for the inner product We assume that ( ) for the dynamics of energy spot prices in the so-called Schwartz dynamics [12].
Here, the spot price ( ) with a corresponding risk-neutral process given by driven by a Lévy process ( ) L t . We assume that ( ) with φ being the logarithm of the moment generating function of ( ) ,.
. If L is a driftless Lévy process, the exponential moment condition on ( )

F t T T w T T T f t T T
where ( )

, , f t T t T
≤ is the forward price for a contract "delivering energy" at the fixed time T, and ( ) ; , w T T T  is a deterministic weight function defined by; ( ) for the forward-style contracts and ( ) Here 0 r > is the risk-free interest rate which we suppose to be constant. In the energy market on NYMEX, say, WTI oil is delivered physically at a location over a given delivery period like month or quarter. We will therefore have the same expression (19) for the oil forward prices as in the case of energy forwards.

F t T T f t T T = ∫
where ( ) where 0 α > is the speed of adjustment, µ is the long-run mean yield, and ( ) dL t is the increment to a standard Lévy process. However, from [13], the , where ρ denotes the correlation coefficient. The variance of the change in the net marginal rate of convenience yield (volatility) is 2 σ representing the measure of jump sizes in the forward driven incomplete market.
Hence, we adopt [14] approach and construct a no-arbitrage portfolio that includes two futures contracts of different maturities and the spot commodity and this approach leads to the following differential equation for futures price; with boundary condition Since convenience yield is non-traded, the differential Equation in (29) depends on investor risk preferences embedded in the market price of risk for convenience yield, λ . To obtain a solution, a version of the Feynman-Kac Theorem is involved and, for tractability, the market price of risk is assumed to be constant. This is equivalent to assuming in a general equilibrium framework that the representative investor has a logarithmic utility function. In this special case, the marginal utility of wealth is independent of wealth; and the market price of risk, which is given by the covariance of the change in the convenience yield with the rate of change in the marginal utility wealth, is constant.
where the expectation is taken with respect to the risk-neutral processes from Equation (17); Then, we define the two-factor theoretical futures and forward prices depend on the current level of the spot price ( ) S t , the current level of the convenience yield, ( ) X t , time to maturity, the parameters of the joint process, and the price of a zero-coupon bond with maturity at time T, ( ) , P t T based on similar approach by [13] such that suppose; The standard Jump-diffusion for ( ) Y t follows the transformation (34), Ito's lemma, and the risk-neutral diffusion for spot prices, and Integrating (35) on both sides with respect to T t ≥ ; since the effect of the interest rate in the drift of the spot process cancels out the effect of the interest rate discount factor. The forward solution is therefore given by ( ) . , The solution for the forward price in Equation (37) is independent of the assumption of stochastic interest rates as long as σ is independent of the spot interest rate, and ( ) , P t T matches the market price of the zero-coupon bond with maturity at T in a no-arbitrage interest model.
For the diffusion assumed for spot price, σ is independent of S and r, the distribution of ( ) The expected value of ( ) To solve the integral in (39), the solution for the risk-neutralized stochastic differential equation for convenience yield is used which is given by Substituting the expected value of Equation (40) into Equation (39) and integrating with respect to T t

Exploring the Forward Curve of Energy Commodity
In this section, we select the West Texas Intermediate (WTI) Oil data for analysis and to evaluate the variation in forward rates in a given time interval. The data contains the daily prices of front month WTI oil price traded by NYMEX (New York Mercantile Exchange).
The front month WTI oil price is a futures contract with the shortest duration that could be purchased in the NYMEX market.
The WTI daily oil price is normalized in Figure 1  However, the spikes level is benchmarked using the properties Hilbert space linear functional of jump-reduction to reduce the price movement because the stochastic price return contains high volatility with massive jumps. This is obvious in the daily change prices for the underlying asset as seen in Figure 2 and risk averse investors tend to drive the market with demand, the daily price change seems to favour the market share capitalization for a long period as seen in the forward curve.     prices of the stochastic model is presented in Figure 4 with 1000 observations.
With the big-jump and small-jump rules, Figure 5 Table 1.
The Log prices as described by Figure 5 and Figure 6 respectively shows that although there appears to be a varying changes in forward rates, with a slight change in volatility, the magnitude of price fluctuation in a stochastic volatility situation is the same with that of a constant volatility which further reinforces the impact of the presence of closed-gap normed spaces in the sequence of forward rates as introduce by the properties of the Banach space (special type of Hilbert space) ( Table 2).
The forward curve without Lévy jumps as described in Figure 7 is estimated with the initial Lévy process ( )

Conclusion and Suggestions
Representing energy forward and futures contracts as a time-changing stochastic process in a Hilbert-space of functions shows clearly, that an arbitrage-free forward price can be derived from the buy-and hold strategy in the energy market thereby enabling investors in the market willing to be salvage from the market uncertainties as well as Arrow-Debreu situations to execute a spot or forward contracts depending on the time and place the market becomes favorable.      However, subsequent evaluation can take into consideration the impact of regime-shifts present in the time-changes as well as measure the sizes of such shifts and their corresponding impact to the price volatility.

Primary Contribution
• The multivariate subordinated processes are deduced as a moment generating function in order to bench mark the time-change process. • The subordinated process in a continuous time is represented as a stochastic integral as Hilbert space linear functional. • The energy forward is represented while incorporating the completeness properties of Banach space (a special type of Hilbert space) for an incomplete energy market.
• The pricing framework for energy forward contracts is used to evaluate the daily change price of an underlying energy contracts.