Definition 1 A return process { X ( t ) } t ≥ 0 used in this study as Lévy process, with non-decreasing paths almost surely, are called subordinators and such processes can be thought of as random models of the evolution.

A wide class of return processes appearing in application is obtained by subordination of Lévy processes with drift. Also, a stochastic time-change to the Lévy process amounts to stochastically altering the clock on which the Lévy process runs.

Meanwhile, for the return process X ( t ) , the subordinated process is such that

X ( t ) = L ( Θ ( t ) ) (1)

where the subordinator Θ ( t ) and L ( t ) are increasing Lévy processes with independent and stationary increments.

According to [9] every semi-martingale X ( t ) can be written as a time-changed Brownian motion (Lévy process), where the random time Θ ( t ) is a positive and increasing semi-martingale. As a result, there exist an incremental and stationary Brownian motion ( B ( u ) , u ≥ 0 ) and a random time-change Θ ( t ) which forms an increasing stochastic process

X ( t ) = B ( Θ ( t ) ) (2)

Hence, every semi-martingale can also be written as a time-changed Lévy process such as

X ( t ) = X t = L ( Θ ( t ) ) (3)

The distribution of increments Δ L ( Θ ( t ) ) is said to be subordinate to the distribution of increment Δ L ( t ) ; where Δ ( t ) is the data frequency or a clock measuring the speed of the evolution.

Meanwhile, in developing a subordinated Lévy process with varying time, the characteristic function of a subordinated process X ( t ) is utilized which is obtained by composition of the Laplace exponent of Θ ( t ) with the characteristics exponent of L ( t ) .

Suppose we let ( Θ t ) t ≥ 0 be a subordinator that is a Lévy process whose trajectories are increasing. Since Θ ( t ) is a positive random variable for all t it is described by its Laplace transform rather than the Fourier transform. Let the characteristic triplet of Θ ( t ) be ( Θ , ρ , α ) . Then the moment generating function of Θ ( t ) is given by;

E [ e u Θ t ] = e t L * ( u ) , ∀ u ≤ 0 (4)

where

L * ( u ) = α u + ∫ 0 ∞ ( e u x − 1 ) ρ ( d x )

and L * ( u ) is the Laplace exponent of Θ ( t ) . Since the time-changed process X t = L Θ t = L ( Θ t ) is a stochastic process evaluated at a stochastic time, its characteristics function involves expectation over two sources of randomness

Φ X t ( u ) = E [ e i u L Θ t ]

Φ X t ( u ) = E [ E [ e i u L ϵ | Θ t = ϵ ] ]

where the inside expectation is taken on L Θ t , conditional on a fixed value of Θ t = ϵ and the outside expectation is on all possible value of Θ t . If the random time Θ t is independent of L t , the randomness due to the Lévy process can be integrated out using moment generating form of L t given by

Φ L t ( u ) = E [ e i u L t ]

Φ L t ( u ) = [ e − i Ψ L ( u ) ]

and a characteristics function of X t , that is

Φ X t ( u ) = E [ e i u L Θ t ]

Φ X t ( u ) = E [ E [ e i u L ϵ | Θ t = ϵ ] ]

Φ X t ( u ) = E e − i Ψ L ( u )

Φ X t ( u ) = L Θ t * ( Ψ L ( u ) )

Under independence, the characteristics function of X t = L Θ t is just the Laplace transform of Θ t evaluated at the characteristics component of L t .

Let ( L t ) t ≥ 0 be a Lévy Process on ℝ with characteristics exponent Ψ ( u ) and triplet ( σ , v , γ ) and let ( Θ t ) t ≥ 0 be a subordinator with Laplace exponent L * ( u ) and triplet ( Θ , ρ , a ) . Then the subordinated process X ( t ) defined by;

X ( t ) = L ( Θ ( t ) ) , (5)

is also a subordinated Lévy process with characteristics function

E [ e i u X t ] = e t L * ( Ψ ( u ) ) . (6)

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.

Since the time-changed return processes X t and Y t 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

Y ( t ) = ∫ 0 t     Ψ ( s ) d L ( s ) , (7)

for a Lévy process L ( t ) with values in a separable Hilbert space U and an integral stochastic process Ψ : ℝ + × U → H , and considering the stochastic partial differential equation

d X ( t ) = ( A X ( t ) + α ( t ) ) d t + σ ( t ) d L ( t ) , t ∈ ℝ (8)

on the interval ℝ = [ 0, T ] with H being a separable Hilbert spaces equipped with some boundary conditions where

· A : H → H is a first order differential operator

· α : ℝ → H is a mapping with Brochner-integrable trajectories

· L : ℝ → H is a square-integrable, H-valued stochastic process with covariance operator Q ∈ L 1 + .

· σ : ℝ → L ( Q 1 / 2 ( H ) , H ) is an operator-valued process.

Suppose for convenience, we are only interested in one-dimensional martingales, i.e., we are interested in

Y ( t ) = ϒ ( Y ( t ) ) , (9)

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;

X ( t ) = ∫ 0 t ( ϒ ∘ Ψ ( s ) ) d L ( s ) . (10)

Suppose we take U as finite dimensional, and some standard (real-valued) Brownian motion { B ( t ) } t ≥ 0 and Itô integral stochastic process σ such that

X ( t ) = ∫ 0 t     σ ( s ) d B ( s ) (11)

This leads us to Theorem 1 which shows that a similar representation holds if U is any separable Hilbert space.

Theorem 1. Let n ∈ ℕ and H, U be separable Hilbert spaces. Let L be a square integrable and mean zero U-valued Wiener process with coveriance Q ∈ L 1 + . Assume that D i m r a n ( Q ) ≥ n and Q is positive definite. Let Ψ ∈ ℵ L 2 , ϒ ∈ L ( H , ℝ n ) and define

X ( t ) = ϒ ( ∫ 0 t     Ψ ( s ) d L ( s ) ) .

Then there is an n-dimensional standard Brownian motion B such that

X ( t ) = ∫ 0 t     σ ( s ) d B ( s ) ,

where σ ( s ) = ( ϒ Ψ ( s ) Q Ψ ( s ) * ϒ * ) 1 / 2 ∈ ℵ B 2 ( ℝ * ) . If σ ( s ) is invertible in ℝ n × n for λ ⊗ P -almost any s ∈ ℝ , then σ − 1 ∈ ℵ X 2 ( ℝ n ) and

B ( t ) = ∫ 0 t ( σ ( s ) ) − 1 d X ( s ) . (12)

Proof: Let Ψ be an elementary random variable such that there exist n ∈ ℕ , 0 ≤ a j ≤ b j < ∞ , and F a j -measurable square integrable random variables Y j and ψ j ∈ L ( U , H ) such that

Ψ = ∑ j = 1 n     Y j 1 [ a j , b j ] ψ j .

By definition of L, we have L ( t ) = W ( Θ ( t ) ) for a U-valued Wiener process W with martingale covariance Q W and a subordinator Θ . Let Γ be the isometric embedding given in Lemma (1). Note that Γ ( ϒ Ψ = ϒ Γ ( Ψ ) ) because this holds if Ψ is elementary. From Lemma (1) and Theorem (1) we have;

ϒ ( ∫ 0 t     Ψ ( s ) d L ( s ) ) = ∫ 0 t     ϒ Ψ ( s ) d L ( s ) = ∫ 0 Θ ( t )     ϒ Γ ( Ψ ) ( s ) d W ( s ) = ∫ 0 Θ ( t )     σ 1 ( s ) d B ( s )

where B is a standard Brownian motion on ℝ n and

σ 1 ( t ) = Γ ( ( ϒ Ψ Q W Ψ * ϒ * ) 1 / 2 ) ( t ) .

Applying Lemma (1) again gives

ϒ ( ∫ 0 t     Ψ ( s ) d L ( s ) ) = ∫ 0 t     σ ( s ) d N ( s ) (13)

where N ( t ) = B ( Θ ( t ) ) , t ∈ ℝ + and

σ ( t ) = ( ϒ Ψ ( t ) Q W Ψ ( t ) * ϒ * ) .

here, N is an n-dimensional Lévy process with the desired properties and the martingale covariance Q L of L and Q W 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);

X ( t ) = ∫ 0 t     Γ ( s ) d L ( s )

where L ∈ M 2 ( U ) and Γ ( s ) = ϒ ∘ Ψ ( s ) are operators from U to another Hilbert space H.

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

L ( t ) = B ( Θ ( t ) ) .

Subordinated Brownian motion L, N are of same type if there are Brownian motions B L , W N and subordinators Θ L , Θ N such that B N , Θ N are independent, B L , Θ L are independent, Θ L , Θ N have the same law and for any t ∈ ℝ + . Subordinated Brownian motions have some similarities with Brownian motion if the subordinator has finite first moment. In particular, the set of integrands can be compared easily. First, we recall the notion of time-changed filtrations:

Definition 3. A time-change is a right-continuous increasing family ( Θ ( t ) ) t ∈ ℝ + of stopping times with respect to some right-continuous filtration ( F t ) t ∈ ℝ + . The time changed filtration is the filtration given by

F t Θ = F Θ ( t ) = ∩ s > t F Θ ( s ) t ∈ ℝ + .

Let X be an F-adapted stochastic process and assume that the time-change is finite valued. The time-changed process by X Θ ( t ) = X ( Θ ( t ) ) , t ∈ ℝ + . Therefore, adding the following results on stochastic integration with respect to subordinated Brownian motions.

Lemma 1. Let B be a mean-zero Brownian motion with values in a separable Hilbert space U relative to some filtration ( F t ) t ∈ ℝ + , H be another separable Hilbert space, ψ ∈ ℵ L 2 ( G , H ) , Θ be a non-zero subordinator with finite moment such that Θ ( t ) is a stopping time for each t ∈ ℝ + . Let ( G t ) t ∈ ℝ + be the time-changed filtration given by G t = F Θ ( t ) and L ( t ) = B ( Θ ( t ) ) , t ∈ ℝ + . Then L is a U-valued square integrable Lévy process and there is an isometric embedding Γ : ℵ L 2 ( G , H ) → ℵ B 2 ( F , H ) such that

∫ 0 t     ψ ( s ) d L ( s ) = ∫ 0 Θ ( t )     Γ ( ψ ) ( s ) d B ( s ) , P -almostsurely (14)

where the left stochastic integral is with respect to the filtration ( G t ) t ∈ ℝ + , the right stochastic integral is with respect to the filtration ( F t ) t ∈ ℝ + .

Proof: See proof in [7].

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.

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.

H1: The functions h ∈ H are continuous and the pointwise evaluation J τ ( h ) = h ( τ ) is a continuous linear functional on H, for all τ ∈ R + .

H2: The semigroup { S ( t ) | t ∈ ℝ + } is strongly continuous in H with infinitesimal generator denoted by A.

Therefore, we use the previously defined class of Hilbert space which follows from the Filipovic space satisfying (H1)-(H2) and which are coherent with economical reasoning about the forward curve τ ↔ h ( τ ) .

Since in practice the forward curve is obtained by smoothing data points using smooth fitting methods it is reasonable to assume;

∫ 0 ∞   h ′ ( τ ) 2 d τ < ∞ (15)

defined on H w of absolutely continuous functions h : ℝ + → ℝ . Moreover, the curve flattens for large time to maturity τ . There is no reason to believe that the forward rate for an instantaneous loan that begins in 10 years differs much from one which one day later. We take this into account by penalizing irregularities of h ( τ ) for large τ by some increasing weighing function w ( τ ) ≥ 1 , that ∫ 0 ∞     w ( τ ) h ′ ( τ ) 2 d τ for a given continuous and increasing weight function w : ℝ + → [ 1, ∞ ) with w ( 0 ) = 1 . The norm of H w is ‖ g ‖ w 2 = 〈 g , g 〉 for the inner product

〈 f , h 〉 = f ( 0 ) h ( 0 ) + ∫ 0 ∞     w ( τ ) h ′ ( τ ) f ′ ( τ ) d τ .

here f , h ∈ H w . We assume that ∫ 0 ∞   w − 1 ( τ ) d τ < ∞ . The typical choice of weight function is that of an exponential function w ( τ ) = e ( w ˜ τ ) for a constant w > 0 , in which case the integrability condition on the inverse of w is trivially satisfied. From [11], we know that H w is a separable Hilbert space. As we shall see, one can realize energy forward prices as linear operators on and in fact interpret energy forward prices as stochastic processes with values in this space.

A simple motivation for the appropriate choice of is the classical model for the dynamics of energy spot prices in the so-called Schwartz dynamics [12].

Here, the spot price at time is given by

for a standardized subordinated return process defined in an Ornstein-Uhlenbeck (OU) process

with a corresponding risk-neutral process given by

driven by a Lévy process. We assume that has exponential moments, , are constants, and. It is simple to see that is continuously differentiable for every t, and

From [12], we get the forward price at time, for a contract delivering at time, is given;

with being the logarithm of the moment generating function of. Suppose we model the spot price directly under the pricing measure Q. Letting, we find (by slightly abusing the notation)

Assuming that the weight function w is such that

Then it follows that from the Cauchy-Schwartz inequality and the assumption. Hence, f is uniformly bounded in since

But then

which shows that. If L is a driftless Lévy process, the exponential moment condition on yields that has the representation

for a constant and Lévy measure. But by the monotone convergence theorem and L’Hopital’s rule, we find that

such that Equation (18) becomes

and therefore when is small. Thus, a sufficient condition for is.

Therefore, we now represent the realization in of general energy forward and futures contracts with a delivery period. Suppose that is the forward price at time t of a contract on energy delivering over the time interval, where [12]. Then we can express this price as

where is the forward price for a contract “delivering energy” at the fixed time T, and is a deterministic weight function defined by;

for the forward-style contracts and

for the futures style.

Here 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. Meanwhile, futures on WTI oil index deliver the money-equivalent from the aggregate index value over a specified period. Hence, the futures can be expressed as

where is the futures price of a contract that delivers the corresponding index at the fixed delivery time, i.e., the weight of the futures contracts can be expressed by (19) with

as the weight function. Aiming at a so-called Museila representation of in (19). Define as the time until start of delivery of the swap and as the length of delivery of the swap. With the notation, we write

for the weight function defined by

where, and. Referring to the different cases of the weight function, we find that for a forward contract and for the forward-style oil swap. Slightly more interesting are the future-style energy swaps, yielding

here, we use the future-style contract as defined in Equation (21). Note that all these cases result in a weight function which is independent of time. Furthermore, the only case that depend on and y is given in Equation (25), which becomes in fact stationary in the sense that depends on.

However, when the asset price is a spot commodity, the solution for forwards price (under deterministic interest rates, futures prices equal forward prices) is given by;

where is the time to expiration of the contract, and with and hence

where is defined by the geometric Lévy process given by;

and

For a Poisson process, measures the sensitivity of the process and at

and we have

Suppose then which violates our sensitivity condition of the process, hence we select, then Equation (26) becomes

Therefore, suppose that the rate of convenience yield follows the mean-reverting process then recall from Equation (17) that SDE;

where is the speed of adjustment, is the long-run mean yield, and is the increment to a standard Lévy process. However, from [13], the, where denotes the correlation coefficient. The variance of the change in the net marginal rate of convenience yield (volatility) is 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,. 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.

Therefore, the Feynman-Kac solution for futures (forwards) price is given by;

where the expectation is taken with respect to the risk-neutral processes from Equation (17);

and

Then, we define the two-factor theoretical futures and forward prices depend on the current level of the spot price, the current level of the convenience yield, , time to maturity, the parameters of the joint process, and the price of a zero-coupon bond with maturity at time T, based on similar approach by [13] such that suppose;

The standard Jump-diffusion for follows the transformation (34), Ito’s lemma, and the risk-neutral diffusion for spot prices, and

Integrating (35) on both sides with respect to;

however, from Equation (34) we take, then;

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 is independent of the spot interest rate, and 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, is independent of S and r, the distribution of is normal, and is the moment-generating function of a normal distribution. In this case, the solution for the forward price is

The expected value of is

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 gives

where

and

The variance of, Equation (36), is

The variance and the covariance terms in Equation (42) are obtained by replacing with the random part of the process (40), giving

Solving for the variance and covariance in Equation (43) gives

The final result for is obtained by using, Equation (41), and, Equation (44) in Equation (38) the forward solution is given by;

where

which gives the pricing framework for estimation of the energy forward dynamics modeled as a subordinated Hilbert space of linear functional.

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 in other to refine the time series properties of the WTI oil data and the weight functions of the forward-style and the futures-style contract to enhance better pricing model for estimation of the forward prices. Between the first 50 monthly price observation, it is clear that noise exist in the data.

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.

Furthermore, it follows from Equation (23) and as seen in Figure 2 that the dynamics of forward price curves can be represent as Hilbert space functional as a result of this assertion we accept that if is an -valued stochastic process, then will be a stochastic process with values in as well.

As the subordinated process, the time change stochastic process as shown in the forward curve in Figure 2 defines how the incomplete market trading options of WTI oil futures and forwards displays the highs and lows of the market situation.

Investors in the energy market are either willing to be risk neutral and trade for high profits or not but the energy forward pricing framework clearly favors the idea that with changing and the forward rates is bound to increase marginal yield over time.

In order to avoid false detection of jumps in the forward curves pricing framework, we employed the following model for simulation;

where is volatility, the drift component and is a Wiener proces that follows a Poisson distribution. In other to estimate the simulation model for a case of constant volatility verses stochastic volatility without Lévy jumps, we integrate both sides of Equation (46);

With a constant volatility and the standard Brownian motion following a Poisson distribution such that, then the simulated stock prices of the stochastic model is presented in Figure 3 with 1000 observations.

Figure 3 clearly indicates that the fluctuation in the stock prices without the presence of Lévy jumps is benchmarked by the volatility value of upwards. However, we notice one spike in price of the stock which is absolute with a 0.05 level of significance and assigned to the variation in price dynamics of the model.

With a stochastic volatility and the standard Brownian motion following a Poisson distribution such that, then the simulated stock prices of the stochastic model is presented in Figure 4 with 1000 observations.

With the big-jump and small-jump rules, Figure 5 clearly shows the detection and arrival of jumps from the small-jumps to big-jumps as the frequency of the observation increases with an initial volatility, given a sensitivity level. Although there exist a drift part, the number of jumps with constant volatility in all 300 observations is seen to be evenly distributed by both rules as can be seen in 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, the average stock price for WTI, and is the drift component for which the price fluctuate with sensitivity and constant volatility of, and respectively where the jumps, are evaluated by the small-jump and big-jump rules accordingly with a convenient yield and correlation coefficient of the forward rates as and respectively.