We have the following technical result on changing from univariate to multivariate polynomial bases:
4.1. Plain-Vanilla Forward Prices
We find the following result, collecting the notation introduced in this section:
Proposition 1. Let . For , it holds,where is given in (16), in Lemma 2 and is the polynomial transition matrix defined in (2). Proof. First note that a polynomial process has finite conditional moments of all orders (see [
8] (Theorem 1)). Hence, the conditional expectation is well-defined. From Lemma 2, we have for
and
,
The result follows from the polynomial process property of X (see Theorem 1). □
If the spot dynamics follows an arithmetic model (
8), then we find the forward price (
10) as
where
is the vector such that
. Of course, if we choose
, then
,
, the
-identity matrix and
Furthermore,
is in this case the matrix mapping
into
, with
being polynomials of order 1 on
. If
and
for
, then
Thus, we have a simple relationship when the monomials are chosen as the basis of
. Indeed, the forward price is
Remark 2. Proposition 1 is a simple extension of the formulas by Kleisinger-Yu et al. [11] who focussed on the case of . Ware [10] also discussed such formulas in his polynomial approach to power spot models. Let us discuss the forward price dynamics in Proposition 1 from an empirical viewpoint. To put our discussion into context, we note that Ware [
10] proposed, among other models, a one-factor polynomial diffusion process combined with a fifth-order polynomial as a spot dynamics, i.e.,
with
and
. Here, we ignore seasonality for simplicity in the discussion. Then, using
, the
time to maturity, we get
where
. Using the monomials as basis, we have
. Moreover, as
X is polynomial, we have that its drift
is in
and diffusion
is in
. If
, it is easy to see that
,
and
for
. Hence,
will be a lower triangular matrix with zero in the first row and diagonal elements
for
. (Note that first diagonal element is zero, the second is
, the next is
, etc. to the last which is
.) The distinct eigenvalues of the matrix then becomes
and
for
. We can find a basis of
of eigenvectors
, where
can be chosen to be the first canonical basis vector of
with 1 in first coordinate and zeros otherwise. From this, we find
In commodity markets, one typically expects forward prices to be flat in the long end of the curve as long as spot prices are expected to possess some stationarity properties. It is evident from above that the forward prices for large
’s tend to
whenever the eigenvalues
are negative. Hence, we obtain a flat forward curve in the long end when
for
. For example, if
, then this is achieved when
. On the other hand, in the Jacobi model suggested by Ware [
10],
, which again implies that
. This is indeed the case in his model.
By an application of Ito’s formula, one furthermore observes that the volatility of will satisfy the Samuelson effect, as the volatility will be determined by the diffusion term from and scaled by exponentials . Whenever , these exponentials will tend to 1 when tends to zero, which gives a Samuelson effect as the forward volatility converges to the spot volatility in this case.
It is also worth noticing that the sum of exponentials in (
17) gives rise to several humps in the forward term structure, that is, the curve
may have several local maxima and minima according to the values of the eigenvalues. A hump shape behaviour of the forward curve is reasonable from an economic viewpoint, indicating differences in risk preferences of the traders along the forward curve. We refer to the work of Benth, Šaltytė Benth and Koekebakker [
4] for a discussion of the various stylised facts of forward curves in power and commodity markets.
The analysis above can be generalised to arbitrary polynomials p, as well as also extending the dimension of the polynomial process beyond .
If the spot follows a geometric model (
7), then, from the Taylor series expansion of the forward price in (
11), one chooses the basis for
to be the monomials, i.e.,
. The basis for
is arbitrary selected. We find:
Proposition 2. Suppose the commodity spot dynamics is given by as in (7). Assume that . Then, the forward price in (11) is given bywhere is the -canonical unit vector in (i.e., the vector with 1 in coordinate and zero otherwise), in Lemma 2, in Lemma 1 and in (2). Proof. From the exponential integrability condition on
, it follows from monotone convergence (see [
38] (Theorem 2.15)) that
Hence, in particular, we find that
for every
. As
therefore is a sequence in
and
, it follows from dominated convergence theorem (see [
38] (Theorem 2.25)) that
Next, by Lemma 2 followed by the polynomial property of
X yield,
The result follows. □
The Proposition above allows for stating the forward price as
where
. In practical computations, one would of course truncate the sum. One could also make use of the (truncated) sum in a regression study, where one empirically could reveal the structure of the
’s by regressing observed forward prices against polynomials of
X. Such a study requires knowledge of the state of
, which can be recovered from the spot prices. Such recovery may involve stochastic filtering if
.
The exponential integrability condition
in Proposition 2 is rather restrictive. Geometric Brownian motion, e.g., does not satisfy this condition for any
. From Example 2 of Filipović and Larsson [
8], the GARCH diffusion process
for constants
is a polynomial process with ergodic solution being inverse Gaussian distributed with shape parameter 2 and
as scale. In the invariant case,
X does not have a finite variance and hence not being exponentially integrable either. By letting
be a geometric Brownian motion, the GARCH diffusion becomes the Pilipović model (see Pilipović [
19]) briefly discussed in
Section 3. This model will not in general be exponentially integrable. Ornstein–Uhlenbeck processes driven by compound Poisson processes having exponential jumps will have a gamma distributed stationary solution. This will also not be exponentially integrable, except under restrictive conditions on the parameters.
4.2. Exotic Forward Prices
Next, consider CDD-forwards on temperature (or a floor electricity forward) with price given in (
12). We need some preparatory material on polynomial expansions of call and put payoff functions. For
, let
denote the
nth Hermite polynomial (known as the “probabilistic” Hermite polynomial) defined as
where
is the density of the standard normal distribution function. We notice that
. Further, define
for
with the usual convention that
. It is known that
is an ONB of the Hilbert space
. Considering
, we readily find that
as
for
and zero for
and
w integrates any polynomial. Moreover, for any
, we find that
with
, a standard normal random variable. Moreover, from elementary functional analysis, we have
The following simple result holds:
Lemma 3. Suppose and denote by . Then, for , converges to in . Moreover, Proof. Obviously,
and
in
as
. The latter means
Hence, the first claim follows. For the second claim, the Cauchy–Schwarz inequality implies
Invoking the first claim proves the Lemma. □
We extend the previous result to more general random variables Y in the next lemma:
Lemma 4. Suppose Y is a random variable with probability density satisfying for , where is a constant and is the normal density function with mean a and variance is with . If then Proof. By the Cauchy–Schwarz inequality,
Consider the positive function
As
, it holds that
and thus
u has a maximum value on
. It follows that
Since and is its truncation in the basis representation, the result follows after passing to the limit. □
We recall that
satisfies the requirement that
. Moreover, CARMA(
)-processes driven by Brownian motion are normally distributed, and hence the above result applies with
being a normal distribution with variance less than 1. We also recall from Ackerer et al. [
9] that the Jacobi volatility process has a distribution which is absolutely continuous with respect to the normal distribution. In addition, rather than using the Taylor series representation
, we may use the Hermite polynomials as series expansion for the exponential function since obviously
.
The condition that the variance
b is strictly less than one is very restrictive. However, one can overcome this by a change in the Hermite basis or by appropriate rescaling the function
f. We provide a thorough discussion of this in
Section 4.3, where we take a more general perspective. For the moment, we note that Ackerer et al. [
9] used an affine transform of the Hermite polynomials as basis.
Remark 3. We notice that the condition on the probability density of Y in Lemma 4 implies that the distribution is absolutely continuous with respect to . By assuming instead that the distribution of Y, is dominated by that of , i.e., in the sense for every Borel set , we find that there exists an non-negative Radon–Nikodym density . In this case, we can define a probability density . However, then, because, if this is not the case there exists a measurable set U with strictly positive mass such , which implies that being a contradiction. Further notice that the constant C must be greater than or equal to 1 simply because we have distributions with total mass 1 on both sides of the bound.
In the next subsection, we take a more general perspective where the distribution of the polynomial process does not need to be bounded by a Gaussian but other suitable classes of distributions for which we can associate polynomials. At the current stage of our exposition, we focus on the Gaussian case as this is the most relevant in connection with temperature forwards, where the underlying dynamics have empirical evidence for being normally distributed (recall discussions in
Section 3).
We next show a polynomial expression for the CDD-temperature forward price: to this end, choose the basis
for
, where
are the normalised Hermite polynomials defined in (
20). For
, we fix an arbitrary basis
.
Proposition 3. Suppose the commodity spot dynamics is given by as in (8), where X is a d-dimensional polynomial process. Assume that the random variable has an -conditional probability density which is bounded by a normal density as in Lemma 4. Then, the forward price in (12) is given bywhere is the -canonical unit vector in (i.e., the vector with 1 in coordinate and zero otherwise), is given in (15), in Lemma 2, in Lemma 1 and in (2). Proof. We find that
, and therefore
From the condition on the density of
given
, it holds from Lemma 4 that the conditional expectation is well-defined as integrability holds, and that we can commute sum and conditional expectation. That is,
By the translation of the monomial basis in Lemma 1, we find that
Furthermore, invoking Lemma 2 gives
Finally, applying the polynomial property of
X, we find
The result follows. □
We remark in passing that the above Proposition could also have been developed for other functions f than the one appearing for the CDD-temperature forwards. In fact, any function would do, with the only difference that the coefficients appearing in the expression for F in Proposition 3 would change (as they depend explicitly on f, of course). For example, using , which defines a function in , Proposition 3 provides an alternative forward price series expression to Proposition 2 for geometric spot price models.
To efficiently compute the CDD-temperature forward price by exploiting the polynomial structure of
X, we truncate the infinite sum. All the matrices and vectors involved are explicitly given, except the coefficient functions
. We recall these to be defined as
for
Y being standard normally distributed. We can compute these coefficients once for a given function
, as they are independent of the polynomial process
X.
Remark 4. In the market for temperature forwards, the contracts are settled over a pre-specified period of time. In that case, a CDD-temperature forward isafter appealing to the Fubini–Tonelli theorem to commute sums. If temperature follows a CARMA-dynamics driven by a Brownian motion, then the dimension
d will indicate the autoregressive order. Moreover,
will be Gaussian and
X a
d-dimensional Ornstein–Uhlenbeck process, and thus the conditions in Proposition 3 hold. We recall that the temperature dynamics is conveniently modelled by a CAR(3)-process (see [
22]), which means that
and
, the canonical unit vector in
with 1 in first coordinate and zero otherwise.
Interestingly, Asian options are closely related to the above CARMA-situation by the following argument: consider an Asian option with payoff
at exercise time
T. Here,
is a
d-dimensional polynomial process and
, with the spot price being
(we ignore seasonality here in this short discussion). Let now
X be the process in
defined as
, where
It follows that X is a polynomial process, and we have that the Asian option payoff can be written as with , the canonical unit vector in with one in the last coordinate and zero otherwise. In particular, assuming that is a multivariate Gaussian Ornstein–Uhlenbeck process, we will have that X is a Gaussian Ornstein–Uhlenbeck process and we find ourselves in a situation which is closely resembling the forward price of a CDD-temperature contract analysed above.
4.3. A General Polynomial Approach to Forward Pricing
In this subsection, we take a general perspective on forward pricing, providing a unifying expression for the forward price in markets with a polynomially based “spot”-process. The approach requires some additional conditions on the polynomial process, but on the other hand gives an attractive treatment of options on forwards, a topic which is analysed in
Section 5.
Suppose that the “spot price” dynamics is given by
for some measurable function
, seasonality function
and
X being a
d-dimensional polynomial process. Examples of relevance can be
for
,
and
being one of the following functions:
(arithmetic spot model),
(geometric spot model) or
(spot for an exotic forward such as temperature futures). Our aim is to compute the forward price, defined as
To achieve this goal, we employ a multivariate generalisation of the space
along with an integrability assumption on the conditional probability distribution of
given
. In fact, without losing any generality for practical purposes in commodity and energy markets, we assume that
X is also a Markovian process. (Note that non-Markovian polynomial jump diffusion processes exist, see., e.g., [
8] (Page 71).)
Let us start by introducing a multi-dimensional generalisation of the space
. To this end, let
be a probability density function on
, and for
, denote by
the Hilbert space of real-valued functions on
for which
with inner product
Assume further that there exists an ONB for of polynomials, given by using the multi-index notation . We use the notation for the order of the multi-index, where it is supposed that . Furthermore, is a basis of polynomials of order N, which we use as . Ranking the basis functions according to their polynomial order is convenient and natural when doing approximations in practical applications of this theory.
Next, denote by
the transition probability distribution on
of
given
. Following Filipović and Larsson [
8] (Sect. 7), introduce the
likelihood ratio as the function
such that
We assume that such a likelihood ratio of with respect to exists. In the next theorem, we state a general series representation in terms of polynomials for the forward price along with a computationally convenient truncation.
Theorem 2. Assume that and for any and , where g and ℓ are defined, respectively, in (21) and (23). Then, we have that (pointwise) when , where F is the forward price in (22) with representationwhile for any , with is such that . We recall , and G given in (2). Proof. Notice first by the Markovian property of
X that
, where
We find from the assumptions
that
Therefore, by Parseval’s identity,
where
and
are the coefficients in the ONB representation of
and
in
, respectively. Tracing back the definitions, we find
By assumption on the polynomial basis,
. Thus, there is a vector with length equal to the dimension of
such that
. We then conclude the desired form,
after appealing to the polynomial property of
X. This proves the representation of
.
Define for each
the approximation
with the notation
We observe that is nothing but projected down on the finite dimensional subspace of spanned by . This gives us .
Notice that by Parseval’s identity,
From the very definitions of
f and
, we find by the Cauchy–Schwarz inequality and Parseval’s identity,
In conclusion, for every when . The proof is complete. □
We notice that the dependency on the seasonality component is merged into the coefficients
and is as such not material in the analysis above. We include it simply because seasonality is present in relevant models, and we prefer to have it explicit. Additionally, it highlights a difference with the other polynomial expansions which we present in this section. Furthermore, we observe that we may compute the coefficients
by numerical integration methods, for example Gaussian quadrature or Monte Carlo simulation. Indeed, we have
where
Z is a
d-dimensional random variable with probability density
.
Remark 5. In the case X is not a polynomial process, we see by inspection of the proof of Theorem 2 that we still have an interesting representation of the forward price in terms of the polynomial moments of conditional on . Indeed, removing the polynomial property, we see that all conclusions in the theorem holds, except that which will not be explicit in terms of polynomials of x. Of course, we still need the regularity assumptions of g and the likelihood ratio ℓ to hold. We can approximate the forward prices by polynomial moments of the process up to a certain order.
The main assumption in our general approach to forward pricing is the existence of a density
admitting a polynomial basis for
, such that there is likelihood ratio function being an element of this space. This problem is classical, and has a long history in probability and physics, where we refer to the works of Asmussen, Goffard and Laub [
39] and Eggers [
40] for some recent applications and studies. One thinks of
as the reference measure, and for a target distribution
the goal is to have a Gram–Charlier series with efficiently computable polynomials. Following the discussion in Asmussen et al. [
39], if
in Dimension 1 has all moments finite, there exists an orthogonal sequence of polynomials, which, moreover, defines a basis in
if
has finite exponential moment. One can easily build up multivariate reference measures in general dimensions
d by tensorising. For example, we may define
. This will provide a
d-dimensional version of the space
based on Hermite polynomials. In Rahman [
41], a general multivariate basis of Hermite polynomials are defined appealing to the Rodrigues formula, i.e., based on the derivatives of the multivariate Gaussian distribution function with mean zero and general covariance. A special case of this, choosing the covariance matrix to be the identity, leads back to the definition of
. Another example could be a reference measure in
defined by the gamma-distribution (see the work of Asmussen et al. [
39], where Laguerre polynomials appear).
We discuss the case
in some more detail. We start by introducing a multi-dimensional version of
. To this end, for
, denote by
the Hilbert space of real-valued functions on
for which
with inner product
where we recall
. An ONB for
is given by
using the multi-index notation
, and
. Here, we recall
to be an ONB of
.
Let us look at some particular cases of the function
g in the case of
, which are of relevance to commodity and energy markets. First, in an exponential spot price model, we have
for
and
. Since
is integrable, we see that
. We compute the coefficients
:
In the arithmetic case, the spot takes the form
with again
and
. Since
is integrable and
. We find the coefficients in this arithmetic case as follows:
Now, for it holds that . Thus, the only non-zero terms in the sum above are those where , where 1 is appearing in coordinate i, in which case . All other will give , except where we get . This is a very unsurprising result, of course.
Next, let us consider the case of a CDD-temperature forward or a floor electricity forward, for which we have that
for
. As the max-function grows at most linearly, we have
, and it follows that
. We may represent the coefficients as
for
with
I being the
identity matrix. By iterated conditional expectation, conditioning on
,
, we can define for
R being a standard normal random variable
and iteratively backwards
,
yielding
.
A more detailed discussion of the condition on the likelihood ratio is in place. We first recall from Ackerer et al. [
9] that the Jacobi volatility model has a likelihood ratio with respect to the Gaussian density which satisfies the condition of square integrability. Hence, the Jacobi volatility model allows itself to a series expansion in terms of the Hermite polynomials, as conducted in detail by Ackerer et al. [
9]. Many of the interesting polynomial models are such that
is Gaussian. For example, we have the two-factor models of Lucia and Schwartz or CARMA-models driven by Brownian motion. This results in a conditional distribution function
being a Gaussian distribution with mean
and covariance matrix
. Here, we collapse the notation to make our discussion more transparent. Hence, we find that the likelihood ratio is
Here, I is the identity matrix. It is evident that the function if and only if , or, equivalently, . This is not always true, as we can have two-factor models with independent Brownian motions having variance each strictly bigger that 2. Then, V is a diagonal matrix with variances on the diagonal which is dominating , and the required integrability of the likelihood ratio fails.
In such cases, we can re-scale the polynomial process
X. To this end, let
C be some
matrix such that
. If we have available such a matrix, we can define a new stochastic process
. Since any matrix transform of a polynomial process again is a polynomial process,
Y is a polynomial process. If further
C is invertible, then
, and we have
In Theorem 2, we assume that
. Furthermore, for the polynomial process
Y we find that the likelihood ratio function is (as a matrix transformation of the Gaussian variable
),
Hence, we have that whenever C is such that .
Here is an example of a re-scaling: Let
be bivariate Gaussian with covariance matrix
where
are two strictly positive constants (being the marginal standard deviations) and
(which is the correlation). For example, this is the situation with the two-factor model of Lucia and Schwartz, or a CARMA-model in
. Observe that a diagonalisation of
V is given by
Let, for some positive constant
Then, we find
since
. In conclusion, we find a scaling
C of the original polynomial process, for which the covariance matrix can be dominated by 2 times the identity. Then, the likelihood ratio has the desired integrability, but we must adjust slightly the integrability condition on
g. For most interesting functions
g, this is not any added restriction, as for example the cases considered above.
Rather than re-scaling, we could use the multivariate Hermite polynomials introduced by Rahman [
41] for a sufficiently big covariance matrix. In the above case of re-scaling, we need to have some knowledge of the matrix
C before doing the computations. However, the advantage then is that one can simply apply the standard one-dimensional Hermite polynomials as basis. An approach using the multivariate Hermite polynomials requires knowledge of a suitable covariance matrix, which essentially is such that the target density
can be dominated by this. The multivariate Hermite polynomials can then be derived, a task that must be tailor-made to the choice of matrix.
Next, we consider a case with non-Gaussian reference probability
, focussing on the one-dimensional situation. We recall that factor models with Ornstein–Uhlenbeck dynamics driven by jump processes are relevant for power price and wind speed modelling. In particular, Ornstein–Uhlenbeck processes with exponential jump processes leading to invariant
-distributions are applied (recall discussion from [
4,
26,
28] in
Section 4, say). In addition, we have CIR-processes as a model for wind speeds as we recall from [
27]. The CIR-process is skewed
-distributed at each time instant, a distribution which is closely related to the
-distribution. Let now
be the density of the
-distribution with scale
and shape
, given as
Suppose we have a target distribution
which behaves as
,
for
and
for
, then the likelihood ratio will be
close to zero, and
for
. However, integrating the square of the likelihood function against
, yields finiteness whenever
and
. Such conditions were found by Asmussen et al. [
39] as well. Thus, tuning the
m to be sufficiently large and
r to be sufficiently small, we can obtain a target
-distribution such that the likelihood ratio is square integrable with respect to
. Furthermore, as is well-known, the basis of orthogonal polynomials for
is the generalised Laguerre polynomials (see [
42]). If we have a two-factor model, with one Gaussian and one exponential jump Ornstein–Uhlenbeck process, we can consider the tensorised space
with
and the canonically generated polynomials from the respective marginal densities.