We finally underline that a proper computation, respectively approximation, of option

prices (for e.g. electricity derivatives) under enlarged filtrations portrays a much more challenging task in contrast to the pure derivation of information premia. This fact immediately becomes clear if we compare the sophisticated derivation methodologies for option price formulas under enlarged filtrations on the one hand, and the less-demanding ones for information premia on the other. Exemplarily, we justify this statement while referring to our proceedings in paragraph 3.3.9: In connection with our derivations in (3.3.133) – (3.3.137) [dedicated to the information premium] we actually used precisely the same techniques that previously had been applied during our option pricing examinations in (3.3.119) – (3.3.131). Hence, option pricing implies the corresponding information premia, obviously. In conclusion, one could regard information premia as an actual by- product of related option prices under forward-looking information. For this reason, future research should rather concentrate on the derivation of option prices under enlarged filtrations, than on information premia. ∎

3.5.3 Optimal electricity futures portfolio selection under forward-looking information

In this subsection we aim to examine the question of how to determine an optimal (in the sense of maximizing a certain utility functional) electricity futures investment strategy – particularly under supplementary knowledge on future price behavior. Right at the beginning, we announce that the current paragraph has been motivated by the sections 8.1, 8.6, 16.5 and 16.6 in [32] which extensively deal with portfolio analysis for an insider in a financial stock market. In what follows, we want to adapt some of the techniques presented in [32] to our electricity market framework. Starting off, we assume that there are two investment possibilities46 in the underlying electricity market, namely:

• a bond/bank account , ∈ 0, , obeying (3.5.19) =

within a constant interest rate > 0 and a deterministic initial value > 0 [recall (3.2.28)].

• an electricity futures ≔ , , ∈ 0, , such as given in Proposition 3.2.2, obeying (3.5.20) = ∑ ℱ,ℚ ,

where ≔ , , ≥ 0 is like defined in (3.2.24).

46

In the electricity market practice there of course are various futures contracts with different delivery periods available. In this regard, note that our model easily can be extended to multiple futures investment possibilities. Nevertheless, we here illustrate the one-bond-one-futures case for the sake of notational simplicity.

84

Similar to above, we introduce the filtrations ℱ ≔ : 0 ≤ ≤ ≔ , … , : 0 ≤ ≤ and ≔ ℱ ∨ , … , while we presume the (non-explicit) intermediate filtration to fulfill (3.5.21) ℱ ⊂ ⊂

for all 0 ≤ < where < . Consequently, the statements of (Condition A and) Lemma 3.3.1 (a) and (d) likewise apply in our current setting – even for indices = 1, … , yet. Next, in accordance to [32], we implement the set of admissible (forward-looking) portfolios due to

≔ = ∈ , à à , − , < ∞ ℚ .

Parallel to the (stock market) setup presented on p.129 in [32], for a portfolio ∈ we suppose the stochastic value to denote the fraction of the total wealth invested in the electricity futures , at time ∈ 0, . In other words, we here think of a fictive electricity market participant (equipped with some additional insider knowledge modeled by the enlarged filtration ) who wants to create an optimal portfolio with respect to his/her individual future information (which other traders do not have). Yet, we suppose such a trader to be a small investor and thus, to act as a price taker; that is, his/her transactions do not have a remarkable impact on the overall price dynamics – compare Remark 8.22 in [32]. For this reason, we presently do not involve the futures price ,ℚ inside (3.5.20), but ≔ ℱ,ℚ _{instead, as the latter designates the reference price for a small investor (even if he/she}

personally has access to some future information). Vice versa, whenever the available future information consists of public knowledge or if we consider a large investor (who may influence prices by his/her individual transactions; also compare Chapter 4 below), then we ought to work with ,ℚ in (3.5.20). Nevertheless, our fictive insider indeed may choose a portfolio with respect to his/her individual future knowledge and hence, the portfolio is allowed to be -adapted. Furthermore, we assume all portfolios ∈ to be self-financing in the spirit of equality “(4.17) in [32]”, i.e. we presume the corresponding wealth process to fulfill the forward SDE (recall Def. 15.7 in [32]) (3.5.22) = + −

with deterministic initial wealth = > 0 and coefficient processes ≔ 1 − and ≔ . At this step, we stress that is ℱ -adapted while is -adapted. Unfortunately, this special case is not captured by Itô’s integration theory. In other words, the object is not well-defined as an Itô integral. Thus, we have to work with forward integration in (3.5.22) whereas we assume the coefficient to be forward-integrable with respect to , respectively ℱ,ℚ = 1, … , , while we understand the symbol in the sense of Definition 15.1 in [32].47 Anyway, if either was ℱ-adapted or if was replaced by ,ℚ, then we would not need forward integration: In fact, for integrands and integrators adapted to the same filtration the forward and Itô integral coincide – see Remark 8.5, Lemma 8.9 and Corollary 8.10 in [32]. Conversely, in our case the integrand is not adapted to the filtration generated by its integrator. Further on, throughout section 8.1 and 8.6 in [32] the underlying risky asset is modeled by a geometric BM, while we are facing an arithmetic pure-jump electricity futures price disposition recently. Hence, our forward equation (3.5.22) slightly deviates from the basic scheme of “(8.1) and (8.27) in [32]”. In addition, the portfolio analysis in [32] is done under the measure ℙ, whereas we work under a risk-neutral measure ℚ [recall (3.5.20)]. Finally, note that our current approach addresses the questions (1) and (3) on p.131 in [32]. However, taking (3.5.19) and (3.5.20) into account, the forward equation (3.5.22) becomes

47

A reader not familiar with forward integration firstly might investigate paragraph 8.2 in [32], particularly Definition 8.3 therein (dedicated to the BM-case), before switching to section 15.1 in [32].

85 (3.5.23)

= 1 − + ℱ,ℚ , .

Herein, we require the integrands to fulfill analogous conditions as in “(16.175) – (16.178) in [32]” with ℍ ≔ , , ≔ . As a consequence of the Itô formula48 (see Th. 15.8 in [32]), we get (3.5.24) =

− − +

+ 1 + ,

wherein we have just used (3.2.20) along with Condition A. By the way, note that (3.5.24) corresponds to the equalities “(8.28) and (16.132) in [32]”. Further on, we introduce a utility function : 0, ∞ ⟶

−∞, ∞ which we presume to be non-decreasing, concave and once continuously differentiable on 0, ∞ . Next, appealing to “(8.29) in [32]”, we examine an optimization problem of the type

(3.5.25) _{∈ ℚ}

which requires us to find an optimal portfolio, say ∗, that maximizes the ℚ-expected utility related to the final total wealth ∗ among all admissible and self-financing portfolios in , in symbols

ℚ ∗ ≥ ℚ ∀ ∈ .

Nevertheless, we refer to subsection 8.6.2 in [32] – particularly, recall (8.50) and (8.52) along with Example 8.33 therein – and penalize large trading volumes in our insider’s portfolio by adding a penalty term to (3.5.25). Hence, instead of the latter, we newly consider the value/utility functional

(3.5.26) ≔ _{∈ ℚ} −

within a deterministic weight function > 0. Merging (3.3.11), (3.3.12) and (3.5.24) into (3.5.26) while presuming a logarithmic utility function, we derive [remind “(16.168) in [32]” at this step] (3.5.27) = + − + ∈ ℚ 1 + − ℚ , − 2 − − . 48

The Itô formula for forward integrals essentially possesses the same structure as its counterpart for common Itô integrals – except from forward integrals appearing at the place of Itô integrals otherwise. We here refer to the very comprehensive (Brownian motion) case studies of section 8.3 in [32]; particularly, see Theorem 8.12 and Remark 8.13 therein. Moreover, Example 8.15 in [32] might be considered in the context of (3.5.24).

86

To find the optimal portfolio ∗ which solves (3.5.27), we maximize (point-wise) the functional (3.5.28)

≔ 1 + ₋ ℚ , − 2 −

−

with respect to (for fixed ∈ 0, ; see the proof of Th. 16.54 in [32]) leading us to the condition (3.5.29)

+ = ℚ_{− 1 +} , − .

As ⁄ < 0 is valid for all self-financing portfolios ∈ , the ( -adapted) solution

∗ _{of (3.5.29) indeed gives the maximum of ∙ . Like in Corollary 16.41, Theorem 16.50 or Theorem}

16.54 in [32], the optimality condition (3.5.29) neither can be analytically solved for . Thus, numerical evaluation methods ought to be used in order to derive the optimal portfolio ∗∈ . However, we leave this topic for future work. Instead, we recall that Chapter 8 in [32] contains portfolio examinations (but for financial stock markets and geometric BM-approaches) wherein the maximization procedures can be done analytically – see “(8.4), (8.5), (8.9), (8.10), (8.44)49 and the sequel of Theorem 8.34 in [32]”. In this regard, we conclude with the following exercise.