To study the effects of price-quantity uncertainties and risk aversion on retailer’s pricing and hedging behaviors, we needed to examine the properties of the pricing
r (and hedging
, if allowed) of our four models across the parametric dimension spaces, including quantity uncertainty
, price uncertainty
, and price-quantity correlation
. However, these properties are high-dimensional and complicated, and may be inverted under different conditions, which makes it difficult to fully characterize these properties. To address the difficulty, we conducted comparative statics in the following two subsections. We plotted pricing r (and hedging
, if allowed) of our four models across quantity uncertainty
in
Section 3.1, and plotted that across price uncertainty
in
Section 3.2, since these two parameters change within a year and the changes are identifiable. Moreover, in
Section 3.1, we varied
and
to see how they affected the relations, and in
Section 3.2, we only varied
to see how they affected the relations, since
made little difference in this subsection. The parameters varied according to
Table 3. In
Table 3, the medium value for
,
, and
were from
Table 2. The low and high value of
were taken from [
3,
23]. The low and high value of
were estimated from [
24], and the high value of
was taken from that during 2002 California electricity crisis [
25], the low value assumed a regulated market where wholesale electricity price changed very little over the year.
3.1. Quantity Uncertainty
To study how retailers set the price and forward position in the benchmark setting when only quantity uncertainty varies, we plot
Figure 4.
Figure 4 shows a graph of retail prices from three models as a function of quantity uncertainty, as well as a graph of forward position from models 2 and 4 as a function of quantity uncertainty. In the left graph, as quantity uncertainty
grew, the risk-neutral retail price
remained the same, but the risk averse price without forward,
, and the risk-averse price with forward,
, both decreased with
. Formally, this is given as:
. This was due to the “price hedge” that reduced the retail price to reduce the quantity induced volatility (QIV), as shown in
Figure 3. The more quantity uncertainty there was, the more “price hedge” that was needed, so the smaller the retail price was. Moreover,
decreased faster than
with
, which is formally given as:
This was due to the “forward hedge” that partially hedged the QIV with a price-based forward contract. With QIV partially hedged, the “price hedge” effect was needed less, so
decreased slower than
with
In the right graph, as the quantity uncertainty
grew, the variance minimization forward position
, and the risk aversion forward position
, both decreased with
, which is formally given as:
. This was due to the forward hedge that partially hedged the QIV with a price-based forward contract. Price uncertainty and quantity uncertainty are correlated, so the retailer will sell a forward contract to hedge QIV. The more quantity uncertainty there was, the more a forward hedge was needed, so the smaller the forward position. Moreover,
decreased faster than
with
, which is formally given as:
. This was due to the “price hedge” that hedged the QIV with the retail price. With QIV partially hedged by the retail price, the forward hedge effect was needed less, so
decreased slower than
.
To study how price uncertainty impacted retailers pricing behaviors with quantity uncertainty, we give
Figure 5.
Figure 5 shows graphs of retail pricing r over the quantity risk axis
for three levels of price uncertainty
. As price uncertainty grew from low (
) to high (
), the risk averse price without forward,
, shifted from below
to above
, the risk neutral price, which is formally given as:
. This is exactly what propostion 1 reveals. This was due to a regime switching from QIV dominant to PIV dominant. When price uncertainty was small, QIV was dominant, and quantity uncertainty was the major concern for risk-averse retailers. Therefore, “price hedge” mostly happened to reduce QIV, leading to a reduction of the profit margin, and a reduction of the retail price. On the other hand, when price uncertainty was large, PIV was dominant, and price uncertainty became a major concern. Therefore, “price hedge” mostly happened to reduce PIV, leading to a reduction of the expected demand, and an increase of the retail price. Another phenemenon was the risk averse price with a forward,
, which remained below
, the risk neutral price, regardless of the price uncertainty, which is formally given as:
. This was due to the forward hedge eliminating all naked positions, and therefore PIV was always small when forward hedge existed, so QIV remained dominant across different price uncertainties.
To study how price uncertainty and correlation impacted retailers’ pricing behaviors with quantity uncertainty, we show
Figure 6 and
Figure 7.
Figure 6 shows graphs of the risk averse price without a forward,
, over the quantity risk axis
for three levels of price uncertainty
and three levels of price-quantity correlation
. As price uncertainty grew from low (
) to high (
),
grew to cause more of a difference in
. In the top-left panel, when price uncertainty was low,
almost overlapped in all levels of correlation, and all decreased with
, which is formally given as:
. This is exactly what proposition 2 reveals. When price uncertainty was small (
), QIV was dominant, therefore retailers decreased the profit margin as quantity uncertainty grew. However, in the bottom panel, when price uncertainty was high,
with high correlation (
) even increased with
when
was smaller than 30, which is formally given as:
. This is exactly what proposition 3 reveals. When the price uncertainty was large (
) and the quantity uncertainty was small (
), PIV had the largest uncertainty, JIV the second largest, and QIV the smallest. Then, using our volatility decomposition method, the PIV–JIV covariance was dominant. As quantity uncertainty grew, JIV grew. With a high correlation (
), the “price hedge” of the PIV–JIV covariance decreased PIV, and with low correlation (
), the “price hedge” of PIV–JIV covariance increased PIV. Therefore, we saw
cause a difference in the
relation when the price uncertainty was large (
).
Figure 7 shows graphs of the risk-averse price with forward,
, over the quantity risk axis
for three levels of price uncertainty
and three levels of price-quantity correlation
. From all three graphs, large- (
) and small- (
) correlated (hereto after named “not uncorrelated”) retail prices seemed to overlap, and both were different from the uncorrelated (
) retail price. This indicates that, in the presence of a forward contract, the effect of price–quantity correlation was not negligible. This phenomenon was due to the partial hedge of quantity uncertainty using a forward contract when the price and quantity uncertainty were not uncorrelated. Such a partial hedge caused the above
relation difference in the large (
) and small (
) correlation case against the uncorrelated (
) case. As price uncertainty grew from low (
) to high (
), the uncorrelated retail price
went from below to above the not uncorrelated one, which is formally given as:
This was still due to a regime switching from QIV-dominant to QIV-JIV-covariance-dominant. When the price uncertainty was small, QIV was dominant. With large () or small () correlations, the forward hedge happened to partially hedge the QIV, so the “price hedge” for QIV was needed less compared to the uncorrelation case. Therefore, both the large and small correlated retail price were higher than the uncorrelated retail price . When the price uncertainty was large, the QIV-JIV covariance was dominant. In all correlation levels, the forward hedge happened to hedge the JIV. Part of the JIV that was not hedged by the forward remained dominant, and the “price hedge” from QIV was needed to hedge JIV. The greater the JIV, the more the “price hedge” was needed to hedge the JIV. The more correlation or anti-correlation that existed, the greater the part of JIV that was hedged by the forward, and the less the “price hedge” was needed, which meant a lower retail price . Therefore, both large and small correlated retail prices were lower than the uncorrelated retail price .
To study how price uncertainty and correlation impacted retailers’ forward hedging behaviors regarding quantity, we show
Figure 8. As price uncertainty grew from low (
) to high (
), the range of forward contracts decreased from 10,000 to less than 150. This decrease indicated the regime switching from quantity uncertainty dominance to price uncertainty dominance. In PIV–QIV dominance, price uncertainty was low, so the forward in PIV was mainly used to partially hedge the quantity uncertainty and the ratio of quantity uncertainty to price uncertainty could be as large as 10,000. In PIV–JIV dominance, price uncertainty was large, and the forward in PIV was mainly used to hedge the naked position, which was around 200. Moreover, from the top-left to bottom graph, the slopes of
against
are inverted for both correlation cases (
) when price uncertainty grew from low to high, which is formally given as:
This was still due to a similar argument of regime switching. In PIV–QIV dominance, where the top-left and the top-right graphs reside, the forward in PIV was mainly used to partially hedge the quantity uncertainty in QIV. Therefore, increasing the quantity uncertainty entailed increasing () or decreasing () the forward position. In PIV–JIV dominance, where the bottom graph resides, the forward mainly hedged the JIV. Therefore, increasing the quantity uncertainty entailed decreasing () or increasing () the forward position, which was totally inverted from the PIV–QIV dominance.
3.2. Price Uncertainty
To study how retailers set the price and forward position in the benchmark setting when only price uncertainty varied, we give
Figure 9.
Figure 9 shows a graph of retail prices from three models as a function of price uncertainty, as well as a graph of the forward position from models 2 and 4 as a function of price uncertainty. In the left graph, as price uncertainty
grows, the risk-neutral retail price
, the risk averse price without forward
, and the risk averse price with the forward
all increased with
, and
grew the fastest,
the second fastest, and
the slowest. This is formally given as:
.
grew faster than
because of the “price hedge” in model 2 that increased the retail price to reduce the price-induced volatility(PIV), as shown in
Figure 3. The greater the price uncertainty, the more “price hedge” that was needed, and therefore the greater the retail price.
grew slower than
because the forward in model 4 hedged all of PIV and part of QIV, and as a result, left QIV dominant. Then, the “price hedge” decreased the retail price to reduce the QIV, and therefore we saw a smaller
compared to
.
In the right graph, as price uncertainty grew, the variance minimization of the forward position and the risk aversion of the forward position both increased with , which is formally given as: This was due to a forward hedge transition from hedging QIV to hedging PIV. Price uncertainty and quantity uncertainty were correlated, and therefore the retailer should sell the forward contract to hedge the QIV. When was small, the greater the price uncertainty, the more hedge each unit of forward contract provided to hedge QIV, and therefore the fewer units of short position forward were needed, which meant an increase in the forward contract position. When was small, PIV was dominant, and therefore the forward position converged to the position of the expected quantity .
To study how correlation affected retailers pricing and hedging behaviors with price uncertainty, we present
Figure 10.
Figure 10 shows graphs of the risk-averse retail price without a forward,
, the risk-averse retail price with a forward,
, and the risk-averse forward position,
, with respect to the price uncertainty axis
. In the top-left graph,
(
) increased the fastest in
,
(
) was second fastest, and
(
) was the slowest, which is formally given as:
This was due to the “price hedge” of the PIV–JIV covariance, as described in propositions 2 and 3. A large correlation(
) led to a positive PIV–JIV covariance such that as
increased, the total profit variance increased the fastest, and the “price hedge” increased the fastest as a result. Meanwhile, a small correlation (
) led to a negative PIV–JIV covariance such that the “price hedge” had the slowest increase.
In the top-right graph, and both increased slower than in . This was due to the forward hedge of JIV. Both large and small correlations made it possible for forward to hedge part of JIV. Then, as increased, JIV increased faster for a medium correlation since no part of JIV could be hedged with a forward. This led to a faster “price hedge” increase as a result.
In the bottom graph, we observe that:
This resulted from a regime switching from PIV–QIV dominance to PIV–JIV dominance. When price uncertainty was much smaller than quantity uncertainty, PIV–QIV correlation was dominant. With a large correlation , PIV and QIV were negatively correlated. Therefore, an additional short forward position was required to hedge JIV compared to the zero correlation. Likewise, with a small correlation , PIV and QIV were correlated; an additional long forward position was required to hedge JIV. When the price uncertainty was much greater than the quantity uncertainty, PIV–JIV correlation was dominant. With a large correlation, PIV and JIV were correlated. Therefore, an additional long forward position was required to hedge JIV compared to the zero-correlation scenario. Likewise, with a small correlation, PIV and JIV were negatively correlated, and an additional short forward position was required to hedge JIV. This result was robust for all three levels of quantity uncertainty.