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Joint optimal ordering and weather hedging decisions:mean-CVaR model
Fei Gao • Frank Y. Chen • Xiuli Chao
Published online: 12 February 2011
� Springer Science+Business Media, LLC 2011
Abstract This paper considers the problem of hedging inventory risk for a seasonal
product whose demand is sensitive to weather conditions, such as the average seasonal
temperature. The newsvendor not only decides the order quantity, but also adopts a
weather hedging strategy. A typical hedging strategy is to use an option (weather
derivative) that is constructed on a weather index before the season begins, which will
compensate the buyer of the option if the actual seasonal weather index is above (or
below) a given strike level. We adopt the risk measure of Conditional-Value at Risk
(CVaR) and explore the joint decision problem in mean-CVaR criterion. We find that
the weather derivative hedging can increase order quantity. Furthermore, it can help
the risk-averse newsvendor improve both expected overall and downside profits.
Keywords Newsvendor � Weather risk hedging � Weather option �Mean-CVaR
1 Introduction
Firms are exposed to a wide variety of risks such as uncertainty about demand and
supply, exchange rate fluctuation, commodity-market volatility, and labor disruption.
F. Gao (&) � F. Y. Chen
Department of Systems Engineering & Engineering Management, Chinese University of Hong
Kong, Shatin, N.T., Hong Kong, China
e-mail: [email protected]
F. Y. Chen
e-mail: [email protected]
X. Chao
Department of Industrial & Operations Engineering, University of Michigan,
Ann Arbor, MI 48109-2117, USA
e-mail: [email protected]
123
Flex Serv Manuf J (2011) 23:1–25
DOI 10.1007/s10696-011-9078-3
This paper is concerned with weather risk, the uncertainty in cash flow and earnings
caused by weather volatility, or the financial exposure that a business firm may have
to non-catastrophic1 weather conditions.
Weather affects the business of many companies. Take the following examples.
(1) To meet customer demand, a propane heating oil distributor must purchase
adequate inventory to cover the expected heating season. If the season is warmer
than normal, then the sales volume will decrease, which leaves the distributor with
excess inventory and the associated storage expenses (Malinow 2002). (2) Demand
for soft drinks is also weather sensitive. As shown in Fig. 1, sales of mineral water
in a German city were highly correlated with the weekly average temperature
(Rathke 2005). (3) A jacket firm located in the North of England reported a third
quarter drop in earnings of 12% compared with the third-quarter of the previous
year. It was also found that three years ago the third quarter earnings were lower by
as much as 14.5%. In both cases the firm believed that the losses were due to milder
than usual winters. Historical data indicates a 91% correlation between the
temperature and the number of items of cold weather clothing sold by the firm
(Speedwell Weather Derivatives Ltd 2003). Although these examples are mostly
related to the sales of consumer goods, the weather risk exists widely in many other
sectors, such as energy, agriculture, construction, outdoor sports, and service
businesses. According to an estimate by the U.S. Department of Commerce, nearly
one-third of the U.S. economy is at risk due to weather, and up to 70% of all
businesses face weather risk of some sort (Baker and Petrillo 2004).
Sophisticated firms, such as the above-mentioned heating oil distributor and
winter jacket manufacturer, have been using weather contracts/derivatives (options,
futures, combinations of both, etc.) to hedge against the financial impact of adverse
(but non-catastrophic) weather conditions, evening out their weather sensitive
earnings. (An example of a weather option will be described in Sect. 3.) Markets for
weather hedging products have emerged in the past decade, and weather risk
R2
15,0 20,0 25,0 30,0
= 0,9549
35,0 40,0
Temperature in °C
Sale
s in
%
Fig. 1 Sales of mineral water as percentage of those at 19�C
1 Examples of catastrophic weather conditions include natural disaster type weathers, such as cyclonic
storms, tornados, blizzards, etc.
2 F. Gao et al.
123
derivatives/contracts have been traded both in exchanges and over-the-counter.
Since its launch on the Chicago Mercantile Exchange in 1997, the weather-
derivative trade has grown to a $45 billion per year industry in the U.S. (WRMA
2006). In addition, weather derivatives trading has spread from the U.S. to Europe
and Asia (Baccardax 2002).
This paper tries to address the joint determination of weather hedging and inventory
decisions in a newsvendor setting, where the newsvendor sells a seasonal product and
faces inventory risk due to weather uncertainty. In our model, a single type of hedge
contract, i.e., a weather call option, is considered. The risk-averse newsvendor needs to
jointly determine the production/ordering quantity and the number of options to buy.
Particularly, we analyze the problem with the risk measure of conditional value-at-risk
(CVaR). The CVaR criterion measures the average value of the profit falling below a
certain quantile level, or value at risk (VaR), which is defined as the maximum profit at
a specified confidence level (Jorion 2006). CVaR is a coherent risk measure for
quantifying risk and has better computational characteristics than VaR does (Artzner
et al. 1999), and thus CVaR has emerged as a practical approach for modeling risk
aversion. A vast literature of financial risk management has been developed based on
CVaR in recent years. However, an objective that is based solely on the conditional
value at risk may be too conservative. Hence, this paper considers the mean-risk
framework, i.e., the newsvendor’s objective is to balance the expected profit and the
conditional value at risk.
The rest of the paper is organized as follows. In the next section, we give a brief
literature review, which is followed by the formulation of the problem. Main
analysis and results are presented in Sect. 4. A summary is given in Sect. 5. All of
the technical proofs are included in the Appendix.
2 Literature Review
For production/inventory risk, as faced by the newsvendor in our problem, one
common hedging method is so-called ‘‘operational hedging’’ on which the research
is enormous. The term operational hedging is set apart from ‘‘financial hedging’’ as
follows. The latter is realized through financial tools, such as options and swaps,
whereas the former manages a firm’s risk operationally, such as delaying production
decisions until more accurate demand information is acquired or building a flexible
capacity to better match supply and demand. In a recent survey, Boyabatli and
Toktay (2004) provide an excellent summary and critique on the existing literature
on operational hedging. There are also some papers discussing the joint use of
operational and financial hedging. For example, Ding et al. (2004) study the
integrated operational and financial hedging decisions of a risk averse global firm
facing demand and exchange rate uncertainties. Chod et al. (2006) consider a two-
product newsvendor and discuss the interaction between the product flexibility and
financial hedging when product demands are correlated. For our problem, in which
the inventory risk is mainly associated with weather sensitive demand, we focus on
financial hedging to investigate the advantages of the newly developed financial
tools for hedging weather risk.
Joint optimal ordering and weather hedging decisions 3
123
In the literature addressing financial hedging in line with operations management,
Caldentey and Haugh (2006) generally investigate the optimal control and hedging
of operations in the presence of financial markets. Their hedging strategy is built
upon the financial asset whose price affects the operating profit. Addressing the
quantity risk in the electricity market, the study of Oum et al. (2006) is also related
to our paper. In Oum et al. (2006), a hedging strategy is developed through a
portfolio of forward contracts and the call and put options of electricity. Their model
does not overlap with ours, because unlike weather, electricity itself has a market
price. Gaur and Seshadri (2005) is the study closest in spirit to our problem. They
address the problem of hedging inventory risk in a newsvendor setting, in which the
product demand is correlated with the price of a financial asset. In their mean-
variance objective model, the newsvendor makes joint decisions on the order
quantity and on the hedging strategy by constructing a portfolio, including the
financial asset, to replicate the newsvendor payoff function. However, in our model,
the weather index is not a tradable asset (Cao and Wei 2004). This makes the
replication approach infeasible, e.g., one cannot short-sell the weather index as a
tradable asset (Hull 2003). In their utility model, they investigate the problem under
general utility function while in our paper, we investigate the weather risk hedging
problem under mean-CVaR criterion.
It is known that there are two main approaches to incorporating risk in risk-
averse supply chain models: one is through utility maximization, and the other is
through the return-risk tradeoff. Lau (1980), Eeckhoudt et al. (1995), Bouakiz and
Sobel (1992), and Chen et al. (2007) study the newsvendor or multi-period
inventory models under the utility framework. In practice, however, utility functions
are too conceptual to identify. To deal with the tradeoff between reward and risk,
mean-variance analysis is an important approach. The mean-variance formulation
was first introduced by Markowitz (1952) for portfolio selection analysis, and
became a common criterion when addressing risk thereafter, not only in the
financial and economic areas, but also for operation and management issues (see
Chen and Federgruen 2000; Gaur and Seshadri 2005; Ding et al. 2004; Caldentey
and Haugh 2006; Seifert et al. 2004). However, it falls to distinguishing between
desirable upside and undesirable downside outcomes. In our paper, we take CVaR
as the risk measure. Compared to utility functions, CVaR is much easier to quantify,
because the only subjective parameter for CVaR is the confidence level, that is the
reason for a wide adoption of VaR and CVaR in the field of risk management. To
reflect the tradeoff between return and risk, we investigate our problem in mean-
CVaR criterion, which is indeed a convex combination of the upside and downside
returns (see Remark 1).
Recently, the CVaR measure has drawn much attention as a coherent and easily-
computed risk criterion in the portfolio management literatures (see Rockafellar and
Uryasev 2000, 2002; Acerbi and Tasche 2002a, b; Szego 2002). In the inventory
literature, a number of recent papers have begun to use the CVaR framework to
analyze the newsvendor and multiple period inventory models, e.g., Gotoh and
Takano (2007), Ahmed et al. (2007), Choi and Ruszczynski (2007), and Chen et al.
(2006). These papers, however, do not consider the risk hedging issue.
4 F. Gao et al.
123
The weather risk management we discussed in this paper is receiving discussion
from other literature (e.g., Dischel 2001; Harrington and Niehaus 2003; Chen and
Yano 2010). Chen and Yano (2010) discuss the use of rebate in improving supply
chain’s profit with and without limiting risk under weather-related demand
uncertainty. Our paper does not consider the supply chain issue and discusses the
weather risk management with weather derivatives under the risk measure of CVaR.
The literature on pricing weather derivatives (e.g., Jewson and Rodrigo 2002;
Jewson 2003; Cao and Wei 2004) provide the foundation for weather risk hedging
from the financial perspective. However, little attention has been devoted to the
interaction between weather risk and operations management, which we address in
this paper.
3 Formulation and notation
Suppose that a newsvendor sells a seasonal product whose demand is contingent on
the seasonal weather, such as the average seasonal temperature. Favorable weather
generates strong sales and hence high revenue, whereas unfavorable weather
significantly shrinks demand, and sometimes even causes severe losses. To protect
his revenue, the newsvendor can buy a number of weather options in a weather
derivatives market that will pay him if the weather is unfavorable and thereby
compensates for weak revenue. (Take the excerpt from a disguised case study,
Dischel and Barrieu (2001), to motivate our model.) The level of compensation
depends on the number of options that the newsvendor purchases. The actual total
profit depends on the initial order quantity, the weather, the actual demand, and the
compensation level. Thus, the newsvendor needs to decide on both the number of
options to purchase (n) and the initial order quantity (Q) to maximize the value of
his objective function.
Let Dð�; tÞ denote the random seasonal demand parameterized by weather index
t and another random source �, which is independent of t. In this paper, t is taken as
the seasonal average temperature. We assume both the random variables and the
demand function are continuous. Assume Dð�; tÞ is stochastically decreasing in t.2
For notational simplicity, we hereafter drop off � in Dð�; tÞ. (Note for given t, D(t) is
still a random variable.) Thus for t1� t2, the random variable D(t1) is less than D(t2)
in a stochastic order. Then for any increasing (decreasing) function w, we have that
ED½wðDðt1ÞÞ� �ED½wðDðt2ÞÞ�: (ED½wðDðt1ÞÞ� �ED½wðDðt2ÞÞ�:) Further denote by
gð�; tÞ and Gð�; tÞ the density and cumulative distribution functions of Dð�; tÞ,respectively, when t is given. We also assume the distribution of t to be common
knowledge to both the option writer and the newsvendor.
The newsvendor can use weather call option to hedge the risk associated with
weather fluctuations. The structure of weather call option is illustrated in Fig. 2. To
2 For example, Dð�; tÞ ¼ a� bt þ �; b [ 0 and Dð�; tÞ ¼ ða� btÞ�; b [ 0�[ 0, for which the correlation
between the demand and temperature t is q ¼ �brtffiffiffiffiffiffiffiffiffiffiffiffiffi
b2r2t þr2
�
p and q ¼ �bl�rt
rDð�;tÞrespectively, where r stands for
standard deviation and l stands for expected value. We see that the demand is negatively correlated with
temperature t.
Joint optimal ordering and weather hedging decisions 5
123
get one unit of such an option, the newsvendor needs to pay a price of K. If the
realized seasonal average temperature is greater (lower) than �t, the strike average
temperature, then the newsvendor receives k for each unit of deviation ðt � �tÞþ(nothing), but the total amount is capped at �k. For simplicity, we let the risk-free
interest rate be zero. Then, the net payoff per option to the newsvendor is
minð�k; kðt � �tÞþÞ � K. Naturally, �k [ K. When demand is stochastically increasing
in the average temperature, a put option can be used to hedge the weather risk, and
the analysis in the sequel can easily be extended to such a case.
Each unit of sales earns the newsvender p. The pre-season purchasing cost is
c per unit of product. Any leftover at the end of the season will be salvaged at s. To
avoid triviality, assume that 0 \ s \ c \ p. As in most of the risk-averse inventory
models, we ignore the shortage cost (other than foregone profit p - c) for ease of
analysis. This assumption is without loss of generality but simplifies the technical
analysis and exposition.
Given Q, the order quantity and n, the number of options bought (selling option is
regarded as purchasing a negative number of options), the net profit of the
newsvendor at the end of the season is a random variable, which can be expressed as
PðQ; n;DðtÞÞ ¼ P1ðQ;DðtÞÞ þ nP2ðtÞ;
where
P1ðQ;DðtÞÞ ¼ � cQþ p minðQ;DðtÞÞ þ sðQ� DðtÞÞþ
¼ ðp� sÞDðtÞ � ðc� sÞQ� ðp� sÞðDðtÞ � QÞþ;
and
P2ðtÞ ¼ minð�k; kðt � �tÞþÞ � K: ð1Þ
Note that PðQ; n;DðtÞÞ is jointly concave in (Q, n). Given the option, denote by
Pl2 ¼ �Kð\0Þ; and Pu
2 ¼ �k � Kð[ 0Þ the minimum and maximum values of
P2ðtÞ, respectively.
Fig. 2 A weather call option
6 F. Gao et al.
123
When the option is charged with price equal to the expected payoff (to the option
buyer), i.e., K ¼ E½minð�k; kðt � �tÞþÞ�, such that E½P2ðtÞ� ¼ 0, the option is called
fairly priced. With fair price, the option writer (seller) can only break even in the
long run. However in practice, the option writer would expect to obtain a risk
premium beyond the fair price for taking the risk of having to pay out. Hence it is
more reasonable that the option price would then be higher than the average payoff
of the option. Nevertheless, in a competitive market, the risk premium should not be
too high, as otherwise, over-supply will drive it to a lower level. A simple method
for determining the risk premium for weather options has been suggested as a
fraction of the standard deviation of the option payoff (see Jewson and Brix 2005).
However, there remains the issue of what fraction is appropriate.
Related literatures on financial hedging usually assume the derivatives analyzed
are fairly priced to draw analytical results for technical reasons (e.g., Gaur and
Seshadri 2005; Ding et al 2004). The fair price is also recommended for analysis on
weather options in Hull (2003). By assuming the option is fairly priced, i.e., the
expected value of the option is zero, the possibility of selling instead of buying options
is eliminated, and the risk reduction effect is emphasized. Therefore, in the sequel, to
draw insights into the impact of weather hedging on the utility value of the decision
maker and the optimal ordering quantity, we often analyze the case in which
E½P2ðtÞ� ¼ 0. This provides a reasonable approximation for cases when the risk
premium of the option is relatively small (as compared with the overall price K).
The model can also be modified for a situation in which the newsvendor is a
manufacturer who has to procure materials with long lead time, but the selling
season is relatively short.
4 Objective of mean-conditional value at risk
We study the problem under mean-CVaR criterion. To present the mean-CVaR
objective function, we first introduce CVaR and relevant VaR notions.
CVaR is defined based on the risk measure of Value-at-Risk (VaR) which is
commonly used by financial institutions and companies that are involved in trading
energy and other commodities, see, e.g., Jorion (2006) for detailed discussions. VaR
allows the decision maker to specify a confidence level b to expect that he can attain
a certain level of wealth. In general, the b-VaR of an investment is the value v such
that there is at least b percent chance that the profit from the investment will be no
less than v. For example, if we know for a confidence level of b = 0.96 that the VaR
or v = $1 million, then there is at least a 96% chance that the investment will gain $1
million or even more. In addition, for a confidence level of b = 0.90, if v =
-$0.5 million, then there is at most a 10% chance that the investment will lose
more than $0.5 million. The formal definition of b-VaR is given as follow.
Denote the probability of PðQ; n;DðtÞÞ not falling below a threshold v by
WðQ; n; vÞ ¼ PðPðQ; n;DðtÞÞ� vÞ: ð2Þ
For a confidence level b 2 ½0; 1Þ, the b-VaR of the profit associated with decisions
Q and n is the value
Joint optimal ordering and weather hedging decisions 7
123
vbðQ; nÞ ¼ maxfvjWðQ; n; vÞ� bg: ð3Þ
The maximum in (3) is attained because WðQ; n; vÞ is nonincreasing and left-
continuous in v.
The b-CVaR then measures the average value of the profit falling below the vb
level. Denote by /b(Q, n) as the b-CVaR of the profit function PðQ; n;DðtÞÞassociated with the decision variables. The larger the b is, the larger the confidence
level for profit is set, hence the more risk averse the newsvendor is.
Rockafellar and Uryasev (2000, 2002) show that the optimization of b-CVaR can
be achieved by optimizing a much simpler auxiliary function without having to first
calculate the b-VaR on which the b-CVaR is defined. And instead, the b-VaR may
be obtained as a byproduct. For our problem setting, the auxiliary function is
FbðQ; n; vÞ ¼ v� 1
1� bE½v�PðQ; n;DðtÞÞ�þ: ð4Þ
It is concave in v, and
/bðQ; nÞ ¼ maxv2R
FbðQ; n; vÞ: ð5Þ
The set fvjv 2 arg maxvFbðQ; n; vÞg is a nonempty, closed, bounded interval
(perhaps reducing to a single point), and the b-VaR, vb(Q, n), is its upper endpoint.
Without option hedging, the b-CVaR of the newsvendor’s profit is
/bðQÞ ¼ maxv
FbðQ; vÞ ¼ maxvfv� 1
1� bE½v�P1ðQ;DðtÞÞ�þg: ð6Þ
As a coherent risk measure, CVaR is widely used in portfolio selection problems
for multiple decision variables. However, there are certain limitations for CVaR, the
major one of which is due to the fact that when being used as a single decision
criterion, it is too conservative for the decision maker because only the worst
outcomes are considered while the part of the profit distribution above VaR are
ignored. For large b, the b-CVaR criterion ‘‘underscores’’ risk-aversion but neglects
a large part of the profit distribution; whereas for small b, the b-CVaR criterion
encompasses a large part of the profit distribution and does not reflect the decision
maker’s real risk attitude. Indeed, as b approaches 0, the b-CVaR criterion will lead
to the same decision as in the risk-neutral case, as to be shown later.
To overcome this weakness of the CVaR criterion, we propose an objective
function which is a convex combination of the expected profit and b-CVaR. Such a
joint objective reflects the desire of the risk-averse retailer to maximize the profit on
the one hand, and minimize the downside risk of his profit on the other. In other
words, the retailer strikes a balance or tradeoff between profit and risk.
Then the objective function for the newsvendor can be expressed as
maxðQ;nÞ
kE½PðQ; n;DðtÞÞ� þ ð1� kÞ/bðQ; nÞ� �
¼ maxðQ;nÞfkE½PðQ; n;DðtÞÞ� þ ð1� kÞmax
vFbðQ; n; vÞg
¼ maxðQ;n;vÞ
fkE½PðQ; n;DðtÞÞ� þ ð1� kÞFbðQ; n; vÞg;
ð7Þ
8 F. Gao et al.
123
where k is a weight, 0� k� 1. If k = 0, the objective function reduces to the
b-CVaR criterion; if k = 1, then the objective function reduces to the risk neutral
newsvendor model. When b = 0, it also reduces to the classical newsvendor model.
The smaller the weight k assigned to the expected profit and the larger the value of
the confidence level b is, the more risk averse the newsvendor is. Note that the risk
factor b also compromises the expected profit: the larger b becomes, the lower the
expected profit the newsvendor can get. Therefore, as a tradeoff objective, the value
of b in (7) should be chosen close to 1 to take account of the tail effect, i.e., to
capture the downside risk, whereas the value of k is chosen to reflect the weight
between the expected profit and risk.
For k 2 ð0; 1Þ, the optimization problem in (7) with the objective as a convex
combination of mean and CVaR can be rewritten as
maxðQ;n;vÞ
E½PðQ; n;DðtÞÞ� � 1� kk
~FbðQ; n; vÞ; ð8Þ
where ~FbðQ; n; vÞ ¼ �FbðQ; n; vÞ. Mathematically, (8) is equivalent to
max E½PðQ; n;DðtÞÞ�s.t.; ~FbðQ; n; vÞ� r;
ð9Þ
or
min ~FbðQ; n; vÞs.t.;E½PðQ; n;DðtÞÞ� � e:
ð10Þ
By optimization theory, if ðQ�; n�; v�Þ solves problem (8), then it solves (9) with
r ¼ ~FbðQ�; n�; v�Þ and it solves (10) with e ¼ E½PðQ�; n�;DðtÞÞ�. Same sort of
argument and formulation are used in Li and Ng (2000), who consider mean-
variance formulation. A frontier of solution can be generated by varying the value of
k. Note in our problem, Fb(Q, n, v) quantifies the downside profit, then ~FbðQ; n; vÞstands for the corresponding downside loss.
Without option hedging, the tradeoff objective of the newsvendor is
maxQ
kE½P1ðQ;DðtÞÞ� þ ð1� kÞ/bðQÞn o
¼ maxðQ;vÞfkE½P1ðQ;DðtÞÞ� þ ð1� kÞFbðQ; vÞg:
ð11Þ
Denote ðQ�; n�Þ and Q as the optimal solutions of (7) and (11), respectively.
Lemma 1 Fb(Q, n, v) is jointly concave in (Q, n, v).
With Lemma 1, the optimization of the mean-CVaR objective, i.e., (7), reduces
to a concave maximization problem by noting that the profit function PðQ; n;DðtÞÞis jointly concave in (Q, n). Without option hedging, it can be similarly proved that
the auxiliary function FbðQ; vÞ is jointly concave in (Q, v), thus (11) reduces to a
concave maximization problem also.
Joint optimal ordering and weather hedging decisions 9
123
To better understand the meaning of the mean-CVaR tradeoff objective, we give
the following remark.
Remark 1 When the distribution of profit function has no probability atom
(Rockafellar and Uryasev 2002), the definition of b-CVaR of a profit function
Pð~x; ~yÞ, where ~x is decision vector, and ~y is random vector, reduces to
/bð~xÞ ¼ E½Pð~x; ~yÞjPð~x; ~yÞ� vbð~xÞ�:
Then the mean-CVaR tradeoff objective can be rewritten in the following way:
kE½Pð~x; ~yÞ� þ ð1� kÞ/bð~xÞ ¼ kð1� bÞE½Pð~x; ~yÞjPð~x; ~yÞ� vbð~xÞ�þ kbE½Pð~x; ~yÞjPð~x; ~yÞ[ vbð~xÞ� þ ð1� kÞE½Pð~x; ~yÞjPð~x; ~yÞ� vbð~xÞ�¼ ð1� kbÞE½Pð~x; ~yÞjPð~x; ~yÞ� vbð~xÞ� þ kbE½Pð~x; ~yÞjPð~x; ~yÞ[ vbð~xÞ�:
Thus, we can say approximately that the mean-CVaR tradeoff is a convex
combination of two conditional expected values: one is conditioned on the lower
profits, and the other is conditioned on the higher profits, which are discriminated by
the vb. The weights represent the retailer’s degree of risk aversion. h
Because the model without weather option hedging is easier to handle than that
with option hedging, and, moreover, its analysis also provides insights into the
analysis of the latter, we first carefully examine the model without options.
4.1 Without option hedging
Proposition 1 Under the objective of the tradeoff between expected profit and the
b-CVaR, the optimal order quantity without option hedging, Q, is determined by
Et½GðQ; tÞ� ¼ 1� c�skðp�sÞ k[ c�s
bðp�sÞ;
Et½GðQ; tÞ� ¼ ð1�bÞðp�cÞð1�kbÞðp�sÞ k� c�s
bðp�sÞ:
(
ð12Þ
Remark 2 From Proposition 1, the following observations can be made.
(a) When k = 0, the tradeoff objective is reduced to the maximization of b-CVaR.
The optimal order quantity, denoted as Q0, is characterized by
Et½GðQ0; tÞ� ¼ ð1� bÞp�cp�s.
(b) When k = 1, the tradeoff objective is reduced to the maximization of expected
profit, i.e., the risk neutral case. The optimal order quantity, denoted as Q1, is
determined by Et½GðQ1; tÞ� ¼ p�cp�s.
(c) For fixed b, the optimal order quantity is increasing in k, i.e., the more weights
the retailer puts on the expected profit, the more he will order since he is less
risk averse. Then Q0� Qk� Q1, where 0 \ k\ 1.
(d) For fixed k, when b [ c�skðp�sÞ, the optimal order quantity is independent of b;
whereas when b� c�skðp�sÞ, the optimal order quantity is decreasing in b.
Furthermore, Q! Q1 as b! 0. h
10 F. Gao et al.
123
It is well known that the optimal ordering quantity for a risk-averse newsvendor
is lower than that for a risk-neutral newsvendor (see Eeckhoudt et al. 1995; Agrawal
and Seshadri 2002). We summarize the monotonicity of the optimal order quantity
in the tradeoff weight k and the risk parameter b in the following Corollary.
Corollary 2 (i) As k increases while b is fixed, i.e., as the newsvendor puts moreweight on the expected profit, a larger quantity will be ordered. (ii) As b. increaseswhile k is fixed, i.e., as the newsvendor becomes more risk-averse, a smallerquantity will be ordered.
With such monotonicity, note that the expected profit is concave in the order
quantity and it attains the maximum value at Q1 when k = 1 or b = 0, we have the
following monotone property for the expected profit in the optimal order quantity Q,
which we denoted as Qðk; bÞ.
Corollary 3 (i) E½P1ðQðk;bÞ;DðtÞÞ� is increasing in k for any given b 2 ð0; 1Þ.(ii) E½P1ðQðk; bÞ;DðtÞÞ� is decreasing in b for any given k 2 ð0; 1Þ.
Corollary 3 indicates that the larger the weight the newsvendor puts on the risk
measure CVaR, or the more the newsvendor is concerned with the downside risk,
the less profit he expects. These two corollaries are intuitive.
4.2 With option hedging
We now consider the mean-CVaR objective with option hedging, which is
represented by (7).
Proposition 4 If the option is fairly priced, i.e., E½P2ðtÞ� ¼ 0, then the optimalnumber of options n� � 0.
When the newsvendor is risk neutral, he has no incentive to buy the fairly priced
option, as it does not create any additional profit. However, when he is risk averse,
although the expected value of the option equals to zero, buying the option can
remove some of the downward risks.
Proposition 5 If G(y; t) is strictly increasing in y for any given t, then Q� � Q ;i.e., the newsvendor orders more in the presence of the weather option.
Proposition 5 can be expected intuitively, because for the maximization of the
expected profit, the newsvendor is risk neutral and will order the same with and
without option hedging, but for CVaR optimization, the newsvendor is risk averse
and will order less without option hedging. Therefore, weather hedging brings the
optimal ordering quantity closer to the risk-neutral profit-maximization quantity.
The technical importance of the result is that in searching for Q*, one can first find
Q, which can be determined almost by a closed form solution, and then starts with
Q ¼ Q upward.
However, for the problem with option hedging, a closed-form solution such as
(12) appears to be impossible. Moreover, an unambiguous monotonicity of the
Joint optimal ordering and weather hedging decisions 11
123
optimal objective function in k and b is difficult to establish. Section 4.3 shows
some counter examples in which the optimal order quantity does not exhibit
monotonicity in k and b, neither does the optimal number of options, n*. This is
because the presence of the option complicates the structure of the optimal objective
function. (We expect such monotonicity properties may exist for some particular
financial instruments, but in this paper we focus on any fairly priced option.) We
also mention that since the correlation between the demand and the temperature is
not a simple parameter (refer to footnote 2), the general analysis of it’s impact on
the hedging effect is not straightforward. Our result that weather hedging increases
the newsvendor’s inventory level is in accordance with that in Gaur and Seshadri
(2005), where they investigate the inventory risk hedging problem using expected
utility. With the use of mean-CVaR criterion, we can observe the hedging effect on
the newsvendor’s risk-averse objective, which is also illustrated in Sect. 4.3.
4.3 Numerical examples
The main objective of the examples is twofold. First, we want to see the magnitude
of the impact of option hedging on the order quantity and objective value. Second,
we provide counterexamples to show non-monotonicity of the optimal solutions in kand b. The demand function is taken as an additive form, DðtÞ ¼ a� bt þ �; where
b [ 0. The average temperature t follows the normal distribution Nðlt; r2t Þ; and
� 2 ð�;��Þ is a uniformly distributed random variable. Let t * N(-5, 2). (Note the
normal distribution is truncated for tractability in our numerical analysis.) The
option is provided as �t ¼ �7; k ¼ 2; �k ¼ 8; with a fair price K = 4. Other
parameters are valued as a ¼ 10; b ¼ 1; � ¼ �1;�� ¼ 1; p ¼ 7; c ¼ 4; and s = 2.
The results for the examples are reported in Table 1a–d. For each combination of
k and b, we calculate the optimal solutions for the cases without/with option
hedging, and report them as QjðQ�; n�Þ in Table 1a. In Table 1b, the corresponding
optimal objective values are presented, from which, we calculate the percentage
gain in the objective value due to option hedging and report the results in Table 1c.
In Table 1d, the corresponding values of the mean and CVaR at the optimal solution
are presented as ðMean;CVaRÞ, where for each value of k, the first row corresponds
to the case without option hedging, while the second row corresponds to the case
with option hedging. Note that when k = 0 (1.0), the objective is reduced to the
maximization of b-CVaR (the risk-neutral newsvendor model, as the option is fairly
priced).
Compared with the case without hedging, the optimal order quantity is increased
with option hedging (see Table 1a). When with b-CVaR alone (i.e., k = 0), the
optimal order quantity can be as much as 50% larger with option hedging (see
Table 1a: the cell (k = 0, b = 0.99)). However, as Table 1a indicates, the optimal
order quantity with hedging is not monotonic in k and b. For example, for k, see the
columns with b = 0.3 and b = 0.9 in Table 1a; and for b, see the row with k = 0.1
in Table 1a.
The option hedging also increases the objective value of the risk averse retailer
(see Table 1b). Furthermore, in Table 1c, we observe that the percentage gain in the
12 F. Gao et al.
123
Tab
le1
Op
tim
also
luti
on
and
op
tim
alo
bje
ctiv
ev
alu
e
kb 0
.10
.30
.90
.95
0.9
9
a.O
pti
mal
solu
tio
n
0.0
15
.15
|(1
5.1
5,
0.0
0)
14
.68
|(1
4.9
9,
0.7
1)
12
.62
|(1
5.9
7,
2.3
9)
12
.13
|(1
6.0
5,
2.5
5)
11
.17
|(1
6.0
8,
3.1
3)
0.1
15
.17
|(1
5.2
4,
0.4
0)
14
.74
|(1
5.0
6,
0.8
0)
12
.69
|(1
5.9
4,
2.3
6)
12
.20
|(1
6.0
4,
2.5
4)
11
.23
|(1
6.0
7,
3.1
2)
0.3
15
.21
|(1
5.2
3,
0.1
0)
14
.85
|(1
5.1
6,
0.9
6)
12
.87
|(1
5.8
4,
2.2
7)
12
.36
|(1
6.0
2,
2.5
2)
11
.37
|(1
6.0
4,
3.1
0)
0.5
15
.26
|(1
5.2
8,
0.1
9)
14
.97
|(1
5.2
5,
1.0
6)
13
.71
|(1
5.6
8,
2.1
4)
13
.71
|(1
5.7
9,
2.3
7)
13
.71
|(1
5.8
0,
2.9
5)
0.7
15
.31
|(1
5.3
3,
0.4
5)
15
.12
|(1
5.3
0,
1.1
3)
14
.72
|(1
5.5
0,
2.0
0)
14
.72
|(1
5.5
6,
2.2
2)
14
.72
|(1
5.5
6,
2.8
0)
0.9
15
.36
|(1
5.3
7,
0.5
3)
15
.29
|(1
5.3
6,
1.1
6)
15
.21
|(1
5.4
6,
2.0
0)
15
.21
|(1
5.4
2,
2.1
9)
15
.21
|(1
5.4
3,
2.7
2)
1.0
15
.39
|(1
5.3
9,
-)1
5.3
9|(
15
.39
,–
)1
5.3
9|(
15
.39
,–)
15
.39
|(1
5.3
9,
–)
15
.39
|(1
5.3
9,
–)
b.
Op
tim
alo
bje
ctiv
ev
alu
e
0.0
41
.61
|41
.61
40
.72
|40
.75
35
.90
|37
.38
34
.62
|36
.60
32
.07
|34
.53
0.1
41
.65
|41
.65
40
.82
|40
.87
36
.09
|37
.83
34
.80
|37
.12
32
.22
|35
.25
0.3
41
.73
|41
.73
41
.04
|41
.12
36
.53
|38
.72
35
.22
|38
.15
32
.59
|36
.70
0.5
41
.81
|41
.81
41
.28
|41
.38
37
.22
|39
.65
36
.05
|39
.21
33
.78
|38
.17
0.7
41
.90
|41
.90
41
.55
|41
.63
38
.87
|40
.59
38
.17
|40
.32
36
.80
|39
.69
0.9
41
.99
|41
.99
41
.86
|41
.90
40
.93
|41
.55
40
.69
|41
.46
40
.24
|41
.25
1.0
42
.03
|42
.03
42
.03
|42
.03
42
.03
|42
.03
42
.03
|42
.03
42
.03
|42
.03
c.P
erce
nta
ge
gai
nin
ob
ject
ive
val
ue
by
hed
gin
g
0.0
0.0
0%
0.0
7%
4.1
2%
5.7
2%
7.6
7%
0.1
0.0
0%
0.1
2%
4.7
4%
6.6
7%
9.4
0%
0.3
0.0
0%
0.1
9%
6.0
0%
8.3
2%
12
.61
%
0.5
0.0
0%
0.2
4%
6.5
3%
8.7
7%
13
.00
%
0.7
0.0
0%
0.1
9%
4.4
3%
5.6
3%
7.8
5%
0.9
0.0
0%
0.1
0%
1.5
1%
1.8
9%
2.5
1%
Joint optimal ordering and weather hedging decisions 13
123
Tab
le1
con
tin
ued
kb 0
.10
.30
.90
.95
0.9
9
1.0
0.0
0%
0.0
0%
0.0
0%
0.0
0%
0.0
0%
d.
Val
ue
of
(Mea
n,
CV
aR)
ato
pti
mal
solu
tio
n
0.0
(42
.00,
41
.61
)(4
1.7
2,
40
.72
)(3
7.6
5,
35
.90
)(3
6.2
9,
34
.62
)(3
3.4
9,
32
.07
)
(42
.00,
41
.61
)(4
1.9
3,
40
.75
)(4
1.8
2,
37
.38
)(4
1.7
7,
36
.60
)(4
1.7
4,
34
.53
)
0.1
(42
.00,
41
.61
)(4
1.7
6,
40
.71
)(3
7.8
5,
35
.89
)(3
6.4
9,
34
.61
)(3
3.6
6,
32
.06
)
(42
.02,
41
.61
)(4
1.9
6,
40
.75
)(4
1.8
4,
37
.38
)(4
1.7
7,
36
.60
)(4
1.7
5,
34
.53
)
0.3
(42
.01,
41
.61
)(4
1.8
5,
40
.69
)(3
8.3
1,
35
.77
)(3
6.9
4,
34
.49
)(3
4.0
7,
31
.95
)
(42
.02,
41
.61
)(4
2.0
0,
40
.74
)(4
1.9
0,
37
.36
)(4
1.7
9,
36
.59
)(4
1.7
7,
34
.53
)
0.5
(42
.02,
41
.60
)(4
1.9
2,
40
.64
)(4
0.2
8,
34
.16
)(4
0.2
8,
31
.83
)(4
0.2
7,
27
.30
)
(42
.02,
41
.60
)(4
2.0
2,
40
.74
)(4
1.9
8,
37
.32
)(4
1.9
3,
36
.49
)(4
1.9
2,
34
.42
)
0.7
(42
.03,
41
.59
)(4
1.9
9,
40
.54
)(4
1.7
4,
32
.16
)(4
1.7
4,
29
.82
)(4
1.7
4,
25
.28
)
(42
.03,
41
.60
)(4
2.0
3,
40
.72
)(4
2.0
2,
37
.25
)(4
2.0
1,
36
.38
)(4
2.0
1,
34
.28
)
0.9
(42
.03,
41
.58
)(4
2.0
3,
40
.38
)(4
2.0
1,
31
.17
)(4
2.0
1,
28
.84
)(4
2.0
1,
24
.30
)
(42
.03,
41
.59
)(4
2.0
3,
40
.70
)(4
2.0
3,
37
.23
)(4
2.0
3,
36
.33
)(4
2.0
3,
34
.23
)
1.0
(42
.03,
–)
(42
.03
,–
)(4
2.0
3,
–)
(42
.03,
–)
(42
.03,
–)
(42
.03,
–)
(42
.03
,–
)(4
2.0
3,
–)
(42
.03,
–)
(42
.03,
–)
14 F. Gao et al.
123
objective value increases in b, which indicates that when the retailer is concerned
with a more adverse scenario, the weather option’s contribution becomes more
extant. However, the percentage profit gain first increases and then decreases in k,
which can be explained as follows. When k increases, the option hedging can
increase the percentage gain in the objective value because it increases the optimal
order quantity and then improves the expected profit, which occupies a larger
fraction of the total objective value with higher value of k. On the other hand, as kcontinues increasing, the problem is more like profit maximization, and the effects
of option hedging on the objective value diminishes.
From the results given in Table 1d, we see that for most examples, the option
hedging not only increases the total expected profit due to the increase in the
ordering quantity, but also improves the value of CVaR, i.e., the average value of
the profit below the vb level. For both cases with and without option hedging, the
frontier of mean-CVaR changes in k and b with same pattern. For fixed k, both the
expected profit and the CVaR value decrease in b, which is intuitive because when
the newsvendor sets a larger confidence level for the profit, he can only expect a
smaller profit value, and the CVaR is taken average below a smaller vb value. For
fixed b, the expected profit is increasing in k while the CVaR is decreasing in k. In
other words, larger total expected profit corresponds to a smaller average downside
profit. However, from the results we also see that with option hedging, either the
mean or the CVaR value is less sensitive to k and b.
In the end of this section, we make a note that when the demand is stochastically
decreasing in temperature t, the option with ‘‘lower’’ strike temperature is seemingly
more preferred, because such an option has more chances of paying out to the
newsvendor. However, the option’s cost, when it is fairly priced, must be higher,
and hence the resulting impact on the newsvendor is unclear. This is substantiated
by the examples shown in Table 2, where Option 1 is �t ¼ �7; k ¼ 2; �k ¼ 8 with fair
price K = 4, and Option 2 is �t ¼ �5; k ¼ 2; �k ¼ 8 with fair price K = 1.13. Other
parameter values are kept unchanged from the basic parameter set. k and b are fixed
as 0.5 and 0.9 respectively. We see that Option 1, which has a lower strike
temperature, has a worse hedging effect than Option 2.
5 Summary
This paper proposes an inventory model with financial hedging by weather
derivatives for seasonal products. The demand for the product is influenced not only
by inherent randomness but also by the weather during the selling season. With the
Table 2 Hedging effects of options with different strike temperatures
Without option With option 1 With option 2
Q Objective value ðQ�; n�1Þ Objective value ðQ�; n�2Þ Objective value
13.71 37.22 (15.68, 2.14) 39.65 (14.43, 2.84) 40.42
Joint optimal ordering and weather hedging decisions 15
123
existence of weather hedging markets, a risk averse newsvendor can use a weather
option to hedge the financial risk caused by unfavorable weather conditions. The
newsvendor’s problem is to make a joint order-quantity and hedging decision to
optimize his objective value.
Analytical results have been established under the mean-CVaR criterion. It has
been shown that the risk averse newsvendor who is conservative with ordering for
fear of large leftovers due to adverse weather will order more with option hedging.
Numerical analysis confirms the analytical results that the optimal order quantity
and objective value are increased with the option hedging. Furthermore, the optimal
order quantity does not exhibit definite monotonicity in the risk-parameters, and the
option hedging may help the risk averse retailer improve both the expected overall
profit and the average downside profit for undesired situation.
Acknowledgments The authors are grateful to the Editor in Chief Professor Hans-Otto Guenther, the
anonymous SE and two reviewers for their constructive comments. The authors also benefited from
discussions with Professors Minghui Xu, Simai He, Janny Leung and Sridhar Seshadri. The research of
Frank Y. Chen was supported in part by the Hong Kong Research Grants Council under grant no.
CUHK411105. The research of Xiuli Chao was partially supported by the NSF under CMMI-0800004
and CMMI-0927631.
Appendix
Proof of Lemma 1 For any k 2 ½0; 1�, and any two different points ðQ1; n1; v1Þand ðQ2; n2; v2Þ,
FbðkðQ1; n1; v1Þ þ ð1� kÞðQ2; n2; v2ÞÞ¼ FbðkQ1 þ ð1� kÞQ2; kn1 þ ð1� kÞn2; kv1 þ ð1� kÞv2Þ¼ kv1 þ ð1� kÞv2 � ð1� bÞ�1
E½kv1 þ ð1� kÞv2
�PðkQ1 þ ð1� kÞQ2; kn1 þ ð1� kÞn2;DðtÞÞ�þ:
As PðQ; n;DðtÞÞ is jointly concave in (Q, n),
PðkQ1 þ ð1� kÞQ2; kn1 þ ð1� kÞn2;DðtÞÞ¼ PðkðQ1; n1Þ þ ð1� kÞðQ2; n2Þ;DðtÞÞ� kPðQ1; n1;DðtÞÞ þ ð1� kÞPðQ2; n2;DðtÞÞ:
Hence,
½kv1 þ ð1� kÞv2 �PðkQ1 þ ð1� kÞQ2; kn1 þ ð1� kÞn2;DðtÞÞ�þ
� ½kv1 þ ð1� kÞv2 � kPðQ1; n1;DðtÞÞ � ð1� kÞPðQ2; n2;DðtÞÞ�þ
¼ ½kðv1 �PðQ1; n1;DðtÞÞÞ þ ð1� kÞðv2 �PðQ2; n2;DðtÞÞÞ�þ
� k½v1 �PðQ1; n1;DðtÞÞ�þ þ ð1� kÞ½v2 �PðQ2; n2;DðtÞÞ�þ;
the last inequality holds due to the convexity of function ½��þ.
Then, substituting the above inequality into (13) yields
16 F. Gao et al.
123
FbðkðQ1; n1; v1Þ þ ð1� kÞðQ2; n2; v2ÞÞ� kv1 þ ð1� kÞv2 � ð1� bÞ�1
Efk½v1 �PðQ1; n1;DðtÞÞ�þ
þ ð1� kÞ½v2 �PðQ2; n2;DðtÞÞ�þg ¼ kfv1 � ð1� bÞ�1E½v1 �PðQ1; n1;DðtÞÞ�þg
þ ð1� kÞfv2 � ð1� bÞ�1E½v2 �PðQ2; n2;DðtÞÞ�þg
¼ kFbðQ1; n1; v1Þ þ ð1� kÞFbðQ2; n2; v2Þ:
Thus, Fb(Q, n, v) is jointly concave in (Q, n, v). h
Proof of Proposition 1 Note the objective in (11) for the case without option
hedging,
maxQfkE½P1ðQ;DðtÞÞ� þ ð1� kÞ/bðQÞg
¼ maxQfkE½P1ðQ;DðtÞÞ� þ ð1� kÞmax
vFbðQ; vÞg
¼ maxQfmax
vfkE½P1ðQ;DðtÞÞ� þ ð1� kÞFbðQ; vÞgg
¼ maxðQ;vÞfkE½P1ðQ;DðtÞÞ� þ ð1� kÞFbðQ; vÞg:
Let Fðb;kÞðQ; vÞ ¼ kE½P1ðQ;DðtÞÞ� þ ð1� kÞFbðQ; vÞ ¼ kE½P1ðQ;DðtÞÞ� þ ð1� kÞfv� ð1� bÞ�1
E½v�P1ðQ;DðtÞÞ�þg: Denote by Dl(t) and Du(t) the lower and
upper bounds of D(t), respectively. For fixed Q,
oFðb;kÞðQ;vÞov
¼ð1�kÞ 1� 1
1�bE½1v[P1ðQ;DðtÞÞ�
� �
¼ð1�kÞ 1� 1
1�bEt
Z
Q
DlðtÞ
1v[ðp�sÞx�ðc�sÞQ �gðx;tÞdxþZ
DuðtÞ
Q
1v[ðp�cÞQ �gðx;tÞdx
2
6
4
3
7
5
8
>
<
>
:
9
>
=
>
;
¼ð1�kÞ 1� 1
1�bEt
Z
minðQ;vþðc�sÞQp�s Þ
DlðtÞ
gðx;tÞdxþZ
DuðtÞ
Q
1v[ðp�cÞQ �gðx;tÞdx
2
6
6
4
3
7
7
5
8
>
>
<
>
>
:
9
>
>
=
>
>
;
¼ð1�kÞ 1� 1
1�bEt
R
vþðc�sÞQp�s
DlðtÞgðx;tÞdx
2
4
3
5
8
<
:
9
=
;
ifv�ðp�cÞQ;
ð1�kÞ � b1�b
h i
ifv[ðp�cÞQ:
8
>
>
>
>
<
>
>
>
>
:
At the point (p - c)Q,
Joint optimal ordering and weather hedging decisions 17
123
o�Fðb;kÞðQ; vÞov
jv¼ðp�cÞQ ¼ ð1� kÞ 1� 1
1� bEt
Z
Q
DlðtÞ
gðx; tÞdx
2
6
4
3
7
5
8
>
<
>
:
9
>
=
>
;
;
oþFðb;kÞðQ; vÞov
jv¼ðp�cÞQ ¼ ð1� kÞ � b1� b
� �
� 0:
(1) Ifo�Fðb;kÞðQ;vÞ
ov jv¼ðp�cÞQ\0, the maximizer v of Fðb;kÞðQ; vÞ satisfies v\ðp� cÞQand is determined by
1� 1
1� bEt
Z
vþðc�sÞQp�s
DlðtÞ
gðx; tÞdx
2
6
6
4
3
7
7
5
¼ 0: ð13Þ
In this case,
Fðb;kÞðQ; vÞ¼kE½p1ðQ;DðtÞÞ�
þð1�kÞ v�ð1�bÞ�1Et
Z
vþðc�sÞQp�s
DlðtÞ
ðvþðc�sÞQ�ðp�sÞxÞgðx;tÞdx
2
6
6
4
3
7
7
5
8
>
>
<
>
>
:
9
>
>
=
>
>
;
dFðb;kÞðQ; vÞdQ
¼oFðb;kÞðQ; vÞ
oQþ
oFðb;kÞðQ; vÞov
ov
oQ
¼kf�cþsþðp�sÞE½1DðtÞ[Q�gþð1�kÞ
�
ð1�bÞ�1ð�cþsÞEt
�
Z
vþðc�sÞQp�s
DlðtÞ
gðx;tÞdx
��
þ0
¼kf�cþsþðp�sÞE½1DðtÞ[Q�gþð1�kÞð�cþsÞ¼kf�cþsþðp�sÞð1�Et½GðQ;tÞ�Þgþð1�kÞð�cþsÞ¼kðp�cÞ�kðp�sÞEt½GðQ;tÞ��ð1�kÞðc�sÞ:
Then Q satisfies Et½GðQ; tÞ� ¼ kðp�cÞ�ð1�kÞðc�sÞkðp�sÞ ¼ 1� c�s
kðp�sÞ, where k[ c�sbðp�sÞ to
make the conditiono�Fðb;kÞðQ;vÞ
ov jv¼ðp�cÞQ\0 hold true.
(2) Ifo�Fðb;kÞðQ;vÞ
ov jv¼ðp�cÞQ� 0, note that the auxiliary function is concave in v, then
v ¼ ðp� cÞQ. In this case,
Fðb;kÞðQ; ðp� cÞQÞ ¼ kE½P1ðQ;DðtÞÞ�
þ ð1� kÞ ðp� cÞQ� ð1� bÞ�1Et
Z
Q
DlðtÞ
ððp� sÞQ� ðp� sÞxÞgðx; tÞdx
2
6
4
3
7
5
8
>
<
>
:
9
>
=
>
;
;
18 F. Gao et al.
123
then
dFðb;kÞðQ; ðp� cÞQÞdQ
¼ kf�cþ sþ ðp� sÞE½1DðtÞ[ Q�g þ ð1� kÞ(
ðp� cÞ
� ð1� bÞ�1ðp� sÞEt
"
Z
Q
DlðtÞ
gðx; tÞdx
#)
¼ kf�cþ sþ ðp� sÞð1� Et½GðQ; tÞ�Þg
þ ð1� kÞfðp� cÞ � ð1� bÞ�1ðp� sÞEt½GðQ; tÞ�g¼ ðp� cÞ � ðkþ ð1� kÞð1� bÞ�1Þðp� sÞEt½GðQ; tÞ�:
Then Q satisfies Et½GðQ; tÞ� ¼ ð1�bÞðp�cÞð1�kbÞðp�sÞ, where k� c�s
bðp�sÞ to make the conditiono�Fðb;kÞðQ;vÞ
ov jv¼ðp�cÞQ� 0 hold true.
From the above analysis, the optimal order quantity without option hedging
under the tradeoff criterion, Q, is determined by (12). h
We give here Lemma 2 which will be repeatedly used in the proofs of the results
to follow.
Lemma 2 Let G1ð�Þ and G2ð�Þ be two integrable functions. If they have contrarymonotonicity, then E½G1ðtÞG2ðtÞ� �E½G1ðtÞ�E½G2ðtÞ�, and if they have samemonotonicity, then E½G1ðtÞG2ðtÞ� �E½G1ðtÞ�E½G2ðtÞ�; where t is a continuousrandom variable.
Proof If G1ð�Þ and G2ð�Þ have contrary monotonicity, for any realizations t and s,
we have ½G1ðtÞ � G1ðsÞ�½G2ðtÞ � G2ðsÞ� � 0) G1ðtÞG2ðtÞ þ G1ðsÞG2ðsÞ�G1ðtÞG2ðsÞ þ G1ðsÞG2ðtÞ. Assuming the density function of t to be f ð�Þ;
E½G1ðtÞ�E½G2ðtÞ� : ¼ E½G1ðtÞ�E½G2ðsÞ�
¼Z
t
G1ðtÞf ðtÞdt
Z
s
G2ðsÞf ðsÞds
¼Z
t
G1ðtÞf ðtÞ�
Z
s
G2ðsÞf ðsÞds
�
dt
¼Z Z
t�s
G1ðtÞG2ðsÞf ðtÞf ðsÞdsdt
¼Z Z
t�s
G1ðtÞG2ðsÞ þ G1ðsÞG2ðtÞ2
f ðtÞf ðsÞdsdt
�Z Z
t�s
G1ðtÞG2ðtÞ þ G1ðsÞG2ðsÞ2
f ðtÞf ðsÞdsdt
¼Z
t
G1ðtÞG2ðtÞf ðtÞdt
¼ E½G1ðtÞG2ðtÞ�;
Joint optimal ordering and weather hedging decisions 19
123
i.e., E½G1ðtÞG2ðtÞ� �E½G1ðtÞ�E½G2ðtÞ�: Similarly, if G1ð�Þ and G2ð�Þ have same
monotonicity, it holds that E½G1ðtÞG2ðtÞ� �E½G1ðtÞ�E½G2ðtÞ�. h
Proof of Proposition 4 Note the objective in (7) for the case with option hedging,
maxðQ;nÞfkE½PðQ; n;DðtÞÞ� þ ð1� kÞ/bðQ; nÞg
¼ maxðQ;nÞfkE½PðQ; n;DðtÞÞ� þ ð1� kÞmax
vFbðQ; n; vÞg
¼ maxðQ;n;vÞ
fkE½PðQ; n;DðtÞÞ� þ ð1� kÞFbðQ; n; vÞg:
Let Fðb;kÞðQ; n; vÞ ¼ kE½PðQ; n;DðtÞÞ�þ ð1� kÞFbðQ; n; vÞ ¼ kE½PðQ; n;DðtÞÞ� þð1� kÞ fv� ð1� bÞ�1
E½v�PðQ; n;DðtÞÞ�þg:oFðb;kÞðQ; n; vÞ
onjn¼0 ¼ kE½P2ðtÞ� þ
1� k1� b
E½P2ðtÞ � 1P1ðQ;DðtÞÞ\v�:
Both P2ðtÞ and ED½1P1ðQ;DðtÞÞ\vjt� are increasing in t (It is easy to check that for
given Q;P1ðQ; �Þ is an increasing function and 1P1ðQ;�Þ\v is a decreasing function
then ED½1P1ðQ;DðtÞÞ\vjt� is increasing in t by the assumption that D(t) is stochastically
decreasing in t.), so by Lemma 2, if E½P2ðtÞ� ¼ 0,
oFðb;kÞðQ; n; vÞon
jn¼0�1� k1� b
E½P2ðtÞ�E½1P1ðQ;DðtÞÞ\v� ¼ 0:
By concavity of Fðb;kÞðQ; n; vÞ; n� � 0. h
Proof of Proposition 5
oFðb;kÞðQ; n; vÞov
¼ ð1� kÞf1� ð1� bÞ�1E½1v [ PðQ;n;DðtÞÞ�g
¼ ð1� kÞ(
1� ð1� bÞ�1Et
"
Z
Q
DlðtÞ
1v [ ðp�sÞx�ðc�sÞQþnP2ðtÞ � gðx; tÞdx
þZ
DuðtÞ
Q
1v [ ðp�cÞQþnP2ðtÞ � gðx; tÞdx
#)
¼ ð1� kÞ(
1� ð1� bÞ�1Et
"
Z
minðQ;vþðc�sÞQ�nP2ðtÞp�s Þ
DlðtÞ
gðx; tÞdx
þZ
DuðtÞ
Q
1v [ ðp�cÞQþnP2ðtÞ � gðx; tÞdx
#)
:
ð14Þ
20 F. Gao et al.
123
If n [ 0;oFðb;kÞðQ;n;vÞ
ov jv¼ð�cþsÞQþnPl2¼ 1� k; and
o�Fðb;kÞðQ; n; vÞov
jv¼ðp�cÞQþnPu2
¼ ð1� kÞ 1� ð1� bÞ�1Et
Z
Q
DlðtÞ
gðx; tÞdxþZ
DuðtÞ
Q
1Pu2 [ P2ðtÞ � gðx; tÞdx
2
6
4
3
7
5
8
>
<
>
:
9
>
=
>
;
¼ ð1� kÞ 1� ð1� bÞ�1Et GðQ; tÞ þ 1Pu
2 [ P2ðtÞð1� GðQ; tÞÞh in o
;
oþFðb;kÞðQ; n; vÞov
jv¼ðp�cÞQþnPu2
¼ ð1� kÞ 1� ð1� bÞ�1Et
Z
Q
DlðtÞ
gðx; tÞdxþZ
DuðtÞ
Q
gðx; tÞdx
2
6
4
3
7
5
8
>
<
>
:
9
>
=
>
;
¼ ð1� kÞð1� ð1� bÞ�1Þ\0:
From Proposition 1,
Et½GðQ; tÞ� ¼ 1� c�skðp�sÞ[ 1� b; k[ c�s
bðp�sÞ;
Et½GðQ; tÞ� ¼ ð1�bÞðp�cÞð1�kbÞðp�sÞ � 1� b; k� c�s
bðp�sÞ:
(
(1) If k [ c�sbðp�sÞ, then
o�Fðb;kÞðQ; n; vÞov
jv¼ðp�cÞQþnPu2�ð1� kÞf1� ð1� bÞ�1
Et½GðQ; tÞ�g\0
by noticing that Et½1Pu2 [ P2ðtÞð1� GðQ; tÞÞ� � 0: Thus, v�ðQ; nÞ 2 ðð�cþ sÞQþ
nPl2; ðp� cÞQþ nPu
2Þ, and by (14),
E½1v�ðQ;nÞ[ PðQ;n;DðtÞÞ� ¼ 1� b;
(2) If k� c�sbðp�sÞ, and if
o�Fðb;kÞðQ;n;vÞov jv¼ðp�cÞQþnPu
2� 0, then v�ðQ; nÞ ¼ ðp� cÞQþ
nPu2, and
Joint optimal ordering and weather hedging decisions 21
123
Fðb;kÞðQ; n; vÞjv¼ðp�cÞQþnPu2¼ kE½PðQ; n;DðtÞÞ� þ ð1� kÞfðp� cÞQþ nPu
2
� ð1� bÞ�1Et
"
Z
Q
DlðtÞ
ððp� sÞðQ� xÞ þ nðPu2 �P2ðtÞÞÞgðx; tÞdx
þZ
DuðtÞ
Q
nðPu2 �P2ðtÞÞgðx; tÞdx
#)
;
oFðb;kÞðQ; n; v�ðQ; nÞÞoQ
jQ¼Q ¼ kfp� c� ðp� sÞEt½GðQ; tÞ�g
þ ð1� kÞ ðp� cÞ � ð1� bÞ�1ðp� sÞEt
Z
Q
DlðtÞ
gðx; tÞdx
2
6
4
3
7
5
8
>
<
>
:
9
>
=
>
;
¼ ðp� cÞ � 1� kb1� b
ðp� sÞEt½GðQ; tÞ� ¼ 0;
(3) If k� c�sbðp�sÞ, and
o�Fðb;kÞðQ;n;vÞov jv¼ðp�cÞQþnPu
2\0, then v�ðQ; nÞ 2 ðð�cþ sÞQþ
nPl2; ðp� cÞQþ nPu
2Þ, and
E½1v�ðQ;nÞ[ PðQ;n;DðtÞÞ� ¼ 1� b:
For the cases (1) and (3), i.e., E½1v�ðQ;nÞ[ PðQ;n;DðtÞÞ� ¼ 1� b;
oFðb;kÞðQ; n; vÞoQ
jQ¼Q ¼ kEoPðQ; n;DðtÞÞ
oQ
" #
þ 1� k1� b
E 1v [PðQ;n;DðtÞÞoPðQ; n;DðtÞÞ
oQ
" #
¼ kfðp� cÞ � ðp� sÞEt½GðQ; tÞ�g
þ 1� k1� b
E½1v [ PðQ;n;DðtÞÞðp� c� ðp� sÞ1DðtÞ\QÞ�
¼ kðp� cÞ � kðp� sÞEt½GðQ; tÞ�
þ 1� k1� b
fðp� cÞE½1v [ PðQ;n;DðtÞÞ�
� ðp� sÞE½1v [PðQ;n;DðtÞÞ1DðtÞ\Q�g:
When k [ c�sbðp�sÞ;Et½GðQ; tÞ� ¼ kðp�cÞ�ð1�kÞðc�sÞ
kðp�sÞ , and
oFðb;kÞðQ; n; v�ðQ; nÞÞoQ
jQ¼Q� kðp� cÞ � kðp� sÞEt½GðQ; tÞ� þ 1� k1� b
fðp� cÞE½1v�ðQ;nÞ[ PðQ;n;DðtÞÞ� � ðp� sÞE½1v�ðQ;nÞ[ PðQ;n;DðtÞÞ�g
¼ ð1� kÞðc� sÞ þ 1� k1� b
fð�cþ sÞE½1v�ðQ;nÞ[ PðQ;n;DðtÞÞ�g ¼ 0:
22 F. Gao et al.
123
When k� c�sbðp�sÞ;Et½GðQ; tÞ� ¼ ð1�bÞðp�cÞ
ð1�bkÞðp�sÞ, and
oFðb;kÞðQ; n; v�ðQ; nÞÞoQ
jQ¼Q� kðp� cÞ � kðp� sÞEt½GðQ; tÞ�
þ 1� k1� b
fðp� cÞE½1v�ðQ;nÞ[PðQ;n;DðtÞÞ� � ðp� sÞE½1DðtÞ\Q�g
¼ kðp� cÞ � 1� bk1� b
ðp� sÞEt½GðQ; tÞ� þ 1� k1� b
ðp� cÞE½1v�ðQ;nÞ[ PðQ;n;DðtÞÞ�
¼ ðk� 1Þðp� cÞ þ 1� k1� b
ðp� cÞE½1v�ðQ;nÞ[ PðQ;n;DðtÞÞ� ¼ 0:
In all, for all cases (1), (2) and (3),oFðb;kÞðQ;n;v�ðQ;nÞÞ
oQ jQ¼Q� 0.
When n \ 0, similar analysis can be conducted by inter-changing the roles of Pl2
and Pu2 in the above proof, and when n = 0,
oFðb;kÞðQ; 0; v�ðQ; 0ÞÞoQ
jQ¼Q ¼oFðb;kÞðQ; vðQÞÞ
oQjQ¼Q ¼ 0:
By concavity of F(b, k)(Q, n, v), we have Q� � Q. This completes the proof.
Note the condition in this proposition, G(y;t) is strictly increasing in y for given t,
is to get a unique solution Q for the unhedged case. Generally without this
condition, assume Q and Q* are the largest solutions for the unhedged and hedged
cases respectively, we still have the result that Q� � Q: h
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Author Biographies
Fei Gao received doctoral degree from the Chinese University of Hong Kong. Her doctoral research was
focused on interfaces between operations and finance/marketing.
24 F. Gao et al.
123
Frank Y. Chen is an associate professor at the Department of Systems Engineering and Engineering
Management, the Chinese University of Hong Kong. His current research is focused on interfaces
between operations and marketing, and inventory models with risk considerations.
Xiuli Chao is a professor in the Department of Industrial and Operations Engineering at the University of
Michigan, Ann Arbor. His research interests include queueing, scheduling, financial engineering,
inventory control, and supply chain management. He received doctoral degree in Operations Research
from Columbia University.
Joint optimal ordering and weather hedging decisions 25
123
