Forward Hedging Under Price and Production Risk of Wheat
Liu Xing and Kyosti Pietola
Paper prepared for presentation at the XI th Congress of the EAAE
(European Association of Agricultural Economists),
'The Future of Rural Europe in the Global Agri-Food System',
Copenhagen, Denmark, August 24-27, 2005
Copyright 2005 by Liu Xing and Kyosti Pietola. All rights reserved. Readers may make
verbatim copies of this document for non-commercial purposes by any means, provided that this
copyright notice appears on all such copies.
FORWARD HEDGING UNDER PRICE AND PRODUCTION RISK OF
This paper estimates optimal hedging ratios for a Finnish spring wheat producer under price and
yield risk. The forward contract available for hedging fixes the price and quantity at the time of sowing for a delivery at harvest. Autoregressive models are used to obtain point forecasts for the conditional mean price and price volatility at harvest. Expected yield and yield volatility are estimated from
the field experiment data. A range of coefficients of absolute risk aversion are used in the computations. The results suggest that yield volatility is large and it dominates the price volatility in the optimal hedging decisions of the Finnish wheat producers. Nevertheless, a potential for large negative
correlation between the price and the yield decreases the optimal hedging ratio since the Finnish farmers do not have access to selling put options when they enter in a forward contract.
Key words: Hedging ratio, risk, forward contract. Q14
The Agenda 2000 reform of European Common Agricultural Policy (CAP) decreased intervention prices for cereals. This allows for the European grain prices to move towards equivalent world
market prices more freely than before. Fluctuations of world market prices will also be transmitted to
the EU market and increase price uncertainty for the producers (Roche & Mcquinn, 2001). Therefore,
appropriate risk management tools such as forward contracts become more attractive to hedge against
the increased price risk.
A quite large literature focuses on optimal hedging problems in the futures markets (see e.g.,
Tomek and Peterson, 2001; Lapan and Moschini, 1994). The standard in the studies on optimal hedging is that market is efficient such that the futures contract is allowed to be offset (liquidated) by an
opposite contract before its maturity. The hedging problem of Myers and Thompson (1989) is also
truly dynamic such that it allows for continuous adjustment in the hedging position.
A problem on generalizing these results to Finland is that the short run grain price movements
may not be fully integrated to other markets, in which also the futures are traded (Kola and Taipale
2000). Another problem is that independent grain producers may not have a direct access to the derivatives market since transactions costs for entering these markets are too high for them. A “full”
hedge, which is a combination of futures contracts and sales of put options is not feasible. Nevertheless, they can usually enter in a forward contract with a local grain dealer. In Finland, for example,
Avena Nordic Grain (Avena) has offered forward contracts for the Finnish grain producers since the
year 2000. In these contracts, the buyer (seller) is obliged to purchase (deliver) grains from (to) Avena
at the agreed maturity date in the future at the fixed price.
The hedging problem of the Finnish grain producer is, at least for two reasons, a special and restricted version of the problems generally studied in the literature. First, the forward contract, they
have access to, is incomplete compared to the liquid and highly standardized futures contracts. The
non-tradable forward contract is irreversible and can only be terminated by the delivery. Therefore,
there is no basis risk and the potential for pure speculation is negligible. Second, in the Northern wheat
producing areas yield uncertainty is large, which moves weight from the price volatility to the yield
volatility in the optimal hedging problem.
The goal of this paper is to estimate optimal hedging ratios for Finish spring wheat producers.
Optimal hedging ratio is defined as the share of the expected yield sold through a certain forward contract. The contract is signed at sowing time for a fixed quantity and price for a delivery at harvest. We
solve the hedging problem by an Mean Variance model (MV) and compare it with the Expected Utility model (EU), first derived into a similar hedging problem by Lapan and Moshini (1994). MV model
is used because it is empirically tractable and it either coincides with the EU-model, or under fairly
general conditions it results in only negligible approximation errors. Sufficient condition for MV
model to coincide with the exact EU-model is that either an investor’s utility function is quadratic, or
the investor’s expectation errors follow the normal distribution (Robison and Barry 1987).
Our empirical examination of optimal hedging focuses on a hypothetical spring wheat producer in
the Southwest of Finland. We assume that the hedging and production decisions are made in April/May once a year, while the execution of the forward contract is performed in August of the same
year. Therefore, we take 21st week of each year as the time at which production commitments are made and the forward contract is signed. Harvest and delivery take place 15 weeks later, at 36th week of
the same year. Termination of the forward contract also occurs at harvest since it can be terminated
only through the delivery.
2. Economic models: MV and EU-model
The goal of a farmer is to maximize his wealth at harvest, W, which is generated from the random
income, y. The MV-approach is to approximate this wealth by the Certainty Equivalence (CE), defined as the difference between the expected income and the risk premium
W = E ( y) −
where E(y) is the expected income and the term
Aσ y2 is the Arrow-Pratt risk premium. It consists
of the constant absolute risk aversion (CARA) coefficient, A, and from the variance of income, σ y
(Pratt 1964; Arrow1971). If the variance of income increases at a given income level, wealth is decreased as A>0.
Let q~ be the random yield per hectare. The expected yield conditional on information (Ω) at
sowing time (t) as
E t [ q~t + s | Ω t ] = µ q
where s is the length of the growing season. Conditional variance of the yield is denoted by σ q . Simi2
larly, we define ~
p as the random spot price at harvest with expected value µ p , defined as
Et [ ~
pt + s | Ω t ] = µ p
and conditional variance as σ p . Information set Ωt includes information available at sowing time,
such as past prices and input use.
We define randomness of ~
p and q~ to be additive such they can also be written as:
p = µ p + v~p
q~ = µ + v~
v~p ~ N (0, σ 2p )
v~q ~ N (0, σ q2 )
where v~p and v~q are jointly normally distributed errors with zero mean, variance σ p and σ q , and
covariance σ pq .
Let h be the forward sale at the harvest per hectare at a fixed price p f . The remaining output
( q~ − h ) is sold at the spot market price at harvest. A hedge occurs when 0 ≤ h ≤ q ; speculation occurs when h > q or when h < 0 . The farmer’s income comes from two sources: the uncertain earn-
ings from the un-hedged output ~
p[ q~ − h] and the certain returns, p f h , from the output sold in the
forward contract. Then, farmer income at harvest is
p[(q~ − h)] + p f h
pq~ + h(p − ~
and the expected income E(y) is:
E ( y) = E ( ~
pq~ ) + h( p f − µ p )
where E ( ~
pq~ ) is the expected value of joint normally distributed price and yield. Variance of income
σ y2 is:
σ y2 = σ pq
+ h 2σ p2 − 2h cov( ~
where σ pq is the variance of the product of the random price and yield, ~
pq~ , and cov( ~
pq~ ) is the
covariance between ~
p and ~
Substituting (7) and (8) into (1), we get:
W = E( ~
pq~ ) + h( p f − µ p ) − A[σ pq
+ h 2σ p2 − 2h cov( ~
The optimal hedge is given by the first order condition:
= p f − µ p − Ahσ p2 + A cov( ~
pq~ ) = 0
and solving for the optimal hedge h we get:
pf − µp
Here, the forward price is deterministic and rest of variables are unknown. Using (4) and (5)
pq~ ) is (See Appendix):
pq~) = µ p rσ pσ q + µ qσ p2
where r is the correlation coefficient between v~p and v~q . Substituting (12) into (11) and dividing both
sides by µ q yields in the optimal hedging ratio:
= (1 +
pf − µp
rσ q / σ p ) +
Aµ qσ p2
which can be separated into two components: “the hedging component” and “the speculative component”. The speculative component is zero and drops out from the problem if either the forward market
is unbiased ( µ p = p f ), or the producer is infinitely risk averse (A → ∞ ). In this case, the optimal
hedge ratio would be identical to the minimum variance hedge ratio. If, in addition, either the price is
independent from the yield (r=0), or the yield risk is zero, the producer sells all of his expected yield
through the forward contract (h=1).
Clearly, when the forward price is an unbiased estimate for the price at harvest, the optimal hedge
under yield uncertainty depends on the conditional forecast for the harvest price ( µ p ), expected yield
( µ q ); volatility of the price and yield at harvest ( σ p and σ q ); and correlation between the price and
the yield (r). Among them, the correlation between the price and the yield is crucial for determining
the optimal hedging strategy. When the correlation between the price and the yield tends to zero, MV
model suggests that the hedge ratio approaches to unit regardless of farmer’s risk attitude (Rolfo 1980,
Newbery et.al., 1981, and Anderson and Danthine 1983). However, Losq (1982) along with Lapan and
Moschini (1994) found in their EU-model that the optimal hedge, in general, is less than the expected
yield even when the price is independent from the yield.
Assuming a Constant Absolute Risk Aversion (CARA) in the utility function, an exact analytical
solution to the hedging problem is possible even under the EU-model, as illustrated by Lapan and
Moschini (1994). In their research, they considered both basis risk and production risk for the future
hedging problem. In our case, the forward price is deterministic and, therefore, the basis risk does not
exist. If the forward price is perceived to be the same as expected price, absent basis risk, the optimal
hedge ratio derived from the EU-model is 1:
µ pσ q2
= (1 +
µq σ p
here we can see that even when the yield and the price are independent (r=0), the optimal hedge is still
less than one. It is reduced by the product of risk aversion coefficient (A), the ratio between expected
price and yield ( µ p / µ q ), and yield volatility.
The weekly data on wheat cash (spot) price and intervention price are obtained from information
centre of the Ministry of Agriculture and Forestry (TIKE). Both price series spans from January 1995
to 21th week of 2002 (Figure 1). The original cash price series consists of 370 observations and 16
missing values. The missing values are replaced by the average of the preceding and following values
in order to keep the continuity of data. The cash price reported in the data is the price at the warehouse
of buyers, i.e. the price of the raw material to the buyer.
Growing seasons are 15 weeks and they are dated from the 21st week of the year to the 36th week
of the year. In Figure 1, growing seasons are marked by vertical lines with marks of s.
The forward contracts are those offered by Avenakauppa and they are available for 2001 and
2002. These contracts are not tradable and they are always terminated by the delivery of the good. No
penalty is imposed if farmer’s yield is lower than expected and he can not deliver the good as much as
agreed in the contract resulting from force majeure. Nevertheless, speculation through deliberate short
selling is not allowed. The terms related to delivering the goods as much as agreed are for the most
part based on trust. Avenakauppa company also publishes its forward prices publicly on internet and
page 747 in the text-tv of MTV3 in Finland. The annual yield data of spring wheat are from Agrifood
Research Finland (MTT) and they span the years from 1995 to 2001. The yield data consist of 40 observations based on experimental trials and they are conditional on different nitrogen applications (70120 kg/ha). The trials have been located in the Southwest Finland, where most of the wheat production
The lengthy derivation of the optimal hedge under the EU-model is given in Lapan and Moschini
(1994) and it is not repeated here.
Conditional mean process for the wheat price
The conditional mean process for the wheat price is modelled as an AutoRegressive (AR) time
series model, which is the standard approach in estimating the parameters for the optimal hedging
model (Dawson et al.1999, Maurice and Kieran 2001). These models are generally found to perform
well in forecasting future commodity prices conditional on current information (e.g. Muth 1961; Beck
A hypothesised relationship is that the Finnish wheat prices are mainly determined by their own
lags, time trend, yield effect at harvest, the intervention price and a disturbance. A general model is
denoted as AR(k) with k representing the number of lagged prices in the estimating equation. The
pt = φ 0 + ∑ φ t − j pt − j + αDt qτ + ϕpti + λt + v pt
where p t is the weekly price quotation for wheat; t is the time index. Dt is yield dummy variable,
and it receives value one at the time of harvest, i.e. at 36th week of each year and zero otherwise. qτ is
the yield in year τ . p ti denotes the intervention prices2 at time t. α , φ t − j , β , ϕ , and λ are parameters, and v p t is the error term.
Time trend (t) is included in the model, because the observed price series show a declining trend
over the estimated years. The yield variable (q) is included to test whether the annual yield has significant effect on the price at harvest. The estimating equation is further augmented by the intervention
price for wheat ( p ti ), since the market prices are expected to, at least partially, respond to the changes
in the intervention price. The intervention price has had the largest changes in summer when the marketing years have changed.
The intervention price is the price at which intervention agencies are obliged to purchase wheat from
the market. The intervention price has gradually decreased between 1999 and 2002 in EU. For instance, at 37th week in 1999 the price was 119 euros per tonne and the same week in 2002 it was 101
euros per tonne.
Figure1. Weekly marketing price and intervention price of wheat in Finland from 1995 to 2002.
One important simplifying assumption for Equation (15) is that of weak stationary of the wheat
price series. For that purpose, we test for unit roots using the Augmented Dickey-Fuller (ADF test)
(Dickey and Fuller, 1981) and follow the sequential procedure of Dickey and Pantula (1987) for all the
estimation period. A linear trend term is added as the wheat price has a noticeable downward trend
over the period. The result, shown in Table 1 suggests that the null hypothesis of unit root is rejected
at 1% risk level. Thus the data are informative enough and the ADF-test has power enough to reject
non-stationarity in favour of stationary process around a deterministic trend.
Table 1. ADF Test statistics. The full sample used in estimation.
With intercept but without drift
With drift and intercept
Note: The number of observations is 401. Mackinnon critical values for rejection of hypothesis of a
unit root at 1% significance are –2.58, -3.47, and –4.01 level for without drift, with drift, and with drift
and trend respectively. A double asterisk (**) denotes significance at 1% level.
Using the procedure of model identification of Box and Jenkins (1976), AR(3) is found to generate a dynamically complete specification such that the error follows a white noise. The parameter estimates in Equation (15) are displayed in Table 2. Noticeably, the p-values for parameters suggest that
the intercept, the autoregressive term and the trend parameter differ from zero at 1% level. The price
response with respect to the intervention price and yield is statistically insignificant. Q-statistic (Ljung
and Box, 1978) serving as a residual test suggests that the residuals yielding from Equation (15) follow the white-noise process. The predictive power of the equation is very high, the R 2 equals to over
90% as is the standard in similar price models.
Table 2. OLS parameter estimates of AR(3) model. Estimated in Equation (15)
Intercept ( φ 0 )
Once lagged price( φ1 )
Twice lagged price ( φ 2 )
Third lagged price ( φ 3 )
Yield ( α )
Intervention price ( ϕ )
Trend ( λ )
Q (10) statistics
Note: (The numbers below the parameter estimates and in the parentheses denote p-values).
Forecasting the conditional mean price at harvest
When the hedge position is opened at 21st week, we look forward and forecast the expected price
15 weeks ahead given the information at 21st week. Because the marginal effects of yield and intervention price on the next period price are insignificant, we can estimate the expected value of price at
harvest simply through the AR(3) price process by excluding the yield and intervention prices. i
We use the minimum mean square error forecast method to predict the wheat price at harvest
(Mills, 1997. p.53-55). A minimum mean square error forecast, denoted by pˆ t + j , made at week t for
the price at period t+j for j=15 can be estimated through the repeated substitution in Equation
(15):Therefore, the forecasting conditional mean price at t+j given the information at t could be generally described as.
⎧⎪ p t + j ,
( Ep t + j | pt , pt −1 , pt − 2 )) = ⎨
⎪⎩ pˆ t + j ,
0 < j ≤ 15
where pˆ t +15 denote the predicted price at harvest. The predicted price at harvest in year 2002 is estimated at 129.84 euros per tonne. The price offered in the forward contract of Avena was for delivery
at 2002 harvest time 135 euros per tonne.
Estimating conditional mean of yield
The expected yield at harvest is estimated using the annual data from 1995 to 2001included in the
MTT experimental trials. The data have 40 observations in total. The yield Equation 5 is estimated
conditional on information that is available for farmers at sowing when the hedging decision is made.
The information includes location, soil quality, time trend and nitrogen fertiliser application. Because
the location of the experimental site and the soil type could not be estimated simultaneously in the
same model we specified two models. The first model includes four dummy variables for location
(five sites) excluding the soil type. The second specification includes five dummy variables for the soil
type (six soil types) but excludes the location effect. The third specification excludes both location and
q ( D i , τ , n ) = a + β D i + γτ + π n + v q
where q is output for the spring wheat ton/per hectare; Di includes dummy variables for location (i=1)
or soil type (i=2); τ is the annual time trend. It equals to 1 in 1995, 2 in 1996, etc: Variable n denotes the nitrogen fertiliser application kg/hectare. a , β , γ and π are parameters, and v q is error with
zero mean and variance σ q .
The parameter estimates suggest that the yield exhibits a slightly decreasing trend and, as expected, the nitrogen application significantly increases yield (Table 3). Nevertheless, nitrogen application and the trend explain only less than 26% of annual yield variation. The dummy variables controlling neither for location of the field experiment nor soil quality significantly improve the fit of the
model. Thus, the results support the view that weather performs a crucial role in Finnish agriculture,
and it contributes to a significant yield uncertainty. The average yield ( µ q ) at 90 kilogram nitrogen
application is estimated in year 2002 at 3.52 tonnes per hectare.
Table 3. Parameters in the yield equation (17).
for soil quality
Time trend ( γ )
Covariance and correlation between the price and the yield
The correlation between the price and the yield is needed in the optimal hedging equations (Equations 13 and 14). Because the data used in estimating the price process in Equation (15) and yield
process in Equation (17) exhibit different time frequencies, the following steps are used in estimating
the correlation coefficient (r). First, we compute the predicted price for each year at harvest and obtain
the prediction error ( v~p ) as a difference between the observed and predicted prices at harvest. Second,
we standardize the expected yield at the nitrogen application of 90 kilograms per hectare. The prediction error for the yield ( v~ q ) is then computed as the difference between the observed and fitted values
in the trials that have applied 90 kilogram of nitrogen per hectare and are the random terms with zero
Because the errors v~p and v~q are jointly distributed by assumption, define
ε = [v p , vq ]' .
The variance-covariance matrix of variation of price and yield can be written as Σ = E [ε ε ´] of the
random vector ε (Lapan and Moschini, 1994):
⎡σ 2p , rσ pσ q ⎤
⎢⎣rσ pσ q , σ q ⎥⎦
where σ p and σ q denote the standard deviation of the price and the yield, and r denotes the correlation coefficient between v p and v q .
5. Optimal hedge ratio
Table 4 summarizes the parameters we need to calculate the optimal hedge ratio as given by
Equations (13) and (14).
Table 4. Summary of the estimated parameters needed in (13) and (14).
Note: (.) is p-value of Pearson’s correlation test.
The point estimate for the correlation coefficient r is -0.36. The Pearson’s correlation test suggests, nevertheless, that the correlation coefficient is not statistically different form zero. P-value for
this test is 0.427. The result indicates that price and yield of wheat in Finland are likely independent
from each other, such that , the yield shock has no effect on the price movement. This result supports
the view that the Finnish market is integrated to the single European market in the extent that the local
(Finnish) yield shocks do not have significant effects on price.
The appropriate value for the risk aversion parameter is very difficult to evaluate, as there has
been little empirical evidence on the magnitude of risk aversion in Finland. Following Newbery and
Stiglitz (p.72-73), it is more usual to express the magnitude of risk aversion as coefficient of relative
risk aversion, Rr , instead of coefficient of absolute risk aversion A. That is because Rr is dimensionless regardless of different level of wealth, while A is not dimensionless3. The relationship between these two coefficients is Rr ( y ) = yA , where y is the end of period income presented in equation (2). Anderson and Dillon (1992) proposed a rough and ready classification of degrees of risk
aversion. Based on the relative risk aversion with respect of wealth, in the range 0.5 indicates hardly
risk averse at all and about 4 very risk averse.
Martinez and Zering (1992) applied constant “Arrow-Pratt” index of absolute risk aversion to
evaluate their optimal dynamic hedging model. The value of the absolute risk aversion were given to
be 0.006, 0.007, and 0.008 and with respect to the total income of average 164.39$/acre (1acre approximately = 0.4 hectare) each year, the corresponding relative absolute risk aversion approximates 1.
Accordingly, If we set up relative absolute risk aversion value between 1 and 10 with respect to minimum and maximum possible incomes per hectare, the corresponding coefficients of constant absolute
risk aversion then vary approximately from 0.002 to 0.02.
Since the coefficient of absolute risk aversion A depends on the units in which income is measured.
Table 5 displays the optimal hedge ratios of Finnish spring wheat producers for values of the absolute risk aversion from 0.002 to 0.02 when the forward market is assumed to be unbiased. Clearly,
increasing negative value of r tends to decrease forward contracts sales. In addition, it appears that
only when the correlation between price and yield tends to zero can make forward contract sales attractive. In fact, MV model yields full hedge ratio when the price and yield are independent regardless
of risk aversion degree, whereas EU model yields the optimal hedge ratios varying significantly for
different values of A. The amount of forward sale clearly declines as risk aversion increases. The more
risk averse, the less forward the farmers is willing to sell.
Table 5. Optimal hedge ratios in MV- and EU-models conditional on different value of r and alternative values for risk aversion.
Correlation coefficient (r)
*MV does not depend on risk attitudes when the futures price is perceived to be unbiased.
Since our empirical results weakly signal for a large negative correlation between the price and
the yield, and yield volatility is relatively high, we simulated the MV-model conditional on alternative
values for yield volatility and the correlation coefficient between the yield and the price. Figure 2
shows that the hedge ratio is a decreasing function of yield volatility and an increasing function of
correlation coefficient. For instance, at the (insignificant) estimate -0.36 for the yield and price correlation (r), the forward contract is attractive to the farmer only if the volatility of yield is less than 6 %
of the mean yield (=3.52 tonnes per hectare). Otherwise forward contract is attractive only if speculation is allowed and market is biased, or the farmer also has an access to options and can complement
the forward contract by selling put options (Sakong et al. 1993, Moshcini and Lapan 1995). Yield risk
increases farmer incentives to trade options even when the price offered in the forward contract is an
unbiased estimate for the harvest price. The reason is that a farmer who has sold his expected yield
forward at a fixed price is still exposed by the risk that lower than expected yield will decrease his
Yield volitility %
Figure 2. Optimal hedge ratio conditional on yield volatility and the correlation coefficient between
price and yield in the MV-model.
Both MV and EU model suggest that the correlation coefficient between the yield and the price
play very crucial role in the optimal hedge ratio. Empirically, the yield and the price of a crop should
be negatively correlated, because total demand food changes only moderately from year to year, while
supply can fluctuate considerably due to weather conditions. (Newbery et al 1981, Rolfo, 1980 Losq
1982, Lapan and Moshini 1994, Tirupattur and Hauser 1996, Mahul 2003). The negative covariance
between price and yield create a partial “natural hedge”4, which weakens the role of income risk reducing from forward contracts (ceteris paribus). When the natural hedge takes place, the optimal
hedge is always less than expected output.
Hedging effectiveness declines further as yield variability increases. Thus, the optimal hedge ratio decreases as the yield variability and production risk increase. The founding is very important in
such country as Finland, where the yield uncertainty dominates price volatility. As a result, forward
contracts could be less attractive in Finland than some other consistent production areas such as
U.S.A, France and Germany.
If the high natural hedge and high yield volatility take place simultaneously, forward sale may not
be an attractive risk instrument any more if the producer does not have access to other derivatives,
such as options. Instead of forward sale, producers would even wish to buy forward as to ensure the
output at harvest. This problem may confront the Finnish wheat growers and traders, because the yield
uncertainty dominates price uncertainty, and farmer access to derivatives markets is likely to incur
high transactions costs.
Even more generally, high yield risk provides a rationale for the use of other risk instruments
along with forward. Mahul (2003) showed futures and crop yield insurance are the complements.
When either option or crop yield insurance is available, forward contracts become more attractive to
It is likely that also the European intervention program that sheds a lower bound for the wheat
price and truncates the price distribution from below, decreases the optimal futures and options posi-
A negative yield-price correlation means that a farmer’s income is less variable from year to year
than it would be otherwise, thus being called “natural hedge”.
tion in a similar fashion as highlighted by Hanson et al. (1999). The truncation effects are left here,
nevertheless, topic for further research.
We conclude that under the Finnish production and under the European intervention programs,
where the yield risk dominates the price risk, forward contracts alone do not provide sufficient means
for efficient hedging. The solution to the problem is to increase farmer access to other risk derivatives
such as options and further develop domestic yield insurance.
Random variables ~
p and q~ can be written as follows:
τ~ ~ N (0, σ p2 )
p = µ p + τ~
q~ = µ + v~
v~ ~ N (0, σ q2 )
where τ~ and v~ are jointly normally distributed errors with zero mean, variance σ p and σ q with
correlation coefficient r.
Therefore, the expected value and ~
pq~ , variance of ~
pq~ and covariance between ~
pq~ and ~
be evaluated as:
pq~ ) = E[( µ p + τ~ )( µ q + v~ )] = µ p µ q + rσ pσ q
Var ( ~
pq~ ) = Var[( µ p + τ~ )( µ q + v~ )] = µ p σ q2 + µ q σ p2 + (1 + r 2 )σ p2σ q2 + 2 pqrσ pσ q
p) = E( ~
p 2 q~ ) − E ( ~
pq~ ) µ p
= E[( µ p + τ~ ) 2 ( µ q + v~ )] − E[( µ p + τ~ )( µ q + v~ )]µ p
= µ p rσ pσ q + µ qσ p2
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