Practical Issues In Pricing (and Using) Asian Basket Options:

A Case of Livestock Gross Margin Insurance

Marin Bozic - University of Minnesota

MFM Seminar, Minneapolis, September 28, 2012

2

3

Room 1: A Barn on Fire

Nature of risk in the dairy sector

Real price risk? Prolonged Period of Margins Much Below Average

20022003

20042005

20062007

20082009

20102011

0.002.004.006.008.00

10.0012.0014.0016.00

Dairy Margin, Foundation for the Future, NMPF

Livestock Gross Margin Insurance for Dairy Cattle (LGM-Dairy)

Jan Feb

Mar

Apr May

Jun Jul Aug

Sep

Oct Nov

Dec

Purchase at End of

Month

No Coverage

1 2 3 4 5 6 7 8 9 10

Insurance Contract Period

Farmer must decide:• Monthly target milk marketings (Mt+i) • expected feed usage (Ct+i, SBMt+i)• Gross Margin Deductible (D)

11 11 11

2 2 2

Margin Guarantee = M C SBMt i t i t i t i t i t i

i i if D M f C f SBM

How is LGM-Dairy priced?

Jan Feb

Mar

Apr May

Jun Jul Aug

Sep

Oct Nov

Dec

Purchase at End of

Month

No Coverage

1 2 3 4 5 6 7 8 9 10

Insurance Contract Period

• Extract information regarding expected prices and volatilities from futures prices and at-the-money options

• Calculate correlations based on historical data• Use Monte Carlo methods to simulate indemnities• Price of the Asian Basket Option set at mark-up over

actuarially fair price (e.g. expected indemnity).

• Identify expected milk marketings, feed amounts

• Choose target IOFC margin to protect• Insure equal percentage of each month’s

production, e.g. flat coverage for 10 months.

A Naïve approach to LGM-Dairy

• Identify expected milk marketings, feed amounts

• Choose target IOFC margin to protect• Find a least-cost profile that protects the

target IOFC.

A (bit less) Naïve approach to LGM-Dairy

A (bit less) Naïve approach to LGM-Dairy

1 2 3 4 5 6 7 8 9 100%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

Insurable Month

Cove

rage

Per

cent

age

Home-feed profile:Insuring 1st-10th month

050

100150200250300350400

024681012141618

LGM Premium PaidActual MarginMargin with LGM Net Indemnity

$/Mg of Milk $/cwt of Milk

Home-feed profile:Insuring 1st-3rd month.

050

100150200250300350400

024681012141618

LGM Premium Paid Actual Margin

Margin with LGM Net Indemnity

$/Mg of Milk $/cwt of Milk

Class III Milk Futures: Open Interest

Sep-12

Oct-12

Nov-12

Dec-12

Jan-1

3

Feb-13

Mar-13

Apr-13

May-13

Jun-1

3Ju

l-13

Aug-13

Sep-13

Oct-13

Nov-13

Dec-13

Jan-1

40

1,000

2,000

3,000

4,000

5,000

6,000

Home-feed profile: Insuring 8-10th month

050

100150200250300350400

024681012141618

LGM Premium Paid Actual Margin

Margin with LGM Net Indemnity

$/Mg of Milk $/cwt of Milk

Why using deferred contracts works the best

Room 2: Mind your Tail

How is LGM-Dairy priced?

Jan Feb

Mar

Apr May

Jun Jul Aug

Sep

Oct Nov

Dec

Purchase at End of

Month

No Coverage

1 2 3 4 5 6 7 8 9 10

Insurance Contract Period

• Extract information regarding expected prices and volatilities from futures prices and at-the-money options

• Calculate correlations based on historical data

• Use Monte Carlo methods to simulate indemnities• Price of the Asian Basket Option set at mark-up over

actuarially fair price (e.g. expected indemnity).

17

Is correlation a good way to think about dependence between variables?

Lower tail dependence

1 20

lim Pr ,Lu

U u U u

Upper tail dependence

1 21

lim Pr ,Uu

U u U u

Copulas: Tool for dealing with nonlinear dependencies

1 2 1 1 1 1 1 1

1 2 1 1 2 2

, ,... ,..., ,

, ,... , ,...,

p p p

p p p

F x x x P X x X x F x P X x

F x x x C F x F x F x

GaussianClayton Gumbel

Comparing Copula Families

Empirical Copula

• Empirical copula replaces unknown distributions with their empirical counterparts:

• Implementation: Bootstrap based on rank-order matrix• Potential shortcomings: Small sample, serial dependency

1 1 1,..., ,...,p p pC u u F F x F x

1 1 1,..., ,...,p n n np pC u u F F x F x

1 1 11

1,..., ,...,T

n p i ip pi

F x x I X x X xT

1

1 T

np p ip pi

F x I X xT

Effect of non-linear dependence on LGM premiums

Home-Feed Market-FeedDeductible $0.00 $1.10 $0.00 $1.10

Official RMA Method $14,569 $7,380 $20,350 $13,308

Rank Correlations $14,998 $7,719 $16,439 $9,504

Empirical Copula $15,286 $8,219 $15,478 $8,246

• Unlike most situations in financial sector, in livestock margin insurance tail dependence decreases portfolio risk.

Room 3: Mr. Black, this drink is flat.

How is LGM-Dairy priced?

Jan Feb

Mar

Apr May

Jun Jul Aug

Sep

Oct Nov

Dec

Purchase at End of

Month

No Coverage

1 2 3 4 5 6 7 8 9 10

Insurance Contract Period

• Extract information regarding expected prices and volatilities from futures prices and at-the-money options

• Calculate correlations based on historical data• Use Monte Carlo methods to simulate indemnities• Price of the Asian Basket Option set at mark-up over

actuarially fair price (e.g. expected indemnity).

Are Futures Prices Unbiased?

Testing for bias in futures prices

t t Tf E p

0t Tt

t

f pEf

, ,

1 ,

1 0N

t i T i

i t i

f pN f

Test Design

22

, ,

1

1ln ln1 2 1

T i t i iN

i i

p f

N

, ,

1 ,

1 0N

t i T i

i t i

f pN f

• Essential assumption: Lognormality

Bootstrap procedure

2exp ln 0.5T t tp z f

Testing for Futures Price Bias

1 2 3 4 5 6 7 8 9-15

-10

-5

0

5

10 Class III Milk

Nearby

Pred

ictio

n Er

ror (

%)

Testing for Futures Price Bias

1 2 3 4 5

-20-15-10

-505

101520

Corn

Nearby

Pred

ictio

n Er

ror (

%)

Testing for Futures Price Bias

1 2 3 4 5 6-15

-10

-5

0

5

10

15Soybean Meal

Nearby

Pred

ictio

n Er

ror (

%)

Testing for Implied Volatility Bias

1 2 3 4 50.70.80.9

11.11.21.31.41.5

Corn

Nearby

Roo

t Mea

n Sq

uare

Sta

ndar

dize

d Pr

edic

tion

Erro

r (%

)

Testing for Implied Volatility Bias

1 2 3 4 5 60.70.80.9

11.11.21.31.41.5

Soybean Meal

Nearby

Roo

t Mea

n Sq

uare

Sta

ndar

dize

d Pr

edic

tion

Erro

r (%

)

Testing for Implied Volatility Bias

1 2 3 4 5 6 7 8 90.70.80.9

11.11.21.31.41.5

Class III Milk

Nearby

Roo

t Mea

n Sq

uare

Sta

ndar

dize

d Pr

edic

tion

Erro

r (%

)

Testing for Implied Volatility Bias

3 4 5 6 7 8 9 10 110.00

0.05

0.10

0.15

0.20

0.25

Mean Implied VolatilityLowest Average IV Consistent with Data

Nearby

Impl

ied

Vola

tility

Effect of biases on LGM premiums

Home-Feed Market-FeedDeductible $0.00 $1.10 $0.00 $1.10

Official RMA Method 9,743 5,191 13,316 8,873

Biased Soymeal Futures

9,744 13,438 5,192 8,992

Biased Milk Volatility 10,972 6,287 14,235 9,686

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Room 4: A reason to smile.

How is LGM-Dairy priced?

Jan Feb

Mar

Apr May

Jun Jul Aug

Sep

Oct Nov

Dec

Purchase at End of

Month

No Coverage

1 2 3 4 5 6 7 8 9 10

Insurance Contract Period

• Extract information regarding expected prices and volatilities from futures prices and at-the-money options

• Calculate correlations based on historical data• Use Monte Carlo methods to simulate indemnities• Price of the Asian Basket Option set at mark-up over

actuarially fair price (e.g. expected indemnity).

Does it matter if marginal distributions are in fact not lognormal?

• In the current RMA ratings method, only at-the-money puts and calls are used to estimate variance of the terminal prices.

15%

20%

25%

30%

35%

40%

Log(Strike/Underlying Futures Price)

Implied Volatility

Date: Jun 26, 2006Contract: Corn, Dec ’06Futures Price: $2.49

2.80 3.30 3.80 4.300.30

0.32

0.34

0.36

0.38

0.40

S=0, K=3 S=0, K=3.5S=0, K=4.5 S=0, K=5.4

Strike Price

Impl

ied

Vol

atili

ty

42

Volatility smiles induced by high kurtosis

$3.00 $3.50 $4.000.30

0.32

0.34

0.36

0.38

0.40

0.42

S=0.3, K=3.5 S=0.6, K=4.5Strike Price

BS: I

mpl

ied

Vola

tility

43

Volatility skews induced by high skewness

-3 -2 -1 0 1 2 3 4 5 60.000.100.200.300.400.500.600.70

S=-1, K=6S=2, K=11S=1, K=6S=0, K=3

431

12

1p pF p

44

Generalized Lambda Distribution (GLD) allows changing one moment at a time

Scenario 1: Corn as the only source of riskCorn skewness boosted 60%

00.

20.

40.

60.

8 11.

21.

41.

61.

8 22.

22.

42.

62.

8 33.

23.

43.

63.

8 44.

24.

44.

64.

8 5

-10.00%-5.00%0.00%5.00%

10.00%15.00%20.00%25.00%30.00%

Skewness Boost

00.

20.

40.

60.

8 11.

21.

41.

61.

8 22.

22.

42.

62.

8 33.

23.

43.

63.

8 44.

24.

44.

64.

8 5

-10.00%-8.00%-6.00%-4.00%-2.00%0.00%2.00%4.00%6.00%8.00%

10.00%Kurtosis Boost

Kurtosis Boost

Scenario 2: Corn as the only source of riskCorn kurtosis boosted 60%

Scenario 3: Corn as the only source of riskBoth skewness and kurtosis boosted

00.

20.

40.

60.

8 11.

21.

41.

61.

8 22.

22.

42.

62.

8 33.

23.

43.

63.

8 44.

24.

44.

64.

8 5

-10.00%-5.00%0.00%5.00%

10.00%15.00%20.00%25.00%30.00%35.00%40.00%

Kurtosis Boost Skewness BoostSkewness & Kurtosis Boost

Scenario 4: Two sources of risk – milk and cornEffect nearly disappears

00.

20.

40.

60.

8 11.

21.

41.

61.

8 22.

22.

42.

62.

8 33.

23.

43.

63.

8 44.

24.

44.

64.

8 5

-10.00%-5.00%0.00%5.00%

10.00%15.00%20.00%25.00%30.00%35.00%40.00%

Skewness & Kurtosis BoostLognormal Milk, S&K Boost

Conclusions

• Modeling dependence using correlations may not suffice – tail dependence matters!

• Simplistic heuristics and CME settlement rules may have rendered dairy options too cheap.

• Volatility smiles may not be important for pricing Asian Basket Options

Practical Issues in Pricing (and Using) Asian Basket Options: A Case of Livestock Gross Margin Insurance

MFM SeminarSeptember 28, 2012

Dr. Marin Bozicmbozic@umn.edu(612) 624-4746Department of Applied EconomicsUniversity of Minnesota-Twin Cities317c Ruttan Hall1994 Buford AvenueSt Paul, MN 55108

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