Simple models for trading climate and weatherrisk

Sebastien Chaumont, Peter Imkeller, Matthias MullerHumboldt-Universitat zu Berlin

Cours Bachelier, Paris, February 17, 2006

Partially supported by the DFG research center MATHEON (FZT 86) in Berlin

SIMPLE MODELS FOR TRADING CLIMATE AND WEATHER RISK 1

El Ni no

phenomenon of anomalously high surface temperatures of eastern SouthPacific (Peru); randomly periodic, every 3-8 years around Christmas

El Nino, Dec’82, SOI=-2.48

Southern Oscillation Index (SOI)SOI: air pressure differenceDarwin (AUS) — TahitiEl Nino: SOI small, east Pacific warmLa Nina: SOI big, east Pacific cold

normal, Dec’83, SOI=-0.17

1880 1900 1920 1940 1960 1980 2000

-1

0

1

2

Southern Oscillation Index 1866-2001

1880 1900 1920 1940 1960 1980 2000

-1

0

1

2

Southern Oscillation Index 1866-2001

SIMPLE MODELS FOR TRADING CLIMATE AND WEATHER RISK 2

Climatic consequences

El Nino conditions

strong precipitation in eastern Pacific(Peru)

normal conditions

strong precipitation in western Pacific(Indonesia)

• cold ocean: food for big animals in surface layer

• warm ocean: food for big animals in deeper layers

SIMPLE MODELS FOR TRADING CLIMATE AND WEATHER RISK 3

Economic consequences: fishing

Peruvian anchoveta catch rates

“Skipjack” tuna catch rate

=⇒ complementary economicinterests: Peru-Indonesia

SIMPLE MODELS FOR TRADING CLIMATE AND WEATHER RISK 4

Economic consequences: farming

Precipitation in Peruand El Nino

82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97

5000

10000

15000

20000

Peru Rainfall data

Forest fires in Indonesiaand El Nino

=⇒ complementary economic interests:fisher- (rice) farmer in Peru

SIMPLE MODELS FOR TRADING CLIMATE AND WEATHER RISK 5

Periodically forced bi-stable temperature

U(t, k) periodic in t, k = −1, k = 1, periodic motion of well depths

U (t)−1

U (t)1

U (t) = K(t+1/2)1U (t) = K(t)−1

1−1

V

v11/2

SIMPLE MODELS FOR TRADING CLIMATE AND WEATHER RISK 6

Trajectories for potential diffusion

σK small

X

t

σK big t

X

σK good t

X

SIMPLE MODELS FOR TRADING CLIMATE AND WEATHER RISK 7

Bi-stable potential diffusion for numerical simulation

bi-stable temperature curve resulting from the non-linear two-dimensionaloscillator by Wang, Bacilon, Fang

bi-stable temperature curve used in our numerical experiments

0.5 1 1.5 2

K1

K2

Temperature. Model 2

SIMPLE MODELS FOR TRADING CLIMATE AND WEATHER RISK 8

Risk exposure of different model agents

fisher’s income

φf(k) = exp(−(k − k)2)

Hf =∫ T

0φf(Ks) ds

K1 K2

Income

farmer’s income

φr(k) = exp(−(k − k)2)

Hr =∫ T

0φr(Ks) ds

K1 K2

Income

SIMPLE MODELS FOR TRADING CLIMATE AND WEATHER RISK 9

farmer’s optimal investment into Y , model B

-0.3

-0.2

-0.1

0

0.1

0.2

-4 -3 -2 -1 0 1 2 3 4

pi_2

,t

K_t ( temperature)

Optimal investment in risk-security, model B, t=0.25

’x049.dat’

-0.3

-0.2

-0.1

0

0.1

0.2

-4 -3 -2 -1 0 1 2 3 4

pi_2

,t

K_t ( temperature)

Optimal investment in risk-security, model B, t=0.75

’x149.dat’

-0.3

-0.2

-0.1

0

0.1

0.2

-4 -3 -2 -1 0 1 2 3 4

pi_2

,t

K_t ( temperature)

Optimal investment in risk-security, model B, t=0.50

’x099.dat’

-0.3

-0.2

-0.1

0

0.1

0.2

-4 -3 -2 -1 0 1 2 3 4

pi_2

,t

K_t ( temperature)

Optimal investment in risk-security, model B, t=1. 0

’x199.dat’

SIMPLE MODELS FOR TRADING CLIMATE AND WEATHER RISK 10

farmer’s optimal investment into Y , model C

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

-4 -3 -2 -1 0 1 2 3 4

pi_2

,t

K_t ( temperature)

Optimal investment in risk-security, model C, t=0.25

’x049.dat’

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

-4 -3 -2 -1 0 1 2 3 4

pi_2

,t

K_t ( temperature)

Optimal investment in risk-security, model C, t=0.75

’x149.dat’

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

-4 -3 -2 -1 0 1 2 3 4

pi_2

,t

K_t ( temperature)

Optimal investment in risk-security, model C, t=0.50

’x099.dat’

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

-4 -3 -2 -1 0 1 2 3 4

pi_2

,t

K_t ( temperature)

Optimal investment in risk-security, model C, t=0.95

’x199.dat’

SIMPLE MODELS FOR TRADING CLIMATE AND WEATHER RISK 11

appreciation rate for trading with Y , models B and C

SIMPLE MODELS FOR TRADING CLIMATE AND WEATHER RISK 12

SourcesTitle page diag.: www.ems.psu.edu/WeatherWorld/features/witn98.html

Slide 1 diag.: www.cdc.noaa.gov/map/clim/sst_olr/old_sst/sst_8283_anim.shtml

SOI-Data: sensorlab.cnu.ac.kr/˜cybercat/soi.dat

Slide 2 diag.: www.jam-web.de/el-nino/k1.php

Slide 3 diag.: www.fao.org/fi/fifacts/plots/LAmer/lat2.asp

diag.: Nature, vol. 389, 16 October 1997, 715-717

Slide 4 Peru Rainfall Data: tao.atmos.washington.edu/data_sets/piura/

diag.:

Slide 6 diag.: B. Wang, A. Barcilon, Z. Fang, J. Atmos. Sciences, 56, 1999, 5-23

Slide 7 diag.: walrus.wr.usgs.gov/globalhydrate/images/pacific.gif

Slide 9 diag.: S. Slide 6

diag.: www.photolib.noaa.gov/fish/images/big/fish2167.jpg

diag.: www.ifad.org/photo/region/PL/PE.htm#, IFAD Photo by Franco Mattioli