© 2001 weather risk advisory. all rights reserved valuation of weather derivatives 13 june 2001

© 2001 Weather Risk Advisory. All rights reserved

Valuation of Weather Derivatives

13 June 2001


Temperature

About a year ago, there was a rumour that Coca-Cola the soft drink company, had plans for a vending machine, which varies the price for a drink according to outside temperature. The company denied the plans but we can see the logic behind the idea. One of the main causes of uncertainties in cash flow is the weather. Temperature in this case. In the hot summers of 1976 and 1995, the machine could have charged a fortune

A regulated power utility company must fix their price for electricity for a certain period. Prices can only be increased with permission of the regulators. Yet, power consumption will be low in warm winters. There is a source of uncertainty of incoming cash flow: temperature


Rain

Agriculture companies have been aware of weather risk for a long time. The major climatic factor which influences crop growth is rainfall

“Excess moisture continues to be a problem, particularly to later seed crops.”

“… but rainfall over the weekend brought haying to a standstill.”

“Excess moisture has taken its toll in some areas as crops in low lying lands have drowned out…”

“…heavy rains causing some localized crop damage”


Wind

One source of alternative energy is on the rise:

Wind farms are being built, not only in this country but also in Europe. Their exposure to lack of wind (and also an excess of wind) is obvious

No wind, no electricity, no money


Weather Derivatives


Example One: Description

Swap

• I shall pay you USD 5,000 for every degree Celsius the average temperature at Heathrow airport is below 18 on any day in the observation period (1 Nov 2001 - 31 Mar 2002)

• In exchange, you pay me a fixed amount

• Of course, these two payments are netted and the difference paid at the end of the contract period

• Netted pay-out is limited to USD 1,000,000


Example One: Description 2

Option

• As above, however, cash flow only occurs when the differential is positive (or negative)

• Plus an up-front premium payment

• Pay-outs limited to USD 1,000,000


Example One: Index

• Heating Degree Days (HDD)

• On each day of the observation period

1. Calculate the number of degrees the average temperature is below 18 degrees (or other reference temperature as defined in contract)

2. Sum them up (and maybe round according to contract details)

• Formula

n

kkTHDD

1

18


Index Calculation: Numerical Example

Date Max T Min T Avg T Daily HDD01-Nov-99 17.3 12.2 14.75 3.2502-Nov-99 13.8 9.9 11.85 6.1503-Nov-99 14.6 5.4 10.00 8.0004-Nov-99 17.2 11.7 14.45 3.5505-Nov-99 21.3 16.3 18.80 0.0006-Nov-99 16.6 12.9 14.75 3.25

… …29-Mar-00 7.1 3.7 5.40 12.6030-Mar-00 8.2 5 6.60 11.4031-Mar-00 10 5 7.50 10.50

=======1664.7


Example One: Trade

Swap Calculate Differential = Fixed swap level - HDD index • When differential is positive, receive USD 5,000 * differential • When negative, pay USD 5,000 * differential

Put OptionCalculate Differential = Fixed swap level - HDD index • When differential is positive, receive USD 5,000 * differential • When negative, no cash flow

Up to the value of the limit


Example One: Payout Formula

Swap

Put Option

limit,limit,minmax strikeHDD

limit,limit,minmax strikeHDD


Example One: Underlying

• Daily average temperature

or

• The index (HDD)

A model is needed for either the temperature process or the index


Example Two: Description

• Pay me USD 500,000 for every day in the observation period (1 Jul 2001 - 31 Aug 2001) on which rainfall is above 0.1 inch at Des Moines, Iowa, International Airport

• In exchange for a fixed up-front premium payment (call option)

• Pay-out limited to USD 2,500,000


Example Two: Index

• Count the number of days in the observation period on which the rainfall is above the specified threshold


Index Calculation: Numerical Example

Trigger 0.10

Date Rainfall Event Cnt1-Jul-00 0.00 02-Jul-00 0.00 03-Jul-00 0.00 04-Jul-00 0.74 15-Jul-00 0.41 16-Jul-00 0.00 07-Jul-00 0.61 18-Jul-00 0.07 09-Jul-00 0.00 0

10-Jul-00 0.00 0=======

3


Example Two: Trade

Call OptionCall option with strike 0 and tick size USD 5,000

• Pay an up-front premium payment• Calculate Differential = rain index – strike

1. When differential is positive, receive USD 5,000 * differential 2. When differential is zero, no cash flow

• Maximum pay-out USD 2,500,000


Example Two: Underlying

• Daily rainfall

or

• The index

Again, a model is needed for either rainfall process or the index


Exotic Structures

Further complications and additions

• Dual trigger event: temperature and rain

• Multi-station trades: baskets or multi-station events

• Event only counted when it occurs on two or three consecutive days


Models


Burn Rate Analysis

• Clean, reconstruct and de-trend the data

• Calculate the index (HDD, etc.) from historical annual observations

• Calculate the resulting trade pay-off for every year

• Calculate the average of the trade pay-offs

• Discount back from settlement date to today

• Add risk premium


Data Modification

• Clean: fill in missing data and correct errors

• Reconstruct: modify data to account for change of equipment, change of location

• De-trend: remove trends due to global warming or urbanisation


Burn Rate Analysis : Numerical Example

Strike 1700

Season Index Swap Option1990/1991 1846 -146 01991/1992 1781 -81 01992/1993 1735 -35 01993/1994 1766 -66 01994/1995 1572 128 1281995/1996 1948 -248 01996/1997 1807 -107 01997/1998 1538 162 1621998/1999 1675 25 251999/2000 1665 35 35

=======================Mean 1733.3 -33.3 35


Burn Rate Analysis : Pros and Cons

Positive

+ Simple to implement

+ Easy to understand

+ Suitable for portfolio valuation

Negative

- Low probability events ignored or unreliably estimated

- General problem is that in such a low sample, a single observation can strongly influence the option value

- Unclear selection and calculation of risk parameters


Stochastic Models for Weather Paths

• Choose a stochastic model for daily observations

• Estimate the model parameters using weather data

• Simulate a weather path

• Calculate index and option value for this path (=simulated observation)

• Repeat many times and calculate mean

• Discount back to today


Stochastic Models: Examples

• Auto-Regression (AR): use the weighted sum of previous days’ temperatures for the estimation of next day’s temperature

• Mean-reverting diffusion (MR): a Markov model with pull-back to the mean

• Cao-Wei: a variation of AR

• Bob Dischel: a variation of MR


Estimators

• Moment estimator

find parameters based on observed and model moments,

e.g. mean and standard deviation

• Maximum-likelihood estimator

find parameters which are most likely, based on observed data


Example: Auto-regression

Equation nnTnTnnT 21 21


Example: Mean-reverting Diffusion

• Equation

or

• alternatively

tdWdttTtadtttdT

tdWdttTtatdT


Stochastic Models: Pros and Cons

Positive

+ One (model) fits all (indexes)

Positive/negative

± Portfolio risk management possible but may have associated problems (multi-location correlation)

Negative

- Slow or inaccurate (in particular for low-event contracts)

- Harder to implement


Weather Forecasts


Long-range Weather Forecasts

• So far, the approach has been to base an estimation of the future on past observations

• With weather, can we do better and use forecasts?

• And how could we integrate them into pricing models?


Ensemble Forecasting - Probability Forecasts

• Ensemble forecasting has been developed over the past decade and has lead to a large increase in the accuracy and usability of medium-range forecasts

• The technique is now being applied to longer range forecasts

• Ensemble forecasting uses the sensitivity of forecasts to the initial conditions to provide probability forecasts

Ensemble Forecasting

By perturbating the initial conditions of a forecast and re-running it multiple times, a range of forecasts for the same time period are produced. These can be used to produce probability distributions of possible outcomes.


Climate Prediction Center Ensemble Forecasts

• They have been found to be reliable in El Nino and La Nina years for predicting the US climate, but have a tendency to be too cold

• CPC predicts whether temperature and rainfall will be above, below or near normal for 30 and 90 day periods

• But El Nino does not effect Europe so this approach cannot be used


Forecast Format

• Forecasts are expressed as the probability anomaly of the observation

• Three classes - above - near - or below normal

• Forecast probability anomaly is the difference between the actual forecast probability of the verifying observation falling in a given category and its climatological value of 33.3%

• Mean temperature and total precipitation


Forecast Format: Example

Below Near AboveCL 33.3% 33.3% 33.3%A 5 28.3% 33.3% 38.3%A 10 23.3% 33.3% 43.3%B 5 38.3% 33.3% 28.3%B 10 43.3% 33.3% 23.3%


Integration of Forecasts into Pricing Models

• Adjust model parameters so that model probability distribution matches forecast

• Read our paper for more details


Summary and Conclusions


Theoretical Considerations of Modelling Approaches

• Arbitrage-free pricing approach (risk-neutral valuation) as used in standard derivatives industry

price = risk-neutral value (+profit margin)

• Actuarial approach

price = statistical value + risk premium (+ profit margin)

Actuarial approach currently used in weather derivatives industry and insurance

• Shadow-price approach


Summary

• A market for weather derivatives is being established

• Typical examples of currently traded products

• Untradable weather risk will probably remain, in particular extremes. They will remain insurance cases

• Two modelling approaches, both have their strengths and weaknesses

• Forecasts

• Actuarial approach


Two Challenges Ahead

• Transition to arbitrage-free approach (when markets become liquid, take future contracts and back out a risk-free temperature curve, take options and work out a volatility surface)

• Portfolio management of mixed books (e.g. power and weather), to give you an idea of that challenge, even portfolio management of weather books is not fully solved


References


References

• M. Cao and Wei, J.: Equilibrium Valuation of Weather Derivatives, Working Paper.

• B. Dischel: At Last: A Model for Weather Risk, EPRM Mar 1999.

• G. Considine: Introduction to Weather Derivatives, Working Paper.

• I. Nelken: Weather Derivatives – Pricing and Hedging, Working Paper.

• L. Zeng: Weather Derivatives and Weather Insurance: Bulletin of the American Meteorological Society, Vol. 81, No.9, Sep 2000.


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© 2001 weather risk advisory. all rights reserved valuation of weather derivatives 13 june 2001

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