Hedging of Weather Derivatives


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U.U.D.M. Project Report 2020:14
Hedging of Weather Derivatives
John Larsson
Examensarbete i matematik, 30 hp Handledare: Maciej Klimek Examinator: Erik Ekström Juni 2020
Department of Mathematics Uppsala University

Abstract
In this thesis, the methods of hedging incomes from crops done in (Hainaut, 2019) are applied to Swedish temperature and crop data. Some background on weather derivatives compared to other financial derivatives is given, as well as the main results needed from the article. In particular, the minimum variance hedging is applied to data from four different Swedish locations while hedging with future contracts on CAT index in Stockholm. The results suggest that this hedging strategy could successfully be utilized in Sweden if weather derivatives became available in Stockholm.

Contents

1 Introduction

3

2 Weather Derivatives

4

2.1 CAT Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2 Pricing Weather Derivatives . . . . . . . . . . . . . . . . . . . 6

2.3 Weather Insurance . . . . . . . . . . . . . . . . . . . . . . . . 7

3 Brownian Sheet

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3.1 Gaussian Processes . . . . . . . . . . . . . . . . . . . . . . . . 8

3.2 Brownian Sheet . . . . . . . . . . . . . . . . . . . . . . . . . . 9

4 A Model for Temperature

10

5 A Model for Crops

12

6 Hedging with CAT Futures

14

6.1 Future CAT Price . . . . . . . . . . . . . . . . . . . . . . . . 14

6.2 Minimum Variance Hedge . . . . . . . . . . . . . . . . . . . . 20

6.3 Maximum Utility Hedge . . . . . . . . . . . . . . . . . . . . . 21

7 Method

22

7.1 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

7.2 Estimating Parameters . . . . . . . . . . . . . . . . . . . . . . 24

7.3 Risk-Free Rate . . . . . . . . . . . . . . . . . . . . . . . . . . 25

7.4 Valuing Hedging Portfolio . . . . . . . . . . . . . . . . . . . . 26

8 Results

27

8.1 Temperature Model . . . . . . . . . . . . . . . . . . . . . . . . 27

8.2 Crop Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

8.3 Minimum Variance Hedge . . . . . . . . . . . . . . . . . . . . 32

8.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

8.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

References

38

Chapter 1
Introduction
Income risk is a natural consideration for any company; the risks associated with their revenue have different causes and as such have different difficulties. In this thesis we will consider weather risks to agriculture and ways to hedge against them. There are different ways to approach this problem and an appropriate solution may vary for different companies depending on their needs and preferences. These hedging strategies, as well as weather insurance, may become even more relevant in the future as there has been recent concern about climate change causing more extreme weather patterns. The models and hedging strategies used here are based on the results from (Hainaut, 2019) where the theory was presented and applied on agricultural data from Belgium using weather derivatives on temperature. The goal will be to investigate how the same models perform for data from Sweden. There are many reasons to expect differences in the results, for example Belgium and Sweden have different climates and other factors such as soil and sun hours may also affect how the models perform.
One of the problems with using weather derivatives for income hedging is that they are only available in some locations. This means that it is not enough to model the weather effects on the base income as there is a discrepancy between the weather there and the weather which governs the value of the weather derivative. In the models that will be presented later, this is modelled by incorporating a geographical component to the temperature and crop models, which allows for both pricing the derivative and predicting the outcomes of crops and derivatives at the same time.
The first chapter is about weather derivatives; what they do and some differences between using weather derivatives and purchasing a weather insurance. Following that is some mathematical background needed to understand the models. Then the models and theory from (Hainaut, 2019) that will be used are restated. In the last two sections, the methods and data sources are discussed as well as the results.
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Chapter 2
Weather Derivatives
A financial derivative is a contract, which has its value derived from underlying assets. Examples of assets could be stocks, interest rates or indices. In turn, an index can be calculated from a basket of stocks or commodities and even weather. An example of a weather index is Heating Degree Days (HDD), which is used to give an indication of the amount of energy used for heating houses. Obviously, heating is only required once the outdoor temperature is low enough and the energy used for heating increases as the outdoor temperature decreases. This is what HDD attempts to capture. If w(ti) denotes the average of the highest and lowest temperature in Celsius on a given day ti, w(ti) = w(tmax,i) + w(tmin,i) , then the HDD for a month
2 is defined as
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HDD = max(d − w(ti), 0),
i=1
where d is a base temperature under the assumption that heating commences once the outdoor temperature falls below the base temperature. For the USA, the base temperature is set to 18 degrees and for the European Union it is 15.5. Note that the days where the temperature is much lower than d will give a larger contribution to the total value of HDD, as is needed in order to capture accumulated energy usage. One might legitimately wonder who uses derivatives on the HDD index and the answer is that energy companies have been using them in the USA since 1996 (Alexandridis & Zapranis, 2012). Since the outdoor temperature dictates energy usage to some extent, their income is also exposed to the weather and derivatives on indices such as HDD can reduce that exposure. As an example, in winter an energy company may want to hedge against warm weather, since less energy would be used for heating. If they take a negative position in the HDD (by selling futures or buying put options on the HDD index) their portfolio would consist of the regular energy incomes which increases when the temperature is low
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and a derivative on the HDD index, which gives a higher payoff when the temperature is high, thus hedging the total income.
The weather derivative market in Europe is much smaller than in the USA (Hainaut, 2019) (Alexandridis & Zapranis, 2012), -the only European cities with weather derivatives traded on the Chicago Mercantile Exchange (CME) are currently London and Amsterdam- it is reasonable to expect the European market to grow in the future. In particular, the demand may increase if, as expected, global warming causes more volatile weather since companies with weather exposure will want to hedge their incomes. Due to this we will, as was also done in (Hainaut, 2019), use future contracts written on the CAT index in locations where this is not currently possible, but where it may become available in the future.
2.1 CAT Index
The weather index on which we will be considering weather derivatives is the Cumulative Average Temperature (CAT) index. In principle, the accumulation between time 0 and T could be calculated using an sum of daily average temperatures,
T
CAT(T ) = w¯(ti)
i=1
where w¯(ti) is the measured average temperature on a given day t. For Swedish temperature data, the Swedish Meteorological and Hydrological Institute (SMHI) provides estimations of average daily temperatures calculated with Ekholm-Modéns formula1. The estimation relies on a few different temperatures recorded during the day defined as follows: T07, T13, T19 are the temperature measurements at times 07:00, 13:00, 19:00 respectively; Tx and Tn are the maximum and minimum temperature recorded between 19:00 the previous day and 19:00 the day of interest. The average daily temperature Tm is then approximated by
Tm ≈ aT07 + bT13 + cT19 + dTx + eTn , 100
where a, b, c, d, e ∈ R are constants dependent on month and longitude. As an example for longitude 11◦ the table of constants is:
1Ekholm-Modéns formula from SMHI.
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a

b

c

d

e

Jan

33

15

32

10

10

Feb

31

18

31

10

10

Mar 32

22

26

10

10

Apr

25

19

27

10

19

May 21

18

25

10

25

Jun

21

18

25

10

26

Jul

19

18

27

10

26

Aug 18

23

22

10

27

Sep

26

24

23

10

17

Oct

31

19

30

10

10

Nov 30

16

34

10

10

Dec

34

15

31

10

10

Table 2.1: Coefficients used for Ekholm-Modéns formula at longitude 11◦ by SMHI. Source: SMHI.

For the financial derivatives on CAT index it is measured from the first day in a month until the last day in a month.

2.2 Pricing Weather Derivatives
The Black-Scholes approach to pricing financial contracts on stocks relies on constructing a replicating portfolio, i.e. a trading strategy that gives the same cash flows as the contract to be priced, then arguing that their prices must be equal as otherwise there would be an arbitrage opportunity. In contrast to contingent claims on stocks, the underlying process for weather derivatives cannot be traded -there is no such thing as trading in precipitation or CAT index. This situation would be described as an incomplete market (Björk, 2009) and the consequence is that the prices need not be unique. As we will see later, this non-uniqueness will be taken into consideration in the pricing approach.
There are several methods that can be used to price weather derivatives, a very comprehensive list of different approaches and descriptions can be found in the book Weather derivatives: modeling and pricing weather-related risk (Alexandridis & Zapranis, 2012). One method of determining a price for a weather derivative is to calculate the historical payoffs from previous years and derive a distribution for the payoffs (Historical Burn Analysis). However, this is not recommended in the book for different reasons; one be-

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ing that it assumes that the weather between different years is independent and identically distributed. A second method is to model an index directly, such as HDD or CAT, using a similar approach to what is commonly used for stocks. This could for example be modelling the index as a geometric Brownian motion, where the distribution of the payoffs is then inferred from the index model. A third possible method is to instead model the temperature process directly. There are many different ways of doing this and the one we will use is presented in Chapter 4.
2.3 Weather Insurance
Why use weather derivatives and not weather insurance? One difference is that insurance is typically used for rare events with a large impact such as drought, floods or extreme temperatures (Alexandridis & Zapranis, 2012). Weather derivatives can easily be used for more common deviations in temperature or precipitation for hedging, as was described at the beginning of Chapter 2. Weather derivatives can also be considered more flexible in since they can be of many different varieties, such as options, futures or compounded, which can be traded as long as the market is liquid.
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Chapter 3
Brownian Sheet

The models for temperature and crop growth rely on time changed Brownian sheets for their stochastic components. This chapter aims to develop some of the theory for understanding Brownian sheets. We begin by discussing Gaussian random variables.

3.1 Gaussian Processes

Definition 3.1.1. A d-dimensional random variable X ∈ Rd is a Gaussian random variable with mean µ and covariance matrix Σ if it has the probability density function

exp(− 1 (x − µ)TΣ−1(x − µ))

fX (x1, . . . , xd) =

2

,

(2π)d|Σ|

where |Σ| denotes the determinant.

Remark 3.1.2. Since the determinant of Σ appears in the denominator, a density function can only exist if Σ is invertible. Also, the covariance matrix is positive semidefinite since, using a standard definition of covariance matrix, for y ∈ R

yTΣy = yTE[(X − E[X])(X − E[X])T]y = E[yTZZTy] = E[yTZ(yTZ)T] ≥ 0

since yTZ(yZ)T ≥ 0.

Definition 3.1.3 (Gaussian Process). A random process (Xt)t ∈ T is a Gaussian Process if for any (t1, . . . , td) ∈ T d and d < ∞, (Xt1, Xt2, . . . Xtd)T is a
Gaussian random variable.

So far, the index set for the processes have been T ⊂ R but in principle there is no reason why the index must be one-dimensional. If instead, we

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Hedging of Weather Derivatives