Temperature derivatives
Added value of using weather derivatives as a hedge for a general insurance company
August 25th, 2014 Faculty of Economics and Business, at University of Amsterdam,
by Tessa van Staten. Student nr: 5733464 Supervisor: Prof. Dr. H.P.Boswijk
Second reader: Prof. Dr. R. Kaas Study Program: Master of Science in Econometrics Specialization: Financial econometrics
Abstract
The main question of this thesis is: In what way can a Dutch insurer use weather options to hedge its claims? While trying to find the answer to this question one of the first insights was that in terms of weather derivatives temperature derivatives are the most promising. Temperature derivatives are contracts just like any other derivative where the underlying accumulates the temperature over a given period. These options are currently traded on the Chicago Mercantile Exchange. For this thesis four
different pricing strategies were examined: the actuarial approach, indifference pricing, general equilibrium pricing and the martingale approach. The martingale approach has the most potential to yield appropriate outcomes.
In order to model the underlying weather process of a temperature derivative the daily average temperature is split in two components: a seasonal component (estimated with a Fourier- truncated series) and a non-seasonal component (estimated with an 𝐴𝑅(3) model with heteroskedastic errors). Together with a claim generating process for wind storm losses in the Netherland the resulting process is used to simulate losses. The temperature derivative is used to hedge these losses. Comparing the temperature derivative with more traditional types of reinsurance leads to the following conclusion.
The temperature derivative underperforms both the excess of loss contract as well as the quota share contract. Even when very rigid assumptions are made the temperature derivative fails to outperform one of the more traditional reinsurance treaties although it is better than no reinsurance. This leads to a clear recommendation to the
marketplaces around the world to develop a more hybrid temperature based product which allows insurance companies to hedge their tail risks better and allows investors to diversify their portfolios further.
Acknowledgements
I am thankful to Professor Peter Boswijk of The University of Amsterdam. He gave me the chance to chase the subject I felt passionate about. Also, I would like to thank him for the valuable support during the writing process.
Advice given by Steve Schmidt has been of great help in exploring the fields of weather based reinsurance and their influence on an insurer. I would like to
acknowledge the support provided by my family and Thomas Akkerhuis during the preparation of my thesis.
My special thanks are extended to the staff of Delta Lloyd N.V for enabling me to visit them during the daily operations and setting up meetings with their broker and other contacts. Assistance by Tjeerd Degenaar was greatly appreciated.
Content
Temperature derivatives ... 1 Abstract ... 2 Acknowledgements ... 3 Content ... 4 Introduction ... 7Part I. Temperature derivatives and pricing mechanisms ... 9
Chapter 1. The temperature derivative ... 9
1.1. A futures contract based on temperature ... 9
1.2. An option on a temperature based futures contract ... 12
Chapter 2. Different pricing mechanisms ... 13
2.1 The actuarial approach or burn-rate model ... 13
2.2 Indifference pricing ... 14
2.3 General equilibrium pricing ... 15
2.4 The equivalent martingale approach ... 17
2.5 Comparison between the four approaches ... 19
2.5.1 Assumptions ... 19
2.5.2 Fit with the underlying (the weather process) ... 20
2.5.3 Outcomes ... 21
Part II. Temperature dynamics ... 23
Chapter 3. Literature review and selection criteria ... 23
Chapter 4. Modelling the temperature data ... 26
4.1 Description of the temperature data ... 26
4.2 Removing seasonality in the mean ... 27
4.2.1 Estimation results ... 28
4.2.2. Back test with non-parametric estimate ... 29
4.2.2.3 Parametric v.s. non-parametric estimate ... 29
4.3 Modeling the non-seasonal component ... 31
4.3.1 The dependent variable ... 31
4.3.2 Is the dependent variable stationary? ... 31
4.3.3 Estimation of the AR(3) model ... 32
4.3.5 Test for normality ... 35
Chapter 5. A continuous-time autoregressive process ... 36
5.1 The explicit form of Xt ... 37
5.2 Stability conditions of Xt ... 38
5.3 The equivalent martingale measure ... 38
Part III. Insurance claims ... 40
Chapter 6. Specification of a loss ... 40
6.1 Relation between temperature and the occurrence of a storm ... 40
6.2 Relation between temperature and the damage given a storm occurs ... 42
Chapter 7. The insurer ... 45
7.1 The minimal variance hedge portfolio with a temperature derivative ... 45
7.2 The excess of loss contract ... 46
7.3 The quota share contract ... 46
Part IV. Simulation, results and conclusions ... 47
Chapter 8. The simulation ... 47
8.1 Simulation of the weather process ... 47
8.2 Simulation of losses ... 49
8.3 Simulation of the insurance contact ... 53
Chapter 9. Results ... 56
9.1 Value at risk ... 56
9.2 Standard deviation ... 59
9.3 Mean and median ... 61
Chapter 10. Conclusions ... 62
Bibliography ... 65
Appendix A- Pricing a CAT futures contract ... 70
A. 1 The Qθt-dynamics of FCATt,τ1,τ2 ... 72
A. 2 A call option on a CAT future... 73
Appendix B- Pricing a CDD futures contract ... 75
B. 1 The Qθt-dynamics of FCDDt,τ1,τ2 ... 76
B. 2 Pricing a call option on a CDD-future ... 77
Appendix C- STATA code used in part II ... 79
Appendix D- From Fourier –truncated series to OLS regression and visa versa and estimation results ... 85
Appendix E – the analytical link between an AR(3) and its continuous version... 87 Appendix F - R code used in part III... 89 Appendix G – List of interviews ... 96
Introduction
The weather influences the economy in many ways: the agricultural, energy and recreational sector have their preference when it comes to the weather. Larsen (2012, p.4) estimates that 16.2% of the aggregate U.S. economy is sensitive to weather conditions while Dutton (2002, p.1306) indicate that “one-third of private industry activities is sensitive to weather”. Weather derivatives could add value by reducing the exposure to weather sensitivity for vulnerable sectors in an economy and mitigate the risk to investors looking for new investment opportunities. One opportunity where weather derivatives might provide protection would be in the insurance sector,
weather conditions determine, at least partially, the claims an insurer has to pay on its non-life insurances. This thesis is aimed at finding a hedging strategy for the weather risk an insurer is taking on its non-life insurances. This leads to the following research question:
Research question: In what way can a Dutch insurer use weather options to hedge its claims?
In order to answer the research question five sub questions are formulated: 1) What is a weather option?
2) How can a weather option be priced?
3) Which model can describe the underlying weather process in the Netherlands the best?
4) How is the weather process related to the claims process of an insurer? 5) What is the hedge efficiency of using a weather option compared to more
traditional reinsurance treaties to hedge the insurance claims?
This thesis is structured as follows. Part 1 contains all details with regard to the temperature derivative. Chapter 1 gives an overview of existing temperature derivatives including payoff schemes. Chapter 2 gives a comparison between the different pricing strategies which could be used in order to find the market price. Four different approaches are investigated: the actuarial approach, indifference pricing, general equilibrium pricing and the equivalent martingale measure approach.
Part 2 outlines the data, methods and modeling choices made in order to estimate the temperature model. Chapter 3 gives a literature overview. The literature review is a starting point from where two selection criteria are formulated. These selection criteria are used in Chapter four to select the temperature model. In Chapter 4 all the modeling choices regarding the temperature model are discussed. In order to capture all relevant characteristics of a temperature process the modeling process is split in two components: a seasonal component and a non-seasonal component. The seasonal component serves the purpose to capture all recurrent temperature phenomena and consists of: constant, trend over the years to capture global warming and a sine function to capture the variation over the year. The non-seasonal component captures all other variation including a seasonally dependent variance. In Chapter 5 the CAR transformation is introduced to transform the daily estimates to a continuous time process, which could be used to price weather derivatives based on the equivalent martingale measure.
Part 3 outlines the data generating process of the claims and the ways an insurer can react to the claims. Chapter 6 will be concerned with estimating the data generating process of the claims surge due to a winter storm. Whereas Chapter 7 is concerned with how an insurer can protect itself against this claims surge. The insurer has many options in this thesis the following options are considered are: status quo, hedge the portfolio with temperature options, an excess of loss treaty or a quota share contract.
Lastly, part 4 will combine all previous parts: the weather derivative, the temperature process and the claims arising due to a winter storm. All these elements are brought together in a simulation. A description of the simulation process can be found in Chapter 8. An analysis of the effects of a temperature derivative on the portfolio of an insurer is given in Chapter 9. The conclusions and a recommendation can be found in Chapter 10.
Part I. Temperature derivatives and pricing mechanisms
Chapter 1. The temperature derivative
In this chapter an overview of the weather derivatives based on temperature is given as well as the characteristics of the options on the weather derivatives. Temperature derivatives are considered because these contracts are traded on the Chicago
Mercantile Exchange (CME). Other weather contracts, for example on hurricanes, frost, snowfall or rainfall, are also traded on this exchange but aren’t based on the Dutch market (CME, 2014). In order to practically do something with a weather option a Dutch insurer is forced to use a temperature derivative.
1.1. A futures contract based on temperature
A futures contract based on temperature measures the daily temperature, the CME defines temperature as follows.
Definition 1 (temperature) let 𝑇𝑖,𝑚𝑎𝑥 (𝑇𝑖,𝑚𝑖𝑛) denote the maximum (minimum) temperature measured at a height of 1,5m in degrees Celsius (℃) on day i. The temperature for day i is defined as: 𝑇𝑖 =1
2(𝑇𝑖,𝑚𝑎𝑥+ 𝑇𝑖,𝑚𝑖𝑛).
A futures contract based on temperature requires a underlying accumulated method of the temperature. There are three different accumulation indices for the futures
contracts traded on the CME exchange: HDD, CDD and CAT (CME, 2013). HDD stands for Heating Degree Days and is a measure for the demand for heat. CAT stands for Cumulative Average Temperature and is a measure for total temperature. CDD stands for Cooling Degree Days and is a measure for the demand for cooling.
Definition 2 (HDD index) The HDD indices accumulates all temperatures below a
certain threshold c, 18°C in case of the CME, over a time period, say, 𝜏1to 𝜏2, where 𝜏2 > 𝜏1and the daily temperature is defined as in Definition 1. The HDD has the following mathematical representation: 𝐻𝐷𝐷(𝜏1, 𝜏2) = ∑𝜏2 𝑚𝑎𝑥(𝑐 − 𝑇𝑖, 0)
Definition 3 (CAT index) The CDD indices accumulates all temperatures over a time
period, say, day 𝜏1to 𝜏2, where 𝜏2 > 𝜏1. The daily temperature is defined as in
Definition 1. The CAT has the following mathematical representation: 𝐶𝐴𝑇(𝜏1, 𝜏2) = ∑𝜏2 𝑇𝑖
𝑖=𝜏1 .
Definition 4 (CDD index) The CDD indices accumulates all temperatures above a
certain threshold c, 18°C in the case of CME, over a time period, say, 𝜏1to 𝜏2, where 𝜏2 > 𝜏1and the daily temperature is defined as in Definition 1. The CDD has the
following mathematical representation: 𝐶𝐷𝐷(𝜏1, 𝜏2) = ∑𝜏𝑖=𝜏2 1𝑚𝑎𝑥(𝑇𝑖− 𝑐, 0).
The CAT and HDD index alternate during the year, the HDD index is used from October to April and the CAT index is used from April to October. The CDD index is not traded on the market due to the HDD-CDD parity. Which states the following: 𝐶𝐷𝐷(𝜏1, 𝜏2) − 𝐻𝐷𝐷(𝜏1, 𝜏2) = 𝐶𝐴𝑇(𝜏1, 𝜏2) − 𝑐(𝜏1, 𝜏2)
Where 𝑐(𝜏1, 𝜏2) = ∑𝜏2 𝑐
𝑖=𝜏1 = 𝑐 × (𝜏2− 𝜏1)
Proof: for any given day t: 𝑚𝑎𝑥(𝑇𝑡− 𝑐, 0) + 𝑚𝑎𝑥(𝑐 − 𝑇𝑖, 0) = 𝑇𝑖− 𝑐. By
definitions 2, 3 and 4 we recognize the accumulation for the CDD, HDD and CAT index. And since this holds for every day in the accumulation period it surely holds for the complete measurement period.
There are two types of maturities traded: a monthly futures contract and a seasonal strip. The seasonal strip consists of at least two, and at most seven, consecutive calendar months in the appropriate index season (CME, 2013). A seasonal strip is a combination of monthly futures. When the seasonal strip contract and the monthly futures contract accumulate the same index over an identical measurement period their payoff is the equal, so by the law of one price their price should be equal.
For a given index, defined above, a futures contract based on temperature obliges the holder to pay or receive: 𝑡𝑖𝑐𝑘 𝑠𝑖𝑧𝑒 ∗ (𝐼𝑁𝐷𝐸𝑋 − 𝑆𝜏0) when a party is on the long side of the market or 𝑡𝑖𝑐𝑘 𝑠𝑖𝑧𝑒 ∗ (𝑆𝜏0− 𝐼𝑁𝐷𝐸𝑋) when a party is on the short side of the market. Where the tick size is predetermined by the both parties and specifies the euro amount for every index point the futures contract ends above or below the strike price.
𝑆𝜏0 is the strike price settle upon on 𝜏0 < 𝜏2 such that the price of this futures contract equal zero at 𝑡 = 0. Since by definition setting up a futures contract does not cost anything (Hull, 2012, p. 22).
An instructive way to look at this futures contract is via an example. All data have been retrieved via First Enercast Financial (2014). If an investor would have bought a HDD futures contract on 15-03-2013 for the month April, he would face the
following bid-ask prices:
Bid Ask Last
211 253 232
The last futures contract created had a strike of 232. Note that the bid-ask spread is very high with 42 HDD index points. Let’s assume he would have agreed on this HDD futures contract for a strike price of 253 and a tick-size of €1.000. The possible payoffs are shown in Figure 1.
Figure 1: payoff of a HDD futures contract position, both the long and short side When the HDD index is exactly equal to the strike price the payoff is equal to zero. What this figure also shows is that the long and short position are each other’s mirror image. When one partly loses, the other wins and the other way around. At 2nd of
-100.000 € -80.000 € -60.000 € -40.000 € -20.000 € 0 € 20.000 € 40.000 € 60.000 € 80.000 € 100.000 € 200 208 216 224 232 240 248 256 264 272 280 288 296 304 312 320 328 Pa yoff HDD index Long Short
May 2013 the HDD index for April was to be settled at 302 HDD points. The payoff for this contract would have been €49.000 with an exposure on the long side and -€49.000 with an exposure on the short size.
1.2. An option on a temperature based futures contract
Next to futures, the CME also organizes a market for call and put options on these futures. A call option gives the owner the right, but not the obligation, to buy the underlying futures contract for a pre-specified price at a certain date. Whereas a put option gives the owner the owner the right, but not the obligation, to sell the
underlying futures contract for a pre-specified price at a certain date (Hull, 2012, p.7-9). The settling price in the contract is known as the exercise price or the strike price, the settling date in a contract is called expiration date or maturity.
A weather option can be formulated with the following characteristics:
The contract type (Call/put)
The contract position (long/short)
The contract period
The underlying index
The official weather station from which the weather data are used
The strike level
The tick size
The maximum payoff, if there is a maximum
On the current moment 𝑡 an option on a futures contract with predetermined parameters selected from the list above gives the holder the right, but not the
obligation, to execute the contract at the date of maturity, 𝜏. The payoff of the contract is specified as: 𝑚𝑎𝑥(𝐹(𝜏, 𝜏1, 𝜏2) − 𝐾, 0). Where 𝜏 ≤ 𝜏1 < 𝜏2. 𝐹(𝜏, 𝜏1, 𝜏2) is the futures contract with specification discussed above. Options on the CME exchange are settled in cash (Chicago Mercantile Exchange, 2013).
Chapter 2. Different pricing mechanisms
Hull (2012) and Etheridge (2002) find the price of a financial claim by using the no-arbitrage argument. When an no-arbitrage exists, an investor can gain a profit without taking any risks. When the markets are efficient, all arbitrage opportunities will be exploited and the market for tradable assets will become arbitrage-free. So, when the price is without arbitrage opportunities, this must be price of the derivative. However, this is only true when the portfolio can be rebalanced at any given point in time. This requires that the assets and its underlying are tradable assets. The weather is a non-tradable asset. Therefore, the market for weather futures is said to be incomplete. In other words, there is no possibility that a portfolio can be created consisting of the underlying and a risk-free asset in such a way that the claim is replicated.
In order to overcome this obstacle four different pricing principles are looked into, these are: the actuarial approach, indifference pricing, general equilibrium pricing and the equivalent martingale approach (Härdle et al., 2012). The methodology, strengths and weaknesses and application to weather derivatives will be elaborated. On order to conclude which of these methods is most preferred the approached are compared with each other on assumptions, fit with the underlying and output performance.
2.1 The actuarial approach or burn-rate model
The actuarial approach is also called a burn rate model. Under the actuarial approach the price of the option is the solution of the expectation under the real physical measure discounted at the riskless rate (Bladt et al., 1998, p.66). In other words, the derivative price is equal to the present value of its expected payoff at expiry, where the expectation is calculated on the basis of historical data (Richards , et al., 2004, p.1008). The value of the option is calculated by: 1) discount deterministic futures contract payoffs created from their expected returns or use the data of the historical realizations; 2) calculate the difference on the actual price and the strike price (in present values) when exercising.
The main advantage of this approach is that it is easy to use. The technique is simple and doesn’t involve a lot of calculations. A disadvantage is that the current
realizations of the weather process are not used as input factor to price the weather derivative. Therefore, there is no way to update the probabilities during an adverse weather event (Richards et al., 2004, p.1008).
2.2 Indifference pricing
Applying this approach to derivatives with incomplete markets was originally proposed by Henderson (2002). The utility indifference price 𝒑𝒃is equal to the price where the investor is indifferent (in the sense that her expected utility is equal whether an investor makes the trade or not) between paying nothing and not having the claim or paying 𝒑𝒃 and receiving the claim, 𝑪
𝑻 on time 𝑻. The number of units bought is
equal to: 𝒌, 𝒌 > 𝟎. We assume that the investor has an initial wealth 𝒙 and zero endowment of the claim. In order to find the indifference price the following equation must be solved: 𝑽(𝒙 − 𝒑𝒃(𝒌), 𝒌) = 𝑽(𝒙, 𝟎). The expectation is made with respect to the minimal martingale measure: the measure under which the discounted trade asset is a martingale.
Henderson (2004, p.12) extended the framework and added a non-traded asset with a constant interest rate. An exponential Brownian motion drives the process of the underlying. Xu et al. (2008, p. 992) compare the price of the option when the
underlying utility function is an exponential function or a power utility function. Their research is aimed at using investors preferences to calculate the price boundaries. Using price boundaries from buyers and sellers in the market they make an
assessment whether a transaction is likely to occur. As robustness check they compare the price boundaries under different initial values for wealth and relative risk
aversion. Musiela and Zaiphopoulou (2004, p.400) examine a multi-period model by a similar framework through a representation of prices as iterative output of a
2.3 General equilibrium pricing
The third principle, the general equilibrium theory. Uses the Lucas’ model of 1978 to model the economy and the temperature as two correlated processes. The set-up of this economy will now be discussed (Mehra and Prescott, 1985 and Bakshi and Chen, 1997).
Consider an economy inhabited by a large number of identical individual consumers with an infinite lifetime horizon. There are three types of assets: a stock, a risk-free bond and a finite number of other contingent claims. The economy behaves according to the following assumptions: 1) the aggregate output of the economy (𝑦𝑡) equals the
dividends (𝛿𝑡) paid by the stock; 2) dividends cannot be stored; 3) the stocks can be sold on for a price 𝑝𝑡, however, when a stock is sold the sale occurs after the existing owner has received the dividends of that period; 4) in a given year each stock
produces exactly the same amount of dividends however over the years this amount fluctuates as a result of the weather; 5) when a consumer is born consumers are
endowed with one share and no risk-free bonds or contingent claims; 6) the net supply of all contingent claims and risk-free bond is zero.
Under these assumptions the decision problem of the consumer to choose an optimal trading strategy to maximize this expected lifetime utility.
In the model of Cao and Wei (2002, p.1072) the agent’s period t utility function is assumed to be of Constant-Relative-Risk-Adversion and can be described as:
𝑈(𝑐𝑡, 𝑡) =𝑐𝑡 𝛾+1
𝛾+1 (2)
with the risk parameter 𝛾 ∈ [−1,0]
In order to model the dividend process we need to make some assumptions about its evolution. In accordance with the original Lucas model and the model of Cao and Wei (2004, p.1073) the growth of the aggregate dividend, 𝛿𝑡, evolves according to the following Markov process:
ln 𝛿𝑡 = 𝛼 + 𝜇 ln 𝛿𝑡−1+ 𝜐𝑡, ∀𝜇 ≤ 1 (3)
where 1 − 𝜇 measures the speed of mean reversion and the error term is dependent on the weather process. The total resources available for consumption by consumer i in period t are the sum of production received from the stocks owned (𝑑𝑡𝑠𝑡), the payoffs
from the risk-free bond (𝑏𝑡) and the contingent claims(𝑞𝑡𝑥⃗⃗⃗ ) plus the potential 𝑡
proceeds if the consumer would sell his shares (𝑝𝑡𝑠𝑡). The consumer uses these recourse for consumption (𝑐𝑡)and purchase of shares (𝑝𝑡𝑠𝑡+1), risk-free bond (𝑟𝑡𝑏𝑡+1) and contingent claims for the next period (𝐹𝑡𝑥⃗⃗⃗⃗⃗⃗⃗⃗ ). Thus we can write the 𝑡+1
budget constraint for an agent as: 𝑐𝑡+ 𝑝𝑡𝑠𝑡+1+ 𝑟𝑡𝑏𝑡+1+ 𝐹𝑡𝑥⃗⃗⃗⃗⃗⃗⃗⃗ = (𝑑𝑡+1 𝑡+ 𝑝𝑡)𝑠𝑡+ 𝑏𝑡+ 𝑞𝑡𝑥⃗⃗⃗ 𝑡
Our representative consumer solves the following problem: v(𝑚𝑡) = max{𝑐𝑡} 𝑡=0 𝑡=∞E𝑡(∑ 𝑒−𝜌𝑡 𝑐𝑡𝛾+1 𝛾 + 1 ∞ 𝑡=0 ) s.t.𝑐𝑡+ 𝑝𝑡𝑠𝑡+1+ 𝑟𝑡𝑏𝑡+1+ 𝐹𝑡⃗⃗⃗⃗⃗⃗⃗⃗ − (𝑑𝑥𝑡+1 𝑡+ 𝑝𝑡)𝑠𝑡− 𝑏𝑡− 𝑞𝑡𝑥⃗⃗⃗ = 0 𝑡
where v(𝑚𝑡) is the value function.
Solving the first order conditions yields that the time t price of a contingent claim is equal to: 𝑋(𝑡, 𝑇) = 1
𝑈𝑐(𝛿𝑡,𝑡)𝐸𝑡[𝑈𝑐(𝛿𝑇, 𝑇) ∗ 𝑞𝑇]. Where 𝑈𝑐(𝛿𝑡, 𝑡) is the first derivative of the period t utility function with respect to consumption and 𝑞𝑇 the payoff at a
future time T.
Consider a futures contract with tick size €1 and a strike price of K like Cao and Wei (1999, p.1074). The underlying index 𝐻𝐷𝐷(𝜏1, 𝜏2). By applying the general pricing condition and the utility function given above, the value of the HDD contract on time
t and strike price K, 𝐻𝐷𝐷(𝑡, 𝜏1, 𝜏2, 𝐾), is given by: 𝐹𝐻𝐷𝐷(𝑡,𝜏1,𝜏2,𝐾)= 𝐸𝑡[𝑈𝑐(𝛿𝜏2, 𝜏2) 𝑈𝑐(𝛿𝑡, 𝑡) ∗ (𝐻𝐷𝐷(𝜏1, 𝜏2) − 𝐾)] = 𝑒−𝜌(𝜏2−𝑡)𝐸 𝑡[ 𝛿𝜏2𝛾 𝛿𝑡𝛾 ∗ (𝐻𝐷𝐷(𝜏1, 𝜏2) − 𝐾)]
Where 𝑒−𝜌(𝜏2−𝑡) is the discount factor. Now consider an European option written on 𝐻𝐷𝐷(𝜏1, 𝜏2) with maturity 𝜏2 and strike price X. Than by applying the general
pricing condition and the utility function given above, the value of the call option on the HDD is given by:
𝐶𝐻𝐷𝐷(𝑡,𝜏1,𝜏2,𝐾) = 𝐸𝑡[𝑈𝑐(𝛿𝜏2, 𝜏2) 𝑈𝑐(𝛿𝑡, 𝑡) ∗ max{𝐻𝐷𝐷(𝜏1, 𝜏2) − 𝑋, 0}] = 𝑒−𝜌(𝜏2−𝑡)∗ 𝛿 𝑡 −𝛾 𝐸𝑡[𝛿𝜏𝛾2 ∗ max{𝐻𝐷𝐷(𝜏1, 𝜏2) − 𝑋, 0}]
An extension of this framework is to change the dividend process from a Markov process to any other process. For example a Mean- Reverting Brownian Motion with
lognormal Jumps and an ARCH specification for the weather process (Richards, 2004, p.1008).
2.4 The equivalent martingale approach
When markets are incomplete there are many price where arbitrage opportunities are eliminated, in fact a whole range of opportunities exists (Etheridge, p.19). In order to select one arbitrage price form this whole range a new parameter is introduced, the market price of risk 𝜃𝑡.This market price of risk can be fixed from fitting the
theoretical values of a futures contract on the historical realizations. By doing so, an equivalent martingale measure, ℚ𝜽𝒕, is developed and the weather based futures contract can be uniquely priced (Härdle and Cabrera, 2012, p.60). The index 𝑡 is added to the market price of risk because the it could be time dependent. The variable can be regarded as a premium that an investor requires for bearing more risk. This risk can arise from the incompleteness of the market, for example Benth and Ekeland (2003).
Now we can start with pricing the temperature based futures contract. The derivations are based on a paper by Benth and Ŝaltytė (2007). We refer to the appendix for more details. Let (Ω, ℱ, 𝑃) be a probability space with a filtration (ℱ𝑡)0≤𝑡≤𝑡𝑚𝑎𝑥, where 𝑡𝑚𝑎𝑥 is the time needed to cover all times of interest in the market. The total payoff of a futures contract is equal to: 𝑌(𝑇𝑡) − 𝐹(𝑡, 𝜏1, 𝜏2). Where 𝑌(𝑇𝑡) is the payoff of the
temperature index (CAT, HDD or CDD) and 𝐹(𝑡, 𝜏1, 𝜏2) is the futures price for 𝑡 ≤ 𝜏1 ≤ 𝜏2. The price of such a contract is zero when the contract is agreed upon so by
convention: 0 = 𝑒−𝑟(𝜏2−𝑡)𝐸ℚ𝜃𝑡[𝑌 − 𝐹(𝑡, 𝜏1, 𝜏2)|ℱ𝑡]. Here ℚ𝜃
𝑡 is the equivalent martingale measure and 𝑟 the risk free rate. Using that 𝐹(𝑡, 𝜏1, 𝜏2) is ℱ𝑡-measureable
we find that 𝐹(𝑡, 𝜏1, 𝜏2) = 𝐸ℚ𝜃𝑡[𝑌|ℱ𝑡]. With this technique we could derive the solutions for the CAT, CDD and HDD accumulation index as well as the prices of the futures contract and the options based on the futures contracts. In order to capture the essence of equivalent martingale pricing the main results for the futures contract for a CAT index as well as this option are given below. The other derivations can be found in Appendix A and B.
The payoff of the CAT index is an accumulation of the temperature so the payoff of a CAT futures contract is equal to: ∫ 𝑇(𝑠)𝑑𝑠𝜏𝜏2
1 and in order to find the price for the CAT futures contract at 0 ≤ 𝑡 ≤ 𝜏1 ≤ 𝜏2 we need to find the expectation of 𝐹𝐶𝐴𝑇(𝑡, 𝜏1, 𝜏2) = 𝐸ℚ𝜃𝑡[∫ 𝑇(𝑠)𝑑𝑠
𝜏2
𝜏1 | ℱ𝑡].
With some intermediate steps shown in Appendix A, we show that price of a futures contract on the CAT index is equal to:
𝐹𝐶𝐴𝑇(𝑡, 𝜏1, 𝜏2) = ∫ Λ𝑠𝑑𝑠 𝜏2 𝜏1 + 𝑎(𝑡, 𝜏1, 𝜏2)𝑿𝑡+ ∫ 𝜎𝑢𝜃𝑢𝑎(𝑢, 𝜏1, 𝜏2)𝑒𝑝𝑑𝑢 𝜏1 𝑡 + ∫ 𝜎𝑢𝜃𝑢𝑒1′𝐴−1(𝑒𝐴(𝜏2−𝑢)− 𝐼 𝑝)𝑒𝑝𝑑𝑢 𝜏2 𝜏1
We see that the price for the futures contract based on a CAT index consists of three components: the aggregated mean temperature (Λ𝑠) over the measurement period, a reversion term (𝑎(𝑡, 𝜏1, 𝜏2)𝑿𝑡) that is dependent on today’s temperature and all lagged values up to 𝑝, as 𝑿𝑡 is a vector containing all the deviations from the seasonal
mean from yesterday until 𝑝 days ago, and the last two terms include a variance term (𝜎𝑢) as well as the market price of risk (𝜃𝑢). They smooth out the price of the temperature based futures contract for the remainder of time till measurement (integral from 𝑡 to 𝜏1) as well as the measurement period itself (integral from 𝜏1 to 𝜏2).
Using Ito’s formula (Øksendal, 1998, p.48) we can find the ℚ𝜃𝑡 dynamic of 𝐹𝐶𝐴𝑇(𝑡, 𝜏1, 𝜏2) for 0 ≤ 𝑡 ≤ 𝜏 ≤ 𝜏1 < 𝜏2can also be written as
𝐹𝐶𝐴𝑇(𝜏, 𝜏1, 𝜏2) = 𝐹𝐶𝐴𝑇(𝑡, 𝜏1, 𝜏2) + ∫ 𝑎(𝑠, 𝜏1, 𝜏2) 𝜏
𝑡
𝑒𝑝𝑑𝐵𝑠𝜃𝑡
So we find that conditional on ℱ𝑡 𝐹𝐶𝐴𝑇(𝜏, 𝜏1, 𝜏2) is normally distributed with
expectation 𝐹𝐶𝐴𝑇(𝑡, 𝜏1, 𝜏2) and by the property named Itô isometry the variance of an
Itô integral is equal to: ∫ 𝜎𝑠2 𝜏 𝑡 𝑒1′𝐴−1(𝑒𝐴(𝜏2−𝑠)− 𝑒𝐴(𝜏1−𝑠))𝑒 𝑝𝑒𝑝′(𝑒𝐴(𝜏2−𝑠)− 𝑒𝐴(𝜏1−𝑠))𝐴−1𝑒1𝑑𝑠 = = ∫ Σ𝐶𝐴𝑇2 (𝑠, 𝜏1, 𝜏2) 𝜏
𝑡 𝑑𝑠. The last term is a matter of notation.
From the definition of an option, the following equation is derived: 𝑒−𝑟(𝜏−𝑡)∙ 𝐸ℚ𝜃𝑡[𝑚𝑎𝑥(𝐹𝐶𝐴𝑇(𝜏, 𝜏1, 𝜏2) − 𝐾, 0)|ℱ𝑡].
With some intermediate steps the price of a call option on a CAT futures contract is 𝐶𝐶𝐴𝑇(𝑡, 𝜏, 𝜏1, 𝜏2) = 𝑒−𝑟(𝜏−𝑡)∙ (𝐹𝐶𝐴𝑇(𝑡, 𝜏1, 𝜏2) − 𝐾) × Φ ( 𝐹𝐶𝐴𝑇(𝑡,𝜏1,𝜏2)−𝐾 √∫ Σ𝑡𝜏 𝐶𝐴𝑇2 (𝑠,𝜏1,𝜏2)𝑑𝑠 ) + ∫ Σ𝐶𝐴𝑇2 (𝑠, 𝜏1, 𝜏2) ∙ φ ( 𝐹𝐶𝐴𝑇(𝑡,𝜏1,𝜏2)−𝐾 √∫ Σ𝐶𝐴𝑇2 (𝑠,𝜏 1,𝜏2) 𝜏 𝑡 𝑑𝑠 ) 𝜏 𝑡 .
We see that the price for the option on the CAT futures contract consists of two parts: the current payoff times the probability the future will end up in the money and secondly a premium for the variance times time the probability density function. The first part can be regarded as a premium for the expected value underlying the option contract and second part as a premium for the underlying volatility.
2.5 Comparison between the four approaches
The four approaches discussed above would be able to price the weather derivatives but which model turns out to be the most preferred one? There are three aspects that can be compared in order to come up with an answer to that question: the
assumptions, the fit with the underlying (the weather process) and outcomes.
2.5.1 Assumptions
What follows are the main assumptions made for each approach: Actuarial approach:
The actuarial approach uses historic realizations which are usually not representative for futures contract realization.
A second disadvantage of this approach is that it uses merely probabilistic and actuarial considerations for pricing options. Economic considerations are not involved (Bladt et al., 1998, p.66).
Indifference pricing:
The underlying preferences of the agents can be described by exponential or power utility functions Henderson (2004, p.13). This is because when the preferences of an investor can be described with an exponential function, the initial endowment of an investor is not relevant while when we use a power utility the initial endowment is relevant and Henderson (2004, p.13) makes a comparison between these two. He shows that it depends on the utility
function whether initial endowment is a driving factor for the price or not. The research shows that initial wealth is a driving factor.
General equilibrium pricing:
This approach uses a lot of assumptions which are described in depth above. Martingale approach:
Assumes that the market price of risk exists and follows a specific structural form (Härdle and Cabrera, 2012, p.60).
Compared with these models, the martingale approach is less demanding in terms of assumptions (Härdle and Cabrera, 2012, p.60). The martingale approach doesn’t require deterministic realizations or a utility function to describe the investor’s preferences. However, the assumption on the market price of risk is still under debate Härdle and Cabrera (2012) show that different specification of the market price of risk are possible and this variable is time varying. Using this approach requires close monitoring of this variable and regular updating which is a clear disadvantage.
2.5.2 Fit with the underlying (the weather process)
What follows are the main concerns regarding the fit with the underlying for each approach:
Actuarial approach:
The current realizations of the weather process is not used as input factor to price the weather derivative. Therefore there is no way to update the
probabilities during an adverse weather event (Richards et al., , 2004, p.1008). Indifference pricing:
The weather process is not used as input factor to price the weather derivative. An extension of the current framework is needed so the weather process plays a part in the expected payoff of the claim.
General equilibrium pricing:
Here we try to estimate the price on an option using a closed (global) economy while the weather process is a local factor by nature.
Martingale approach:
The martingale approach uses the weather process as basis to price the option and no extension is needed.
In terms of the fit with the underlying, the martingale performs best.
2.5.3 Outcomes
What follows are the main concerns regarding the outcomes for each approach: Actuarial approach:
The actuarial approach uses historical realizations. The risk of using historical information is that it implies that the futures contract behaves exactly the same as the past which is de facto not true. According to Richards et al. (2004, p.1009) burn rate model will trade infrequently, if at all, due to the fit with the underlying.
Indifference pricing:
The actual investment, and therefore also the realized price, will depend on the initial wealth of an investor (Henderson, 2004, p.13).
Xu et al. (2008) apply this technique to show the willingness to pay for weather insurance in Germany. They find that willingness depends on the production program as well as the region. The large geographical risk and production risk erode the potential benefits of weather derivatives over traditional crop insurance. To overcome this problem, they suggest that weather derivatives should be made to suit the individual wishes of the farmers.
General equilibrium pricing:
Both Cao and Wei (1999) as well as Richards et al. (2004) used the Lucas model to estimate the price of a weather option. Cao and Wei (1999)
investigated the impact of various correlation structures between the weather dynamics and the dividend process and the impact of mean reversion in the weather process. However, they didn’t compare their outcomes with observed market prices. Richards et al. (2004, p.1015) estimated the hedge effectiveness for weather impact on nectarine yield. They found that the optimal strategy to hedge the weather risk is a straddle: simultaneously buying a call and a put option where the strike value is set by the production.
Martingale approach:
Härdle and Cabrera (2012, p.83-84) show that the proposed framework had a low RMSE for different specification for the market price of risk.
Alaton et al. (2002, p.18-19) use to martingale approach to compare the outcome under the closed-end formula and simulations with the actual
outcome and find a difference of less that 2% for the price of an option on the future.
In all three aspects that have been investigated, the martingale approach has the most potential to yield appropriate outcomes. In terms of assumption it poses the least restrictive assumptions. In terms of fit with the underlying, the martingale approach uses the weather dynamic as its starting point. As for outcomes Härdle and Cabrera (2012, p.83-84) and Alaton et al. (2002, p.18-19) show completing evidence this method has appropriate outcomes.
Part II. Temperature dynamics
This part starts with an overview of the literature on the subject of temperature modelling. Here two selection criteria a formulated by which the model is estimated in Chapter 4. In Chapter 5 the discrete model is transformed towards a continuous time model which could be used as input for the equivalent martingale approach.
Chapter 3. Literature review and selection criteria
The first researchers using a model to capture the underlying weather process such that it could be used to price a weather derivative were Dornier and Querel (2000). They used an Ornstein-Uhlenbeck stochastic process to model the temperature observations with constant variance. An Ornstein-Uhlenbeck process can be considered as the continuous time analogue of a discrete-time 𝐴𝑅(1) process (Barndorff-Nielsen et al., 1998). Thus the underlying process, the temperature, is modeled as a 𝐴𝑅(1) process and used as an input for the Ornstein-Uhlenbeck process. Alaton et al. (2002) extended this Ornstein-Uhlenbeck model with a monthly varying variance and derived two formulas. The first is an approximation formula for the price of the derivative and the second is an equation that could be used as an input for a Monte Carlo simulation. The results showed that the two methods yield similar results.
Campbell and Diebold (2005) looked deeper into weather forecasting. They were the first to divided the temperature regression for the mean temperature into three parts: a trend (𝛽0+ 𝛽1× 𝑦𝑒𝑎𝑟), a seasonal part (which for the first time includes 3
Fourier-truncated series (FTS), ∑ (𝑐1,𝑝𝑐𝑜𝑠 (2𝜋𝑝 𝑡 365) + 𝑐2,𝑝𝑠𝑖𝑛 (2𝜋𝑝 𝑡 365)) 𝑃=3 𝑝=1 ) and a cyclical
part (an autoregressive part up till the third lag, ∑𝐿=3𝑙=1𝜌𝑡−𝑙𝑇𝑡−𝑙 ). Where 𝑡 is the day of the year, for example January first is the first day and December 31th the 365 (366 in case of a leap year). Next to this, they discovered not only a strong seasonality exists in the mean but also in the volatility. They modeled this as a 𝐺𝐴𝑅𝐶𝐻(1,1) with 3 FTS as additional regressors. They showed that this model produced better temperature forecasts than the model used in previous papers.
In order to incorporate this weather model in the pricing mechanics Benth et al. (2007) create a weather model that includes an autoregressive component (𝐴𝑅(3)) for the mean. As for the seasonal volatility four FTS are included but no 𝐺𝐴𝑅𝐶𝐻- effects. The resulting residuals are assumed to be close enough to white noise to use the Wiener process as noise process. The discrete outcomes are used as input for the continuous time stochastic process, a vectorial Ornstein-Uhlenbeck. This process is rewritten in such a way that an equation to price the weather derivatives by means of a Monte Carlo simulations emerges. A similar approach is used by Härdle and Cabrera (2012), their research is aimed at examining the differences between historical and risk neutral behavior of temperature and gives an insight in to the market price of risk. They show that the risk premium associated with the market price of risk is significantly different from zero, changing over time but not over location. One way the model of Härdle and Cabrera (2012) differs from that of Benth et al. (2007) is by the estimation of the seasonality in the mean and volatility. Härdle and Cabrera (2012) extend model of Benth et al .(2007) by looking at nonparametric estimation techniques to capture the seasonality.
Using these insights in this research, it becomes clear that modelling the temperature process as two separate components, each with a different dynamic, is a good starting point. The seasonal component should (Λ𝑡) consist of a constant, a trend to reflect the
global warming and Fourier-truncated series (FTS) to capture the seasonal variation over the year.
Subtracting the seasonal component from the daily temperatures yields the Fourier-truncated daily average corrected temperatures (FTDACT). On this corrected
temperature we fit a model to capture the mean-reverting and long memory property. In order to do so we take the FTDACT and model an 𝐴𝑅(𝑝) model with
heteroskedastic errors. All estimations in this Chapter are done in STATA ( see appendix C for the code). As a back test both the seasonal component as well as the variance of the non-seasonal part are compared with a non-parametric technique, the Lowess estimator.
Before we can start estimating we must write down the restrictions such that the modeled temperature process is useable for a martingale approach as well as the selection criteria by which on specification is preferred over another.
The first selection criterion is that we use Akaike Information Criteria (AIC) in order to find the most preferred model. For any model the AIC is calculated as: 𝐴𝐼𝐶 = 2𝑘 − 2ln (𝐿) where k is the number of parameters in the model and L is the log-likelihood. A decrease in AIC when adding an additional regressor, means that the 2ln (𝐿) has decreased more than the penalty for adding an addition regressor has increased, and thus a model with added regressor is preferred.
A second selection criterion is that we move from small-to-big. We start out with the simplest of models: regressing the dependent variable on a constant. From here we move to more complex models by adding one regressor at the time and check the AIC criteria, when this AIC no longer increased, we remove the insignificant regressors until we are left with a model containing only significant regressors.
Chapter 4. Modelling the temperature data
First a description of the temperature data is given. The temperature model comprises two elements: a seasonal and a non-seasonal component. First the seasonal
component is modeled, thereafter the non-seasonal component. Both components are back tested with a non-parametric estimation technique.
4.1 Description of the temperature data
The weather station used is Schiphol Airport, the main airport in the center of the Netherlands. The weather data originates from the Royal Netherlands Meteorological Institute (KNMI, 2014). The daily temperature, 𝑇𝑖, is measured by the Royal
Netherlands Metrology Institute (KNMI) and is defined in Definition 1.
The data are available from 01-01-1951 to 01-03-2014 and can be updated daily. The resulting dataset contains 23,071 temperature observations. Table 1 shows the most important summary statistics while Figure 2 shows the path of the daily average temperature from 2000 to 2013. From these two we can see that while the average temperature is around 10℃ it’s not stable over the year. Looking at the empirical distribution function of the temperature in Figure 3, we can see that the daily temperature is not distributed as a normal distribution. It looks like the daily temperature is created from a bimodal distribution.
Figure 3: density of the temperatures Figure 2: monthly average temperatures
Statistic Schiphol Mean 9.76 Median 9.9 Mode Bimodal Standard deviation 6.16 Minimum -12.9 Maximum 26.7 Skewness -.21 Kurtosis 2.61
Table 1: summary statistics for the daily temperature
4.2 Removing seasonality in the mean
Before we can remove the seasonality in the mean, we need a model to describe it. Campbell and Diebold (2005) observe that over several decades there has been an increase in the daily mean of temperatures in Swedish and American cities. There can be many reasons for this, for example global warming. Another explanation is due to urbanization, the so-called heat island effect, which indicates that the temperature tends to rise in areas near a big city since the city is warming its surroundings. To catch this trend it is assumed that the warming trend is linear. Other properties of the temperature are that the seasonality process revolves around a mean and vary over the year. In order to capture all these features we use a constant, a trend and Fourier-truncated series (FTS). The variable 𝑦𝑒𝑎𝑟 constructed to be equal to 1 of the observed year is 1950, 2 if the observed year is equal to 1952, etc. In order to establish the optimal number of Fourier functions, L, we compare the Akaike information criteria (AIC) (Selection criteria with the first 40 FTS the AIC no longer increases when
This leads to the following specification: Λ𝑡= 𝛼0+ 𝛼1∗ 𝑦𝑒𝑎𝑟 + ∑ 𝛼𝑙𝑠𝑖𝑛(2𝜋∗𝑙
365(𝑡−𝜙𝑙))
24
𝑙=1 ,
𝛼0 and 𝛼1 are parameters describing the average level and the global warming trend in the mean temperature. 𝛼𝑙 is the amplitude of the sine function, while 𝜙𝑙 is the phase angle that is introduced since the yearly minimum and maximum mean
temperatures do not usually coincide with the first of January and July. We divide by 365 days because the 29th of February is removed from the dataset for each leap year.
The equation above cannot be estimated straightforwardly by OLS. Therefore we use the properties of a sine function to rewrite it in terms of a sine and cosine function. Appendix D shows the details of this derivation.
For this regression the Gauss-Markov assumptions that the residuals have a
homoscedastic error term and are uncorrelated does not hold. The temperature process is divided in two parts such that the long-memory process can be captured by the non-seasonal component. By estimating the non-seasonal component with OLS the error terms might be too small. In order to resolve this problem the Huber-White sandwich estimator for the variance is used to create robust standard errors. Estimation outcomes can be found in Appendix D.
4.2.1 Estimation results
The constant and the first two Fourier series are significant at the 1% level when using the Huber-White sandwich estimator to create robust standard errors. Using robust standard errors for this estimation is necessary because the residuals display autocorrelation. The 𝑅2 is equal to 0.735 meaning that around 73% of all variation in
the temperature is explained by the seasonal component. 𝛼̂ = 8.91 is close to the 0 average daily temperature (table 1 shows a mean of 9.74), while the amplitude of the sine function is about 7.5℃, which means that the difference in temperature between a typical winter day and a summer day is about 15℃. The linear trend (𝛼̂ × year =2 .0271 × year ) is statistically significant at the 1% level. The estimate would imply a change in the mean temperature over the complete period of 1,72℃, a little bit higher than the global average of 0,65℃ (University corp., 2013). The difference could be explained by the heat-island effect.
A plot of this function (the solid line) together with the daily temperatures for the year 1968 is shown in Figure 5. From this graph we can deduct that it the seasonal
component follows the daily temperatures nicely as a sort of average. However there are outliers that are not captured by the seasonal component.
4.2.2. Back test with non-parametric estimate
Another way to estimate the seasonal component is by looking at non-parametric methods, as done before by Härdle and Cabrera (2012). The advantage of using nonparametric methods is that it makes very minimal assumptions regarding the process that generated the data and it places a flexible curve on a (𝑥, 𝑦) scatter plot with no parametric restrictions. The local regression estimator that we use is a locally weighted scatterplot smoothing or Lowess estimator. This is a type of nonparametric local regression that provides a smoother estimate of the seasonality mean, Λ𝑡, as it uses a tricubic kernel, 𝐾(𝑧) =70
81(1−|𝑧|
3)3𝑥 𝟏(|𝑧|<1). It is attractive to use in this case because it uses a variable bandwidth and is robust against outliers (Cameron and Trivedi, 2005). We use a multivariate Lowess smoother to smooth over both the day of the year as well as over the years, such that we can correct for the climate change as well as the seasonal variation. The left side of Figure 2 shows the correction over the days of the year while the figure on the right shows the correction over the years. From the right panel it is clear that using a linear trend for the global warming is appropriate.
4.2.2.3 Parametric vs. non-parametric estimate
Figure 5 sums up the efforts made to remove the seasonality in the mean, it shows the daily temperatures, the estimate of the Fourier series and the Lowess estimator for the year 1968. From the figure it is clear that the Lowess estimator is smoother than the Fourier series even though they are very close to each other, the correlation between them is 0.9792. Using a non-parametric estimation technique instead of a parametric
estimation technique doesn’t capture widely different information, the only difference is that the result is more smooth. Comparing the goodness of fit, the 𝑅2 the Lowess estimator has a 𝑅𝐿𝑜𝑤𝑒𝑠𝑠2 = 0.7255 while the parametric method has a 𝑅𝑝𝑎𝑟2 = 0.7342.
Therefore we can conclude that using a completely different estimation technique would yield similar results.
Figure 2: the Lowess smoother over the day of the year (left) and the years (right)
Figure 3: Plot of daily temperatures, seasonal component modeled as Fourier-truncated series and Lowless estimator for 1968
Figure 4: scatter plots from regressing today’s temperature against
yesterday’s. FTDACT (left) and today’s FTDACT against yesterday’s FTDACT (right).
4.3 Modeling the non-seasonal component
To model the cyclicality in the non-seasonal component we have to determine our dependent variable, check weather this dependent variable is a stationary process of exhibits a unit root. Next we estimate an 𝐴𝑅(3) model and compare the outcomes with a non-parametric technique.
4.3.1 The dependent variable
To model the cyclicality in the FTDACT we could follow the approach from Campbell and Diebold (2005) who are regressing today’s temperature against yesterday’s FTDACT instead of the approach by Benth and Ŝaltytė-Benth (2005, 2007) who estimate todays FTDACT on yesterday’s FTDACT. The best way to solve this problem is by looking at Figure 6. These scatter plots shows the two options. By eyeballing we can see that the second option of regressing today’s FTDACT against those of yesterday is better since the relationship between them is more linear.
4.3.2 Is the dependent variable stationary?
We start with the FTDACT and check whether this is a stationary process. In order to do that we use a Dickey-Fuller test and Kwiatkowsky, Phillips, Schmidt and Shin
(KPSS) test. The Dickey-Fuller unit root test has a test-statistic of -17.83 (p<0.01). Therefore the test-statistic rejects the null-hypothesis that our FTDACT have a unit root. The KPSS test has a test-statistics of 0.186 (p<0.01) meaning that we cannot reject our null-hypothesis that the process is stationary. Both tests indicate that stationary is achieved.
Figure 5: The autocorrelation function (left) and partial autocorrelation function (right) of the FTDACT
4.3.3 Estimation of the 𝐴𝑅(3) model
The ACF in Figure 5 shows an exponentially decaying autocorrelation with a long memory. While the PACF (Figure 5) shows a clear cut-off point, after the third partial autocorrelation, the PACF is stable around zero. This suggests that the 𝐴𝑅(3) model would suit this type of data.
To capture the seasonal dependency, we calibrate a Fourier-truncated series together with the 𝐴𝑅(3). This results in the following model for the FTDACT.
𝑦𝑡= 𝛼1𝑦𝑡−1+ 𝛼2𝑦𝑡−2+ 𝛼3𝑦𝑡−3+ 𝜎𝑡𝜀𝑡, 𝜀𝑡~𝑁(0,1) 𝜎𝑡2 = 𝛾0+ ∑ (𝛾𝑙,1𝑠𝑖𝑛 ( 2𝜋∗𝑙∗𝑡 365 ) + 𝛾𝑙,2𝑐𝑜𝑠 ( 2𝜋∗𝑙∗𝑡 365 )) 𝐿 𝑙=1
where 𝑦𝑡 = Tt− Λt, the FTDACT and εt are the residuals. The fitted parameters and standard deviation are given in Table 2
Table 2: Coeffients of fitted AR(3) with heteroskedasticity (1) (2) VARIABLES 𝑦𝑡 𝜎𝑡2 𝑦𝑡−1 0.895*** (0.00624) 𝑦𝑡−2 -0.194*** (0.00854) 𝑦𝑡−3 0.0863*** (0.00639) 𝑠𝑖𝑛 (2𝜋∗𝑡 365) 0.123*** (0.0121) 𝑠𝑖𝑛 (4𝜋∗𝑡 365) -0.103*** (0.0120) 𝑐𝑜𝑠 (2𝜋∗𝑡 365) 0.108*** (0.0122) 𝑐𝑜𝑠 (4𝜋∗𝑡 365) 0.151*** (0.0122) Constant 1.263*** (0.00867) Observations 23,055 23,055
Standard errors in parentheses *** p<0.01, ** p<0.05, * p<0.1
In order to establish that optimal number of lag’s in the AR process is indeed 𝑃 = 3 we compared the AIC for different choices 𝑃. The AIC decreases up to the 9th lag.
The first three lags are always significant at the 10% level even in the 𝐴𝑅(9) model. Lags four up till nine are no longer significant when adding a new lag. Therefore we believe the 𝐴𝑅(3) is the most parsimonious way to model the FTDACT. Looking at the residuals from the 𝐴𝑅(3) model, the Box-Ljung test is insignificant for the first to 9th lag. The ACF of residuals,𝜀𝑡, are close to zero as can be seen from Figure 6. In order to establish the optimal number of Fourier functions, L, to include in the conditional heteroskedasticity, we compare the AIC the first 40 models with ascending number of FTS. The AIC was minimalized when L=2.
This figure also shows the ACF of the squared residuals. These squared residuals still contain some sort of seasonal pattern and are very large the first lags. Therefore the Lung-Box test of white noise is rejected for the squared residuals.
Figure 6: The ACF of the residuals 𝜺𝒕 (on the left) and the squared residuals 𝜺𝒕𝟐 (on
the right)
4.3.4 Back test with a non-parametric method
Alternatively we could use a Lowess regressor. As mentioned above, using a non-parametric estimation technique has many advantages. To estimate the Lowess regressor for the variance we estimate an 𝐴𝑅(3) model, predict the residuals and use these as inputs for the Lowess regressor. We use tricubic kernel, 𝐾(𝑧) =
70 81(1−|𝑧|
3)3𝑥 𝟏(|𝑧|<1) and a bandwidth of 0.15.
Figure 7 shows the fitted functions when estimated by a Fourier series 𝜎𝑡2𝐹𝑇𝑆 as well
as the non-parametric method, the Lowess estimator 𝜎𝑡2
𝐿𝑜𝑤𝑒𝑠𝑠 for each day of the
year. We observe a high variance in the winter, and in the late spring/early summer. This result is similar to the observations from Benth, Ŝaltytė-Benth, and Koekebakker (2007) for Swedish data. The correlation between the two estimators is very high, with a correlation coefficient of .9866. Using a non-parametric estimation technique instead of a parametric estimation technique doesn’t capture widely different
information, the only difference is that the result is more smooth. Therefore we can conclude that using a different estimation technique would yield similar results.
4.3.5 Test for normality
After dividing out the seasonal volatility 𝜎𝑡2
𝐹𝑇𝑆 and 𝜎𝑡2𝐿𝑜𝑤𝑒𝑠𝑠 we take a look the
residuals, 𝜀𝑡 and the squared residuals, 𝜀𝑡2. Table 3 shows that both estimators contain
kurtosis and skewness for the residuals 𝜀𝑡 and the squared residuals, 𝜀𝑡2 after
removing the seasonal volatility. The Jarque- Bera test indicates that Fourier series of conditional heteroskedasticity gives results which are closer to the normal distribution although both estimators reject the Jarque-Bera test of normality.
Figure 7: Daily empirical variance for the Lowess estimator (dashed line) and the Fourier Truncated AR model (solid line)
Table 3: Kurtosis and skewness of the residuals after removing the seasonal volatility
(1) (2)
VARIABLES 𝜎𝑡2𝐹𝑇𝑆 𝜎𝑡2𝐿𝑜𝑤𝑒𝑠𝑠
Jarque-Bera 189,29 201,23
Kurtosis 3,4315 3,4458
Chapter 5. A continuous-time autoregressive process
In the last part of this part we move the estimate from Chapter 4 from discrete time steps (days) to a continuous time model which could serve as input for the martingale approach of pricing the weather option as we have seen in Chapter 2. This
transformation is called a Continuous-time autoregressive model or CAR. We will also investigate the stability since this model is used in Part III to simulate time dependent paths for the weather process.
The seasonal component, Λ𝑡, is deterministic through time and is estimated as in Chapter 4, recall that the estimation results can be found in Appendix D. For the non-seasonal component, 𝑦𝑡, it is more complicated as the realization of the next period depends on the realizations of the previous periods. There exists an analytical link between an 𝐴𝑅(3) and its continuous version.
Let (Ω, ℱ, 𝑃) be a probability space with a filtration (ℱ𝑡)0≤𝑡≤𝑡𝑚𝑎𝑥, where 𝑡𝑚𝑎𝑥 is the time needed to cover all times of interest on the market. Let 𝑿𝑡be a state vector that is
defined in the following way: 𝑿𝒕 = ( 𝑋1𝑡 𝑋2𝑡 𝑋3𝑡 ) = ( 𝑦𝑡−2 ∆𝑦𝑡−1 ∆2𝑦𝑡 ) = [ 0 0 1 0 1 −1 1 −2 1 ] ( 𝑦𝑡 𝑦𝑡−1 𝑦𝑡−2)
Here 𝑦𝑡 is equal to the daily temperature after removing the seasonality at time 𝑡, the non-seasonal component. Let 𝑋𝑞(𝑡) be the qth coordinate of the vector 𝑿𝑡 where 𝑞 = 1, … , 𝑝. In our model 𝑝 = 3. Following this termology, the temperature series at t (𝑞 = 1) is equal to: 𝑇𝑡= Λ𝑡+ 𝑋3(𝑡)+ 2𝑋2(𝑡)+ 𝑋1(𝑡), Since 𝑋3(𝑡)+ 2𝑋2(𝑡)+ 𝑋1(𝑡)=
∆2𝑦
𝑡+ 2 ∗ ∆𝑦𝑡−1+ 𝑦𝑡−2= 𝑦𝑡− 2𝑦𝑡−1+ 𝑦𝑡−2+ 2(𝑦𝑡−1− 𝑦𝑡−2) + 𝑦𝑡−2= 𝑦𝑡. The
equation says that the temperature today is equal to its seasonal and non-seasonal component. Define a 𝑝 × 𝑝 –matrix A as A= ( 0 1 0 0 0 1 ⋮ ⋮ ⋯ 0 0 ⋯ ⋮ ⋮ ⋱ 0 0 𝛼𝑝 𝛼𝑝−1 ⋯ 0 0 1 𝛼2 𝛼1) .
This is part of the vectorial Ornstein-Uhlenbeck process given by
𝑑𝑿𝑡 = 𝐴𝑿𝑡𝑑𝑡 + 𝑒𝑝𝜎𝑡𝑑𝐵𝑡. Where 𝑿𝑡 ∈ ℝ𝒑 for 𝑝 ≥ 1, 𝑒𝑝 denotes the pth unit vector in ℝ𝒑 filled with zero’s and a 1 op the pth entry, 𝜎
𝑡 is the temperature volatility, 𝐵𝑡 is a
Brownian motion and 𝛼𝑝 are negative constants. Appendix E elaborates on how to find these constants 𝛼𝑝. In combination with Table 2, the estimation results of the non-seasonal component, we define matrix A as:
∆𝑋𝑡= [
0 1 0
0 0 1
−.2127 −1.4026 −1.1029
] 𝑋𝑡−1𝑑𝑡 + 𝑒3𝜎𝑡𝑑𝐵𝑡.
5.1 The explicit form of 𝑿𝒕
For the price derivations we need the explicit form of 𝑋𝑡. In order to obtain the
explicit form we will use Itô’s formula as described by Øksendal (1998). We start with the vectorial Ornstein-Uhlenbeck process as described above and an alternative process 𝑌𝑡.
𝑑𝑿𝑡 = 𝐴𝑿𝑡𝑑𝑡 + 𝑒3𝜎𝑡𝑑𝐵𝑡 𝑌𝑡 = −𝑒−𝐴𝑡𝑿𝑡
We can obtain the Stochastic Differential Equation (SDE) of 𝑌𝑡by applying Itô’s
formula: 𝑓(𝑿𝑡, 𝑡) = −𝑒−𝐴𝑡𝑿𝑡, 𝑓̇(𝑿𝑡, 𝑡) = 𝐴𝑒−𝐴𝑡𝑿 𝑡 = −𝐴𝑌𝑡, 𝑓′(𝑿𝑡, 𝑡) = −𝑒−𝐴𝑡, 𝑓′′(𝑿𝑡, 𝑡) = 0,
So the SDE of 𝑌𝑡 is equal to 𝑑𝑌𝑡 = 𝑓̇𝑑𝑡 + 𝑓′𝑑𝑿𝑡+12𝑓
′′(𝑑𝑿 𝑡)2 =
−𝐴𝑌𝑡𝑑𝑡−𝑒−𝐴𝑡 𝑑𝑿𝑡+12∙ 0 ∙ (𝑑𝑿𝑡)2=𝐴𝑒−𝐴𝑡𝑿
𝑡𝑑𝑡−𝑒−𝐴𝑡(𝐴𝑿𝑡𝑑𝑡 + 𝑒3𝜎𝑡𝑑𝐵𝑡) =
−𝑒−𝐴𝑡𝑒3𝜎𝑡𝑑𝐵𝑡.
So the explicit form of 𝑌𝑡 is equal to 𝑌𝑠 = 𝑌𝑡− ∫ 𝑒−𝐴∙𝑢𝑒3𝜎𝑢𝑑𝐵𝑢 𝑠
𝑡 .
Our definition 𝑌𝑡 = −𝑒−𝐴𝑡𝑿𝑡 implies that 𝑌𝑠 = −𝑒−𝐴𝑠𝑿𝑠 such that 𝑿𝑠 = −𝑒𝐴𝑠𝑌𝑠. Resulting in 𝑿𝑠 = −𝑒𝐴𝑠(𝑌𝑡− ∫ 𝑒−𝐴∙𝑢𝑒3𝜎𝑢𝑑𝐵𝑢 𝑠 𝑡 ) = −𝑒𝐴𝑠𝑌𝑡+ ∫ 𝑒𝐴∙(𝑠−𝑢)𝑒3𝜎𝑢𝑑𝐵𝑢 𝑠 𝑡 = 𝑒 𝐴(𝑠−𝑡)𝑿 𝑡+ ∫ 𝑒𝐴∙(𝑠−𝑢)𝑒3𝜎𝑢𝑑𝐵𝑢 𝑠 𝑡 .
5.2 Stability conditions of 𝑿𝒕
Even though the process contains a time dependent heteroskedasticity the process should be stable such that it will not explode to ±∞. Therefore we are interested in the conditions that must hold such that the process of 𝑿𝑡 is stable. We use the
property that an Itô integral 𝑿𝑡 is stable when its variance converges when 𝑡 → ∞. In order to calculate the variance of 𝑿𝑡 , because 𝑿𝑡 is an Itô integral we can use Itô isometry. The variance of 𝑿𝑠 is equal to: 𝑣𝑎𝑟(𝑿𝑡) = 𝐸[𝑿𝑡2] =
𝐸 [(∫ 𝑒𝐴∙(𝑠−𝑢)𝑒3𝜎𝑢𝑑𝐵𝑢 𝑠 𝑡 ) 2 ] = 𝐸[(∫ 𝑒𝐴∙(𝑠−𝑢)𝑒3𝜎𝑢𝜎𝑢𝑒′3𝑒𝐴′∙(𝑠−𝑢)𝑑𝑢 𝑠 𝑡 )] = ∫ ∫ 𝑒𝐴∙(𝑠−𝑢)𝑒 3𝜎𝑢𝜎𝑢𝑒′3𝑒𝐴′∙(𝑠−𝑢)𝑑𝑢𝑑𝑠 𝑠 𝑡 𝑡 0 = ∫ 𝜎𝑡−𝑠 2 𝑒𝐴∙𝑠𝑒 3𝑒′3𝑒𝐴′𝑠 𝑡 0 𝑑𝑠.
Since our variance function is a scalar and known in advance, we can write
𝜎𝑡−𝑠𝜎𝑡−𝑠 = 𝜎𝑡−𝑠2. By using the definition of an integral and integrating the inner part,
we find the variance matrix.
From this variance matrix it is clear that the stochastic process 𝑿𝑡 is stable when:
𝜎𝑡−𝑠2 is bounded, this is the case because 𝜎𝑡−𝑠2 is a deterministic function. The variance ∫ 𝜎𝑡−𝑠2 𝑒𝐴∙𝑠𝑒3𝑒′3𝑒𝐴′𝑠
𝑡
0 𝑑𝑠 should converge as 𝑡 → ∞. Which is
trivial when the eigenvalues of the matrix 𝐴 has negative real parts. The eigenvalues of our matrix 𝐴 are equal to {−0.4669 + 1.013i, −0.4669 − 1.013i, −0.1711 }.
So our continuous weather process is stable.
5.3 The equivalent martingale measure
Usually when we can replicate a claim we use the risk-neutral risk measure ℚ to find the price of a call option in the following way: 𝐶𝑡= 𝐹(𝑡, 𝑥) = 𝐸ℚ[𝑒−𝑟(𝑇−𝑦)(𝑋
𝑡−
𝐾)+|𝑋𝑡 = 𝑥], where 𝑑𝑋𝑡 = 𝑟𝑋𝑡𝑑𝑡 + 𝜎𝑋𝑡𝑑𝑊𝑡.
The temperature itself is not a tradable asset and therefore we cannot find such a portfolio. However, temperature futures and options are traded so in that respect they must be free of arbitrage. We will choose a parameterized equivalent martingale pricing measure ℚ = ℚ𝜃𝑡 based on historic realization. In such a way we can find the
market price of the temperature derivatives and their expected value with respect to our equivalent martingale pricing measure. Note that 𝜃𝑡 is the time dependent market
price of risk, which should be real-valued, bounded and piecewise continuous on [0, 𝑡𝑚𝑎𝑥]. By the use of Girsanov’s theorem we have the following:
𝑑𝑿𝑡 = 𝐴𝑿𝑡𝑑𝑡 + 𝑒𝑝𝑡𝜎𝑡𝑑𝐵𝑡 ⇔ 𝑿𝑡 = (𝐴𝑿𝑡+ 𝑒𝑝𝜎𝑡𝜃𝑡)𝑑𝑡 + 𝑒𝑝𝜎𝑡(−𝜃𝑡𝑑𝑡 + 𝑑𝐵𝑡). Where −𝜃𝑡𝑑𝑡 + 𝑑𝐵𝑡 is 𝑑𝐵𝑡
𝜃𝑡, a Brownian motion under ℚ
𝜃𝑡.
Under ℚ𝜃𝑡 the stochastic differential equation (SDE) for 𝑿𝑡 is equal to 𝑑𝑿𝑡 = (𝐴𝑿𝑡+ 𝑒𝑝𝜎𝑡𝜃𝑡)𝑑𝑡 + 𝑒𝑝𝜎𝑡𝑑𝐵𝑡
𝜃𝑡 and its implicit function will be equal to: 𝑿
𝑠 = 𝑒𝐴(𝑠−𝑡)𝑿 𝑡+ ∫ 𝑒𝐴∙(𝑠−𝑢)𝑒𝑝𝜎𝑢𝜃𝑢𝑑𝑢 𝑠 𝑡 + ∫ 𝑒 𝐴∙(𝑠−𝑢)𝑒 𝑝𝜎𝑢𝑑𝐵𝑢 𝜃𝑡 𝑠
𝑡 for 𝑠 ≥ 𝑡 ≥ 0. This can be
easily verified by using the same technique as was used above. We derive the SDE of 𝑌𝑡 = −𝑒−𝐴𝑡𝑿𝑡 with the help of Itô’s lemma, solve the SDE of 𝑌𝑡 and substitute 𝑿𝑡
back.
By using historical realizations we can solve for the market price of risk, 𝜃𝑢. All other variables are defined by the temperature process specified in Chapter 4 and its
continuous counterparty we have derived above. Using the estimated market price of risk the equivalent martingale approach can be used in pricing temperature options. Appendix A and B shows the derivations on how to price of temperature contracts using the equivalent martingale measure.
