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We propose a continuous-time autoregressive model for the temperature dynamics with volatility being the product of a seasonal function and a stochastic process. We use the Barndorff-Nielsen and Shephard model for the stochastic volatility. The proposed temperature dynamics is flexible enough to model temperature data accurately, and at the same time being analytically tractable. Futures prices for commonly traded contracts at the Chicago Mercantile Exchange on indices like cooling- and heating-degree days and cumulative average temperatures are computed, as well as option prices on them.

Protection against undesirable weather events, like, for instance, hurricanes and droughts, has been offered by insurance companies. In the recent decades, the securitization of such weather insurances has taken place, and nowadays one can trade weather derivative contracts at the Chicago Mercantile Exchange (CME), or more specialized weather contracts in the OTC market. At the CME, one finds derivatives written on temperature and snowfall indices, measured at different locations worldwide. The market for weather derivatives is emerging and gaining importance as a tool for hedging weather risk.

In this paper we propose a new model for the dynamics of temperature which forms the basis for pricing weather derivatives. The model generalizes the continuous-time autoregressive models suggested in Benth et al. [

As it turns out, one may derive reasonably explicit expressions for futures prices of commonly traded contracts at the CME. Our proposed model is flexible in capturing the stylized features of temperature data, as well as being analytically tractable. To account for seasonality, we introduce a multiplicative structure, which is much simpler to analyse than the complex model suggested by Campbell and Diebold [

The risk premium in derivatives prices is parametrized by a time-dependent market price of risk. In addition, we include a market price of volatility risk. Thus, the risk loading in derivatives prices, interpreted as the risk premium in the financial setting, includes both temperature and volatility. In mathematical terms, the pricing measure is obtained by a combination of a Girsanov and Esscher transform.

We discuss the estimation of the proposed temperature model on data from Stockholm, Sweden. The purpose of the study is not to give a full-blown fitting of the model, but to outline the steps and to justify the model. Next, we discuss some issues related to the mean reversion of the model, where in particular we derive the so-called

We present our findings as follows. In the next section we review the basics of the CME market for weather derivatives. Next, in Section

The organized market for temperature derivatives at the CME offers trade in futures contracts "delivering" various temperature indices measured at different locations in the world, including the US, Canada, Australia, Japan, and some countries in Europe. In addition, there are contracts written on snowfall in New York and frost conditions at Schiphol airport in Amsterdam. On the exchange one may also buy plain vanilla European call and put options on the futures.

The main temperature indices are so-called

An HDD index is defined similarly to the CDD as the cumulative amount of temperatures below a threshold

The CAT index is simply the accumulated average temperature over the measurement period defined as

Temperature futures may be used by energy producers to hedge their demand risk (see Perez-Gonzales and Yun [

For mathematical convenience, we will use integration rather than summation and define the three indices CDD, HDD, and CAT over a measurement period

In this paper we are concerned with modelling the temperature dynamics

Brockett et al. [

Let

We suppose that the temperature

The seasonal mean function

Empirical analysis of temperature data in Norway, Sweden, and Lithuania (see Benth et al. [

For

In this paper we are concerned with an extension of the

Hence, we propose the following

In Benth et al. [

A class of stochastic volatility processes providing a great deal of flexibility in precise modelling of residual characteristics is given by the Barndorff-Nielsen and Shephard [

In the BNS model a dependency on

In the rest of this paper we suppose that the deseasonalized temperature

In this section we derive the futures price dynamics based on the CDD/HDD and CAT indices. This will be done with respect to some pricing measure

We use the intrinsic notation

One may ask why we do not consider stochastic interest rates in this setup. In fact, it would not be any problem to do so, by, for example, assuming a risk-free spot rate

Note that the pricing measure

To restrict the class of pricing probabilities, we consider a deterministic combination of a Girsanov and Esscher transform. The Girsanov transform introduces a market price of risk with respect to the Brownian motion, while the Esscher transform will model a market price of volatility risk. In mathematical terms, we define the probability

There has been some work trying to reveal the existence and structure of the risk premium, or the market price of risk, for temperature derivatives. The theoretical benchmark approach by Platen and West [

In the following proposition we compute the CAT futures price.

The CAT futures price at time

We know that the

Observe that the market price of volatility risk does not enter the CAT futures price explicitly, only the market price of risk

The

This follows by a straightforward application of the multidimensional Itô Formula, where the observation that

As is evident from this dynamics, the CAT futures price will have a seasonal stochastic volatility

We next move our attention to CDD futures, which will become nonlinearly dependent on the temperature. Thus, we will not get as explicit results on their prices as for the CAT futures. It turns out to be useful to apply Fourier transform techniques (see Carr and Madan [

Recall from Folland [

Relevant to the analysis of CDD futures, consider the function

For any

We have

We apply this in our further analysis of the CDD futures price.

The CDD futures price at time

First, from the definition of the CDD index and the

Although seemingly very complex, the price dynamics of a CDD futures can be simulated by using the fast Fourier transform (FFT) technique (see Carr and Madan [

The CDD futures price depends explicitly on the current states of the volatility

Since

The HDD futures price dynamics

The technique using Fourier transform is well adapted for option pricing as well, and we will briefly discuss this in connection with European call and put options written on CAT and CDD futures.

We start with a call option on a CAT futures, with exercise time

The price at time

Options on CDD futures are far more technically complicated to valuate. However, we may here as well resort to Fourier techniques and in principle obtain transparent price formulas. Considering a call option at time

We refer to an empirical estimation which can be found in Benth et al. [

Our main concern in this paper is the precise modelling of the residuals. We have proposed a model where the volatility

The residuals after removing the effect of the function

In Figure

The empirical autocorrelation function on a log-scale for de-seasonalized squared residuals together with a fitted line for the first 10 lags.

Admittedly, the linear fit to the log-autocorrelation function in Figure

The final step in estimating our temperature model is to fit a subordinator process

Based on this, we

One may ask whether some or all of the parameters in the specified temperature model may be time dependent. In our empirical analysis of Stockholm data, we have not detected any structural changes in the seasonality function

To understand how fast the temperature dynamics is reverting back to its long-term average

The half-life of

First, we have after using the tower property of conditional expectations that

Note that the first coordinate of

For a temperature dynamics based on the Stockholm estimates referred to above, we find the solution to the half-life equation in Lemma

Let us investigate the contribution from the various terms in the CAT futures price dynamics in order to get a feeling for their relative importance in the context of weather derivatives. For illustrative purposes, we choose a June contract, which starts measurement at time

Let us investigate the stochastic volatility contribution to the CDD price. As we see from the pricing formula in Proposition

We are not aware of any empirical studies of the potential existence of a market price of volatility risk. One may estimate

In view of the existence of European-style call and put options written on the temperature futures, precise knowledge of the volatility is important. The volatility of the underlying temperature futures will determine the price of the option and play a crucial role in a hedging strategy of the option. In particular, the Samuelson effect of the volatility of the temperature futures makes the options particularly sensitive to the volatility close to measurement. Hence, a precise model for the stochastic volatility is important. The nonlinearity in the payoff of options also makes second-order effects in the underlying dynamics more pronounced, and thus even small stochastic volatility variations may become significant in the option dynamics.

The authors are grateful for the comments and critics from two anonymous referees, which led to a significant improvement of the original paper. Fred Espen Benth acknowledges financial support from the project Energy markets: modelling, optimization and simulation (emmos), funded by the Norwegian Research Council under Grant 205328/v30.