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:: Modelling Temperature ::

Our previous article (What are weather derivatives?) explained the basic concept behind weather derivatives, the history of their upcoming, and uses as hedging instruments particular to the energy industry.  In this article, we shall concern ourselves with modelling temperature as the underlying variable for the weather derivatives.  Most common weather derivatives are written on temperature.

To aid the development of a reasonable model, one needs to look at recorded temperatures over a long period of time from different locations.  In the figure below, the lower plot shows the daily mean temperature recorded in central England (CET) between 1990 to 1999.  It is evident from this plot that there is a strong seasonal cycle from in the summers and during the winters.  The upper plot shows a simulation of a particular family of Fractional Brownian Motion (FBM), known as a Ornstein-Uhlenback processes.  We shall come back to this point later in the article.

One can approximate this seasonal cycle by some form of a sinusoid functions since the period of the cycle is fixed.  So,

where denotes the time, measured in days, and the period of oscillation neglecting leap years.  The phase function is necessary if we let be the first, second, and so on, days of the year - since the seasonal cycle is out of phase with the western calendar.  Another trend that is often seen on some datasets of temperature is a slight annual increase in mean temperature.  Reasons for this can be the global warming trend, or, urbanisation near big cities, with cities growing in size causing a net warming of the surroundings.  This trend is much weaker than that of the seasonal cycle, so, to first approximation on can approximate this to a polynomial expression dominated by the linear term.  Using this and the seasonal cycle, one can write the mean temperature, as a function of time ,

where parameters A, B, C, are constants.

So far we have just a deterministic model for the mean temperature.  However, we know that temperature is a stochastic process.

Naturally, as one does very often in financial mathematics, a Wiener process , is put forward as a first approximation to the stochastic part of the model.  The figure below show the daily changes in average temperature agree very well to a gaussian distribution.

Thus, the 'noise' term in our stochastic differential equation (SDE) will be of the form , where is the time varying standard deviation.  Sometimes the standard deviation is found to be approximately constant for each month, so is chosen from the set , where is the month of the year.  This deals with our stochastic term.

Now the deterministic part - already found for the mean daily temperature . We present the following form for the change in temperature

The term comes from the mean reversion property of temperature.  i.e. cannot deviate away from the mean temperature on long time scales. Constant a (real and positive) determines the speed of mean reversion.  A better physical explaination for this may be found in the thermodynamic conditions of the system (atmosphere), though this is not trivial to see.  This equation describes an Ornstein-Uhlenbeck processes, and is found to agree well with empirical data, as shown above.

The only problem with our SDE so far is that the long term mean is not the same as (this can be shown when solving the SDE).  By adding the differential of the mean ,

to the deterministic drift term the solution of the SDE gives our desired long term mean .  Thus,

The solution is

where

We have now developed a physically representable stochastic model of temperature which can be used to price temperature derivatives.

Recently, modelling wind has become of importance since it is predicted that within 5 year 10% of the UK energy will be produced by this renewable source.  The problem with measuring wind is that, unlike temperature, local variations are very large.

Written by Alessio Farhadi

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