Evaluating weather derivatives requires a different approach from that used for evaluating common financial products. One reason is the difficulty of replication, as temperature, rainfall or wind is not a traded asset. Consequently delta neutral techniques cannot be used and, in addition, there is a lack of liquidity in some temperature contracts. Therefore a number of market participants have started to use an actuarial approach when dealing with weather derivatives. Extracting and de trending heating degree days or cooling degree days from data, and then fitting a distribution to the events, makes valuation possible based on the expectation of the loss plus a given risk premium that reflects the sensitivity to risk. However, in doing so, a number of problems arise. These stem from the fact from that, in most cases, a maximum of 40 years of data is available. Some of these issues include:
* How many years should be taken into account?
* Should the temperature be detrended before or after extracting heating or cooling degree days?
* Were the data correctly detrended and was the forward accurately extracted?
* Is the fitted distribution accurate or appropriate?
* Has there been any change in the distribution in recent years?
Example of Actuarial Pricing
Actuarial pricing methodology is based on extracting the distribution of risk from historical values. The three necessary steps to obtain the fair value of a weather contract are illustrated below using Heathrow as a reference site.
Step 1 - Filtration:
First the data are filtered taking into account any trend and extrapolated to the next year.
Step 2 Fitting distribution:
Then, a parametric distribution is fitted to the discrete distribution of risk like the normal distribution in the next figure:
Step 3 Evaluation:
Finally, the price is evaluated using closed formulae. Because an HDD is a cumulative index, it is possible to break down the price of a call option into an up and out call and a digital call. This makes them easy to value. Note that in the weather market, all call options are capped, and therefore a call option refers to a call spread.
So, assuming a normal distribution, the price of an up and out call is:
and a digital call is :
where HDDValue is the value for each recorded degree day below the temperature reference (usually 65°F), HDDinf and HDDsup are the strikes of the call option, Cap = (HDDsup HDDinf) * HDDValue is the maximum money value of the pay off, *=((HDD inf -µ )/*); ß=((HDD Sup-µ )/*); *=(µ -HDDinf )*; µ & * are the mean and the standard deviation of the HDD distribution, N(X;0;1) the standard normal cumulative distribution function evaluated in X.
Of course, the evaluation strongly depends on the treatment of the data (filtration) and the fitted distribution. All these referred problems and pitfalls are exacerbated in portfolio management due to the requirement to estimate multivariate distributions.
Simulating the underlying index (temperature, rainfall, etc) becomes extremely important because of the constraints above. Instead of merely having 40 data points available, one can recreate the temperature process extracted from 40 * 365 = 14,600 data points. The process can then be simulated thousands of times to simulate the real distribution of the pay offs of the weather deal.
One can simulate the temperature using the following process:
Ti =mi + si +* i AR(p)
where Ti is the temperature value at time i, si the seasonality of the temperature at time i, mi the trend of the temperature, *i a sinusoid function of i and AR(p) an autoregressive process of order p with non identically distributed noise.
Fitting it to London data for the period starting the 1st of November and ending the 31st of March (simulation started the 30th October) we obtain the following distribution:
The main information that we extract from this distribution is that the forward is different from the one extracted using detrended HDD (1739 instead of 1702) and that the volatility is lower than the one extracted (100 instead of 130). There are also some limited differences in the skewness and kurtosis.
Some advantages of the simulation are obvious. When the option has started, there is no need to value any conditional distribution, as one needs to do in "actuarial" analysis; the information is contained in the process itself. Other benefits of temperature simulation lie in portfolio risk management as explained below.
Let's look at several examples. First, consider two sites, London and Manchester. These are approximately 180 miles (roughly 300 km) apart. The correlation one should model for the combined profit and loss balance of a portfolio of temperature contracts based on these cities is not necessarily that based on the correlation of the HDD indices. Supposing, for example, that the portfolio consists of two critical temperature (CTD) options, one referring to London and the other one to Manchester. In this example, the critical temperature reference is set to 34 C (93.2 F). It can be seen that over the last 40 years the maximum temperature measured in Manchester is 33.7 C and in London 36.5 C. Therefore, on a descriptive point of view, one can conclude that the Pearson's correlation between these two indices is zero. In reality we know this cannot be the case: the underlying processes in both are temperature processes and we know these to be correlated.
In this example, simulation is key since one can extract from the previous equation the residues for both sites and separate out the seasonal effects. Clearly, when it is winter in London it is winter in Manchester and so the temperature is correlated. But looking at the residues the view is somewhat different. If we know that the temperature is below its average in London does this give us any information about the temperature value in Manchester relative to its own average? With CTD structures this becomes an interesting correlation since we may not want them to happen simultaneously in both locations.
For our second example, suppose now that the portfolio contains two options related to the same location, one based on CTD, the other one based on HDD. We can see that the correlation between these two indices may be weak even though both indices are derived from the same temperature processes. This is shown in the figure below:
Therefore one needs to use the same temperature values for both locations and not consider them separately. No convexity would be apparent here if using Pearson's correlation.
This week's Learning Curve was written by Michael Moreno, associate director ofSpeedwell Weather Derivativesand lecturing teacher at French management & actuarial schoolISFA.