# "AND NOW THE WEATHER ... " - PART III: AN INTRODUCTION TO WEATHER DERIVATIVES

Weather derivatives markets have traditionally been characterized as having a small number of participants and large bid-ask spreads, but this is rapidly changing.

Weather derivatives markets have traditionally been characterized as having a small number of participants and large bid-ask spreads, but this is rapidly changing. During the last 18 months, the market has increased dramatically, and in the first semester of 1999 it is estimated that 800-1,800 transactions have taken place in the United States, which represent a sharp increase from 100-150 transactions in 1997. As new firms join the market, and the volume of transactions rises, liquidity is improving and the bid-ask spreads are getting tighter, although they remain high compared with more developed markets.

The main players in the weather derivatives markets can be grouped in four main categories: Market Makers, Brokers, Insurance and Reinsurance Companies, and End-Users such as Gas and Power Marketers, Utilities and perhaps other market participants from retail, agriculture, travel, transport and distribution, leisure and tourism firms.

TRENDS AND SEASONALITIES IN WEATHER AND CHOICE OF LOOKBACK PERIOD

Financial contracts derived from weather-specific measures, such as the expected future value of a local temperature, require the ability to predict regional weather conditions, months into the future. Thus, an effective model of the variations of a given weather-specific measure over the course of many months is essential for the accurate pricing of a weather derivative.

Meteorologists employ a variety of sophisticated models, involving many tens of parameters, to make both long and short range predictions about evolving weather conditions, both regionally and globally. Whatever the assumptions employed to model the behavior of weather-specific measures, one is confronted with inescapable truisms. Weather dynamics are governed by the laws of physics, from which yesterday's cold and damp overcast unfolds into today's warm, clear-blue sky--albeit with a measure of uncertainty. Uncertainty prevails when forecasting the weather, although trends exist. In practice, this means that today's weather is more predictable than tomorrow's, and tomorrow's is more predictable than that for next week.

Understanding past weather patterns, including seasonal effects, is an important part of long-range weather forecasting. It is quite evident, for example, that many weather stations have experienced long-term warming, whether this arises from increasing urbanization, more fossil fuel usage, or global warming. In the U.S. the National Weather Service (NWS) usually provides the data for such purposes.

The most important decision to be made at the time of analyzing prior data used to price a weather derivative is the choice of the "lookback" period. This is the period of time in which to estimate average temperatures and volatilities. Common wisdom holds that 10-20 years of weather data may be required, and that accounting for trends and seasonalities is
*essentiale rigor*.

PRICING MODELS FOR WEATHER DERIVATIVES

The
**Black-Scholes**
model, which has proven quite useful for financial options, is most probably inadequate for weather derivatives, for several reasons:

*
Weather does not "walk" quite like an asset price "random

walk," which can in principle wander off to zero (think of

degrees Kelvin) or infinity (hotter than the sun). Instead,

variables such as temperature tend to remain within

relatively narrow bands, probably because of a

mean-reverting tendency, i.e. a tendency to come back

to their historical levels.

*
Weather is not "random" quite like an asset price random

walk. Because of its chaotic nature, weather is

approximately predictable in the short run and

approximately random only in the long run. This means

that short-dated weather derivatives may behave

fundamentally different than their long-dated counterparts.

*
Weather derivatives usually provide for averaging over a

period of time, and are therefore are more akin to "Asian"

options, i.e. have a non-Black-Scholes payoffs.

*
Many weather derivatives are also capped in payoff,

unlike the standard Black-Scholes option.

*
The underlying variables (e.g. temperature or

precipitation) are not tradeable prices, and so pricing

cannot be free of economy risk aversion factors, unlike

the Black-Scholes model.

SIMULATIONS BASED ON HISTORICAL DATA OR "BURN ANALYSIS"

This approach is very simple to implement and tries to answer the question: What would have been the average payoff of the option in the past
*x*
years? The main objection is that it does not incorporate temperature forecasts in the pricing.

MONTE CARLO BASED SIMULATIONS

Monte Carlo simulations provide a flexible way to price different weather derivatives structures. Various types of averaging periods, such as those based on HDDs or CDDs, can be specified easily. Similarly, and as easily, a contractual cap placed on the price of the derivative can be taken into account. One can study a variety of stochastic processes of a given underlying weather-specific measure, say local temperature, whose parameters must necessarily be calibrated to the "lookback" period of available weather data.

For example, consider the relatively simple stochastic process described by:

which can be applied to the underlying of a weather derivative and subsequently studied using Monte Carlo techniques. Here,
*T*
is the temperature of the locale of interest, say Las Vegas, which varies with the passage of time*t*. (The function
*(*t*) is determined by imposing a constraint that the expected value of
*T*
at time
*t*
is consistent with the temperature forecasted for time
*t*.) The mean reversion parameter
*a(t)*
is synonymous with the duration of a seasonal average weather system, the order of a few days, which tends to pass through the locale. Put another way, a typical weather system tends to displace the locale's temperature from its seasonal norm as it enters the locale, but as the system leaves the locale, the temperature subsequently reverts to it seasonal norm. The Weiner process
*dz*
and seasonal volatility
*(*t*) describe the seasonal variations of
*T*
as normally distributed about the mean-reverting mean temperature. Mean reversion parameters and the term structure of volatilities can be fitted to historical data spanning a "lookback" period of many years.

*This week's Learning Curve was written by**Mark Garman**, president for**Financial Engineering Associates**,**Carlos Blanco**, manager of global support and educational services for**FEA**, and**Robert Erickson**, senior financial engineer for FEA.*