Weather Derivatives and Seasonal Forecast
Shiyong Yoo∗ This version: January 2003 Second version: November 2002 First version: June 2002
Abstract: The purpose of this paper is to incorporate the CPC (Climate Prediction Center) seasonal forecast in the temperature process so that conditional means and conditional variances of both the temperature and the CDD (Cooling Degree Day) may be redetermined in accordance with the seasonal forecasting probabilities. Under the Gaussian property of the underlying process, the prices of the CDD options conditioned on the seasonal forecasting can be calculated by both the pricing formula and the Monte Carlo simulation. Using the temperature data of five cities in the east coast of the United States, first in the case where there is no truncation in the temperature process, the Monte Carlo simulation shows the appropriate accuracy, which means that the CDD option values obtained through both the pricing formula and the Monte Carlo simulation are close enough. And the magnitude of changes in option values conditional on the seasonal forecast are relatively small. In cases where temperature paths less than 65◦ F are truncated, however, the option values obtained by the Monte Carlo simulation are very sensitive to the seasonal forecast probabilities and the magnitude of their variations is very substantial. This is because the density of the CDD conditional on the seasonal forecast shifts a large amount as a result of both truncation and aggregation.
JEL Classification: C15, G13. Keywords: weather derivatives, seasonal forecast, incomplete market, OrnsteinUhlenbeck process, cooling degree days, Monte Carlo simulation.
∗ Department of Applied Economics and Management, Cornell University, Ithaca, NY 14853, Tel: (607) 255-5467; E-mail: sy61@cornell.edu
1
Introduction
In recent years, there has been a large increase in the traded volume of weather derivatives at various locations in the United States as well as in Europe as the weather risk is significantly recognized. These weather derivatives are different in some respects to traditional financial assets. Unlike financial derivatives that are useful for price hedging but not for quantity hedging, weather derivatives are mainly used to hedge the volumetric risks, even though weather-related quantity and price are closely related to each other. Weather derivatives are also different from weather insurance. Whereas weather insurance covers high-risk, low-probability events, weather derivatives protect revenue against lower-risk, high-probability events. Weather derivatives can also have benefits over weather insurance in that they remove moral hazard. The weather derivatives market has grown rapidly. The history of weather derivatives started in 1996 when deregulated energy markets were exposed by the weather risk. Energy companies like Enron, Koch Industries, and Aquila were the first parties to arrange for and issue weather derivatives in 1996, and one of the publicized trades was executed between Koch Energy and Enron in 1997. Although the weather derivatives market started in the US energy market in 1997, it has been getting more liquid and global. In September 1999, the Chicago Mercantile Exchange (CME) started an electronic market place for weather derivatives to increase the size of the market and to remove credit risk from trading weather contracts. However, it will not likely be as good as traditional price-hedging markets, since weather is by its nature a very location-specific and non-standardized commodity, unlike a specific grade of crude oil. And the over-the-count (OTC) market is still more active than the exchange, so the bid/ask spreads are quite large. Weather derivatives depend on the evolution of meteorological variables such as heating degree days, cooling degree days, average temperature, maximum temperature, minimum temperature, precipitation (rainfall, snowfall), among others. These contracts are widely used by many economic agents whose economic outcome depends on those meteorological variables. The electricity sector is especially very sensitive to the temperature. Not only the electricity load but also the electricity price are affected by meteorological variables. In Li and Sailor (1995), and Sailor and Mu˜oz n (1997), temperature is the most significant weather factor explaining electricity and gas demand in the United States. The impact of the temperature in both electricity demand and price has been considered in many other papers, including Peirson and Henley (1994), Henley and Peirson (1988), Engle et al. (1992), and Pardo et al. (2000). In general, a strong relationship between electricity load and temperature has been observed in the PJM (Pennsylvania–New Jersey–Maryland), the NYISO (New York Independent Operating System), and the ISO New England market (see Figure 1). The dependence of electricity load on temperature is significant, and the relation is non-linear, showing an increasing electricity demand both for low and high temperatures, corresponding to winter and summer, respectively. Weather risk is volumetric risk and is a separate issue from price risk that may also be present. There are many 1
derivatives instruments at utilities’ disposal to hedge price risks, however weather derivatives are unique in enabling agents to hedge their volume risks. Weather derivatives are a classic incomplete market, because underlying weather variables are not tradable. When the market is incomplete, prices cannot be derived from the no-arbitrage condition, since it is not possible to replicate the payoff of a given contingent claim by a controlled portfolio of the basic securities. And this is why the classical Black-Sholes-Merton methodology cannot be directly applied. There are several approaches for dealing with incomplete markets.1 One of them is to introduce the “market price of risk” for the particular type of incomplete market, namely a “factor model,” a market where there are some non-traded underlying objects. Weather derivatives are also path-dependent. They are very similar to the average price Asian option. Geman and Yor (1993) used Bessel processes, which have the merit of being stable by additivity and of being related to the geometric Brownian motion through a time change, to obtain an exact analytical expression of the Laplace transformation in time of the option price. The greeks are obtained with the same accuracy due to the linearity of the operators derivation and Laplace transformation. Another possible way of pricing the Asian option is to use a Monte Carlo simulation. Geman and Eydeland (1995) show that because of the smoothness of the Asian payout, a good approximation is obtained by a relatively low number of runs (e.g., 10,000). The same accuracy for the greeks, however, requires a higher number of simulations. Alaton et al. (2002) derived the closed form pricing formula for the weather derivative using the Gaussianity property of the underlying distribution of the temperature process. Since the sum of the temperature processes over a certain period also follows the Gaussian process, if the individual temperature process follows the Ornstein-Uhlenbeck (O-U) process, which is the Gaussian process. The price of the weather derivatives heavily depends on the expectation about the future temperature behavior. Without any information about future temperature patterns, all the risk and the uncertainty about future temperature are embedded in the market price of risk. However, since the dynamics of the climate system are chaotic (Wilks, 2002), seasonal forecasts (long-term weather forecast) are necessarily less specific than weather forecasts. Furthermore, since the chaotic dynamics of the system preclude even approaching exact specifications of some seasonally aggregated variables, the forecasts are expressed as probability distributions rather than deterministic point values. But this information about these probability distributions can be crucial to determine the price of weather derivatives.
1 In incomplete markets, exact replication is impossible and holding an option is a risky business, meaning that no preference-free pricing formula is possible. There are some approaches to deal with incomplete markets, even though none are quite satisfactory. After all, to get exactly one price for the derivative is to pick one equivalent martingale measure (EMM). Using utility-maximization or risk-minimization methodology, one EMM can be determined so that the exact price can be derived, because the potential option purchaser’s attitude to risk is specified in the utility function and from it the unique EMM (pricing kernel) is derived naturally. However, there can be more than one EMM. In that case, the bid/ask price or the no-arbitrage interval can be obtained using coherent risk measure(See Carr et al. (2001) for the coherent risk measure approach) or super/sub-replication.
2
In this paper, the effects of the CPC (Climate Prediction Center) seasonal forecast probabilities on the price of weather derivatives are explored. This paper is organized as follows. In section 2, the temperature process is modelled and estimated. specifically, the mean temperature process is described to reflect the CPC seasonal forecast probabilities under the assumption that conditional variance is constant. In section 3, the closed form pricing formula for the weather derivatives in an incomplete market is derived, following Alaton et al. (2002). In section 4, the long-term weather forecast is incorporated into pricing the weather derivatives. Section 5 presents the results from both the formula and the Monte Carlo simulation, and section 6 concludes.
2
2.1
Seasonal Forecast and Temperature Process
Seasonal Forecast
Recent advances in understanding the climate system have allowed successful forecasts of seasonal temperature and precipitation at lead times up to one year in advance. At least two groups currently produce operational seasonal forecast: the Climate Prediction Center (CPC) of the US National Centers for Environmental Prediction, and International Research Institute (IRI) for Climate Prediction. While much of their predictive ability is thought to derive from the effects of ENSO (El Ni˜o-Southern Osn cillation) on other parts of the climate system, the operational forecasts have demonstrated predictive skill during non-ENSO periods as well. Because the dynamics of the climate system are chaotic, seasonal forecasts are necessarily less specific than weather forecasts. In particular, the evolution of individual weather events cannot be explicitly forecasted at these timescales with any credibility. Rather, the predictands in existing operational seasonal forecast are seasonal average temperature and seasonal total precipitation. Furthermore, since the chaotic dynamics of the system preclude even approaching exact specifications of these seasonally aggregated variables, the forecasts are expressed as probability distributions rather than deterministic point values. Briggs and Wilks (1996) proposed that subseasonal statistics consistent with a given probabilistic seasonal forecast could be estimated by resampling the observed climate record for a location according to the probabilities in that forecast. Essentially, the procedure produces climatological statistics by weighting the contributions of the data from a particular year according to the probabilities in a forecast and the seasonal mean in that year, rather than weighting all years equally. One class of subseasonal statistics that can be estimated in this way (i.e., conditionally, on a seasonal forecast) is the parameter set of a stochastic weather generator. The NOAA (National Oceanic and Atmospheric Administration) has been issuing long-term weather forecasts for many years. The forecast, which is often called seasonal outlook, has a lead time of from 2 weeks to as long as one year. It is issued in the form of probability anomaly of above-normal (pA ), near-normal (pN ), and below-normal (pB ). The probability of above-normal is determined as pA = 2/3 − pB , whereas the probability of the near-normal usually remains unchanged at pN = 1/3. 3
The three categories are defined by the two terciles of the relevant climatological probability distribution, q1/3 and q2/3 , which divide the climatological distribution into three equal parts: q1/3 1 f (x) dx = , 3 −∞ and
q2/3 −∞
2 f (x) dx = , 3
where f (x) is the climatological probability density function (assumed to be Gaussian) for the climatic variable x, i.e., the sum of the temperature over the summer. The summer is below normal (or cool) if the sum of the whole summer temperature, x, falls into the region of (−∞, q1/3 ], near normal (or normal) if x ∈ (q1/3 , q2/3 ], and above normal (or warm) if x ∈ (q2/3 , ∞).
2.2
Temperature Process
Modelling and forecasting the temperature process is crucial to determine the price of the weather derivatives because it decides the amounts of predictable and unpredictable components of weather fluctuations, respectively. There are some basic statistical features of the daily temperature that help to find the appropriate statistical process for the daily temperature. Firstly, the daily temperature process shows a significant seasonal behavior that is not shown by other financial variables such as bond prices, stock indices, and currencies.2 A strong seasonal behavior with an annual period can be seen in Figure 2, and this should be taken into account in modelling the temperature process. The temperature process shows also the mean-reversion property. This means that the level of the daily temperature does not deviate too far from a long run equilibrium value. There exits heteroskedasticity in the temperature process across seasons for the whole year, however in case of only one season it is not clear. Even though it is very common in financial assets, it is not considered here since only the summer season is considered in this paper. Instead, the constant seasonal volatility is used to reflect seasonally different volatilities.3 Consequently, the temperature process basically incorporates seasonality and mean reversion.
The seasonality also observed in lots of commodity prices. Torr´ et al. (2001) found some characteristics observed in the Spanish temperature index fluco tuations: a mean reversion to the seasonal trend, an autoregressive behavior in the temperature conditional volatility, low sensitivity of the volatility to the temperature level and inverse relation between volatility and temperature level. And it shows that the temperature model is significantly improved by incorporating a mean reversion to a seasonal trend and generalized autoregressive conditional heteroskedasticity (GARCH) structure. Moreno (2000) showed that even though residuals from the estimated process have constant mean and variance through time, higher moments differ though time. It was therefore suggested that estimated temperature processes should not be used to simulate the temperature because the distribution of the noise is not identical through time.
3 2
4
The deterministic cyclical part of the unconditional temperature fluctuation can be explained by the sinusoidal function: Ttm = β0 + β1 sin(ωt) + β2 cos(ωt), (1)
where t denotes the date, and ω = 2π/365, since the period is one year (neglecting leap years).4 To incorporate the seasonal forecast, the unconditional mean temperature process is assumed to be explained by the linear combination of above-normal (warm), near-normal, and below-normal (cool) mean temperature processes (see Section 4). Ttm = pA Ttw + pN Ttn + pB Ttc ,
w w w n n n where Ttw = β0 + β1 sin(ωt) + β2 cos(ωt), Ttn = β0 + β1 sin(ωt) + β2 cos(ωt), and c c c Ttc = β0 + β1 sin(ωt) + β2 cos(ωt). A standard Wiener process (Wt , t ≥ 0) with respect to the natural probability measure P defined on the fundamental probability space (Ω, F, P) is appropriate not only with regard to the mathematical tractability of the model, but also because the residual of daily deterministic temperature fits the normal distribution well (see section 2.3). It is assumed that the variance σt of the residual process is a piecewise constant function, with a constant value during each three-month season, denoted by σ1 , σ2 , σ3 , and σ4 , for December through February, March through May, June through August, and September through November, respectively. Thus, the residual process of the temperature would be (σWt , t ≥ 0, σ = σs , s = 1, 2, 3, 4). The residual process of the temperature also has the mean-reverting property. Putting all these assumptions together, the temperature process can be modelled as a stochastic process solution of the following stochastic differential equation (SDE):
dTt =
dTtm + α(Ttm − Tt ) dt + σdWt , dt
(2)
where α ∈ R determines the speed of the mean-reversion, and σ = σs , s = 1, 2, 3, 4. The solution of such an equation is called an Ornstein-Uhlenbeck process. As the mean temperature Ttm is not constant, this term will adjust the drift so that the solution of the SDE has the long run mean Ttm . For 0 ≤ s ≤ t, the solution of the above SDE is as follows:
t
Tt = e−α(t−s) (Ts − Tsm ) + Ttm +
s
σe−α(t−τ ) dWτ
where Ttm = β0 + β1 sin(ωt) + β2 cos(ωt). The temperature process is a Gaussian process and has the conditional mean and the conditional variance: E[Tt |Fs ] = e−α(t−s) (Ts − Tsm ) + Ttm ,
4 Some temperature data reveal a positive trend, even though it is very weak and almost zero, because of global warming and urban heating effect. The time trend is therefore modelled to incorporate that kind of the trend in time. However, the data in this paper show that the estimate of the time trend is zero significantly. So the time trend is not included.
5
t
Var[Tt |Fs ] = σ 2
s
e−α(t−u) du =
σ2 1 − e−α(t−u) . 2α
σ2 . 2α
Therefore, as t → ∞, the limiting conditional variance becomes Finally, cooling degree-days (CDD) are defined as follows:
n
CDDn ≡
i=1
max{Ti − 65, 0}.
2.3
Parameter Estimation
The daily average temperature data from Boston, New York City, Allentown, Philadelphia and Washington DC for the 53-year period of January 1949 through December 2001 are analyzed (see Table 1 and 2). Table 1: Summer (June-July-August) Temperature (1949 ∼ 2001)
City Boston New York City Allentown Philadelphia DC #(obs) 4876 4876 4876 4876 4876 Mean 71.3 74.7 71.9 75.0 77.2 Median 71.5 75.0 72.0 75.5 77.5 Min 48.5 53.5 52.0 54.0 55.5 Max 92.5 93.0 88.5 92.0 93.0 STD 6.7 6.0 5.9 5.9 5.5 Skewness -0.3 -0.3 -0.4 -0.4 -0.5 Kurtosis 0.1 0.1 -0.1 -0.1 0.1
Table 2: Summer (June-July-August) Cooling Degree Days (1949 ∼ 2001)
City Boston New York City Allentown Philadelphia DC #(obs) 53 53 53 53 53 Mean 638.0 906.5 669.4 934.5 1131.3 Median 635 884 640 929 1118.5 Min 410.0 677.5 469.5 648.0 899.0 Max 872.0 1144.0 966.5 1249.0 1383.5 STD 112.4 126.2 124.5 144.7 127.7 Skewness 0.2 0.2 0.7 0.3 0.0 Kurtosis -0.7 -1.1 -0.6 -0.5 -0.8
The mean-reversion parameter (α) and seasonal volatility (σs , s = 1, 2, 3, 4) can be estimated by discretizing equation (2) for each season:
m m Ti = Tim − Ti−1 + αTi−1 + (1 − α)Ti−1 + σs εi−1 , m where εi is i.i.d. N(0, 1). Putting Ti ≡ Ti − (Tim − Ti−1 ), the following equation can be regarded as a regression equation: m Ti = αTi−1 + (1 − α)Ti−1 + σs εi−1 ,
that is, a regression of today’s temperature on yesterday’s temperature. And then 2 the efficient estimator of σs is obtained as follows: σs ˆ2 1 = Ns − 2
Ns
Ti − αTi−1 − (1 − α)Ti−1 ˆ m ˆ
i=1
2
,
6
where Ns is the number of days in the season. The estimates of the mean-reversion parameters and seasonal volatility are listed in Table 3. The goodness-of-fit tests for normality of the residuals of temperature processes are done (D’Agostino and Stephens, 1986). Anderson-Darling test (Stephens, 1974) is considered here. This test statistic (A2 ) belongs the quadratic class of EDF (empirical distribution function) statistics:
∞
A2 = n
−∞
[Fn (x) − F (x)]2 [F (x)(1 − F (x))]−1 dF (x),
and is calculated as follows: 1 A = −n − n
2 n
(2i − 1) [ln F (xi ) + ln(1 − F (xn+1−i ))] ,
i=1
where F is the cumulative normal distribution function, Fn the sample cumulative distribution function, and xi are ordered data of daily summer temperature. The pvalues for Anderson-Darling statistic are smaller than the usual cutoff values of 0.05 and 0.10, indicating that all residuals are normally distributed (see Table 3).5 Table 3: Mean-Reversion Parameter (ˆ ), Seasonal Volatility (ˆ ), Normality Test Statistic (A2 ) α σ
Parameter α ˆ σ1 ˆ σ2 ˆ σ3 ˆ σ4 ˆ A2 (p-value) Boston 0.3732 6.919 5.655 4.796 5.551 4.033 (0.005) New York 0.3254 6.487 5.740 4.052 5.185 12.362 (0.005) Allentown 0.3134 6.443 5.611 4.081 5.330 9.664 (0.005) Philadelphia 0.3140 6.416 5.618 3.886 5.271 16.832 (0.005) DC 0.3060 6.297 5.789 3.621 5.076 30.570 (0.005)
3
Pricing Weather Derivatives
The market for weather derivatives is a typical example of an incomplete market, because the underlying variable–the temperature–is not tradable. Therefore the market price of risk λ is introduced in order to obtain unique prices for such contracts. And for simplicity, it is assumed that the market price of risk is constant. Furthermore, a risk free interest rate r is assumed to be constant and the tick price is given as $1 per degree day. Thus, under a martingale measure Q, characterized by the market price of risk λ, the price process also denoted by Tt satisfies the following dynamics: dTt =
5
dTtm + α(Ttm − Tt ) − λσ dt + σdWtQ , dt
(3)
Other test statistics such as Kolmogorov-Smirnov and Cram´r-von Mises show the same result. e
7
where (WtQ , t ≥ 0) is a Q-Wiener process. For 0 ≤ s ≤ t, its solution under Q-measure is: t λσ Tt = e−α(t−s) (Ts − Tsm ) + Ttm − 1 − e−α(t−s) + σe−α(t−τ ) dWτQ . α s This follows the Gaussian process with conditional mean and conditional variance, respectively: EQ [Tt |Fs ] = e−α(t−s) (Ts − Tsm ) + Ttm − VarQ [Tt |Fs ] = λσ 1 − e−α(t−s) , α (4)
σ2 (5) 1 − e−2α(t−s) . 2α The covariance of the temperature between two different days, for 0 ≤ s ≤ t ≤ u, CovQ [Tt , Tu |Fs ] = e−α(u−t) VarQ [Tt |Fs ] = e−α(u−t) σ2 1 − e−2α(t−s) . 2α
Since the price of a derivative is expressed as a discounted value under the martingale measure Q, the expected value and the variance of the price process Tt can be easily computed. Indeed, as a Girsanov transformation only changes the drift term, the variance of the price process Tt is the same under both measures. Suppose now that t1 and t2 denote the first and last day of a season and start the process at some time s from the season before t ∈ [t1 , t2 ]. To compute the expected value and variance of Tt in this case, we split the integrals in equation (4) and equation (5) into two integrals where σ is constant in each one of them. For s t1 and t ∈ [t1 , t2 ], the conditional mean is µt ≡ EQ [Tt |Fs ] = e−α(t−s) (Ts − Tsm ) + Ttm − and the conditional variance is υt ≡ VarQ [Tt |Fs ] 2 2 2 2 σi − σj −2α(t−t1 ) σi −2α(t−s) σj = e − e + 2α 2α 2α
2 2 where σi is the variance of the season where s is and σj is that of the season of [t1 , t2 ]. Therefore, under Q and the given information set at time s, each temperature value process follows the Gaussian process:
λ λσi −α(t−s) λσj (σi − σj )e−α(t−t1 ) + e − , α α α
Tt |Fs ∼ N(µt , υt ).
8
3.1
Pricing Cooling Degree Day Option
X = κ max(CDDn − K, 0),
The payout of the CDD European call option is of the form
where, for simplicity, κ = 1, unit of currency per degree day and K is the strike, and
n
CDDn =
i=1
max(Tti − 65, 0).
And, for simplicity, suppose that max(Tti − 65, 0) > 0.6 The contract above is very similar to the Asian option, which depends on the arithmetic average of the stock price during some periods. Not like the case of a log-normally distributed underlying asset, in which the closed form of pricing formula is not available, the cooling degree days option depends on the summation of the CDD over the summer in which each temperature process follow the Gaussian process, that is, Tt ∼ N(µt , υt ). Then we can get the conditional mean and conditional variance of the CDD that is the sum of the each Gaussian process over the summer. Therefore,
n
CDDn =
i=1
Tti − 65n,
and, for t < t1 , the conditional mean and the conditional variance of the CDDn can be obtained as follows: CDDn ∼ N(mn , s2 ), n where,
n
mn ≡ E [CDDn |Ft ] =
Q i=1 n
EQ [Tti |Ft ] − 65n, CovQ [Tti , Ttj |Ft ].
i