Weather Derivatives Pricing Using
Regime Switching Models
Emanuel Evarest, Fredrik Berntsson, Martin Singull
and Xiangfeng Yang
Switching Models
Emanuel Evarest
1,2, Fredrik Berntsson
1, Martin Singull
1and
Xiangfeng Yang
11
Department of Mathematics
Linköping University
SE-581 83 Linköping, Sweden
emanuel.evarest@liu.se, fredrik.berntsson@liu.se,
martin.singull@liu.se, xiangfeng.yang@liu.se
2
Department of Mathematics
University of Dar Es Salaam
P.O.Box 35062, Dar Es Salaam, Tanzania
Abstract
In this study we discuss the pricing of weather derivatives whose underlying weather variable is temperature. The dynamics of temperature in this study follows a two state regime switching model with a heteroskedastic mean reverting process as the base regime and a shifted regime defined by Brownian motion with mean different from zero. We develop the mathematical formulas for pricing futures contract on heating degree days (HDDs), cooling degree days (CDDs) and cumulative average temperature (CAT) indices. We also present the mathematical expressions for pricing the corresponding options on futures contracts for the same temperature indices. The local volatility nature of the model in the base regime captures very well the dynamics of the underlying process, thus leading to a better pricing processes for temperature derivatives contracts written on various index variables. We provide the description of Montecarlo simulation method for pricing weather derivatives under this model and use it to price a few weather derivatives call option contracts.
Keywords:Weather derivatives, Arbitrage-free pricing, Regime switching, Monte Carlo simulation, Option pricing
1
Introduction
During the 20th century, weather derivatives emerged in the financial market as an
in-strument for hedging various weather related risks in the energy sector and other closely related sectors. Weather derivatives were officially introduced in the major financial mar-kets like Chicago Mercantile Exchange (CME) in 1999 [9, 8]. Over the recent years, weather Derivatives have gained popularity where apart from energy and power indus-tries who were the first indusindus-tries in the business, other indusindus-tries and sectors joined the business. These industries include agricultural, tourism, insurance and reinsurance companies, brokers as well as retail business, which are now among the many partici-pants in the weather derivatives markets [18]. Currently, CME offers weather derivatives futures contracts and options based on daily average temperature indices in over 46 dif-ferent cities around the world, see [19, 25]. The key factor in using weather derivatives as an instrument for hedging risk is a reliable pricing method that could accelerate the development of weather derivatives which seems to be impended by lack of standard pricing method [16]. The situation has been contributed by its nature, where the under-lying weather variable is not a tradable asset. Also there is little liquidity in the market, implying that the weather derivatives market is an incomplete market. Due to the na-ture of incomplete market pricing models in acknowledging the presence of both hedge-able and unhedgehedge-able risks, they are considered as more appropriate for pricing weather derivatives [6]. Weather derivatives market comprises of derivatives written on various weather variables like temperature, precipitation, wind and Hurricanes. Temperature derivatives contract are written based on heating degree days (HDDs), cooling degree days (CDDs), cumulative average temperature (CAT) and Pacific Rim (PRIM) indices [3, 18].
For a daily average temperature process Td(t), these temperature indices are defined
as AccHDDs =Rt2 t1 max Tref− ˜Td(t) dt, AccCDDs = Rt2 t1 max ˜Td(t) − Tref dt. Also, CAT = Rt2 t1 ˜ Td(t)dt and PRIM = t2−t11 R t2 t1 ˜
Td(t)dt, where Tref is the reference
temperature for regulating domestic heating and cooling demands. For weather deriva-tives based on other weather variables like precipitation and wind see [18, 25]. Generally, a standardized weather derivative contract comprises of a weather measurement station, a weather variable (s), a contract period, a payoff function, and a premium depending on the nature of contract. The variables defining the payoff function also varies according to the type of contract that might be swaps, options, collars, straddles, strangles and binaries, see [23, 18].
In this study we focus on temperature derivatives pricing because the largest proportion ( about 98%) of weather derivatives currently traded in the market are based on temper-ature [14]. Specifically, we focus on pricing these derivatives whose underlying
tem-perature process is governed by a regime switching model with a heteroskedastic base regime. The model allows the volatility of the underlying process in the base regime to vary with changes in temperature process. In general regime switching models allow the underlying process to switch between states of the process over time. That means there are periods where the stochastic process is under the base regime and periods when it is under the shifted regime. The switching process is caused by changes in volatility of the underlying financial variables like interest rate, energy prices, weather variables etc. As a result, regime switching models have attracted many applications in various problems in economics and finance particularly in the area of option valuation, see [5, 4, 10, 20, 24]. For temperature dynamics, factors like deforestation, urbanization, changes of weather station and clear skies contribute to the discrete shifts of temperature stochastic process from one state to another [15].
The regime switching models represent temperature dynamics relatively better than single regime models due to its ability to capture the discrete shifts in the process. Then pric-ing of weather derivatives based on regime switchpric-ing models will also provide relatively good pricing method since the underlying temperature models captures most of the nec-essary features of the underlying variables. Most of the research on weather derivatives pricing has been done on single regime models for different noise driving processes like Brownian motion, fractional Brownian and Levy process see [7, 1, 2, 21, 22]. The model for pricing weather derivatives used in this study is adopted from our previous article [13]. The model allows the volatility of temperature process to vary locally under its base regime temperature process. Thus, the price process of weather derivatives contracts un-der the equivalent measure will have the mean and variance as functions of the unun-derlying variable dynamics.
The contribution of this study is as follows: We develop mathematical expressions for pricing temperature derivatives contracts written on CAT, HDDs and CDDs futures. Then we formulate the dynamics of these futures prices under the equivalent probability mea-sure. Also pricing formulas for European call option price written on futures contracts are developed. Due to the complicated nature of the pricing formulas developed for weather derivatives contracts, they dont give explicit formulas for the expected payoff of the HDDs, CAT and CDDs call option contracts. Therefore, we describe the Monte Carlo simulation approach for the underlying temperature dynamics model and then use it to price the call option contracts.
The remaining part of the article is organized as follows: In Section 2, we present a regime switching model for temperature derivatives pricing, and transformed to an equiv-alent probability measure Q by introducing the market price of risk. Also the closed form
solution of the regime switching model under the Q measure is presented. In Section 3 we provide closed form expressions for arbitrage free pricing of CAT, HDDs and CDDs fu-tures contracts together with the dynamics of these fufu-tures with respect to the underlying regime switching process. In Section 4 we describe a Monte Carlo simulations method for pricing temperature derivatives for European call options and then compute the expected payoff of these contracts under the market price of risk. Finally, in Section 5 we give concluding remarks and possible future work.
2
Regime Switching Model for Temperature Derivatives
In this section, we consider the regime switching model for temperature dynamics for pricing weather derivatives contracts. The model allows us to reflect different states of weather dynamics contributed by various factors. We adopt the model from [13], defining temperature dynamics by two states regime switching model, comprising of mean revert-ing heteroskedastic process as base regime and Brownian motion as the shifted regime.
The model represent the dynamics of the deseasonalized temperature ˜Td(t), after
remov-ing the seasonality Sd(t) from the daily average temperature Td(t). The model is given
as ˜ Td(t) = ˜ Tt,1 : d ˜Tt,1= (µ1− β ˜Tt,1)dt + σ1T˜t,1dWt, with probability q1, ˜ Tt,2 : d ˜Tt,2= µ2dt+ σ2dWt, with probability q2, (2.1)
where β is the mean reversion speed,µ1
β is the long-term mean and σ1is the volatility of
the mean-reverting heteroskedastic proces in base regime while µ2and σ2 are the mean
and volatility of the shifted regime process respectively. The seasonality is defined by
Sd(t) = A1sin
2π
365(t − A2)
+ A3t+ A4, (2.2)
where A1is the amplitude, A2is the phase angle, A3and A4are constants defining the
linear trend. The movement of the regime switching process from one state to another is driven by the transition probabilities given by
Pij = P (St= j|St−1= i) for i, j = 1, 2. (2.3)
Using the Ito formula, an integral representation to (2.1) can be derived and written as ˜ Td(t) = ˜ Tt,1 : ˜Tt,1= µβ1 + ( ˜T0,1−µβ1)e−βt+R0tσ1T˜s,1e−β(t−s)dWs ˜ Tt,2 : ˜Tt,2= ˜T0,2+ µ2t+R0tσ2dWs. (2.4)
Note that both volatilities σ1and σ2are positive numbers.
The regime switching temperature dynamics model is calibrated using the Expectation Maximization algorithm, see [11, 17]. In this algorithm the whole vector of unknown parameters
θ= {q, µ1, µ2, β, σ1, σ2}
is estimated by two steps iterative algorithm. The two-step iterative procedure alternates between the conditional expectation computation and solving the unconstrained optimiza-tion problem with respect to the set of unknown model parameters using the historical daily average temperature data from Malmslätt, Linköping in Sweden for the period of January 1998 to December 2001. In the expectation step, the expectation of likelihood function is computed by considering the missing variables as observable ones. In the max-imization step, the maximum likelihood estimation of the unknown parameters are com-puted by maximizing its expected likelihood function obtained in the expectation step, for more details see [13]. The estimation process begins with estimation of unknown parame-ters from the seasonality process given by (2.2) by using the Gauss-Newton Least squares method. The parameter estimates from the seasonality process given in Table 2 are in turn used to produce the deseasonalized temperature data set that is used for parameter estimation for the model given by (2.1). The estimates for unknown parameters for (2.1) are shown in Table 1 and simulated temperature values for estimated set of parameters is given by Figure 1.
Parameter q1 q2 µ1 β σ1 µ2 σ2
Estimates 0.9630 0.0370 3.0104 6.4060 2.2420 0.0185 0.0651
Table 1: Parameter estimates for the two states regime switching model based on Malm-slätt historical data from January 1998 to December 2001
Parameter A1 A2 A3 A3
Estimates 9.5786 78.6415 −4.4 × 10−5 6.9360
Table 2: Parameter estimates for the seasonality process based on Malmslätt historical data from January 1998 to December 2001.
For the model given in (2.1) to be used for pricing temperature derivatives, a Girsanov theorem given by Thomas et al [12] is needed to transform the model for temperature dynamics from its existing probability measure P to an equivalent probability measure Q, where the solution of resulting process is a martingale under the new probability measure
Q. The two probability measures P and Q are related by
dQ(ω) = L(ω)dP(ω), (2.5)
where the function L(t) is called the Radon-Nikodym derivative of Q with respect to P. Theorem 2.1 (Girsanov) The stochastic process
L(t) = exp − Z t 0 γdW(s) − 1 2 Z t 0 γs2ds , t∈ [0, T ], (2.6)
is a martingale process with respect to the natural Wiener filtrationFt= σ(W (s), s ≤ t),
fort∈ [0, T ], under the probability measure P. The relation
Q(A) = Z
A
LT(ω)dP(ω), A∈ FT, (2.7)
defines an equivalent probability measure Q∼ P on FT in such a way that under Q,
Vt= Wt+
Z t
0
γsds, t∈ [0, T ],
is a Wiener process on(Ω, Ft, Q).
Using Girsanov’s theorem under the equivalent measure Q, we have
dVt= dWt+ γtdt, (2.8)
where γtis a real-valued function representing the market price of risk. Combining (2.1)
and (2.8), a stochastic process for temperature dynamics under the risk-neutral probability measure Q is obtained and given by
˜ Td(t) = ˜ Tt,1 : d ˜Tt,1= (µ1− β ˜Tt,1− σ1T˜t,1γt)dt + σ1T˜t,1dVt, with probability q1, ˜ Tt,2 : d ˜Tt,2= (µ2− σ2γt)dt + σ2dVt, with probability q2. (2.9)
Recall that an Itˆo stochastic process with variable X is defined by
dX= a(X, t)dt + b(X, t)dWt, (2.10)
where dWtare increments of a Wiener process. Suppose f (t, X) ∈ C2(R) is a twice
continuous differentiable function, then f (t, X) also follows an Itˆo process
df = ∂f ∂t + a ∂f ∂X + 1 2b 2∂2f ∂X2 dt+ b∂f ∂XdWt, (2.11)
where a and b2are the mean and variance of the stochastic process (2.10), while
∂f ∂t + a ∂f ∂X + 1 2b 2∂2f ∂X2 and b∂f ∂X 2
are the mean and variance of the process (2.11), respectively.
Using Itˆo’s formula the solution to (2.9) at any time x ≥ t under the probability space
(Ω, Ft, Q) is derived as follows: For the base regime, let
Yx= ˜Tx,1−
µ1
β . (2.12)
Differentiating (2.12) and comparing with the base regime of (2.9) we obtain
dYx= d ˜Tx,1= −βYxdx− σ1γxT˜x,1dx+ σ1 Yx+ µ1 β dVx. (2.13)
Using the exponential term eβtY
x, we have d eβxYx = βeβxYxdtx+ eβxdYx= −σ1γxT˜x,1dx+ eβxσ1 Yx+ µ1 β dVx. (2.14)
The solution of (2.14) is given by
Yx= Yte−βx− Z x t σ1γsT˜s,1e−βxds+ e−βx Z x t eβsσ1 Ys+ µ1 β dVs. (2.15)
Substituting (2.12) into (2.15) we obtain the solution for the base regime in (2.9) as the expression ˜ Tx,1 = µ1 β + ˜ Tt,1− µ1 β e−βx− Z x t σ1γsT˜s,1e−βxds+ Z x t e−β(x−s)σ 1T˜s,1dVs. (2.16) Similarly, the solution of the shifted regime is given by
˜ Tx,2= ˜Tt,2+ µ2x− Z x t σ2γsds+ Z x t σ2dVs. (2.17)
Hence, the integral form for (2.9) for any time x ≥ t is given by
˜ Td(x) = ˜ Tx,1=µβ1+ ˜Tt,1−µβ1 e−βx−Rx t σ1γsT˜s,1e −βxds+Rx t e −β(x−s)σ 1T˜s,1dVs ˜ Tx,2= ˜Tt,2+ µ2x− Rx t σ2γsds+ Rx t σ2dVs. (2.18)
Therefore, it can be observed that under the probability measure Q, the process ˜Td(x)
conditioned to filtration Ft, t≤ x is normally distributed with mean
EQh ˜Td|Ft i = 2 X i=1 qiEQh ˜Tx,i|Ft i , (2.19) and variance V arQh ˜Td|Ft i = 2 X i=1 qiV arQh ˜Tx,i|Ft i + 2 X i=1 qi EQh ˜Tx,i|Ft i2 − 2 X i=1 qiEQh ˜Tx,i|Ft i !2 , (2.20)
where EQh ˜Tx,i|Ft i = µi β (2−i) + ˜Tt,i (i−1) + (µix)(i−1)+ ˜ Tt,i− µ1 β e−βx (2−i) − 2 − Z x t σiγs ˜Ts,ie −βx(2−i) ds, (2.21) and V arQh ˜Tx,i|Ft i = Z x t σi2e−2β(x−s)T˜2 s,i (2−i) ds, (2.22)
for the process ˜Td(x) being in regime i, with probability qi. The EQh ˜Td|Ft
i and
varQh ˜Td|Ft
i
are obtained using the idea of weighted mixture of the regimes. The simu-lated daily average temperature under the real world measure P and equivalent probability measure Q is shown on Figure 1.
Time(days) 0 200 400 600 800 1000 1200 1400 1600 1800 2000 Temperature -20 -10 0 10 20 30 40 Time(days) 0 200 400 600 800 1000 1200 1400 1600 1800 2000 Temperature -20 -10 0 10 20 30 40
Figure 1: Simulated daily average temperature under real world measure P (left) and equivalent probability measure Q (right), showing drift adjustment in the dynamics of temperature process.
3
Pricing of Temperature Derivatives
In this section, the regime switching model for temperature dynamics presented in (2.9), is used for pricing weather derivatives contracts written on temperature indices under the equivalent probability measure Q. Theoretical mathematical expressions for pricing temperature derivatives contracts written on CAT, HDDs and CDDs indices are presented.
3.1
Futures and Options pricing for the CAT index
Given the CAT future weather derivatives contract for a period [x1, x2] in days. The
arbitrage free future price at time t ≤ x1 < x2, under a risk-free probability measure Q
with risk-free interest rate r is given by
er(x2−t)E Q Z x2 x1 ˜ Td(x)dx − FCAT(t, x1, x2)|Ft = 0 (3.23)
Since an arbitrary stochastic process Y = (Yt, t≥ 0) is always adapted to the natural
filtration generated by Y , then we can say that FCAT is Ftadapted. Therefore the CAT
future price FCAT(t, x1, x2) for a temperature derivative contract is given by
FCAT(t, x1, x2) = EQ Z x2 x1 ˜ Td(x)dx|Ft . (3.24)
In a similar fashion, the HDDs future price at time t ≤ x1< x2is given by
FHDD(t, x1, x2) = EQ Z x2 x1 max0, Tref − ˜Td(x) dx|Ft . (3.25)
Using (2.18) and (3.24), the price of future temperature derivative contract for CAT index
at time t such that t ≤ x1< x2can be derived as follows: For the base regime we have
FCAT(t, x1, x2) = EQ Z x2 x1 ˜ Td(x)dx|Ft = EQ Z x2 x1 µ1 β + ˜ Tt,1− µ1 β e−βx−Z x t σ1γsT˜s,1e −βx ds+ Zx t e−β(x−s) σ1T˜s,1dVs dx|Ft = Z x2 x1 µ1 βdx+ Z x2 x1 ˜ Tt,1− µ1 β e−βx dx− Z x2 x1 Z x t σ1γzT˜z,1e −βz dzdx = Z x2 x1 µ1 βdx+ Z x2 x1 ˜ Tt,1− µ1 β e−βx dx− Z x2 x1 Z x2 t χ[t,x](z)σ1γzT˜z,1e −βz dzdx (3.26)
where χ[t,x]is the indicator function. Since χ[t,x]is zero outside the interval [t, x], then
we can change the order of integration for the second integral containing χ[t,x]. Splitting
them, we get FCAT(t, x1, x2) = Z x2 x1 µ1 β dx+ Z x2 x1 ˜ Tt,1− µ1 β e−βxdx − Z x1 t Z x2 x1 χ[t,x](z)σ1γzT˜z,1e−βzdxdz − Z x2 x1 Z x2 x1 χ[t,x](z)σ1γzT˜z,1e−βzdxdz. (3.27)
Similarly for shifted regime, FCAT(t, x1, x2) = Z x2 x1 ˜ Tt,2dx+ Z x2 x1 µ2xdx− Z x1 t Z x2 x1 χ[t,x](z)σ2γzdxdz − Z x2 x1 Z x2 x1 χ[t,x](z)σ2γzdxdz. (3.28)
Hence, the price of CAT future contract at time t ≤ x1< x2is given by
FCAT(t, x1, x2) = 2 X i=1 qi " µi β(x2− x1) 2−i + 1 β2e −β(x2−x1)˜ Tt,i− µi β 2−i + h(t) + I1 # , (3.29) where h(t) = ˜Tt,i(x2− x1) i−1 +µi 2 (x2− x1) 2i−1− 2, and I1= Z x1 t Z x2 x1 σiγz ˜Tz,ie−βz (2−i) dxdz+ Z x2 x1 Z x2 x1 σiγz ˜Tz,ie−βz (2−i) dxdz.
The dynamics dFCAT(t, x1, x2) of future prices under the equivalent probability measure
Qfor temperature process ˜Td(x) is obtained by applying Itˆo formula to the process (3.29).
Since FCAT is a martingale process under the measure Q, then
dFCAT(t, x1, x2) = ˜VCAT(t, x1, x2, ˜Td(x))dVt, (3.30) where ˜ VCAT(t, x1, x2, ˜Td(x)) = σi dFCAT d ˜Td . (3.31)
The term ˜VCAT(t, x1, x2, ˜Td(x)) is interpreted as the volatility of the CAT future
dynam-ics.
For a call option written on CAT futures for any given contract period, the price can be estimated as follows: By definition, the call option price at exercise time c with strike level K is given by
CCAT(t, c, x1, x2) = e−r(c−t)EQ[max (FCAT(c, x1, x2) − K, 0) |Ft] . (3.32)
But the dynamics of FCAT given by (3.30) can also be given as
FCAT(c, x1, x2) = FCAT(t, x1, x2) +
Z c
t ˜
VCAT(s, x1, x2, ˜Td(x))dVs. (3.33)
Thus, FCAT(c, x1, x2) conditioned on FCAT(t, x1, x2) under the probability measure Q
Rc t V˜
2
CAT(s, x1, x2, ˜Td(x))ds. Therefore,
CCAT(t, c, x1, x2) = e−r(c−t)EQ[max (FCAT(c, x1, x2) − K, 0) |Ft]
= e−r(c−t) Z ∞ K (y − K)fFCAT(y)dy = e−r(c−t) (FCAT(t, x1, x2) − K) Φ FCAT(t, x1, x2) − K q Rc t V˜ 2 CAT(s, x1, x2, ˜Td(x))ds + e−r(c−t) Z c t ˜ VCAT2 (s, x1, x2, ˜Td(x))dsφ FCAT(t, x1, x2) − K q Rc t V˜CAT2 (s, x1, x2, ˜Td(x))ds , (3.34)
where Φ is the cumulative standard normal distribution function and φ(y) = Φ′(y) is the
density function.
3.2
Futures and Option Pricing for HDDs and CDDs indices
Also, the future price for the weather derivative contract written on HDDs and CDDs can be developed in a similar fashion as that of the CAT using (2.18) and (3.25) respectively.
Therefore, future price for HDDs contract at time t ≤ x1< x2is given by
FHDD(t, x1, x2) = EQ Z x2 x1 maxTref− ˜Td(x), 0 dx|Ft = Z x2 x1 EQ h maxTref− ˜Td(x), 0 |Ft i dx. (3.35)
But the process ˜Td(x) in (2.18) is normally distributed with mean and variance given
by (2.19) and 2.20 respectively. Then, it follows that, Tref − ˜Td(x) will be normally
distributed with mean Z(t, x, ˜Td(x)) and variance V2(t, x, ˜Td(x)) given by
Z(t, x, ˜Td(x)) = Tref− EQh ˜Td(x)|Ft i (3.36) and V2(t, x, ˜Td(x)) = V arQh ˜Td(x)|Ft i . (3.37)
Using the properties of normal distribution, it follows that the arbitrage free price of HDDs futures is given by FHDD(t, x1, x2) = Z x2 x1 V(t, x, ˜Td(x))Π Zt, x, ˜Td(x) V(t, x, ˜Td(x)) dx, (3.38)
where Π(z) is the function defined in terms of cumulative standard normal probability distribution Φ defined as Π(z) = zΦ(z)+φ(z) and φ(z) is the probability density function
(i.e Φ′(z)). The future price for the weather derivative contract written on CDDs F
CDD,
follows in similar way as in FHDD.
Since FHDD(t, x1, x2) is a martingale process under the measure Q, the dynamics of
FHDD(t, x1, x2) is obtained by using the Itˆo formula. Considering the diffusion part
since the drift term is zero, we have
dFHDD(t, x1, x2) = ˜VHDD t, x1, x2, ˜Td(x) dVt. (3.39) But ˜ VHDD t, x1, x2, ˜Td(x) =σi Z x2 x1 V′ (t, x, ˜Td(x))Π Zt, x, ˜Td(x) V(t, x, ˜Td(x)) dx + σi Zx2 x1 V(t, x, ˜Td(x))Π ′ Z t, x, ˜Td(x) V(t, x, ˜Td(x)) ˜U t, x, ˜Td(x) dx, (3.40) where Π′(z) = Φ(z) and ˜ Ut, x, ˜Td(x) = V(t, x, ˜Td(x))Z′ t, x, ˜Td(x) − Zt, x, ˜Td(x) V′(t, x, ˜Td(x)) V2(t, x, ˜T d(x)) .
For a weather derivative call option written on HDDs indices with contract period [x1, x2]
and strike level K, the arbitrage free price at exercise time c is given by
CHDD(t, c, x1, x2) = e−r(c−t)EQ[max (FHDD(c, x1, x2) − K, 0) |Ft] , (3.41)
where FHDD(c, x1, x2) is obtained from Equation (3.38) at exercise time c. The
Euro-pean call option price for contract written on CDDs index can be expressed in similar way as with HDDs index prices.
The presented mathematical expressions for pricing European call option weather deriva-tives contracts are complicated, in such a way that they do not provide explicit computa-tions of HDDs, CAT and CDDs opcomputa-tions prices. Thus, to achieve the explicit computacomputa-tions, we use Monte Carrlo simulation approach to compute the European call option prices for the weather derivatives contracts written on temperature.
4
Weather derivatives Option price by Monte Carlo
Sim-ulations
Due to the complexity nature of the dynamics of future prices, we choose to use numerical approach for computing the arbitrage free prices of weather derivative contracts written on CAT, HDDs and CDDs futures under the equivalent probability measure Q. The Monte Carlo Simulation technique is used to approximate the expected value of some function
h(Y (t)), where Y is the solution of some given stochastic differential equation. The
approximate value of expectation is given by the sample average
E[h (Y (t))] ≅ 1 n n X k=1 h(Yk(t)) . (4.42)
Ideally, the number of samples or simulations n has to be large enough to make the com-puted sample average approximately equal to the expectation value of the intended func-tion or random variable.
Consider the European call option for the contract period [x1, x2], the price of weather
derivative contract at time t ≤ x1 < x2written on HDDs index with strike level K is
given by
CHDD(t, x1, x2) = e−r(x2−t)ℵpEQ[max(HDD(x1, x2) − K, 0)] , (4.43)
where ℵpis the nominal price. The Monte Carlo simulation approach for arbitrage free
pricing of a European call option for temperature derivatives contracts proceed as fol-lows: We begin by simulating n independent and identically distributed daily average
temperature time series ˜Tdk(t), k = 1, 2, . . . , n under the probability measure Q. This is
followed by computation of accumulated HDDs, CDDs and CAT over the contract period
for each ˜Tdk(t) and their corresponding payoff given by
accHDDk= Z x2 x1 Tref − ˜Tdk(t) dx (4.44) and CHDDk(t, x1, x2) = e−r(x2−t)ℵpEQ[max(accHDDk− K, 0)] , (4.45)
respectively. Then, the expected payoff is computed by
CHDD(t, x1, x2) = 1 n n X k=1 CHDDk. (4.46)
Finally, the 100(1 − α)% confidence interval is constructed for CHDD by
Time(days) 0 200 400 600 800 1000 1200 1400 1600 1800 2000 Temperature -20 -10 0 10 20 30 40 Time(days) 0 200 400 600 800 1000 1200 1400 1600 1800 2000 Temperature -20 -10 0 10 20 30 40
Figure 2: Two different samples of daily average temperature obtained from Monte Carlo simulations of the dynamics of temperature process.
for given significance level α and standard error of estimate σn = √snn, where sn is the
sample standard deviation. The sample simulated daily average temperature for n = 1, 2 are shown in Figure 2.
Contract variables Contract1 Contract2 Contract3
Contract period 01 Dec - 30 Jany 01 July - 30 Aug 01 July - 30 Aug
Underlying index HDD CDD CAT
Contract type Call Option Call Option Call Option
Maturity date 30 January 30 August 30 August
Strike level 200 HDDs 50 CDDs 300 CAT
Risk-free rate 4% 4% 4%
Nominal price 400 400 400
Table 3: Weather derivatives contract specifications for some chosen measurement station. Monte Carlo simulations are performed for the weather derivatives contracts specified as shown in Table 3. The computed expected payoff of the weather derivatives contracts, together with their stochastic errors of the payoff and confidence interval for the given significance level α are given in Table 4 and Table 5.
The results of Monte Carlo simulations in Table 4 and Figure 3 shows that the expected payoff of the weather derivative call option contract for both HDDs and CAT converges to
3.1174 × 104and 2.5720 × 104respectively. These expected payoffs for HDDs and CAT
call options are shown by red straight lines in Figure 3 are obtained after 9000 iterations
with 95% confidence intervals of (3.1149, 3.1198) × 104 and (2.5693, 2.5748) × 104
HDD Call Option CAT Call Option Sample Expected 95% C.I. S.E Expected 95% C.I S.E
size Payoff Payoff
×104 ×104 ×104 ×104 100 3.1297 (3.1085, 3.1509) 108.1585 2.5726 (2.5451, 2.6002) 140.4832 500 3.1200 (3.1099, 3.1300) 51.4709 2.5772 (2.5647, 2.5896) 63.4158 1000 3.1173 (3.1101, 3.1246) 37.1917 2.5738 (2.5653, 2.5822) 43.0303 2000 3.1196 (3.1144, 3.1248) 26.5485 2.5723 (2.5664, 2.5783) 30.3881 3000 3.1187 (3.1145, 3.1229) 21.3834 2.5717 (2.5669, 2.5766) 24.7329 4000 3.1178 (3.1141, 3.1214) 18.7907 2.5728 (2.5687, 2.5770) 21.0518 5000 3.1166 (3.1133, 3.1199) 16.8149 2.5738 (2.5701, 2.5775) 18.9075 6000 3.1166 (3.1136, 3.1197) 15.4477 2.5734 (2.5700, 2.5768) 17.2214 7000 3.1165 (3.1137, 3.1193) 14.4276 2.5725 (2.5694, 2.5757) 16.0061 8000 3.1167 (3.1141, 3.1194) 13.4385 2.5721 (2.5691, 2.5750) 14.9390 9000 3.1174 (3.1150, 3.1199) 12.5690 2.5721 (2.5694, 2.5749) 14.0413 9010 3.1174 (3.1149, 3.1198) 12.5617 2.5720 (2.5693, 2.5748) 14.0346 9011 3.1174 (3.1149, 3.1198) 12.5603 2.5720 (2.5693, 2.5748) 14.0330 9012 3.1174 (3.1149, 3.1198) 12.5589 2.5720 (2.5693, 2.5748) 14.0320
Table 4: Monte Carlo Simulations results for HDDs and CAT Call option expected prices for the contract specification given by Table 3.
expected payoff of approximately 334.8852 after 9002 iterations with 95% confidence interval of (317.2751, 352.4953). From these results it shows that the expected payoff for these three call option contracts converges to their respective values with almost the same number of iterations. Also the CDDs expected payoff are very small compared to the corresponding CAT expected payoff due to the fact that summer temperature for European cities is not very much higher than the reference temperature. Therefore, our model agrees with the use of CAT weather derivatives contracts for European cities during summer period as it is done at CME market.
Also, different values of market price of risk (MPR) leads to different values of expected payoff as shown in Figure 3 (right). In this study we have done pricing of the call op-tion contracts under 0% market price of risk. Changing to 2% MPR, the expected pay-off changes accordingly. HDDs expected paypay-off increases with increase in MPR, while CDDs and CAT expected payoff decreases with increase in MPR. The change in the ex-pected payoff is the outcome of the change in drift term by the MPR.
Sample size Expected Payoff 95% C. I. S.E. 100 359.5794 (192.5575, 526.6013) 85.2153 500 376.4120 (296.6862, 456.1379) 40.6765 1000 343.2759 (291.5862, 394.9656) 26.3723 2000 339.8532 (301.0024, 378.7041) 19.8219 3000 340.5983 (309.2875, 371.9091) 15.9749 4000 336.2268 (310.1673, 362.2864) 13.2957 5000 336.8163 (313.0729, 360.5596) 12.1140 6000 336.2336 (314.8321, 357.6351) 10.9191 7000 336.6038 (316.6739, 356.5337) 10.1683 8000 334.0722 (315.5229, 352.6215) 9.4639 9000 334.9596 (317.3459, 352.5734) 8.9866 9001 334.9224 (317.3105, 352.5343) 8.9857 9002 334.8852 (317.2751, 352.4953) 8.9847 9003 334.8480 (317.2397, 352.4563) 8.9838
Table 5: Monte Carlo Simulations results for CDDs Call option expected prices for the contract specifications given by Table 3.
5
Conclusion
Weather derivatives plays an important role for risk management for various businesses that are directly influenced by unpredictable weather dynamics. The effectiveness of weather derivatives for minimizing weather related risks depends on the reliability of the model for the underlying weather variable. Weather variables are local with some com-mon features like mean reversion about a long term mean of the process. In this study we have presented the pricing of weather derivatives contracts based on temperature us-ing regime switchus-ing model with mean revertus-ing heteroskedastic base regime. We have developed mathematical expressions for HDDs, CDDs and CAT future contracts together their corresponding call option contracts on these futures. The developed option pricing formula are based on the local volatility of the base regime of the temperature dynamics model, thus capturing most of the local variations of the underlying temperature process. Also, the Montecarlo simulation approach for temperature derivative pricing presented in this study demonstrated a good convergence speed for expected payoff for the call option contracts written on HDDs, CDDs and CAT indices. The pricing of these contracts were based on some arbitrarily chosen constant for market price of risk due to the lack of real market prices for estimating it. Therefore, for realistic contract payoff, it is important
Number of Simulations
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000
HDDs Log Expected payoff
10.345 10.346 10.347 10.348 10.349 10.35 10.351 10.352 Number of Simulations 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000
HDDs Log Expected payoff
10.344 10.346 10.348 10.35 10.352 10.354 10.356 Number of Simulations 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000
CAT Log Expected payoff
10.151 10.152 10.153 10.154 10.155 10.156 10.157 10.158 10.159 10.16 10.161 Number of Simulations 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000
CAT Log Expected payoff
10.135 10.14 10.145 10.15 10.155 10.16 10.165 0% mpr 2% mpr Number of Simulations 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000
CDDs Log Expected payoff
5.45 5.5 5.55 5.6 5.65 5.7 5.75 5.8 5.85 5.9 5.95 Number of Simulations 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000
CDDs Log Expected payoff
5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 6 0% mpr 2% mpr
Figure 3: Convergence of expected payoff (left) and response of contract payoff to dif-ferent market price of risk (MPR) (right) for HDDs, CAT and CDDs indices call option contract.
to estimate the MPR based on the available market prices and hence make comparison between the market prices and expected payoff from the model.
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