Weather Derivatives Pricing Using Regime Switching Models

Department of Mathematics

Emanuel Evarest, Fredrik Berntsson, Martin Singull and Xiangfeng Yang

LiTH-MAT-R--2017/04--SE Department of Mathematics Link¨oping University S-581 83 Link¨oping Weather Derivatives Pricing Using Regime Switching Models

Emanuel Evarest1,2, Fredrik Berntsson1, Martin Singull1 and Xiangfeng Yang1

1Department of Mathematics Linköping University SE-581 83 Linköping, Sweden [email protected], [email protected], [email protected], [email protected]

2Department 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

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1 Introduction

During the 20th century, weather derivatives emerged in the financial market as an instrument for hedging various weather related risks in the energy sector and other closely related sectors. Weather derivatives were officially introduced in the major financial markets like Chicago Mercantile Exchange (CME) in 1999 [9, 8]. Over the recent years, weather Derivatives have gained popularity where apart from energy and power industries who were the first industries in the business, other industries 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 different 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 underlying 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 nature of incomplete market pricing models in acknowledging the presence of both hedge- able and unhedgeable 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 t2 T T˜ t dt, AccCDDs t2 T˜ t T dt. = t1 max ref − d( ) = t1 max d( ) − ref Also, CAT tR2 T˜ t dt and PRIM 1 t2 T˜ t dt,R where T is the reference = t1 d( ) = t2 t1 t1 d( ) ref temperature forR regulating domestic heating− andR cooling demands. For weather derivatives 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 temperature [14]. Specifically, we focus on pricing these derivatives whose underlying tem- 3 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 pricing of weather derivatives based on regime switching 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 under the equivalent measure will have the mean and variance as functions of the underlying 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 measure. 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 equivalent probability measure Q by introducing the market price of risk. Also the closed form 4 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 futures contracts together with the dynamics of these futures 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 reverting heteroskedastic process as base regime and Brownian motion as the shifted regime.

The model represent the dynamics of the deseasonalized temperature T˜d(t), after remov- ing the seasonality Sd(t) from the daily average temperature Td(t). The model is given as

T˜t,1 : dT˜t,1 = (µ1 − βT˜t,1)dt + σ1T˜t,1dWt, with probability q1, T˜d(t)=  (2.1) T˜t,2 : dT˜t,2 = µ2dt + σ2dWt, with probability q2,

 µ1 where β is the mean reversion speed, β is the long-term mean and σ1 is the volatility of the mean-reverting heteroskedastic proces in base regime while µ2 and σ2 are the mean and volatility of the shifted regime process respectively. The seasonality is deﬁned by

2π S (t)= A sin (t − A ) + A t + A , (2.2) d 1 365 2 3 4 where A1 is the amplitude, A2 is the phase angle, A3 and A4 are 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

µ1 µ1 βt t β(t s) T˜t,1 : T˜t,1 = + (T˜0,1 − )e− + σ1T˜s,1e− − dWs T˜ (t)=  β β 0 (2.4) d ˜ ˜ ˜ t R Tt,2 : Tt,2 = T0,2 + µ2t + 0 σ2dWs.  R 5

Note that both volatilities σ1 and σ2 are 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 maximization step, the maximum likelihood estimation of the unknown parameters are computed by maximizing its expected likelihood function obtained in the expectation step, for more details see [13]. The estimation process begins with estimation of unknown parameters 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 5 Estimates 9.5786 78.6415 −4.4 × 10− 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 6

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

t t 1 2 L(t) = exp − γdW (s) − γs ds , t ∈ [0,T ], (2.6) Z0 2 Z0 is a martingale process with respect to the natural Wiener ﬁltration Ft = σ(W (s),s ≤ t), for t ∈ [0,T ], under the probability measure P. The relation

Q(A)= LT (ω)dP(ω), A ∈FT , (2.7) ZA deﬁnes an equivalent probability measure Q ∼ P on FT in such a way that under Q, t Vt = Wt + γsds, t ∈ [0,T ], Z0 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 γt is 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

T˜t,1 : dT˜t,1 =(µ1 − βT˜t,1 − σ1T˜t,1γt)dt + σ1T˜t,1dVt, with probability q1, T˜d(t)= T˜ : dT˜ =(µ − σ γ )dt + σ dV , with probability q .  t,2 t,2 2 2 t 2 t 2 (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)

2 where dWt are increments of a Wiener process. Suppose f(t,X) ∈ C (R) is a twice continuous differentiable function, then f(t,X) also follows an Itˆo process ∂f ∂f 1 ∂2f ∂f df = + a + b2 dt + b dW , (2.11) ∂t ∂X 2 ∂X2 ∂X t where a and b2 are the mean and variance of the stochastic process (2.10), while

∂f ∂f 1 ∂2f ∂f 2 + a + b2 and b ∂t ∂X 2 ∂X2 ∂X 7 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 µ Y = T˜ − 1 . (2.12) x x,1 β Differentiating (2.12) and comparing with the base regime of (2.9) we obtain µ dY = dT˜ = −βY dx − σ γ T˜ dx + σ Y + 1 dV . (2.13) x x,1 x 1 x x,1 1 x β x

βt Using the exponential term e Yx, we have µ d eβxY = βeβxY dtx + eβxdY = −σ γ T˜ dx + eβxσ Y + 1 dV . (2.14) x x x 1 x x,1 1 x β x The solution of (2.14) is given by

x x βx βx βx βs µ1 Yx = Yte− − σ1γsT˜s,1e− ds + e− e σ1 Ys + dVs. (2.15) Zt Zt β Substituting (2.12) into (2.15) we obtain the solution for the base regime in (2.9) as the expression

x x µ1 µ1 βx βx β(x s) T˜x,1 = + T˜t,1 − e− − σ1γsT˜s,1e− ds + e− − σ1T˜s,1dVs. β β Zt Zt (2.16) Similarly, the solution of the shifted regime is given by x x T˜x,2 = T˜t,2 + µ2x − σ2γsds + σ2dVs. (2.17) Zt Zt Hence, the integral form for (2.9) for any time x ≥ t is given by

˜ µ1 ˜ µ1 −βx x ˜ −βx x −β(x−s) ˜ Tx,1 = β + Tt,1 − β e − t σ1γsTs,1e ds + t e σ1Ts,1dVs T˜d(x)=  x x T˜x, = T˜t, +µ x − σ γsds +R σ dVs. R  2 2 2 t 2 t 2 (2.18)  R R Therefore, it can be observed that under the probability measure Q, the process T˜d(x) conditioned to ﬁltration Ft,t ≤ x is normally distributed with mean

2 EQ T˜d|Ft = qiEQ T˜x,i|Ft , (2.19) h i Xi=1 h i and variance

2 2 2 2 2 V arQ T˜d|Ft = qiV arQ T˜x,i|Ft + qi EQ T˜x,i|Ft − qiEQ T˜x,i|Ft , i=1 i=1 i=1 ! h i X h i X h i X h (2.20)i 8

where − − (2 i) (i−1) (2 i) µi (i−1) µ1 −βx EQ T˜ |F = + T˜ +(µ x) + T˜ − e − 2 x,i t β t,i i t,i β h i x (2−i) −βx − σiγs T˜s,ie ds, (2.21) t Z and x (2 i) ˜ 2 2β(x s) ˜2 − V arQ Tx,i|Ft = σi e− − Ts,i ds, (2.22) h i Zt for the process T˜d(x) being in regime i, with probability qi. The EQ T˜d|Ft and h i varQ T˜d|Ft are obtained using the idea of weighted mixture of the regimes. The simulated dailyh averagei temperature under the real world measure P and equivalent probability measure Q is shown on Figure 1.

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10 10 Temperature Temperature 0 0

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-20 -20 0 200 400 600 800 1000 1200 1400 1600 1800 2000 0 200 400 600 800 1000 1200 1400 1600 1800 2000 Time(days) Time(days)

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. 9

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

x2 r(x2 t) e − EQ T˜d(x)dx − FCAT (t, x1, x2)|Ft =0 (3.23) Zx1

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 Ft adapted. Therefore the CAT future price FCAT (t, x1, x2) for a temperature derivative contract is given by

x2 FCAT (t, x1, x2)= EQ T˜d(x)dx|Ft . (3.24) Zx1

In a similar fashion, the HDDs future price at time t ≤ x1 < x2 is given by

x2 FHDD (t, x1, x2)= EQ max 0,Tref − T˜d(x) dx|Ft . (3.25) Zx1 Using (2.18) and (3.24), the price of future temperature derivative contract for CAT index at time t such that t ≤ x1 < x2 can be derived as follows: For the base regime we have

x2 FCAT (t,x1,x2)= EQ T˜d(x)dx|Ft Zx1 x2 x x µ1 µ1 −βx −βx −β(x−s) = EQ + T˜ − e − σ γ T˜ e ds + e σ T˜ dV dx|F β t,1 β 1 s s,1 1 s,1 s t Zx1 Zt Zt

x2 x2 x2 x µ µ − − = 1 dx + T˜ − 1 e βxdx − σ γ T˜ e βzdzdx β t,1 β 1 z z,1 Zx1 Zx1 Zx1 Zt

x2 x2 x2 x2 µ µ − − = 1 dx + T˜ − 1 e βxdx − χ (z)σ γ T˜ e βzdzdx (3.26) β t,1 β [t,x] 1 z z,1 Zx1 Zx1 Zx1 Zt

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

x2 x2 µ1 µ1 βx FCAT (t, x1, x2) = dx + T˜t,1 − e− dx Zx1 β Zx1 β x1 x2 ˜ βz − χ[t,x](z)σ1γzTz,1e− dxdz Zt Zx1 x2 x2 ˜ βz − χ[t,x](z)σ1γzTz,1e− dxdz. (3.27) Zx1 Zx1 10

Similarly for shifted regime,

x2 x2 x1 x2 ˜ FCAT (t, x1, x2) = Tt,2dx + µ2xdx − χ[t,x](z)σ2γzdxdz Zx1 Zx1 Zt Zx1 x2 x2 − χ[t,x](z)σ2γzdxdz. (3.28) Zx1 Zx1

Hence, the price of CAT future contract at time t ≤ x1 < x2 is given by

2 2−i 2−i µ 1 − − µ F (t,x ,x )= q i (x − x ) + e β(x2 x1) T˜ − i + h(t)+ I , CAT 1 2 i β 2 1 β2 t,i β 1 i=1 " # X (3.29) where i 1 i 1 − µi 2 − h(t)= T˜t,i(x2 − x1) + (x2 − x1) − 2, 2 and

x1 x2 (2 i) x2 x2 (2 i) βz − βz − I1 = σiγz T˜z,ie− dxdz + σiγz T˜z,ie− dxdz. Zt Zx1 Zx1 Zx1

The dynamics dFCAT (t, x1, x2) of future prices under the equivalent probability measure

Q for temperature process T˜d(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)= V˜CAT (t, x1, x2, T˜d(x))dVt, (3.30) where dFCAT V˜CAT (t, x1, x2, T˜d(x)) = σi . (3.31) dT˜d

The term V˜CAT (t, x1, x2, T˜d(x)) is interpreted as the volatility of the CAT future dynamics.

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

r(c t) CCAT (t,c,x1, x2)= e− − 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

c FCAT (c, x1, x2)= FCAT (t, x1, x2)+ V˜CAT (s, x1, x2, T˜d(x))dVs. (3.33) Zt

Thus, FCAT (c, x1, x2) conditioned on FCAT (t, x1, x2) under the probability measure Q follows a normal distribution with mean FCAT (t, x1, x2) and variance 11

c ˜ 2 ˜ t VCAT (s, x1, x2, Td(x))ds. Therefore, R −r(c−t) CCAT (t,c,x1,x2)= e EQ [max (FCAT (c, x1,x2) − K, 0) |Ft] ∞ −r(c−t) = e (y − K)fFCAT (y)dy ZK

−r(c−t) FCAT (t,x1,x2) − K = e (FCAT (t,x1,x2) − K) Φ   c V˜ 2 s,x ,x , T˜ x ds  t CAT ( 1 2 d( ))   q  c R −r(c−t)  2 FCAT (t,x1,x2) − K  + e V˜CAT (s,x1,x2, T˜d(x))dsφ , (3.34)  t  c V˜ 2 s,x ,x , T˜ x ds Z t CAT ( 1 2 d( ))    qR 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 < x2 is given by

x2 FHDD(t, x1, x2) = EQ max Tref − T˜d(x), 0 dx|Ft Zx1 x2 = EQ max Tref − T˜d(x), 0 |Ft dx. (3.35) Zx1 h i

But the process T˜d(x) in (2.18) is normally distributed with mean and variance given by (2.19) and 2.20 respectively. Then, it follows that, Tref − T˜d(x) will be normally 2 distributed with mean Z(t, x, T˜d(x)) and variance V (t, x, T˜d(x)) given by

Z(t, x, T˜d(x)) = Tref − EQ T˜d(x)|Ft (3.36) h i and 2 V (t, x, T˜d(x)) = V arQ T˜d(x)|Ft . (3.37) h i Using the properties of normal distribution, it follows that the arbitragefree price of HDDs futures is given by

x2 Z t, x, T˜d(x) F (t, x , x )= V (t, x, T˜ (x))Π dx, (3.38) HDD 1 2 d  ˜  Zx1 V (t, x, Td(x))   12 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 FCDD, 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)= V˜HDD t, x1, x2, T˜d(x) dVt. (3.39) But

x2 Z t,x, T˜ x ′ d( ) V˜HDD t,x1,x2, T˜d(x) =σi V (t,x, T˜d(x))Π dx x1  V(t,x, T˜d(x))  Z   x2 Z t,x, T˜ x ′ d( ) + σi V (t,x, T˜d(x))Π U˜ t,x, T˜d(x) dx, x1  V(t,x, T˜d(x)) Z   (3.40)

where Π′(z)=Φ(z) and

V (t, x, T˜d(x))Z′ t, x, T˜d(x) − Z t, x, T˜d(x) V ′(t, x, T˜d(x)) U˜ t, x, T˜d(x) = . 2 ˜ V (t, x, Td(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

r(c t) CHDD (t,c,x1, x2)= e− − 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 derivatives contracts are complicated, in such a way that they do not provide explicit computations of HDDs, CATand CDDs optionsprices. Thus, to achieve the explicit computations, we use Monte Carrlo simulation approach to compute the European call option prices for the weather derivatives contracts written on temperature. 13

4 Weather derivatives Option price by Monte Carlo Sim- ulations

Due to the complexitynature of the dynamicsof 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. TheMonte 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 n 1 E [h (Y (t))] ≅ h (Y (t)) . (4.42) n k Xk=1 Ideally, the number of samples or simulations n has to be large enough to make the computed sample average approximately equal to the expectation value of the intended function or random variable.

Consider the European call option for the contract period [x1, x2], the price of weather derivative contract at time t ≤ x1 < x2 written on HDDs index with strike level K is given by

r(x2 t) CHDD (t, x1, x2)= e− − ℵpEQ [max(HDD(x1, x2) − K, 0)] , (4.43) where ℵp is the nominal price. The Monte Carlo simulation approach for arbitrage free pricing of a European call option for temperature derivatives contracts proceed as follows: 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

x2 accHDDk = Tref − T˜dk(t) dx (4.44) Zx1 and r(x2 t) CHDDk (t, x1, x2)= e− − ℵpEQ [max(accHDDk − K, 0)] , (4.45) respectively. Then, the expected payoff is computed by n 1 C (t, x , x )= C . (4.46) HDD 1 2 n HDDk Xk=1

Finally, the 100(1 − α)% conﬁdence interval is constructed for CHDD by

CHDDn − Zα/2σn, CHDDn + Zα/2σn , 14

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10 10 Temperature Temperature 0 0

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-20 -20 0 200 400 600 800 1000 1200 1400 1600 1800 2000 0 200 400 600 800 1000 1200 1400 1600 1800 2000 Time(days) Time(days)

Figure 2: Two different samples of daily average temperature obtained from Monte Carlo simulations of the dynamics of temperature process.

sn for given signiﬁcance level α and standard error of estimate σn = √n , where sn is the sample standard deviation. The sample simulated daily average temperature for n =1, 2 are shown in Figure 2.

Contract variables Contract 1 Contract 2 Contract 3 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 convergesto 3.1174 × 104 and 2.5720 × 104 respectively. 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 respectively. On the other hand, CDDs expected payoffs in Table 5 converges to the 15

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 option contracts under 0% market price of risk. Changing to 2% MPR, the expected payoff changes accordingly. HDDs expected payoff increases with increase in MPR, while CDDs and CAT expected payoff decreases with increase in MPR. The change in the expected payoff is the outcome of the change in drift term by the MPR. 16

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 using regime switching model with mean reverting 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 17

10.352 10.356

10.351 10.354

10.35 10.352

10.349

10.35

10.348

10.348 10.347 HDDs Log Expected payoff HDDs Log Expected payoff

10.346 10.346

10.345 10.344 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 Number of Simulations Number of Simulations

10.161 10.165 0% mpr 2% mpr 10.16

10.16 10.159

10.158 10.155

10.157

10.156 10.15

10.155

10.145 10.154 CAT Log Expected payoff CAT Log Expected payoff

10.153 10.14

10.152

10.151 10.135 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 Number of Simulations Number of Simulations

6 5.95 0% mpr 2% mpr 5.9 5.9

5.85 5.8

5.8 5.7

5.75 5.6

5.7 5.5 5.65

5.4 5.6 CDDs Log Expected payoff CDDs Log Expected payoff 5.3 5.55

5.5 5.2

5.45 5.1 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 Number of Simulations Number of Simulations

Figure 3: Convergence of expected payoff (left) and response of contract payoff to different 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. 18

References

[1] Peter Alaton, Boualem Djehiche, and David Stillberger. On modelling and pricing weather derivatives. Applied Mathematical Finance, 9(1):1–20, 2002.

[2] Fred Espen Benth. On arbitrage-free pricing of weather derivatives based on fractional brownian motion. Applied Mathematical Finance, 10(4):303–324, 2003.

[3] Fred Espen Benth and Jurate Saltyte Benth. The volatility of temperature and pricing of weather derivatives. Quantitative Finance, 7(5):553–561, 2007.

[4] Nicolas Bollen, Stephen Gray, and Robert E. Whaley. Regime switching in foreign exchange rates: Evidence from currency option prices. Journal of Econometrics, 94(1):239–276, 2000.

[5] Nicolas P. B. Bollen. Valuing options in regime-switching models. Derivatives, 6:38–49, 1998.

[6] Patrick L. Brockett, Linda L. Goldens, Min-Ming Wen, and Charles C. Yang. Pric- ing weather derivatives using the indifference pricing approach. North American Actuarial Journal, 13(3):303–315, 2012.

[7] Dorje C. Brody, Joanna Syroka, and Mihail Zervos. Dynamical pricing of weather derivatives. Quantitative Finance, 2(3):189–198, 2002.

[8] Brenda Lopez Cabrera, Martin Odening, and Mattias Ritter. Pricing rainfall derivatives at the cme. Sonderforschungsbereich 649: Ökonomisches Risiko. Wirtschaftswissenschaftliche Fakultät, 2013.

[9] Adam Clements, A.S. Hurn, and K.A. Lindsay. Estimating the payoffs of temperature-based weather derivatives. Technical report, National Centre for Econo- metric Research, 2008.

[10] Mark Davis. Pricing weather derivatives by marginal value. 2001.

[11] Arthur P.Dempster, Nan M. Laird, and Donald B. Rubin. Maximum likelihood from incomplete data via the em algorithm. Journal of the royal statistical society. Series B (methodological), pages 1–38, 1977.

[12] Thomas W. Epps. Pricing Derivative Securities. World Scientiﬁc Publishing Co. Pte. Ltd, second edition edition, 2007. 19

[13] EmanuelEvarest, Fredrik Berntsson, Martin Singull, and Wilson M Charles. Regime switching models on temperature dynamics. International Journal of Applied Math- ematics and Statistics, 56(2):19–36, 2017.

[14] Mark Garman, carlos Blanco, and Robert Erickson. Seeking a standard pricing model. Environmental Finance, 2000.

[15] James D. Hamilton. Regime-switching models. The new palgrave dictionary of economics, 2, 2008.

[16] Wolfgang Karl Härdle and Brenda López Cabrera. The implied market price of weather risk. Applied Mathematical Finance, 19(1):59–95, 2012.

[17] Joanna Janczura and Rafał Weron. Efﬁcient estimation of markov regime-switching models: An application to electricity spot prices. AStA Advances in Statistical Anal- ysis, 96(3):385–407, 2012.

[18] Stephen Jewson and Anders Brix. Weather derivative valuation: the meteorological, statistical, ﬁnancial and mathematical foundations. Cambridge University Press, 2005.

[19] Travis L. Jones. Agricultural applications of weather derivatives. International Business & Economics Research, 6(6):53–59, 2007.

[20] RH. Liu and D. Nguyen. A tree approach to options pricing under regime- switching jump diffusion models. International Journal of computer Mathematics, 92(12):2575–2595, 2015.

[21] Mohammed Mraoua. Temperature stochastic modeling and weather derivatives pricing: empirical study with moroccan data. Afrika Statistika, 2(1), 2007.

[22] Anatoliy Swishchuk and Kaijie Cui. Weather derivatives with applications to cana- dian data. Journal of Mathematical Finance, 3(1), 2013.

[23] Calum G. Turvey. The pricing of degree-dayweather options. Agricultural Financial Review, 65(1), 2005.

[24] M. Yousuf, AQM. Khaliq, and RH. Liu. Pricing american options under multi- stage regime switching with an efﬁcient l-stable method. International Journal of computer Mathematics, 92(12):2530–2550, 2015.

[25] Achilleas Zapranis and Antonis K. Alexandridis. Weather derivatives Modeling and Pricing weather related risk. Springer New York., 2013.