disasters not only cause the jump movement in the DAT, but also bring out the volatility increasing sharply and rapidly. Under the S-MR-S-SV model, the volatility can only moves gradually, and it cannot measure the rapid shock in the volatility.
3.5 Seasonal Mean-Reversion and Jump Diffusion Model with Seasonal Stochastic Volatility
(S-MR-JD-S-SV)
Owing to the global warming, the greenhouse effect, the climate change, the jump move-ments of the DAT occur with more probability. The daily temperature increasing or decreasing sharply becomes widespread at everywhere. The investors face the more and more weather risk. For this reason, the jump diffusion must be considered while pricing the derivatives.
Consider a probability space, {Ω, F , P}, generated by two Wiener processes, WX,t ∼ N (0, t), WV,t ∼ N (0, t), Corr (WX,t, WV,t) = ρ, and the compound Poisson process, PNt
j=0Yj, where Ω is the universal set of all outcomes, F is the σ-field of the subset be-long to Ω, and P is the physical probability measure. The Wiener processes {WX,t}Tt=0, {WV,t}Tt=0, the jump frequency {Nt}Tt=0, and the jump amplitude {Yj}∞j=0 are indepen-dent. Let {Ft}Tt=0 be the filtration including the series of the Wiener processes and the compound Poisson process at time t. According to above definition, the DAT, Xt, follows the S-MR-JD-S-SV model, then its dynamic process under the P measure is
dXt=
where, for the part of the temperature process, β > 0 is the speed of reversion. Xtα is the long-run mean level where the second term is the trend factor and the third and fourth terms are the seasonal factors. For the part of the variance process, Vt is the
‧
instantaneous volatility. κ > 0 is the speed of reversion. Vtφ is the long-run mean level where φ0 > 0 is the based level, φ1 ≥ 0 is the seasonal factor of the volatility, and φ2 is the location parameter. Also, the sine function plus one ensures the positive volatility.
σ√
Vt is the volatility of the variance.
The shocks of the DAT coming from the natural disasters not only cause the jump movement in the DAT, but also bring out the volatility increasing sharply and rapidly.
Under the S-MR-JD-S-SV model, the volatility can only moves gradually, and it cannot measure the rapid shock in the volatility. Thus, the next model will modify this drawback.
3.6 Seasonal Mean-Reversion and Jump Diffusion Model with Seasonal Stochastic Volatility
and Jump Risk (S-MR-JD-S-SVJ)
The shocks of the DAT coming from the natural disasters not only cause the jump move-ment in the DAT, but also bring out the volatility increasing sharply and rapidly. Eraker Johannes, and Polson (2003) argued the diffusive volatility can only move gradually and connot measure the shock in short term. Thus, jump in volatility revises this drawback, and captures the rapid movements of jumps.
Consider a probability space, {Ω, F , P}, generated by two Wiener processes, WX,t ∼ N (0, t), WV,t ∼ N (0, t), Corr (WX,t, WV,t) = ρ, where Ω is the universal set of all outcomes, F is the σ-field of the subset belong to Ω, and P is the physical probability measure. Let {Ft}Tt=0 be the filtration including the series of the Wiener process at time t. According to above definition, the DAT, Xt, follows the S-MR-S-SV model, then its dynamic process under the P measure is
dXt=
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where, for the part of the temperature process, β > 0 is the speed of reversion. Xtα is the long-run mean level where the second term is the trend factor and the third and fourth terms are the seasonal factors. For the part of the variance process, Vt is the instantaneous volatility. κ > 0 is the speed of reversion. Vtφ is the long-run mean level where φ0 > 0 is the based level, φ1 ≥ 0 is the seasonal factor of the volatility, and φ2 is the location parameter. Also, the sine function plus one ensures the positive volatility.
σ√
Vt is the volatility of the variance.
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Chapter 4
Temperature Derivatives Pricing Formula
In this chapter, we narrate the underlying asset of the temperature derivatives contract at first. As its name implies, the payoff of the temperature derivatives is linked to the DAT.
However, the heating degree day (HDD) and cooling degree day (CDD) is used generally instead of the DAT. in the section 4.1, the definitions of the HDD, the CDD, and the un-derlying asset of the temperature derivatives contract will be shown in detail. Second, the temperature derivatives are usually traded at the Chicago Mercantile Exchange (CME) and the over-the-counter (OTC). The payoff patterns of the temperature derivatives at maturity are different in each market. We aim these different payoffs to discuss in the section 4.2. Finally, as in Bates (1996), Heston (1993), Duffie, Pan, and Singleton (2000), Wong and Lo (2009), Pillay and O’Hara (2011), we derive the characteristic function of each model by the partial integro-differential equation (PIDE) which is also called as Kolmogorov backward equation (KBE) or Fokker-Planck equation (FPE). Then, the probability density function (PDF) can be computed by the inverse Fourier transform.
Given the payoff at maturity, the CDD/HDD futures can be priced. The pricing formula is reported in the section 4.3. Unfortunately, there is no closed-form solution for the CDD/HDD futures options and the Monte-Carlo simulation pricing method is used to value the premium of the options.
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4.1 Underlying Asset
The HDD is computed by the difference between the daily average temperature and the reference index only if the daily average temperature is less than the reference index.
The larger HDD implies the more demand of the heater in the winter and the more temperature risk for the energy companies, the insurance companies, the agricultural companies, and so on. On the other hand, the CDD is calculated by the difference between the daily temperature and the reference index only if the daily temperature is greater than the reference index. The higher CDD means the more demand of the refrigerant in the summer and the more temperature risk for the climate-related companies. In which, the daily average temperature is the average of the minimum level and the maximum level in one day, and the reference index is usually stipulated by 18°C (or 65°F). In this study, we use the degree of Centigrade (°C) as the unit. According to above description, the formula of the CDD and the HDD at time Ti can be expressed by
CDDTi = max (XTi − 18, 0) , HDDTi = max (18 − XTi, 0) . (4.1) And, the relationship between the CDD and the HDD indexes is
CDDTi− HDDTi = XTi− 18. (4.2) The temperature derivatives is the contract whose underlying asset is the sum of the HDD or the CDD index over a period. Let the notations H (T1, Tn) and C (T1, Tn) be the sum of the HDD or the CDD index over the period [T1, Tn], respectively. Consider a temperature derivatives contract with maturity Tn and the releasing period [T1, Tn]. The payoff of the CDD and HDD derivatives at maturity are
C (T1, Tn) = δ
Tn
X
k=T1
CDDk = δ
Tn
X
k=T1
max (Xk− 18, 0) , (4.3)
H (T1, Tn) = δ
Tn
X
k=T1
HDDk = δ
Tn
X
k=T1
max (18 − Xk, 0) ,
where δ is the dollars per 1°C ($/) or the tick size. According to the temperature derivatives issued by the CME, the tick size is δ = $20/. The equation (4.3) is the non-linear function, and it is difficult to deduce the distribution and the dynamic process of the cumulative value. However, Alaton, Djehince, and Stillberg (2002) pointed out the
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zero level of the CDD and HDD in the specific with much lower probability. They assumed that the cumulative value, H (T1, Tn) and C (T1, Tn), is the linear function. Therefore, The equation (4.3) can be rewritten as
H (T1, Tn) = δ 18n −
Tn
X
k=T1
Xk
!
, C (T1, Tn) = δ
Tn
X
k=T1
Xk− 18n
!
. (4.4) This assumption is the more straightforward to depict the distribution of the cumulative value. However, there are many days for the DAT greater than 18°C in the winter and less than 18°C in the summer. The zero or negative level for the CDD/HDD index cannot be ignored. Therefore, this study modifies Alaton et al. (2002)’s assumption and derive the CDD/HDD derivatives pricing method.