6.3 Model Performance
6.3.3 Comparison of Performances cross Regions
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the S-SV model, 0.2731 for the JD-S-SV model, and 0.2575 for the S-MR-JD-S-SVJ model, respectively. The S-MR-S-SV model performs the better forecasting capability cross six models. Moreover, the S-MR-JD-S-SV model is the better out-of-sample pricing performance for the CDD/HDD futures options. The results are shown in Table 28.
Summarily, the type of the stochastic volatility model is provided with the better forecasting performance. The investors can observe the past market price of risk and predict the future path of the weather derivatives. Then, the traders can use it to determine the trading/hedging strategy for the CDD/HDD derivatives.
6.3.3 Comparison of Performances cross Regions
This section investigates the model performances cross four regions like in Figure 1 in-stead of the fitting and pricing performances cross six models as previous sections. Table 29 reports the total number of the best fitting for each region. For the fitting performance of the historical estimation, as shown in Panel (a), the model with jump risks in the tem-perature index and the stochastic volatility is better in the Midwest, the Northeast, and the South regions, and the stochastic volatility model with jump risks of the temperature index is better in the West region. Panel (b) shows the in-sample pricing performance.
For the CDD/HDD futures, the model with jump risks in the temperature index and the stochastic volatility exists the better performance in the South and the West regions, and the stochastic volatility model with jump risks of the temperature index is better in the Midwest and the Northeast regions. For the CDD/HDD futures options, the jump diffusion model is the better performance in the Midwest region, and others regions do not obey the specific model significantly. Panel (c) shows the out-of-sample pricing performance. For the out-of-sample pricing performance of the CDD/HDD futures, the stochastic volatility model is better in the Midwest, the Northeast, and the West regions, and the jump diffusion model is better in the South region. For the out-of-sample pricing performance of the CDD/HDD futures options, the stochastic volatility model with jump risks of the temperature index exists the better performance in the Midwest, the North-east, and the West regions, and the model with jump risks in the temperature index and the stochastic volatility exists the better performance in the South region.
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Chapter 7 Conclusions
This study supports the S-MR-JD-S-SVJ model to depict the path of the DAT and to value the CDD/HDD futures and futures options. This model can measure the seasonality and the jump movements of the temperature and its volatility. The sample includes the DAT and market price of the CDD/HDD derivatives for each city from the CME group.
There are 18 cities in sample such as Atlanta, Baltimore, Boston, Chicago, Cincinnati, Dallas, Des Moines, Detroit, Houston, Kansas City, Las Vegas, Minneapolis, New York, Philadelphia, Portland, Sacramento, Salt Lake City, Tucson, in the U.S.
There are some contributions in this study: (i) we extend the Alaton, Djehince, and Stillberg (2002)’s framework by employing the Merton (1976)’s jump diffusion model, the Heston (1993)’s stochastic volatility model, the Bates (1996)’s stochastic volatility model with jump risks, and Eraker, Johannes, and Polson (2003)’s jump risk in volatility and return model. (ii) The model parameters are estimated by the MLE for the S-MR model and the S-MR-S model, the EM algorithm for the S-MR-JD-S model, and the PF algorithm for the S-MR-S-SV model, the S-SV model, and the S-MR-JD-S-SVJ model. And, the complex model exists the better goodness-of-fit for the path of the temperature index. (iii) We employ the Esscher transform to change the probability measure and derive the probability density function of each model by the Feynman-Kac formula and the Fourier transform. (iv) The semi-closed form of the CDD/HDD futures pricing formula is derived, and we use the Monte-Carlo simulation to value the CDD/HDD futures options due to no closed-form solution. (v) Whatever in-sample and out-of-sample pricing performance, the type of the stochastic volatility performs the better fitting for the temperature derivatives. (vi) The market price of risk (the parameter of the Esscher
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transform) differs to zero significantly (most are negative), so the investors require the positive weather risk premium for the derivatives.
The theoretical and empirical results in this study can provide the guide of fitting model and pricing derivatives to the authorities, the energy companies, the insurance companies, and others weather-linked institutions in the future.
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The S-MR-JD-S model is the example of the deterministic volatility model like the equa-tion (3.7). The DAT, Xt, follows the S-MR-JD-S model, then its dynamic process under the P measure is
dXt=
dPt be the Radon-Nikod´ym derivatives with the parameters h1 and h2. dQ
Owing to independence between the Brownian motion (denoted as 1) and the jump term (denoted as 2), they can be changed probability measure separately. For the standard
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Obviously, the distribution of the standard Brownian motion under the Q measure is dWtQ = dWt− h1σtdt ∼ N (0, dt) . (A.4) For the jump term, given the number of occurrences of jumps Nt = n, then
dQ2|N
The distribution of the compound Poisson process under the Q measure is Nt∼ P oi Therefore, the dynamic processes of the DAT for the deterministic volatility model (S-MR-JD-S model) under the Q measure is
dXt =
where h1 is the temperature risk premium and h2 is the jump risk premium.
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The S-MR-JD-S-SVJ model is the example of the stochastic volatility model like the equation (3.10). The DAT, Xt, follows the S-MR-JD-S-SVJ model, then its dynamic process under the P measure is
dXt=
where B1,t ∼ N (0, t) and B2,t ∼ N (0, t) are the independent Brownian motions generated by the correlated Brownian motions and the Cholesky decomposition. That is,
dWX,t = dB1,t, dWV,t = ρdB1,t+p
1 − ρ2dB2,t. (B.2) And, NX,t ∼ P oi (λXt) and NV,t ∼ P oi (λVt) are the number of occurrences of jumps for the DAT and the volatility, and those jump amplitudes, YX,j i.i.d.∼ N (θX, νX2), YV,j i.i.d.∼ exp (θV) j = 1, 2, · · · , ∞. The jump amplitudes of the DAT and the volatility are inde-pendent. Let dQdPt
t be the Radon-Nikod´ym derivatives with the parameters h1, h2, h3, and
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Owing to the Brownian motion for the DAT (denoted as 1), the Brownian motion for the volatility (denoted as 2), the jump term for the DAT (denoted as 3), and the jump term for the volatility (denoted as 4), they can be changed probability measure separately. For the standard Brownian motion for the DAT,
dQ1 =e−12h21R0tVudu+h1R0t
Obviously, the distributions of two standard Brownian motions under the Q measure are dB1,tQ = dB1,t− h1p
Vtdt ∼ N (0, dt) , (B.6) dB2,tQ = dB2,t− h2p
Vtdt ∼ N (0, dt) . (B.7)
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For the jump terms, given the number of occurrences of jumps NX,t = nX, then
dQ3|N
and, given the number of occurrences of jumps NV,t = nV, dQ4|N
The distributions of the compound Poisson processes for the DAT and the volatility under the Q measure are
NX,t∼ P oi Therefore, the dynamic processes of the DAT for the stochastic volatility model (S-MR-JD-S-SVJ model) under the Q measure is
dXt=
where h1 is the temperature risk premium, h2 is the volatility risk premium, and h3, h4 are the jump risk premiums for the temperature and the volatility.
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The S-MR-JD-S model is the example of the deterministic volatility model. Given the filtration Ft, if the dynamic process of the DAT follows the equation (4.12) or (A.7) under the Q measure, the characteristic function of the DAT Xk at time k,
g (t, x; k, u) = EtQ eiuXk
Xt= x , t < k. (C.1) By the Feynman-Kac formula, the characteristic function must satisfy following partial integro-differential equation (PIDE). density function of the jump amplitude, then
λeh2θ+
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(C.2) is rewritten by−∂g
with the boundary condition g (0, x; u) = eiux. Guess that the solution of the equation (C.4) has the form,
g (τ, x; u) = eA(τ ;u)+B(τ ;u)x+iux, (C.5) and its partial derivatives are
∂g
Substituting these partial derivatives into the equation (C.4), then
The PIDE is reduced to solve two ordinary differential equations (ODEs),
−∂A For the first ODE, the solution is
A (τ ; u) =
The integral can be computed by some numerical methods like the trapezium rule and the fast Fourier transform (FFT).
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Appendix D
Characteristic Function: Stochastic Volatility Model
The S-MR-JD-S-SVJ model is the example of the deterministic volatility model. Given the filtration Ft, if the dynamic process of the DAT follows the equation (4.18) or (B.12) under the Q measure, the characteristic function of the DAT Xk at time k,
g (t, x, v; k, uX, uV) = EtQ eiuXXk+iuVVk
Xt = x, Vt= v , t < k. (D.1) By the generalized Feynman-Kac formula, the characteristic function must satisfy follow-ing partial integro-differential equation (PIDE).
∂g
∂t +
β (Xtα− x) + dXtα
dt + h1v − λθ ∂g
∂x (D.2) +
"
˜
κ κVtφ
˜ κ − v
!
+ dVtφ dt
#∂g
∂v + 1 2v∂2g
∂x2 + 1
2σ2v∂2g
∂v2 + ρσv ∂2g
∂x∂v +λXeh3θX+
h23ν2 X 2
Z ∞
−∞
[g (t, x + yX, v; k, uX, uV) − g (t, x, v; k, uX, uV)] qX(yX) dyX
+ λVθV θV − h4
Z ∞
−∞
[g (t, x, v + yV; k, uX, uV) − g (t, x, v; k, uX, uV)] qV (yV) dyV = 0, with the boundary condition g (k, Xk, Vk; k, uX, uV) = eiuXx+iuVv. In which, qX(yX) and qV (yV) are the probabilities density function of the jump amplitude for the DAT and the
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the equation (D.2) is rewritten by−∂g the equation (D.5) has the form,
g (τ, x, v; uX, uV) = eA(τ ;uX,uV)+B(τ ;uX,uV)x+C(τ ;uX,uV)v+iuXx+iuVv, (D.6) and its partial derivatives are
∂g
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Substituting these partial derivatives into the equation (D.5), then
"
The PIDE is reduced to solve three ordinary differential equations (ODEs),
−∂A
For the second ODE, the solution is
B (τ ; uX, uV) = iuX e−βτ − 1 .
For solving above Riccati equation, we assume the form of the function ˜C (z; uX, uV) to find its particular solution as follow:
C (z; u˜ X, uV) = 2βzU0(z; uX, uV)
σ2U (z; uX, uV) , (D.10) then the third ODE can be rewritten again by
∂2U
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The equation (D.11) is called as Whittaker’s equation (1904) and its general solution is U (z; uX, uV) =c1z
where c1 and c2 are the constants,
µ1 = −ρ (˜κ − β) + h1σ
In which, K (·) is the hyper-geometric function (or the Kummer’s series expression), U (·) is the confluent hyper-geometric function, and Γ (·) is the gamma function. As Wong and Lo (2009), let c1 = 1 and c2 = 0 in the equation (D.12), then we can get the particular solution of the equation (D.11) as follow:
U (z; uX, uV) = z
and its first-order derivative,
∂U
Substituting the equations (D.13) and (D.14) into the equation (D.10), we can get C (z; u˜ X, uV) =κ − β − iu˜ Vσ2+ 2β µ2+ 12
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Thus, the particular solution of the third ODE is
Cp(τ ; uX, uV) =κ − β − iu˜ Vσ2+ 2β µ2+12 For the first ODE, the solution is
A (τ ; uX, uV) =iuX
The integral can be computed by some numerical methods like the trapezium rule and the fast Fourier transform (FFT).
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Table 1: Location of each Weather Station in the U.S.
This Table lists the locations of 18 cities in the U.S. The name of each city is in the first column. The first three letters of the name is used as the abbreviation and is shown in the second column. The region which the city locates is in the third column. As shown in Figure 1, the U.S. is decomposed into four regions whose sequence is Midwest, Northeast, South, and West. The state which the city locates is in the fourth column. The longitude and the latitude which are obtained from Wikipedia are reported in the fifth and sixth columns. In which, ”W” is west and ”N” is north.
City Abbreviation Region State Longitude Latitude
Atlanta ATL South Georgia 84°23’W 33°45’N
Baltimore BAL South Maryland 76°37’W 39°17’N
Boston BOS Northeast Massachusetts 71°03’W 42°21’N
Chicago CHI Midwest Illinois 87°41’W 41°50’N
Cincinnati CIN Midwest Ohio 84°31’W 39°60’N
Dallas DAL South Texas 96°47’W 32°46’N
Des Moines DES Midwest Iowa 93°37’W 41°35’N
Detroit DET Midwest Michigan 83°02’W 42°19’N
Houston HOU South Texas 95°22’W 29°45’N
Kansas City KAN Midwest Missouri 94°34’W 39°05’N
Las Vegas LAS West Nevada 115°08’W 36°10’N
Minneapolis MIN Midwest Minnesota 93°16’W 44°59’N
New York NEW Northeast New York 74°00’W 40°42’N
Philadelphia PHI Northeast Pennsylvania 75°10’W 39°57’N
Portland POR West Oregon 122°40’W 45°31’N
Sacramento SAC West California 121°28’W 38°33’N
Salt Lake City SAL West Utah 111°53’W 40°45’N
Tucson TUC West Arizona 110°55’W 32°13’N
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Table 2: Descriptive Statistics of DAT Increments
Using the DAT of 18 cities obtained from CME group, such as Atlanta, Baltimore, Boston, Chicago, Cincinnati, Dallas, Des Moines, Detroit, Houston, Kansas City, Las Vegas, Minneapolis, New York, Philadelphia, Portland, Sacramento, Salt Lake City, Tucson, in the U.S. over the period January 01, 2002 to December 31, 2011 as sample, the number of the available sample is 3,652 days. Mean is the average level. S.D. is the stnadard deviation. Min is the minimum level. Med is the median level. Max is the maximum level. Skew is the skewness level. The distribution which is said to be left-skewed/bell-shaped/right-skewed is the negative/zero/positive level. Kurt is the kurtosis level. The distribution which is said to be platykurtic/normal/leptokurtic is over/equal to/below 3. J(+) is the number of the temperature without seasonal adjustment which is greater than the mean plus three standard deviations.
J(−) is the number of the temperature without seasonal adjustment which is greater than the mean minus three standard deviations.
City Mean S.D. Min Med Max Skew Kurt J(+) J(−)
ATL 0.0033 2.7639 −11.9444 0.2778 10.8333 −0.4823??? 4.2968??? 9 29 BAL 0.0038 3.2599 −13.3333 0.0000 12.5000 −0.1805??? 3.7877??? 9 13 BOS 0.0012 3.4122 −14.7222 0.0000 12.2222 −0.0779? 3.3793??? 7 9 CHI 0.0026 3.4993 −12.5000 0.0000 14.4444 −0.1239??? 3.7926??? 10 18 CIN 0.0034 3.4848 −13.3333 0.2778 13.8889 −0.1955??? 4.0395??? 10 23 DAL 0.0028 3.2930 −14.7222 0.2778 12.7778 −0.5590??? 4.6338??? 6 31 DES 0.0051 3.6736 −13.8889 0.0000 15.8333 −0.1439??? 3.7394??? 11 14 DET 0.0027 3.2604 −13.8889 0.0000 11.9444 −0.0876?? 3.7293??? 10 10 HOU 0.0037 2.9381 −12.7778 0.0000 11.3889 −0.4670??? 5.0982??? 14 46 KAN 0.0048 3.7925 −14.7222 0.2778 15.5556 −0.2533??? 3.7029??? 9 15 LAS 0.0007 2.1154 −10.8333 0.2778 6.3889 −0.7993??? 4.6417??? 1 38 MIN 0.0037 3.4414 −13.3333 0.0000 14.1667 −0.0910?? 3.7409??? 9 15 NEW 0.0030 3.0739 −13.6111 0.2778 11.1111 −0.1446??? 3.4127??? 10 10 PHI 0.0036 3.0844 −13.8889 0.2778 12.5000 −0.2229??? 3.7183??? 5 14 POR −0.0012 2.1446 −9.7222 0.0000 7.7778 −0.1294??? 3.5379??? 9 14 SAC −0.0011 1.9376 −9.1667 0.0000 6.1111 −0.2979??? 3.4494??? 5 20 SAL 0.0006 3.0010 −13.6111 0.2778 9.7222 −0.6186??? 4.1967??? 3 26 TUC 0.0015 2.2046 −10.2778 0.2778 8.3333 −0.4451??? 4.1776??? 11 33
???,??, and? are, respectively, the statistical significance levels at 1%, 5%, and 10%.
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Table 3: Mean of DAT in each Month
Using the DAT of 18 cities obtained from CME group, such as Atlanta, Baltimore, Boston, Chicago, Cincinnati, Dallas, Des Moines, Detroit, Houston, Kansas City, Las Vegas, Minneapolis, New York, Philadelphia, Portland, Sacramento, Salt Lake City, Tucson, in the U.S. over the period January 01, 2002 to December 31, 2011 as sample, the number of the available sample is 3,652 days. The mean is measured by the average of the DAT in each month.
City Jan. Fed. Mar. Apr. May. Jun. Jul. Aug. Sep. Oct. Nov. Dec.
ATL 6.50a 7.73 12.81 17.08 21.24 25.61 26.63 26.95b 23.55 17.68 12.31 7.44 BAL 0.93a 1.67 7.03 12.98 17.67 23.11 25.58b 24.77 20.62 13.77 8.89 2.79 BOS −1.51a −0.38 3.72 9.52 14.52 19.69 23.48b 22.83 19.10 12.54 7.70 1.75 CHI −4.25a −3.02 3.37 9.94 14.82 20.90 23.69b 22.83 18.66 11.72 5.46 −2.08 CIN −0.70a 0.31 6.83 13.10 17.36 22.40 24.30 24.43b 20.39 13.14 7.56 1.25 DAL 8.26b 9.49 15.13 19.44 23.77 28.36 30.13 30.68b 26.10 20.21 14.74 9.04 DES −5.28a −3.62 4.26 11.55 16.84 22.52 24.88b 23.67 19.05 11.86 4.93 −2.56 DET −3.60a −2.87 2.87 10.14 15.05 21.08 23.46b 22.64 18.76 11.56 5.84 −0.99 HOU 11.75a 12.90d 17.51d 21.33d 25.40d 28.53 29.20 29.71b 26.92 22.31d 17.04d 12.47d KAN −1.66a 0.08 7.20 13.48 18.19 23.58 26.08b 25.68 20.28 13.52 7.33 0.39 LAS 9.46 11.17 15.58 19.29 25.26 30.78d 34.65b,d32.90d 28.58d 20.99d 14.03 8.66a MIN −8.80a,c−6.68c 0.38c 9.41c 14.86 20.94 24.21b 22.27 17.74c 9.91c 2.36c −5.90c NEW 0.92a 1.81 6.11 12.15 17.05 22.70 25.97b 25.31 21.66 14.85 9.91 3.78 PHI 0.66a 1.61 6.90 13.00 18.02 23.29 26.08b 25.29 21.55 14.37 9.26 3.08 POR 5.29 6.49 8.68 10.89 14.42c 17.61c 21.19b,c20.91c 18.19 12.82 7.87 4.69a SAC 8.19b 10.33 12.63 14.46 18.53 22.06 24.57b 23.67 22.23 17.32 11.78 8.42 SAL −1.79a 0.80 6.13 10.00 15.12 20.96 27.23b 25.12 19.34 11.79 4.42 −0.70 TUC 11.83d 12.79 16.29 19.93 24.79 30.04 31.25b 30.10 28.03 22.16 16.26 11.17a a and b, respectively, are the minimum and maximum of the mean cross 12 months for each city. c and d, respectively, are the minimum and maximum of the mean cross 18 cities for each month.
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l C h engchi U ni ve rs it y
Table 4: Volatility of DAT in each Month
Using the DAT of 18 cities obtained from CME group, such as Atlanta, Baltimore, Boston, Chicago, Cincinnati, Dallas, Des Moines, Detroit, Houston, Kansas City, Las Vegas, Minneapolis, New York, Philadelphia, Portland, Sacramento, Salt Lake City, Tucson, in the U.S. over the period January 01, 2002 to December 31, 2011 as sample, the number of the available sample is 3,652 days. The volatility is measured by the standard deviation of the increments of the DAT in each month.
City Jan. Fed. Mar. Apr. May. Jun. Jul. Aug. Sep. Oct. Nov. Dec.
ATL 5.56b 4.56 4.73 3.99 3.46 2.31 1.74a 2.17 2.70 4.08 4.35 4.66 BAL 5.49b 4.59 4.90 4.92 4.46 3.46 2.64a 2.78 3.16 4.51 4.39 4.34 BOS 5.73b 4.40 4.51 4.56 4.25 4.12 3.13a,d 3.26 3.37 4.15 4.27 4.60 CHI 6.16b 5.54 5.76 5.44 4.85 3.98 3.05 2.98a 3.97 4.89 4.86 5.40 CIN 6.59b 5.80 5.99 5.25 4.45 2.93 2.60a 2.86 3.81 4.89 5.18 5.27 DAL 5.66 6.06b 5.12 4.11 3.73 2.40 2.16a 2.59 3.02 4.31 4.93 5.01 DES 6.82b,d 6.66d 6.37 5.65 4.51 3.11 2.85a 3.12 4.12 5.46 5.41d 5.77d DET 5.76b 4.81 5.51 5.34 4.54 3.52 2.95 2.87a 3.74 4.65 4.63 4.58 HOU 5.22 5.35b 4.45 3.63 2.93c 1.83c 1.59a,c 1.79c 2.22c 3.99 4.59 5.02 KAN 6.72b 6.48 6.19 5.57 4.46 3.02 3.00a 3.50d 4.04 5.24 5.35 5.66 LAS 3.12 3.03 4.00 3.76 4.48b 3.36 2.22a 2.32 3.30 3.87 3.81 3.04 MIN 6.61b 6.24 6.60d 5.70d 4.91 3.71 3.05a 3.12 4.70d 5.67d 5.40 5.74 NEW 5.60b 4.27 4.53 4.64 3.88 3.76 2.77a 2.95 3.05 4.27 4.03 4.37 PHI 5.45b 4.55 4.76 4.87 4.14 3.43 2.51a 2.77 3.02 4.48 4.19 4.30 POR 3.26 2.50a 2.67c 2.84c 3.39b 3.17 2.99 2.68 2.93 2.88 3.21 3.39 SAC 2.37c 2.27a,c 2.95 3.11 3.28b 3.06 2.84 2.35 3.01 2.84c 2.89c 2.77c SAL 4.73 4.47 4.25 4.30 5.21b,d 4.66d 2.50a 3.21 4.16 4.68 4.69 4.44 TUC 3.33 3.55 4.00b 3.43 3.33 2.54 2.48 2.27a 2.34 3.62 3.95 3.60 a and b, respectively, are the minimum and maximum of the volatility cross 12 months for each city. c
ATL 5.56b 4.56 4.73 3.99 3.46 2.31 1.74a 2.17 2.70 4.08 4.35 4.66 BAL 5.49b 4.59 4.90 4.92 4.46 3.46 2.64a 2.78 3.16 4.51 4.39 4.34 BOS 5.73b 4.40 4.51 4.56 4.25 4.12 3.13a,d 3.26 3.37 4.15 4.27 4.60 CHI 6.16b 5.54 5.76 5.44 4.85 3.98 3.05 2.98a 3.97 4.89 4.86 5.40 CIN 6.59b 5.80 5.99 5.25 4.45 2.93 2.60a 2.86 3.81 4.89 5.18 5.27 DAL 5.66 6.06b 5.12 4.11 3.73 2.40 2.16a 2.59 3.02 4.31 4.93 5.01 DES 6.82b,d 6.66d 6.37 5.65 4.51 3.11 2.85a 3.12 4.12 5.46 5.41d 5.77d DET 5.76b 4.81 5.51 5.34 4.54 3.52 2.95 2.87a 3.74 4.65 4.63 4.58 HOU 5.22 5.35b 4.45 3.63 2.93c 1.83c 1.59a,c 1.79c 2.22c 3.99 4.59 5.02 KAN 6.72b 6.48 6.19 5.57 4.46 3.02 3.00a 3.50d 4.04 5.24 5.35 5.66 LAS 3.12 3.03 4.00 3.76 4.48b 3.36 2.22a 2.32 3.30 3.87 3.81 3.04 MIN 6.61b 6.24 6.60d 5.70d 4.91 3.71 3.05a 3.12 4.70d 5.67d 5.40 5.74 NEW 5.60b 4.27 4.53 4.64 3.88 3.76 2.77a 2.95 3.05 4.27 4.03 4.37 PHI 5.45b 4.55 4.76 4.87 4.14 3.43 2.51a 2.77 3.02 4.48 4.19 4.30 POR 3.26 2.50a 2.67c 2.84c 3.39b 3.17 2.99 2.68 2.93 2.88 3.21 3.39 SAC 2.37c 2.27a,c 2.95 3.11 3.28b 3.06 2.84 2.35 3.01 2.84c 2.89c 2.77c SAL 4.73 4.47 4.25 4.30 5.21b,d 4.66d 2.50a 3.21 4.16 4.68 4.69 4.44 TUC 3.33 3.55 4.00b 3.43 3.33 2.54 2.48 2.27a 2.34 3.62 3.95 3.60 a and b, respectively, are the minimum and maximum of the volatility cross 12 months for each city. c