Localizing Temperature Risk

Transcription

1 Localizing Temperature Risk Wolfgang Karl Härdle, Brenda López Cabrera Ostap Okhrin, Weining Wang Ladislaus von Bortkiewicz Chair of Statistics C.A.S.E. - Center for Applied Statistics and Economics Humboldt-Universität zu Berlin

2 Motivation 1-1 Weather PricewaterhouseCoopers Survey 2005 releases the Top 5 sectors in need of financial instruments to hedge weather risk. 7% 5% 4% 69% 2% 13% Energy Agriculture Retail Construction Transportation Other Figure 1: PwC survey 2005 for Weather Risk Management Association

3 Motivation 1-2 Weather Influences our daily lives and choices Impact on corporate revenues and earnings Meteorological institutions: business activity is weather dependent British Met Office: daily beer consumption gain 10% if temperature increases by 3 C If temperature in Chicago is less than 0 C consumption of orange juice declines 10% on average

4 Motivation 1-3 Examples Natural gas company suffers negative impact in mild winter Construction companies buy weather derivatives (rain period) Cloth retailers sell fewer clothes in hot summer Salmon fishery suffer losses by increase of sea temperatures Ice cream producers hedge against cold summers Disney World (rain period)

5 Motivation 1-4 What are Weather Derivatives (WD)? Hedge weather related risk exposures Payments based on weather related measurements Underlying: temperature, rainfall, wind, snow, frost Chicago Mercantile Exchange (CME) Monthly/seasonal/weekly temperature Futures/Options 24 US, 6 Canadian, 9 European and 3 Asian-Pacific cities From 2.2 billion USD in 2004 to 15 billion USD through March 2009

6 Motivation 1-5 WD market CME offers weather contracts on 42 cities throughout the world

7 Motivation 1-6 Weather Derivatives CME products HDD(τ 1, τ 2 )= τ 2 τ 1 CDD(τ 1, τ 2 )= τ 2 τ 1 CAT(τ 1, τ 2 )= τ 2 τ 1 max(18 C T t, 0)dt max(t t 18 C, 0)dt T t dt, where T t = Tt,max +T t,min T ti dt i and T ti AAT(τ 1, τ 2 )= τ 2 τ Tt 1 dt, where T t = 1 24 denotes the temperature of hour t i, (also referred to as C24AT index). Go to pricing approaches

8 Motivation 1-7 Algorithm Econometrics Fin. Mathematics. T t CAR(3) X t = T t Λ t F CAT (t,τ1,τ 2 ) X t+p = a X t + σ t ε t MPR ˆε t = ˆX t ˆσ t N(0, 1)

9 Motivation 1-8 How to smooth the seasonal variance curve? How close are the residuals to N(0, 1)? How to price no CME listed cities? 8 7 Seasonal Variance Beijing Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Time Go to details Figure 2: Daily empirical variance, seasonal variance ˆσ t,ftsg 2, ˆσ2 t,llr using Epanechnikov Kernel and bandwidth h = 4.49 for Beijing

10 Motivation 1-9 Outline 1. Motivation 2. Weather Dynamics 3. Stochastic Pricing of WD 4. Local Temperature Risk

11 Weather Dynamics 2-1 Weather Dynamics: Asian Data Temperature Market (CME): Tokyo and Osaka Beijing Tokyo Osaka Taipei Kaohsiung

12 Weather Dynamics 2-2 AAT Index Can we make money? Trading Period Measurement Period Index Code First-trade Last-trade τ 1 τ 2 CME 1 AAT 2 F G H J K Table 1: Osaka AAT contracts listed at CME on Source: Bloomberg. CME 1 prices of AAT Futures as listed on CME, 2 AAT index values computed from the realized temperature data.

13 Weather Dynamics 2-3 Temperature: T t = X t + Λ t Seasonal function with trend: L { } 2π(t dl ) Λ t = a + bt + c l cos l 365 l=1 â: average temperature, ˆb: global Warming. (1) City Period Berlin Osaka Tokyo Beijing Taipei Kaohsiung Table 2: Daily average temperature data. Source: Bloomberg, Deutsche Wetter Dienst.

14 Seasonality Seasonal function with trend: { 3 ˆΛ t = πi(t t + ĉ i cos ˆd } i ) 365 i=1 { 6 2π(i 4)(t + I(t ω) ĉ i cos ˆd } i ), 365 i=4 with I(t ω) an indicator for December, January and February. Seasonal function with a Local linear Regression (LLNR), t = days: arg min e,f 365 t=1 { T s e s f s (t s) } ) 2 (t s K h (2) Go to details

15 Temperature in C Tokyo Temperature in C Osaka Time Time Temperature in C Beijing Time Temperature in C Taipei Time Figure 3: The Fourier truncated, the corrected Fourier and the the local linear seasonal component for daily average temperatures in Tokyo Narita International Airport (upper left), Osaka Kansai International Airport (upper right), Beijing (lower left), Taipei (lower right).

16 Go to details AR(p): X t+p = p i=1 β ix t+p i + σ t ε t 5 60 ε t Beijing 0 5 ε 2 t Beijing Time Time 5 60 ε t Taipei Time ε 2 t Taipei Time Figure 4: Residuals ˆε t (left) and squared residuals ˆε 2 t (right) of the AR(p) (Beijing (up), Taipei (down)). No rejection of H 0 that the residuals are uncorrelated at 0% significance level, according to the modified Li-McLeod Portmanteau test

17 Seasonal Volatility: Highly seasonal ACF for squared residuals of AR(p) ACF ε t Beijing Lag 0.05 ACF ε 2 t Beijing Lag 0.1 ACF ε t Taipei 0 ACF ε 2 t Taipei Lag Lag Figure 5: ACF for residuals ˆε t (left) and squared residuals ˆε 2 t Beijing (up), Taipei (down). (right) of the AR(p) for

18 Calibration of Seasonal Variance: σ 2 t Calibration of daily variances of residuals AR(3) for 36 years: 2 Steps: Fourier truncated series + GARCH(p,q): ˆσ 2 t,ftsg ˆσ 2 t,ftsg = c 1 + L l=1 { c 2l cos ( ) 2lπt + c 2l+1 sin 365 ( )} 2lπt α 1 (σ 2 t 1ε t 1 ) 2 + β 1 σ 2 t 1 (3) 1 Step: Local linear Regression (LLR): ˆσ 2 t,llr arg min g,h 365 t=1 2 {ˆε t g s h s (t s) } ) 2 (t s K h (4) return

19 Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Seasonal Variance Tokyo Seasonal Variance Osaka Time 0 Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Time Seasonal Variance Beijing Seasonal Variance Taipei Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Time 0 Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Time Figure 6: Daily empirical variance, seasonal variance ˆσ t,ftsg 2, ˆσ2 t,llr using Epanechnikov Kernel and bandwidth h = 4.49 for Tokyo (upper left), Osaka (upper right), Beijing (lower left), Taipei (lower right). Go to details

20 ACF of (Squared) Residuals after Correcting Seasonal Volatility ACF ε t Beijing ACF ε 2 t Beijing Lag Lag 0.04 ACF ε t Taipei ACF ε 2 t Taipei Lag Lag Figure 7: (Left) Right: ACF for temperature (squared) residuals (up), Taipei (down) ˆε t ˆσ t,llr for Beijing

21 Fitting ˆσ t: 1-2 Step 3-4 City JB Kurt Skew KG AD Tokyo ˆε t ˆσ t,ftsg ˆε t ˆσ t,llr Osaka ˆε t ˆσ t,ftsg ˆε t ˆσ t,llr Beijing ˆε t ˆσ t,ftsg ˆε t ˆσ t,llr Taipei ˆε t ˆσ t,ftsg ˆε t ˆσ t,llr Kaohsiung ˆε t ˆσ t,ftsg ˆε t ˆσ t,llr Table 3: Skewness, kurtosis, Jarque Bera (JB), Kolmogorov Smirnov (KS) and Anderson Darling (AD) test statistics (365 days). Critical value at at 5% significance level is 5.99, at 1% 9.21.

22 Fitting ˆσ t: 1-2 Step 3-5 Residuals ( ˆεt ˆσ t ) become normal: QQ Plot of Sample Data versus Standard Normal QQ Plot of Sample Data versus Standard Normal Normal Quantiles Normal Quantiles QQ Plot of Sample Data versus Standard Normal QQ Plot of Sample Data versus Standard Normal Normal Quantiles Normal Quantiles Figure 8: QQ-plot of residuals ˆε t ˆσ t,ftsg (upper) ˆε t ˆσ t,llr (lower) and Normal residuals of Tokyo (upper left), Osaka (upper right), Beijing (lower left), Taipei (lower right)

23 Stochastic Pricing Model for temperature 4-1 Temperature Dynamics Temperature time series: T t = Λ t + X t with seasonal function Λ t. X t can be seen as a discretization of a continuous-time process AR(p) (CAR(p)). This stochastic model allows CAR(p) futures/options pricing.

24 Stochastic Pricing Model for temperature 4-2 Stochastic Pricing Ornstein-Uhlenbeck process X t R p : dx t = AX t dt + e p σ t db t e k : kth unit vector in R p for k = 1,...p, σ t > 0, A: (p p)-matrix A = α p α p 1... α 1 Go to proof details

25 Stochastic Pricing Model for temperature 4-3 Stationarity condition Solution of X t = x R p, s t 0: X s = exp {A(s t)}x + s t exp {A(s u)}e p σ u db u is stationarity as long as all the eigenvalues λ 1,..., λ p of A have negative real parts, i.e. the variance matrix: t converges as t. 0 { } σt s 2 exp {A(s)}e p e p exp A (s) ds

26 Stochastic Pricing Model for temperature 4-4 Temperature Futures Price Q θ pricing so that: F (t,τ1,τ 2 ) = E Q θ [Y F t ] (5) where Y equals the payoff of the temperature index and by Girsanov theorem: t Bt θ = B t θ u du 0 is a Brownian motion for t τ max. θ: a real valued, bounded and piecewise continuous function (market price of risk)

27 Stochastic Pricing Model for temperature 4-5 Temperature Dynamics under Q θ Under Q θ : dx t = (AX t + e p σ t θ t )dt + e p σ t dbt θ (6) with explicit dynamics, for s t 0: X s = exp {A(s t)}x + + s t s t exp {A(s u)}e p σ u θ u du exp {A(s u)}e p σ u db θ u (7)

28 Stochastic Pricing Model for temperature 4-6 CAT Futures For 0 t τ 1 < τ 2 : F CAT (t,τ1,τ 2) = E Q θ = + [ τ2 τ 1 τ2 ] T s ds F t τ 1 Λ u du + a t,τ1,τ 2 X t + τ1 t τ2 θ u σ u a t,τ1,τ 2 e p du τ 1 θ u σ u e 1 A 1 [exp {A(τ 2 u)} I p ] e p du (8) with a t,τ1,τ 2 = e 1 A 1 [exp {A(τ 2 t)} exp {A(τ 1 t)}], I p : p p identity matrix Benth et al. (2007)

29 Localized Temperature Risk Analysis 5-1 Local Temperature Risk Normality of ε t requires estimating the function θ(t) = { Λ t, σt 2 } with t = days, j = 0 J years. Recall: X t,j = T t,j Λ t L X t,j = β l X t l,j + σ t ε t,j l=1 ε t,j N(0, 1), i.i.d. (9) where N(0, 1) is the standard normal cdf.

30 Localized Temperature Risk Analysis 5-2 Adaptation Scale Fix s 1, 2,..., 365, sequence of ordered weights: W s (k) = (w(s, 1, h k ), w(s, 2, h k ),..., w(s, 365, h k )). Define w(s, t, h k ) = K hk (s t), (h 1 < h 2 <... < h K ). Local likelihood: L ˆε 365j+t = X 365j+t ˆβ l X 365j+t l θ k (s) = arg min θ Θ def = arg min θ Θ 365 l=1 t=1 j=0 J {log(2πθ)/2 + ˆε 2 t,j/2θ}w(s, t, h k ), L(W (k) s, θ)

31 Localized Temperature Risk Analysis 5-3 Local Temperature Residuals θ k (s) = t,j ˆε 2 t,jw(s, t, h k )/ t,j w(s, t, h k ) = t ˆε 2 t w(s, t, h k )/ t w(s, t, h k ) with ˆε t def = (J + 1) 1 J j=0 ˆε 2 t,j. For Taipei, daily average temperature , J = 36.

32 Localized Temperature Risk Analysis 5-4 Mirror Observations To avoid the boundary problem, use mirrored observations: Assume h K < 365/2, then the observations look like ˆε 2 364, ˆε2 363,..., ˆε2 0, ˆε2 1,..., ˆε2 730, where ˆε 2 t ˆε 2 t def = ˆε t, 364 t 0 def = ˆε 2 t 365, 366 t 730

33 Localized Temperature Risk Analysis 5-5 Parametric Exponential Bounds L( W (k), θ, θ ) def = L(W (k), θ) L(W (k), θ ) = NK(W (k), θ, θ ) K(W (k), θ, θ ) = {log(θ/θ ) + 1 θ /θ}/2 where K{θ, θ } is the Kullback Leibler divergence between θ and θ and N = 365 J. For any z > 0, P θ {L(W (k), θ, θ ) > z} 2 exp( z) E θ L(W (k), θ, θ ) r r r, where r r = 2r z 0 zr 1 exp( z)dz.

34 Localized Temperature Risk Analysis 5-6 LMS Procedure Construct an estimate ˆθ = ˆθ(s), on the base of θ 1 (s), θ 2 (s),..., θ K (s). Start with ˆθ 1 = θ 1. For k 2, θ k is accepted and ˆθ k = θ k if θ k 1 was accepted and L(W (k), θ l, θ k ) z l, l = 1,..., k 1 ˆθ k is the the latest accepted estimate after the first k steps.

35 Localized Temperature Risk Analysis 5-7 Illustration CS k*+1 Stop

36 Localized Temperature Risk Analysis 5-8 "Propagation" Condition A bound for the risk associated with first kind error: E θ L(W (k), θ k, ˆθ k ) r r r α (10) where k = 1,..., K and r r is the parametric risk bound.

37 Localized Temperature Risk Analysis 5-9 Sequential Choice of Critical Values Consider first only z 1 letting z 2 =... = z K 1 =. Leads to the estimates ˆθ k (z 1 ) for k = 2,..., K. The value z 1 is selected as the minimal one for which L{W (k), θ k, ˆθ k (z 1 )} r sup E θ α, k = 2,..., K. θ r r K 1 Set z k+1 =... = z K 1 = and fix z k lead the set of parameters z 1,..., z k,,..., and the estimates ˆθ m (z 1,..., z k ) for m = k + 1,..., K. Select z k s.t. L{W (k), θ m, ˆθ m (z 1, z 2,..., z k )} r sup E θ kα θ r r K 1, m = k + 1,..., K.

38 Localized Temperature Risk Analysis 5-10 Critical Values Figure 9: Simulated CV with θ = 1, r = 0.5, MC = 5000 with α = 0.3, 0.5, 0.8 (left), with different bandwidth sequences (right).

39 Localized Temperature Risk Analysis 5-11 Bound for Critical Values Suppose 0 < µ h k 1 /h k µ 0 < 1. Let θ(.) = θ, for all t (0, 365). There is a constant a 0 > 0 depending on r and µ 0, µ, s.t. ensures (10). z k = a 0 log K + 2 log(nh k /α) + 2r log(h K /h k )

40 Localized Temperature Risk Analysis 5-12 "Small Modeling Bias" (SMB) Condition: (W (k), θ) = 365 K{θ(t), θ}1{w(s, t, h k ) > 0} t=1, k < k (11) k is the maximum k satisfying the SMB condition. Property: For any estimate θ k and θ satisfying SMB, it holds: E θ(.) log{1 + L( θ k, θ) r /r r } + 1

41 Localized Temperature Risk Analysis 5-13 "Stability" Property Stability: the attained quality of estimation during "propagation" can not get lost at further steps. L(W (k ), θ k, ˆθˆk)1{ˆk > k } z k

42 Localized Temperature Risk Analysis 5-14 "Oracle" Property Theorem Let (W (k), θ) for some θ Θ and k k. Then E θ(.) log {1 + L(W (k ), θ k, θ) r } r r (W (k ) ) E θ(.) log {1 + L(W (k ), θ k, ˆθ) r } r r (W (k ) ) α + log{1 + z k r r }

43 Localized Temperature Risk Analysis 5-15 averdaysqresid fix.band.param - adapt.para Bandwidth Figure 10: Bandwidth sequences (upper), fixed bandwidth curve; adaptive bandwidth curve for squared residuals (blue middle); difference between adaptive and fixed bandwidth (lower), Beijing, α = 0.3, r = 0.5

44 Localized Temperature Risk Analysis 5-16 averdaysqresid fix.band.param - adapt.para Bandwidth Figure 11: Bandwidth sequences (upper), fixed bandwidth curve; adaptive bandwidth curve for squared residuals (blue middle); difference between adaptive and fixed bandwidth (lower), Kaoshiung, α = 0.3, r = 0.5

45 averdaysqresid Bandwidth ix.band.param - adapt Figure 12: Bandwidth sequences (upper), fixed bandwidth curve; adaptive bandwidth curve for squared residuals (blue middle); difference between adaptive and fixed bandwidth (lower), Berlin, α = 0.3, r = 0.5 Go to details

46 Localized Temperature Risk Analysis 5-18 Adapt Bandwidth (p=0.6184) Fixed Bandwidth (p= ) Adapt Bandwidth (p= ) Fixed Bandwidth (p= ) Adapt Bandwidth (p= ) Fixed Bandwidth (p= ) Figure 13: Q-Q Plot for squared residuals from Berlin, Kaoshiung, Beijing

47 Localized Temperature Risk Analysis 5-19 Normalized Residual city bandwidth KS JB AD Beijing FB AB ABS Kaoshiung FB e e-06 AB e e-03 ABS e e-03 Berlin FB AB Table 4: P-value of normality tests for fixed bandwidth curve (FB), adaptive bandwidth curve (AB), adaptive smoothed bandwidth curve (ABS)

48 Localized Temperature Risk Analysis 5-20 Tokyo & Osaka AAT Future Prices Trading Period Measurement Period Index City Code First-trade Last-trade τ 1 τ 2 CME 1 F AAT F AAT,loc Osaka F G H J K Tokyo J K Table 5: Osaka AAT contracts listed at CME on Source: Bloomberg. CME 1 prices of AAT Futures as listed on CME, 2 AAT index values computed from the realized temperature data and with adaptive bandwidth F AAT,loc

49 Localized Temperature Risk Analysis 5-21 Conclusions and further work Temperature risk stochastics closer to Wiener process when applying adaptive methods Better estimates of λ t and σ t lead to fair price -> pure MPR

50 Localized Temperature Risk Analysis 5-22 References F.E. Benth and J.S. Benth and S. Koekebakker Putting a Price on Temperature Scandinavian Journal of Statistics 34: , 2007 F.E. Benth and W.K. Härdle and B. López Cabrera Pricing of Asian Temperature Risk To appear in Statistics Tools of Finance and Insurance, W.K. Härdle, R. Weron editors, 2nd. edition, 2011 P.J. Brockwell Continuous-time ARMA Processes Handbook of Statistics 19: , 2001

51 Localized Temperature Risk Analysis 5-23 References W.K. Härdle and B.López Cabrera Implied Market Price of Weather Risk Submitted to Journal of Applied Mathematical Finance, 2010 U. Horst and M. Müller On the Spanning Property of Risk Bonds Priced by Equilibrium Mathematics of Operations Research 4: , 2007 W.K. Härdle, J. Franke, C.M. Hafner Statistics of Financial Markets: 3rd edition Springer Verlag, Heidelberg, 2011

52 Localizing Temperature Risk Wolfgang Karl Härdle, Brenda López Cabrera Ostap Okhrin, Weining Wang Ladislaus von Bortkiewicz Chair of Statistics C.A.S.E. - Center for Applied Statistics and Economics Humboldt-Universität zu Berlin

53 Appendix 6-1 Appendix A Li-McLeod Portmanteau Test modified Portmanteau test statistic Q L to check the uncorrelatedness of the residuals: Q L = n L k=1 r 2 k L(L + 1) (ˆε) +, 2n where r k, k = 1,..., L are values of residuals ACF up to the first L lags and n is the sample size. Then, Q L χ 2 (L p q) Q L is χ 2 distributed on (L p q) degrees of freedom where p,q denote AR and MA order respectively and L is a given value of considered lags.

54 Appendix 6-2 WD pricing models 1. General Equilibrium Theory for incomplete markets: Indifference, Marginal, Market Clearing 2. Pricing via no arbitrage arguments: adequate equivalent martingale measure return

55 Appendix 6-3 X t can be written as a Continuous-time AR(p) (CAR(p)): For p = 1, dx 1t = α 1 X 1t dt + σ t db t For p = 2, For p = 3, X 1(t+2) (2 α 1 )X 1(t+1) + (α 1 α 2 1)X 1t + σ t (B t 1 B t ) X 1(t+3) (3 α 1 )X 1(t+2) + (2α 1 α 2 3)X 1(t+1) + ( α 1 + α 2 α 3 + 1)X 1t + σ t (B t 1 B t )

56 Appendix 6-4 Proof CAR(3) AR(3) Let A = α 3 α 2 α 1 use B t+1 B t = ε t assume a time step of length one dt = 1 substitute iteratively into X 1 dynamics

57 Appendix 6-5 Proof CAR(3) AR(3): X 1(t+1) X 1(t) = X 2(t) dt X 2(t+1) X 2(t) = X 3(t) dt X 3(t+1) X 3(t) = α 1 X 1(t) dt α 2 X 2(t) dt α 3 X 3(t) dt + σ t ε t X 1(t+2) X 1(t+1) = X 2(t+1) dt X 2(t+2) X 2(t+1) = X 3(t+1) dt X 3(t+2) X 3(t+1) = α 1 X 1(t+1) dt α 2 X 2(t+1) dt X 1(t+3) X 1(t+2) = X 2(t+2) dt X 2(t+3) X 2(t+2) = X 3(t+2) dt α 3 X 3(t+1) dt + σ t+1 ε t+1 X 3(t+3) X 3(t+2) = α 1 X 1(t+2) dt α 2 X 2(t+2) dt α 3 X 3(t+2) dt + σ t+2 ε t+2 return

58 return City(Period) â ˆb ĉ1 ˆd1 ĉ 2 ˆd2 ĉ 3 ˆd3 Tokyo ( ) ( ) ( ) ( ) ( ) Taipei ( ) ( ) ( ) Osaka ( ) ( ) ( ) ( ) ( ) Kaohsiung ( ) ( ) ( ) ( ) Beijing ( ) ( ) ( ) ( ) ( ) Table 6: Seasonality estimates ˆλ t of daily average temperatures in Asia. All coefficients are nonzero at 1% significance level. Data source: Bloomberg.

59 Appendix 6-7 AR(p): X t+p = p i=1 β ix t+p i + σ t ε t City β 1 β 2 β 3 Tokyo(p=3) Osaka(p=3) Beijing(p=3) Taipei(p=3) Kaoshiung(p=3) Table 7: Coefficients of AR(p), Model selection: AIC The long memory diagnosis can be replicated by a short memory process with structural breaks! return

60 Appendix 6-8 Calibration of Seasonal Variance: σ 2 t ĉ 1 ĉ 2 ĉ 3 ĉ 4 ĉ 5 ĉ 6 ĉ 7 α β Tokyo Osaka Beijing Taipei Table 8: First 7 Coefficients of σt 2 black are significant at 1% level. and GARCH(p = 1, q = 1). The coefficients in return

61 Appendix 6-9 Laverdaytemp Bandwidth fix.band.param - adapt.para Figure 14: Bandwidth sequences (upper), fixed bandwidth curve; adaptive bandwidth curve for daily temperature difference between adaptive and fixed bandwidth (lower), Beijing, α = 0.3, r = 0.5 return

62 Appendix 6-10 Laverdaytemp ix.band.param - adapt Bandwidth Figure 15: Bandwidth sequences (upper), fixed bandwidth curve; adaptive bandwidth curve for daily temperature ; difference between adaptive and fixed bandwidth (lower), Kaoshiung, α = 0.3, r = 0.5

63 Appendix 6-11 Laverdaytemp x.band.param - adap Bandwidth Figure 16: Bandwidth sequences (upper), fixed bandwidth curve; adaptive bandwidth curve for daily temperature ; difference between adaptive and fixed bandwidth (lower), Berlin, α = 0.3, r = 0.5