Calibrating Weather Derivatives

Transcription

1 Calibrating Weather Derivatives Brenda López Cabrera Wolfgang Karl Härdle Institut für Statistik and Ökonometrie CASE-Center for Applied Statistics and Economics Humboldt-Universität zu Berlin

2 1-2 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 temperature Future/Option contracts 18 US, 9 European and 2 Asian-Pacific cities From 2.2 billion USD in 2004 to 22 billion USD through September 2005

3 1-3 Weather Derivative Figure 1: A WD table quoting prices of May 2005 contracts. Mercantile Exchange s Weather-i Source: Chicago

4 1-4 Pricing Methods Price of a contingent claim F : F = exp { rt } E Q [ψ(i)] (1) I: weather index, ψ(i): payoff of the derivative at expiration, r: risk free interest rate, Q: risk neutral probability measure Burn analysis: F = exp { rt }n 1 n t=1 ψ(i t) Stochastic Model/Daily simulation Market ist Incomplete: need of a equivalent measure Q as a pricing measure

5 Stochastic Pricing Model for Temperature Derivatives 1-5 Mean reversion model: Dornier et al. (2000), Alaton et al. (2002) Fractional Brownian Motion: Brody et al. (2002) ARMA model with seasonal ARCH innovations: Campbell and Diebold (2005) CAR model with seasonal volatility: Benth et al. (2007) AIM: Calibrate WD from weather data and CME data

6 1-6 CME WD data HDD-WD for Berlin (289 days). 451 contracts: prices (0, ), maturity (-35, 112)

7 1-7 Outline 1. Motivation 2. Weather Derivatives Indices 3. Stochastic Pricing Model: CAR(p) 4. Application to Berlin data

8 Weather Derivatives (WD) 2-1 Temperature Indices Heating degree day (HDD): over a period [τ 1, τ 2 ] τ2 τ 1 max(k T u, 0)du (2) Cooling degree day CDD): over a period [τ 1, τ 2 ] τ2 τ 1 max(t u K, 0)du (3) K is the baseline temperature (typically 18 C or 65 F), T u is the average temperature on day u.

9 Weather Derivatives (WD) 2-2 Weather indices: temperature Average of average temperature (AAT): measure the excess or deficit of temperature. The average of average temperatures over [τ 1, τ 2 ] days is: 1 τ 1 τ 2 τ2 τ 1 T u du (4) Cumulative averages (CAT): The accumulated average temperature over [τ 1, τ 2 ] days is: τ2 τ 1 T u du (5)

10 Weather Derivatives (WD) 2-3 Weather indices: temperature Event indices: number of times a certain meteorological event occurs in the contract period Frost days: temperature at 7: local time is less than or equal to 3.5 C HDD-CDD parity: CDD(τ 1, τ 2 ) HDD(τ 1, τ 2 ) = CAT (τ 1, τ 2 ) K Sufficient to analyse only CDD and CAT futures

11 Weather Derivatives (WD) 2-4 Weather indices: temperature Figure 2: HDDs for Berlin over 57 years. Stations: Berlin Tempelhof (black line), Berlin Dahlem (blue line), Postdam (red line)

12 Weather Derivatives (WD) 2-5 Stochastic Model for temperature Define the vectorial Ornstein-Uhlenbleck process X t R p : dx t = AX t dt + e pt σ t db t e k : k th unit vector in R p for k = 1,...p, σ t > 0: temperature volatility, A: p p-matrix, B t : Wiener Process, α k : constant A = α p α p 1... α 1 Solution of X t = x R p : X s = exp (A(s t))x + s t exp (A(s u))e p σ u db u

13 Weather Derivatives (WD) 2-6 X 1t is a CAR (p) model Resulting discrete-time dynamics: For p = 1, then X t = X 1t: dx 1t = α 1X 1tdt + σ tdb t For p = 2, X 1(t+2) (2 α 1)X 1(t+1) + (α 1 α 2 1)X 1(t) + σ t(b t 1 B t) For p = 3, 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 1(t) + σ t(b t 1 B t)

14 Weather Derivatives (WD) 2-7 Temperature Dynamics Continuous-time AR(p) (CAR(p)) model for Temperature: T t = Λ t + X 1t (6) X q : q th coordinate of vector X with q = 1,.., p Λ t : seasonality function X 1t is a CAR (p) model Stationarity holds when the variance matrix: t converges as t o σ 2 (t s) exp(a(s))e pe p exp(a s )ds (7)

15 Weather Derivatives (WD) 2-8 Girsanov theorem: an equivalent probability measure Q θ : t Bt θ = B t θ u du 0 is a Brownian montion for t τ max. θ t : a real valued, bounded and piecewise continous function (market price of risk). Under Q θ : dx t = (AX t + e p σ t θ t )dt + e p σ t db θ t (8) 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 (9)

16 Weather Derivatives (WD) 2-9 Temperature futures price Under the Q risk neutral probability: 0 = exp { r(τ 2 t)} E Q [ Y F (t,τ1,τ 2 ) F t ] (10) Under the Q θ pricing probability: F (t,τ1,τ 2 ) = E Qθ [Y F t ] (11) where Y may be equal to the payoff from the CAT/HDD/CDD future

17 Weather Derivatives (WD) 2-10 CAT futures price [ τ2 ] F CAT (t,τ1,τ 2 ) = E Qθ max(t s )ds F t = + τ 1 τ2 τ 1 Λ u du + a t,τ1,τ 2 X t + τ2 τ 1 τ2 τ 1 θ u σ u a t,τ1,τ 2 e p du θ u σ u e 1 A 1 {exp (A(τ 2 u)) I p } e p du 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 Time Q θ -dynamics of F CAT : df CAT (t,τ1,τ 2 ) = σ t a t,τ1,τ 2 e p db θ t

18 Weather Derivatives (WD) 2-11 CAT call option written on a CAT future during the period [τ 1, τ 2 ] is: where and C CAT (t,t,τ1,τ 2 ) = exp { r(t t)} { (F CAT (t,τ1,τ 2 ) K)Φ(d (t, T, τ 1, τ 2 )) T } + Σ 2 CAT (s,τ 1,τ 2 ) dsφ(d (t, T, τ 1, τ 2 )) (12) d (t, T, τ 1, τ 2 ) = t T t F CAT (t,τ 1,τ 2 ) K Σ 2 CAT (s,τ 1,τ 2 ) ds Σ CAT (s,τ1,τ 2 ) = σ t a t,τ1,τ 2 e p and Φ denotes the standard normal cdf.

19 Weather Derivatives (WD) 2-12 Hedging strategy for CAT call option Delta of the call option: Φ(d (t, T, τ 1, τ 2 )) = C CAT (t,t,τ 1,τ 2 ) F CAT (t,τ1,τ 2 ) (13) Hold: close to zero CAT futures when the option is far out of the money, otherwise close to 1.

20 Weather Derivatives (WD) 2-13 CDD futures price [ τ2 ] F CDD(t,τ1,τ 2 ) = E Qθ max(t u K, 0)du F t τ 1 τ2 [ m{t,s,e ] = υ t,s ψ 1 exp (A(s t))x t} ds τ 1 υ t,s where m {t,s,x} = Λ s c + s τ 1 σ u θ u e 1 exp (A s t)e p du + x υt,s 2 = s { } t σ2 u e 2 1 exp (A s t )e p du ψ(x) = xφ(x) + φ(x) with x = e 1 exp (A(s t))x t Φ is the standard normal cdf

21 Weather Derivatives (WD) 2-14 CDD futures dynamics τ2 } df CDD(t,τ1,τ 2 ) = σ t {e 1 exp(a(s t))e p Φ τ 1 [ { }] m t, s, e 1 exp (A(s t))x t v t,s CDD volatility Σ CDD(s,τ1,τ 2 ) recovers CAT volatility dsdb θ t

22 Weather Derivatives (WD) 2-15 CDD call options C CDD(t,T,τ1,τ 2 ) = exp { r(τ t)} [ ( τ2 ( ) )] mindex E max υ τ,s ψ ds K, 0 (14) τ 1 υ τ,s x=x t ( τ index = τ, s, e 1 exp (A(s t))x + e 1 exp (A(s u))e p σ u θ u du t + Σ s,t,τ Y ) Y is a std. normal variable, Σ 2 s,t,t = T { } t e 2 1 exp (A(s u))e p σ 2 u du

23 Weather Derivatives (WD) 2-16 Hedging strategies CDD call options Let C = max(f CDD(τ,τ1,τ 2 ) K, 0) be the payoff of the option, its Clark Ocone representation is: C = E Qθ [C] + τ Then, the hedging strategy in CDD-futures: 0 E Qθ [D, C F t ]db θ t (15) H CDD(t,τ1,τ 2 ) = Σ 1 CDD(t,τ 1,τ 2 ) EQθ [D, C F t ] (16) where D t is the Malliavin derivative

24 Applications 3-1 Berlin temperature Daily average temperatures: 1950/1/1-2006/7/24 Station: BERLIN-TEMP.(FLUGWEWA) 29 February removed recordings

25 Applications 3-2 Seasonality Suppose seasonal function with trend: ( ) 2π(t a3 ) Λ t = a 0 + a 1 t + a 2 cos 365 Estimates: â 0 = 91.52(90.47, 92.56), â 1 = 0.00(0.00, 0.00), â 2 = 97.96(97.22, 98.69), â 3 = 165.1( 165.5, 164.6) with 95% confidence bounds RMSE = , R 2 :

26 Applications 3-3 Seasonality Figure 3: Temperature in Berlin

27 Applications 3-4 Temporal dependence Remove seasonality: Y t = T t Λ t ADF-Test: (1 L)y = c 1 +c 2 trend+τly+α 1 (1 L)Ly+... α p (1 L)L p y+u τ = , with 1% critical value equal to Reject H 0 (τ = 0), hence Y i is a stationary process I(0)

28 Applications 3-5 PACF AR(3): Y i+3 = 0.91Y i Y i Y i + (510.63) 1 2 ε i CAR(3)-parameters: α 1 = 2.09, α 2 = 1.38, α 3 = 0.22 Stationarity condition for the CAR(3) is fulfilled: λ 1 = , λ 2,3 = ± i. Figure 4: Partial autocorrelation function (PACF)

29 Applications 3-6 Figure 5: Residuals (up) and squared residuals (down) of the AR(3). Rejection of H 0 for zero-mean residuals at 1% significance level

30 Applications 3-7 Seasonal volatility Close to zero ACF for residuals of AR(3) and according to Box-Ljung statistic the first few lags are insignificant. Figure 6: ACF for residuals AR(3)

31 Applications 3-8 Seasonal volatility Highly seasonal ACF for squared residuals of AR(3) Figure 7: ACF for squared residuals AR(3)

32 Applications 3-9 Calibration of daily variances of residuals AR(3) for 56 years: 4 { ( ) ( )} 2iπt 2iπt σt 2 = c 1 + c 2k cos + c 2i+1 sin i=1 Figure 8: Seasonal variance: daily empirical variance (blue line), fitted squared volatility function (red line) at 10% significance level

33 Applications 3-10 Figure 9: ACF for residuals (up) and squared residuals (down) after dividing out the seasonal volatility

34 Applications 3-11 Residuals become normal T-test: Accept H 0 of normality with p= , Skewness= , Kurtosis= Figure 10: Left: pdf for residuals (black line) and a normal pdf (red line).

35 Applications 3-12 Samuelson Effect Figure 11: The CAT term structure of volatility

36 Applications 3-13 Samuelson and Autoregressive effect Figure 12: CAT volatility prior of 2 contracts in June: one with measurement period of 1 month (blue line) and the other of 1 week (red line)

37 Applications 3-14 AR(3)-contribution to CAT volatility Figure 13: AR(3) contribution to the CAT volatility prior of 2 contracts in June.

38 Applications 3-15 to do.. Compute market price risk θ u from WD data: [ τ2 ] F CAT (t,τ1,τ 2 ) = E Qθ max(t s )ds F t = + τ 1 τ2 τ 1 Λ u du + a t,τ1,τ 2 X t + τ2 τ 1 τ2 τ 1 θ u σ u a t,τ1,τ 2 e p du θ u σ u e 1 A 1 {exp (A(τ 2 u)) I p } e p du θ u is a real valued piecewise linear function: { } θ1, u (u θ(u) = 1, u 2 ) θ 2, u (u 1, u 2 )

39 Research 4-16 Questions Explicit prices/hedging strategies of WD traded at CME Spatial dependence in temperature dynamics: DSFM? Random internal climate/urbanisation variability Role of the strike value

40 Conclusion 5-1 Conclusion CAR(3) model for the temperature dynamics Samuelson effect and autoregressive effect observed in Berlin data

41 Conclusion 5-2 Reference F.E. Benth (2004) Option Theory with Stochastic Analysis: An Introduction to Mathematical Finance Berlin: Springer. F.E Benth and J.S. Benth and S. Koekebakker (2007) Putting a price on temperature Scandinavian Journal of Statistics F.E. Benth and J.S. Benth (2005) Stochastic Modelling of temperature variations with a view towards weather derivatives Appl.Math. Finance.

42 Conclusion 5-3 K. Burnecki (2004) Weather derivatives Warsaw. M. Cao, A. Li, J. Wei (2003) Weather Derivatives: A new class of Financial Instruments Working Paper, Schulich School of Business, York University, Canada, 2003 S. Campbell, F. Diebold (2005) Weather Forecasting for weather derivatives J.American Stat. Assoc. J.C. Hull (2006) Option, Futures and other Derivatives 6th ed. New Jersey: Prentice Hall International

43 Conclusion 5-4 S. Jewson, A. Brix (2005) Weather Derivatives Valuation: The Meteorological, Statistical, Financial and Mathematical Foundations Cambridge: Cambridge University Press, M. Odening (2004) Analysis of Rainfall Derivatives using Daily precipitation models: opportunities and pitfalls C. Turvey (1999) The Essentials of Rainfall Derivatives and Insurance Working Paper WP99/06, Department of Agricultural Economics and Business, University of Guelph, Ontario.

44 Conclusion 5-5 Appendix Residuals with and without seasonal volatility: Lag Qstat res QSIG res Qstat res1 QSIG res Table 1: Q test using Ljung-Box s for residuals with (res) and without seasonality in the variance (res1)

45 Conclusion 5-6 Appendix Proof CAR(3) AR(3): Let A = α 3 α 2 α 1 - Use B t+1 B t = ɛ t - Substitute iteratively in X 1 dynamics: X 1(t+1) X 1(t) = X 1(t) dt + σ t ɛ t X 2(t+1) X 2(t) = X 3(t) dt + σ t ɛ t X 3(t+1) X 3(t) = α 3 X 1(t) dt α 2 X 2(t) dt α 1 X 3(t) dt + σ t ɛ t X 1(t+2) X 1(t+1) = X 1(t+1) dt + σ t+1 ɛ t+1 X 2(t+2) X 2(t+1) = X 3(t+1) dt + σ t+1 ɛ t+1 X 3(t+2) X 3(t+1) = α 3 X 1(t+1) dt α 2 X 2(t+1) dt α 1 X 3(t+1) dt + σ t+1 ɛ t+1 X 1(t+3) X 1(t+2) = X 1(t+2) dt + σ t+2 ɛ t+2 X 2(t+3) X 2(t+2) = X 3(t+2) dt + σ t+2 ɛ t+2 X 3(t+3) X 3(t+2) = α 3 X 1(t+2) dt α 2 X 2(t+2) dt α 1 X 3(t+2) dt + σ t+2 ɛ t+2