The Volatility of Temperature and Pricing of Weather Derivatives

Transcription

1 The Volatility of Temperature and Pricing of Weather Derivatives Fred Espen Benth Work in collaboration with J. Saltyte Benth and S. Koekebakker Centre of Mathematics for Applications (CMA) University of Oslo, Norway Quantitative Finance Seminar Humboldt-Universität zu Berlin, 11 December, 2006

2 Overview of the presentation 1. The temperature market 2. A stochastic model for daily temperature Continuous-time AR(p) model with seasonal volatility 3. HDD, CDD and CAT Explicit futures prices Options on temperature forwards

3 The temperature market

4 The temperature market Chicago Mercantile Exchange (CME) organizes trade in temperature derivatives: Futures contracts on monthly and seasonal temperatures European call and put options on these futures Contracts on 18 US and 2 Japanese cities 9 European cities London, Paris, Amsterdam, Rome, Essen, Barcelona, Madrid Stockholm Berlin

5 HDD and CDD HDD (heating-degree days) over a period [τ 1, τ 2 ] τ2 τ 1 max (18 T (u), 0) du HDD is the accumulated degrees when temperature T (u) is below 18 CDD (cooling-degree days) is correspondingly the accumulated degrees when temperature T (u) is above 18 τ2 τ 1 max (T (u) 18, 0) du

6 CAT and PRIM CAT = cumulative average temperature Average temperature here meaning the daily average τ2 τ 1 T (u) du PRIM = Pacific Rim, the average temperature 1 τ 2 τ 1 τ2 τ 1 T (u) du

7 At the CME... Futures written on HDD, CDD, CAT and PRIM as index HDD and CDD is the index for US temperature futures CAT index for European temperature futures, along with HDD and CDD PRIM only for Japan Discrete (daily) measurement of HDD, CDD, CAT and PRIM All futures are cash settled 1 trade unit=20 Currency (trade unit being HDD, CDD or CAT) Currency equal to USD for US futures and GBP for European Call and put options written on the different futures

8 A continuous-time AR(p)-process Related models Stockholm temperature data

9 A continuous-time AR(p)-process Related models Stockholm temperature data A continuous-time AR(p)-process Define the Ornstein-Uhlenbeck process X(t) R p dx(t) = AX(t) dt + e p (t)σ(t) db(t), e k : k th unit vector in R p σ(t): temperature volatility A: p p-matrix [ A = 0 I α p α 1 ]

10 A continuous-time AR(p)-process Related models Stockholm temperature data Explicit solution of X(t): s X(s) = exp (A(s t)) x + exp (A(s u)) e p σ(u) db(u), t Temperature dynamics T (t) defined as T (t) = Λ(t) + X 1 (t) X 1 (t) CAR(p) model, Λ(t) seasonality function Temperature will be normally distributed at each time

11 A continuous-time AR(p)-process Related models Stockholm temperature data Why is X 1 a CAR(p) process? Consider p = 3 Do an Euler approximation of the X(t)-dynamics with time step 1 Substitute iteratively in X 1 (t)-dynamics Use B(t + 1) B(t) = ɛ(t) Resulting discrete-time dynamics X 1 (t + 3) (3 α 1 )X 1 (t + 2) + (2α 1 α 2 1)X 1 (t + 1) + (α (α 1 + α 3 ))X 1 (t) + σ(t)ɛ(t).

12 A continuous-time AR(p)-process Related models Stockholm temperature data Related models Mean-reverting Ornstein-Uhlenbeck process Let p = 1, implying X(t) = X 1 (t) dx 1 (t) = α 1 X 1 (t) dt + σ(t) db(t) Benth and Saltyte-Benth (2005,2006) Seasonal volatility σ(t) Empirical analysis of Stockholm and cities in Norway Pricing of HDD/CDD/CAT temperature futures and options Spatial model for temperature in Lithuania

13 A continuous-time AR(p)-process Related models Stockholm temperature data Dornier and Querel (2000): Constant temperature volatility σ Empirical analysis of Chicago, O Hare airport temperatures Alaton, Djehiche and Stillberger (2002): Monthly varying σ Empirical analysis of Stockholm, Bromma airport temperatures Pricing of various temperature futures and options Brody, Syroka and Zervos (2002): Constant σ, but B is fractional Brownian motion Empirical analysis of London temperatures Pricing of various temperature futures and options

14 A continuous-time AR(p)-process Related models Stockholm temperature data Campbell and Diebold (2005): ARMA time series model for the temperature Seasonal ARCH-model for the volatility σ Empirical analysis for temperature in US cities The model seems to fit data very well However, difficult to do analysis with it Futures and option pricing by simulation

15 A continuous-time AR(p)-process Related models Stockholm temperature data Stockholm temperature data Daily average temperatures from 1 Jan 1961 till 25 May February removed in every leap year 16,570 recordings Last 11 years snapshot with seasonal function

16 A continuous-time AR(p)-process Related models Stockholm temperature data Fitting of model goes stepwise: 1. Fit seasonal function Λ(t) with least squares 2. Fit AR(p)-model on deseasonalized temperatures 3. Fit seasonal volatility σ(t) to residuals

17 A continuous-time AR(p)-process Related models Stockholm temperature data 1. Seasonal function Suppose seasonal function with trend Λ(t) = a 0 + a 1 t + a 2 cos (2π(t a 3 )/365) Use least squares to fit parameters May use higher order truncated Fourier series Estimates: a 0 = 6.4, a 1 = , a 2 = 10.4, a 3 = 166 Average temperature increases over sample period by 1.6 C

18 A continuous-time AR(p)-process Related models Stockholm temperature data 2. Fitting an auto-regressive model Remove the effect of Λ(t) from the data Y i := T (i) Λ(i), i = 0, 1,... Claim that AR(3) is a good model for Y i : Y i+3 = β 1 Y i+2 + β 2 Y i+1 + β 3 Y i + σ i ɛ i,

19 A continuous-time AR(p)-process Related models Stockholm temperature data The partial autocorrelation function Suggests AR(3) autocorrelation lag

20 A continuous-time AR(p)-process Related models Stockholm temperature data Estimation gives β 1 = 0.957, β 2 = 0.253, β 3 = All parameters are significant at the 1% level R 2 is 94.1% Higher-order AR-models did not increase R 2 significantly

21 A continuous-time AR(p)-process Related models Stockholm temperature data 3. Seasonal volatility Consider the residuals from the auto-regressive model Autocorrelation function for residuals and their squares Close to zero ACF for residuals Highly seasonal ACF for squared residuals autocorrelation autocorrelation lag lag

22 A continuous-time AR(p)-process Related models Stockholm temperature data Suppose the volatility is a truncated Fourier series 4 4 σ 2 (t) = c + c i sin(2iπt/365) + d j cos(2jπt/365) i=1 j=1 This is calibrated to the daily variances 45 years of daily residuals Line up each year next to each other Calculate the variance for each day in the year

23 A continuous-time AR(p)-process Related models Stockholm temperature data A plot of the daily empirical variance with the fitted squared volatility function High variance in winter, and early summer Low variance in spring and late summer/autumn seasonal variance days

24 A continuous-time AR(p)-process Related models Stockholm temperature data Same observation for other cities Several cities in Norway and Lithuania Seasonality in ACF for squared residuals observed in Campbell and Diebold s paper Example from Alta, northern Norway City on the coast close to North Cape Climate influenced from the warm Gulf Stream Different weather systems as in Stockholm

25 A continuous-time AR(p)-process Related models Stockholm temperature data Dividing out the seasonal volatility from the regression residuals ACF for squared residuals non-seasonal ACF for residuals unchanged Residuals become normally distributed autocorrelation lag

26 Some generalities Risk neutral probabilities CDD futures CAT futures

27 Some generalities Risk neutral probabilities CDD futures CAT futures Some generalities on temperature futures HDD-futures price F HDD (t, τ 1, τ 2 ) at time t τ 1 No trade in settlement period [ 0 = e r(τ2 t) τ 2 ] E Q max(c T (u), 0) du F HDD (t, τ 1, τ 2 ) F t. τ 1 Constant interest rate r, and settlement at the end of index period, τ 2 Q is a risk-neutral probability Not unique since market is incomplete Temperature (nor HDD) is not tradeable c is equal to 65 F or 18 C

28 Some generalities Risk neutral probabilities CDD futures CAT futures Adaptedness of F HDD (t, τ 1, τ 2 ) yields [ τ 2 ] F HDD (t, τ 1, τ 2 ) = E Q max(c T (u), 0) du F t Analogously, the CDD and CAT futures price is τ 1 [ τ 2 ] F CDD (t, τ 1, τ 2 ) = E Q max(t (u) c, 0) du F t [ τ 2 ] F CAT (t, τ 1, τ 2 ) = E Q T (u) du F t τ 1 τ 1

29 Some generalities Risk neutral probabilities CDD futures CAT futures A class of risk neutral probabilities Parametric sub-class of risk-neutral probabilities Q θ Defined by Girsanov transformation of B(t) db θ (t) = db(t) θ(t) dt θ(t) time-dependent market price of risk Density of Q θ ( Z θ t (t) = exp θ(s) db(s) 1 t θ 2 (s) ) ) ds 0 2 0

30 Some generalities Risk neutral probabilities CDD futures CAT futures Change of dynamics of X(t) under Q θ : dx(t) = (AX(t) + e p σ(t)θ(t)) dt + e p σ(t) db θ (t). or, explicitly s X(s) = exp (A(s t)) x + + s t t exp (A(s u)) e p σ(u)θ(u) du exp (A(s u)) e p σ(u) db θ (u), Feasible dynamics for explicit calculations

31 Some generalities Risk neutral probabilities CDD futures CAT futures CDD futures CDD-futures price F CDD (t, τ 1, τ 2 ) = where τ2 τ 1 m(t, s, x) = Λ(s) c + v 2 (t, s) = s t ( ) m(t, s, e v(t, s)ψ 1 exp(a(s t))x(t)) ds v(t, s) s t σ(u)θ(u)e 1 exp(a(s u))e p du + x σ 2 (u) ( e 1 exp(a(s u))e p ) 2 du Ψ(x) = xφ(x) + Φ (x), Φ being the cumulative standard normal distribution function.

32 Some generalities Risk neutral probabilities CDD futures CAT futures The futures price is dependent on X(t)... and not only on current temperature T (t) In discrete-time, the futures price is a function of The lagged temperatures T (t), T (t 1),..., T (t p)

33 Some generalities Risk neutral probabilities CDD futures CAT futures Time-dynamics of the CDD-futures price τ2 df CDD (t, τ 1, τ 2 ) = σ(t) e 1 exp(a(s t))e p τ ( 1 ) m(t, s, e Φ 1 exp(a(s t))x(t) ds db θ (t) v(t, s) Follows from the martingale property and Itô s Formula

34 Some generalities Risk neutral probabilities CDD futures CAT futures Pricing options on CDD-futures CDD-futures dynamics does not allow for analytical option pricing Monte Carlo simulation of F CDD is the natural approach Simulate X(τ) at exercise time τ Derive F CDD by numerical integration Alternatively, simulate the dynamics of the CDD-futures

35 Some generalities Risk neutral probabilities CDD futures CAT futures CAT futures CAT-futures price F CAT (t, τ 1, τ 2 ) = τ2 + + τ 1 τ1 t τ2 τ 1 Λ(u) du + a(t, τ 1, τ 2 )X(t) θ(u)σ(u)a(t, τ 1, τ 2 )e p du θ(u)σ(u)e 1A 1 (exp (A(τ 2 u)) I p ) e p du with I p being the p p identity matrix and a(t, τ 1, τ 2 ) = e 1A 1 (exp (A(τ 2 t)) exp (A(τ 1 t)))

36 Some generalities Risk neutral probabilities CDD futures CAT futures Time-dynamics of F CAT df CAT (t, τ 1, τ 2 ) = Σ CAT (t, τ 1, τ 2 ) db θ (t) where Σ CAT (t, τ 1, τ 2 ) = σ(t)e 1A 1 (exp (A(τ 2 t)) exp (A(τ 1 t))) e p Σ CAT is the CAT volatility term structure

37 Some generalities Risk neutral probabilities CDD futures CAT futures Seasonal volatility, with maturity effect Plot of the volatility term structure as a function of t up till delivery Monthly contracts Parameters taken from Stockholm for CAR(3) 9 CAT volatility term structure CAT volatility Jan Feb Mar Apr May June July Aug Sept Oct Nov Dec measurement month

38 Some generalities Risk neutral probabilities CDD futures CAT futures The Samuelson effect The volatility is decreasing with time to delivery Typical in mean-reverting markets AR(3) has memory Implies a modification of this effect Plot shows volatility of CAT with monthly vs. weekly measurement period CAT volatility Trading days prior to measurement period

39 Some generalities Risk neutral probabilities CDD futures CAT futures Call option prices easily calculated (strike K at exercise time τ) where { C(t) = e r(τ t) (F CAT (t, τ 1, τ 2 ) K) Φ(d) τ } + Σ 2 CAT (s, τ 1, τ 2 ) ds Φ (d) t d = F CAT(t, τ 1, τ 2 ) K τ t Σ2 CAT (s, τ 1, τ 2 ) ds Φ cumulative normal distribution

40 CAR(p) model for the temperature dynamics Auto-regressive process, with Seasonal mean seasonal volatility Allows for analytical futures prices for the traded contracts on CME HDD/CDD, CAT and PRIM futures Futures contracts with delivery over months or seasons Seasonal volatility with a modified Samuelson effect Options on HDD-futures require numerical pricing Black-formula for options on CAT-futures

41 Coordinates fredb

42 References Alaton, Djehiche, and Stillberger. On modelling and pricing weather derivatives. Appl. Math. Finance, 9, 2002 Benth and Saltyte-Benth. Stochastic modelling of temperature variations with a view towards weather derivatives. Appl. Math. Finance, 12, 2005 Benth, Saltyte-Benth and Jalinskas. A spatial-temporal model for temperature with seasonal variance. Preprint To appear in J. Appl. Statist. Benth and Saltyte-Benth. The volatility of temperature and pricing of weather derivatives. Preprint To appear in Quantit. Finance Benth, Saltyte Benth and Koekebakker. Putting a price on temperature. Preprint Brody, Syroka and Zervos. Dynamical pricing of weather derivatives. Quantit. Finance, 3, 2002 Campbell and Diebold. Weather forecasting for weather derivatives. J. Amer. Stat. Assoc., 100, 2005 Dornier and Querel. Caution to the wind. Energy Power Risk Manag., August, 2000