Information Premium and Weather Forecasts

Transcription

1 Informaion Premium and Weaher Forecass Energy Finance Workshop of HUB and Uni DuE Solberg, Harz, Richard Biegler-König Chair for Energy Trading and Finance Universiy of Duisburg-Essen

2 Seie 2/35 Table of conens The Informaion Approach Moivaion The Informaion Premium The Informaion Premium wih Weaher Forecass The Spo Model A simple Temperaure Model Enlargemen of Filraions The Info Premium wih Weaher Forecas Sylised Examples and Muliple Informaion Sylised Examples Muliple Fuure Informaion

3 Seie 3/35 The Informaion Approach The Informaion Approach

4 Seie 4/35 The Informaion Approach Moivaion Spo-Forward Relaionship: Classical heory Wha is he relaionship beween spo and forward prices on elecriciy markes? The classical spo-forward relaionship is where F (, T ) = E Q [S(T ) F ] F is informaion generaed by he spo price up unil ime (i.e. he hisorical filraion) Q is a risk-neural probabiliy The underlying elecriciy is non-sorable Buy-and-hold argumen does no work

5 Seie 4/35 The Informaion Approach Moivaion Spo-Forward Relaionship: Classical heory Wha is he relaionship beween spo and forward prices on elecriciy markes? The classical spo-forward relaionship is where F (, T ) = E Q [S(T ) F ] F is informaion generaed by he spo price up unil ime (i.e. he hisorical filraion) Q is a risk-neural probabiliy The underlying elecriciy is non-sorable Buy-and-hold argumen does no work

6 Seie 5/35 The Informaion Approach Moivaion Sylised and Marke Examples Fuure predicions abou he marke will no affec he curren spo price... bu will affec forward prices Sylized examples: planned ouage of a power plan in one monh weaher forecass Inroducion of CO 2 cerificaes in 2008 wih anicipaed seep increase due o exra coss German Aom Moraorium following he Tōhoku earhquake March 2011 immediae shu-down of 7 nuclear power plans for 3 monhs possible complee shu-down!

7 Seie 5/35 The Informaion Approach Moivaion Sylised and Marke Examples Fuure predicions abou he marke will no affec he curren spo price... bu will affec forward prices Sylized examples: planned ouage of a power plan in one monh weaher forecass Inroducion of CO 2 cerificaes in 2008 wih anicipaed seep increase due o exra coss German Aom Moraorium following he Tōhoku earhquake March 2011 immediae shu-down of 7 nuclear power plans for 3 monhs possible complee shu-down!

8 Seie 6/35 The Informaion Approach Moivaion Example: 2008 CO 2 Emission Coss Inroducion of CO2 fees Spo on Ok. 06 Nov. 06 Dez. 06 Jan. 07 Feb. 07 Mrz. 07 Apr Ok. 07 Nov. 07 Dez. 07 Jan. 08 Feb. 08 Mrz. 08 Apr. 08 Figure: EEX Forward prices observed on 01/10/06 (lef) and 01/10/07 (righ) Typical winer and bank holidays behaviour in boh graphs General upward shif in 2008

9 Seie 7/35 The Informaion Approach Moivaion Example: 2011 German Moraorium - Spo 65,00 EEX Spo (7 days MA) Earhquake ( ) "Moraorium" ( ) Decision o shu down he "7" ( ) End of "Moraorium" ( ) 60,00 55,00 50,00 45,00 40,00 35,00 30, Figure: EEX spo prices 2010/2011 No significan impac on spo prices

10 Seie 8/35 The Informaion Approach Moivaion Example: 2011 German Moraorium - Forward 65 May 2011 Forward Earhquake ( ) "Moraorium" ( ) Decision o shu down he "7" ( ) Figure: EEX May 2011 forward price Permanen increase from 14/03/2011

11 Seie 9/35 The Informaion Approach The Informaion Premium Informaion Approach For elecriciy he hisorical filraion is no sufficien F = σ(s s, s ) Idea: enlarge he filraion!... by addiional fuure informaion a ime T Υ Noe: This echnique has been used o modell insider rading on sock exchanges

12 Seie 9/35 The Informaion Approach The Informaion Premium Informaion Approach For elecriciy he hisorical filraion is no sufficien F = σ(s s, s ) Idea: enlarge he filraion!... by addiional fuure informaion a ime T Υ Noe: This echnique has been used o modell insider rading on sock exchanges

13 Seie 9/35 The Informaion Approach The Informaion Premium Informaion Approach For elecriciy he hisorical filraion is no sufficien F = σ(s s, s ) Idea: enlarge he filraion!... by addiional fuure informaion a ime T Υ Noe: This echnique has been used o modell insider rading on sock exchanges

14 Seie 10/35 The Informaion Approach The Informaion Premium Filraions and Spo-Forward-Relaionship Filraions F - he hisorical filraion H - complee informaion, i.e. H = F σ(s(t Υ )) G - he filraion of all informaion publicly available o he marke Hence, we have he relaion F G H New Spo-Forward-Relaionship: F (, T ) = E Q [S(T ) G ]

15 Seie 10/35 The Informaion Approach The Informaion Premium Filraions and Spo-Forward-Relaionship Filraions F - he hisorical filraion H - complee informaion, i.e. H = F σ(s(t Υ )) G - he filraion of all informaion publicly available o he marke Hence, we have he relaion F G H New Spo-Forward-Relaionship: F (, T ) = E Q [S(T ) G ]

16 Seie 11/35 The Informaion Approach The Informaion Premium The Informaion Premium - Definiion Quanify he influence of fuure informaion using: Informaion Premium The informaion premium is defined o be I(, T ) = E[S T G ] E[S T F ] = F G (, T ) F F (, T ) i.e. he difference beween he prices of he forward under G and F. Noe: no delivery period considered (ye)

17 Seie 12/35 The Informaion Approach The Informaion Premium The Informaion Premium - Propery Lemma The informaion premium is orhogonal o L 2 (F, P). Proof: E[I G (, T ) F ] = E[ E[S(T ) G ] E[S(T ) F ] F ] = 0 In paricular, he resul is valid for all measures equivalen o P Usual mehod o aain he marke price of risk is a measure change This is no possible for he Informaion Premium

18 Seie 12/35 The Informaion Approach The Informaion Premium The Informaion Premium - Propery Lemma The informaion premium is orhogonal o L 2 (F, P). Proof: E[I G (, T ) F ] = E[ E[S(T ) G ] E[S(T ) F ] F ] = 0 In paricular, he resul is valid for all measures equivalen o P Usual mehod o aain he marke price of risk is a measure change This is no possible for he Informaion Premium

19 Seie 13/35 The Informaion Approach The Informaion Premium Empirical Tes for he Informaion Premium Showing he exisence of he info premium is problemaic due o orhogonaliy We developed a es based on Hilber space represenaion... and isolaed a non-measurable premium for he inroducion of cerificaes as well as he Moraorium... ha makes sense in shape and size This is discussed in deail in our paper (o appear)

20 Seie 14/35 The Informaion Premium wih Weaher Forecass The Informaion Premium wih Weaher Forecass

21 Seie 15/35 The Informaion Premium wih Weaher Forecass The Spo Model Spo Price Model We will model he spo using a simple, well known wo-facor model: S() = Λ() + X() + Y () where for a BM W () and Lévy process L() T X(T ) = e α(t ) X() + σ Y (T ) = e β(t ) Y () + T e α(t s) dw (s) e β(t s) dl(s) i.e. a seasonal funcion Λ and wo OU processes Base componen X, spike componen Y

22 Seie 16/35 The Informaion Premium wih Weaher Forecass The Spo Model Forward Price wih Delivery under P and F The forward price wih delivery in [T 1, T 2 ] is hen given by [ ] 1 T2 F (, T 1, T 2 ) = E S(u)du F (App.I) T 2 T 1 and furher calculaions yield T 1 1 F(, T 1, T 2 ) = T 2 T 1 ( T2 Λ(u)du + ᾱ(, T 1, T 2 )X() T 1 + β(, T 1, T 2 )Y () + φ (0) ˆβ(, ) T 1, T 2 ) φ(u) is he log-momen generaing funcion of L ᾱ, β, ˆβ are deerminisic funcions (see App. II)

23 Seie 17/35 The Informaion Premium wih Weaher Forecass A simple Temperaure Model Temperaure Model Given weaher forecass in T Υ > T 2, wha is he informaion premium? We will modell emperaure as a Gaussian OU process where W Z T Z T = e α Z (T ) Z + is a F -BM T + σ Z e α Z (T s) dws Z α Z µ Z (s)e α Z (T s) ds Model fis reasonably well (especially wih seasonal α Z, see various Benh papers)

24 Seie 18/35 The Informaion Premium wih Weaher Forecass A simple Temperaure Model Temperaure and Elecriciy We will assume dw dw Z = ρd Or, in oher words where W S dw = ρdw Z is he spo-specific BM + 1 ρ 2 dw S The spo s base componen is correlaed wih emperaure F is now generaed by W S, W Z and L

25 Seie 19/35 The Informaion Premium wih Weaher Forecass Enlargemen of Filraions Enlargemen of Filraion We will model he addiional informaion using he Theory of Enlargemen of Filraion mainly developed in he 1980s by French mahemaicians (Jeulin, Yor, Jacod) bu also by Proer, Imkeller Iniiaing heorem by Iō (1976) (see App. IV) Iō s heorem provides he decomposiion of he corresponding Lévy process under enlargemen of filraion by is own fuure value Sill, we observe prices/emperaures, no heir random shocks We wan o enlarge by emperaure, i.e. a funcional of a BM

26 Seie 19/35 The Informaion Premium wih Weaher Forecass Enlargemen of Filraions Enlargemen of Filraion We will model he addiional informaion using he Theory of Enlargemen of Filraion mainly developed in he 1980s by French mahemaicians (Jeulin, Yor, Jacod) bu also by Proer, Imkeller Iniiaing heorem by Iō (1976) (see App. IV) Iō s heorem provides he decomposiion of he corresponding Lévy process under enlargemen of filraion by is own fuure value Sill, we observe prices/emperaures, no heir random shocks We wan o enlarge by emperaure, i.e. a funcional of a BM

27 Seie 20/35 The Informaion Premium wih Weaher Forecass Enlargemen of Filraions Yor s Crierion Theorem Le F be he Brownian filraion and G a random variable and define H = F σ(g). Le P G (dl) be he law of G and P G (dl) he condiional law. If here exiss p such ha P G (dl) = p (l)p G (dl) hen he H decomposiion of BM W is where ξ is a H BM. d < p. (G), W. > s W = ξ + 0 p s (G) Noe: laws needed in conrary o Iō s heorem!

28 Seie 21/35 The Informaion Premium wih Weaher Forecass Enlargemen of Filraions Imkeller s Mehod To calculae he quadraic covariaion of p and W we normally need a edious applicaion of Iō s lemma Imkeller proposes o use he Malliavin derivaive: Theorem Wih he same seup as before he H decomposiion of BM W is D s Ps G (G) W = ξ + 0 Ps G (G) ds where D denoes he Malliavin derivaive.

29 Seie 22/35 The Informaion Premium wih Weaher Forecass Enlargemen of Filraions P G and P G We have H = F σ(z TΥ ) = F σ( T Υ e α Z (T Υ s) dw Z s ) We define G = m = s 2 = TΥ e α Z (T Υ s) dw Z s e α Z (T Υ s) dw Z s e 2α Z (T Υ s) ds = 1 2α Z (e 2α Z (T Υ ) e 2α Z T Υ ) so ha P G 1 (dl) = exp( 2πs 1 )dl 2TΥ 2 st 2 Υ P G (dl) = 1 exp( 2π(s 1 (l m ) 2 )dl 2TΥ s 2 ) 2 st 2 Υ s 2 l 2

30 Seie 22/35 The Informaion Premium wih Weaher Forecass Enlargemen of Filraions P G and P G We have H = F σ(z TΥ ) = F σ( T Υ e α Z (T Υ s) dw Z s ) We define G = m = s 2 = TΥ e α Z (T Υ s) dw Z s e α Z (T Υ s) dw Z s e 2α Z (T Υ s) ds = 1 2α Z (e 2α Z (T Υ ) e 2α Z T Υ ) so ha P G 1 (dl) = exp( 2πs 1 )dl 2TΥ 2 st 2 Υ P G (dl) = 1 exp( 2π(s 1 (l m ) 2 )dl 2TΥ s 2 ) 2 st 2 Υ s 2 l 2

31 Seie 22/35 The Informaion Premium wih Weaher Forecass Enlargemen of Filraions P G and P G We have H = F σ(z TΥ ) = F σ( T Υ e α Z (T Υ s) dw Z s ) We define G = m = s 2 = TΥ e α Z (T Υ s) dw Z s e α Z (T Υ s) dw Z s e 2α Z (T Υ s) ds = 1 2α Z (e 2α Z (T Υ ) e 2α Z T Υ ) so ha P G 1 (dl) = exp( 2πs 1 )dl 2TΥ 2 st 2 Υ P G (dl) = 1 exp( 2π(s 1 (l m ) 2 )dl 2TΥ s 2 ) 2 st 2 Υ s 2 l 2

32 Seie 23/35 The Informaion Premium wih Weaher Forecass Enlargemen of Filraions The Informaion Drif We can easily calculae he Malliavin derivaive D P G (l) = P G (l)d ( 1 (l m ) 2 ) 2 st 2 Υ s 2 = P G (l) l m D ( e α Z (TΥ s) dw st 2 Υ s 2 s Z ) 0 = P G (l) l m e α Z (T Υ ) st 2 Υ s 2... by he Malliavin chain rule and properies of Wiener polynomials Thus, using Imkeller s mehod he decomposiion is ξ = W Z = W Z G m e α st 2 s 2 Z (T Υ ) ds Υ 0 0 2α Z e α Z s e 2α Z T Υ e 2α Z s }{{} =a(s) TΥ s e α Z u dw Z u ds

33 Seie 23/35 The Informaion Premium wih Weaher Forecass Enlargemen of Filraions The Informaion Drif We can easily calculae he Malliavin derivaive D P G (l) = P G (l)d ( 1 (l m ) 2 ) 2 st 2 Υ s 2 = P G (l) l m D ( e α Z (TΥ s) dw st 2 Υ s 2 s Z ) 0 = P G (l) l m e α Z (T Υ ) st 2 Υ s 2... by he Malliavin chain rule and properies of Wiener polynomials Thus, using Imkeller s mehod he decomposiion is ξ = W Z = W Z G m e α st 2 s 2 Z (T Υ ) ds Υ 0 0 2α Z e α Z s e 2α Z T Υ e 2α Z s }{{} =a(s) TΥ s e α Z u dw Z u ds

34 Seie 24/35 The Informaion Premium wih Weaher Forecass The Info Premium wih Weaher Forecas Calculaing he Informaion Premium 1/3 The info premium is I G(, T 1, T 2 ; T Υ ) = F G(, T 1, T 2 ; T Υ ) F F(, T 1, T 2 ) Λ(), X, Y erms cancel Lévy erms cancels as well (boh filraions coincide) (16) [ T2 ( u I G(, T 1, T 2 ; T Υ ) = 1 E σρ e α(u s) dw T 2 T 1 s Z T 1 +σ u ) ] 1 ρ 2 e α(u s) dws S du G [ T2 = 1 E σρ T 2 T 1 T 1 u... because expecaion of Iō-inegral is zero e α(u s) dw Z s du G ]

35 Seie 24/35 The Informaion Premium wih Weaher Forecass The Info Premium wih Weaher Forecas Calculaing he Informaion Premium 1/3 The info premium is I G(, T 1, T 2 ; T Υ ) = F G(, T 1, T 2 ; T Υ ) F F(, T 1, T 2 ) Λ(), X, Y erms cancel Lévy erms cancels as well (boh filraions coincide) (16) [ T2 ( u I G(, T 1, T 2 ; T Υ ) = 1 E σρ e α(u s) dw T 2 T 1 s Z T 1 +σ u ) ] 1 ρ 2 e α(u s) dws S du G [ T2 = 1 E σρ T 2 T 1 T 1 u... because expecaion of Iō-inegral is zero e α(u s) dw Z s du G ]

36 Seie 25/35 The Informaion Premium wih Weaher Forecass The Info Premium wih Weaher Forecas Calculaing he Informaion Premium 2/3 We plug in he informaion drif [ T2 E T 1 σρ = E = u e α(u s) dw Z s du G ] [ T2 T2 T 1 T 1 σρ σρ u u ( e α(u s) d ξ s + e α(u s) a(s)e [ TΥ s s 0 TΥ ] a(v) e α Z w dww Z dv )du G v e α Z v dw Z v G ] ds du... an auxiliary resul allows moving he expecaion ou of he double-inegral (i.e. replacing s wih, see App. V) We can hen wrie [ TΥ ] TΥ E e α Z v dwv Z G = E[Z TΥ G ] e α Z (TΥ ) Z α Z µ Z (s)e α Z (TΥ s) ds

37 Seie 25/35 The Informaion Premium wih Weaher Forecass The Info Premium wih Weaher Forecas Calculaing he Informaion Premium 2/3 We plug in he informaion drif [ T2 E T 1 σρ = E = u e α(u s) dw Z s du G ] [ T2 T2 T 1 T 1 σρ σρ u u ( e α(u s) d ξ s + e α(u s) a(s)e [ TΥ s s 0 TΥ ] a(v) e α Z w dww Z dv )du G v e α Z v dw Z v G ] ds du... an auxiliary resul allows moving he expecaion ou of he double-inegral (i.e. replacing s wih, see App. V) We can hen wrie [ TΥ ] TΥ E e α Z v dwv Z G = E[Z TΥ G ] e α Z (TΥ ) Z α Z µ Z (s)e α Z (TΥ s) ds

38 Seie 26/35 The Informaion Premium wih Weaher Forecass The Info Premium wih Weaher Forecas Calculaing he Informaion Premium 3/3 Afer a lenghy calculaion we arrive a ( TΥ ) I G( ) = ρ A( ) T 2 T 1 E[Z TΥ G ] e α Z (TΥ ) Z α Z µ Z (s)e α Z (TΥ s) ds where A( ) is a deerminisic funcion A( ) = 2α Z σe α Z (T Υ+) ( 1 α Z (e α Z (T 2 ) e α Z (T 1 ) ) + 1 α (e α(t 2 ) e α(t 1 ) )) σ Z (α + α Z )(e 2α Z T Υ e 2α Z ) Hence, knowing he curren emperaure and our forecas we can easily calculae he info premium Sign of premium depends on ρ and emperaure erms ρ < 0 Germany, ρ > 0 California, ρ = 0 Qaar

39 Seie 26/35 The Informaion Premium wih Weaher Forecass The Info Premium wih Weaher Forecas Calculaing he Informaion Premium 3/3 Afer a lenghy calculaion we arrive a ( TΥ ) I G( ) = ρ A( ) T 2 T 1 E[Z TΥ G ] e α Z (TΥ ) Z α Z µ Z (s)e α Z (TΥ s) ds where A( ) is a deerminisic funcion A( ) = 2α Z σe α Z (T Υ+) ( 1 α Z (e α Z (T 2 ) e α Z (T 1 ) ) + 1 α (e α(t 2 ) e α(t 1 ) )) σ Z (α + α Z )(e 2α Z T Υ e 2α Z ) Hence, knowing he curren emperaure and our forecas we can easily calculae he info premium Sign of premium depends on ρ and emperaure erms ρ < 0 Germany, ρ > 0 California, ρ = 0 Qaar

40 Seie 27/35 Sylised Examples and Muliple Informaion Sylised Examples and Muliple Informaion

41 Seie 28/35 Sylised Examples and Muliple Informaion Sylised Examples Examples - Seup We find relaed formulae for T Υ T 1 and T 1 < T Υ < T 2 (see App. VI) We wan o check our formula and develop some inuiion Firs, wha is he influence of ime and magniude of weaher forecas? Le s assume we have X = 0 as well as Z = µ Z = 0 Tha means we have zero degrees oday and expec zero degrees Furhermore, le ρ = 0.5, σ = σ Z = and = 0, T 1 = 10 and T 2 = 20 We will le α Z = α and check wo possible values

42 Seie 29/35 Sylised Examples and Muliple Informaion Sylised Examples Info Premium wih α Z = α = Info Premium (Euro) Time T Y Weaher forecas (degrees) Figure: Info Premium for differen values of T Υ and E[Z TΥ G ]

43 Seie 30/35 Sylised Examples and Muliple Informaion Sylised Examples Info Premium wih α Z = α = ,8 0,6 Info Premium (Euro) 0,4 0,2 0 0,2 0,4 04 0,6 0, Time T Y Weaher forecas (degrees) Figure: Info Premium for differen values of T Υ and E[Z TΥ G ]

44 Seie 31/35 Sylised Examples and Muliple Informaion Sylised Examples Realisic Parameers Wha happens if we ry more realisic values? Le s assume again X = 0 as well as Z = µ Z = 0 Again, ρ = 0.5 and = 0, T 1 = 10 and T 2 = 20 Now α = 0.5, σ = and α Z = 0.1 and σ Z = 2.0 Our forecas on he oher hand is E[Z TΥ G ] = 10

45 Seie 32/35 Sylised Examples and Muliple Informaion Sylised Examples Info Premium wih α Z α, σ Z σ 4 Info Premium Delivery Period Value of T Y Figure: Info Premium for differen values of T Υ, E[Z TΥ G ] = 10

46 Seie 33/35 Sylised Examples and Muliple Informaion Muliple Fuure Informaion Muliple Fuure Informaion Normally, we would have a whole series of forecass Especially wih long-delivery conracs his is imporan Playing around wih filraions and spliing inegrals we can incorporae muliple fuure informaion a imes T Υ1, T Υ2,... For his, we have o assume precise (i.e. H = G ) informaion Becomes messy even wih wo pieces

47 Seie 34/35 References and Conac References R. B-K, F. E. Benh, R. Kiesel An empirical sudy of he informaion premium on elecriciy markes, o appear R. B-K, F. E. Benh, R. Kiesel Elecriciy opions and addiional informaion, o appear F. E. Benh and Th. Meyer-Brandis The informaion premium for non-sorable commodiies, Journal of Energy Markes, 2009

48 Seie 35/35 References and Conac Conac phone: +49 (0) Chair for Energy Trading and Finance Universiy Duisburg-Essen web: Thank you for your aenion...

49 Seie 35/35 Appendix I Appendix I Risk-neural valuaion formula yields: [ ] T2 0 = e r E Q e r(u ) (S(u) F(, T 1, T 2 ))du F T 1 If selemens only ake place a he final dae T 2 one ges [ ] T2 0 = e r E Q (S(u) F(, T 1, T 2 ))du F T 1 and finally for he fuures price: [ ] T2 F(, T 1, T 2 ) = E Q 1 S(u)du F T 1 T 2 T 1 17

50 Seie 35/35 Appendix II Appendix II Deerminisic Funcions: { 1 ᾱ(, T 1, T 2 ) = α (e α(t2 ) e α(t1 ) ) T 1 1 α (e α(t2 ) 1) > T 1 { 1 β β(, T 1, T 2 ) = (e β(t2 ) e β(t1 ) ) T 1 1 β (e β(t2 ) 1) > T 1 17 ˆβ(, T 1, T 2 ) = { 1 β (T 2 T β (e β(t 2 ) e β(t 1 ) )) T 1 1 β (T β (e β(t 2 ) 1)) > T 1

51 Seie 35/35 Appendix III Appendix III Remember he Lévy par of he spo For he empirics, we used dy () = βy ()d + dl() N L = where N is Poisson process, inensiy λ, D i are i.i.d jump sizes We used double-exponenially disribued (i.e. he Kou model) wih densiy i=1 D i f D (x) = pη 1 e η 1x 1 x 0 + qη 2 e η 2 x 1 x 0 where p + q = 1 and η 1, η 2 0.

52 Seie 35/35 Appendix IV Appendix IV Theorem Iō s heorem for Lévy processes and addiional incomplee informaion. Le L be a Lévy process and G H = F σ(l TΥ ). Then 1. L is sill a semimaringale wih respec o G 2. if E[ L ] < hen is a G -maringale. TΥ E[L TΥ L s G s ] ξ() = L ds 0 T Υ s 20

53 Seie 35/35 Appendix V Appendix V Theorem Le L() be a Lévy process, F he hisorical filraion. Le G H = F σ(l(t Υ )). If he informaion drif is of he form [ ] TΥ µ G s = g(s)e f (u)dl(u) G s where g and f are con. funcion on [0, T Υ ]. Then [ ] [ TΥ ] TΥ E f (u)dl(u) G = E f (u)dl(u) G s for ime poins s T Υ. 27 s e s f (u)g(u)du

54 Seie 35/35 Appendix VI Appendix VI If T Υ T 1 hen I G(, T 1, T 2 ; T Υ ) = 1 T 2 T 1 ᾱ(t Υ, T 1, T 2 )I G(, T Υ ; T Υ ) where I G (, T Υ ; T Υ ) is he premium wih a delivery ime raher han period If T 1 < T Υ < T 2 hen I G( ) = 1 T 2 T 1 ((T Υ T 1 )I G(, T 1, T Υ ; T Υ ) + (T 2 T Υ )I G(, T Υ, T 2 ; T Υ )) 32