Risk-Neutral Modeling of Emission Allowance Prices

Transcription

1 Risk-Neutral Modeling of Emission Allowance Prices Juri Hinz /01/2009, Singapore

2 1 Emission trading 2 Risk-neutral modeling 3 Passage to continuous time

3 Greenhouse GLOBAL gas effect WARMING SIX MAIN GREENHOUSE GASES (GHGs) GREENHOUSE EFFECT HFCS CARBON DIOXIDE METHANE SULPHUR HEXAFLUORIDE PFCS NITROUS OXIDE

4 Reduction by cap-and-trade mechanism=emission trading scheme central authority allocates credits (allowances) to polluters sets penalty for each unit of pollutant not covered by credits defines compliance dates within a time period polluters reduce or avoid penalty by applying abatement measures technological changes replacement of input/output products, trading allowances physically (spot) financially (forwards/futurues) Example EU ETS Phase I and II credits are called EUA

5 Financial products on EUA EUA: (European Union Allowance) is the emission certificate which covers the emission of one tonne of carbon dioxide equivalent within EU ETS. Futures on EUA s are traded. ECX lists EUA futures with expiry date on the first Monday of March, June, September and December.

6 Financial products on EUA European style plain vanilla options written on EUA futures are listed. At the ECX: maturity: three business days before the expiry of the underlying strikes range from 1 to 55 EURO with an interval of 0.5. volume: about 450,000 tonnes.

7 Financial products on CER CERs: (Certified Emission Reductions) are emissions certificates issued for the successful completion of CDM climate protection projects. within EU ETS, installations are allowed to cover their emissions by CERs, (with upper bound on the total number of CERs valid for compliance) Futures on CER s are traded. ECX lists CER futures with expiry date on the first Monday of March, June, September and December.

8 Complexity No-Arbitrage: EUA2012 future is a martingale with resect to spot martingale measure, finishing at EUA spot price 12/2012: Zero + next-period EUA?, if emissions < EUAs. CER price, if EUAs + CERs < emissions penalty + next-period EUA? if EUAs + CERs emissions Depending on banking and withdrawal rule

9 Futures from ECX Settlement Price [EUR] EUA 2007 EUA 2009 EUA 2012 CER 2009 CER

10 Simplest situation one period only (no time inter-connection) one scheme only (no space inter-connection) interest rate zero (spot=future=forward) Model allowance price (A t ) t [0,T] as a digital martingale A t = πe Q (1 N F t ), t [0, T] Distinguish between reduced-form model hybrid model

11 Both types of risk neutral models describe allowance price (A t ) t [0,T] as a digital martingale A t = πe Q (1 N F t ), t [0, T] reduced-form model: Non-compliance event N is modeled exogenously. (in a flexible way, to match observed price properties, and option prices) hybrid model: stylized fundamental factors (emission, reduction intensities + market mechanisms) determine N.

12 In this talk: hybrid model Idea Analyze equilibrium of a stylized market. Derive a relation between allowance price, stochastic drivers, abatement activity Conclude implications for risk-neutral dynamics non-risk averse setting: optimal stochastic control risk-averse setting: fixed-point equalities for martingales References With co-workers: 1 Optimal stochastic control and carbon price formation. SIAM Journal on Control and Optimization, Market designs for emissions trading schemes. SIAM Review (to appear) 3 On fair pricing of emission-related derivatives Bernoulli (to appear) 4 Jump-diffusion modeling in emission markets Preprint

13 Dynamical model compliance date T action times t = 0,...,T all processes on (Ω,F, P,(F t ) T t=0 ) are adapted finite number of agents i I abstract from production, focus on abatement activity Revenue of agent i for strategy (ξ i,ϑ i ), given prices A = (A t ) T t=0 T 1 L A,i (ϑ i,ξ i ) = (ϑ i t A t + C i (ξt i )) ϑi T A T t=0 T 1 }{{}( penalty t=0 π (Et i ξi t ϑi t ) ϑi T γi ) +

14 Model ingredients T 1 L A,i (ϑ i,ξ i ) = (ϑ i t A t + Ct i (ξi t )) ϑi T A T t=0 T 1 }{{}( penalty t=0 π (Et i ξi t ϑi t ) ϑi T γi ) + Business-as-usual emissions (Et i)t 1 t=0 of the agents i I Abatement policy ξ i = (ξt i)t 1 t=0 of the agent i I Costs of abatement policy (ξt i)t 1 t=0 are T 1 t=0 Ci t (ξi t ) ϑ i t change of allowance number by trade at time t T 1 t=0 A tϑ i t costs of trading at allowance prices (A t) T t=0 γ i [0, [ endowment less unpredictable emission

15 Model ingredients Costs can be random (ω, x) C t (x)(ω) F t B-measurable due to stochastic fuel prices Abatement activity ξ i = (ξ i t )T 1 t=0 must be feasible ξ i U i := {(ϑ i,ξ i ) : 0 ξ i t E i t t = 0,...,T 1}. since abatement can not exceed emission.

16 Model ingredients Risk aversion of agent i I is described by agent-specific utility function U i Rational behavior Given prices A = (A t ) T t=0, each agent i I maximizes ) ( (ϑ i,ξ i ) E over all admissible policies U i. U i (L A,i (ϑ i,ξ i )) } {{ } =u i (L A,i (ϑ i,ξ i ))

17 Equilibrium state Definition A = (A t )T t=0 is an equilibrium allowance price process, if there exist agent s policies (ϑ i,ξ i ) U i, i I such that: (i) Each agent i I is satisfied by the own policy (ϑ i,ξ i ) is maximizer to (ϑ i,ξ i ) u i (L A,i (ϑ i,ξ i )) on U i, furthermore u i (L A,i (ϑ i,ξ i )) <. (ii) Changes in allowance positions are in zero net supply = 0, for all t = 0,...,T. i I ϑ i t

18 Three equilibrium properties (under additional assumptions) It turns out that in the equilibrium: a) No arbitrage opportunities for allowance trading b) Allowance price instantaneously triggers all abatement measures whose costs are below allowance price c) There are merely two final outcomes for allowance price A T = 0 in the case of allowance excess = π in the case of allowance shortage A T

19 Formal characterization (under slight assumptions) Theorem If (A t )T t=0 is an equilibrium price and (ξi corresponding abatement policies, then t ) T 1 t=0 for i I are (a) (A t )T t=0 is a martingale with respect to some Q P (b) For each i I holds ξt i = c i (A t ), t = 0,...,T 1, with abatement volume function c i t (a) = argmax(x Ci t (x)+ax) (c) The terminal allowance price is given by A T = π1 { i I ( T 1 t=0 (Ei t ξi t ) γ i ) 0}

20 From risk-neutral perspective, allowance price is a Q-martingale, whose terminal value A T = π1 {E T T 1 t=0 c t(a t ) 0} depends on the intermediate values through Cap-adjusted BAU emission E T = i I T 1 t=0 E i t i I γ i and market abatement volume function c t (a) := i I ct i (a), t = 0,..., T 1

21 Hybrid modeling given measure Q P, random variable E T, and abatement volume functions (c t ) T 1 t=0, determine a Q-martingale (A t )T t=0 with A T = π1 {E T T 1 t=0 c t(a ) 0}. t

22 Illustration for one time step from 0 to T = 1 π A 0 = πeq 0 (1 {E 1 c(a 0 ) 0} ) A 0 A 0 π

23 Solution to the problem of hybrid modeling To obtain (A t )T t=0 from Q, E T, and (c t ) T 1 t=0 follow the intuition that the allowance price is a function of A t (ω) = α t(g t (ω))(ω) recent time t current situation ω reduction demand G t = Et Q (E T ) t 1 s=0 }{{} c(a s) E t

24 Reduced-form approach Current allowance price is a function of time to maturity current situation saved pollutant A t (ω) = α t(g t (ω))(ω) G t (ω) = E t (ω) t 1 s=0 c s(a s)(ω) for t = T obviously α T (g) = π1 [0, [ (g), for all g R for t = T 1,...,0 hypothetically α t : R Ω [0,π], B(R) F t -measurable

25 Guess a recursion from martingale property Idea α t (g)(ω) = E Q t [α t+1(g c t (α t (g))+ε t+1 )](ω), for all g R, ω Ω where ε t+1 = E t+1 E t Indeed: α t (G t ) = A t(ω) = E Q t [A t+1 ] = EQ t [α t+1(g t+1 )] = E Q t (α t+1(g t c t (A t )+ε t+1)) = E Q t [α t+1(g t c t (α t (G t ))+ε t+1 )]

26 Recursion for (α t ) T t=0 Idea α t (g)(ω) = E Q t (α t+1(g c t (α t (g)(ω))+ε t+1 ))(ω), for all g R, ω Ω start with α T (g) = π1 [0, [ (g), for all g R proceed recursively for t = T 1,...,1, determining α t (g)(ω) as the unique solution to the fix point equation a = E Q t (α t+1(g c t (a)+ε t+1 ))(ω)

27 Formal result Theorem i) Given measure Q P there exist functionals α t : R Ω [0,π], B(R) F t -measurable, for t = 0,...T which fulfill for all g R α T (g) = π1 [0, [ (g), α t (g) = E Q t (α t+1(g c(α t (g))+ε t+1 )), t = 0,.., T 1 ii) There exists a Q martingale (A t )T t=0 which satisfies A T A t = π1 {Et T 1 t=0 c t(a t ) 0} recursively obtained by t 1 := α t (E t c(a s)), t = 0,.., T s=0

28 A numerical example: constant and deterministic c t = c Suppose that ε t+1 and F t are independent under Q for all t = 0,...,T 1. which makes calculations easier, since the randomness enters allowance price through the present up-to-day emissions only. More precisely one verifies (recursively!) that ω α t (g)(ω) = α t (g) is constant on Ω. Hence, allowance price A t+1 is just Borel function of the present up-to-day emission G t+1 and the condition F t can be replaced by the condition σ(g t ): α t (G t ) = E Q (α t+1 (G t c(α t (G t ))+ε t+1 ) σ(g t )).

29 A numerical example: least-square Monte-Carlo method Given the fixed point equation for Borel measurable function α t α t (G t ) = E Q (α t+1 (G t c(α t (G t ))+ε t+1 ) σ(g t )), try to obtain a solution as limit α t = lim n α n t of iterations α n+1 t (G t ) = E Q (α t+1 (G t c(α n t (G t ))+ε t+1 ) σ(g t )), n N started at α 0 t = α t+1. For numerical calculations, we suggest to use the least-square Monte-Carlo method. The idea here is to consider functions within a linear space spanned by basis functions and to replace the integration by a sum over finite sample.

30 A numerical example (ε t ) T t=1 are i. i. d. 1 Initialization: Given sample S = (e k, g k ) K k=1 R2 (of i.i.d realizations of (ε t+1, G t )) and a set of basis functions Ψ = (ψ i ) J j=1 on R, define M = ( ψ j (g k ) ) K,J k=1,j=1 Set α T (g) = 1 [0, ] (g) for all g R, and proceed in the next step with t := T 1. 2 Iteration: Define α 0 t = α t, and proceed in the next step with n := 0. 2a) Calculate φ n+1 (S) := (α t+1 (g k c(α n t (g k))+e k )) K k=1 2b) Determine a solution q n+1 R J to M Mq n+1 = M φ n+1 (S). 2c) Define α n+1 t 2d) If max K k=1 αn+1 := J j=1 qn+1 j ψ j. (g k ) α n t (g k) ε, then put n := n+1 and t continue with the step 2a). If max K k=1 αn+1 t (g k ) α n t (g k) < ε then set t := t 1. If t > 0, go to the step 2, otherwise finish.

31 e Illustration 1 to maturity 2 to maturity 3 to maturity 4 to maturity 5 to maturity 6 to maturity i rp c relative demand Parameters penalty π = 100, martingale increments (ε t ) T t=1 i.i.d, ε t = N(0.5, 1), K = 1000 basis functions (Ψ j ) J j=1 piecewise linear, J = 16 abatement volume function c : R R, a 0.1 (a) +

32 Pricing European Call 1 Given maturity time τ {1,...,T} of the European call, determine its payoff fτ τ := (α τ K) +. Calculate least-square projections recursively processing for u = τ,...,t 2 Calculate least-square projections recursively processing for u = τ,..., t a) put φ(s) = (fu τ (g k c u (α u (g k ))+e k )) K k=1 b) obtain q as solution to M Mq = Mφ c) set fu 1 τ = J j=1 q jψ j d) if u 1 = t finish, else set u := u 1 and go to a). 3 Given recent allowance price a, calculate the state variable g as solution to a = α t (g) 4 Plug in the state variable g and into function f τ (t, ) to obtain the price of the European call as as f τ (t, g).

33 Hybrid modeling = continuous time Allowance price is martingale A t = E Q t (A T ), t [0, T] Allowance price is digital at compliance date T A T = π1 N Allowance price triggers abatement T N = {E T c(a u )du} 0

34 Problem Given on a probability space (Ω,F, P,(F t ) t [0,T] ) an equivalent measure Q P, random variable E T, and a family of abatement functions (c t ) t [0,T], determine a Q-martingale (A t ) t [0,T] with A T = π1 {E T T 0 c t(a t )dt 0}. If increments of (E t ) t [0,T] are independent then solution can be claimed (?) as A t = α(t, G t ) t [0, T] with expected current allowance shortage G t = Et Q (E T ) }{{} E t t 0 c(a u )du

35 In the diffusion framework we have da t = dα(t, G t ) de t = σ t dw t (σ t ) t [0,T] is deterministic = (1,0) α(t, G t )dt (0,1) α(t, G t )c(α(t, G t ))dt (0,2)α(t, G t )σ 2 t dt }{{} =0 + (0,1) α(t, G t )σ t dw t which yields PDE on [0, T] R (1,0) α(t, g) (0,1) α(t, g)c(α(t, g)) (0,2)α(t, g)σ 2 t = 0 with boundary condition α(t, g) = π1 [0, [ (g) for all g R

36 Call on allowance price with strike price K and maturity τ [0, T] C τ (t) = E Q t ((A τ K) + ) = f τ (t, G t ) t [0,τ] is obtained by solution the same PDE, on [0,τ] R (1,0) f τ (t, g) (0,1) f τ (t, g)c(α(t, g)) (0,2)f τ (t, g)σ 2 t = 0 but with other boundary condition f τ (τ, g) = (α(τ, g) K) + for all g R

37 For linear abatement function a c a there is solution, which is almost explicit, obtained by numerical integrations. Here an example: for initial allowance price a = A 0 = 25 time compliance date T = 2 diffusion coefficient σ = 4 penalty π = 100 linear abatement function with proportionality c = 0.02.

38 The functions α(t, ) displayed for t = 1.9, 1.6, 1.3, 1.0, 0.7, 0.4. price to maturity 2 to maturity 3 to maturity 4 to maturity 5 to maturity 6 to maturity relative demand

39 Family of calls with same K = 25 but different maturity times τ [0, T]. With contract s maturity, price increases form from zero 0 = C 0 (0) (Because at-the-money situation) to = A 0 = (π K)/π (Call is equivalent to 0.75 allowances) option s maturity option s price

40 In the jump-diffusion settings The allowance price is modeled in the same way, A t = α(t, G t ), t [0, T] but now we assume that the martingale may have jumps. In view of modeled by E t := E Q [E T F t ], t [0, T] dg t = c(a t )d t + de t dg t = c(α(t, G t ))dt+ σ(t, G t )dw t + a(t, G t, y)(p ν q ν )(dy dt), R } 0 {{} de t

41 Modeling with jumps Expected allowance shortage (G t ) t [0,T] follows dg t = c(α(t, G t ))dt +σ(t, G t )dw t + a(t, G t, y)(p ν q ν )(dy dt), R 0 with Poisson random measure p ν and its compensator q ν. q ν (dt, dy) = λdtν(dy), λ is the jump intensity ν jump distribution on R 0 = R\{0} Jumps size state and time dependent through a(, ) volatility is state and time dependent through σ(, )

42 Integro-PDE instead of PDE Due to martingale property of (A t ) t [0,T] conditions of α are (1,0) α(t, g) = c(α(t, g)) (0,1) β(t, g)+ 1 2 σ2 (t, x) (0,2) α(t, g) +λ [α(τ, x + a(t, g, y)) β(τ, x) a(t, g, y) (0,1) α(t, g) ] ν(dy) for all (t, g) ]0, T[ R subject to α(t, g) = π1 [0, [, g R

43 Numerical solution Appropriate discretization of integro-pde is available. Boils down to numerical integral approximation and solutions of tri-diagonal linear equations Conditions for existence and uniqueness of discredited solutions are determined Numerics is implemented (speed needs to be increased!)

44 Example time T = 1, penalty π = 1, volatility σ(, ) = 1, jump intensity: λ = 5 jump size: N(0, 1)-distributed Functions α(t, ): for t = 0.8, 0.6, 0.4, 0.2, price to maturity 0.4 to maturity 0.6 to maturity 0.8 to maturity 1 to maturity

45 Example A typical realization of (G t ) t [0,T] :

46 Example The corresponding realization of (A t = α(t, G t )) t [0,T] : 1 Allowance price

47 Multi periods markets So far, we focused on one compliance period. This is a simplification since real-world markets are operating in a multi-period framework Usually, periods are connected by regulations. Three regulatory mechanisms Borrowing Banking Withdrawal

48 Three regulatory mechanisms Borrowing allows for the transfer of a (limited) number of allowances from the next period into the present one; Banking allows for the transfer of a (limited) number of (unused) allowances from the present period into the next; Withdrawal penalizes firms which fail to comply in two ways: by penalty payment for each unit of pollutant which is not covered by credits and by withdrawal of the missing allowances from their allocation for the next period. Transfer rule opens the national market to the international credits. For instance, within EU ETS, CER are accepted.

49 Two period model Assumptions: two periods [0, T] and [T, T ] two processes (A t ) t [0,T], (A t) t [0,T] for futures contracts with maturities at compliance dates T, T written on allowance prices from the first and the second period respectively. Both prices fulfill A t = α(t, G t ), t [0, T] A t = α (t, G t), t [T, T ] Inter-connection rules define a relation between α(t, ) and α (T, )

50 Two period model Banking + Penalty { π if g 0 α(t, g) = e r(t T) α (T, g + G 0 ) if g < 0

51 Two period model Banking + Penalty+ Withdrawal α(t, g) = { π + e r(t T) α (T, g + G 0 ) if g 0 e r(t T) α (T, g + G 0 ) if g < 0

52 Two period model Banking + Penalty+ Withdrawal+ Transfer Transfer of Q allowances at price C α(t, g Q) if g x + Q α(t, g) = C if x < g < x + Q α(t, g) if g < x where x = inf{g R : α(t, g) C}

53 What now, how to price carbon options? We need to know the rules on multi-period connection. For instance, suppose certain rules are applied forever iterate the procedure α (T, ) α (T, ) α(t, ) =: α (T, ) infinitely many times obtain so the boundary conation α(t, ) calculate option prices as explained Option price depends on: Inter-connection rules, reduction target G 0, long-term interest rates.

54 Conclusion within EU ETS, beyond physical allowances, a large volume of allowance futures is traded. trading volume of European options written on these futures is increasing, although there is no clear pricing principle we present merely basic cornerstones of EUA option pricing more work is required to reflect the reality: multi-period scheme operation connection to international emission markets