modeling carbon market using forward-backward sdesfrank011/mpenl/econclim/chassagneux.pdf ·...

IntroductionOne-period Model

Multi-period model

Modeling Carbon Market using Forward-BackwardSDEs

J-F Chassagneux (Universite Paris Diderot)joint works with H. Chotai (Citibank), D. Crisan (Imperial College)

and M. Muuls (Imperial College)

Workshop: Mathematics of the Economy and Climate,Soesterberg, the Netherlands,

15-17 July 2019

J-F Chassagneux Modeling Carbon Market using Forward-Backward SDEs

Multi-period model

IntroductionEmission Trading SchemeContributions

One-period ModelFrameworkFBSDE in a nutshellMain results

Multi-period modelFinite number of periodsInfinite number of periodsNumerics

Multi-period model

Emission Trading SchemeContributions

Outline

Multi-period model

Carbon markets

I Caused mainly by GHG emission, global warming is a challenge for society

I The main part comes from carbon dioxide (CO2) emission.

I This negative impact of emission has no ”priced” cost.

I Emission reduction could be achieved by a right carbon price

I To do so, one can implement of cap-and-trade schemes.

I China, whose carbon emissions make up approximately one quarter of the globaltotal, is considering introducing a national emissions trading scheme (withvarious pilot schemes already running)

I Since 2005, the EU has had its own emissions trading system (ETS).↪→ This is our main example in this talk.

Multi-period model

EU ETS

I The EU ETS is an example of cap and trade scheme:

- A central authority set a limit on pollutant emission during a given period.Allowances are allocated to participating installations.

- At the end of the period, each participating installation has to surrender anallowance for each unit of emission. Emissions above the cap leads to apenalty to be paid.

- During the period, participants can trade the allowances.

I The EU ETS has started in 2005. It governs more than 11000 power andmanufacturing plants and 31 countries, it accounts for 45% of European GHGemission.

I It has been divided in 4 phases (Phase I: 2005-2007, II: 2008-2012, III:2013-2020, IV: 2021 - 2028) with various changes in mechanism. Carbon priceobserved on the market has fluctuated a lot.

Multi-period model

EU ETS

Multi-period model

EU ETS

Multi-period model

EUA price (P. MacDonald 2016 sandbag .org .uk)

Multi-period model

Related literature & some motivations

Some literature

I Equilibrium approach (discrete time model): Carmona, Fehr, Hinz & Porchet(2010)

I risk neutral pricing, reduced form approach: Carmona & hinz (2011)

I risk neutral pricing, structural approach: Howison & Schwarz (2012)

I FBSDEs related papers: Carmona, Delarue, Espinosa & Touzi (2013), Carmona& Delarue (2013)

Some motivations

1. Modelisation of allowance price, allows for pricing of emission derivatives (cleanspread options: has been done in risk neutral but reduced-form model)

2. Analyse the impact of multiple period setting

3. Quantify the dependency on various model parameter: cap, demand etc.

4. Answer question like: what level of allowance price would reduce the emissionwith a given probability?

Multi-period model

Related literature & some motivations

Some literature

I Equilibrium approach (discrete time model): Carmona, Fehr, Hinz & Porchet(2010)

I risk neutral pricing, reduced form approach: Carmona & hinz (2011)

I risk neutral pricing, structural approach: Howison & Schwarz (2012)

I FBSDEs related papers: Carmona, Delarue, Espinosa & Touzi (2013), Carmona& Delarue (2013)

Some motivations

1. Modelisation of allowance price, allows for pricing of emission derivatives (cleanspread options: has been done in risk neutral but reduced-form model)

2. Analyse the impact of multiple period setting

3. Quantify the dependency on various model parameter: cap, demand etc.

4. Answer question like: what level of allowance price would reduce the emissionwith a given probability?

Multi-period model

Approach and Results

I We work at an aggregate level and follows risk neutral pricing

I We take into account feedback of the allowance price in the global emissionprocess↪→ this leads to an intricate forward-backward coupling

I We rely on the theory of singular Forward-Backward Stochastic DifferentialEquation (Carmona & Delarue 2013)

I We obtain well-posedness of a market price in the setting of

1. a market with a given number of period linked through realistic mechanisms

2. a market with no end date and an infinite number of period

I To do so, we extend the previous results on singular FBSDE and we study anovel PDE problem for the case of an infinite number of period.

I We are working towards efficient probabilistic numerical methods to compute theallowance price.

Multi-period model

FrameworkFBSDE in a nutshellMain results

Outline

Multi-period model

Main features

Three main processes for one period [0,T ].

1. The spot allowance price Y : we assume that the market is frictionless andarbitrage-free and that there is a probability such that (e−rtYt)0≤t≤T is amartingale, namely

dYt = rYtdt + ZtdWt

r is the interest rate, Z is a square integrable process.

2. Emission process E : cumulative process with impact from the allowance price

dEt = µ(Pt ,Yt)dt

↪→ µ should be decreasing in Y

3. Auxiliary process P:

dPt = b(Pt)dt + σ(Pt)dWt

Represent state variables that trigger the emission process ( Electricity price ordemand & fuel prices etc.) Fundamentals that are linked the good emitting CO2.

Multi-period model

Main features

dYt = rYtdt + ZtdWt

dEt = µ(Pt ,Yt)dt

Multi-period model

Main features

dYt = rYtdt + ZtdWt

dEt = µ(Pt ,Yt)dt

Multi-period model

Associated FBSDE

I System of Equations: 0 ≤ t ≤ T

dPt = b(Pt)dt + σ(Pt) dWt ,

dEt = µ(Pt ,Yt) dt,

dYt = rYt dt + Zt dWt ,

E0, P0 is known but Y0 is unknown!

I The terminal condition for the Allowance price Y . There is a cap Λ on the total

emission set by the regulator

1. If non-compliance i.e. ET > Λ then the penalty ρ is paid so YT = ρ

2. If compliance i.e. ET < Λ then the Allowance is worth nothing (Emission

regulation stops at the end of the period) so YT = 0

↪→ YT = φ(ET ) := ρ1{ET>Λ} and Yt = e−r(T−t)E[YT |Ft ]

Multi-period model

Associated FBSDE

I System of Equations: 0 ≤ t ≤ T

E0, P0 is known but Y0 is unknown!

I The terminal condition for the Allowance price Y . There is a cap Λ on the total

emission set by the regulator

1. If non-compliance i.e. ET > Λ then the penalty ρ is paid so YT = ρ

2. If compliance i.e. ET < Λ then the Allowance is worth nothing (Emission

regulation stops at the end of the period) so YT = 0

↪→ YT = φ(ET ) := ρ1{ET>Λ} and Yt = e−r(T−t)E[YT |Ft ]

Multi-period model

Definition

I In a Markovian setting:

Xt = X0 +

b(Xs)ds +

σ(Xs)dWs

Yt = g(XT ) +

f (Ys)ds −∫ T

I Linked with semi-linear PDE: Yt = u(t,XT ) where u satisfies

∂tu + b(x)∂xu +1

2σ2(x)∂xxu + f (u) = 0 and u(T , ·) = g(·)

I The difficult fully coupled case:

Xt = X0 +

b(Xs ,Ys)ds +

σ(Xs)dWs

Yt = g(XT ) +

f (Ys)ds −∫ T

Still Yt = u(t,Xt) decoupling field (price function here)

Multi-period model

Definition

Xt = X0 +

b(Xs)ds +

σ(Xs)dWs

Yt = g(XT ) +

f (Ys)ds −∫ T

∂tu + b(x)∂xu +1

2σ2(x)∂xxu + f (u) = 0 and u(T , ·) = g(·)

Xt = X0 +

b(Xs ,Ys)ds +

σ(Xs)dWs

Yt = g(XT ) +

f (Ys)ds −∫ T

Multi-period model

Definition

Xt = X0 +

b(Xs)ds +

σ(Xs)dWs

Yt = g(XT ) +

f (Ys)ds −∫ T

∂tu + b(x)∂xu +1

2σ2(x)∂xxu + f (u) = 0 and u(T , ·) = g(·)

Xt = X0 +

b(Xs ,Ys)ds +

σ(Xs)dWs

Yt = g(XT ) +

f (Ys)ds −∫ T

Multi-period model

Results for Fully Coupled case

I Fully coupled case:

Xt = X0 +

b(Xs ,Ys)ds +

σ(Xs)dWs

Yt = g(XT ) +

f (Ys)ds −∫ T

and associated quasi-linear PDE:

∂tu + b(x , u)∂xu +1

2σ2(x)∂xxu + f (u) = 0 and u(T , ·) = g(·)

I Lipschitz coefficients and small time parameter T : existence and uniqueness (via“classical” fixed point argument) and the function u is Lipschitz-continuous

I Extension to arbitrary time if non-degeneracy of the noise

I Here: no noise in the emission process and singularity in the terminal condition!

Multi-period model

Xt = X0 +

b(Xs ,Ys)ds +

σ(Xs)dWs

Yt = g(XT ) +

f (Ys)ds −∫ T

∂tu + b(x , u)∂xu +1

2σ2(x)∂xxu + f (u) = 0 and u(T , ·) = g(·)

Multi-period model

Xt = X0 +

b(Xs ,Ys)ds +

σ(Xs)dWs

Yt = g(XT ) +

f (Ys)ds −∫ T

∂tu + b(x , u)∂xu +1

2σ2(x)∂xxu + f (u) = 0 and u(T , ·) = g(·)

Multi-period model

Xt = X0 +

b(Xs ,Ys)ds +

σ(Xs)dWs

Yt = g(XT ) +

f (Ys)ds −∫ T

∂tu + b(x , u)∂xu +1

2σ2(x)∂xxu + f (u) = 0 and u(T , ·) = g(·)

Multi-period model

Results for one-period model

I From Carmona and Delarue (2013), there exists a unique solution to the pricingequations:

with terminal condition: φ−(ET ) ≤ YT ≤ φ+(ET ) !

where φ−(e) = ρ1{e>Λ} and φ+(e) = ρ1{e≥Λ}

I The pricing function is well defined on [0,T )× Rd × R and satisfies

φ−(e) ≤ limt→T

v(t, p, e) ≤ φ+(e)

On [0,T ): Yt = v(t,Pt ,Et).

I Key ingredients: monotonicity of φ and y 7→ µ(e, y). Links with scalarconservation law.

I Technical extension: OK for terminal condition (p, e) 7→ φ(p, e) (depending onthe fundamentals P)

Multi-period model

v(t, p, e) ≤ φ+(e)

On [0,T ): Yt = v(t,Pt ,Et).

Multi-period model

v(t, p, e) ≤ φ+(e)

On [0,T ): Yt = v(t,Pt ,Et).

Multi-period model

v(t, p, e) ≤ φ+(e)

On [0,T ): Yt = v(t,Pt ,Et).

Multi-period model

Main step of the proof

I Approximation procedure via regularisation:

1. Introduce noise into the equation (εn)n2. Mollify the terminal condition (φk)k

I Get uniform estimates in terms of regularisation parameters:

|∂pvn,k(t, p, e)| ≤ C and |∂evn,k(t, p, e)| ≤ C

T − t(1)

I Let εn → 0 and φk → φ: get existence via compactness argument thanks to theprevious estimates

I Existence of v with property (1) allows to get existence for (P,E ,Y ).

Multi-period model

Finite number of periodsInfinite number of periodsNumerics

Outline

Multi-period model

Market mechanisms

1. The one-period model describes quite well the ETS phase I

2. Most cap-and-trade market have many periods:

[T0,T1], [T1,T2] . . . [Tq−1,Tq]

3. The connection between periods: Borrowing, Banking and Withdrawal.

I Banking: allowances that are not used in one period can be carried forwardfor compliance in the next period.

I Withdrawal: for any 1 ≤ i ≤ q − 1, if the cap on emissions is exceeded atTi , then the regulator removes a quantity of allowances from the [Ti ,Ti+1]market allocation. The quantity of allowances removed is equal to the levelof excess emissions at Ti .

I Borrowing: for any 1 ≤ i ≤ q − 1, firms may trade some of the allowances

to be released at Ti during [Ti−1,Ti ]. If each trading period represents a

year, this means that firms can, in a particular year that is not the final

year, use the following year’s allowance allocation for compliance.

Multi-period model

Example

3 period example

I Suppose that the regulator releases ck ≥ 0 allowances at Tk−1 for k = 1, 2, 3

I The cap on emission for the period [T0,T1] is Γ1 = c1 + c2 (borrowing)

I For [T1,T2], taking into account banking, borrowing and withdrawal:

Γ2(ET1 ) = c1 + c2 + c3 − ET1 , (2)

I For the last period, Γ3(ET2 ) = c1 + c2 + c3 − ET2

I Notation: Cap on emission on [T0,Tk ], Λk(ETk−1 ) = Γk(ETk−1 ) + ETk−1

(constant in the previous example)

Multi-period model

Pricing problem

Pricing is given by a backward induction

I On each period [Tk−1,Tk ], we still have:

I E0, P0 are known and P and E are continuous process (no jump between theperiod).↪→ But this might not be the case for Y as emission ”control” occurs.

I Link between the period: {non-compliance} = {ETk − ETk−1 ≥ Γk(ETk−1 )}

YTk− = ρ1{non-compliance} + YTk+ 1{compliance}

and for the last period: YTq = ρ1{non−compliance}

Multi-period model

Pricing problem

Multi-period model

Pricing problem

Multi-period model

Main results

I There exists a unique solution to the pricing system (P,E ,Y ,Z) on [0,T ]

I For t ∈ [Tk−1,Tk ] (period k), Yt = vk(t,Pt ,Et)

I On each period, (t, p, e) 7→ vk(t, p, e) solution to

∂tv + µ(p, v)∂ev + b(p)∂pv +1

2σ2(p)∂ppv = rv

with terminal condition φk(p, e) = ρ1{e≥Λk} + vk+1(t, p, e)1{e<Λk}

I Proof: main point is to make sure we can recursively use the results of the oneperiod model.↪→ Prove in particular that e 7→ φk(p, e) is non-decreasing.

Multi-period model

Main results

∂tv + µ(p, v)∂ev + b(p)∂pv +1

2σ2(p)∂ppv = rv

Multi-period model

Main results

∂tv + µ(p, v)∂ev + b(p)∂pv +1

2σ2(p)∂ppv = rv

Multi-period model

Setting

I We consider a scheme running without specified end date:Tk+1 − Tk = ∆T , the cap at each period is Λk = k ×∆Λ

↪→ At Tk , we have {non-compliance} = {ETk ≥ k ×∆Λ}

I In a stationary regime, we expect that the market is just a repetition of similarpricing period↪→ Decoupling between the period for the pricing function but...

I Focus on the first period: [T0,T1]

1. Within the period, usual system of equation.2. At the end

YT1− = ρ1{non-compliance} + v2(T1,PT1 ,ET1 )1{compliance}

3. Note: v2 is almost the same as v1, the cap sequence has increased by ∆Λ.

↪→ v2(T1, p, e) = v1(T0, p, e −∆Λ)

Multi-period model

Setting

↪→ At Tk , we have {non-compliance} = {ETk ≥ k ×∆Λ}I In a stationary regime, we expect that the market is just a repetition of similar

pricing period↪→ Decoupling between the period for the pricing function but...

↪→ v2(T1, p, e) = v1(T0, p, e −∆Λ)

Multi-period model

Setting

↪→ At Tk , we have {non-compliance} = {ETk ≥ k ×∆Λ}I In a stationary regime, we expect that the market is just a repetition of similar

pricing period↪→ Decoupling between the period for the pricing function but...

↪→ v2(T1, p, e) = v1(T0, p, e −∆Λ)

Multi-period model

Main results

I On each period [Tk ,Tk+1], the pricing function satisfies:

∂tv + µ(p, v)∂ev + b(p)∂pv +1

2σ2(p)∂ppv = rv

with terminal condition φ(p, e) = ρ1{e≥k∆Λ} + v(Tk , p, e −∆Λ)1{e<k∆Λ}

I The processes (P,E ,Y ) exist on [0,∞) with Yt = v(t,Pt ,Et), for t 6= Tk , k ≥ 1.

I If r > 0, the function v is unique

I The function v is given by the monotone limit of the q-period model whenq →∞.

Multi-period model

Main results

∂tv + µ(p, v)∂ev + b(p)∂pv +1

2σ2(p)∂ppv = rv

Multi-period model

Main results

∂tv + µ(p, v)∂ev + b(p)∂pv +1

2σ2(p)∂ppv = rv

Multi-period model

Main results

∂tv + µ(p, v)∂ev + b(p)∂pv +1

2σ2(p)∂ppv = rv

Multi-period model

Setting

I One period Toy model (r = 0):

dPt = dWt , P0 = p,

dEt = (Pt − Yt)dt, E0 = e,

dYt = ZtdWt , YT = 1[Λ,∞)(ET ), t ∈ [0,T ],

I Two probabilistic methods

1. Bender & Zhang method (generic)

2. Particle system method (for this model)

I Note: there are other probabilistic methods that could be considered (and in thislow dimensional example deterministic method as well, see Howison & Schwarz(2010) )

Multi-period model

Setting

dPt = dWt , P0 = p,

Multi-period model

Setting

dPt = dWt , P0 = p,

Multi-period model

Bender & Zhang method

I “Nothing to do” for P. Picard iteration for the coupled system (E ,Y ) (startwith E 0

t = e)

dY jt = Z j

tdWt , Y jT = 1[Λ,∞)(E

j−1T ), t ∈ [0,T ],

dE jt = (Pt − Y j

t )dt, E0 = e,

I Discretisation in time on grid (ti )1≤i≤N :

= v j(ti ,Pti ,Ej−1ti

) = E[Y j

ti+1|Pti ,E

j−1ti

](we need v j)

E jti+1

= E jti

+ (Pti − v j(ti ,Pti ,Ejti

))(ti+1 − ti )

I Compute the conditional expectation on some basis function via regressionmethod e.g.

Multi-period model

t = e)

dY jt = Z j

j−1T ), t ∈ [0,T ],

dE jt = (Pt − Y j

t )dt, E0 = e,

) = E[Y j

ti+1|Pti ,E

j−1ti

](we need v j)

E jti+1

= E jti

))(ti+1 − ti )

Multi-period model

t = e)

dY jt = Z j

j−1T ), t ∈ [0,T ],

dE jt = (Pt − Y j

t )dt, E0 = e,

) = E[Y j

ti+1|Pti ,E

j−1ti

](we need v j)

E jti+1

= E jti

))(ti+1 − ti )

Multi-period model

Numerical results for Bender & Zhang (FAIL)

Multi-period model

Bender & Zhang + Regularisation (OK)

Multi-period model

Particle system approach (Bossy & Talay 97)

I Previous method requires regularisation, difficult to tune...

I Here, one can simplify the model by considering Et = Et + (T − t)Pt :

dEt = −Ytdt + (T − t)dWt

dYt = ZtdWt

I Decoupling field v satisfies:

∂t v − v∂e v +(T − t)2

2∂ee v = 0 and v(T , e) = 1e≥Λ

I Set w(t, e) = ∂e v(T − t, e) then w solves the (forward) Fokker-Planck equation:

∂tw + ∂e [vw ]− t2

2∂eew = 0 and w(0, e) = δΛ(e)

I Associated McKean-Vlasov equation: v(t, e) =∫

1{y≤e}dw(t, y)

dXt = v(t,Xt)dt + tdWt where Xt has density w(t, ·)

Multi-period model

dYt = ZtdWt

∂t v − v∂e v +(T − t)2

2∂ee v = 0 and v(T , e) = 1e≥Λ

∂tw + ∂e [vw ]− t2

2∂eew = 0 and w(0, e) = δΛ(e)

1{y≤e}dw(t, y)

Multi-period model

dYt = ZtdWt

∂t v − v∂e v +(T − t)2

2∂ee v = 0 and v(T , e) = 1e≥Λ

∂tw + ∂e [vw ]− t2

2∂eew = 0 and w(0, e) = δΛ(e)

1{y≤e}dw(t, y)

Multi-period model

Numerical result for infinite period model

From 2 to 4000 periods... (left:price function at t=0 / right: zoom to check monotonicity)

modeling carbon market using forward-backward sdesfrank011/mpenl/econclim/chassagneux.pdf ·...

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