Valuation of energy storages: a numerical approach based ...sfb649.wiwi.hu-berlin.de/fedc/...Energy_Storage.pdf · stochastic control Energy Finance Workshop 2014 Christian Kellermann

Valuation of energy storages: a numerical approach based onstochastic control

Energy Finance Workshop 2014

Christian Kellermann | Chair for Energy Trading and Finance | University of Duisburg-Essen

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Outline

Motivation

Model

Numerics

Christian Kellermann | Stolberg, Harz | May 8, 2014

Page 3/22 | Motivation

StoBeS Project

Our project "Stochastic Methods for Management and Valuation ofCentralized and Decentralized Energy Storages in the Context of theFuture German Energy System" is part of



Research interest

We aim to develop methods to value different types of storages.

The value of the storage for the remaining time window as a function of current input numbers.


Value

Inventory Price

Value in 10^7 $


Status Quo

The standard example is a gas storage.

I Forward or spot?I Intrinsic, rolling intrinsic or extrinsic value?



Literature

Once we have an objective function for the extrinsic value, there aretwo approaches:

I develop the HJB equations and use FD-Methods to solve theP(I)DE (see Davison et al.),

I use the Longstaff-Schwartz approach (see Carmona/Ludkovski orBoogert/de Jong). This consists of

1. Monte Carlo simulation,2. Least Squares regression.


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Advantages

We choose the approach by Carmona and Ludkovski because

I it is easy to implement,I it is applicable to various settings,I various extensions can be implemented.


Page 8/22 | Model

Value function

V (t ,p, c) := supu∈U

E(p,c)

[∫ T

tf (s,us,Cs,Ps)ds − K (u) + g(PT ,CT ,uT )

],

where we haveI the time t ∈ [0,T],I a price process P,I the strategy u for our storage, i.e. the decision which amount of

our commodity we would like to inject or to withdraw,I the current inventory level C ∈ [Cmin,Cmax] with

dCs = a(us,Cs)ds.


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Payoff and penalty

V (t ,p, c) := supu∈U

E(p,c)

[∫ T

tf (s,us,Cs,Ps)ds − K (u) + g(PT ,CT ,uT )

],

where we haveI the payoff f (s,us,Cs,Ps), which depends linear on a(.);I the penalty term g(PT ,CT ,uT );I a sum of switching costs K (u).


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Impulse Control

Let U := U(t ,p, c, i) be the set of admissible controls with ut = i . Wespecify our control as

u := ((v1, τ1), (v2, τ2), ...),

where vi ∈ {1,−1,0} - "in, out, store" - and τi is the optimal stoppingtime or rather the optimal switching time. The simplified impulse set isa result of the so-called bang-bang property.

Furthermore, we can make profitable use of the iterative scheme forimpulse control!


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Multiple Switching Problem

Let X = (P,C) be the state of our system. We consider the case,where at most m switches are allowed, i.e. Um(t , x). We define fork = 1, . . . ,m

I V 0(t , x , i) = E[∫ T

t f (s, i , xs)ds + g(T ,XT )|Xt = x],

I V k (t , x , i) = supΘ≤T E[∫ Θ

t f (s, i , xs)ds +Mk,i (Θ, x)],

I whereMk,i (Θ, x) = maxj 6=i{−Ki,j + V k−1(Θ, x , j)} is theintervention operator.


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ε-optimal Control

From Carmona/Ludkovski or Øksendal/Sulem we find

I τm−k+1 := inf{

s ≥ τm−k : V k (s, x , i) =Mk,i (s, x)}

,I V m(t , x , i) = supu∈Um V (t , x ,u) andI limm→∞ V m(t , x , i) = V (t , x , i).


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Conditions

Alltogehter, we make use of

I the fact, that we get an initial value problem, because V (T ,p, c)is deterministic w.r.t. g(.),

I the bang-bang property,I a time grid plus the Bellman principle,I the extended Longstaff and Schwartz approach.



Rewriting our objective

For a fixed time point t1 we get

V (t1,p, c, i) = supu∈U

E(p,c)

[∫ T

t1f (s,us,Cs,Ps)dsK (u) + g(PT ,CT ,uT )

]↓

supτ≤T

E(p,c)

[∫ τ

t1f (s,ut ,Cs,Ps)ds−K (i ,uτ ) + V (τ,Pτ ,Cτ ,uτ )

]↓

f (t1,ut1 , c,p) + maxj∈{−1,0,1}

(− K (i , j) + E(p,c)

[V (t2,Pt2 ,Ct2 , j)

])



Parameter

For our computations we use the gas price process

d log Pt = 17.1(log 3− log Pt )dt + 1.33dWt

(with parameters from Carmona/Ludkovski) andI a time interval of 1 year with 200 trading days,I inventory bounds [0,8],I loading rates ain = .06 and aout = .25,I continuous cost Kus = 0.1c/365 and switching costs of

Kswitch = 0.25,I the penalty V (T ,p, c, i) = −2p(4− c)+.



First Algorithm

We

I discretize the inventory using a grid C0 with 80 equidistantintervals and

I simulate N = 10.000 price paths.

At T we know the N × 80 different values. Besides the standard gridC0 we consider also the shifted ones C1 and C−1, where the shiftdepends on ain or aout .



First Algorithm

Starting with the initial value(s) in T , we go backwards. For twoconsecutive points in time t1 < t2, we know V (t2, .) (on C0).

1. We interpolate V (t2, .) on C−1 and C1.2. For all c ∈ C0 and each strategy i ∈ {−1,0,1} we carry out a

linear regression for V (t2, c + ai ,pnt2 ) on the first 4 monomials of

pnt1 , where 0 ≤ n ≤ N.



First Algorithm

3. We compute the three estimators for the "continuation value" anddetermine w.r.t. to the payoff function the maximal value.

Value

The storage value for a fixed inventory as a function of the price if we switch to INJECTION,

STORE or WITHDRAWAL.


INJ

STO

WITH

Control before:

Price


Second Algorithm

Now we simulate also C and thus we reduce the computation by onefor-loop. For each path and for each node we get a tuple (Pn

t ,Cnt ) for

each strategy i ∈ {1,0,−1}.

1. In T we pick CnT from an uniform distribution on [Cmin,Cmax ].

2. In the regression step we consider V (t2, .) and the monomials inPn

t1 and Cnt2 for each i ∈ {−1,0,1}.

3. Before the computation of the estimators we have to determineCn

t1 : we choose a distribution on {1,0,−1} so that E[a] = 0. Thatleads to Cn

t1 (i) = Cnt2 (j(ω))− aj(ω).

4. If j(i) ≡ j(ω), we take V (.) instead of V (.).



Second Algorithm

The value of the storage for the remaining time window as a function of current input numbers.


Value in 10^7 $

Value

Inventory Price


Second Algorithm

At a fixed time point the optimal decision for INJECTION, STORE or WITHDRAWAL depending on

price and inventory at a certain .


Inventory

Price

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References

A.Boogert, C.De Jong: Gas Storage Valuation Using a MonteCarle Method, 2008

R.Carmona, M.Ludkovski: Valuation of energy storage: anoptimal switching approach, 2010

B.Øksendal, A.Sulem: Applied Stochastic Control of JumpDiffusions, 2007

M.Thompson, M.Davison, H.Rasmussen: Natural Gas StorageValuation and Optimization: A Real Options Application, 2009

Thank you for your attention...


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Valuation of energy storages: a numerical approach based ...sfb649.wiwi.hu-berlin.de/fedc/...Energy_Storage.pdf · stochastic control Energy Finance Workshop 2014 Christian Kellermann