valuation of energy storages: a numerical approach based...
TRANSCRIPT
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
Page 2/22 |
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
Christian Kellermann | Stolberg, Harz | May 8, 2014
Page 4/22 | Motivation
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.
Christian Kellermann | Stolberg, Harz | May 8, 2014
Value
Inventory Price
Value in 10^7 $
Page 5/22 | Motivation
Status Quo
The standard example is a gas storage.
I Forward or spot?I Intrinsic, rolling intrinsic or extrinsic value?
Christian Kellermann | Stolberg, Harz | May 8, 2014
Page 6/22 | Motivation
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.
Christian Kellermann | Stolberg, Harz | May 8, 2014
Page 7/22 | Model
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.
Christian Kellermann | Stolberg, Harz | May 8, 2014
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.
Christian Kellermann | Stolberg, Harz | May 8, 2014
Page 9/22 | Model
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).
Christian Kellermann | Stolberg, Harz | May 8, 2014
Page 10/22 | Model
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!
Christian Kellermann | Stolberg, Harz | May 8, 2014
Page 11/22 | Model
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.
Christian Kellermann | Stolberg, Harz | May 8, 2014
Page 12/22 | Model
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).
Christian Kellermann | Stolberg, Harz | May 8, 2014
Page 13/22 | Numerics
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.
Christian Kellermann | Stolberg, Harz | May 8, 2014
Page 14/22 | Numerics
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)
])
Christian Kellermann | Stolberg, Harz | May 8, 2014
Page 15/22 | Numerics
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)+.
Christian Kellermann | Stolberg, Harz | May 8, 2014
Page 16/22 | Numerics
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 .
Christian Kellermann | Stolberg, Harz | May 8, 2014
Page 17/22 | Numerics
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.
Christian Kellermann | Stolberg, Harz | May 8, 2014
Page 18/22 | Numerics
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.
Christian Kellermann | Stolberg, Harz | May 8, 2014
INJ
STO
WITH
Control before:
Price
Page 19/22 | Numerics
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 (.).
Christian Kellermann | Stolberg, Harz | May 8, 2014
Page 20/22 | Numerics
Second Algorithm
The value of the storage for the remaining time window as a function of current input numbers.
Christian Kellermann | Stolberg, Harz | May 8, 2014
Value in 10^7 $
Value
Inventory Price
Page 21/22 | Numerics
Second Algorithm
At a fixed time point the optimal decision for INJECTION, STORE or WITHDRAWAL depending on
price and inventory at a certain .
Christian Kellermann | Stolberg, Harz | May 8, 2014
Inventory
Price
Page 22/22 |
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...
Christian Kellermann | Stolberg, Harz | May 8, 2014