Valuation of energy storages: a numerical approach based on - - PowerPoint PPT Presentation

Valuation of energy storages: a numerical approach based on stochastic 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

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StoBeS Project

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

Christian Kellermann | Stolberg, Harz | May 8, 2014

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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 $

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Status Quo

The standard example is a gas storage.

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

Christian Kellermann | Stolberg, Harz | May 8, 2014

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Literature

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

◮ develop the HJB equations and use FD-Methods to solve the

P(I)DE (see Davison et al.),

◮ use the Longstaff-Schwartz approach (see Carmona/Ludkovski or

Boogert/de Jong). This consists of

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

Christian Kellermann | Stolberg, Harz | May 8, 2014

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Advantages

We choose the approach by Carmona and Ludkovski because

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

Christian Kellermann | Stolberg, Harz | May 8, 2014

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Value function

V(t, p, c) := sup

u∈U

E(p,c) T

t

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

• ,

where we have

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

• ur commodity we would like to inject or to withdraw,

◮ the current inventory level C ∈ [Cmin, Cmax] with

dCs = a(us, Cs)ds.

Christian Kellermann | Stolberg, Harz | May 8, 2014

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Payoff and penalty

V(t, p, c) := sup

u∈U

E(p,c) T

t

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

• ,

where we have

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

Christian Kellermann | Stolberg, Harz | May 8, 2014

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Impulse Control

Let U := U(t, p, c, i) be the set of admissible controls with ut = i. We specify our control as u := ((v1, τ1), (v2, τ2), ...), where vi ∈ {1, −1, 0} - "in, out, store" - and τi is the optimal stopping time or rather the optimal switching time. The simplified impulse set is a result of the so-called bang-bang property. Furthermore, we can make profitable use of the iterative scheme for impulse control!

Christian Kellermann | Stolberg, Harz | May 8, 2014

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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 for k = 1, . . . , m

◮ V 0(t, x, i) = E

T

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

• ,

◮ V k(t, x, i) = supΘ≤T E

Θ

t f(s, i, xs)ds + Mk,i(Θ, x)

• ,

◮ where Mk,i(Θ, x) = maxj=i{−Ki,j + V k−1(Θ, x, j)} is the

intervention operator.

Christian Kellermann | Stolberg, Harz | May 8, 2014

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ǫ-optimal Control

From Carmona/Ludkovski or Øksendal/Sulem we find

◮ τm−k+1 := inf

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

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

Christian Kellermann | Stolberg, Harz | May 8, 2014

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Conditions

Alltogehter, we make use of

◮ the fact, that we get an initial value problem, because V(T, p, c)

is deterministic w.r.t. g(.),

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

Christian Kellermann | Stolberg, Harz | May 8, 2014

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Rewriting our objective

For a fixed time point t1 we get V(t1, p, c, i) = sup

u∈U

E(p,c) T

t1

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

• ↓

sup

τ≤T

E(p,c) τ

t1

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

• ↓

f(t1, ut1, c, p) + max

j∈{−1,0,1}

• − K(i, j) + E(p,c)
• V(t2, Pt2, Ct2, j)
• Christian Kellermann |

Stolberg, Harz | May 8, 2014

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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) and

◮ a time interval of 1 year with 200 trading days, ◮ inventory bounds [0, 8], ◮ loading rates ain = .06 and aout = .25, ◮ continuous cost Kus = 0.1c/365 and switching costs of

Kswitch = 0.25,

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

Christian Kellermann | Stolberg, Harz | May 8, 2014

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First Algorithm

We

◮ discretize the inventory using a grid C0 with 80 equidistant

intervals and

◮ simulate N = 10.000 price paths.

At T we know the N × 80 different values. Besides the standard grid C0 we consider also the shifted ones C1 and C−1, where the shift depends on ain or aout.

Christian Kellermann | Stolberg, Harz | May 8, 2014

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First Algorithm

Starting with the initial value(s) in T, we go backwards. For two consecutive 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, pn

t2) on the first 4 monomials of

pn

t1, where 0 ≤ n ≤ N.

Christian Kellermann | Stolberg, Harz | May 8, 2014

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First Algorithm

• 3. We compute the three estimators for the "continuation value" and

determine 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

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Second Algorithm

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

t , Cn t ) for

each strategy i ∈ {1, 0, −1}.

• 1. In T we pick Cn

T from an uniform distribution on [Cmin, Cmax].

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

Pn

t1 and Cn t2 for each i ∈ {−1, 0, 1}.

• 3. Before the computation of the estimators we have to determine

Cn

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

leads to Cn

t1(i) = Cn t2(j(ω)) − aj(ω).

• 4. If ˆ

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

Christian Kellermann | Stolberg, Harz | May 8, 2014

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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

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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

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References

A.Boogert, C.De Jong: Gas Storage Valuation Using a Monte Carle Method, 2008 R.Carmona, M.Ludkovski: Valuation of energy storage: an

• ptimal switching approach, 2010

B.Øksendal, A.Sulem: Applied Stochastic Control of Jump Diffusions, 2007 M.Thompson, M.Davison, H.Rasmussen: Natural Gas Storage Valuation and Optimization: A Real Options Application, 2009

Thank you for your attention...

Christian Kellermann | Stolberg, Harz | May 8, 2014