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
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 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
Page 4/22 | Motivation Research interest We aim to develop methods to value different types of storages. Value Value in 10^7 $ Inventory Price The value of the storage for the remaining time window as a function of current input numbers. Christian Kellermann | Stolberg, Harz | May 8, 2014
Page 5/22 | Motivation 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
Page 6/22 | Motivation 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
Page 7/22 | Model 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
Page 8/22 | Model Value function �� T � V ( t , p , c ) := sup E ( p , c ) f ( s , u s , C s , P s ) ds − K ( u ) + g ( P T , C T , u T ) , u ∈U t 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 our commodity we would like to inject or to withdraw, ◮ the current inventory level C ∈ [ C min , C max ] with dC s = a ( u s , C s ) ds . Christian Kellermann | Stolberg, Harz | May 8, 2014
Page 9/22 | Model Payoff and penalty �� T � V ( t , p , c ) := sup f ( s , u s , C s , P s ) ds − K ( u ) + g ( P T , C T , u T ) , E ( p , c ) u ∈U t where we have ◮ the payoff f ( s , u s , C s , P s ) , which depends linear on a ( . ) ; ◮ the penalty term g ( P T , C T , u T ) ; ◮ 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 u t = i . We specify our control as u := (( v 1 , τ 1 ) , ( v 2 , τ 2 ) , ... ) , where v i ∈ { 1 , − 1 , 0 } - "in, out, store" - and τ i 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
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. U m ( t , x ) . We define for k = 1 , . . . , m �� T � ◮ V 0 ( t , x , i ) = E t f ( s , i , x s ) ds + g ( T , X T ) | X t = x , �� Θ � ◮ V k ( t , x , i ) = sup Θ ≤ T E t f ( s , i , x s ) ds + M k , i (Θ , x ) , ◮ where M k , i (Θ , x ) = max j � = i {− K i , j + V k − 1 (Θ , x , j ) } is the intervention operator. Christian Kellermann | Stolberg, Harz | May 8, 2014
Page 12/22 | Model ǫ -optimal Control From Carmona/Ludkovski or Øksendal/Sulem we find ◮ τ m − k + 1 := inf � s ≥ τ m − k : V k ( s , x , i ) = M k , i ( s , x ) � , ◮ V m ( t , x , i ) = sup u ∈U m V ( t , x , u ) and ◮ lim m →∞ 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 ◮ 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
Page 14/22 | Numerics Rewriting our objective For a fixed time point t 1 we get � � T � V ( t 1 , p , c , i ) = sup E ( p , c ) f ( s , u s , C s , P s ) ds K ( u ) + g ( P T , C T , u T ) u ∈U t 1 ↓ � � τ � sup f ( s , u t , C s , P s ) ds − K ( i , u τ ) + V ( τ, P τ , C τ , u τ ) E ( p , c ) τ ≤ T t 1 ↓ � � �� f ( t 1 , u t 1 , c , p ) + max − K ( i , j ) + E ( p , c ) V ( t 2 , P t 2 , C t 2 , j ) j ∈{− 1 , 0 , 1 } Christian Kellermann | Stolberg, Harz | May 8, 2014
Page 15/22 | Numerics Parameter For our computations we use the gas price process d log P t = 17 . 1 ( log 3 − log P t ) dt + 1 . 33 dW t (with parameters from Carmona/Ludkovski ) and ◮ a time interval of 1 year with 200 trading days, ◮ inventory bounds [ 0 , 8 ] , ◮ loading rates a in = . 06 and a out = . 25, ◮ continuous cost K u s = 0 . 1 c / 365 and switching costs of K switch = 0 . 25, ◮ the penalty V ( T , p , c , i ) = − 2 p ( 4 − c ) + . Christian Kellermann | Stolberg, Harz | May 8, 2014
Page 16/22 | Numerics First Algorithm We ◮ discretize the inventory using a grid C 0 with 80 equidistant intervals and ◮ simulate N = 10 . 000 price paths. At T we know the N × 80 different values. Besides the standard grid C 0 we consider also the shifted ones C 1 and C − 1 , where the shift depends on a in or a out . 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 two consecutive points in time t 1 < t 2 , we know V ( t 2 , . ) (on C 0 ). 1. We interpolate V ( t 2 , . ) on C − 1 and C 1 . 2. For all c ∈ C 0 and each strategy i ∈ {− 1 , 0 , 1 } we carry out a linear regression for V ( t 2 , c + a i , p n t 2 ) on the first 4 monomials of p n t 1 , 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" and determine w.r.t. to the payoff function the maximal value. Control before: INJ STO WITH Price 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
Page 19/22 | Numerics 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 ( P n t , C n t ) for each strategy i ∈ { 1 , 0 , − 1 } . 1. In T we pick C n T from an uniform distribution on [ C min , C max ] . 2. In the regression step we consider V ( t 2 , . ) and the monomials in P n t 1 and C n t 2 for each i ∈ {− 1 , 0 , 1 } . 3. Before the computation of the estimators we have to determine C n t 1 : we choose a distribution on { 1 , 0 , − 1 } so that E [ a ] = 0. That leads to C n t 1 ( i ) = C n t 2 ( j ( ω )) − a j ( ω ) . 4. If ˆ j ( i ) ≡ j ( ω ) , we take V ( . ) instead of ˆ V ( . ) . Christian Kellermann | Stolberg, Harz | May 8, 2014
Page 20/22 | Numerics Second Algorithm Value Value in 10^7 $ Inventory Price The value of the storage for the remaining time window as a function of current input numbers. Christian Kellermann | Stolberg, Harz | May 8, 2014
Page 21/22 | Numerics Second Algorithm Inventory Price 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
Page 22/22 | References A.Boogert, C.De Jong: Gas Storage Valuation Using a Monte Carle Method , 2008 R.Carmona, M.Ludkovski: Valuation of energy storage: an optimal 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
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