Analysis and Enhancement of Prac4ce- based Methods for the Real Op4on Management of Commodity Storage Assets

Transcription

1 Analysis and Enhancement of Prac4ce- based Methods for the Real Op4on Management of Commodity Storage Assets Nicola Secomandi Carnegie Mellon Tepper School of Business Interna4onal Conference on Stochas4c Programming, Bergamo July 9, 2013 Research supported by NSF grant CMMI Nicola Secomandi - CMU Tepper School 1

2 Natural Gas (NG) Storage Assets Storage Facili/es Markets for the Commodity NYMEX NG Futures Curves (November 2006) A Salt Caverns B Aquifers C Depleted Reservoirs $/MMBtu DEC.06 MAY.07 OCT.07 MAR.08 AUG.08 JAN.09 JUN.09 NOV.09 APR.10 SEP.10 FEB.11 JUL.11 DEC.11 Maturi4es 11/01/ /02/ /03/ /06/ /07/ /08/ /09/ /10/2006 Nicola Secomandi - CMU Tepper School 2

3 Storage Asset Management and Valua4on Merchants manage and value commodity storage assets as real op/ons on the commodity price evolu4on In principle, the idea is simple: Buy low, inject, store, withdraw, and sell high but there are prac4cal difficul/es Constraints on minimum/maximum storage space and injec4on/withdrawal capacity (speed of the asset) Inventory adjustment costs and losses (fric/ons) Modeling the evolu/on of the commodity price A real op/on model is needed Nicola Secomandi - CMU Tepper School 3

4 Exact but Intractable Markov Decision Process (MDP) Stages: inventory trading dates Finite horizon: Finite number of trading dates, N States: (Endogenous informa/on, Exogenous informa/on) Endogenous informa/on: Asset inventory, x Exogenous informa/on: Forward curve on date T n (N n)- dimensional vector: F n,m : Date T n price of the futures contract with maturity T m Ac/on: Inventory traded per stage in the wholesale spot market, a, at price s Reward: Trading payoff p(a,s) Determining an op/mal policy requires, in general, solving a stochas/c dynamic program (DP): Risk- free discount factor & expecta4on Computa/onally intractable model (in general) Nicola Secomandi - CMU Tepper School 4

5 Heuris4c Solu4on Approach Prac4ce and academia Determine a heuris/c policy (Boogert and de Jong 2008, 2011, Carmona and Ludkovski 2010, Lai et al. 2010, Boogert and Mazieres 2011, Thompson 2012, Wu et al. 2012, Nadarajah et al. 2013) Es4mate a lower bound (LB) on the value of storage by Monte Carlo simula4on of this policy Academia Also es4mate an upper bound (UB) on the value of storage using approximate dynamic programming, duality methods for MDPs (Brown et al. 2010), and Monte Carlo simula4on (Lai et al. 2010, Nadarajah et al. 2013) Let s focus on prac4ce- based math (linear) programming reop4miza4on methods Rolling intrinsic policy Rolling basket of spread op4ons policy Nicola Secomandi - CMU Tepper School 5

6 Research Ques4ons These reop4miza4on- based prac4ce- based policies are known to be near op4mal (Lai et al. 2010) Can we provide more numerical and some structural support for this observed behavior? Can we enhance these policies with an effec4ve dual upper bound that can be es4mated by solving, within Monte Carlo simula4on, simple modifica4ons of the linear programs used by these policies? Nicola Secomandi - CMU Tepper School 6

7 Rolling Intrinsic Policy Intrinsic (I) model This model computes the intrinsic value of storage (the value of seasonality) This model can be equivalently reformulated as a linear program (LP) Rolling intrinsic (RI) policy: Reop4mize the intrinsic model (or its equivalent LP) in every stage with updated price and inventory informa4on The value of this policy can be es4mated by Monte Carlo simula4on Nicola Secomandi - CMU Tepper School 7

8 Spread Op4ons Date T 0 value of an op/on on the spread between prices F m,n and s m modified to account for discoun/ng and fric/ons Modified prices A spread op/on approximates the value of a two- date, T m < T n, storage asset with 1 unit of space and 1 unit of capacity Spread op4on valued at date T 0 Buy 1 unit of commodity at date T m if advantageous rela4ve to selling it at date T n Sell any commodity bought at date T n 0 m n Stages Nicola Secomandi - CMU Tepper School 8

9 Rolling Basket of Spread Op4ons Policy Think of a storage asset as a constrained portolio of spread op/ons (SOs) q m,n : No/onal amount of spread op/on m- n Injec/on Withdrawal Stage Stage 1 n N 1 0 q 0,1 q 0,n q 0,N - 1 m q m,n q m,n - 1 N 2 q N 2,N - 1 Basket of spread op/ons model: LP that op/mizes the value of this portolio subject to space and capacity constraints (there are also forward sales) Rolling basket of spread op/ons (RSO) policy: Obtained by reop4mizing the basket of spread op4on model The value of this policy can be es4mated by Monte Carlo simula4on Nicola Secomandi - CMU Tepper School 9

10 Benefit of Reop4miza4on - Reop/miza/on useful - Prac/ce- based policies near op/mal Extended Lai et al. (2010) NG assets w/ fric/ons => Nicola Secomandi - CMU Tepper School 10

11 Structural Reop4miza4on Results Reop4miza4on is provably useful when rolling the intrinsic model Unaware of a similar result in other applica4ons Intui/on: (i) The op4mal basis obtained in the previous op4miza4on is always feasible in the current op4miza4on (ii) The expected discounted value of the objec4ve func4on of the current op4miza4on is the objec4ve func4on of the previous op4miza4on (for every solu4on) => Upda4ng the op4mal basis cannot yield a worse policy Reop4miza4on is provably useful when rolling the basket of spread op4ons LP If the asset is fric4onless Or, with fric4ons, when the comparison is against either (a) the intrinsic value or (b) the op4mal objec4ve func4on of the basket of spread op4ons LP op4mized once Nicola Secomandi - CMU Tepper School 11

12 Policy Structure Results Op4mal policy structure (S. et al. 2012) Stage and forward curve dependent double base stock (buy- and- inject and withdraw- and- sell) targets Finite number of possible op4mal targets (under a mild assump4on) The RI policy obviously has this structure The RSO policy also has this structure Proof based on reformula4ng the underlying LP in a recursive manner as an approximate dynamic program and as a network flow model Nicola Secomandi - CMU Tepper School 12

13 A Tractable Case Fast storage asset: Asset can be filled up or emp4ed in between two successive trading dates Fric/onless storage asset: No inventory adjustment costs and losses Op4mal decision rule in stage n Buy and fill up if δf n,n+1 > s n Do nothing if δf n,n+1 = s n => One- stage look- ahead policy is op/mal Empty and sell δf n,n+1 < s n Op4mal value func4on: Marginal value of inventory equal to the spot price, marginal value of space equal to a sum of exchange op4on (EO) values max space Nicola Secomandi - CMU Tepper School 13

14 Simplified Dual Upper Bounding Approach A perfect informa4on UB can be es4mated by solving determinis4c hindsight DPs within Monte Carlo simula4on A 4ghter bound can typically be es4mated by solving within Monte Carlo simula4on determinis4c dual DPs obtained by penalizing their trading payoffs for knowledge of the future (dual UB; Brown et al. 2010) Current approaches define dual penal/es using value func4ons obtained via approximate dynamic programming (Lai et al. 2010, Nadarajah et al. 2013) Simpler approach: Use the value func4on of the fast and fric4onless asset to define penal4es The resul4ng penal4es are based on price spreads and are propor/onal to the current inventory and the inventory change: δ(s n+1 F n,n+1 ) (x a) + zero mean intercept term (control variate) Enhancement to prac4ce- based policies: The resul4ng dual DPs can be formulated as simple modifica/ons of the LPs used to determine the RI and RSO policies Nicola Secomandi - CMU Tepper School 14

15 Is This Simpler Approach Any Good? EO: Value func4on of the fast and fric4onless asset based on Exchange Op4ons DS: Dual UB based on price Spreads Property: DS <= EO EO is loose DS is compe//ve Average CPU Seconds Nadarajah et al. (2010) 444 DS 7 Nicola Secomandi - CMU Tepper School 15

16 Conclusions Provided new numerical and structural support for the near op4mal performance of the prac4ce- based RI and RSO policies Numerical results on extended NG instances Structural benefit of reop4miza4on Structure of these policies consistent with op4mal policy structure (trivial for the RI policy) Enhanced the RI and RSO policies with near op4mal dual upper bound based on solving within Monte Carlo simula4on simple modifica4ons of the LPs used by these policies Provable improvement of the exchange op4on upper bound Promising numerical results on extended NG instances Linear programming and Monte Carlo simula4on appear to be effec4ve at managing and valuing commodity (NG) storage assets Nicola Secomandi - CMU Tepper School 16