Notes. Cases on Static Optimization. Chapter 6 Algorithms Comparison: The Swing Case

Transcription

1 Notes Chapter 2 Optimization Methods 1. Stationary points are those points where the partial derivatives of are zero. Chapter 3 Cases on Static Optimization 1. For the interested reader, we used a multivariate Gaussian for all the relevant risk variables, and we performed a calibration using a four-years sample ( ) with market data coming from the Italian power market, the same sample used by Bonacina (2013). Chapter 6 Algorithms Comparison: The Swing Case 1. We could also take into account that l 1,l 2 should be functions of T n because the length of the month is not equal for every month. However, this is barely an improvement, and to avoid a further cumbersome notation we do not model this. Chapter 7 Storage Contracts 1. Of course, this situation and the situation in Example 1 are mutually exclusive, otherwise one would consider this gas loss two times, both in the physical gas quantity as well as in the financial part. 2. In fact, as already noticed, the cost of pumping gas in the reservoir is paid either as a financial cost (e.g., in electricity or gas consumed) but leaving the gas reservoir intact, or as gas spent to produce energy without financial expenses. 3. The term σ 2 /2 is an Ito correction, which however turns out to be negligible. 185

2 186 Notes Chapter 8 Optimal Trading Strategies in Intraday Power Markets 1. Or calculate via analytical calculations, which however turn out to be quite cumbersome in this case; see [3] for details.

3 Index algorithms comparison, algorithm, algorithm and reduction to one dimension, boundary conditions, dynamic programming, finite differences method, 114, 122 3, indexed strike price modelling for gas swing contracts, Least Square Monte Carlo method, 114, 128 9, Naïve Monte Carlo with linear programming, 131 numerical experiments, one-year contract, stochastic control problem, swing contracts, 115 annual contract quantity (ACQ), 119 annual expected return, 6 the art of linearization, 1 asset allocation with capital constraints, asset allocation under risk capital constraints, 18 binomial tree for price evolution, 19 asset s return, 4 Awerbuch, S., 6, 79 Basak, S., 18 Bellman equation, 44 Bellman s optimality principle, 121 Berger, M., 6 binary variable, 5 Binomial tree, 45 Bonacina, F., 79, 81 boundary conditions algorithms comparison, deterministic numerical methods, 57 finite difference algorithm, storage contracts, 154 capital allocation, 15 capital/risk optimization, 15 co-dependency, 3 computationally simple trees in dimension 1, 60 3 with general diffusion, 62 consistent pricing technique, 12 constraints, 28 continuous time intraday trading, 178 contract price, 107, 116 control variables, 13 Courant-Friedrichs-Lewy (CFL) condition, 123 Cuoco, D., 18 cycling, 150 Dantzing, George, 27 decision tree approach, 10 decision variable, 14 deterministic dynamic programming, 35 8 deterministic numerical methods boundary conditions, 57 finite difference method for HJB equation, 55 7 deterministic optimization, 48 deterministic part of transition function, 14 deterministic problems, 2, 107 versus stochastic problems, 2 diagrammatic approach, valuing project s flexibilities using, 92 description of investment problem, 92 4 discounted cash flow (DCF) methodologies, 94 electricity price trinomial tree, 97 evaluation diagram, 100, 102,

4 188 Index diagrammatic approach, valuing project s flexibilities using (Continued) modelling electricity price dynamics, 95 6 static DCF analysis of investment alternatives, 95 synoptic representation of investment alternatives, 93 traditional evaluation methods, 94 5 valuing investment flexibilities by means of lattice approach, discounted cash flow (DCF) methodologies, 94 static DCF analysis of investment alternatives, 95 discretization, 21 dynamic problems, 2, 16 versus static problems, 2 3 dynamic programming, 35 deterministic, 35 8 principle, 40, 47, 129 stochastic: continuous time, 48 9; discrete time, 38 swing contracts, Dynamic Programming algorithm (DPA), 36 8, 42 economic flow, 15 energy asset optimization, 7 decision alternatives and project s value probabilistic evolution of simple investment project in generic energy asset, 11 generation, transportation and storage asset operational optimization and valuation, generation asset investment valuation with real option methodology, 7 10 energy markets, optimization in asset allocation with capital constraints, classification of optimization problems, 1 deterministic versus stochastic problems, 2 energy asset optimization, 7 energy trading and optimization, generation, transportation and storage asset operational optimization and valuation, generation asset investment valuation with real option methodology, 7 11 intraday trading, 22 4 linear versus nonlinear problems, 1 optimal portfolio selection among different investment alternatives, 3 7 static versus dynamic problems, 2 3 energy markets, pricing in, 32 energy producers, 12 energy trading and optimization, equivalent martingale measure, 31 Euler scheme, 67 European Power Exchange (EPEX), 22 explicit scheme, 56 financial assets, 32 finite difference (FD) algorithm, 114, 122 3, 137 algorithm, boundary conditions, firm storage service (FSS), 147 flexibility options, 8 forest of trees, 59, 63 7 framing, 9 gas cave, gas spot price, 116, 153 gas storage contract, 149 generation asset investment valuation with real option methodology, 7 11 generic intertemporal asset allocation, 17 Geometric Brownian Motion (GBM), 83 cash flow modelling, 83 cash flows, 85 costs, 84 revenues, 83 Glensk, B., 7 Hamilton-Jacobi-Bellman equation, 48, 50 5, 114, 122, 138, 180

5 Index 189 finite difference method for HJB equation, 55 7 Hessian matrix, 29 high deverabilitymultiple-cycle (HDMC), 150 1, 153 incomplete markets, pricing in, 32 3 industrial initiatives, 4 injection capacity, 150 internal rate of return (IRR), 4, 72, 74, 81 intraday power markets, optimal trading strategies in, absolute spread between day-ahead and intraday markets, 163, 164 evidence regard liquidity, forecast error reduction for wind generation, 162 high-low spread in EPEX Intraday market, 164 intraday power price features, intraday price in continuous market, 165 Italian intraday market, liquidity, 166 optimal algorithmic trading: in auction-based intraday power markets, ; in continuous time power markets, 178 9; EPEX Spot market, 181 4; optimization problem, percentage of transactions already done with respect to time left to delivery, 167 qualitative tests, structure and organization of intraday markets in Europe, 162 trend identification, 168 intraday trading, 22 4 Italian intraday market, Karush-Kuhn-Tucker conditions (KKT conditions), 30 Kushner scheme, 56 Lagrange multipliers, 29, 31 lattice methods, 44 lattice of trees, see forest of trees Least Square Monte Carlo (LSMC) algorithm, 128 9, adapted process, 143 algorithm and reduction to one dimension, approximation procedure, NMC output, 141 optimal control, performance, usability, value function in discrete time, 128 Least Square Monte Carlo methods, 46 7 Levy, H., 16 liberalization of energy markets, 11 linear optimization, 26 7 LP problems, 27 8 linear versus nonlinear problems, 1 Longstaff, F. A., 46 Madlener, R., 6 Markov chain, 58, 61 Markov process, 64 Markowitz, H. M., 3, 16, 76 Markowitz s problem, 3 4 Mathematical Finance, 31 MATLAB, 73 Mean Variance Portfolio (MVP) theory, 81 Merton, C., 17, 20, 38 Merton problem, 17, 41 minimization of risk measures, 33 minimum annual quantity (maq), 119 Mixed Integer Linear Problems (MILP), 27 modelling electricity price dynamics, 95 6 Monte Carlo method, 43 Munoz, I., 81, 83 Naïve Monte Carlo (NMC) with linear programming, 47 8, 115, 131 Nelson, D. B., 60, 62 Net Present Value (NPV), 5

6 190 Index nonlinear optimization, 28 constrained problems: with equality constraints, 29 30; with inequalities constraints, 30 1 unconstrained problem, 28 9 numerical experiments, finite differences, Least Square Monte Carlo, one-year contract, operation and maintenance costs (O&M), 81 optimal generation fuel mix, 6 optimal generation mix for electricity producer, optimal investment alternatives, 4 optimal portfolio selection among different investment alternatives, 3 7 optimal selection problem, 6 optimal stopping problems, 43 optimal strategy, optimal trading strategies in intraday power markets, average risk of strategy versus benchmark, 177 cumulated performance of strategy, 176 forecast error reduction for wind generation, 162 intraday power price features, Italian intraday market, modelling portfolio dynamics, 180 OMIE market, 170 optimal algorithmic trading: in auction-based intraday power markets, ; in continuous time power markets, 178 9; EPEX Spot market, 181 4; optimization problem, performance OU, 183 ratio between expected P&L and expected risk, 178 structure and organization of intraday markets in Europe, 162 trading strategy, 170 optimal way, 15 optimization methods, 26 computationally simple trees in dimension 1, 60 3 constrained problems with equality constraints, constrained problems with inequalities constraints, 30 1 deterministic dynamic programming, 35 8 deterministic numerical methods: boundary conditions, 57; finite difference method for HJB equation, 55 7 general case, 41 3 Hamilton-Jacobi-Bellman equation, 50 5 lattice of trees, 63 7 Least Square Monte Carlo methods, 46 7 linear optimization, 26 7; LP problems, 27 8 Monte Carlo methods, 67 motivating example, Naïve Monte Carlo with linear programming, 47 8 nonlinear optimization, 28 pricing financial assets, 31 2 pricing in energy markets, 32 pricing in incomplete markets, 32 3 probabilistic numerical methods, 57 9 stochastic dynamic programming: continuous time, 48 9; discrete time, 38 tree methods, 43 6; continuous time, unconstrained problem, 28 9 utility indifference pricing, 33 5 optimization problems, classification of, 1 deterministic versus stochastic problems, 2 linear versus nonlinear problems, 1 static versus dynamic problems, 2 3 Ornstein-Uhlenbeck process, 153, 179 portfolio selection, 3 power generation assets, 105 pricing financial assets, 31 2

7 Index 191 pricing measure, 31, 148 project/asset valuation, correct, 9 Rawaswamy, 60 real gas caves, 150 real option framing process, 10 real options, 7 8 recombining tree methods, see lattice methods Renewable Energy Sources (RES), 161 return, 81 risk-neutral measure, see equivalent martingale measure risk/return optimization, 15 Schwartz S., 46 set of admissible controls, 42 Shapiro, A., 18 spot price, 118 state variables, 13 static optimization, cases on cash flow simulations, 72 efficient frontier, 76 efficient MV frontier, 88; for different levels of interest rates, 89 electricity production by source, 80 expectedirr and standard deviations, 76 free cash flow, 70, 71 GBM parameters, 83 independence among cash flows variability, 71 input data used for solution of problem, 82 investment alternatives, 69 logical implementation scheme of optimization problem, 73 mean-variance portfolio, 82 minimum risk portfolio with min target expected IRR, 77, 78 model parameters, 71 modelling of free cash flow uncertainty, 71 normalized NPV, 90; interest, 86 optimal generation mix for electricity producer, optimal portfolio allocation, 75 optimal portfolio s weights, 89, 90 optimization problem, 87 serial dependence, 71 standard deviation of portfolio, 87 static problems, 2, 16 versus dynamic problems, 2 3 stochastic differential equation, 59 Stochastic Dynamic Programming (SDP), 114 continuous time, 48 9; Hamilton-Jacobi-Bellman equation, 50 5 discrete time, 38; general case, 41 3; Least Square Monte Carlo methods, 46 7; motivating example, 39 41; Naïve Monte Carlo with linear programming, 47 8; tree methods, 43 6 optimization methods: continuous time, 48 9; discrete time, 38 stochastic optimal control problem., stochastic problems, 2 stochastic state variable, 14 storage contracts boundary conditions, 154 contract, evaluation problem, gas cave, gas spot price, 153 implementation, marginal profit and loss of, 147 numerical experiment: no-penalty case, 154 6; penalty case, optimal control as function of cumulated gas quantity, 155, 156; and spot price, 157, 158 optimal strategy, value function as function of cumulated gas quantity, 156; and spot price, 158, 159 strategy variable, 13 superreplication, 32 3 swing contracts, 115, see also algorithms comparison contract value, 134

8 192 Index swing contracts (Continued) contract value and execution time with finite differences, 133 discretization on binomial tree of admissible cumulated quantity for, 130 dynamic programming, indexed strike price modelling for gas swing contracts, Monte Carlo algorithms, 135 one-year contract, sensitivities of LSMC algorithm, 136 stochastic control problem, take-or-pay, see swing contracts terminal cost, 148 traded financial assets, 32 transition function, 15 tree methods, 43 6, see also lattice methods continuous time, utility-based pricing, 33 utility functions, 16, 171 utility indifference pricing, 33 5 value function, 13, 36, 49 valuing project s flexibilities using diagrammatic approach, 92 description of investment problem, 92 4 modelling electricity price dynamics, 95 6 traditional evaluation methods, 94 5 valuing investment flexibilities by means of lattice approach, VaR (Value at Risk) risk limit, 17, 18 20, 21 variable operating and maintenance (VOM), 94 verification theorem, 50, 53 virtual power plant contracts, Brent index formula on historical values, 117 constraint of VPP contract, 110 contract price, 107 deterministic problem, 107 energy and gas spot price Monte Carlo simulations, 112 flat hourly curves used for intrinsic valuation, 111 forward term structure used in example, 111 Gaussian stochastic model, 112 intrinsic valuation, 110 optimizing flexibility, 115 payoff function, 106 power generation assets, 105 spot price, 118 state space, online/offline, 108 state transitions, 108 state transitions, given operational constraints, 109 technical point of view, valuation problem, weak convergence, 58