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Prelims

George Levy (RWE npower, UK)

Energy Power Risk

ISBN: 978-1-78743-528-5, eISBN: 978-1-78743-527-8

Publication date: 10 December 2018

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Citation

Levy, G. (2018), "Prelims", Energy Power Risk, Emerald Publishing Limited, Leeds, pp. i-xvii. https://doi.org/10.1108/978-1-78743-527-820181012

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Emerald Publishing Limited

Copyright © 2019 George Levy

Half Title Page

ENERGY POWER RISK

Title Page

ENERGY POWER RISK: DERIVATIVES, COMPUTATION AND OPTIMIZATION

GEORGE LEVY

RWE npower, UK

United Kingdom – North America – Japan – India – Malaysia – China

Copyright Page

Emerald Publishing Limited

Howard House, Wagon Lane, Bingley BD16 1WA, UK

First edition 2019

Copyright © 2019 George Levy.

Published under exclusive license.

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No part of this book may be reproduced, stored in a retrieval system, transmitted in any form or by any means electronic, mechanical, photocopying, recording or otherwise without either the prior written permission of the publisher or a licence permitting restricted copying issued in the UK by The Copyright Licensing Agency and in the USA by The Copyright Clearance Center. Any opinions expressed in the chapters are those of the authors. Whilst Emerald makes every effort to ensure the quality and accuracy of its content, Emerald makes no representation implied or otherwise, as to the chapters’ suitability and application and disclaims any warranties, express or implied, to their use.

British Library Cataloguing in Publication Data

A catalogue record for this book is available from the British Library

ISBN: 978-1-78743-528-5 (Print)

ISBN: 978-1-78743-527-8 (Online)

ISBN: 978-1-78743-956-6 (Epub)

Dedication

To Kathy, Matthew, Claire and Rachel

List of Figures

Chapter 3
Figure 3.1 Scatter Plot of Current and Previous Day Wind Generation Load Factors. 37
Figure 3.2 Scatter Plot of Current and Previous Half Hour Wind Generation Load Factors. 37
Figure 3.3 Scatter Plot of Current and Previous Daily Average UK Solar PV Generation in MW During July 2016. 38
Figure 3.4 Scatter Plot of Current and Previous Half Hour UK Solar PV Generation in MW During July 2016. 38
Figure 3.5 Actual Half Hourly UK Wind Generation. 39
Figure 3.6 Simulated Half Hourly UK Wind Generation. 40
Figure 3.7 Actual Half Hourly UK Summer Wind Generation, for One Week. 40
Figure 3.8 Actual Half Hourly UK Winter Wind Generation, for One Week. 40
Figure 3.9 Actual Half Hourly UK Summer Solar PV Generation, for One Week. 41
Figure 3.10 Simulated Half Hourly UK Summer Solar PV Generation, for One Week. 41
Figure 3.11 Actual Half Hourly UK Winter Solar PV Generation, for One Week. 42
Figure 3.12 Simulated Half Hourly UK Winter Solar PV Generation, for One Week. 42
Figure 3.13 Actual UK Solar PV Generation and APX Prices for a Day in August 2016. 43
Figure 3.14 Simulated Summer Power Price Distribution from the Power Fundamentals Model With 1,000 Scenarios. 43
Figure 3.15 Actual UK Solar PV Generation and APX Prices for a Day in December 2016. 44
Figure 3.16 Simulated Winter Power Price Distribution from the Power Fundamentals Model with 1,000 Scenarios. 44
Chapter 4
Figure 4.1 Using the Function bs_opt Interactively Within Excel. 69
Figure 4.2 Excel Worksheet Before Calculation of the European Option Values. 70
Figure 4.3 Excel Worksheet After Calculation of the European Option Values. 71
Figure 4.4 The Johnson Distribution Parameter Estimates Obtained Using the VBA in Code Excerpts 4.7–4.11. 93
Chapter 5
Figure 5.1 A Standard Binomial Lattice Consisting of Five Time Steps. 121
Figure 5.2 The Error in the Estimated Value, est _val, of an American Put Using a Standard Binomial Lattice. 129
Figure 5.3 Mean Reverting Trinomial Lattice. 138
Figure 5.4 Branching Types for Nodes in the Trinomial Lattice: Normal Branching (i), Upward Branching (ii), and Downward Branching (iii). 139
Figure 5.5 An Example of Uniform Grid, Which Could be Used to Estimate the Value of a Vanilla Option Which Matures in Two Years’ Time. 156
Chapter 7
Figure 7.1 Half Hour Daily Electricity Consumption Profiles for a Low-forecast Error Import Customer. 189
Figure 7.2 Half Hour Daily Electricity Generation Profiles for a (High-forecast Error) Wind-generation Site. 190
Figure 7.3 Shaped Half Hourly Within Day Price. 196
Figure 7.4 Linear Regression Spread Against MIP. 197
Figure 7.5 Regression of Spread Against MIP and Previous Spread. 197
Figure 7.6 Customer Risk Against Correlation, £300 SBP Cap. 198
Figure 7.7 Customer Risk Against Correlation, £150 SBP Cap. 199
Figure 7.8 Customer Risk Against Correlation, £100 SBP Cap. 199
Figure 7.9 Risk Distributions for Stand-alone Customer, and Portfolio Correlations of 10 and 30 Percent. 200
Figure 7.10 Simulated Risk Probability Distributions When Reference Price is APX, and Fixed (Forward Curve). 200
Figure 7.11 Percentile Risks with Reference Price Fixed. 201
Figure 7.12 Percentile Risks with Reference Price APX. 201
Figure 7.13 Probability Density Function for Half Hour Load Factors for an Example Wind Farm. 203
Figure 7.14 Regression Coefficients and p-Values for Winter 2014 Weekdays. 205
Figure 7.15 Call Options, Value per Half Hour. 208
Figure 7.16 Put Options, Value per Half Hour. 208
Figure 7.17 Collar Percentile Values for a CFD Wind Contract. 209
Figure 7.18 Annual Value of a 1-MW h Swing Contract with Two Upswing and Two Downswing Rights per Day on the Half Hourly Spot Price, £/MW h. 213
Figure 7.19 Annual Value of a 1-MW h Swing Contract with Two Upswing and Two Downswing Rights per Day on the Hourly Average Spot Price, £/MW h. 214
Figure 7.20 Annual Value of a 1-MW h Swing Contract with Two Upswing and Two Downswing Rights per Day on the Three Hourly Average Spot Price, £/MW h. 214
Figure 7.21 Annual Value ofa1 MWh Battery with Daily Optimization Using Mixed Integer Linear Programming (MILP) and Least Squares Monte Carlo (Longstaff–Schwartz). 215
Figure 7.22 Annual Value ofa1 MWh Battery with Daily Optimization Using Mixed Integer Linear Programming (MILP) and Least Squares Monte Carlo (Longstaff– Schwartz). 215
Figure 7.23 A Scenario Showing the Simulated Intraday Power Price and the Battery Storage Levels. 216
Figure 7.24 An Example of a Typical Intraday Simulated Scenario Showing the Power Price, Solar PV Generation, and the Battery Storage Level. 217
Figure 7.25 The Annual Value ofa1 MWh Battery Installed on an Import Site with PV Generation, and a Bid Offer Spread of £100/MW h. 218
Figure 7.26 Intraday Generation Schedule for a Single Monte Carlo Scenario Without Startup or Shutdown Costs. 220
Figure 7.27 Intraday Generation Schedule for the Same Monte Carlo Scenario as in Figure 7.26, but with Startup and Shutdown Costs. 221
Figure 7.28 Intraday Generation Schedule with Startup and Shutdown Costs. 221
Chapter 8
Figure 8.1 The Portfolio Data in the Excel Worksheet Markowitz_Example_Data. 229
Figure 8.2 The Computed Efficient Frontier, Transaction Costs Are Not Included, and Short Selling Is Not Allowed. 230
Figure 8.3 The Portfolio Data in the Excel Worksheet Markowitz_Example_Data. 239
Figure 8.4 The Computed Efficient Frontier, for a Benchmark Portfolio with Proportional Transaction Costs. Short Selling is Allowed. 239

List of Tables

Chapter 4
Table 4.1 European Put: Option Values and Greeks. The Parameters Are S = 100.0, K = 100.0, r = 0.10, σ = 0.30, q = 0.06. 62
Table 4.2 European Call: Option Values and Greeks. The Parameters Are S = 100.0, K = 100.0, r = 0.10, σ = 0.30, q = 0.06. 62
Table 4.3 Calculated Option Values and Implied Volatilities from Code Excerpt 4.4. 67
Chapter 5
Table 5.1 Lattice Node Values in the Vicinity of the Root Node R. 126
Table 5.2 Valuation Results and Pricing Errors for a Vanilla American Put Option Using a Uniform Grid With and Without a Logarithmic Transformation; the Implicit Method and Crank–Nicolson Method Are Used. 162
Chapter 6
Table 6.1 The Computed Values and Absolute Errors, in Brackets, for European Options on the Maximum of Three Assets. 168
Table 6.2 The Computed Values and Absolute Errors, in Brackets, for European Options on the Minimum of Three Assets. 168
Table 6.3 The Computed Values and Absolute Errors for European Put and Call Options on the Maximum of Two Assets. 179
Table 6.4 The Computed Values and Absolute Errors for European Put and Call Options on the Minimum of Two Assets. 179
Table 6.5 The Computed Values and Absolute Errors for European Options on the Maximum of Three Assets. 186
Table 6.6 The Computed Values and Absolute Errors for European Options on the Minimum of Three Assets. 186
Table 6.7 The Computed Values and Absolute Errors for European Options on the Maximum of Three Assets. 187
Table 6.8 The Computed Values and Absolute Errors for European Options on the Minimum of Three Assets. 187
Chapter 7
Table 7.1 Table Showing the Relationship Between the Binary Variables in the MILP. 220
Chapter 8
Table 8.1 The Asset Standard Deviations and Average Returns. 228
Table 8.2 The Correlation Matrix for the Assets. 228
Table 8.3 The Covariance Matrix for the Assets. 229

Notations

The notation used is as follows:

GB M

Geometric Brownian motion

BM

Brownian motion

Wt

Brownian motion at time t – the term may be nonzero

ρ

The correlation coefficient

E[x]

The expectation value of X

Var[X]

The variance of X

Cov[X, Y ]

The covariance between X and Y

Cov[X]

The covariance between the variates contained in the vector X

σ

The volatility. Since assets are assumed to follow GBM, it is com- puted as the annualized standard deviation of the n continuously compounded returns

N 1 (a)

The univariate cumulative normal distribution function. It gives the cumulative probability, in a standardized univariate normal distribution, that the variable x 1 satisfied x 1 ≤ a

N 2 (a, b, ρ)

The bivariate cumulative normal distribution. It gives the cumu- lative probability, in a standardized bivariate normal distribution, that the variables x 1 and x 2 satisfy x 1 ≤ a and x 2 ≤ b when with correlation coefficient between x 1 and x 2 is ρ

r

The risk free interest rate

q

The continuously compounded dividend yield

Sit

The ith asset price at time t

Inn

The n by n unit matrix

A(μ, σ 2)

A lognormal distribution with parameters μ and σ 2 . If y = log (x) and y ∼ N (μ, σ 2), then the distribution for x = e y is x ∼ A(μ, σ 2 ). We have E[x] = exp (μ + (σ 2 /2)) and V ar [x] = exp (2μ + σ 2) (exp (σ 2) − 1)

log (x)

The natural logarithm of x

N (a, b)

Normal distribution, with mean a and variance b

dWt

A normal variate (sampled at time t ) from the distribution N (0, dt ), where dt specified time interval e.g., dx = μd t + dWt

dZ t

A normal variate (sampled at time t ) from the distribution N (0, 1). Note: The variate d ψ = d t d Z t has the same distribution as dWt

IID

Independently and identically distributed

U (a, b)

The uniform distribution, with lower limit a and upper limit b

|x|

The absolute value of the variable x

P DF

The probability density function of a given distribution

x ∧ y

The minimum of x and y, that is, min (x, y)

Preface

The aim of this book is to provide readers with sufficient knowledge to under- stand and create quantitative models for energy/power risk and derivative valuations. The topics covered include the mathematics of stochastic processes, assets optimization, Markowitz portfolio optimization, derivative valuation, and financial engineering using C++. One could write a separate book on each of these subjects, and therefore of necessity their coverage in this book could be considered to be an introduction. However, I trust that readers will find the book a useful reference and building block for their projects.

I would like to thank my wife Kathy for her support, and also my daughter Rachel for her expert help in creating some of the figures.

In addition, I am grateful to Pete Baker and Katy Mathers of Emerald

Publishing Ltd. for all their hard work, patience, and help with the book.

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