Index
George Levy
(RWE npower, UK)
ISBN: 978-1-78743-528-5, eISBN: 978-1-78743-527-8
Publication date: 10 December 2018
This content is currently only available as a PDF
Citation
Levy, G. (2018), "Index", Energy Power Risk, Emerald Publishing Limited, Leeds, pp. 321-326. https://doi.org/10.1108/978-1-78743-527-820181011
Publisher
:Emerald Publishing Limited
Copyright © 2019 George Levy
INDEX
American options
, 178–182
for pricing
, 255–256
American/European vanilla options
, 120
Analytic option pricing formulae
, 105–113
Annualized standard deviation
, 63
APX power exchange (APX)
, 192, 199
Arithmetic progressions
, 293
Arrow–Debreu prices
, 143
Basic Linear Algebra Subprograms (BLAS)
, 245
Basket options. See Multi-asset options
Battery storage
, 214–217
BEGKR method
, 169
Benchmark portfolio
, 234–244
Binomial distribution
, 284–289
for large n and low probability p
, 285–286
for large number
, 286–288
normalized binomial variates
, 288–289
Binomial expansion
, 294–295
Binomial lattice
, 115
techniques
, 115
Binomial model
, 171
Bivariate cumulative normal
, 174
Black–Scholes equation
, 146, 259
Black–Scholes model
, 47
formula
, 52
Girsanov’s theorem
, 53
Greeks
, 60–61, 62
historical and implied volatility
, 61–67
inclusion of continuous dividends
, 57–60
Microsoft Excel
, 67–70
multi-asset option pricing partial differential equation
, 50–52
option pricing partial differential equation
, 47–50
probability density function
, 54
univariate cumulative normal function
, 55
Black–Scholes pricing framework
, 163
Brownian Bridge
, 15–17
Brownian motion
, 1, 3
model of asset price movements
, 5
with one source of randomness
, 11–12
properties
, 3–5
C++
, 1
classes
, 245
option pricing
, 254–256
random number class
, 250–254
risk percentiles
, 256–258
vector class
, 245–250
Call options on maximum and minimum of two assets
, 174
Central Limit Theorem
, 265–266
Confidence Intervals
, 267–268
Continuous dividends
, 46–47
inclusion
, 57–60
Contract risk distributions
, 198–200
Contract valuation
, 206–209
Correlation coefficient
, 164
Correlation matrix
, 9, 83, 223–224, 272
Covariance
, 270–272
covariance of n variables
, 271–272
three variables
, 271
two variables
, 270–271
Cox, Ross, and Rubinstein binomial lattice (CRR lattice)
, 118, 169
Crank–Nicolson method
, 150
Cumulative distribution function (CDF)
, 22, 39, 68, 193, 282, 289
Cumulative normal distribution function
, 292–293
Delta function
, 60, 126–127, 261
Demand side response (DSR)
, 2
Dependent variables
, 165
Doubly truncated Weibull distribution
, 98–102, 202–203
Downward branching node
, 139, 142–143, 145
Dual cash-out price method
, 190–191
Early exercise
, 155–159
Efficient frontier
, 225, 226
with no transaction costs
, 225–233
with transaction costs and benchmark portfolio
, 234–244
Empirical CDFs
, 193, 289–290
European options
, 173
call options on maximum and minimum of two assets
, 174
for pricing
, 254–255
put options on maximum of two assets
, 175–178
put options on minimum of two assets
, 174–175
Excel Visual Basic code
, 223
Exponential distribution
, 34, 281–282
Fast response generation units
, 206, 210
Financial risk distributions
, 189
Finite difference
approximation
, 146–150
lattices
, 169
method
, 160–162
Forward price
, 76–77
Four-state Markov Chain
, 195
Fubini’s theorem
, 22–23, 297
Fully implicit method
, 149–150
Gamma function
, 60, 95, 127, 260–261
Gaussian Copula
, 194
Gaussian distribution
, 7, 16
GBP/EUR exchange rate
, 34
GBP/USD exchange rate
, 34
Geometric Brownian motion (GBM)
, 1, 5, 47, 115
Geometric progressions
, 293
Girsanov’s theorem
, 8
Greeks
, 60–61, 125–129
Delta
, 261
European call
, 62
European put
, 62
Gamma
, 260–261
Rho
, 263
Theta
, 262–263
for Vanilla European options
, 259–260
Vega
, 263–264
Grid methods for Vanilla options
, 145
log-transformed grids
, 159–162
standard grids
, 146–159
stochastic process
, 145
Half hourly marginal system price benefit
, 194
Half hourly power price
, 42–44
simulation
, 36
Historic portfolio volume forecast error distribution
, 192–193
Imbalance risk
, 189
benefit of including customer into portfolio
, 194–195
contract risk distributions
, 198–200
contracted energy
, 190–191
MIP
, 195–197
risk distribution
, 192–193
SBP and SSP
, 197–198
single cash-out price
, 191–192
stand-alone cost
, 193–194
Independent, identically distributed random variables (IID random variables)
, 265, 269–270, 273
IID lognormal distribution
, 102
Intraday generation
, 217–221
Intraday power storage and demand optionality
, 210
battery storage
, 214–217
import site with storage and solar PV
, 212–213
storage connected to grid with/without solar PV generation
, 211–212
swing contract
, 213–214
Ito product rule
, 11
Brownian Motion with one source of randomness
, 11–12
in n dimensions
, 14–15
Ito quotient rule
, 12–14
Ito’s formula. See Ito’s lemma
Ito’s isometry–correlated processes
, 25–27
Ito’s isometry–single process
, 23–25
Ito’s lemma
, 5–8, 297
for multi-asset geometric Brownian motion
, 9–11
Johnson binomial lattice
, 133–137
Johnson distribution
, 84
option pricing formula
, 84–88
parameter estimation
, 89–93
Jump diffusion process
, 103
Kalman filter
, 82
Lattice methods
, 145
constructing and using standard binomial lattice
, 121–129
Johnson binomial lattice
, 133–137
log transformed binomial lattice
, 129–133
standard binomial lattice
, 115–121
trinomial lattice
, 137–145
for Vanilla options
, 115
Least squares Monte Carlo optimization
, 211–212, 213
Lithium ion batteries
, 210
Load management
, 35
Log transformed binomial lattice
, 129–133
Log-transformed grids
, 159
derivation of equation
, 159–160
finite difference method
, 160–162
see also Standard grids
Lognormal distribution
, 117, 280–281
Longstaff Schwartz regression approach
, 2
see also Least squares Monte Carlo optimization
Marginal power price
, 33
Market index price (MIP)
, 190, 195–197
Markov process
, 3
Markowitz efficient frontier
, 225
Markowitz mean–variance portfolio selection problem
, 223
Markowitz portfolio optimization
, 1
Martingale measure, pricing derivatives using
, 45–46
Martingale process
, 3
Mathematical reference
arithmetic and geometric progressions
, 293
cumulative normal distribution function
, 292–293
series expansions
, 294–295
standard integrals
, 291–292
Mersenne Twister uniform random number generator
, 250, 252
Merton jump diffusion model
, 102, 255
analytic option pricing formulae
, 105–113
jump diffusion process
, 103
Monte Carlo simulation
, 104–105
parameter estimation
, 113–114
Microsoft Excel
, 67–70
Microsoft Excel VBA code
, 1, 90–93, 223, 230, 240–242
Mixed integer linear programming (MILP)
, 211
Moment generating functions
, 272–273
Monte Carlo
fundamental power stack model
, 2
lattice approach
, 165
methods
, 163
simulation
, 33, 104–105, 192, 250
Multi-asset
Black–Scholes equation
, 163–164
derivative
, 50
geometric Brownian motion
, 1, 9–11
Multi-asset options
, 165
multidimensional lattice methods
, 163, 169–171
multidimensional Monte Carlo methods
, 165–169
pricing partial differential equation
, 50–52
three asset options
, 183–187
two asset options
, 171–182
see also Single asset American style options; Single asset European options
Multifactor forward curve model
, 82–83
N2EX
, 192
National Grid
, 210
Newton’s method
, 64, 66
Normal (Gaussian) distribution
, 277–280
Normal cumulative distribution
, 67
Normalized binomial variates
, 133, 288–289
NORMDIST function
, 67
Numeraires
, 45
Object-based programming languages
, 248
Objective Function
, 1, 230, 233, 244
One factor forward curve model
, 72, 137
exponential factor
, 72
forward price and spot price
, 76–77
option pricing formula
, 77–80
spot price process
, 73–76
One-factor spot model
, 70–72
Option payoff at terminal nodes
, 123–124
Option pricing
, 254–256
American options
, 255–256
European options
, 254–255
formula
, 77–80, 84–88
partial differential equation
, 47–50
Option values computation at given time instant
, 151–155
Ornstein–Uhlenbeck process
, 1, 17
mean
, 18
variance
, 19–21
Payoff
, 45, 46, 54, 123–124, 172, 213
Poisson distribution
, 282–283
Poisson process
, 21–22, 34, 102
Poisson random number generator functions
, 253
Portfolio optimization
, 224
covariance matrix
, 223–224
efficient frontier with no transaction costs
, 225–233
efficient frontier with transaction costs and benchmark portfolio
, 234–244
optimum asset allocation
, 224–225
Portfolio volume forecast error distributions
, 194
Power contracts
imbalance risk
, 189–200
intraday generation
, 217–221
intraday power storage and demand optionality
, 210–217
wind contracts
, 200–209
Power price model
modeling wind and solar generation
, 36–42
power stack model
, 33–36
simulated half hourly power price
, 42–44
Power spot price, stochastic process for
, 205–206
Power stack model
, 33–36
Price European exchange options
, 171
Probability
density function
, 23, 54, 58, 275, 277
distribution
, 15
measure
, 53
Problems
, 27–32
answers to
, 297–313
Put options
on maximum of two assets
, 175–178
on minimum of two assets
, 174–175
Put–call parity
, 46–47
Radon–Nikodym derivative
, 8
Random number class
, 250–254
Random walk
, 3
Reasonable approximation
, 119
Relative contract maturities
, 83
Renewable energy
, 2
generators
, 189
Risk percentiles
, 256–258
Simulation
, 204–205
Monte Carlo simulation
, 33, 104–105, 192, 250
Single asset American style options
Grid methods for Vanilla options
, 145
lattice methods for Vanilla options
, 115–145
see also Multi-asset options
Single asset European options
Johnson distribution
, 84–93
Merton jump diffusion model
, 102–114
multifactor forward curve model
, 82–83
one factor forward curve model
, 72–80
one-factor spot model
, 70–72
pricing derivatives using martingale measure
, 45–46
put–call parity
, 46–47
two-factor spot model
, 81–82
Vanilla options and Black–Scholes model
, 47–70
Weibull distribution
, 93–102
see also Multi-asset options
Single cash-out price method
, 190–192
Single-factor forward curve model
, 82
Solar generation
, 35, 36
actual half hourly UK summer solar PV generation
, 41
actual half hourly UK winter solar PV generation
, 42
current and previous daily average UK solar PV generation
, 38
current and previous half hour UK solar PV generation
, 38
simulated half hourly UK summer solar PV generation
, 41
simulated half hourly UK winter solar PV generation
, 42
Solar PV generation
import site with storage and
, 212–213
storage connected to grid with/without
, 211–212
Spot price process
, 73–77
Standard Binomial lattice
, 115
asset values to lattice nodes
, 122–123
computing Greeks
, 125–129
constructing and using
, 121
iterate backwards through lattice
, 124–125
lognormal mean
, 116
lognormal variance
, 116–121
option payoff at terminal nodes
, 123–124
values of constants by lattice
, 122
Standard Brownian motion
, 4
Standard grids
, 146
backwards iteration and early exercise
, 155–159
boundary conditions
, 150–151
computation of option values at given time instant
, 151–155
finite difference approximation
, 146–150
see also Log-transformed grids
Standard integrals
, 291–292
Standard normal distribution
, 4
Standard statistical results
, 265–273
Central Limit Theorem
, 265–266
Confidence Intervals
, 267–268
covariance
, 270–272
covariance matrix
, 272
law of large numbers
, 265
moment generating functions
, 272–273
variance
, 268–270
Standard Template Library (STL)
, 245
Standard Weibull distribution
, 93–98
Statistical distribution functions
Binomial distribution
, 284–289
empirical CDF
, 289–290
exponential distribution
, 281–282
lognormal distribution
, 280–281
normal (Gaussian) distribution
, 277–280
Poisson distribution
, 282–283
uniform distribution
, 275–277
Stochastic integral expectation
, 27
Stochastic integrals
, 22
expectation of stochastic integral
, 27
Fubini’s theorem
, 22–23
Ito’s isometry–correlated processes
, 25–27
Ito’s isometry–single process
, 23–25
Stochastic processes
, 1, 8, 145
for power spot price
, 205–206
Swing contract
, 213–214
System buy price (SBP)
, 190, 197–198
System long
, 190
System price
, 189
System sell price (SSP)
, 190, 197–198
System short
, 190
Taylor Series
, 294
Terminal nodes, option payoff at
, 123–124
Tesla
, 210
Three asset options
, 183–187
Tidal power
, 2
Time varying drift and volatility
, 8
Time-varying mean
, 76
Trinomial lattice
, 137–145
branching types for nodes in trinomial lattice
, 139
downward branching node
, 142–143, 145
mean reverting trinomial lattice
, 138
normal branching node
, 140–141, 145
pricing using lattice
, 144
upward branching node
, 141–142, 144–145
Two asset options
, 171
American options
, 178–182
European exchange options
, 171–173
European options on maximum or minimum
, 173–178
Two dimensions, Ito product and quotient rules in
, 11–14
Two-factor spot model
, 81–82
Uniform distribution
, 275–277
Uniform grid
, 156
Univariate cumulative normal function
, 55
Value at risk (VAR)
, 192
Vanilla European options, Greeks for
, 259–264
Vanilla options
, 47–70
Vanilla options, grid methods for
, 145
log-transformed grids
, 159–162
standard grids
, 146–159
stochastic process
, 145
Vanilla options, lattice methods for
, 115
constructing and using standard binomial lattice
, 121–129
Johnson binomial lattice
, 133–137
log transformed binomial lattice
, 129–133
standard binomial lattice
, 115–121
trinomial lattice
, 137–145
Variance
, 268
of n variables
, 269–270
one variable
, 268
Ornstein–Uhlenbeck process
, 19–21
three variables
, 269
two variables
, 268–269
Vector class
, 245–250
Vega function
, 60, 128–129, 263–264
Visual Basic
, 67
Volatility
, 5
historical
, 61–64
implied
, 61, 64–67
smile
, 64
Weibull distribution
, 93, 202–204
doubly truncated Weibull distribution
, 98–102
standard Weibull distribution
, 93–98
Wind contracts
, 200
contract valuation
, 206–209
simulation and calibration
, 204–205
stochastic process for power spot price
, 205–206
Weibull distribution
, 202–204
Wind generation
, 35–36
actual half hourly UK
, 39
actual half hourly UK summer
, 40
actual half hourly UK winter
, 40
current and previous day wind generation load factors
, 37
current and previous half hour wind generation load factors
, 37
simulated half hourly UK
, 40
- Prelims
- Chapter 2 Overview
- Chapter 2 Brownian Motion and Stochastic Processes
- Chapter 3 Fundamental Power Price Model
- Chapter 4 Single Asset European Options
- Chapter 5 Single Asset American Style Options
- Chapter 6 Multi-asset Options
- Chapter 7 Power Contracts
- Chapter 8 Portfolio Optimization
- Chapter 9 Example C++ Classes
- Appendix A The Greeks for Vanilla European Options
- Appendix B Standard Statistical Results
- Appendix C Statistical Distribution Functions
- Appendix D Mathematical Reference
- Appendix E Answers to Problems
- References
- Index