Mat-2.108 Independent Research Project in Applied Mathematics

Analytical delta-gamma VaR methods for portfolios of electricity derivatives

2 October 2004

Helsinki University of Technology

Department of Engineering Physics and Mathematics

Systems Analysis Laboratory

Timo Javanainen

51625C

Department of Engineering Physics and Mathematics

timo.javanainen@hut.fi

Table of contents

1. Introduction................................................................................................................. 1

2. Theoretical background .............................................................................................. 2

2.1. Definition of Value at Risk ................................................................................. 2

2.2. Linear model ....................................................................................................... 3

2.3. Quadratic model.................................................................................................. 5

2.3.1. Moment matching approaches .................................................................... 7

2.3.2. Cornish Fisher expansion............................................................................ 8

2.3.3. Moment generating functions and Fourier Transforms .............................. 9

2.3.4. Comparison of methods ............................................................................ 13

2.4. Backtesting........................................................................................................ 15

3. Applying analytical delta-gamma methods in Nord Pool’s financial market........... 17

3.1. Nord Pool’s financial market ............................................................................ 17

3.2. Backtesting set up ............................................................................................. 19

3.3. Backtesting results ............................................................................................ 20

4. Conclusions............................................................................................................... 22

References......................................................................................................................... 24

1. Introduction

Value at Risk (VaR) is the most widely used tool to measure and control market risk.

VaR quantifies the maximum loss over a given period of time with some statistical

confidence level. In other words, VaR figure is a quantile of the portfolio profit and loss

distribution. VaR was introduced and popularized in 1994 by J.P. Morgan's RiskMetrics

software and since then VaR has become a standard concept in risk management. (Pichler

and Selitch 1999; Jorion 2001.)

Although the basic concept of VaR is simple, the ways of implementing VaR calculators

differ greatly. VaR methods can be divided into analytical and simulation methods.

Analytical methods include methods like delta and delta-gamma VaR. The method

introduced by J.P. Morgan's RiskMetrics (often called delta-normal VaR) is linear and is

insufficient to measure the risk of portfolios consisting of nonlinear instruments, like

options. Delta-gamma methods (often called quadratic methods) try to tackle the problem

of nonlinear payoffs by using a quadratic approximation of the portfolio profit and loss

distribution. Simulation based VaR methods include full Monte Carlo, partial (or delta-

gamma) Monte Carlo and historical simulation.

A VaR estimate is often calculated on a daily basis. In these situations, computation time

is crucial and thus, simulation methods are out of question. Quadratic VaR methods have

proven to give accurate results in many cases (Zangari 1996b; Mina and Ulmer 1999) and

are straightforward to implement.

In this paper analytical quadratic VaR methodologies - such as delta-gamma-normal and

delta-gamma approximation using Cornish Fisher expansion and Fast Fourier

Transform - are described in detail. These methods are then evaluated by a backtesting

procedure on two portfolios consisting of Nord Pool traded electricity derivatives. The

study is conducted for Fortum Power and Heat Oy.1

1 Fortum Power and Heat Oy, http://www.fortum.com

1

The outline of this paper is as follows. In Chapter 2 the theoretical framework is

provided. The concept of VaR and the linear model are briefly discussed in Chapters 2.1

and 2.2. This is followed by discussion on the quadratic model and different quadratic

methods in Chapter 2.3 and VaR backtesting theory in Chapter 2.4. Chapter 3 deals with

the practical problem setting of applying analytical delta-gamma VaR methods. Nord

Pool’s financial market and its features are discussed in Chapter 3.1. The main results of

this paper, i.e. backtesting set up and results with two imaginary electricity trading

portfolios, are described in Chapters 3.2 and 3.3. Chapter 4 gives the conclusions.

2. Theoretical background

2.1. Definition of Value at Risk

Let the V denote the (random) market value of a portfolio after selected time period T.

The absolute VaR with confidence level α can then be defined by (Jorion 2001)

, (1) ∫∞−

=VaR

dxxf )(α

where f(∆V) is the profit and loss distribution of the portfolio over time T. Equivalently,

one can use the inverse of the cumulative density function F(∆V) to calculate the absolute

VaR

) . (2) (1 α−= FVaR

Time period T, known as the holding period, can be defined as the period required for

liquidation or the "normal" holding period for assets in the portfolio. Confidence level α

is usually 0.01 or 0.05 reflecting probability levels of 99 % and 95 %.

Usually it is more informative to relate the maximum possible change in portfolio value

to its expected value, E(∆V). For this purpose, relative VaR can be defined (Jorion 2001)

. (3) )()()( absoluteVaRVErelativeVaR −∆=

2

In essence, the interest in estimating VaR lies in the α-quantile of the portfolio profit and

loss distribution. This is illustrated in Figure 1.

Figure 1. Definition of VaR. Absolute VaR is the α-quantile of the portfolio profit and loss distribution (left) and can be equivalently calculated using the inverse of the cumulative density function (right).

2.2. Linear model

The delta-normal approach originally introduced by J.P. Morgan's RiskMetrics software

is based on two important assumptions (J.P. Morgan 1996):

1. Linearity: The change in the value of the portfolio over a given interval of time is

linear in the returns of N < ∞ risk factors.

2. Normality: The returns of the risk factors1 follow a multivariate normal

distribution.

1 Risk (or market) factors used in a VaR methodology can be e.g. interest rates, foreign exchange rates or underlying asset prices.

3

Define the logarithmic return1 of the ith risk factor on day t as

)ln(1,

,,

−

=ti

titi S

Sr , (4)

where Si,t and Si,t-1 denote the value of the risk factor on day t and on day t-1. Based on

the first assumption (linearity), change in the market value of a portfolio over one period,

∆V, can be written as (omitting the time symbol t)

∑=

=∆N

iii rV

1δ , i

ii S

SV

∂∂

=δ , (5)

where δi denotes sensitivity of the portfolio value with respect to ith risk factor.2 This can

be written in matrix form as

, (6) RV Tδ=∆

where δ is N × 1 vector of factor sensitivities and R is N × 1 vector of factor returns.

The second assumption (normality) can be formalized as R ~ N(µ, Σ), where µ is N × 1

vector of expected factor returns and Σ is the estimated N × N covariance matrix for

holding period T. Since the holding period is usually small, it is often assumed that µ=0.

This assumption is made in this paper as well.

As the linear combination of normal variables is normal, it follows from the assumptions

of linearity and normality that the distribution of ∆V is itself normal. Therefore,

∆V ~ N(0, δTΣδ) and VaR can be easily estimated as the α-quantile of the normal

distribution (Jorion 2001)

Vdelta zVaR ∆= σα , (7)

1 Instead of logarithmic returns, arithmetic returns rit = (Si,t-Si,t-1) / Si,t-1 can be used as well when defining factor returns. By using a Taylor series expansion it can be shown that for small rit both expressions are approximately equal. All calculations in this paper are based on logarithmic returns. 2 The delta of a portfolio respect to the ith risk factor is usually defined as δi = ∂V/∂Si. Delta defined in ( ) is often called return adjusted delta. The same applies to gamma defined in Chapter 2.3.

5

4

where zα denotes the α-quantile of standard normal distribution and σ∆V denotes the

standard deviation of ∆V. VaR methods based on the linear relationship (6) between risk

factor returns and change in the market value of a portfolio are generally called delta

methods. If normality of factor returns is assumed - and thus, normal distribution is used

in calculating the desired quantile of portfolio profit and loss distribution - the methods

are generally called delta-normal.

2.3. Quadratic model

The linearity assumption is crucial for guaranteeing the normality of the distribution of

∆V. If the relationship between factor returns and change in the market value of a

portfolio is nonlinear, the distribution of ∆V will generally be non-normal. Numerical

examples with portfolios containing positions in one single option show that distribution

of ∆V can show extreme skewness and curtosis. This makes the linearity assumption and

the use of delta-normal method questionable with portfolios including options. (Pichler

and Selitsch 1999.)

The gamma terms of a portfolio can be defined as (Hull 2000)

ji

ji SSV∂∂

∂=Γ

2

, . (8)

These gamma terms, i.e. the rate of change of the ith delta with respect to the jth

underlying risk factor, are neglected in the linear model (Hull 2000).

Consider a portfolio consisting of derivatives on one risk factor. The profit and loss

distribution of a portfolio with positive gamma tends to be positively skewed and of a

portfolio with negative gamma negatively skewed. As a result, using linear model and

assuming the distribution to be normal when estimating VaR will probably result in too

high (low) VaR figure in the case of portfolios with positive (negative) gamma. (Hull

2000.)

5

In quadratic VaR methods1 change in the market value of a portfolio is approximated by

a second order Taylor series expansion instead of a first order expansion used in (6). This

results in equation

RRRV TT Γ+=∆21δ , (9)

where Γ denotes the N × N return adjusted gamma matrix with terms

jiji

ji SSSS

V∂∂

∂=Γ

2

, . (10)

Equation (9) takes into account the deltas and gammas of instruments and is thus

reasonable to be used in the case of nonlinear instruments.2 Nevertheless, it should be

kept in mind that the approximation behind equation (9) is only local, and thus, if the

value of a portfolio is not a smooth and continuous function of the underlying risk

factors, the approximation can provide inaccurate results (Mina and Ulmer 1999). This is

also the case, when VaR is estimated for a relatively long holding period and for a market

where the daily volatility is high (Liu 2000).

Estimating VaR based on quadratic approximation (9) requires calculating the desired

quantile of the distribution of ∆V. As the relation in (9) is nonlinear, the distribution of

∆V is no longer normal and the desired quantile cannot be directly calculated in closed

form. Nevertheless, the moments of the distribution can be calculated. For details on

calculating the moments of the distribution, see Mathai and Provost (1992).

Given the portfolio structure (δ and Γ) and normal distribution of factor returns (Σ, µ=0),

the expected value and variance of ∆V can be calculated as

[ΓΣ=∆ trVE21)( ] [ ]2

21)( ΓΣ+Σ=∆ trVVar T δδ , (11)

1 Quadratic VaR methods are also called delta-gamma methods. In this paper these names are used interchangeably. 2 Even the theta of portfolio of derivatives, θ, i.e. the rate of change in portfolio value with respect to the passage of time, can be included in the model. This would imply addition of term θ∆t in equation ( ). 9

6

where tr denotes the trace of a matrix. Defining a new variable X as the standardized

value of ∆V

)()(

VVarVEVX

∆∆−∆

= , (12)

the higher moments of X (with r ≥ 3) can be calculated by

[ ] [ ]

2

2

)(

)!1(21!

21

)( r

rrT

r

VVar

trrrXE

∆

ΓΣ−+ΓΣΣ=

− δδδ. (13)

Using r=3 and r=4 in (13) gives the skewness and curtosis of ∆V, respectively. (Pichler

and Selitch 1999.)

Three analytical approaches to calculate the desired quantile are well covered in the

literature (Pichler and Selitch 1999; Mina and Ulmer 1999). Two of these approaches are

based on the moments of distribution, i.e. moment matching approaches and direct

quantile approximation (Pichler and Selitch 1999). Moment matching approaches are

described in Chapter 2.3.1 below and direct quantile approximation with Cornish-Fisher

expansion is described in Chapter 2.3.2. The third approach, i.e. calculating the desired

quantile numerically with Fast Fourier Transform, is described in Chapter 2.3.3.

Several other analytical approaches have been proposed as well. Britten-Jones and

Schaefer (1997) propose the use of Solomon-Stephens approximation and Rogers and

Zane (1999) present the use of saddle point approximation to calculate the desired

quantile.

2.3.1. Moment matching approaches

In moment matching approaches the distribution of ∆V is approximated by finding a

distribution the quantiles of which can be calculated. Simplest form of moment matching

7

approach is to approximate the distribution of ∆V by a normal distribution with expected

value and variance given in (11). This method is often called delta-gamma-normal.1

Zangari (1996b) was first to present the use of Johnson transformation to match the

moments of quadratic form. He used the first four moments of ∆V to fit a member of the

Johnson family of distributions to approximate the original distribution. An algorithm to

fit a Johnson distribution given the first four moments can be found in (Hill, Hill and

Holder 1976). Based on their empirical studies, Mina and Ulmer (1999) conclude that the

use of Johnson transformation is not robust choice for implementing delta-gamma VaR.

Therefore, the Johnson transformation was not implemented in this study.

Currently there is no moment matching approach established in the financial literature to

take into account higher moments than curtosis (Pichler and Selitsch 1999).

2.3.2. Cornish Fisher expansion

Zangari (1996a) was the first to use the Cornish Fisher expansion to directly approximate

the desired quantile of ∆V. The idea is to express the quantile in terms of the known

cumulants of the standard normal distribution N(0,1). This approach leads to an analytic

approximation of the quantile as long as the moments of distribution are known. For

details on Cornish-Fisher expansion, see (Johnson and Kotz 1970).

For example, using the Cornish-Fisher expansion with 5 moments gives the α-quantile as

( ) ( )

( ) ( )

( ) ( ) 33424342

542233

4332

)(34241066481)()(12630

1441

)(36120

1)(52361

)(3241)(1

61~

XEzzXEXEzz

XEzzXEzz

XEzzXEzzz

++−+−−

+++−+−

+−+−+=α

(14)

1 The naming convention is not unambiguous. Also methods using delta-gamma approximation and assuming normality of factor returns are somewhere called delta-gamma-normal methods, instead of the convention used here.

8

where z is the α-quantile of standard normal distribution. Originally Zangari used the

Cornish Fisher expansion with four moments, but the method can be expanded to take

into account higher moments as well.

2.3.3. Moment generating functions and Fourier Transforms

Rouvinez (1997) showed that an exact formulation for the quadratic form could be found

using moment generating functions and Fourier inversion. Given portfolio structure (δ, Γ

and Σ and assuming normality of factor returns) an explicit formula for the moment

generating function of ∆V can be written.

The moment generating function M(u) and the probability density function f(V) of a

distribution are closely related by equation (see, e.g. Mina and Ulmer 1999)

, (15) ∫∞

∞−

= dVVfeiuM iuV )()(

where i is 1− . Inverting (15) and knowing the moment generating function, the

probability density function can be calculated from

∫∞

∞−

−= )(21)( iuMeVf iuV

π. (16)

In practice, (16) is approximated by Fast Fourier Transform (FFT). Before that the

formula for the moment generating function must be derived. The representation below

follows closely the one in Volmar (2002).

The derivation of the moment generating function is based on two essential steps.

1. Transformation of coupled random variables ri into independent random variables

using the Cholesky decomposition of the covariance matrix.

2. Diagonalization of the gamma matrix to decouple the contributions of the risk

factors.

9

In step 1, Cholesky decomposition of covariance matrix is done, i.e. a lower triangular

matrix A such that Σ=AAT is calculated. Matrix A can be used to transform correlated

random variables to uncorrelated and vice versa (see, e.g. Sharma 1996). Thus, given

correlated normally distributed variables Y ~ N(0, Σ), YAX 1−= is uncorrelated standard

normally distributed, i.e. X ~ N(0, 1). Introducing unity matrix AAI 1−= into equation

(9) gives

RRRV TT Γ+=∆21δ

)()(21)( 111 RAAARARAAV TTT −−− Γ+=∆ δ (17)

In step 2, the gamma matrix is diagonalized using orthogonal matrix O consisting of the

eigenvalues of ATΣA. Introducing unity matrix I = OOT into equation (17) gives

)()(21)( 111 RAAOOAOORARAAOOV TTTTTT −−− Γ+=∆ δ

)()(21)()( 111 RAOAOAORAORAOAOV TTTTTTTTT −−− Γ+=∆ δ . (18)

Comparing equations (9) and (18) the analogy can be seen: defining new variable

and new portfolio sensitivities RAOx T 1−=

δδ TT AO=~ , AOAO TT Γ=Γ~ (19)

equation (18) can be written as

xxxV TT Γ+=∆ ~21~δ , (20)

where the new gamma matrix is diagonal and the new variables are normal and

independent. This may be written as

10

∑ ∑= =

Γ+=∆=∆N

i

N

iiiiiii xxVV

1 1

2, )~

21~(δ . (21)

Completing the square, for 0~, ≠Γ ii ∆Vi can be written as

ii

i

ii

ii

ii

i xVΓ

−⎟⎟⎠

⎞⎜⎜⎝

⎛

Γ+Γ=∆ ~2

~~~

~21 22

,

δδ. (22)

As squared normal variables are χ2 -distributed, the term in brackets is χ2 -distributed

with one degree of freedom and with non-centrality parameter 2)~/~( iii Γδ . Consequently, it

has been now shown that the distribution of ∆V can be written as a sum of χ2 -distributed

variables (cases with 0~, ≠Γ ii ) and normally distributed variables (cases with 0~

, =Γ ii ). As

sum of normal variables is normal, the normal variables can be combined into one

variable 00Qλ and equation (21) can be written as

, (23) KQQVn

iii ++=∆ ∑

=00

1

λλ

where coefficients are chosen to match those of equation (22), i.e.

∑≠Γ∈ Γ−=

}0~|{

2

~2

~

iiii ii

iKδ

∑=Γ∈

=}0~|{

20

~

iiiiiδλ

iii Γ= ~21λ . (24)

As xi's are independent, Qi's in equation (23) are independent as well, and thus, ∆V is sum

of independent random variables. It is known from probability theory that the moment

generating function of sum of independent random variables is product of these

11

individual moment generating functions (Milton and Arnold 2002). Therefore, the

moment generating function of ∆V can be written as

, (25) ∏=∆

n

iiiQQV uMuMuKuM

i)()()exp()(

0λ

where

)21exp()( 22

00uuM Q λ= (26)

)~~

21exp(

211)(

2

⎟⎟⎠

⎞⎜⎜⎝

⎛

Γ−−=

ii

iQ u

uu

uMi

δ. (27)

With the method described above, the probability distribution function of the quadratic

form (9) can be calculated numerically with Fast Fourier Transform. Thereafter, area

under the probability density function can be calculated using some numerical integration

method (e.g. trapezoidal rule or Simpson's rule) and the desired VaR figure can be

interpolated from the cumulative distribution function. Mina and Ulmer (1999) conclude

that relatively high accuracy is achieved with FFT with only N = 28 points equally spread

in the integration range of ±Nσ2/10, where σ2 is the variance given in equation (11).

Example of the method is presented in Figure 2. The quadratic profit and loss distribution

of an option portfolio was calculated using the Fast Fourier Transform with N = 212

points and approximated also with the delta-normal method.1 As seen from Figure 2, the

delta-normal method amply underestimates risk as it neglects the negative skewness and

excess curtosis of the distribution. Delta-normal method results in absolute VaR equal to

-3030 as the exact VaR of the quadratic approximation calculated with FFT is -5950. For

reference, Cornish-Fisher expansion with three moments results in absolute VaR equal

1 The option portfolio OPTPF and the estimation of parameters are more closely described in Chapter 3.2. The example depicts the situation in 14.11.2003 and is calculated using real Nord Pool data. At that time OPTPF consists of 22 options and one forward contract.

12

to -6570 and thus overestimates risk. The quadratic form has skewness -2.33 and curtosis

7.73.

Figure 2. Example of estimating (absolute) VaR of an option portfolio with Fast Fourier Transform and with delta-normal method

2.3.4. Comparison of methods

When comparing the accuracy of analytical quadratic VaR methods two effects mix up.

The first is the adequacy of the delta-gamma approximation of the portfolio profit and

loss distribution. The magnitude of this effect can be measured by using full Monte Carlo

simulation to estimate VaR figures. In full Monte Carlo simulation risk factor returns are

sampled from estimated probability distribution and the value of portfolio is revaluated in

every scenario of factor returns. Full Monte Carlo VaR is generally thought to give the

most accurate VaR estimates.

The second effect is the accuracy of estimating the desired quantile of the delta-gamma

distribution. This effect can be measured or avoided by using Fourier inversion or partial

13

(delta-gamma) Monte Carlo simulation. Partial Monte Carlo simulation uses the

quadratic approximation (9) to evaluate the change in portfolio value and is thus

computationally less intensive than full Monte Carlo simulation.

Plenty of literature compares the quadratic VaR methods discussed in this paper in the

context of financial markets. Zangari (1996b) examines the accuracy of Johnson

transformation compared to full simulation with single put and call options. Zangari

concludes that unless the option is near expiration and at-the-money, the method

performs well in terms of accuracy and speed compared to full Monte Carlo simulation.

Pichler and Selitsch (1999) use random δ, Γ and Σ to compare five different VaR

methods: Johnson transformations, delta-normal approach, delta-gamma-normal and

Cornish Fisher approximations with four and six moments. Comparison is done relative

to the solution from partial Monte Carlo simulation. Pichler and Selitsch point out that

using delta-gamma-normal method leads to very inaccurate results in the case of small

number of risk factors. For distributions with negative skewness (i.e. portfolios with

negative gamma) they find all other methodologies performing equally well. In the case

of positive skewness accuracy of all methods is worse due to the effect of higher

moments. Since the method is very easy to implement, Pichler and Selitch recommend

using Cornish-Fisher expansion with at least six moments.

Mina and Ulmer (1999) compare Fourier inversion, Cornish-Fisher expansion, partial

Monte Carlo and Johnson transformation in the case of four test portfolios. They

conclude that delta-gamma approximation is very close to the results from full Monte

Carlo simulation even for the extreme portfolios covered in the study. According to Mina

and Ulmer, the best methods are partial Monte Carlo and Fourier inversion; Cornish-

Fisher expansion is extremely fast, but less accurate and gives "unacceptable results" in

the case of one test portfolio. Mina and Ulmer recommend the latter only for quick

checks and remark that Fourier inversion is the best choice unless the number of risk

factors is large (1000-5000 depending on the VaR confidence level).

Jaschke (2002) examines Cornish-Fisher expansion in more detail and concludes that the

expansion works well if the distribution is close to normal. Jaschke points out that

14

Cornish-Fisher expansion has some qualitative shortcomings and bad worst-case

behavior but achieves sufficient accuracy faster and simpler than Fourier inversion and

partial Monte Carlo. He recommends frequent use of full Monte Carlo simulation to

check the suitability of the quadratic approximation.

Volmar (2002) compares Cornish Fisher approximations of different order (from 4 to 12)

to the exact quadratic solution calculated by Fourier inversion in the case of standard

option strategies. Volmar's results are similar to the ones by Jaschke: Cornish Fisher

expansion works well if the delta-gamma approximation is good, and if this condition

does not hold the Cornish Fisher expansion with six or more moments can show

unreasonable behavior. According to Volmar, the Fourier inversion does not show this

unreasonable behavior, but can result in worthless results when the integration range or

the number of points for the FFT, are chosen too small.

Castellacci and Siclari (2003) implement five different VaR methods and compare results

with the results from full Monte Carlo simulation. In their comparisons, Castellacci and

Siclari use five test portfolios consisting of 1-4 options. They conclude that delta-gamma-

normal VaR can even produce less accurate VaR figures than simpler delta-normal

method. They remark that delta-normal, delta-gamma-normal and delta-gamma using

Cornish Fisher expansion tend to overpredict VaR.

2.4. Backtesting

VaR methods are useful only insofar as they predict market risk reasonably well.

Therefore, a VaR method should be accompanied by a validity test in order to check the

reliability of the risk measure. Backtesting is such a formal statistical framework. In

backtesting actual losses are compared to those predicted by the VaR method. (Jorion

2001.)

Consider a VaR metric with holding period of T days. Market value of a portfolio could

change during the period of T days in two ways: either the market prices of the

instruments change or the contents of the portfolio changes. When estimating VaR, one is

only interested in the changes in the market prices. Therefore, the portfolio content must

15

be held constant (or "frozen"). To minimize the effect of change in portfolio contents,

backtesting horizon should be short, for example one day. (Jorion 2001.)

Let Pi,t denote the position in instrument i on day t and Si,t the price of the respective

instrument. Define change in the value of a portfolio consisting of N instruments on day t

as

. (28) ∑=

+ −=∆N

itititit SSPV

1,1,, )(

In effect, in equation (28) the portfolio content is held constant and only changes in the

market prices of instruments affect the change in portfolio value. Now, VaR figure on

day t (VaRt) and the change in portfolio value (∆Vt) can be compared. Let zt denote a

binomial variable that is equal to 0, if ∆Vt is greater than VaRt, and equal to 1, if ∆Vt is

less than VaRt. Let Z denote the sum of zt over the backtesting horizon of M days, i.e. the

number of outliers (or exceptions) when actual losses happen to be greater than the

estimated VaR figures. Ideally, the failure rate Z/M should converge to the confidence

level α as M increases. If failure rate is far from the confidence level, the validity of VaR

method should be questioned. (Jorion 2001.)

If the daily backtests are assumed independent, the tests are a sequence of independent

Bernoulli trials and thus the number of outliers X follow a Binomial distribution,

X ~ Bin(M,α) (Jorion 2001). The expected value and variance of Binomial distribution are

E(X)=Mα and Var(X)=α(1-α)M. Kupiec (1995) has developed a statistical test based on

log-likelihood ratio to test for the unconditional coverage of a VaR method. The test

gives confidence regions for not rejecting the null hypothesis that α is the true probability

of an outlier. For example, with M=255 and α=0.05 one could observe from 7 to 20

outliers before rejecting the null hypothesis with 95 % confidence level.1

1 M=255 equals the number of trading days in one year. Confidence level refers here to the confidence level in statistical testing, i.e. in rejecting null hypothesis or not, and not to the confidence level in the VaR method.

16

Kupiec's test assumes independence of trials and thus ignores the time variation in the

data. A VaR method could have correct unconditional coverage, but the outliers could be

clustered closely in time and thus invalidate the model. Christoffersen (1998) has

developed a practical statistical test for both conditional and unconditional coverage. For

details, see Christoffersen (1998).

3. Applying analytical delta-gamma methods in Nord Pool’s financial market

The VaR methods presented in Chapters 2.2 and 2.3 were implemented to estimate VaR

figures for two portfolios consisting of electricity derivatives traded in the Nordic

financial power market. These VaR estimates were then backtested with real Nord Pool

market data. Below Nord Pool's financial market is briefly discussed and backtesting

results are presented and discussed.

3.1. Nord Pool’s financial market

Nord Pool, the Nordic power exchange, was established in 1993. Norway acted as a

pioneer as it decided to deregulate its domestic power market in 1991. Sweden joined

Nord Pool in 1996, Finland in 1998 and Denmark in 1999. The Nordic countries are

forerunners in electricity trading as Nord Pool is the first international power exchange in

the world.

In Nord Pool, electricity and related financial instruments like forwards, futures and

options are traded in the physical and financial market respectively. In the physical

market hourly demand and supply curves of all market participants are matched day-

ahead. The equilibrium of these curves sets the hourly spot price which functions as a

reference index for trading with derivatives.

Nord Pool's standardized financial derivatives are used for price hedging, risk

management and trading purposes. More exotic derivatives are traded actively in the

OTC-market. Since the collapse of the major American trader Enron in 2001, the

17

participants in Nord Pool's financial market have been mostly hedgers, i.e. producers,

retailers and end-users.

Excluding the decline in 2003, the volume of power contracts traded in Nord Pool's

financial market has grown from year to year. This can be seen in Figure 3 below. The

traded volume in the financial market outgrew the volume in the physical market in 1997

and has been greater ever since. In 2003, the volume of power contracts traded in Nord

Pool's financial market was 550 TWh, as the volume in the physical market was 120

TWh. The decline in the traded volume in 2003 was due to uncertainty in the physical

market and high spot prices causing several market participants to meet their risk limits

and reduce their market exposure. (Nord Pool 2003.)

Yearly volume of power contracts traded in Nord Pool's financial market

0

200

400

600

800

1000

1200

1996 1997 1998 1999 2000 2001 2002 2003

TWh

Figure 3. Yearly volume of power contracts traded (in TWh) in Nord Pool's financial market (Nord Pool 2003)

Products in Nord Pool's financial market are all cash-settled contracts: futures, forwards,

contracts for price area differences and European options. A forward contract is an

agreement to buy or sell an asset at a certain future time for a certain price (Hull 2000). In

Nord Pool, the underlying asset of futures and forward contracts is spot price. With

futures and forward contracts market participants can hedge against fluctuations in

18

electricity price. Contracts for price area difference (CFDs) can be used to hedge against

the risk due to differences in area prices. Such differences are caused by physical

constraints in transmission grid. (Nord Pool 2003.)

The non-storability of electricity makes the electricity market different from the financial

markets and other commodity markets. First of all, electricity forwards differ from

conventional commodity forwards in the way that electricity forwards have a delivery

period instead of a specific delivery time. Currently in Nord Pool the delivery periods of

futures and forwards range from one day to one year. The payoff of a contract is

determined by the arithmetic average of spot price during the delivery period.

In addition, available forward pricing models differ between electricity and other

markets. Forward pricing models based on non-arbitrage conditions and convenience

yield are available in the financial and conventional commodity markets. Electricity

forward and futures prices reflect the expectations of future spot prices, but unlike in

financial and other commodity markets, no analytical expression has been established

between the spot prices and forward prices. This is due the fact that electricity delivered

now and at any given time in the future are separate assets. (Vehviläinen 2002.)

A European option gives the holder the right to buy or sell the underlying asset at a

predefined date for a predefined price (Hull 2000). Underlying assets of Nord Pool traded

options are season and year forwards. Currently, options are mostly traded bilaterally and

options in Nord Pool are quite illiquid (Nord Pool 2003). As electricity forwards are

tradable instruments, a replicating hedging portfolio for a European option on electricity

forwards can be created. Thus, standard electricity options on forwards can be valued

with the Black-76 model (see Black 1976) in a rather straightforward way (Vehviläinen

2002).

3.2. Backtesting set up

For backtesting purposes one artificial portfolio consisting of electricity derivatives was

generated. Although being mostly random, the portfolio (denoted PF below) was

designed to act as a realistic trading portfolio of Nord Pool exchanged electricity

19

derivatives. Another test-portfolio (denoted OPTPF) was created as a sub-portfolio of the

original portfolio: all options and option strategies from PF were included in OPTPF. In

this way, two different but not unique random portfolios were generated.

On average, PF includes 1-35 different option contracts and 40-60 different electricity

futures and forward contracts. The average skewness and curtosis of PF - assuming the

quadratic relation (9) - are 0.00 (with standard deviation of 0.11) and 0.03 (with standard

deviation of 0.08). OPTPF includes 1-35 different option and 1-4 different forward

contracts. OPTPF has average skewness of -0.18 (with standard deviation of 0.85) and

curtosis of 1.18 (with standard deviation of 2.33). Thus, PF is a large portfolio with profit

and loss distribution close to normal and OPTPF a small portfolio with high skewness

and curtosis in the profit and loss distribution.

Delta-normal, delta-gamma-normal and two different Cornish-Fisher expansions were

used in backtesting with the two portfolios. Cornish-Fisher expansion with three

moments was implemented according to method presented in Hull (2000) and Cornish-

Fisher expansion with five moments according to the matrix formulation described in

Chapter 2.3. Fourier inversion was used with OPTPF to check the adequacy of the

quadratic approximation with a highly skewed portfolio like OPTPF. When estimating

VaR figures, covariance matrix Σ was estimated from past returns with Exponentially

Weighted Moving Average (for details, see Hull 2000).

3.3. Backtesting results

The backtesting period was 255 trading days and relative 1-day VaR was estimated with

0.05 confidence level. As stated in Chapter 2.4, Kupiec's (unconditional coverage)

confidence region for not rejecting the null hypothesis is from 7 to 20 outliers. Besides

the number of outliers, Mean Average Percentage Error (MAPE) was calculated with

OPTPF as (Zangari 1996b)

∑=

−=

255

12551

iiFFT

iFFT

iA

VaRVaRVaRMAPE , (29)

20

where VaRiA denotes VaR figure approximated by a quadratic method and VaRi

FFT

denotes the exact quadratic VaR figure calculated by Fast Fourier Transform. The results

are shown in Table 1.

Table 1. Backtesting results for different VaR methods with M = 255 and α = 0.05

MAPEPF OPTPF OPTPF

Delta-normal 15 30 15,4 %Delta-gamma-normal 15 28 12,8 %Cornish-Fisher, 3 moments 15 28 5,5 %Cornish-Fisher, 5 moments 15 28 5,4 %FFT - 28 -

Number of outliers

Three important observations can be made from the results in Table 1. First, all the

methods perform equally well with the relatively large portfolio PF and have the number

of outliers within the 95 % confidence region. As the profit and loss distribution of PF is

close to normal, it is reasonable that even the delta-normal method performs well.

Secondly, all the methods underestimate risk in the case of the more skewed portfolio

OPTPF. The number of outliers is outside the 95 % confidence region with all methods

and thus, the methods are inadequate to measure the market risk OPTPF is exposed to.

Delta-gamma-normal and Cornish Fisher expansion with three and five moments perform

equally poorly in terms of number of outliers. In terms of Mean Average Percentage

Error, Cornish-Fisher expansions with three and five moments have only little difference

but outperform clearly the delta-normal and delta-gamma-normal methods.

Thirdly, and most importantly, even the Fast Fourier Transform, which gives the exact

solution to the quadratic approximation, does not work well with OPTPF. Therefore,

given the portfolio structure (δ, Γ) and normally distributed factor returns (Σ, µ=0), the

quadratic approximation (9) is not adequate to be used with portfolio like OPTPF.

Volmar's (2002) conclusion that Cornish-Fisher expansion with six or more moments can

show unrealistic results when the quadratic approximation is bad, realized in this study.

Cornish-Fisher expansion with six moments resulted in highly unrealistic VaR figures

and was therefore not included in the results in Table 1. Volmar also pointed out the

21

possibility of numerical difficulties in implementing FFT. Occasional difficulties in

choosing the integration range and number of points to calculate the FFT properly were

encountered in this study as well.

4. Conclusions

In this paper analytical quadratic, i.e. delta-gamma, VaR methods with electricity

derivatives were considered. VaR methods based on simulation (including delta-

gamma/partial simulation) were not included in the study due to too long computation

time.

Quadratic methods take into account the nonlinear relationship in the portfolio payoff

function and thus, instead of linear delta methods, they can generally be used with

portfolios consisting of nonlinear instruments. The assumption of normally distributed

factor returns is usually made with delta-gamma methods as well. As the portfolio payoff

function is quadratic, the distribution of ∆V, the change in portfolio value, is non-normal.

Thus, the desired quantile of ∆V can not be directly calculated in closed form.

Three methods for calculating the desired quantile were described: 1) moment matching

approaches (including delta-gamma-normal method), 2) direct quantile approximation

with Cornish-Fisher expansion and 3) the use of moment generating functions and Fast

Fourier Transform. As the quadratic payoff function is an approximation itself, the first

two methods mentioned are approximations of an approximation. Fast Fourier Transform

calculates numerically exact solution to the quadratic approximation.

These three methods were implemented and backtested with two imaginary trading

portfolios consisting of Nord Pool traded electricity derivatives. Delta-gamma-normal

and Cornish-Fisher expansion proved to be computationally fast methods and very easy

to implement. Implementing FFT was more complex and some numerical problems

occurred.

In the case of a large portfolio like PF with profit and loss distribution close to normal,

even the linear delta-normal method performs well; no differences in linear and quadratic

22

methods can be seen in the backtesting results. In addition, backtesting results indicate

that the analytical quadratic methods are not proper in the case of an option portfolio with

high skewness and curtosis like OPTPF.

The analytical delta-gamma methods could have failed due to couple of issues not

covered in this paper and worth further consideration. Firstly, the failure could be due to

too abrupt changes in risk factor values invalidating the local quadratic approximation.

Secondly, realism behind the assumption of normally distributed factor returns in the case

of electricity derivatives can be questioned. In full and partial Monte Carlo simulation the

assumption of normally distributed factor returns can be relaxed and thus, with these

simulation methods further insight to the failure of analytical delta-gamma methods could

be gained. Nevertheless, full or partial Monte Carlo simulation should be implemented to

properly capture the market risk portfolio like OPTPF is exposed to.

23

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