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During the 1990s, practitioners faced a difficult choice in selecting the form of transformation procedure to use with value-at-risk (VaR) measures. Linear transformations were exact and ran in real time, but they applied only to linear portfolios. Monte Carlo and historical transformations were more generally applicable, but they entailed standard error and ran more slowly. Practitioners sought a compromise—some middle ground between the accuracy, speed and limited applicability of linear transformations and the general applicability, standard error and slowness of Monte Carlo and historical transformations. An obvious avenue of research was to explore VaR measures that applied to quadratic portfolios. Even before transformation procedures were formalized for them, such measures were called delta-gamma VaR measures. A better name—one that makes no tacit assumption that a delta-gamma remapping must be used—is quadratic VaR measures.
Let time 0 correspond to the current time, and let time 1
correspond to the end of the VaR
horizon. Mathematically, a portfolio is defined by a current value
, which is a known constant, and a future value 1P,
which is a random variable (see the
notation conventions documentation). Typically, 1P is specified
as a function
of a
random vector 1R called a
key vector. Its components
, called
key factors, represent market variables such as prices, interest
rates, spreads or
implied
volatilities as of time 1. Current values of key factors are indicated
with a constant vector 0r. The relationship
1P =
(1R)
is called a portfolio mapping,
and the function
is
called a portfolio mapping function.
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We say that a portfolio is quadratic if its portfolio mapping
function
is a quadratic polynomial:
 |
[1] |
Here, c is a symmetric square matrix, b is a row vector and a is a scalar. A prime ' indicates transposition.
Many researchers worked to develop a transformation
procedure applicable to quadratic portfolios—what we call a
quadratic transformation. An
obvious solution is to simply apply the
Monte Carlo method. With
a
quadratic polynomial, realizations 1p[k]
= (1r[k])
can be valued rapidly. Because it depends upon the Monte Carlo method,
this solution entails standard error. Depending upon the number of
realizations used, it imposes a modest tradeoff between run-time and
standard error.
Wilson (1994) published an innovative quadratic transformation based upon an optimizing search routine. This limits 1R to some pre-defined region and searches for the maximum portfolio loss within that region. The VaR metric—maximum portfolio loss within a pre-specified set of possible values for 1R —is non-standard and is of limited usefulness.
Researchers sought a general quadratic transformation that might offer the accuracy and calculation speed of linear transformations. Such a solution does exist based upon the established mathematics of quadratic polynomials of joint-normal random vectors. By 1996, a number of researchers had found the solution. In January of that year, Fallon (1996) circulated a working paper with a solution. The description of his quadratic transformation comprised only a small portion of the paper. It attracted little attention.
Eleven months later, a fourth edition of the RiskMetrics Technical Document was published. Zangari (1996) contributed a cursory description of a quadratic transformation similar to Fallon’s. By this time, the RiskMetrics group was struggling to justify its existence within JP Morgan. They were earning fees through consulting and would soon be spun off as a separate consulting firm. For one reason or another, the solution Zangari published was incomplete. It offered tantalizing clues that the RiskMetrics group possessed a solution, but a complete derivation or information on how to implement a general solution were not provided.
The first complete published solution was by Rouvinez (1997).
He detailed how, if
is a
quadratic polynomial, and 1R is joint-normal with
positive definite covariance matrix, the characteristic function of 1P
can be calculated. Since a characteristic function fully specifies the
cumulative distribution function (CDF) of a random variable, this makes it
theoretically possible to evaluate any VaR metric.
Eight months after Rouvinez published his paper, Cárdenas, et al. (1997) published a similar solution. Working papers by Britten-Jones and Schaefer (1997) and Jahel, Perraudin and Sellin (1997) offered similar solutions.
The quadratic transformations described in these papers
differ in various respects, but they all employ the mathematics of
quadratic polynomials of joint-normal random vectors. Their collective
solution can reasonably be called the quadratic transformation.
Because of its association with RiskMetrics, Zangari’s solution attracted much attention. Being incomplete, it was not useful. The other papers attracted little attention. All were technical papers that were released with little fanfare. A reader would have to have strong mathematical skills and devote several hours to deciphering any of them in order to realize that the sought-after quadratic solution had indeed been found. Apparently, not many people did.
The popular literature on value-at-risk largely ignored the new quadratic measures. Discussions of value-at-risk continued to focus on linear, Monte Carlo or historical transformations. Books on value-at-risk—including Best (1998), Dowd (1998), Butler (1999), and Jorion (2000)—either failed to mention quadratic VaR measures or focused on crude solutions published prior to 1997. Six years after Fallon first circulated his solution, the financial risk management community remained largely unaware of the mathematics of quadratic VaR measures.
Example: Quadratic Value-at-Risk in One-Dimension
Let’s start with a one-dimensional example. This will illustrate a
number of important concepts underlying the mathematics of quadratic VaR
measures.
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Measure time in trading days. A Canadian investor is short a call option on JPY 100MM. The option is struck at 0.0120 CAD/JPY and expires in one month. With a current exchange rate of 0.0115 CAD/JPY, the option is out-of-the-money. The position’s current value 0p is –5,386.67 CAD. The investor wants to calculate the position’s one-day 97.5% CAD VaR. sources of risk include:
the CAD/JPY exchange rate,
the
implied volatility of the JPY/CAD exchange rate,
applicable CAD and JPY interest rates.
To limit the problem to a single dimension, treat the implied
volatility and interest rates as constant. Specifically, assume the
implied volatility equals 16% and applicable CAD and JPY interest rates
are .05 and .02, respectively. Model only the CAD/JPY exchange rate as a
key factor, which is denoted 1R1. Based upon
time series analysis of
recent market data, 1R1 is assumed normal with mean
equal to the
current exchange rate 0.0115 CAD/JPY and standard deviation
of 0.00012
CAD/JPY (the superscripts 1|0 indicate that these are parameters for time
1 conditional on information available at the current time 0).
A portfolio mapping function
is specified based upon the
Garman and
Kohlhagen (1983) modified Black-Scholes option pricing formula. This is
remapped (approximated) by valuing
at three different values for 1R1 and
quadratically interpolating between the results. Values used are centered
at
and are spaced
apart.
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Interpolation
Points for the Quadratic Remapping |
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1R1 Value |
Option Value |
 |
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–3,141.24 |
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–5,007.52 |
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–7,633.18 |
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Interpolation points used to apply a quadratic remapping. |
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Interpolating between these points yields the quadratic remapping
 |
[2] |
Assume
=
= –5,386.67.
Exhibit 2 compares quadratic polynomial
with the Garman and Kohlhagen formula
, which
it approximates. The three interpolation points are shown. The light green
region indicates plus or minus 2 standard deviations for 1R1.
Events outside that region are unlikely to significantly impact the
position’s 97.5% VaR. Inside that region,
appears to be a reasonable approximation for the Garman and Kohlhagen
formula .
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Quadratic
Remapping |
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 |
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In the foreign exchange example, a portfolio mapping function is constructed from the Garman and Kohlhagen option pricing formula. This is remapped (approximated) with a quadratic polynomial, which is obtained by interpolating between the three indicated points. The light green region indicates plus or minus 2 standard deviations in the key factor 1R1. |
Next, introduce a change of variables
~ N(0,1) with
|
1R1 = .00012
|
[3] |
to obtain
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[4] |
This represents
as a linear polynomial of two random variables:
~ N(0,1)
Because one is the square of the other, the two random variables are not independent. By "completing the squares," we rewrite [4] as
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[5] |
This represents
as a linear polynomial of a single random variable:
This fully characterizes the probability distribution of
in terms of a single random variable with a standard probability
distribution. Based upon this, any VaR metric for
—including
the desired 97.5% VaR metric—can be valued. There are various ways to
proceed. Before addressing these, lets extend the mathematics illustrated
in this example to multiple dimensions.
Quadratic
Transformation in Multiple Dimensions
Consider a quadratic portfolio
|
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[6] |
where 1R is a joint normal random
vector with mean vector
and
covariance matrix
.
Apply change of variables
 |
[7] |
where
,
z is the
Cholesky matrix of
,
and u is a matrix whose rows are the orthonormal
eigenvectors of
.
With this change of variables, we obtain a new expression for 1P:
 |
[8] |
where:
It can be shown—see Holton (2003)—that this change of variables achieves four conditions:
is joint normal,
the mean vector of
is the 0 vector,
the covariance matrix of
is the identity matrix I, and
v
is a diagonal matrix.
The fourth item means that 1P can depend upon each of the variables
in one of four ways:
no dependence:
and
;
linear dependence:
and
;
central quadratic dependence:
and
;
or
vnon-central quadratic dependence:
and
.
In the last case 1P has a dependence of
the form
.
Completing the squares, this becomes
.
Consequently, 1P is a linear polynomial
of independent random variables, each of which is either
standard normal,
central chi-squared
with one degree of freedom, or
non-central chi-squared
with one degree of freedom and non centrality parameter
.
Since a linear polynomial of independent normal random variables is itself normal, all normal terms can be combined into one. A general expression for 1P is
 |
[9] |
where the
are chi-squared with one degree of freedom and non-centrality parameters
.
is standard normal.
We are now in much the same position we were in with our one-dimensional example. We have characterized the distribution of 1P. We wish to value a desired VaR metric. How we do this depends, of course, on the specific VaR metric. In this article, I will focus on the popular quantile-of-loss VaR metrics.
Various solutions have been proposed. Zangari (1996) approximates a solution using Johnson (1949) curves. Fallon (1996) and Pichler and Selitsch (2000) recommend approximate solutions based on the Cornish-Fisher expansion. Rouvinez (1997) uses the trapezoidal rule to invert the characteristic function. Britten-Jones and Schaefer (1997) use an approximation due to Solomon and Stephens (1977). Cárdenas et al. (1997) use the fast Fourier transform (FFT).
Of these, solutions based upon the Cornish-Fisher expansion, trapezoidal rule and FFT are the most effective. Holton (2003) covers the first two in detail, so I will discuss the third here. Because the method of Johnson curves was mentioned in the RiskMetrics Technical Document, it is of some historical interest. The method is inferior to others, but I will describe it briefly at the end of this article.
Quantile of Loss from the Fast
Fourier Transform
The inversion theorem of probability theory provides the following
expression for the probability density function
of a random variable in terms of its characteristic function
.
 |
[10] |
Substituting
w = 2
t,
this becomes
 |
[11] |
which is a Fourier transform. It can be approximated with the FFT.
It can be shown—see Holton (2003)—that the characteristic function of a random variable of form [9] is
 |
[12] |
where
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[12] | |
 |
[13] | |
 |
[14] | |
 |
[15] |
and tan–1 denotes the inverse tangent function with output in radians.
For example, suppose a quadratic portfolio's value 1P is expressed in form [9] as
 |
[16] |
which is a linear polynomial of three independent random
variables. Each is non-centrally chi-squared with one degree of freedom
and non-centrality parameters
of 9.871, 4,773.841 and 7,261.351, respectively. The characteristic
function of 1P is
 |
[17] |
We substitute w =
2t
and take the FFT, sampling n = 64 values with sample spacing
=
.00003/64. Note that input values are
complex numbers. However, output is
real. Results are indicated in Exhibit 3.
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Probability
Density Function of Portfolio Value 1P Obtained
with the FFT |
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 |
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The probability density function for portfolio value 1P was obtained using the FFT. The graph has the shape of a delta-hedged positive gamma portfolio. |
Let's use the FFT results to calculate the portfolio's 95% VaR. For this, we need the .05-quantile of 1P. We can calculate area under the PDF using Simpson’s rule. The evenly spaced output from the FFT is excellent for this purpose; however, the .05-quantile does not fall precisely at any of the values obtained from the FFT. We employ Simpson’s rule to find four values of the cumulative distribution function (CDF) that bracket the desired quantile and then interpolate with a cubic polynomial to obtain the .05-quantile of 1P. Subtracting this from the portfolio's current value 0p yields the portfolio's 95% VaR.
Approximating a
Quantile-of-Loss with Johnson Curves
Zangari (1996) proposed that a quantile-of-loss VaR
metric might be approximated using Johnson (1949)
curves. Because other solutions provide superior results, I cover this
solution only for historical interest.
When faced with a body of statistical data, researchers often try to fit some standard probability distribution to the data. For this purpose, various families of probability distributions—called families of curves—have been defined. These include the Pearson (1895), Edgeworth (1898) and Johnson (1949) families.
Johnson curves are constructed through translation of variables. In this context, a translation of variables specifies a probability distribution for a random variable X through an equation of the form:
 |
[18] |
where
is a
monotone function and Z is standard normal. Because Z is
standard normal,
imparts a probability distribution to X. This is not an unfamiliar
concept. Recall that a random variable X is said to be
lognormal if:
 |
[19] |
is standard normal for some m and s. This defines a family of curves parameterized by m and s. The more general Johnson family of curves is defined with the family of translation functions comprising any of the forms
 |
[20] | |
 |
[21] | |
 |
[22] |
where
are parameters—similar to m and s for the lognormal family.
There are various ways to fit a distribution from a family of curves to a particular random variable. Two methods are:
find that curve that matches certain moments of the
random variable;
find that curve that matches certain quantiles of the
random variable.
Hill, Hill and Holder (1976) extended the family of Johnson curves to include normal distributions:
 |
[23] |
They also provided an algorithm for fitting a Johnson curve based upon
the first four moments of a random variable. The algorithm determines the
appropriate form of translation function from [20],
[21], [22] and
[23] and then determines values
for that function.
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Given a quadratic portfolio of form [9], you can calculate the first four moments—see Holton (2003). Fit a Johnson curve to 1P using the Hill, Hill and Holder algorithm to obtain
 |
[24] |
where Z
~ N(0,1) and
is a
translation function. Since translation functions are monotone, they are
invertible. You obtain the approximation
 |
[25] |
Because
is monotone,
–1
is also monotone. Monotone functions map quantiles to quantiles, so you
can calculate any quantile of 1P from the corresponding
quantile of Z. For example, the .05-quantile of a standard normal
random variable is –1.645. Accordingly, the .05-quantile of 1P
is approximately
–1(–1.645).
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