Analytical Approaches to Non- Linear Value at Risk Simon Hubbert, Birkbeck College London.
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Transcript of Analytical Approaches to Non- Linear Value at Risk Simon Hubbert, Birkbeck College London.
Analytical Approaches to Non-Linear Value at Risk
Simon Hubbert, Birkbeck College London
Overview
Review Value at Risk approaches for linear portfolios.
Consider the case for a portfolio of derivatives.
Use Taylor approximations to derive closed form solutions.
Based on: Non-linear Value at Risk : Britten-Jones and Schaeffer: European Finance Review 2. 1999.
Portfolio Monitoring
Invest in n Risky Assets:
Portfolio value today:
Potential future loss/profit:
where
nSS ,...,1
n
iii tStV
1
)()(
n
iii SV
1
.,...,1)()( nitSttSS iii
Normal Value at Risk
Q. How much are we likely to lose 100 % of the time over the future period?
A. The number that satisfies:
If then
Typically or
VaR
VaRVPr
),(~ 2NV );(1 VaR
33.2)01.0(1 65.1)05.0(1
Non-linear Portfolio
Invest in n derivatives:
Each is a non-linear function of and
Potential future loss/profit:
we cannot assume the are normally distributed.
We need to approximate…
)(),...,( 11 nn SgSg
ig iS .t
n
iii gV
1
ig
Simple Approximations
1st Order – Delta Approximation:
2nd Order – Gamma Approximation:
ii
iiii S
S
gt
t
ggg
22
2
2
1i
i
ii
i
iiii S
S
gS
S
gt
t
ggg
22
2
2
1i
i
ii S
S
gg
Coping with high dimensionality
A large number of derivatives in the portfolio creates high computational demands.
Eg. The covariance structure:
requires numbers.
We introduce a factor model:
where .
),cov( ji gg )1( nji 2
2n
kikiii fbfbaS 11 nk
Employing Delta Approximation
With the Factor model we consider:
The delta approximation then becomes:
In vector notation:
),...,(),...,,...,( 111 knk ffgffg
V
n
ii
1
ig
k
jj
j
ii ff
gt
t
g
1
n
ij
k
j
n
i j
ii
ii f
f
gt
t
g
1 1 1
k
jjjt f
1
fV Tt
j
k
j
n
i j
iit ff
g
1 1
Delta Normal VaR
Suppose that:
where
Then..
Given a small we have:
),0(~ ANf ),cov( jiij ffA
),(~ ANfV Tt
Tt
.)(1 AVaR T
t
Employing Gamma Approximation
The Delta VaR is known to be a weak estimate (see BJ&S 1999).
We turn to the gamma approximation:
With a single factor this becomes:
fV t 21
2
2
2
1f
f
gn
i
ii
22
1f
Gamma approx cont’d
How is distributed ?
It is a quadratic:
Complete the square - consider:
Expand and match:
and
V
22
1ffV t
22
1efV t
2
2
1
tt
e
Towards Gamma VaR
If we assume that then:
Furthermore:
Use statistical tables to find
),0(~ 2Nf
1,~e
Nef
1,~2
22
eef
Z
)(z
)(Pr zZ
Gamma VaR
Since
We see that is equivalent to:
Thus we can read off VaR estimates:
2/2
2
tVefZ
)(Pr zZ
2)(
2
1Pr zV t
2)(2
1 zVaR t
Gamma VaR: the multi-factor case
BJ & S (1999) show that gamma VaR provides a much more accurate estimates than the delta approach applied to long options.
We want to modify our analysis to cover multi-factor modelling:
Idea: Use the approach used in the single factor case to develop a strategy.
Multi-factor Gamma approximation
The approximate profit/loss is given by
Where:
:
fV Tt
n
i
k
r
k
ss
sr
iri f
ff
gf
1 1 1
2
2
1
k
rs
k
s
n
i sr
iir f
ff
gf
1 1 1
2
2
1 s
k
r
k
srsr ff
1 12
1
kkR
n
i sr
iirs ff
g
1
2
ff T 2
1
Distribution of gamma approx
Recall – Single factor case we considered:
and found that .
In the multi-factor case we set and analogously we can show:
where,
2)( ef 1e
1e
,2
1 11 ffVT
t
1
2
1 Ttt
Variable Transformations
We assume that
where is positive definite:
and
To make simplifications we set:
and
ANf ,0~ A
)()( 2/12/1 AAA T )()( 2/12/11 AAA T
)( 12/1 fAy )()( 2/12/1 AAB T
Towards Gamma approx
With these transformations we can neatly write:
One step further – spectral decomposition of B:
Orthonormal matrix of Eigenvectors.
Eigenvalues of B.
yByV Tt
2
1
Tk CdiagCB ),...,( 1
kkRC
k ,...,1
The Gamma Approximation
One final transformation:
Yields:
A sum of squares of normal random variables each with unit variance, i.e., a sum of non-central chi-squared random variables.
yCz T
k
jjjt zV
1
2
2
1
),(~ 1k
T ICN
Approximating the distribution
What more can we say about ?
We can write down analytic expressions for its moments:
1st
2nd
3rd
V
)(2
11 ATrm t
AATrmm T 2212 )(
2
1
)(3)()(3 32121
313 AAATrmmmmm T
An idea..
The distribution of is not known. However..
We have expressions for the integer moments.
Idea: Fit the moments to a more tractable distribution
Hope for a good approximation to .
V
V
A candidate random variable
Britten-Jones and Schaeffer (1999) consider a chi-squared random variable:
where with p degrees of freedom.
Such a random variable was proposed by Solomon and Stephens (1977) - showed that it can provide a good approximation to a sum of chi-square variables.
kpaY 2~ p
A distributional approximation
The integer moments of the random variable
are given by…
where denotes the gamma function.
kpaY
...3,2,1)2/(
)2/(2
rp
prka krr
r
)(
Moment matching
We have analytic expressions for the integer moments of both
:
and
:
Matching moments gives values for and
V ,...,, 321 mmm
kpaY ,...,, 321
pa, .k
Gamma VaR
Using the approximation
We can read, from a table of , values such that
We then set:
For an appropriate confidence level .
VY
2 )(y
)(Pr yY
VaRy )(
Overview
How to compute analytical non-linear VaR:
Set up a factor model Employ first or second order Taylor approximations. Assume a distribution for the risk factors (eg, normal). Using the first order approximation with multi-factors
Analytical solution – Delta VaR. Using the second order approximation with single factor
Analytical solution – Gamma VaR Using the second order approximation with multi-factors
Semi-analytical solution – Approximate Gamma VaR
Numerical Findings
Numerical tests (BJ-S 1999) against Monte Carlo approach, suggest that:
Delta approximations provide weak estimates of VaR. Gamma approximation (with a single factor) improves the
VaR estimates – however a single factor assumption may not be realistic.
Success of the approximate gamma VaR (with many factors) to VaR estimates is very dependent upon the curvature of the derivatives. Encouraging results are reported for portfolios of long European options.
Bibliography
Britten-Jones, M and S. M. Schaeffer: (1999)
Non-Linear Value at Risk
Economic Finance Review 2: pp 161 – 187.
Solomon, H and M. A. Stephens (1977)
Distribution of a sum of weighted chi-square variables
Journal of American Statistical Association 72: 881-885.