Download Management of Financial Risk

Computational Finance
Zvi Wiener
02-588-3049
http://pluto.mscc.huji.ac.il/~mswiener/zvi.html
CF-3
Bank Hapoalim
Jun-2001
Plan
1. Hypothesis test
2. Maximal Likelihood estimate
3. Multidimensional VaR, contour plots
4. Monte Carlo type methods
Zvi Wiener
CF3
slide 2
Hypothesis tests
Given a population, we would like to perform
a test in order to accept or reject the claim that
it is distributed according to some rule (for
example normal or normal with some mean
and standard deviation).
Zvi Wiener
CF3
slide 3
Pearson goodness of fit test
Is based on calculating the sample moments and then
comparing the number of observations in more or
less equally full bins.
The comparison of actual number of points versus
the expected frequency allows to estimate the
likelihood of the distribution to belong to the
suspected class.
Zvi Wiener
CF3
slide 4
Pearson goodness of fit test
For large samples the following Q statistics
follows a Chi Square distribution with m-1
degrees of freedom.
Here m – the number of bins.
fk – is the actual number of points in bin k,
ek is the expected number of points.
( f k  ek ) 2
Q
ek
k 1
m
Zvi Wiener
CF3
slide 5
The Kolmogorov-Smirnov test
Here we compare the maximal distance
between actual and proposed cumulative
distribution.
1. Numerical Recepies in C, second edition, p. 623-625.
2. Mathematica in Education and Research Journal, vol. 5:2,
1996, p. 23-30 by David K. Neal.
Zvi Wiener
CF3
slide 6
The Kolmogorov-Smirnov test
Define test statistics by
D  n sup F ( x)  G( x)
Here n is the number of sample points, F(x)
the sample and G(x) the expected cumulative
distribution. For large n distribution of D
converges to a distribution Y (see Degroot 1986).

P(Y  t )  2 (1) e
i  2i 2t 2
i 1
Zvi Wiener
CF3
slide 7
The Kolmogorov-Smirnov test
data
-2
-1
hypothesis
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
1
2
-2
-1
1
2
difference
0.08
0.06
0.04
0.02
-2
-1
1
2
-0.02
-0.04
Zvi Wiener
CF3
slide 8
Maximal Likelihood
Zvi Wiener
CF3
slide 9
Example
Your portfolio is exposed to two independent
(correlation =0) risk factors.
Each one is uniformly distributed between –1
and 1 for a given time horizon.
What is your VaR95% for the same horizon?
Zvi Wiener
CF3
slide 10
Example
Probability density
B
A
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slide 11
2 dimensional risk
B
1
-1
1
A
-1
Probability of 5%
Zvi Wiener
CF3
slide 12
Example
The total probability is 1, the area of the
rectangle is 4, so the height is 0.25.
We are looking for x, such that
1 21
x  1  0.95
2 4
x=0.6325, and VaR95%=2-x=1.3675
Zvi Wiener
CF3
slide 13
Ovals
Consider a portfolio managed versus benchmark.
The benchmark has duration T and includes only
government bonds (no credit risk).
A manager has two degrees of freedom. He can
choose non-government bonds and have a
duration mismatch.
Denote the actual duration by T+q and by a – %
of the assets invested in non-government bonds.
Zvi Wiener
CF3
slide 14
Ovals
Denote by r – the current yield on treasuries
Denote by L – LIBOR
(for simplicity we assume a flat term structure).
Denote by dr and dL the possible change in each
risk factor during a short period of time.
Zvi Wiener
CF3
slide 15
Ovals
The value of the benchmark today is 1.
rT  rT
Benchmark 0  e e
quantity discount factor
The value of the benchmark tomorrow will be
rT  ( r  dr )(T  dT )
Benchmark1  e e
For short time intervals we ignore dT.
Zvi Wiener
CF3
slide 16
Ovals
Similarly the dollar P&L of the portfolio will be
P1  P0  ae
 dL (T  q )
 (1  a)e
 dr (T  q )
We ignore convexity and carry effects.
To measure relative performance we use
P1  P0 ae

B1  B0
Zvi Wiener
 dL (T  q )
CF3
 (1  a)e
 drT
e
 dr (T  q )
slide 17
Assume that dr and dL are jointly normal and use delta
approach. Then the gradient vector is
 T  (1  a)(T  q) 

grad  
  a(T  q)

One should also check the impact of the
second derivative.
For a given variance covariance matrix of the
risk factors one can easily construct the level
curves of the total risk on the q, a plane.
Zvi Wiener
CF3
slide 18
Assuming correlation of 26% between dr and dL we
have:
 portf / benchm  grad ..grad
The resulting contour plot shows the levels of
risk for any potential position as seen
according to today’s data.
Zvi Wiener
CF3
slide 19
VaR=1bp
Ovals
0.6
VaR=2bp
0.4
0.2
spread
0
-0.2
-0.4
-0.6
0
Zvi Wiener
0.1
0.2
0.3
CF3
0.4
0.5
slide 20
Plan
1. Monte Carlo Method
2. Variance Reduction Methods
3. Quasi Monte Carlo
4. Permuting QMC sequences
5. Dimension reduction
6. Financial Applications
simple and exotic options
American type
prepayments
Zvi Wiener
CF3
slide 21
Monte Carlo
1
0.5
-1
0.5
-0.5
1
-0.5
-1
Zvi Wiener
CF3
slide 22
Monte Carlo Simulation
15
10
5
10
20
30
40
-5
-10
-15
Zvi Wiener
CF3
slide 23
Introduction to MC
The idea is very simple
1

0
N
1
f ( x)dx   f ( xi )
N i 1
Hopefully due to the strong law of large
numbers the approximation is good.
Zvi Wiener
CF3
slide 24
Introduction to MC
1

0
N
1
f ( x)dx   f ( xi )
N i 1
How to determine a well distributed sequence?
How one can generate such a sequence?
How to measure precision?
Zvi Wiener
CF3
slide 25
Speed of Convergence
I
 f ( x)dx
[ 0 ,1]s
From the central limit theorem the error of
approximation is distributed normal with

mean 0 and standard deviation
N
 
2
  f ( x)  I  dx
2
[ 0 ,1]s
Zvi Wiener
CF3
slide 26
Regular Grid
An alternative to MC is using a regular grid to
approximate the integral.
Advantages:
The speed of convergence is error~1/N.
All areas are covered more uniformly.
There is no need to generate random numbers.
Disadvantages:
One can’t improve it a little bit.
It is more difficult to use it with a measure.
Zvi Wiener
CF3
slide 27
Variance Reduction
Let X() be an option.
Let Y be a similar option which is correlated
with X but for which we have an analytic
formula.
Introduce a new random variable
X  ( )  X ( )   Y ( )  Y 
Zvi Wiener
CF3
slide 28
Variance Reduction
The variance of the new variable is
var[ X  ]  var[ X ]  2 cov[ X , Y ]   var[Y ]
2
If 2cov[X,Y] > 2var[Y] we have reduced
the variance.
Zvi Wiener
CF3
slide 29
Variance Reduction
The optimal value of  is
cov[ X , Y ]
 
var[Y ]
*
Then the variance of the estimator becomes:
var[ X  * ]  (1   XY )var[ X ]
2
Zvi Wiener
CF3
slide 30
Variance Reduction
Note that we do not have to use the optimal
* in order to get a significant variance
reduction.
Zvi Wiener
CF3
slide 31
Multidimensional Variance Reduction
A simple generalization of the method can be
used when there are several correlated
variables with known expected values.
Let Y1, …, Yn be variables with known means.
Denote by Y the covariance matrix of
variables Y and by XY the n-dimensional
vector of covariances between X and Yi.
Zvi Wiener
CF3
slide 32
Multidimensional Variance Reduction
Then the optimal projection on the Y plane is
given by vector:  *   TXY Y1
The resulting minimum variance is
var[ X  * ]  (1  RXY )var[ X ]
2
where
RXY 
2
Zvi Wiener
   XY
T
XY
1
Y
var[ X ]
CF3
slide 33
Variance Reduction
• Antithetic sampling
• Moment matching/calibration
• Control variate
• Importance sampling
• Stratification
Zvi Wiener
CF3
slide 34
Monte Carlo
• Distribution of market factors
• Simulation of a large number of events
• P&L for each scenario
• Order the results
• VaR = lowest quantile
Zvi Wiener
CF3
slide 35