cf-3 bank hapoalim jun-2001 zvi wiener 02-588-3049 mswiener/zvi.html computational finance

CF-3 Bank Hapoalim Jun-2001

Zvi Wiener

02-588-3049http://pluto.mscc.huji.ac.il/~mswiener/zvi.html

Computational Finance

CF3 slide 2Zvi Wiener

Plan

1. Hypothesis test

2. Maximal Likelihood estimate

3. Multidimensional VaR, contour plots

4. Monte Carlo type methods


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).


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.


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.

m

k k

kk

e

efQ

1

2)(


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.



Define test statistics by

)()(sup xGxFnD 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).

222

1

)1(2)( ti

i

ietYP



-2 -1 1 2

0.2

0.4

0.6

0.8

1

-2 -1 1 2

0.2

0.4

0.6

0.8

1

-2 -1 1 2

-0.04

-0.02

0.02

0.04

0.06

0.08

data hypothesis

difference


Maximal Likelihood


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?


Example

A

B

Probability density


2 dimensional risk

-1 1 A

B

1

-1Probability of 5%


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

95.014

1

2

1 2 x

x=0.6325, and VaR95%=2-x=1.3675


OvalsConsider 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.


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.


Ovals

The value of the benchmark today is 1.rTrT eeBenchmark 0

quantity discount factor

The value of the benchmark tomorrow will be

))((1

dTTdrrrT eeBenchmark

For short time intervals we ignore dT.


Ovals

Similarly the dollar P&L of the portfolio will be

)()(01 )1( qTdrqTdL eaaePP

We ignore convexity and carry effects.

To measure relative performance we use

drT

qTdrqTdL

e

eaae

BB

PP

)()(

01

01 )1(


Assume that dr and dL are jointly normal and use delta approach. Then the gradient vector is

)(

))(1(

qTa

qTaTgrad

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.


Assuming correlation of 26% between dr and dL we have:

gradgradbenchmportf ../

The resulting contour plot shows the levels of risk for any potential position as seen according to today’s data.


0 0.1 0.2 0.3 0.4 0.5

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

spread

OvalsVaR=1bp

VaR=2bp


Plan1. 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


Monte Carlo

-1 -0.5 0.5 1

-1

-0.5

0.5

1


Monte Carlo Simulation

10 20 30 40

-15

-10

-5

5

10

15


Introduction to MC

N

iixf

Ndxxf

1

1

0

)(1

)(

Hopefully due to the strong law of large

numbers the approximation is good.

The idea is very simple


Introduction to MC

N

iixf

Ndxxf

1

1

0

)(1

)(

How to determine a well distributed sequence?

How one can generate such a sequence?

How to measure precision?


Speed of Convergence

From the central limit theorem the error of approximation is distributed normal with mean 0 and standard deviation

s

dxIxf]1,0[

22 )(

s

dxxfI]1,0[

)(

N


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.


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

YYXX )()()(


Variance Reduction

The variance of the new variable is

]var[],cov[2]var[]var[ 2 YYXXX

If 2cov[X,Y] > 2var[Y] we have reduced

the variance.


Variance Reduction

The optimal value of is

Then the variance of the estimator becomes:

]var[

],cov[*

Y

YX

]var[)1(]var[ 2* XX XY


Variance Reduction

Note that we do not have to use the optimal

* in order to get a significant variance

reduction.


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.


Multidimensional Variance Reduction

Then the optimal projection on the Y plane is given by vector: 1* Y

TXY

The resulting minimum variance is

]var[)1(]var[ 2* XRX XY

where

]var[

12

XR XYY

TXY

XY


Variance Reduction

• Antithetic sampling

• Moment matching/calibration

• Control variate

• Importance sampling

• Stratification


Monte Carlo

• Distribution of market factors

• Simulation of a large number of events

• P&L for each scenario

• Order the results

• VaR = lowest quantile

cf-3 bank hapoalim jun-2001 zvi wiener 02-588-3049 mswiener/zvi.html computational finance

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