cf-4 bank hapoalim jun-2001 zvi wiener 02-588-3049 mswiener/zvi.html computational finance

CF-4 Bank Hapoalim Jun-2001

Zvi Wiener

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

Computational Finance

CF4 slide 2Zvi Wiener

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


Monte Carlo

-1 -0.5 0.5 1

-1

-0.5

0.5

1


20 40 60 80 100

-0.04

-0.02

0.02

0.04

0.06

0.08

Speed of convergence

Whole circle

Upper triangle


Smart Sampling


Spectral Truncation


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 in Risk Management

• Distribution of market factors

• Simulation of a large number of events

• P&L for each scenario

• Order the results

• VaR = lowest quantile


How to design MC

The central point is to model the

distribution of relevant risk factors.

For example, in pricing you should use the

risk-neutral distribution.

For risk measurement use true distribution.

What should be used for an estimate of

frequency of hedge?


Geometrical Brownian Motion

SdBrSdtdS

)()( SpayoffEeoption neutralrisktTr


Lognormal process

SdBrSdtdS

dBdtrSd

2)(log

2

tZtr N

eStS)1,0(

2

2)0()(


Euler Scheme

SdBrSdtdS

)()1,0( tOtSZtrSS N


Milstein Scheme

SdBrSdtdS

)()1(2

1 22)1,0(

22

)1,0(

tOtZS

tSZtrSS

N

N


MC for simple options

Needs["Statistics`NormalDistribution`"]

Clear[MCEuropean, MCEuropeanCall, MCEuropeanPut]

nor[mu_,sig_]:=Random[NormalDistribution[mu,sig]];


MC for simple options

MCEuropean[s_, T_, r_, _, n_, exercise_Function]:=

Module[{m = N[Log[s]+(r - 0.52)*T], sg=N[ Sqrt[T] ], tbl},

tbl= Table[nor[m, sg], {i, n}];

Exp[-r*T]*Map[exercise, Exp[Join[tbl, 2*m - tbl]]]//

{Mean[#], StandardErrorOfSampleMean[#]}&

]


MC for simple optionsMCEuropeanCall[s_, x_, T_, r_, _, n_]:=

MCEuropean[s, T, r, , n, Max[#-x,0]&]

MCEuropeanPut[s_, x_, T_, r_, _, n_]:=

MCEuropean[s, T, r, , n, Max[x-#,0]&]


MC for path dependent options

RandomWalk[n_Integer] :=

FoldList[Plus, 0, Table[Random[] - 1/2, {n}]];

ListPlot[ RandomWalk[500], PlotJoined -> True];



The function paths generates a random sample of price

paths for the averaging period. It returns a list of

numberPaths random paths, each consisting of

numberPrices prices over the period from time T1 to time

T. The prices at the start of the period are given by the

appropriate lognormal distribution for time T1.



paths[s_,sigma_,T1_,T_,r_,numberPrices_,numberPaths_]:=

Module[{meanAtT1=Log[s]+(r-sigma^2/2)*T1,

sigmaAtT1 = sigma*Sqrt[T1],

meanPath = 1+ r*(T-T1)/(numberPrices-1),

sigmaPath = sigma*Sqrt[(T-T1)/(numberPrices-1)]

},

Table[NestList[# nor[meanPath,sigmaPath]&,

Exp[nor[meanAtT1,sigmaAtT1]],

numberPrices - 1], {i,numberPaths}]

]


MC for Asian options

MCAsianCall[s_,x_,sigma_,T1_,T_,r_,numberPrices_,numberPaths_]:=

Module[{ t1, t2, t3},

t1 = paths[s,sigma,T1,T,r,numberPrices,numberPaths] ;

t2 = Map[Max[0,Mean[#] - x]&, t1];

t3 = Exp[-T*r]*t2;

{Mean[t3], StandardErrorOfSampleMean[t3]}

]


Quasi Monte Carlo

• Van der Corput

• Halton

• Haber

• Sobol

• Faure

• Niederreiter

• Permutations

• Nets


Quasi Monte Carlo

Are efficient in low (1-2) dimensions.

Sobol sequences can be used for small

dimensions as well.

As an alternative one can create a fixed set of

well-distributed paths.


Do not use free sequences

0.2 0.4 0.6 0.8 1

0.2

0.4

0.6

0.8

1


Other MC applications

• Pricing

• Optimal hedging

• Impact of dividends

• Bounds on a basket

• Prepayments

• Tranches of MBS


Other MC related topics

Use of analytical approximations

Richardson extrapolation

Ratchets example

American properties

Bundling

Modeling Fat tails

CF-4 Bank Hapoalim Jun-2001

Zvi Wiener

The Hebrew University of Jerusalem

[email protected]

Value of Value-at-Risk


P&L

VaR

1 day

1% probability 1d

1 week

1% probability 1w


-1-3 -2

0.5%1%

100%

VaRA

VaR is not sub-additive.Cumulative distribution of assets A and B.

0


Joint distribution of A and BThe integral over the gray triangle is

0.99*0.99 + 0.99*0.005*2/2 = 0.98505, which means thatthere are more than 1% chances to be outside the

gray area and VaRA+B > VaRA+VaRB=$2.

-1-3 -2

VaRA+B

AB

-1

-2

-3

Total loss $1

Total loss $2

0.005 0.005 0.99

0.99

0.005

0.005

0.98010.00495

0.004950.000025

0.0000250.000025

0.000025

0.00495

0.00495


Model

• Bank’s choice of an optimal system

• Depends on the available capital

• Current and potential capital needs

• Queuing model as a base


Required Capital

Let A be total assets

C – capital of a bank

- percentage of qualified assets

k – capital required for traded assets

kAAC 08.0)1(


Maximal Risk (Assets)

The coefficient k varies among systems, but a better

(more expensive) system provides more precise risk

measurement, thus lower k.

Cost of a system is p, paid as a rent (pdt during dt).

Amax is a function of C and p.

k

CA

08.0)1(max


Risky Projects

Deposits arrive and are withdrawn randomly.

All deposits are of the same size.

Invested according to bank’s policy.

Can not be used if capital requirements are

not satisfied.


Arrival of Risky ProjectsWe assume that risky projects arrive randomly (as a Poisson process with density ).

This means that there is a probability dt that during dt one new project arrives.


Arrival of Risky ProjectsA new project is undertaken if the bank has enough capital (according to the existing risk measuring system).

We assume that one can NOT raise capital or change systems quickly.


Termination of Risky Projects

We assume that each risky project disappears randomly (as a Poisson process with density ).


Termination of Risky Projects

We assume that each risky project disappears randomly (as a Poisson process with density ).

This means that there is a probability ndt that during dt one out of n existing projects terminates.

With probability (1-ndt) all existing projects will be active after dt.


ProfitWe assume that each existing risky project gives a profit of dt during dt.

Thus when there are n active projects the bank has instantaneous profit (n-p)dt.


States

After C and p are chosen, the maximal number of active projects is given by s=Amax(C,p).

0 1 2 s-1 s

0 1 2 s-1 s

2 s


States

0 1 2 s-1 s

0 1 2 s-1 s

2 s

Stable distribution:

0 = 1

1 = 2 2

…

s-1 = s s

where

i

n

i

ns

i

i

n

s

ii

i

n

n

n ,

!

!

!

!

00


Probabilities

• Probability of losing a new project due to capital requirements is equal to the probability of being in state s, i. e. s.

• Termination of projects does not have to be Poissonian, only mean and variance matter.

),1()1(

)1(

!

!

0

sn

se

i

nn

s

i

i

n

n


Expected Profit

An optimal p (risk measurement system) can be found by maximizing the expected profit stream.

ppprofitE ps )(1)( ps

ss

),1(

),(


Example

• Capital requirement as a function of p (price) and q (scaling factor), varies between 1.5% and 8%.

q

p

epk

)015.008.0(015.0)(


Example

q

p

epk

)015.008.0(015.0)(

1 2 3 4

12.5

12.75

13.25

13.5

13.75

14

14.25q=0.5

q=1

q=3

p

Amax


Example of a bank

• Capital $200M

• Average project is $20K

• On average 200 new projects arrive each day

• Average life of a project is 2 years

• 15% of assets are traded and q=1

• spread =1.25%


Bank’s profit as a function of cost p. C=$200M, arrival rate 200/d,size $20K, average life 2 yr.,

spread 1.25%, q=1, 15% of assets are traded.

1 2 3 4 5 629.5

30.5

31

31.5

32

32.5

33

rent p

Expected profit


rent p

Expected profit

1 2 3 4 5 6

21

22

23

24

25

Bank’s profit as a function of cost p. C=$200M, arrival rate 200/d,size $20K, average life 2 yr.,

spread 1%, q=1, 5% of assets are traded.


Conclusion• Expensive systems are appropriate for banks with

• low capitalization

• operating in an unstable environment

• Cheaper methods (like the standard approach) should be appropriate for banks with

• high capitalization

• small trading book

• operating in a stable environment• many small uncorrelated, long living projects


A simple intuitive and flexible model of

optimal choice of risk measuring method.

cf-4 bank hapoalim jun-2001 zvi wiener 02-588-3049 mswiener/zvi.html computational finance

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