murex slide show

AMERICAN MONTE CARLO FOR PORTFOLIO

CVA AND PFE HPCFinance Conference

May, 2013 Alexandre Morali

Copyright © 2012 Murex S.A.S. All rights reserved 2

CVA/PFE: New analytics challenges

The CVA/PFE computation for exotics brings new challenges for financial quantitative analysts:

Given a counterparty, full product coverage including both Vanillas and Exotics is fundamental as PFE/CVA figures must

consider the total exposure.

For trade aggregation at Counterparty level, there is need for consistent multi-asset joint calibration & diffusion

models.

CVA/PFE scenarios are generated through Monte Carlo simulations. Nested Monte Carlo is not an option for

performance reasons, as a result Front-office libraries can not be used in a straight forward manner especially for

exotics.

Exotics require complex modelling and heavy computations leading to time consuming calculation. Performance is

always key to have consistent figures in a timely manner.

In this context there is a need for a generic, accurate and efficient resolution method.

American Monte Carlo method seems to be a good candidate for PFE/CVA computation. This

presentation highlights some of the challenges related to the method in this specific context.


Computation of the Potential Future Exposure of a counterpart at t = 𝝉𝒊

A brief reminder of PFE/CVA computation principle

𝑷𝑭𝑬𝒕(𝒑) = 𝒊𝒏𝒇 𝑴:𝑷𝒓 𝑴𝒕𝑴(𝒕) > 𝑴 𝒙(𝒕) = 𝒑%

𝒑 : Quantile value

𝒙 : Risk factor

MTM

(current exposure)

Liability

Worst-case (unexpected)

Exposure

Expected Exposure

Exposure

Liability

.

MTM1 (𝝉𝒊)

MtM2 (𝝉𝒊)

MtMN (𝝉𝒊)

.

.

.

.

.

How to compute those MtM prices?

Horizon

𝑪𝑽𝑨 = 𝟏 − 𝑹 ∗ [𝑬𝑷𝑬(𝒕) ∗ 𝑷𝑫 𝒕, 𝒕 + 𝒅𝒕 ]𝒅𝒕∞

𝟎

𝑹 : Recovery rate

𝑬𝑷𝑬(𝒕) : Expected Positive Exposure at t

𝑷𝑫 𝒕, 𝒕 + 𝒅𝒕 : Counterpart Default Probability between t & t+dt


1

Forward Phase

2

Backward Phase

3

Forward Phase (CVA specific)

Model Diffusion

Storage of deal cash flows

Construction of regression basis

N Monte Carlo paths

Payoff raw evaluation

Regression function building

N MC paths

Payoff evaluation using regression

functions and path dependency

P Monte Carlo paths

Goal :

• Compute at a given date the payoff conditional expectation as a function of observables at that date.

• These observables are called the regression basis.

• The function is called the regression function. It can be parametric (polynomial form) or not.

MTMs evaluation American Monte Carlo: Regression method

Important Remark: In « Phase 3 », MC diffusion can be completly different from « Phase 1 »


MTMs evaluation American Monte Carlo

T1 T2 T

Let us illustrate this with a Bermuda swaption which can be exercised at 2 dates T1 or T2:

Libor

6M



Forward phase: we diffuse risk factors and store in memory contract cash flows

and payoff variables as they will be used during the Backward phase.

T1 T2 T

Libor

6M



T1 T2 T

Backward phase: we discount payoff contract and add cash flows up to T2

U(T)=0 and O(T)=0. We obtain U(T2+) cloud of points. O(T2+) worth 0.

U(t)/O(t) represents the value of the « Underlying Swap »/ « Option » contract at time t

Libor

6M

U(T2+)

O(T2+)



T1 T2 T

At T2+ we obtain U(T2+) cloud of points. O(T2+)=0.

Regression is performed to obtain U(T2-) and O(T2-) as a function of regression basis.

Libor

6M

U(T2+) U(T2-)

O(T2+) O(T2-)



T1 T2 T

We discount Underlying and Option variables again to get U(T1+) and O(T1+)

adding cash flows from table.

Libor

6M O(T2-) O(T1+)

U(T2+) U(T2-) U(T1+)

O(T2+)



T1 T2 T

Regression is performed to obtain U(T2-) and O(T2-) as a function of regression basis

Libor

6M

U(T2+) U(T2-) U(T1+) U(T1-)

O(T2-) O(T1+) O(T1-) O(T2+)



T1 T2 T

Forward phase: We then move forward again and invoke regression functions with the new set of MC paths.

We update the trade state “S” at any “Call date” by comparing regression function values.

Libor

6M U(T1) U(T2)

O(T2) O(T1)

S(T1) S(T2)



T1 T2 T

We can compute distribution of MTMs, exposure, PFE, etc…

MtM1 (T2)

MtM2 (T2)

MtM8 (T2)

.

.

.

.

.

.

Libor

6M

American Monte Carlo for Portfolio CVA and PFE

1 Local versus parametric regression

2 Regression basis choice

3 Convergence to digital features

4 Performance


Parametric versus Local regression Limitation of parametric approach

Let us study and check on a PFE profile the accuracy of the AMC when a closed-form solution exists.

We plot PFE(95) and PFE(99) profiles for the following European swaption.

Swaption characteristics Model framework

1Y european swaption in delivery mode (1M notional)

Underlying wap Tenor: 4y, K = 3% (Forward swap rate: 2.44%)

Frequency Fixed/Floating : Semi Annual

1 Factor Hull and White

Calibration Basket: coterminal swaptions

Monte Carlo number of paths= 8K MC paths

Parametric Regression: Polynomial Form Degree 2

0,00%

10,00%

20,00%

30,00%

40,00%

50,00%

60,00%

70,00%

80,00%

US

D (

%N

oti

on

al)

Horizon

European Swaption USD: Profiles

PFE(5): Parametric PFE(5): Closed Form

PFE(1): Parametric PFE(1): Closed Form



The following table expresses the maximum difference observed on one of the 15 PFE dates for

PFE(95%) and PFE(99%) between closed-form formula and American Monte Carlo

Observations:

The higher the quantile, the higher the difference

The more ITM, the higher the difference for Hull and White model

Higher volatility factors leads to higher differences

Test Case Max error (PFE 95%) Max error (PFE 99%)

HW: K = 2% 9.62% 12.24%

HW: K = 3% 6.20% 9.04%

HW: K = 4% 1.48% 5.74%

HW: K = 2%; High vol factors (*5) 38.69% 48.45%

HW: K = 3%; High vol factors (*5) 35.76% 45.82%

HW: K = 4%; High vol factorsl (*5) 32.83% 43.19%

LMM: K = 2% 3.63% 5.60%

LMM: K = 3% 2.67% 4.70%

LMM: K = 4% 1.41% 3.01%



Comparing parametric function shape induced by the American Monte Carlo when regressing on

the Underlying swap rate highlights differences especially on tail events impacting PFE

computations.

Parametric Regression Results:

Option Value at Expiry 1y (Second order Polynomial Form)

(150 000)

(100 000)

(50 000)

-

50 000

100 000

150 000

200 000

250 000

-1% 0% 1% 2% 3% 4% 5% 6% 7% 8%

Op

tio

n V

alu

e

Swap Rate

MC Realization

Parametric Regression

Closed Form


Parametric versus Local regression Introduction of Local regression approach

Local Regression (LOWESS: LOcally Weighted Scatterplot Smoothing) approach is based on a

definition of subspaces for the cluster of points generated through Monte Carlo method where we

perform locally regression.

Number of MC paths: N

Number of subspaces

Size of subcloud: α

Link function between subcoulds

The method depends on:

Let us apply this method to American Monte Carlo and our previous test case.

Bandwidth(α)


Parametric versus Local regression Results of the local regression approach

Here are the results on the previous cluster of points:

Conclusion: Local regression method allows us to fit the theoretical value of the payoff profile

through the cloud of points.

(150 000)

(100 000)

(50 000)

-

50 000

100 000

150 000

200 000

-1% 0% 1% 2% 3% 4% 5% 6% 7% 8%Op

tio

n V

alu

e

Swap Rate

MC Realization

Closed Form

Local Regression



New profiles with the local regression method on the same case test case:

Conclusion: Visual inspection shows that discrepancies between closed form and American Monte

Carlo have been significantly reduced.

0,00%

10,00%

20,00%

30,00%

40,00%

50,00%

60,00%

70,00%

80,00%

US

D (

%N

om

inal)

Horizon

European Swaption USD: Profiles

PFE(5): Closed Form PFE(1): Closed Form

PFE(1): Local PFE(5): Local



The following table expresses the new results on the European swaption test case

Conclusions: In all cases the local regression method allows a significant reduction of the maximum

error between American Monte Carlo and Closed form expression.

Test Case Max error (PFE 95%) Max error (PFE 99%)

Parametric vs Local Parametric vs Local

HW: K = 2% 9.62% 0.40% 12.24% 0.28%

HW: K = 3% 6.20% 0.34% 9.04% 0.28%

HW: K = 4% 1.48% 0.18% 5.74% 0.27%

HW: K = 2%; High vol factors (*5) 38.69% 2.43% 48.45% 1.17%

HW: K = 3%; High vol factors (*5) 35.76% 2.48% 45.82% 1.16%

HW: K = 4%; High vol factorsl (*5) 32.83% 2.53% 43.19% 1.33%

LMM: K = 2% 3.63% 3.44% 5.60% 4.35%

LMM: K = 3% 2.67% 2.41% 4.70% 3.82%

LMM: K = 4% 1.41% 1.17% 3.01% 2.55%


Parametric versus Local regression Results on CVA

The following graph illustrates the difference between CVA results using local regression versus

parametric

Conclusion:

While impacts on PFE are very significant, impacts on CVA are lower.

PFE considers extreme market data scenario whereas CVA computing is based on a integral of average exposure and

consequently it is less impacted by the lack of accuracy of regression functions at distribution tails.

-

0,50

1,00

1,50

2,00

2,50

3,00

3,50

4,00

4,50

5,00

CV

A (

bp

s)

CVA (bps)

𝑪𝑽𝑨 = 𝟏 − 𝑹 ∗ [𝑬𝑷𝑬(𝒕) ∗ 𝑷𝑫 𝒕, 𝒕 + 𝒅𝒕 ]𝒅𝒕∞

𝟎

𝑹 : Recovery rate

𝑬𝑷𝑬(𝒕) : Expected Positive Exposure at t

𝑷𝑫 𝒕, 𝒕 + 𝒅𝒕 : Counterpart Default Probability between t & t+dt

(150 000)

(100 000)

(50 000)

-

50 000

100 000

150 000

200 000

250 000

-2% 0% 2% 4% 6% 8%

MC Realization


Closed Form





4 Performance


Choice of the regression basis Limitation of using only market observables

We consider a 3y FX TARN product paying annual coupons and submitted to early redemption

Fx Tarn characteristics Model framework

1M USD Nominal EUR/USD 3y Fx Tarn

Coupon Frequency: Annual;

Strike=1.3 Target=8%

Coupon Type: Max(Fx-Strike;0)

1 Factor Hull and White

Fx log normal calibrated on ATM Fx Options

Monte Carlo number of paths = 130K

Regression Type: Local

Conclusions: • While getting closer to maturity, probability to reach the Target gets higher leading an increasing proportion of

« Dead » Scenario No Future Cash Flow Expected on those paths (i.e. No exposure).

• Hence, the regression function does not correctly explain the shape of the cluster of points. This is quite obvious when

very close to maturity.

• Some extreme exposures are not taken into account in regression results leading to PFE underestimation.

The following four graphs illustrate the shape of the cloud of points and the local regression

function at dates T=0.5y, T= 1.5y, T=2.5y, T=2.99y as functions of Fx spot.

EUR/USD EUR/USD EUR/USD EUR/USD


Choice of the regression basis Using payoff observable variable

Adding one additional element in the regression basis such as the cumulated coupon leads to the

following 2 dimension profiles at T=1.5y and T=2.99y (cluster of points is in blue and regression

function in red)

T=1.5y T=2.99y


Choice of the regression basis Using payoff observable variable

Adding one additional element in the regression basis such as the cumulated coupon leads to the

following 2 dimension profiles at T=1.5y and T=2.99y (cluster of points is in blue and regression

function in red).

Adding this dimension allows us to capture simultaneously:

Accumulated

coupon

Fx Spot

“Dead” Scenario: points close to zero.

“Alive” Scenario : Non zero exposure contributing to PFE quantiles estimation.

Fx Spot

Accumulated

coupon

T=1.5y T=2.99y


Choice of the regression basis Numerical results

Looking at the PFE 95% and PFE 99% at each of the four dates considered:

If we look the Rsquare, Regression quality has increased significantly with additional element in the

regression basis.

As expected the exposure is increased when adding the second regression basis so this method

allows a better accuracy of the PFE results.

CVA moves from 4.78 bps to 6.04 bps so is impacted by a bit more than 1 bp.

PFE Date Tarn probability

PFE 95% PFE 99% Rsquare

FX spot FX + Accum FX spot FX + Accum FX spot FX + Accum

0.50 0% 19% 19% 25% 25% 29% 29%

1.50 37% 5% 12% 5% 17% 9% 46%

2.50 53% 2% 5% 3% 10% 8% 51%

2.99 53% 3% 6% 3% 13% 11% 81%





4 Performance


Convergence to digital features Limitation of a generic approach

We consider a 1y Worst-of digital on two equity assets

We plot PFE 95% and PFE 99% profiles:

Worst of characteristics Model framework

1M USD Nominal 1y Digital Equity Worst-Of

Spot 1 = Spot 2 = 10,000. Zero correlation.

Strike = 10,000

S1 & S2 log normal model: Calibration on ATM vol

Monte Carlo number of paths = 16K

Regression Type: Local

0%

20%

40%

60%

80%

100%

120%

0 0,2 0,4 0,6 0,8 1

USD

(%

No

min

al)

Horizon

Digital On WorstOf PFE profile

PFE(99%) Alpha=30% PFE(95%) Alpha=30%

Final 99% Final 95%

~=87%

~=105%

The method highlights obvious

flaws

At maturity PFE 99% is greater

(105%) than the engaged nominal

which is not possible.

Knowing that 11% of the paths are in

the money at maturity PFE (95%)

should also be equal to nominal.


Convergence to digital features Limitation of a generic approach

Spot 1

Spot 2

Plotting close to Option Expiry, the regression result as a function of [Spot 1,Spot 2] indicates

that the angle is not captured accurately.

Moreover for high spot values the regression function overshoots the digital “expected” profile

explaining why PFE(99%) is above the maximum possible exposition of 100%.

The sub cloud of points (alpha = 30% of the total cloud) for each sub space is too big to capture

this shape

T=1y


Convergence to digital features Reduction of the size of the sub clouds of points

Reducing the size of each sub-cloud of points allows us to better capture the shape of the digital

worst-of payoff

Spot 1

Spot 2

T=1y


Convergence to digital features Reduction of the size of the sub clouds of points

Obvious flaws found in the first place are solved by reducing the size of the cloud from 30% to 3%

The smaller the size of the sub clouds of points, the better the convergence.

0%

20%

40%

60%

80%

100%

120%

0 0,2 0,4 0,6 0,8 1

USD

(%

No

min

al)

Digital On Worst-Of PFE profile

PFE(99%) Alpha=30% PFE(95%) Alpha=30%

PFE(99): Alpha = 3% PFE(95): Alpha = 3%





4 Performance


Performance CPU vs GPU computation time

“Limits pre-deal check” context

Check no PFE limit break

Compute CVA fee and reflect it on price

Arbitrate between different counterparts to optimize CVA consumption

Swaption characteristics Model framework

Bermuda Swaption: 10y; First call date: 1y (36 call dates total)

Underlying Frequency Fixed/Floating : Quaterly

Call frequency: Quaterly

1 Factor LMM or HW model

Calibration Basket: Underlying Swaptions

Regression on 65,000 paths

Model CPU (sec)

HW 1.81

LMM 65.51

GPU (sec) Speed up

0.5 4x

2.7 25x


Regression Basis Dimension is a headache

0,032

0,078

0,125

0,359

0

0,1

0,2

0,3

0,4

100 000 250 000 500 000 1 000 000

1D

0,063 0,172

0,609

2,062

0

0,5

1

1,5

2

2,5

100 000 250 000 500 000 1 000 000

2D

0,235 0,985

3,109

12,203

0

2

4

6

8

10

12

14

100 000 250 000 500 000 1 000 000

3D

0,0218

0,0624

0,125

0,253

0

0,05

0,1

0,15

0,2

0,25

0,3

100 000 250 000 500 000 1 000 000

0,0624

0,2312

0,472

0,953

0

0,2

0,4

0,6

0,8

1

1,2

100 000 250 000 500 000 1 000 000

0,2468

0,75

1,5782

3,1124

0

0,5

1

1,5

2

2,5

3

3,5

100 000 250 000 500 000 1 000 000

Local Regression


1 CPU CORE

Single regression timings as a funtion of MC paths & Regression basis dimension


Regression Basis Dimension is a headache

0,0015

0,0032

0,0047

0,0078

0

0,001

0,002

0,003

0,004

0,005

0,006

0,007

0,008

0,009

100 000 250 000 500 000 1 000 000

0,0031 0,0046

0,0078

0,0157

0

0,002

0,004

0,006

0,008

0,01

0,012

0,014

0,016

0,018

100 000 250 000 500 000 1 000 000

x40

x20

x80

0,0344 0,0406

0,0484

0,0687

0

0,01

0,02

0,03

0,04

0,05

0,06

0,07

0,08

100 000 250 000 500 000 1 000 000

x25

x10

x65

GPU

… and for a 10 year Bermuda swaptions on a quaterly swap and 160 PFE dates, (160+36)*2 regressions are

needed.

Most of Exotics will require high dimension regression basis.

0,032

0,078

0,125

0,359

0

0,1

0,2

0,3

0,4

100 000 250 000 500 000 1 000 000

1D

0,063 0,172

0,609

2,062

0

0,5

1

1,5

2

2,5

100 000 250 000 500 000 1 000 000

2D

0,235 0,985

3,109

12,203

0

2

4

6

8

10

12

14

100 000 250 000 500 000 1 000 000

3D

CPU: 1 core

x25

x20

x25

x45 x130 x180


Conclusion

American Monte Carlo is an interesting technique allowing us to model an extensive set of payoffs

in a CVA/PFE context which is absolutely crucial.

But it has many critical parameters when it comes to compute extreme quantiles exposure.

Choosing the right regression function to match extreme quantiles exposure

Choosing the right regression basis to explain properly the payoff

Finding the right way of matching Payoff profile

Performance is a real challenge and switching to GPU allows close to « Real time » computing

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