don’t overexpose yourself with risk neutral pfes and epes

Don’toverexposeyourself

Harvey Stein

Introduction

Exposuredefinitions

PFE Math

Constructingnumeraires

Equity examples

Fixed incomeexamples

More fixedincome

Canonicalpricing measure

Basel impact

Back to the realworld

Summary

References

Don’t overexpose yourself with risk neutral

PFEs and EPEs

Harvey [email protected]

Head, Regulation and Credit ModelingStrategic Risk Research

Bloomberg LP

QWAFAFEWMay 2014

Id: UnderexposedRiskSnapshots-body.latex.tex 44309 2014-05-27 19:59:06Z hjstein

1 / 51


Harvey Stein

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Outline1 Introduction

2 Exposure definitions

3 PFE Math

4 Constructing numeraires

5 Equity examples

6 Fixed income examples

7 More fixed income

8 Canonical pricing measure

9 Basel impact

10 Back to the real world

11 Summary

Bibliography

2 / 51


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PFE Math


Equity examples


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Basel impact


Summary

References

Exposure calculations

Counterparty credit risk management uses various credit exposuremeasures to manage credit risk [Basel Committee on BankingSupervision, 2005, 2011, Canabarro and Duffie, 2003]

Bank credit limits are set based on

• Potential future exposure (PFE)

• Expected exposures (EE)

• Expected positive exposures (EPE)

Basel II and III capital charges are set based on

• Effective expected positive exposure (EEPE)

• Expected exposures

• Effective expected exposures (EEE)

It is important to know what these are and how they behave.

3 / 51


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Underexposure or overexposure?

CVA calculations are also needed for risk management.

It’s become common to try to kill two birds with one stone:

• People try to use CVA calculation infrastructures to computeexposures.

The problem is that this computes the exposures under the riskneutral measure used for the CVA calculation, not under the realworld measure.

Here we will look at how badly this impacts the exposure numbers[see Stein, 2013b].

4 / 51


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Potential Future Exposure

A PFE at time T of X with a confidence level of p means that p ofthe time, the value of the portfolio at time T will be X or less.

In other words, a PFE of X for a confidence level of p satisfies:

P[VT < X ] = EP [1VT<X ] = p

where P is the real world probability measure, and VT is the time T

value of the portfolio.

For example, a 1 year 95% PFE of $1,000,000 means that there is a95% chance that the portfolio will be worth $1,000,000 or less in ayear, or in other words, a 95% chance that the exposure will notexceed $1,000,000.

5 / 51


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Expected exposures

The expected exposure (EE) at time T for a counterparty is theexpected loss due default of the counterparty1:

EE (V ,T ) = EP [max(VT , 0)] = EP [VT1VT≥0]

The expected positive exposure (EPE) is the average loss over a timeperiod:

EPE (V , t1, t2) =1

t2 − t1

∫ t2

t1

EE (V , s)ds

1Exposure measures assume zero recovery6 / 51


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Effective exposures

The effective expected exposure (EEE) for a time period is the worstexpected exposure that occurs in that period:

EEE (V , t1, t2) = maxt1≤s≤t2

(EE (V , s))

The effective expected positive exposure (EEPE) is the average of theeffective expected exposures over a time period:

EEPE (V , t1, t2) =1

t2 − t1

∫ t2

t1

EEE (V , t1, s)ds

Basel II and III use EEPE (V , 0, 1) and EE to determine capitalcharges. The EEE comes into play as well.

7 / 51


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CVA calculations

For CVA, we need to calculate the current market value of the loss ifa default occurs at time t.

The loss at time t with zero recovery rates would be:

max(Vt , 0)

The current market value of this loss is:

EQ

[

max(Vt , 0)

Nt

]

N0

where Q is the risk neutral measure with respect to numeraire N.

This is the value of the call option on the portfolio that matures attime t. The CVA is the value of the above for t being the defaulttime [Stein and Lee, 2011, Pykhtin and Zhu, 2007, Brigo andCapponi, 2009].

8 / 51


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PFE Math


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Shortcut

If one has built systems that can compute the CVA, it is seductive touse the same system to compute credit exposures.

• With minor additional work, CVA calculations can yield riskneutral credit exposures.

But the measure used depends on the numeraire used in the CVAcalculation. Different CVA systems can use different measures andnumeraires even when using the same models calibrated to the samedata.

Question: How much of a difference does the choice of numerairemake?

9 / 51


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Introduction

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PFE Math


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Summary

References

PFEs as CDFs

The PFE at time T for a portfolio with value VT is the value X

satisfyingP[VT < X ] = EP [1VT<X ] = p.

ThenX = F−1

P (p)

whereFP(X ) = EP [1VT<X ]

is the cumulative distribution function for V under the measure P .

So, the question of how the PFE varies as the risk neutral measurechanges is really a question of how the distribution of the portfoliovalue changes as the measure changes.

10 / 51


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Numeraireology

What is a numeraire?

• A numeraire is just an asset or self financing strategy that isalways positive.

• We price derivatives by dividing by a numeraire and computingexpectations.

What is the intuition?

• Dollars are not tradeable in the market, so quoting the prices ofassets in terms of dollars is arbitrary.

• Multiplying all the prices by arbitrary functions of time wouldyield the same market.

• Dividing by the price of an asset in the market expresses prices interms of shares of a tradeable asset, thus normalizing everything.

11 / 51


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Numeraireology

How does dividing by the numeraire simplify calculations?

• The market is arbitrage free iff prices are given by integratingagainst measures in a consistent way [Harrison and Kreps, 1979,Harrison and Pliska, 1981, Delbaen and Schachermayer, 1994].

• After dividing by a numeraire, the numeraire becomes an assetwith a constant value of 1.

• These measures become probability measures and integratingagainst them becomes computing expectations.

• Each asset divided by the numeraire becomes a martingale:

St/Nt = EQ [ST/NT ]

This greatly simplifies calculations [Stein, 2007].

Note — This can be done with any selection of numeraire, butdifferent choices of numeraires will necessarily correspond to differentmeasures.

12 / 51


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PFEs and measure change.Consider two risk neutral measures Q and Q ′ with respect tonumeraires N and N ′, respectively. If these are for the same model,then for all contingent claims X paid at time T , we have that

S0 = EQ [X/NT ]N0 = EQ′

[X/N ′T ]N

′0

ThenEQ [X ] = EQ′

[XNT/N′T ]N

′0/N0

The CDF of VT , FQ(X ) satisfies:

FQ(X ) = EQ [1VT<X ] = EQ′

[

1VT<X

NT/N0

N ′T/N

′0

]

So, the impact on the PFE of the measure change from Q ′ to Q isthe difference between computing

EQ′

[1VT<X ]

and computing

EQ′

[

1VT<X

NT/N0

N ′T/N

′0

]

13 / 51


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Numeraire hackingTo see the impact of this numeraire change, we will consider theT -forward measure and construct a numeraire change.

Let Q ′ be the T forward measure. Then N ′t = Z (t,T ), the time t

value of the zero coupon bond maturing at time T . Consider the riskneutral measure Q with respect to numeraire N.

Under Q ′, Nt/N′t is a martingale, so

N0 = EQ′

[NT/N′T ]N

′0 = EQ′

[NT ]Z (0,T )

So, the change of measure from Q ′ to the risk neutral measure Q

with respect to numeraire N is then

dQ

dQ ′=

NT

EQ′ [NT ]

and the percentile level for an exposure of X under Q is

EQ [1VT<X ] = EQ′

[

1VT<X

NT

EQ′ [NT ]

]

14 / 51


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Numeraire hackingConsider 1VT<X and 1VT≥X . We have that

E [1VT<X ] + E [1VT≥X ] = 1

and each expectation is nonnegative. Pick positive numbers A and B

such thatAE [1VT<X ] + BE [1VT≥X ] = 1

Define NT byNT = A1VT<X + B1VT≥X

ThenNt = EQ′

t [NT/N′T ]N

′t

is a numeraire. Let its measure be Q. Under this numeraire,

EQ [1VT<X ] = EQ′

[

1VT<X

NT

EQ′ [NT ]

]

= EQ′

[1VT<X (A1VT<X + B1VT≥X )]

= AEQ′

[1VT<X ]

Using this numeraire, we can set the level X PFE to anything wewant.15 / 51


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Numeraire hacking

Let’s get a PFE of no more than X at a confidence level of p underthe risk neutral measure Q.

If EQ′

[1VT<X ] ≥ p, just use the forward measure Q ′.

If EQ′

[1VT<X ] < p, pick

A =p

EQ′ [1VT<X ]

Then A > 0 and B = 1−p

EQ′ [1VT<X ]> 0, so NT > 0 and is the value of a

legal numeraire at time T . Then

EQ [1VT<X ] = AEQ′

[1VT<X ] = p

so under the risk neutral measure Q for numeraire N, the PFE is X .

16 / 51


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Pick a PFE, any PFE

More generally, the full PFE profile under Q ′ is given by F−1Q′ (p),

whereFQ′(X ) = EQ′

[1VT<X ]

is the CDF for VT .

Consider a desired target PFE profile F . If FQ′ is an increasing,continuous function of X , then by the same approach, along with theinverse function theorem, we can find a numeraire N and acorresponding measure Q such that

F (X ) = EQ′

[1VT<XNT/EQ′

[NT ]] = EQ [1VT<X ]

thus rendering any desired PFE profile.

In fact, by doing this at T1, and then at T2 > T1 conditional on T1,etc, we can match an arbitrary set of PFE profiles at the discretetimes Ti .

17 / 51


Harvey Stein

Introduction

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PFE Math


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Basel impact


Summary

References

Risk neutral exposures?

The same game can be played for any exposure measure (EE, EPE,etc).

So, what is meant by a risk neutral exposure?

• Anything we want!

But, for firms which aren’t so nefarious as to pick a numeraire to settheir exposures to anything they desire, how much can the exposuremeasures be affected?

18 / 51


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Equity examples

Consider the Black-Scholes model:

dSt = µStdt + σStdWt

dBt = rBtdt

Under the risk neutral measure with respect to the money marketaccount, St/Bt must be a martingale, so:

St = e(r−σ2/2)t+σW ′t

We can also use the stock process itself as our numeraire. Then, it’sBt/St that must be a martingale, and:

St = e(r+σ2/2)t−σW ′′t

19 / 51


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PFE profiles

Consider a portfolio consisting of a share of the stock. Under themoney market numeraire, our time T CDF is

FB(X ) = Φ

(

lnX − (r − σ2/2)T

σ√T

)

With the stock numeraire, we have:

FS(X ) = Φ

(

lnX − (r + σ2/2)T

σ√T

)

20 / 51


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Stock PFEThe difference in drift can have a significant impact. Consider thePFE of one share of stock currently priced at $1 as a function ofhorizon time with 30% volatility, 5% interest rates and no dividends.The stock measure yields PFEs as much as 50% higher than themoney market numeraire.

1.5

2

2.5

3

3.5

4

4.5

5

1 1.5 2 2.5 3 3.5 4 4.5 5 0

20

40

60

80

100

PFE

Perc

enta

ge

Horizon

Stock Potential Future Exposure

Money MarketStock Measure

Relative change

21 / 51


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Stock EEUnder the same conditions, the stock EE can also be over 50% higherunder the stock measure versus the money market measure.

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0

20

40

60

80

100

EE

Perc

enta

ge

Horizon

Stock Expected Exposure


Relative change

22 / 51


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Stock Call PFESince stock option prices are monotonic in the stock price, the p levelPFE of a call option is the value of the call option at the p level PFEof the stock. The 1 year PFE can be over 20% higher in the stockmarket measure.

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 8

10

12

14

16

18

20

22

24

26

PFE

Perc

enta

ge

Strike

1 year PFE, 5 year call


Relative change

23 / 51


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Stock Put PFE

The put prices exhibit large relative differences as the strike gets small.

0

0.2

0.4

0.6

0.8

1

1.2

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0

20

40

60

80

100PF

E

Perc

enta

ge

Strike

1 year PFE, 5 year put


Relative change

24 / 51


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Stock Call EE

The EE of the call option shows similar differences to the PFE.

0

0.2

0.4

0.6

0.8

1

1.2

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 5

10

15

20

25

30PF

E

Perc

enta

ge

Strike

1 year EE, 5 year Call


Relative change

25 / 51


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Stock Put EE

The EE of the put options are similarly affected.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 15

20

25

30

35

40

45

50

55

60

65PF

E

Perc

enta

ge

Strike

1 year EE, 5 year Put


Relative change

26 / 51


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Fixed income examples

For examples from fixed income, we will consider the Hull-Whitemodel with constant parameters.

The Hull-White model is commonly used with three differentnumeraires:

• The money market numeraire

• The T -forward measure

• The LGM numeraire.

We will consider some fixed income portfolios and how their PFEscompare.

27 / 51


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Hull-White, money marketUnder the Hull-White model with respect to the money marketnumeraire, the short rate follows a mean reverting normal process:

drt = (θ(t)− ar)dt + σdWt

The time dependent parameter θ is chosen to calibrate to the yieldcurve, and a and σ can be chosen to fit some observed option prices.

Solving for r :

rT = ea(t−T )rt + e−aT

∫ T

t

eauθ(u)du + e−aT

∫ T

t

eauσdWu

so conditional on rt , rT is Gaussian, with mean

ea(t−T )rt + e−aT

∫ T

t

eauθ(u)du

and variance∫ T

t

(e−aT eauσ)2du =σ2

2ae2aT

(

e2aT − e2at)

28 / 51


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Hull-White, money market

In the Hull-White model under the money market measure, the moneymarket account itself is

Bt = e∫

T

0rsds

The zero coupon bond price processes are given by:

Z (t,T ) = Et [1/BT ]Bt

Their SDEs are:

dZ (t,T )/Z (t,T ) = rtdt +σ

a(ea(t−T ) − 1)dWt

This allows us to write a formula for the zero coupon bond prices attime t in terms of the short rate at time t.

29 / 51


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Hull-White, forward measure

Under the T -forward measure, Z (t, S)/Z (t,T ) must be a martingale.Computing its SDE, we see that to be a martingale, we must be inthe measure under which W ′

t is Brownian motion, where

dW ′t = dWt +

σ

a

(

1− ea(t−T ))

dt

in which case

drt =

(

θ(t)− ar +σ2

a(ea(t−T ) − 1)

)

dt + σdW ′t

which we can similarly solve rt to find that it’s Gaussian with thesame variance, but a different mean.

30 / 51


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Hull-White, LGM style

Our third numeraire of choice is the LGM version of the Hull-Whitemodel. This is given by choosing α(t) and H(t) and setting

dXt = α(t)dWt

η(t) =

∫ t

0

α2(s)ds

Nt =1

D(t)eH(t)Xt+

12H

2(t)η(t)

Under the measure for which Wt is a Brownian motion, andD(t) = Z (0, t), the current discount factor for time t, then Nt is ournumeraire, and the zero coupon bond prices are given by:

Z (t,T ) = E [1/NT ]N0

31 / 51


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LGM is Hull-White

We had the LGM model:

dXt = α(t)dWt

η(t) =

∫ t

0

α2(s)ds

Nt =1

D(t)eH(t)Xt+

12H

2(t)η(t)

The equivalent to the Hull-White model by choosing H and α tosatisfy:

a = −H ′′(t)/H ′(t)

σ(t) = H ′(t)α(t)

θ(t) = −(logD(t))′H ′′(t)/H ′(t) + (logD(t))′′ − (H ′(t))2η(t).

32 / 51


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Hull-White PFEs and EEs

To compute 95% time T PFEs under Hull-White for zero couponbonds, since ZCB prices are monotonic in r and X , we need only findthe 5th percentile level of r and the 95th percentile level of X , anddetermine the value of the ZCB at this level.

For EEs, we need to compute expectations of max(Vt , 0) under thedifferent models. This can be done for bonds and swaps by notingthat their horizon prices are monotonic in the short rate, so we needonly integrate Vt from the rt such that Vt = 0 out to plus or minusinfinity.

33 / 51


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Hull-White ZCB PFEEven though PFEs and EEs are most important for OTC derivatives,it is illustrative to look at how they vary for simple instruments, suchas zero coupon bonds and coupon bonds. Here we see that at the 5year horizon long zero coupon bond PFEs can differ as much as 30%depending on the choice of measure.

0.4

0.5

0.6

0.7

0.8

0.9

1

5 10 15 20 25 30 0

5

10

15

20

25

30

35

PFE

Perc

enta

ge

Bond Maturity

5 year 95% PFEs of zero coupon bonds

Money market30 yr fwd60 yr fwd

LGMRelative change

34 / 51


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Hull-White ZCB EE

We see similar difference for the EEs.

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

5 10 15 20 25 30-5

0

5

10

15

20

25

30

35E

E

Perc

enta

ge

Bond Maturity

5 year EE of zero coupon bonds


LGMRelative change

35 / 51


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Swap PFEsNow let’s consider Swap PFEs. The relative differences blow up aseach crosses through zero. Away from zero, we see that the PFEseasily differ by a factor of 2 or more, with an absolute difference inPFEs of as much as 1.8 million.

-6e+06

-4e+06

-2e+06

0

2e+06

4e+06

6e+06

8e+06

1e+07

0 2 4 6 8 10 400000

600000

800000

1e+06

1.2e+06

1.4e+06

1.6e+06

1.8e+06

PFE

Dif

fere

nce

Coupon

5 yr 95% PFE, 10 yr pay fix swap, 10m notional


LGMMax diff

36 / 51


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Swap PFEs

We have a similar result for receive fixed swaps.

-1e+06

0

1e+06

2e+06

3e+06

4e+06

5e+06

6e+06

0 2 4 6 8 10 660000 680000 700000 720000 740000 760000 780000 800000 820000 840000 860000 880000

PFE

Dif

fere

nce

Coupon

5 yr 95% PFE, 10 yr rec fix swap, 10m notional


LGMMax diff

37 / 51


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Swap EEsThe swap EEs show the profile we would expect from their call optionpayoff. With different drifts, they turn at different points, yieldingsubstantial differences, with the EE under the money market measurebeing 3 times that under the LGM measure for an at the money swap.

0

500000

1e+06

1.5e+06

2e+06

2.5e+06

0 2 4 6 8 10 0

100

200

300

400

500

EE

Perc

enta

ge

Coupon

5 yr EE, 10 yr pay fix swap, 10m notional


LGMRelative change

38 / 51


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Swap EEsFor an at the money receive fixed swap, the EE under the moneymarket measure being double that under the LGM measure.

0

500000

1e+06

1.5e+06

2e+06

2.5e+06

3e+06

3.5e+06

0 2 4 6 8 10 0

100

200

300

400

500

EE

Perc

enta

ge

Coupon

5 yr EE, 10 yr rec fix swap, 10m notional


LGMRelative change

39 / 51


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More fixed income

A Markov functional model is given by:

dXt = σ(t)dWt

Nt = f (t,Xt)

Z (t,T ) = Et [1/f (T ,XT )]f (t,Xt)

It is similar to the LGM model, in that the numeraire is a function ofa driftless diffusion X . The difference lies in that the function f ischosen to calibrate to the market.

This gives us yet another numeraire used in practice, and furtherdifferences in exposures.

40 / 51


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Canonical pricing measure

In a complete model, while numeraires are not unique, the pricingmeasure itself is unique. Rescaling the pricing measure to define aprobability measure gives us a canonical risk neutral measure.

With a risk neutral measure Q and numeraire N, the pricing functionon a contingent claim paying X at time T is

V(X ) = EQ [X/NT ]N0.

The pricing measure is given by

Q∗(A) =

∫

A

N0/NTdQ

and is independent of choice of numeraire.

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Harvey Stein

Introduction

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PFE Math


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Basel impact


Summary

References

Canonical pricing probability

To make Q∗ a probability measure, we divide by the measure of thewhole space

Q∗(Ω) =

∫

Ω

dQ∗ = V(1Ω) = Z (0,T )

This then yields a canonical probabability measure:

Q = Q∗/Q∗(Ω) = Q∗/Z (0,T )

Dividing by Q∗(Ω) is the only way to make the pricing measure aprobability measure while preserving the measure’s partial order.

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Harvey Stein

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PFE Math


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Under Q, PFEs are given by

E Q [1VT<X ] = EQ∗

[1VT<X/Z (0,T )] = V(1VT<X/Z (0,T ))

so under the canonical pricing probability, PFEs are the price of thecontingent claim

1VT<X/Z (0,T )

which is another way of seeing that it is independent of choice ofnumeraire and easy to compute under any risk neutral measure.

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Harvey Stein

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Summary

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Surprisingly, the canonical pricing measure Q is actually theT -forward measure!

V(X ) = EQ [X ]Z (0,T ) =

∫

XdQ∗ = E Q [X ]Z (0,T )

so PFEs under Q are PFEs under the T -forward measure.

However, this means we must use the T -forward measure for time T

exposures, not just choose a terminal T -forward measure to computePFEs for arbitrary horizons.

Also, while this is a canonical choice, it is still different from the realworld measure:

• Exposures will differ from the real world measure

• Risk management implications are unclear

Moreover, this does not resolve the multi-currency case, as there’s achoice of pricing measure, one for each currency.

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Harvey Stein

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Basel impact


Summary

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Basel impactRisk weighed assets (RWA) under Basel:

RWA = 12.5× K × EAD

K = LGD ×

(

Φ

(

Φ−1(PD) +√RΦ−1(0.999)√

1− R

)

− PD

)

× 1 + (M − 2.5)b

1− 1.5b

b = (0.11852− 0.05478 log(PD))2

R = 0.121− e−50PD

1− e−50+ 0.24

(

1− 1− e−50PD

1− e−50

)

Exposure at default (EAD) when using the Internal Model Method:

EAD = α× EEPE

α = max(Economic capital (EC)/EC based on EPE, 1.2)

If you can pick the numeraire, then you can set your RWA toanything you’d like.

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Harvey Stein

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Basel III CVA

For IMM banks, Basel III stipulates a formula for the standard CVAcharge:

CVA = LGD∑

i

max(0, e−s(ti−1)ti−1

LGD − e−s(ti )ti

LGD )

× EE (ti−1)D(ti−1) + EE (ti )D(ti )

2

where si is the credit spread at time ti .

Here we go the other way — the EE is assumed to be under the realworld measure even though this is intended to approximate the CVAcomputation.

It can be argued that one should use a “risk neutral” EE, but then,we are not normalizing by the numeraire, leading once again to theability to adjust one’s capital.

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Harvey Stein

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Back to the real world

Salvaging the CVA computational framework without suffering fromrisk neutral measures is possible. Options include:

• Make the change of measure to the real world measure explicit.

• Use the above techniques with the change of measure tocompute real world exposures using the risk neutral framework.

• Compute horizon prices using the CVA framework, but evaluatethe prices at the real world measure.

The first option potentially requires modifying one’s risk neutralmodel. The second option does not.

For more details on such approaches, see Stein [2013a].

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Harvey Stein

Introduction

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PFE Math


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Basel impact


Summary

References

Summary

When computing exposures using a risk neutral measure, one obtainsdifferent results depending on the choice of numeraire. We looked atthe impact of the numeraire choice and saw that:

• One can pick a numeraire to make the exposure match anydesired profile.

• As such, exposures under risk neutral measures are arbitrary.

• Even if one restricts oneself to commonly used numeraires, thereare still substantial differences in computed exposures, with PFEsand EEs on swaps differing between common numeraires by largefactors.

• EEs can vary by a factor of 2 or more.

• Changing one’s pricing methodology (not one’s pricing model)can substantially change computed exposures.

We recommend using techniques such as in Stein [2013a] to use a riskneutral CVA framework to compute real world exposures.

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Harvey Stein

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References I

Basel Committee on Banking Supervision. The application of Basel II totrading activities and the treatment of double default effects. Technicalreport, Bank of International Settlement, June 2005. URLhttp://www.bis.org/publ/bcbs116.pdf.

Basel Committee on Banking Supervision. Basel III: A global regulatoryframework for more resilient banks and banking systems. Technicalreport, Bank of International Settlement, June 2011. URLhttp://www.bis.org/publ/bcbs189.htm.

D. Brigo and A. Capponi. Bilateral Counterparty Risk Valuation withStochastic Dynamical Models and Application to Credit Default Swaps.SSRN, 2009. URL http://ssrn.com/abstract=1318024.

Eduardo Canabarro and Darrell Duffie. Measuring and markingcounterparty risk. Asset/Liability Management of Financial Institutions,ed. Leo M. Tilman, Institutional Investor Books, 2003. URLhttp://www.darrellduffie.com/uploads/surveys/DuffieCanabarro2004.pd

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http://www.bis.org/publ/bcbs116.pdf

http://www.bis.org/publ/bcbs189.htm

http://ssrn.com/abstract=1318024

http://www.darrellduffie.com/uploads/surveys/DuffieCanabarro2004.pdf⤮


Harvey Stein

Introduction

Exposuredefinitions

PFE Math


Equity examples


More fixedincome


Basel impact


Summary

References

References II

Freddy Delbaen and Walter Schachermayer. A general version of thefundamental theorem of asset pricing. Mathematische annalen, 300(1):463–520, 1994.

J Michael Harrison and David M Kreps. Martingales and arbitrage inmultiperiod securities markets. Journal of Economic theory, 20(3):381–408, 1979.

J Michael Harrison and Stanley R Pliska. Martingales and stochasticintegrals in the theory of continuous trading. Stochastic processes andtheir applications, 11(3):215–260, 1981.

M. Pykhtin and S. Zhu. A guide to modelling counterparty credit risk.Global Association of Risk Professionals, July/August 2007.

Harvey J. Stein. Valuation of Exotic Interest Rate Derivatives - Bermudansand Range Accruals. SSRN, December 2007. URLhttp://ssrn.com/abstract=1068985.

Harvey J. Stein. Joining risks and rewards. Technical report, BloombergLP, December 2013a. URL http://ssrn.com/abstract=2368905.

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Harvey Stein

Introduction

Exposuredefinitions

PFE Math


Equity examples


More fixedincome


Basel impact


Summary

References

References IIIHarvey J. Stein. Fixing underexposured snapshots: Proper computation of

credit exposures under the real world and risk neutral measures.Technical report, Bloomberg LP, December 2013b. URLhttp://ssrn.com/abstract=2365540.

Harvey J. Stein and K. P. Lee. Counterparty valuation adjustments. InTomasz Bielecki, Damiano Brigo, and Frederic Patras, editors, CreditRisk Frontiers: Subprime Crisis, Pricing and Hedging, CVA, MBS,Ratings, and Liquidity, volume 138. Wiley.com, New York, 2011.

c© Bloomberg Finance L.P. All rights reserved.

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don’t overexpose yourself with risk neutral pfes and epes

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