From CVA to EPE - Chirikhin · CVA and EPE Introduction CVA VAR: spread risk component (Basel III)...

41
BUILDING TOMORROW rbs.com/gbm From CVA to EPE Comparisons and contrasts Andrey Chirikhin, Han Lee Royal Bank of Scotland 3 February 2012

Transcript of From CVA to EPE - Chirikhin · CVA and EPE Introduction CVA VAR: spread risk component (Basel III)...

BUILDING TOMORROW™

rbs.com/gbm

From CVA to EPEComparisons and contrasts

Andrey Chirikhin, Han Lee

Royal Bank of Scotland

3 February 2012

Outline

CVA and EPE Introduction

Comparisons and contrasts

Future work

Conclusion

Outline

CVA and EPE Introduction

Comparisons and contrasts

Future work

Conclusion

CVA and EPE IntroductionFundamental questions

I Are CVA and EPE scopes overlapping or complimentary?

I Should modelling approaches be similar or different?

I Can CVA infrastructure be reused for EPE?

I Can EPE infrastructure be reused for CVA?

I What are the implications for

I active counterparty risk management,

I regulatory capital management,

I optimization of funding?

CVA and EPE IntroductionComparison and scope

Despite aiming at seemingly same goals, the scopes of EPE and CVA

systems are quite different.

EPE CVA

Generates FMTM yes yes

Main output EEPE, CVA VAR (EE) CVA, DVA, FVA, ...

Application whole book actively hedged

Measure historical or risk neutral risk neutral

Collateral logic yes, approximate yes, part of payoff

Scenarios/deltas stress test, ad-hoc deltas, PNL predict

Pre-deal yes yes, including deltas

Backtest regular, statistical ad-hoc, PNL explain

Calibration/estimation at least quarterly driven by hedging

CVA and EPE IntroductionCVA Definition

I CVA is the floating leg of an FTD with stochastic notional

CVA = CcyR

∫ T

0

EQ(

MtM+t (1− Rc

t )λct e−

∫ t

0(λc

u+λsu)duDt

)dt

≈ CcyRN

∑i=1

EQ(MtM+

t i (1− Rcti))

Dti ∆pcFTD(ti)

= CcyRN

∑i=1

ELti Dti ∆pcFTD(ti).

I Thus, the essential component in pricing is the strip of terminal

expected losses (ELti ), which is to be computed using the same

mega-model that will also diffuse all other factors showing in the

pricing equation.

CVA and EPE IntroductionCVA VAR: spread risk component (Basel III)

I Since ultimately CVA VAR is computed by applying VAR model to

regulatory CVA, its own definition depends on the type of the VAR

model used by the bank.

I For banks with full IMM approval for bonds and advance method

for collateral, pretty much the intuitive definition of CVA:

CVA = LGD ∑i

(EEi−1Di−1 + EEiDi

2

)(e−si−1ti−1/LGD − e−si ti /LGD

)+I Banks with different scope of IMM model are provided with other

formulas, e.g. CS01 CVA approximation.

I EEs beyond 1 year are relevant under Basel III.

CVA and EPE IntroductionEEPE: counterparty risk component (Basel II+)

I Denote MtMt the netted set’s mark-to-market at time t , computed

by in-the future by the EPE evolution model.

I Then the regulation defines

Expected Exposure: EEt = E(MtM+

t

)Expected Positive Exposure: EPEt =

1t

∫ t∧1

0EEudu

Effective EE EEEt = max (EEt , EEEt−)

Effective EPE EEPEt =1t

∫ t∧1

0EEEudu

I "Effective" exposures are defined for the shortest of the longest

maturity in the netting set and one year.

I EEPEt is defined as EAD for deals in trading book.

Counterparty risk: exposure type examples

CVA and EPE IntroductionHistorical EPE vs risk neutral CVA?

I Regulators do not require CCR model to be historical. This, in

principle, opens the door for the banks with strong risk neutral

CVA infrastructure to use it also for EPE.

I The issue is that it is not easy to make a risk neutral model a

"sufficiently" risky.

I Adding risk premia to the drifts will not make dynamics of the

market data similar to the one observed in the past.

I Essential risk factors will be missing:

I To price CVA on an equity option, one can use Black-Scholes (BS)

economy. Only stock needs to be risk neutrally stochastic.

I For CCR in BS economy, one wants to make not only stock

historically stochastic, but also implied vol and evolve both stock

and vol in a correlated fashion.

CVA and EPE IntroductionEPE: historical vs risk neutral

I Consider BS economy with a single stock as tradeable.

I Risk neutral with historical drift:

St = S0 exp

(µt + σ

√tXt

),

Xt ∼ N(0, 1).

Risk neutral drift would be µ = r − σ2/2.I Historical:

St = S0 exp

(µt + σ0 exp

[µσt + σσ

√t

(ρXt +

√1− ρ2Yt

)]),

Xt , Yt ∼ N(0, 1).

Outline

CVA and EPE Introduction

Comparisons and contrasts

Future work

Conclusion

Comparisons and contrastsRates, Inflation and FX

I As we will show CVA and EPE analysis can be complementary, in

particular

I to benchmark one model against the other,

I to identify the risk-neutral model’s shortcuts.

I We consider a realistic setup of running CVA and EPE kind of

calculations for

I 10 year IRS,

I 10 year RPI swap,

I 10 year FX option.

I Limit outputs to EPE, ENE produced by CVA and EPE models.

I Initial data for IR and Inflation models as of January 24, 2011.

Comparisons and contrastsCVA Model for IR/Inflation

I Hybrid rates-inflation risk-neutral model (HW/HW/BS):

dr = (fr − a · r(t)) dt + σr (t)dW rt ,

di = (fi − a · i(t)) dt + σi(t)dW it ,

dY /Y = i(t)dt + σY dW Yt .

I Calibrated to ATM IR caplets and YOY Inflation caplets

I Projection curve = discount curve (LIBOR)

I IR/Break even correlation marked

I Other correlations set to zero (for this exercise)

Comparisons and contrastsEPE Modelling: arbitrage-free curves

I We work with "flat instantaneous forward curves" (easier to rule

out arbitrage).

I Select an array of "break" times {t0, t1,...,tn} define

s(t) =btc−1

∏k=0

e−hk (tk+1−tk )e−hbtc(t−tbtc),

where btc = min(i : t − tk ≥ 0),is segment selection function

and |{h0, ..., hn} is the array of "forward rates":

d

dt[− ln s(t)] = hbtc.

Comparisons and contrastsEPE Modelling: stochastic Nelson-Siegel model

I The vector of {h0, ..., hn} is driven by a stochastized version of

Nelson-Siegel (NS) model:

hn,t = ν1,t f1(tn) + ν2,t f2(tn) + ν3,t f3(tn)

f1(τ) = 1,

f2(τ) =1− exp(−λτ)

λτ,

f3(τ) =1− exp(−λτ)

λτ− exp(−λτ),

νi ,t+1 = αi + βi νi ,t+1 + σi εi ,t , εi ,t ∼ N(0, 1).

Comparisons and contrastsEPE Modelling: stochastic Nelson-Siegel model

NS postulates the "smoothed" versions of the typical principal

components, which makes it much handier for (stressed) scenario

analysis.

I NS factors represent the

components building up the

curve.

I The second and the third

factors are related to the

Laguerre polynomials.

I Similar shapes are typically

extracted by PCA analysis of

the individual tenor evolution.

Comparisons and contrastsEPE Modelling: NS versus "market standard"

I A more standard approach is to model each hi directly. For

example, if non-negativity is essential then

νi ,t+1 = αi + βi νi ,t+1 + σi εi ,t , εi ,t ∼ N(0, 1)

hi ,t = exp(νi ,t)

εi ,t dependent

I Typically this requires 10-20 tenors to be modelled.

I Correlation matrix of εi ,t is usually PCA’ed to produce 3-5

principal factors.

I The key advantage is that this can fit the initial data better at the

expense much heavier evolution.

Comparisons and contrastsEPE Modelling: equivalence

Note however that if βi = β in NS, then

∆hn,t =3

∑i=1

∆νi ,t fi(tn) =3

∑i=1

(αi + (βi − 1) νi ,t + σi εi ,t) fi(tn)

=3

∑i=1

(αi + (β− 1) νi ,t + σi εi ,t) fi(tn)

=3

∑i=1

αi fi(tn) + (β− 1)3

∑i=1

νi ,t fi(tn) +3

∑i=1

fi(tn)σi εi ,t

= An + (β− 1) hn,t + Σεn,t ,

which is equivalent to "standard" approach with constant β.

ComparisonsEPE Model: specification

I Rates and break-even curves: 3 factor Nelson-Siegel.

I Inflation index:

dIt = (α+ βX (t , 1)) dt + σdWt ,

where X (t , 1) is the one-year tenor point of the inflation break even

curve

I All estimated from history:

I 5 year history,

I daily returns where available,

I monthly data for inflation index.

I Each marginal model is estimated first. Correlations between the

diffusion terms are then determined.

Comparisons and contrastsInflation models: calibration/Estimation

CVA ModelTenor 31/12/2012 01/01/2013 31/12/2013 01/01/2014 31/12/2014 01/01/2015 31/12/2015 01/01/2016 +Rate vol, bp 56 84 84 109 109 119 119 116BE vol, bp 179 179 179 179 179 179 202 202Index vol 1.00% 1.00% 1.00% 1.00% 1.00% 1.00% 1.00% 1.00%

MR Speeds: Rate 0.05 BE 0.12 Rate/BE corr 0.25

EPE ModelRate Break even IndexParameter Level Curvature Tilt Level Curvature TiltLT level, bp 460 ­110 40 332 ­65 ­35MR speed 0.250 0.250 0.250 0.445 0.730 1.224Sigma (ann), bp 89 113 260 45 180 277 2.906%

CorrelationsRate Level/Curvature ­0.70Rate Level/Titl ­0.50Rate/BE ­0.20… +0.20BE Level/Curvature 0.10

Comparisons and contrasts10 year IRS: broadly same

Comparisons and contrasts10 year RPI: CVA riskier?

Comparisons and contrastsRPI swap history: curve titling in 2008

ComparisonsRPI models

I NS-based EPE model correctly picks up 2008 curve tilting by

assigning higher vol to the second factor and making its

long-term level negative.

I CVA model we chose cannot reproduce this dynamics, as

1-factor HW essentially produces parallel shifts.

I Thus future evolution of CVA model will move the whole curve in

parallel, resulting in higher exposure.

I NS model will mostly tilt the curve, by moving the short end.

I That’s why CCR EPE graph is so flat: shortening maturity is

offset by higher short term volatility induced by NS dynamics.

I Raising vol of the first NS factor to the level of second NS factor,

makes NS model much more similar to HW.

Comparisons and contrasts10 year RPI: CVA vol brought to EPE level

Comparisons and contrasts10 year RPI: NS first factor vol = NS second factor vol

Comparisons and contrastsFX option models

I CVA

I Black’ 76 setup, rates set to zero

F = F0 exp

(−σ2

F t

2+ σF WF

), σBS = σF .

I EPE

I In addition to the above

σBS = σF exp

(−σ2

σt

2+ σσ

(WF ρ+Wσ

√1− ρ2

)).

I Correlation kept constant through EPE evolution

I 3 cases of correlation: ρ ∈ {−0.5, 0, 0.5}.

Comparisons and contrastsFX option: WWR in EPE

More variance in the model implies higher EPE.

F0 = 10

Strike= 15

σF = 0.3

σσ = 0.3

Comparisons and contrastsSummary

I Difference may be material.

I Differences to be more pronounced for non-linear products,

where WWR can be present because of correlations not present

in the CVA model.

I EPE model can be a useful tool to identify pitfalls of risk neutral

models, trying to just fit the market.

Outline

CVA and EPE Introduction

Comparisons and contrasts

Future work

Conclusion

Future workOpen topics

I Going CVA way: dimensionality reduction; analytical approach

I Going EPE way: capital optimization problem; numerical

approach

I We have seen in some explicit model examples the effect on

CVA/EPE based upon on choice of risk factors and

dimensionality. How can these be looked at from the same

perspective especially from a practical implementation point of

view ?

Future workLocal measure change

I CVA modelling is based on American Monte Carlo (AMC). The

choice of a Risk-Neutral measure in CVA has an implication for

feasibility of this extended use of AMC within EPE.

I A connection between the two approaches is to use a local

measure change (historical to risk-neutral), equivalent to the

Delta-Gamma approximation (Duffie and Pan, Finance and

Stochastics, 5, 155-180; 2001) and then numerically making use

of a bundling of paths to sample efficiently the additional dynamic

parameters (e.g. volatility)

I For accuracy of callability, can make use of similar result that

dimensionality of exercise boundary can be dimensionality of

underlying risk factors (Hunt and Kennedy 2005; SSRN:

http://ssrn.com/abstract=627921 )

Future workAMC for CVA motivation

I We specifically refer to adaptation of LS rollback to CVA.

I Effectively it implements MC within MC

I Risk neutral assumption is essential to justify the rollback.

I Given the future MtM (FMTM) distribution at tn+1, AMC rollback

automatically (in the limit) generates the continuation values,

which can be used to construct FMTMt .I Continuation values are provided in terms of projection on some

basis functions of the time t−measurable variables, typically,

easily computable.

I Provides huge performance gains not only for exotics, but also for

the aggregated portfolios of vanillas (especially if long/short

positions are present, which allows to offset AMC rollback bias).

Future workAMC for EPE challenges

I Risk-neutral AMC does not automatically extend to EPE, because

EPE MC will evolve not only observables in historical measure,

but also model parameters.

I Essentially CVA (pricing) rollback is the one with the parameters

fixed once, during the model calibration.

I Therefore CVA projections (strictly speaking) cannot be used for

EPE, as they are conditioned on the fixed model parameters.

I Without AMC, adding exotics to EPE methodology implies

considerably higher hardware requirements, mostly driven by

valuation routines.

I EPE AMC would thus be "MC within MC within MC".

I Given regulators’ requirements, EPE pricers need to be

sufficiently accurate for the trades to receive IMM treatment.

Future workAMC for EPE: setup

I Given time slices tn and tn+1, in CVA AMC we would discount and

roll back the vectors of continuation values V (tn+1, X tn+1|ξRN

),I the functions of the projection basis X tn+1

,

I conditioned on the fixed risk-neutral model parameters ξRN

.

I In EPE MC, assume we know V (tn+1, X tn+1, ξ

RN

tn+1|ζEPE

), i.e.

I (per path) realizations of the values continuation values V ,I as functions of the "market" variables X tn+1

and time-tn+1 values of

the valuation model parameters ξRN

tn+1on the same EPE path,

I conditioned on the fixed risk-neutral model parameters ζRN

.

I We need to project V (tn+1, X tn+1, ξ

RN

tn+1|ζEPE

) risk neutrally onto

X tn+1, conditioned on both ξ

RN

tnand ζ

EPE.

Future workAMC for EPE: extension

A possible solution:

1. Put an interpolator on V (tn+1, X tn+1, ξ

RN

tn+1|ζEPE

) in terms of

X tn+1, ξ

RN

tn+1. Often such interpolator can be a low dimensional

correction to some analytical "Black" formula, which will

effectively serve as a "control variate".

2. Perform a single risk-neutral sub-sampling step from tn into this

interpolator, using per-path values of ξRN

tn.

I In the worst case one would sample from each path at tn, but with

smaller number of path per "sub"-sample.

I Bundling paths with close ξRN

tnallows doing traditional AMC

projection within each bundle, thus avoiding direct sub-sampling.

Outline

CVA and EPE Introduction

Comparisons and contrasts

Future work

Conclusion

Conclusion

I We need both CVA and EPE methodologies. They are

complimentary.

I We want EPE methodology to be richer in terms of the risk factor

coverage (or at least to be easily extendable to include new risk

factors).

I A uniform architecture based on AMC may be possible,

containing CVA AMC as a special case.

I This may yield a holistic solution to both

I local "risk neutral" hedging,

I capital/term funding optimization problem.

Disclaimer

This communication has been prepared by The Royal Bank of Scotland N.V., The Royal Bank of Scotland plc or an affiliatedentity (’RBS’). This material should be regarded as a marketing communication and has not been prepared in accordance withthe legal and regulatory requirements to promote the independence of research and may have been produced in conjunction withthe RBS trading desks that trade as principal in the instruments mentioned herein. This commentary is therefore not independentfrom the proprietary interests of RBS, which may conflict with your interests. Opinions expressed may differ from the opinionsexpressed by other divisions of RBS including our investment research department. This material includes references tosecurities and related derivatives that the firm’s trading desk may make a market in, and in which it is likely as principal to have along or short position at any time, including possibly a position that was accumulated on the basis of this analysis material prior toits dissemination. Trading desks may also have or take positions inconsistent with this material. This material may have beenmade available to other clients of RBS before it has been made available to you and regulatory restrictions on RBS dealing in anyfinancial instruments mentioned at any time before is distributed to you do not apply. This document has been prepared forinformation purposes only. It shall not be construed as, and does not form part of an offer, nor invitation to offer, nor a solicitationor recommendation to enter into any transaction or an offer to sell or a solicitation to buy any security or other financialinstrument. This document has been prepared on the basis of publicly available information believed to be reliable but norepresentation, warranty or assurance of any kind, express or implied, is made as to the accuracy or completeness of theinformation contained herein and RBS and each of their respective affiliates disclaim all liability for any use you or any other partymay make of the contents of this document. This document is current as of the indicated date and the contents of this documentare subject to change without notice. RBS does not accept any obligation to any recipient to update or correct any suchinformation. Views expressed herein are not intended to be and should not be viewed as advice or as a recommendation. RBSmakes no representation and gives no advice in respect of any tax, legal or accounting matters in any applicable jurisdiction. Youshould make your own independent evaluation of the relevance and adequacy of the information contained in this document andmake such other investigations as you deem necessary, including obtaining independent financial advice, before participating inany transaction in respect of the securities referred to in this document.

Disclaimer

This document is not intended for distribution to, or use by any person or entity in any jurisdiction or country where suchdistribution or use would be contrary to local law or regulation. The information contained herein is proprietary to RBS and isbeing provided to selected recipients and may not be given (in whole or in part) or otherwise distributed to any other third partywithout the prior written consent of RBS. RBS and its respective affiliates, connected companies, employees or clients may havean interest in financial instruments of the type described in this document and/or in related financial instruments. Such interestmay include dealing in, trading, holding or acting as market-makers in such instruments and may include providing banking,credit and other financial services to any company or issuer of securities or financial instruments referred to herein. Thismarketing communication is intended for distribution only to major institutional investors as defined in Rule 15a-6(a)(2) of the U.S.Securities Act 1934. Any U.S. recipient wanting further information or to effect any transaction related to this trade idea mustcontact RBS Securities Inc., 600 Washington Boulevard, Stamford, CT, USA. Telephone: +1 203 897 2700.The Royal Bank of Scotland plc. Registered in Scotland No. 90312. Registered Office: 36 St Andrew Square, Edinburgh EH22YB. The Royal Bank of Scotland plc is authorised and regulated by the Financial Services Authority.The Royal Bank of Scotland N.V., established in Amsterdam, The Netherlands. Registered with the Chamber of Commerce inThe Netherlands, No. 33002587. Authorised by De Nederlandsche Bank N.V. and regulated by the Authority for the FinancialMarkets in The Netherlands.The Royal Bank of Scotland plc is in certain jurisdictions an authorised agent of The Royal Bank of Scotland N.V. and The RoyalBank of Scotland N.V. is in certain jurisdictions an authorised agent of The Royal Bank of Scotland plc.c© Copyright 2011 The Royal Bank of Scotland plc. All rights reserved. This communication is for the use of intended recipients

only and the contents may not be reproduced, redistributed, or copied in whole or in part for any purpose without The Royal Bankof Scotland plc’s prior express consent.