From CVA to EPE - Chirikhin · CVA and EPE Introduction CVA VAR: spread risk component (Basel III)...
Transcript of From CVA to EPE - Chirikhin · CVA and EPE Introduction CVA VAR: spread risk component (Basel III)...
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From CVA to EPEComparisons and contrasts
Andrey Chirikhin, Han Lee
Royal Bank of Scotland
3 February 2012
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.
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).
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
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 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.
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.
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.
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