Post on 29-Mar-2015
Etienne KoehlerBarclays CapitalEtienne.Koehler@univ-paris1.fr
CVA - VaR
January 2012
Shahram AlavianRoyal Bank of ScotlandShahram_Alavian@yahoo.com
2
Preliminaries Credit Value Adjustment (CVA ) Value at Risk (VaR) VaR for CVA
The Proposed Approach VaR properties VaR and Monte Carlo simulation VaR for CVA
Examples A non-linear payoff Application to Regulatory VaR
Concluding Remarks Limits of the approach Challenges
In Nutshell
Statement Calculate VaR on CVA using a single batch run instead of multiple runs; once for each sensitivity.
Benefit Faster execution of VaR
Approach Insert a 1-day time horizon in the simulation timeline. Model the distribution of the change in value, at 1-day horizon, as a function of the underlying factors. Use this function to reprice the CVA changes in the same way a trade’s pricing formula is used to calculate its VaR.
What we say … How we go about it ….
3
Credit Value Adjustment (CVA ) Value at Risk (VaR) VaR for CVA
Preliminaries
4
Definition It is a Fair Value Adjustment to an OTC Trade. It is equivalent to the risk that a dealer has taken on its counterparty’s credit by entering into an OTC trade
Objective Accounting, risk mitigation, and regulatory Unilateral versus bilateral
Management Accounting – Managing its volatility in the balance sheet. Risk Mitigation – Selling protection to other Proposed business functions (desks) Regulatory Capital – Minimizing the cost of regulatory capital
Preliminaries - CVA
Dealer Counterparty
=Dealer Counterparty
+Dealer Counterparty
Defaultable Cash flow
Default Free Cash flow
Option to Default
5
Main Components of CVA Market component Credit component (including default) Cross gamma component
Hedging CVA Inconsistencies in hedging from different objectives Hedging for Default? Hedging for Capital Hedging for volatility
The hedged CVA book, П is
Preliminaries – CVA Continued …
HCVACVA
6
Value at Risk (VaR) Is A measure of unexpected loss due to market move. tool for calculating capital qualitative representation of the volatility of the balance-sheet. risk-limit measure for trading desks
Sensitivity Based Sensitivities are sent to market risk department. VaR and its relevant break-down numbers are then returned. Not all sensitivities are included
Historical Based Recent (3 year) daily time series “stressed period “
Preliminaries – VaR
7
CVA Volatility leads to: CVA Capital
CVA as a fair value adjustment brings volatility to the balance sheet For firms using unilateral CVA (no DVA), the unexpected loss due to a rise in credit spread of the market can be very large. Newly introduced regulatory CVA capital (BASEL III)
Limit on CVA-VaR Level - how much? Bucket – Region, industry, currency, …
Preliminaries – VaR for CVA
8
VaR properties VaR and Monte Carlo simulation VaR for CVA
Proposed Approach
9
1. Instantaneous: It excludes the duration of the portfolio, omits any cash flow during the time horizon and limits the change of value to the instantaneous change in the underlying risk factors, only.
2. Conditional: It is conditional on its initial value at current time.
3. Functional: Similar to a pricing function, it is a function of the risk factors driving its value. This feature provides the means for generating VaR from any arbitrary distribution.
Proposed Approach – VaR Properties
10
Monte Carlo Approach N risk factors are simulated under a joint process. For each path j and time t, we have (dependence on t is implied)
Proposed Approach – VaR and Monte Carlo
j
Njjjj RFRFRF )()2()1( ,,, RF
tt jj ,RF
Valuation of an asset π conditional on each time t and path j will result
Therefore, we have a distribution of π, as a function of its underlying risk factors, for each time t.
Defining a 1-day Implied VaR. Calculate the conditional values of the asset at δ = 1-day time horizon for each path j.
j
In future slides, we replace π with expected exposure (EE) and CVA itself.
11
1. Create an Instantaneous Change
Proposed Approach – VaR and Monte Carlo
00, jj
0'0,' jj
'
0,'0,',
jj
jjjj
This is not an instantaneous change. Therefore, we pick up another path j’.
and create an instantaneous change by
Calculate the change in values of the asset at 1-day time horizon for each path j.
2. Create a Conditional Change
0' j
So we have a conditional change from j’ which given the above assumption, we a have conditional on spot.
Having simulated a large number of paths, one can find a path for which
<= This change is from any j’ to any j
12
3. Create a Functional Change
Proposed Approach – VaR and Monte Carlo
',', jjjj RF M
Assuming there exists a model, M, representing the conditional changes as a function of the changes in the risk factors
with
jjjj RFRFRF '',
0,22
0,120,110',', ; RFRFRFβRF k
x
kkkjjjj M
Using a linear regression model, for example,
Putting It All Together ][%99,1 jday RFVaR M
with
)()()1()1(11 ,, N
tN
tttj jjjj RFRFRFRF RF
simulated. represents and historical represents that Note jj RFRF
13
Proposed Approach – VaR for CVA
Change of Value of a Hedged CVA Book
Re-writing the above in the same notation as previous section
HCVACVA
000 Hjjj CVACVA
Following our prescription, we need to calculate
jCVA
which we can do using a rollback method like Least Squared MC, under two different approaches.
14
Proposed Approach – Method I
. of percentile-99 thebe wouldVaR 9.
. changes, (daily) historical theCalculate 8.
)( of form
functional ain represent order toin of iessensitivitrelevant allObtain 7.
;fix order toin calculate to and Use6.
; model regression theof spolynomialin termsrelevant all Include 5.
]notation simplify to[ factorsrisk
impliedmarket theofeach for path from )( changes 1 ofset a Generate 4.
smallest. theis 0 when path thefind and paths all through Go 3.
method.rollback a requires This .path every for at lconditiona theCalculate 2.
simulationjoint aunder factorsrisk market andcredit theSimulate 1.
0
0 0
00
j
j
jH
HH
jCVAjj
jCVA
j,jj
j,j
j
j
CVA
CVACVA
CVA
jN-
- CV ACVA j: j j
jtCVA
RF
RF
βRFβRF
βRF
RFRF
RF
M
M
Joint Simulation of the Credit and Market
Step-by-Step
βRF ;00 ,, jjCVAjjCVA M
CVA
15
Proposed Approach – Method II
1
,,,,
k
kckkdk EExEExCVA
Simulation of the market, only CVA (bilateral case)
EE
SimulatedNot
discounted and simulation From
1
1,
1,
x
EE
PQ
tQtPtPLGDx
tQtPtPLGDx
kc
kd
kdddk
kd
kc
kccck
16
Proposed Approach – Method II
to from ofback -roll requires
change historicaldaily from is
;
spot theis
change historicaldaily from is
spot theis
luecurrent va its fromCVA of change
Terms Cross
Credit
Market
,,
,,
,
,
,
,,,,
,,,,
1
,,,,
ttttEEEE
EE
EE
x
x
CVA
EExEEx
xEExEE
EExEExCVA
kkkk
j
jkEE
kj
k
dkj
dk
j
kj
ckj
kj
dkj
ckj
kdkj
k
k
kj
ckkj
dkj
RF
βRFM
Simulation of the market, only Change in CVA (bilateral case)
17
Proposed Approach – Method II
. of percentile-99 thebe wouldVaR 7.
in
calculate ,each for and series, timehistorical in the every For 6.
; calculate ,each for and series, timehistorical in the every For 5.
. and ,for series timehistorical Obtain the 4.
of form functional
ain represent order toin of iessensitivitrelevant all Calculate 3.
; fix order toin Calculate (d)
]notation simplify to[ Calculate (c)
smallest theis
0 when :path thefind and paths all through Go (b)
alogrithmback -roll a using Calculate (a)
:horizon every timeFor 2.
factorsrisk market all Simulate 1.
,
,,
,
,,
,,00
,
00
k
kjj
kj
jkEE
dkj
dkjj
jH
Hj
H
jkEE
kk
jjjjj
kkkk
k
CVACVA
CVAkj
kj
xx
CVA
CVACVA
EE
EEEEjjj
EE
k
kk
βRF
RF
RF
βRFβ
RFRFRF
M
M
Step-by-Step
18
A Power Option An Interest Rate Swap and Its Application to Regulatory VaR
Examples
19
Examples – A Power Option
Motivation For This Example To show the effectiveness of linear regression when modelling a non-linear trade
Motivation For This Trade The power option is a highly non-linear trade with an analytical pricing formula.
750 elements historical ofNumber
5,000Paths ofNumber
0.5. and both for ies volatilitHistorical
1
005.0
02.0)0(3.0
100)0(25.0
0,maxpayoff 2
HS
LGD
CVAr
dwdwE
HdwdtH
dH
SdwdtrS
dS
SK
c
H
HS
H
S
T
Setup Assumption Rates and Vols are not
stochastic. Counterparty has sold
us a put on its stock
20
Examples – A Power Option [ ρ = 0 ]
55
2210
82
765
52
210
;
II-Method
;,,
I-Method
SSSS
SHHHSSSSH
kkkk
EE
kkkkkkkCVA
k
β
β
M
M
For every strike, with ρ = 0, both S and H were simulated. First, the VaR using benchmark(BM) was produced by pricing the CVA with current market data and then pricing the CVA with each of the 750 scenarios. The VaR of this distribution, for this strike, produces one point under BM label. Separately, Method-I, and Method-II were used to calculate their corresponding VaR numbers. Each VaR makes a data point under the Method-I and Method-II labels, respectively. This process was then repeated for strikes ranging from 4,000 to 40,000. The longest part of the exercise was the generation of the VaR using BM, since it had to reprice the CVA 751 times for each strike value.
21
Examples – A Power Option [ ρ = -0.95 ]
SHHHSSSSH kkkkkkkCVA 8
276
55
2210;,,
I-Method
βM
The same set of simulations was repeated, for the same range of strikes,using ρ =-0.95, incorporating a large WWR. In this case, however, only Method-I was calculated. This figure illustrates the effect of WWR to VaRusing Method-I.
22
Examples – An Interest Rate Swap
Motivation For This Example Sensitivity of the approach to various volatility values To compare the BASEL –III CVA regulatory capital using method-I.
Motivation For This Trade Trade’s value can be negative creating the exposure non-linear and making
the rollback process challenging.
750 elements historical ofNumber
8,000Paths ofNumber
1. to0 from varies
1.1. to0 from aries , , of ies volatilitHistorical
1
.2 and ofy volatilitImplied
05.0)0(02.0)0(
vH
LGD
dwdwE
Hr
rH
H
c
HS
Setup Assumption Current period is also
the “stress Period” There are no hedging
instruments.
23
Examples – Effect of Volatility on Approach
rHHHrrrrH kkkkkkkCVA 8
276
55
2210;,,
I-Method
βM
Since the regulatory VaR does not take any contribution from the un-hedged exposure variations, the first test would be to compare the credit component of the VaR for various historical spread volatilities. This is done by generating a time series for H while keeping the historical volatility of the rates to zero. The results are then compared with the regulatory VaR. Using the regulatory VaR as the benchmark, one can also observe that the VaR methodology proposed here can generate, from a single implied volatility of 20%, the correct VaR for different historical volatilities generated from various time series.
24
Examples – Regulatory VaR and WWR
rHHHrrrrH kkkkkkkCVA 8
276
55
2210;,,
I-Method
βM
The next objective is to compare the regulatory and the proposed VaR for a given historical volatility of market (both, hazard rate and the rates) and for various correlation. This is done by performing a Monte Carlo simulation for each level of ρ, generating the time series matching the corresponding correlation and calculating the VaR as prescribed in method-I. Since the regulatory VaR depends only on the historical volatility of the hazard spread, it produces a flat line. Note that for ρ=0, method-I includes variations in the market component and regulatory VaR does not.
25
Concluding Remarks
Limits of the Approach Implied Volatility – In cases where rollback methods were used to obtain the conditional prices of option, there will be no volatility to regress against.
Challenges A Robust rollback algorithm – Obviously, there is no analytical formula to calculate the conditional CVA or exposures. This means we need a robust rollback method in order to obtain convergent conditional values . Nonlinearity of the exposures – Even when using a rollback one still needs a substantial number of paths for the exposures and CVA values to converge as they are highly nonlinear. A different VaR platform – Almost in all cases, all desks use the same VaR platform. This method now requires a different one.
26
Questions and Comments
Thank you