A simple model for mapping default probability estimation into risk neutral world and it's application to CVA pricing and Risk management.

1. A Default Probability Mapping Model for Pricing and
Managing CVA
A.Gigli, E. Renzetti August 2014

2. 2
DISCLAIMER
The views expressed in this presentation are those of the speaker and do not necessarily represent those
of current employers.
Additional information is available upon request.
Information has been obtained from public sources believed to be reliable but the authors does not
warrant its completeness or accuracy.
All information contained herein is as of the date referenced.
This material is for informational purposes only, should be viewed solely in conjunction with the oral
briefing provided at the time of circulation, and is not intended as an offer or solicitation for the purchase
or sale of any financial instrument.

3. DERIVATIVE PRICING (NO CTP RISK)
• Derivatives are financial contracts which allow to bet on the future value
of a given underlying without holding it physically.
V
• Without taking into account Counterparty Risk, the value of the
contract overtime depends on the moneyness of the bet and on the
expected value of future cash flow(s)
( ) 1 t C) ( 2 t C) ( 3 t C) ( T t C n Expected cash flow 
Payement Dates
1t 2t 3t t T n 
• Martingale pricing theory tells us that under a risk-neutral measure Q it
is possible to price a derivative contract according to
V(t) E V(T) Q
t 
Q  
t E
where is the expectation operator under the Q measure
conditioned to the information available in t.
3

4. DERIVATIVE PRICING (CTP RISK)
• If the counterparty of a derivative contract defaults in then no
future payments will be received by the derivative buyer after
( ) 1 C t ( ) 2 t C) ( 3 t C) ( T t C n Expected cash flow 
Payement Dates
1t 2t 3t T tn 
• Introducing Counterparty risk, the value of the contract overtime
depends also on the probability of actually receiving positive future
cash flow(s) from the counterparty and
• CVA is defined as
T  


Vˆ
Vˆ (t)  V (t)
CVA(t) V(t) Vˆ (t)  0
4

5. CVA
• Credit value adjustment is the price (i.e. the market value) of
counterparty credit risk, that is the premium to take into account of
losses upon counterparty default when pricing a derivative contract.
• CVA can be calculated as the risk neutral expectation of the discounted
loss over the life of the transactions with a given counterparty
where

B
    

Q
t
CVA t E R t
 EE
    ( ) 1 ( )

B

T
is the counterparty-level exposure at the time of default
is the counterparty time of default
is the recovery rate
is the value of the money market account at time t
5
EE( )

R
t B

6. CVA
• Assuming constant recovery rate R, we can write
where:
T
 
*
CVA EE t dQ t
(0) 1-R ( ) ( )



 

EE t
B
0
* 0
EE t E
Q
( ) |
 


B
t
t
 is the risk-neutral cumulative probability of default (PD)
between time 0 and time t
* EE(t)
 is the risk-neutral discounted expected exposure (EE) at
time t conditional on the counterparty default at time t.
• If both exposure and money market account are independent then

EE t EE t E * * 0 ( ) ( )   


B
  t

t
EE
B
T
* (0) 1-R ( )
t CVA EE dQ t
0
Q(t)
6

7. CVA: CREDIT VALUATION ADJUSTMENT
Looking at CVA formula in a naïve way
where:
   
* (0) 1-R ( )
- LGD (loss given default) is the percentage exposure we loose in
the case of counterparty default;
- EAD (exposure at default) is the expected derivative transaction
value at the time of default;
- DP (default probability) is the default probability assigned to
counterparty
• This relation holds only assuming independence between exposure
and counterparty’s credit risk
T
t CVA EE dQ t
0
LGD * EAD * DP
7

8. CVA AND RISK MANAGEMENT
• Counterparty risk implies
Vˆ (t)  V (t)
) ( ˆ t V) (t V
• It also implies that changes in and for changes in the
underlying value are different
• where
CVA t
underlying
V t
underlying
V t
underlying
V t
underlying










 ˆ( ) ( ) ˆ( ) ( )
EAD
underlying
LGD DP
CVA
underlying





* *
8

9. CVA AND RISK MANAGEMENT
Assuming no collateral agreement in place and assuming we can
approximate EAD(0) with
then
ˆ (0) 0
ˆ(0)



V



EAD

V underlying
underlying
ˆ (0) 0
ˆ(0)
* *
( ) ˆ( )



V



V t



V t

V underlying
LGD DP
underlying
underlying
MaxVˆ (0)  Addon(0),0
9

10. LGD, EAD, DP
In order to compute
And
we need:
CVA  LGD  EAD  DP
V
V t
V t
- EAD, which depends on a specific model assumptions;
- LGD, which depends on several factors, both transaction specific
(eg risk mitigation instruments in place) or not (eg country, sector or
counterparty);
- DP, which is derived from a market measure.
Several choices have to be made, few ones are both feasible and
effective in order to price and manage CVA risk.
10
ˆ (0) 0
ˆ(0)
* *
( ) ˆ( )










V underlying
LGD DP
underlying
underlying

11. DP(0,T)
• Risk Neutral DP(0,T) for a specific counterparty can be recovered from
market prices, provided CDS quotes for that counterparty exists.
• If market data are not available, banks have at their disposal a
statistical estimation of the default probabilities for homogeneous
Rating groups, DP(0,T)*
• If a function mapping Statistical (or Real) probabilities into Risk-Neutral
probabilities exists it would allow banks to price and manage CVA
under a Measure closer to the Risk-Neutral one.
DP(0,T)* DP(0,T)
11

12. RISK NEUTRAL VS REAL PROBABLILITIES
• Assuming the Black-Scholes-Merton setting holds, the relation between
Real measure P and the Risk Neutral measure Q can be defined by
where
Q  P  T  T T     1
is the Real measure for maturity T
is the RN measure for maturity T
is the cumulative normal distribution function,
is the market price of risk
is the correlation between issuer asset return and market return.
12
T P
T Q
N()



13. RECOVERING RN PROBABILITIES FOR
• We propose to estimate the relation
Q  P  T  T T     1
using CDS implied default probabilities (as the Market RN probabilities)
and Rating agencies default probabilities (as Statistical Default
Probability)
• Once an estimate for and is obtained, we use the estimated
function to recover Market RN probabilities starting from Bank XYZ
statistical estimates of counterparty default probabilities
• In the following section
• We fit the model
• We show an application of the mapping function
13
 

14. MODEL FITTING
14

15. S&P MATRIX
The S&P default probability Matrix estimated over 1981-2013 period:
15

16. 5Y CDS SPREAD
16
200
150
100
50
0
AA
200
150
100
50
0
A
250
200
150
100
50
0
BBB
600
500
400
300
200
100
0
BB
1000
800
600
400
200
0
B
3000
2500
2000
1500
1000
500
0
CCC

17. RECOVERING
T T T   
• We estimate for qualitatively homogeneous issuers (labeled
“R” in the following) accordingly to the S&P Rating scale.
• Using daily Risk Neutral DP recovered from CDS spread and Real
DP* available in S&P matrix we can solve for the following
q ˆ R
    
1 m ˆ R
  ˆ R
T  T T
T
for maturity T = {1y, 2y, 4y, 5y, 7y, 10y} where
is the statistical default probability for issuers belonging to
rating group “R” from S&P matrix, for maturity T
is the risk-neutral default probability observed for issuers
belonging to rating group “R” on the market, for maturity T
R
T mˆ
R
T qˆ
17
R
T ˆ
T T T    

18. BOX-PLOT FOR DAILY ESTIMATES OF
18
R
T 

19. A TERM STRUCTURE FOR R
T ˆ
19

20. R
In the following table we report the average over the period October
2010 –July 2014 for each maturity/rating group
Rating 1Y 2Y 4Y 5Y 7Y 10Y
AA 0.97 0.70 0.63 0.57 0.54 0.55
A 0.70 0.57 0.57 0.53 0.50 0.49
BBB 0.49 0.38 0.40 0.38 0.39 0.40
BB 0.40 0.29 0.33 0.33 0.35 0.38
B 0.04 0.08 0.22 0.26 0.32 0.38
CCC -0.35 -0.07 0.20 0.25 0.35 0.45
20
AVERAGE ESTIMATE FOR
R
T ˆ

T

21. MODEL EXTRAPOLATION
21

22. RECOVERING ACTIONABLE DP(0,T) FROM
Let’s assume Bank XYZ has estimated the following internal default matrix
We can estimate the Risk Neutral DP(0,T), for any counterparty C
belonging to rating group R, starting from the statistical estimate of the
R
Real DP(0,T)* using estimates of
ˆR
DP (0, T ) R
    
1 DPˆ  0, T *   ˆ R
T  T T
T
R
T ˆ
22
Rating 1Y 2Y 4Y 5Y 7Y 10Y
AA 0.034% 0.100% 0.388% 0.626% 1.304% 2.763%
A 0.131% 0.357% 1.135% 1.685% 3.048% 5.545%
BBB 0.605% 1.458% 3.760% 5.100% 7.933% 12.205%
BB 1.667% 3.734% 7.922% 9.801% 13.050% 16.943%
B 6.458% 11.788% 18.856% 21.195% 24.584% 27.985%
CCC 16.458% 23.944% 30.245% 31.808% 33.790% 35.548%

T

23. RISK NEUTRAL MEASURE DP(0,T)
R
Using CDS data from July 28th 2014 for estimating we obtain the
following Risk Neutral probability estimates
80%
70%
60%
50%
40%
30%
20%
10%
0%
0 2 4 6 8 10
AA A BBB BB B CCC
23
Rating 1Y 2Y 4Y 5Y 7Y 10Y
AA 0.29% 0.77% 3.84% 6.65% 12.65% 21.77%
A 0.43% 1.42% 7.22% 11.73% 19.26% 28.40%
BBB 1.21% 3.02% 11.88% 17.68% 27.25% 37.85%
BB 2.30% 4.80% 15.91% 22.47% 31.06% 41.03%
B 3.62% 7.51% 21.44% 28.69% 38.30% 49.21%
CCC 6.86% 16.66% 36.61% 43.85% 56.71% 70.99%

T

24. RISK NEUTRAL MEASURE DP(0,T)
R

Using the average estimates of we obtain the following Risk Neutral
probability estimates
90%
80%
70%
60%
50%
40%
30%
20%
10%
0%
T
0 1 2 3 4 5 6 7 8 9 10
AA A BBB BB B CCC
24
Rating 1Y 2Y 4Y 5Y 7Y 10Y
AA 0.75% 1.80% 8.08% 11.06% 21.64% 42.53%
A 1.06% 3.00% 12.80% 17.34% 29.06% 48.22%
BBB 2.16% 4.97% 16.36% 21.91% 34.93% 53.73%
BB 4.20% 8.48% 22.54% 29.09% 41.72% 59.18%
B 7.01% 14.33% 33.21% 41.69% 55.89% 73.02%
CCC 9.32% 21.15% 44.97% 53.82% 69.28% 85.20%

25. RECAP
25
S&P DP
matrix
Rating 1Y 2Y 4Y 5Y 7Y 10Y
AA 0.034% 0.100% 0.388% 0.626% 1.304% 2.763%
A 0.131% 0.357% 1.135% 1.685% 3.048% 5.545%
BBB 0.605% 1.458% 3.760% 5.100% 7.933%
12.205
%
BB 1.667% 3.734% 7.922% 9.801%
13.050
%
16.943
%
B 6.458%
11.788
%
18.856
%
21.195
%
24.584
%
27.985
%
CCC
16.458
%
23.944
%
30.245
%
31.808
%
33.790
%
35.548
%
Q  P  T  CCC T T T T      1
Rating 1Y 2Y 4Y 5Y 7Y 10Y
AA 0.64 0.47 0.45 0.44 0.41 0.36
A 0.38 0.35 0.41 0.42 0.38 0.32
BBB 0.26 0.22 0.30 0.32 0.30 0.27
BB 0.13 0.08 0.21 0.24 0.24 0.23
B -0.28 -0.18 0.05 0.11 0.15 0.18
CCC -0.51 -0.18 0.09 0.14 0.22 0.29
600
500
400
300
200
100
3000
2500
2000
1500
1000
500
Rating 1Y 2Y 4Y 5Y 7Y 10Y
AA 0.29% 0.77% 3.84% 6.65% 12.65% 21.77%
A 0.43% 1.42% 7.22% 11.73% 19.26% 28.40%
R
T ˆ(0, ) 1 ˆ 0, * ˆ      DP for pricing and
BBB 1.21% 3.02% 11.88% 17.68% 27.25% 37.85%
BB 2.30% 4.80% 15.91% 22.47% 31.06% 41.03%
B 3.62% 7.51% 21.44% 28.69% 38.30% 49.21%
CCC 6.86% 16.66% 36.61% 43.85% 56.71% 70.99%
250
200
150
100
50
0
10/29/2009 10/29/2010 10/29/2011 10/29/2012 10/29/2013
BBB
0
10/29/2009 10/29/2010 10/29/2011 10/29/2012 10/29/2013
BB
0
10/29/2009 10/29/2010 10/29/2011 10/29/2012 10/29/2013
T T T ˆ ˆ ˆ
DP T  DP T R
 R
T  T
T
managing CVA
Bank XYZ
matrix
CDS implied
DP

26. CONCLUSIONS
26
• CVA pricing and management requires Default Probability under a Risk
Neutral Measure, which can be recovered from market data. If market data
are not available for a specific counterparty, usually banks extrapolate a DP
from the statistical estimation of the Default Probabilities at their disposal.
• If a model to map Real probabilities in the Risk Neutral space exists, it is
possible to price CVA and compute CVA DV01 accordingly, using a RN
measure
T T T   
• In our application, we calibrated a term structure for for each
rating class starting from CDS data and S&P default probabilities.
T T T   
• Once we’ve estimated we use it to compute the risk neutral
probabilities starting from statistical estimation of default probabilities for Bank
XYZ customers.

27. 27
THANK YOU
For feedbacks please contact:
Andrea Gigli
andrea.gigli@mpscs.it
Eros Renzetti
eros.renzetti@valuecuberesearch.com
The views expressed in this presentation are those of the speakers only.
Additional information is available upon request. Information has been obtained from public sources believed to be reliable
but the authors does not warrant its completeness or accuracy.
All information contained herein is as of the date referenced. This material is for informational purposes only, should be
viewed solely in conjunction with the oral briefing provided at the time of circulation, and is not intended as an offer or
solicitation for the purchase or sale of any financial instrument.