We will show how to calibrate the main parameter of the model and how we have used it in order to evaluate the CVA and the CVAW of a one derivative portfolio with the possibility of early exercise.

1. Project 9
CVA in presence of WWR and Early Exercise
Chiara Annicchiarico, Michele Beretta, Alessandro Canevisio
24
th
June 2017
Financial Engineering
Final Project
A.A. 2016-2017
1

2. Contents
1 Introduction 3
2 Calibration of the Model 5
2.1 Model Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 a(t) Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2.1 Data Mining . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2.2 Payo American Put Option - Binomial Tree . . . . . . . . . . . . . 7
2.2.3 Calibration Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3 CVAW pricing and WWR impact 10
3.1 CVA computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3.2 CVAW computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3.3 WWR impact: results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
4 Comparison with other derivatives 13
4.1 European Call . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
4.1.1 a(t) Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
4.1.2 WWR Impact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
4.1.3 No dividends case . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
4.2 American Call Options with parallel shift in the Cost of Carry . . . . . . . 15
4.2.1 a(t) Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
4.2.2 WWR Impact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
5 CVAW dierence between European and
American Options varying the moneyness 19
5.1 Put . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
5.2 Call . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
6 Appendix 22
2

3. Abstract
In the following sections of this paper we'll deal with the Hull and White approach [1]of
Credit Value Adjustment (CVA) in presence of Wrong Way Risk (WWR).
We will show how to calibrate the main parameter of the model and how we have used it
in order to evaluate the CVA and the CVAW of a one derivative portfolio with the possi-
bility of early exercise. In the end we'll propose some numerical results for the variation
in CVAW after a change in the derivative title and features.
1 Introduction
Let us consider the case of a derivatives portfolio in USD of the bank XX versus Cater-
pillar as counterparty, it is just composed by a long position on an ATM-Spot 2 years
American Put written on the EUR-USD exchange whose dynamics are described by a
Garman-Kohlhagen model.
Due to the presence of Caterpillar as counterparty we have to take into consideration
that the latter could default, and so we focus our attention on the estimation of two main
quantities: the CVA and the CVAW .
Denition 1. The Credit Value Adjustment (CVA) can be dened as the dierence
between the risk-free portfolio value and the real one that takes into account the possibility
of a counterparty's default.
Denition 2. The Wrong Way Risk (WWR) is the non-negligible dependency between
the value of the derivatives portfolio and the counterparty default probability. In partic-
ular WWR is usually faced when the creditworthness of a party falls down whenever the
portfolio exposure increases.
We consider as CVA the, so called, unilateral CVA which is obtained considering one
of the two counterparties default-free, and has the following formula:
CV A = (1 − R)
T
t0
B(t0, t) · EE(t) · PD(dt)
where:
• R = recovery value (in percentage)
• B(t0, t) = discount factor between t0 and t
• EE(t) = expected exposure of the derivatives portfolio at time t
• PD(dt) = probability density of counterparty default between t and t + dt
3

4. We obtain the recovery value R from the credit analysts that calibrate it according to
credit market situation, the discount factors can be obtained via the bootstrap of the
most liquid IB products and then via a linear interpolation in the corresponding zero
rates r(t0, t) it is easy to nd all the needed B(t0, t) (In this case we were already given
the discount curve).
The Expected Exposure EE(t) is computed as the maximum between the value of the
portfolio derivatives and 0, so it is equal to zero only when we have nothing or we are
losing money, while the default probabilities PD(dt) are computed as the dierence of the
survival ones in t + dt and in t, this is due to the fact that we assume there is no default
before the beginning of the considered time interval, so before the generic time t.
In the next sections we will analyze the model written by Hull White [1] for the stochas-
tic survival probabilities and their approach for the computation of the CVA considering
the WWR.
4

5. 2 Calibration of the Model
2.1 Model Introduction
The rst quantity that we have to compute are the default probabilities, and hence,
the survival ones; generally they are linked to an intensity based model, so they can be
computed as:
SPi = exp −
ti
t0
h(t)dt , (1)
where h(t) is the hazard rate function of the intensity based model. Tipically in order to
have simpler computations the time line is divided into N steps {ti}i=1,..N , and between
two consecutive times of this new grid, the hazard rate is assumed to be constant, so it is a
piecewise constant function of time. What Hull White introduced is a stochastic hazard
rate ˜h(t) because in the WWR framework the Exposure and the Default Probabilities are
no more independent, so it is modeled [1] as a stochastic function of the portfolio value
at time t (v(t)) with the following expression:
˜h(t) = exp a(t) + bv(t) . (2)
Where b ∈ R+
is dened as the WWR parameter while a(t) is a deterministic function of
time that has to be calibrated for each time ti of the grid and for each value of the range
in which the parameter b varys, using the following relation:
SPi = E exp −
ti
t0
˜h(t)dt ∀i = 1, ..., N.
Once we have chosen b and found the corresponding a(t) the CVAW can be easily com-
puted rewriting the general formula for the CVA in which the WWR was neglected.
2.2 a(t) Calibration
In the considered Case Study we were given an American Put option, so in order to get its
own value at each time ti we have decided to use a Binomial Tree, so the CRR Method. But
since the given derivative is path-dependent we have implemented a recursive approach,
presented by Hull White[2] to calibrate the deterministic function of time a(t), that
avoids path-dependency. In this way the stochastic Survival Probability between ti−1 and
ti becomes:
SPi(j) ≡ exp −(ti − ti−1)˜h(j) ≡ Piηi(j) ∀i = 1, ..., N. (3)
Where:
• ηi is the vector of weights of the binomial tree nodes at time ti, it is the driver of
stochasticity and is fundamental in avoiding path dependency.
• the indexes i and j take into consideration respectively the time grid and the generic
jth
node of the tree at time ti.
• Pi ≡ SPi
SPi−1
represents the survival probability between ti−1 and ti evalued in t0.
5

6. The recursive approach is dened by the following:
Proposition 1. The calibration problem, under our conditions, becomes:
ji
pi(ji)ηi(ji) = 1 (4)
Where pi(ji) are the nodes probabilities and can be obtained via the recursive equation:
pi(ji) =
ji−1
qi(ji−1, ji)ηi−1(ji−1)pi−1(ji−1) (5)
with the initial condition p0(j0) = η0(j0) = 1. qi(ji−1, ji) is the transition probability from
a node to the following one.
In our case the transition probabilities are supposed to be constant for each time step,
in fact they have been computed as the probability of going upward (or downward) in the
binomial tree.
2.2.1 Data Mining
Since we have to build a binomial tree to be able to price the option and to nd the weights
η in order to have all the ingredients to calibrate the function a(t), we have divided the
time interval (T=2years) into N(=750) steps, nding all the possible TreeDates in which
the American option could be exercised.
What we need are the Survival Probabilities at each time step of the grid, the domestic
and the foreign interest rates since the derivative is written on the EUR-USD exchange,
and the dynamics of the latter.
Starting from the provided market data we have computed the survival probabilities us-
ing the CDS rates via a recursive bootstrap for the hazard rates (provided by the Matlab
function Prob_Hazard.m), in doing the bootstrap we have not considered the 6months
CDS because we were told that the payments were annual, while the 6months survival
probability has been computed independently from the others.
Once we have obtained these values, in order to nd the survival probabilities in each
point of the time grid we have considered the hazard rate as a piecewise constant function
(as suggested) [?] and then we have computed them using (1).
In the given market data, the bootstrap for the discount factors was already done, so
we have been able to nd all the domestic interest rates (r(t0, t)) and then having either
the values of the Forward F(t0,t) we have found the foreign ones (d(t0, t)) inverting the
formula of the dynamics described by a Garman-Kohlhagen process, in fact:
F(t0, t) = S0 · e(r(t0,t)−d(t0,t))(t−t0)
.
And then we have interpolated both the domestic and foreign rates linearly to be able to
nd them at each time of the discrete grid.
6

7. 2.2.2 Payo American Put Option - Binomial Tree
Now that we have found all the ingredients, we can compute the value of the American
Put Option at each time step and for every node (see Matlab function PayoOption-
CRR.m). Starting from the last time (tN ) and going backward we can nd the values
in the previous nodes as the discounted sum of the two linked nodes values weighted by
the probabilities of going upward and downward, but since we could have early exercise
in each node (American Option) the nal value v(t) is the maximum between the just
explained weighted sum and the payo of the option exercised in the considered time.
Mathematically speaking what we get is the following:
vi(ji) =
Φi(ji) for i = N
max(ci(ji), Φi(ji)) for i = 1, .., N − 1
Where:
• Φi(j) = K − F0 · uξi
+
is the vector of the payos of the option exercised in the
generic node at time ti.
• ξi = [i : −2 : −i] is the vector of powers that denes the value of the underling in each
node at time ti (for the jth
node we choose the jth
component, with j = 1, ..., i + 1).
• K is the strike price of the given option.
• u = eσ
√
∆t
is the increasing factor in the payo at each time step, where σ is the
volatility of the derivatives' portfolio, and ∆t is the length of each time interval.
• ci(ji) is the weighted sum of the connected nodes' values presented above, i.e.:
ci(ji) = B(t0, ti, ti+1)
ji+1
q(ji, ji+1)vi+1(ji+1).
• B(t0, ti, ti+1) are the forward discount factors.
• q(ji, ji+1) is the transition probabilitiy from a node to another.
q(ji, ji+1)
q
F(t0, ti|ji) · u
1 − q
F(t0, ti|ji) · d
• q =
eCoC(tN )
− d
u − d
• d = 1/u
• CoC(ti) = rti
− dti
The transition probability q is supposed to be constant in order to avoid path-
dependency as suggested in the paper [2]
Once we have computed the tree of the values, we need to consider the possibility of Early
Exercise. In fact, if it is convenient one exercises the option before the maturity and in
this way some nodes could become unattainable, indeed we have written the functions
Check_1.m and Check_2.m.
7

8. • The rst one checks the tree time by time and in particular considers each consec-
utive pair of nodes and if early exercise happens in both sets equal to 0 the value
of the following node in between, moreover it builds a matrix where the generic
element could assume just three values: 1 if it is reachable but early exercise does
not happen, 3 if the latter happens and 0 if the node is unattainable.
0
3 1 − q
3 q
• The second works on this last matrix, rstly considers all the upper branches and,
when early exercise condition is satised, i.e. when the element is equal to 3, it
nullies al the consecutive upper elements (in Matlab all the elements on the same
row of the Payo_Early' matrix), then the function does the same thing starting
from the time ti where early exercise happened in the rst node, but considering the
second node of each time step, and so on. Secondly it follows the same algorithm
for the upper branches but considering the lower ones.
Upper Check: Lower Check:
Figure 1: Two examples of how the algorithm explained in Check_2.m works, in particular
in the rst graph is represented the control done on the upper branches, while in the
second, the one done on the lower branches
2.2.3 Calibration Algorithm
Now that we have found all the quantities that appear in the equation (3) we can write
the weights ηi(j) in function of a(ti) in the following way:
ηi(j) =
1
Pi
· e−(ti−ti−1)ea(ti)+bvi(j)
After computing the probabilities of each node of the tree at time t1, p1(j1) (in Matlab
P_Tree(1:2,2)), using (5), we can obtain the value of the deterministic function a(t1)
imposing the normalizing equation (4) in ti = t1 (in Matlab we have used the function
fzero.m), then we can iterate the process for all the following time steps and hence recon-
struct the function a(t) for a given b, ∀t = t1, ..., tN = T.
8

9. 2.2.4 Results
• From the bootstrap of the hazard rate and survival probabilities starting from the
CDS market data, we get the following quantities:
t=6m t=1y t=2y t=3y t=4y t=5y
sCDS(t) 0.51% 0.87% 1.21 % 1.53% 1.78% 1.99%
h(t) 0.0085 0.0145 0.0201 0.0255 0.0298 0.0335
SP(t) 0.9957 0.9854 0.9601 0.9254 0.8860 0.8437
• Observing the formula that links a(t) to b and to the survival probabilities (SPi),
we can see that if b is equal to 0, the exponential of a(t) must be equal to the
hazard rate h(t) and so the unknown function must be piecewise constant. While,
as b grows, a(t) must vary in order to keep constant its sum with the payo' impact.
Therefore we expect a jump in the value of a(t) in correspondence of the dates used
to calibrate the hazard rates. In fact, we get, plotting the unknown deterministic
function:
Figure 2: Calibration of a(t) with three dierent values of the parameter b which is
dened as a percentage value per million USD, so the values written in the title should
be multiplied by a 10−6
factor. Data: ATM-Spot American Put Option, S0=1.1414,
σ=10.48%, T=2years, N=750, Notional=100 Mln USD.
9

10. 3 CVAW pricing and WWR impact
After a(t) calibration, we proceed with what is the main aim of this analysis: CVAW
computation and WWR impact on CVA itself.
In order to get the required results we divide our problem into three easier steps.
3.1 CVA computation
As we have already seen, the CVA can be easily estimated as the integral between t0 and
maturity T of the discounted expected exposure B(t0, t)EE(t) multiplied by the proba-
bility density of counterparty default between t and t + dt, PD(dt), and the Loss Given
Default 1 − R.
Thanks to the tree structure described above, we have been able to approximate the
pricing problem in the integral form by a nite sum according to the following expression.
CV A = (1 − R)
N
i=1
BiEEi + Bi−1EEi−1
2
PDi (6)
where:
• N represent the dimension of the discretized set of times
• Bi stands for the deterministic discount B(t0, ti)
• PDi is the default probability between ti−1 and ti
In order to get the CVA estimation, we have, at rst, computed each EEi as the weigthed
sum of the ith tree payo vector obtained as explained above, by the respective tree prob-
ability modelled as a binomial distribution. Then, by setting each PDi as the dierence
between SPi−1 and SPi, we have just applied the formula above and found that, in our
case study, the portfolio CVA is 49'665 USD i.e less than 0.05% of the Notional Value.
3.2 CVAW computation
We have already said that the CVA general formula holds whenever there is no dependence
between the exposure of an option and the counterparty probability to default, however,
when we suppose this dependence to exist and in particular to be positive, we have the
presence of the so called Wrong Way Risk: the tendency of the counterparty's default
probability to be higher when the exposure of the portfolio grows.
The formula for CVAW estimation can be implemented from the CVA's one via a simple
generalization:
CV AW = (1 − R)
N
i=1
E
DiEi + Di−1Ei−1
2
PDi (7)
where:
• Di is the stochastic discount D(t0,ti)
• PDi is the stochastic default probability between ti−1 and ti written in terms of ˜h(t)
10

11. Thanks to the unstochasticity of the provided discount factors and thanks to the relation
between the stochastic survival probability SP(t) and the hazard rate function ˜h(t) al-
ready shown in (3), we have been able to rewrite the expectations as follows: at each time
ti the expectation becomes the weighted sum, over dierent couples of consecutive states
(ji−1,ji) belonging to the tree structure, of single contributions to CVAW (a better and
more detailed explanation of this formula will be provided at the end of the document).
Using the same data and the same payos already computed for the standard CVA, con-
sidering parameter b equal to 0,05% per Mln USD and parameter a(t) equal to the one
obtained from the calibration part, nally we have been able to compute the whole CVAW
of our derivatives' portfolio. The total amount of CVAW turned out to be 57'971 USD i.e
0.056% of the Notional.
As we expected the CVAW turns out to be greater than the standard CVA due to the
presence of a positive dependency between the counterparty default probability and the
option exposure. However, thanks to the possibility of early exercise, which reduces the
Option future exposure nullifying some of its future payos, the dierence between the
two approaches stays small, which means that WWR is, in a certain sense, taken under
control.
The concept can be better claried in the next sections.
3.3 WWR impact: results
In order to better understand the role of WWR in the Credit Value Adjustment, we have
computed both CVA and CVAW considering WWR parameter b ∈ (0,0.1%) per Mln of
USD, getting the following results:
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
b 10-7
1
1.05
1.1
1.15
1.2
1.25
1.3
1.35
1.4
CVAW
/CVA
American Put : CVAW
/CVA
Figure 3: Ratio between CVA computed in WWR framework and the standard one.
11

12. From the gure above it is interesting to notice two important behaviours of the WWR
impact curve:
• When parameter b is equal to zero, which means that the stochastic hazard rate
function ˜h(t) and the deterministic one h(t) coincide, there is no WWR and so the
two quantities are equal and the ratio is unitary.
• Even for higher value of parameter b, as 0.05% or 0.1% the impact of WWR is
mitigated by the presence of early exercise.
Here we have a small tabular containing some of the outcomes just discussed. For sim-
plicity, one can consider all the CVA values equal to the rst one with parameter b equal
to zero. This simplication is allowed since the changes in the value (10−6
for the worse
case) are smaller than the market bid-ask spread (1bp = 104
)
b (% per Mln USD) CVA (USD) CVAW (USD) % of Notional WWR impact
0
49'665
49'665 0.049 1
0.05 57'971 0.058 1.168
0.1 67'957 0.068 1.367
12

13. 4 Comparison with other derivatives
In this section we are going to compare the results obtained above for the American Put
with some other derivatives:
• European Call with same features
• American Call with a parallel shift in the Cost of Carry, respectively of the order
[-2%, -1%, +1%, +2%]
Everything varying the parameter b into the range (0, 0.10%) per Million USD.
4.1 European Call
The Procedure for this computation is the same as the one for the American Put, using
the function Computation_VaryingWWRparameter.m with dierent ag input:
• ag1=1 : Call
• ag2=1 : European
The main dierence with the American case is related to the function PayoOption-
CRR.m: in case of ag2=1 each node will be the expected value of the forwards nodes
with no possibility of early exercise.
This is a big issue, it means that CVA related to European Options will have no null
exposure up to the expiry, i.e., concerning (6), CVA takes into account terms that have
no possibility to vanish.
4.1.1 a(t) Calibration
In this section we will show the results coming from the calibration of the function a(t)
for dierent b in (0, 0.10%) per Million USD with step 0.02%.
Jul 2016 Jan 2017 Jul 2017 Jan 2018 Jul 2018
time
-5
-4.5
-4
-3.5
a(t)
b = 0.00
Jul 2016 Jan 2017 Jul 2017 Jan 2018 Jul 2018
time
-5
-4.5
-4
-3.5
a(t)
b = 0.02
Jul 2016 Jan 2017 Jul 2017 Jan 2018 Jul 2018
time
-5
-4.5
-4
-3.5
a(t)
b = 0.04
Jul 2016 Jan 2017 Jul 2017 Jan 2018 Jul 2018
time
-6
-5
-4
-3
a(t)
b = 0.06
Jul 2016 Jan 2017 Jul 2017 Jan 2018 Jul 2018
time
-6
-5
-4
-3
a(t)
b = 0.08
Jul 2016 Jan 2017 Jul 2017 Jan 2018 Jul 2018
time
-6
-5
-4
-3
a(t)
b = 0.10
American Put
European Call
Figure 4: Calibration of a(t) American Put and European Call
13

14. For b=0 the shapes are clearly overlapped according to (2). As b grows, we observe
that the shape of a(t) in the European Call starts to decrease while in the American Put
increases, this is related to the fact that the value of the portfolio v(t) in the American
option, in some scenarios, will vanish due to early exercise, otherwise in the European
Call it will not, so that a(t) has to decrease in order to calibrate the model.
4.1.2 WWR Impact
What we expect is that CVAW for the European Call option will be greater than the one
for the American Put option, in fact we get the following:
Figure 5:
CV AW
CV A
for European Call and American Put
As we expected, we observe an increasing CVAW with respect to b, both for European
and American, this is due to the fact that an increment on the hazard factor obviously
increases the exposure at risk.
Further, as we have forecast, the shape of the European is way higher than the American
one.
4.1.3 No dividends case
We know that an European call is equal to the American one in case of no dividends,
which could imply that CVAW would be the same, let us check this thesis.
In order not to have dividends, the Cost of Carry has to be equal to the domestic interest
rate; once we have imposed this condition, the computation will be the same as before.
14

15. Graphically we get the following:
Figure 6:
CV AW
CV A
for European Call and American Call in case of no dividends
Our thesis perfectly matches with the results, in fact, the shapes perfectly overlap, in par-
ticular we notice a jump from American Put option to the Call one: this can be related
both to dierent features option and shift of Cost of Carry.
It could be interesting to analyze the behaviour of the WWR impact due to a parallel
shift in the Cost of Carry: this will be the topic of the next section.
4.2 American Call Options with parallel shift in the Cost of Carry
The computation is the same as above with three dierences:
• ag1=1 : Call.
• The curve of Cost of Carry ('CoC2Year') will be shifted by a parallel increment or
decrement of the order of 1% and 2%.
• The forward F(t0, t) (in Matlab F0_t) will be dierent since it depends on the Cost
of Carry over the time horizon.
15

16. 4.2.1 a(t) Calibration
We will plot the dierent calibrations of the function a(t) for the American Call with
dierent shifts.
Jul 2016 Jan 2017 Jul 2017 Jan 2018 Jul 2018
time
-5
-4.5
-4
-3.5
a(t)
b = 0.00
Jul 2016 Jan 2017 Jul 2017 Jan 2018 Jul 2018
time
-5
-4.5
-4
-3.5
a(t)
b = 0.02
Jul 2016 Jan 2017 Jul 2017 Jan 2018 Jul 2018
time
-6
-5
-4
-3
a(t)
b = 0.04
Jul 2016 Jan 2017 Jul 2017 Jan 2018 Jul 2018
time
-6
-5
-4
-3
a(t)
b = 0.06
Jul 2016 Jan 2017 Jul 2017 Jan 2018 Jul 2018
time
-6
-5
-4
-3
a(t)
b = 0.08
Jul 2016 Jan 2017 Jul 2017 Jan 2018 Jul 2018
time
-6
-5
-4
-3
a(t) b = 0.10
A.C CoC+2% A.C CoC+1% A.P. A.C CoC-1% A.C CoC-2% A.C.
Figure 7: a(t) Calibration American Put option for dierent shifts
We notice that the function a(t) for Call option has lower values for positive shifts while
for negative ones tends to the Put one; this has a coherent nancial meaning: for positive
values of d(t0, t), American Option admits early exercise and so v(t) of (2) is less or equal
than the corresponding European.
Since the values of d_2y (Matlab variable name for foreign interest rates) are negative with
minimum values -0.0034, for negative shifts less than -0.0034 we have that the American
Call Options have optimal exercise before expiry; that is exactly what we observe in the
shapes of a(t) and probably later on with WWR impact associated.
4.2.2 WWR Impact
In order to obtain dierent CVAW for dierent shift of Cost of Carry we use again the
function Computation_VaryingWWRparameter.m with two dierent inputs:
• Cost of Carry will be shifted
• F(t0, t) will be dierent according to its dynamics that is a function of Cost of Carry
16

17. The results are shown in the following gure:
Figure 8:
CV AW
CV A
for American Put and Call with Shift in CoC of +2% , +1% , -2% and
-1%
What is more interesting is the plot of the shapes all together, in fact:
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
b 10-7
1
1.2
1.4
1.6
1.8
2
2.2
2.4
2.6
2.8
3
CVAW
/CVA
Comparison for different Options varying CoC
American Put
American Call CoC +2%
American Call CoC +1%
American Call No Shift%
American Call CoC -1%
American Call CoC -2%
European Call
Figure 9:
CV AW
CV A
for European Call and American Call with dierent shifts
17

18. What we observe perfectly agrees with our hypothesis from a(t) calibration: CVAW is
decreasing for positive shifts and we observe a huge jump for shift of -2%. The shift of
-1% produces an increment in WWR impact, but it does not mean that this is wrong, it
depends on the Payo.
What really matters is the dierence between the corresponding European, as shown in
the following:
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
b 10-7
1
1.2
1.4
1.6
1.8
2
2.2
2.4
2.6
2.8
3
CVAW
/CVA
Comparison for different Options varying CoC
American Call CoC +2%
American Call CoC +1%
American Call No Shift%
American Call CoC -1%
American Call CoC -2%
European Call CoC +2%
European Call CoC +1%
European Call
European Call CoC -1%
European Call CoC -2%
Figure 10:
CV AW
CV A
for European Call and American Call with dierent shifts
These outcomes conrm all our assumptions: the shapes are equal for no shift and for
positive shifts, whereas, the curves corresponding to the American Options are always
above the European ones.
18

19. 5 CVAW dierence between European and
American Options varying the moneyness
In this section we will focus on the dierence in CVAW between American and European
options, not only for dierent values of the parameter b, but also for dierent moneyness.
The option will be ATM forward and we will have moneyness shifted from -30% to +30%.
Basically we use the functions implemented before and we build a matrix of the dimension
of the vector b times the dimension of the vector moneyness: for each row we have a xed
b and dierent values of the moneyness over the columns. We have decided to use the
step 0.01% for the parameter b and the step 0.06% for the moneyness,so we will have a
matrix (11×11).
Given the features of the option, we build the vector of moneyness from F0 in the range
of [K×70%, K×130%], then for each component of the vector of moneyness we use the
function Computation_VaryingWWRparameter.m.
5.1 Put
Using a Laptop of 12 GB RAM this process takes 379.90 seconds; what we obtain is
represented in Figure 11:
Figure 11: Dierence in CVA between European and American Put options
We expect that the CVAW associated to American Option will be less or equal the Euro-
pean one, so the blanket (we will refer to the 3D plot in this way) has to be less or equal
zero; further, if the blanket is decreasing it means that we are in the case of American
Option with early exercise.
19

20. Given the dynamics for the underlying, as we expected, we observe that the Put option is
more likely to be In the Money for higher value of the moneyness, that is why the blanket
is decreasing as Strikes increase; further, as stressed in the other sections, CVAW is an
increasing function of the parameter b, that is what we observe here.
WWR plays dierent roles for European and American Options, and early exercise is a
very important feature when we are dealing with credit risk over a time maturity.
5.2 Call
It is interesting to see what happens in the case of Call Option; we have proceeded exactly
in the same way, the only dierence is ag1=1 and the results are shown in Figure 12.
Figure 12: Dierence in CVAW between European and American Call options
What we obtain has an interesting nancial meaning: early exercise is never optimal, so
European and American are the same thing, with the same tree of payos.
This result was predictable from what we have obtained in the last section, in particular
in Figure 10 we have observed that American and European Options are the same without
and for positive shifts, but they are not for negative ones. This is due to the fact that in
case of negative or zero foreign rates the options lose their feature of being American.
That is why we have decided to shift the curve of Cost of Carry of the order of -2%
(the same could have been obtained with a parallel shift at least bigger than d_2year)
obtaining the following reasonable results.
20

21. Figure 13: Dierence in CVAW between European and American Call options with shift
of CoC of -2%
As we expected the blanket is a decreasing function of b and it is increasing in the Strike,
this is due to the fact that the Payo of the call is decreasing in the Strike, so everything
is coherent.
21

22. 6 Appendix
In this section we are going to show how we have developed formula (7) in order to
compute the CVAW of our options.
Let us recall the general formula for CVA in presence of WWR.
CV AW = (1 − R)
N
i=1
E
DiEi + Di−1Ei−1
2
PDi
Where we have immediately substituted
PDi = (1 − ˜Pi)SPi−1
Since is known that, in our case,the discount factors are deterministic and that
˜Pi = ηiPi where Pi =
SPi
SPi−1
the formula above, after some semplications, can be written as:
CV AW = (1 − R)
N
i=1
E
BiEi + Bi−1Ei−1
2
(SPi−1 − ηiSPi)
Reasoning on couple of nodes (ji−1,ji) the CVAW becomes:
CV AW = (1−R)
N
i=1 j−1
q ·p(ji−1)η(ji−1)
BiEE(jup
i ) + Bi−1EE(ji−1)
2
(SPi−1 −η(jup
i )SPi)
+(1−q)·p(ji−1)η(ji−1)
BiEE(jdown
i ) + Bi−1EE(ji−1)
2
(SPi−1−η(jdown
i )SPi)
where:
• jup
i and jdown
i represents respectively the nodes of the tree that can be reached at
time ti, starting from the node ji−1 at time ti−1 with probability q and (1-q)
• p(ji−1)η(ji−1) can be seen as the probability to be in the particular node ji−1 at
time i-1
22

23. References
[1] J. Hull, A. White CVA and Wrong Way Risk 2012
[2] R.Baviera, P. Pellicioli, G. La Bua CVA with Wrong-Way Risk in the Presence of
Early Exercise 2016
23