# CVA Probability of default

I have to estimate CVA for an exotic option. I used Monte Carlo method to price the option with 1000 number of simulation, maturity = 1 year, and 360 time steps. So I have two questions:

1. I've read in many papers that counterparty's probability of default can be estimated by CDS. How can I do it? I'm working in Matlab
2. Is the probability of default costant over the time steps?

Simplistically, CDS = implied probability of default * loss given default. For one year maturity, you can assume flat CDS term-structure, therefore constant PD. Choose LGD (usually 50%), and you can back out the implied PD:

PD = CDS / LGD.

You could use Poisson process to model the PD, but I think the simple model above is a sufficient approximation.

If you ever work with longer maturities than 1 year, you need to take CDS term structure into account: say your option maturity is 3 years. Then you will have 1 year CDS, 2 year CDS, and 3 year CDS. You will need to bootstrap the CDS curve (similarly to any other bootstrapping), to get the forward CDS spreads (i.e. 1y1y, 2y1y): then your PD will no longer be constant.

Finally, your PD is also proportional to the time step length you choose. Your time steps are constant I assume as you split your year into 360 equal time steps. If you choose non-constant granularity, your PD will be proportional to your time-steps.

• Thanks a lot for your answer. I understand the formula that you suggest me but I don't understand the last part of your answer. So I have two new questions. Where can I find CDS data? Then, if I have a 1 year CDS, I have to use bootstrapping to calculate 365 daily probabilty of default? I never used this method, so If you have some references in MATLAB, it will be a great help for me. Aug 4 '20 at 6:47
• Maybe I'm Wrong Aug 4 '20 at 7:56
• For 1-year CDS, you don't need to bootstrap anything. Say your 1-year CDS=500bps. Assume LGD=50%. Then your 1-year PD is 500bps/50%=10%. If you are familiar with poisson processes, then split your year into 260 time steps (there are only 260 trading days in 1 year): assume that PD in the first time step is 10%/260, and calibrate your poisson process default rate to that. CVA in your first time step is EAD •PD •LGD. In all subsequent time steps, your CVA is EAD •P(survived) •PD(time-step)*LGD. If you don't like Poisson process, assume PD(1st time step)=10%/260 & PD(survive_t)=1-PD(t). Aug 4 '20 at 8:01
• Your total CVA is the sum of all incremental CVAs, discounted to today. Aug 4 '20 at 8:03
• Maybe I'm wrong with the definition of CVA. I have constructed this code: I have 10000 simulations (rows) and 360 time steps (columns) and I have calculated the value of my exotic option in each point. So at the end I have a Matrix 10000×360 of values. Then I have calculated the credit exposure at each point by this formula: max(V(t),0) where V(t) Is the market value at times t. Then I have calculated the Expected exposure at time t by mean of the value of the columns. So finally I have 360 Expected exposure. Is this algorithm wrong? Aug 4 '20 at 8:06

That is really extensive question. I am no longer part of CVA desk, but I'll answer on this question, based on my experience.

So, let's take a look at your first question. The CDS is a credit derivative, which price demonstrates a spread (sometimes with a multiplier) between bond/debt/protected asset buyer and seller, and it can be interpreted as a probability of the default. There are index products, like a «pan-American» CDX or «pan-European» iTraxx and a single-names protection.

At this point let's take a look a this website called WorldGovernmentBonds

I am not 100% sure that its data represent the actual market. I don't take any responsibility and so on. Anyway, it's not IHS Markit, BUT, to be honest we don't need to an actual CDS data to answer first question, and this site does represent the perfect example situation.

So as you may see, the CDS price is usually correlated with SP rating, and PD can be evaluated from CDS price & VaR metrics, but it's kinda.. well, we calculating the underlying event probability, based on price of derivative, which price and volatility shows us the % change of underlying (credit) event*.

We should always remember, that default !== credit event, the responsibility of the protection seller, is a different problem in another field and a very complicated question. Such as amount of Recovery and so on.

You might also saw a relevant question on QuantFinance about it How to compute the implied probability of default from a CDS spread? and you may already found Nomura's PDF with necessary formulas, which is an answer to your question.

1. Is the probability of default constant over the time steps?

If we are talking about CDS prices, — no. It depends on tenor and it's volatility. But it's a bit more complicated than that. For example, imagine yourself an N axis matrix, like:

• SP rating [AAA -> SD]
• Industry [Mining -> HealthCare]
• Operation Region [EMEA -> APAC]
• Any other relevant axis, like T (time)

This awfully long 145 page *.pdf might help you to understanding it. As I mentioned above, and you may seen it by yourself, in most cases CDS price is correlated with SP rating (and with PD itself). So for each single name (company) PD is always compared to other company in the operating region / industry sector / (relevance criteria) and so on. So it's constant, not over time steps, but over each other.

This 45 page *.pdf is a bit irrelevant, but you might find it useful in your case. Cause it's demonstrates rating change within different time periods.

Might, you find my answer useful.

• Thanks a lot. Any information is important for me. Aug 4 '20 at 6:51
• Do you have any information about Wrong way risk? I read many books and papers but I don't understand to estimate it Aug 12 '20 at 18:15