Converting Monte Carlo CVA into a Running Spread
Summary
The document asks how to express a dollar-valued CVA from Monte Carlo simulation as an annual running spread. The example uses future interest-rate term structures simulated with a one-factor Hull–White model. It describes dividing the CVA by a CDS risky annuity, and then by notional to express the result as a percentage. The difficulty is applying the annuity formula when the initial risk-free curve slopes rather than staying constant.
The answer cautions that this conversion is approximate: adding a spread changes the contract’s CVA, so a fully consistent spread requires iterative repricing until the risky mark-to-market is zero. It points to a simpler method in cited literature and tentatively suggests using a risk-free rate, such as OIS, for the relevant maturity. That rate guidance is explicitly uncertain, and the document supplies no worked calculation or comparison of methods.
Key ideas
- A Monte Carlo CVA can be expressed approximately as a running spread by dividing by a risky annuity and notional.
- A constant-rate annuity formula does not directly settle the calculation for a sloping initial curve.
- Adding a spread affects CVA, making the exact conversion nonlinear.
- An iterative calculation can seek a spread that brings the risky mark-to-market to zero.
- The suggested choice of a maturity-matched OIS rate is tentative.
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# CVA as a running spread - risk annuity calculation in the Monte Carlo framework # CVA as a running spread - risk annuity calculation in the Monte Carlo framework I have simulated future term structures in the one-factor Hull-White model and calculated the CVA of a particular trade (let's say, now I have it in absolute value, in dollars). However, I want to represent this CVA value as a running spread. As far as I understood from the Gregory's book (2011) "Counterparty credit risk - The new challenge for global financial markets", to get the annual spread value I have to divide my absolute CVA value by the CDS risky annuity (and then also by the par value to get the value in %). This is something where I am stuck. The formula of the risk annuity given in the book is (1-exp(-(r+h)(T-t)))/(r+h) where r is the constant continiously compounded interest rate and h is the hazard rate. For my case T-t = 12 years, and the yearly CDS spread is 300 bps. However, in my example r is not constant (I have the upward-sloping initial interest rate term structure). Can anyone advice me how to estimate CVA as a running spread in this case? If u also can offer a good paper as a reference, I would also be very grateful. Thanks in advance. ## Answer by AfterWorkGuinness (score 2, accepted) https://quant.stackexchange.com/a/21019 This only produces an approximation. As per Gregory (page 256) > However, adding a spread to a contract such as a swap, the problem is non-linear since the spread itself will have an impact on the CVA. The correct value should be calculated recursively (since the spread is risky too) until the risky MTM of the contract is zero. He points to a simpler solution that doesn't require a recursive solution in Vrins and Gregory (2011) In regards to what value of r to use for an upward sloping curve, I do believe (but am not certain) you want a risk free rate (such as OIS) for the same maturity as T-t.
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