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# CVA valuation using Monte Carlo

#### Plirts

##### New Member
Hi!
Probably it will be explained in video and my apologies for the question. But I'd like to understand 2nd step. "If A default, it is gain/loss for B and B's losses/gains are positive/negative"? How can loss be positive? And if cpt default, the only gain I could think of is the case where B should pay to A and A defaults before payment. I do not understand...
Also Study Notes page 40 CDS settlement methods: cash, physical, or digital. Below you only explain cash and physical. What is a digital settlement?
Thanks,
Plirts

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi Plirts,

I assume you realize it's directly from the assigned reading (full text copied below)?

For me, the key part to this is the idea that: from "Counterparty A's" perspective, a GAIN in the value (market-risk only; not credit risk) creates an exposure to the counterparty, such that a simulated future DEFAULT on the positive market-priced position becomes a simulated LOSS to Counterparty A. For example, you enter a swap with me, value = 0 to both of us at inception, where you pay fixed. The floating rate increases, an you move to "in the money" with a gain. But I default, you LOST the higher floating coupon that I owed to you. (Meanwhile, I am out of the money, with no credit exposure: if you default, I am no worse off).

Canabarro gives:
CVA [where positive --> B has greater exposure; negative --> A has greater exposure] = E(A)*LGD(A) - E(B)*LGD(B)
In Canabarro's example, E(A) = 200, LGD(A) = 2%; E(B) = 100, LGD(B) = 5%, such that:
CVA [from A's perspective] = E(A)*LGD(A) - E(B)*LGD(B) = 200*2% - 100*5%; we can abstract this for better understanding maybe:
CVA [from A's perspective] = Expected Loss (from B's perspective) - Expected Loss (from A's perspective)

If we simulate out a trial into the future, I like to imagine an interest rate swap:
• If, in a trial, A's position improves such that A becomes "in the money," A will have a gain on the position which creates positive future exposure: If B defaults with A is ITM, A experiences a credit LOSS. In the CVA above, this is a negative; i.e., to the extent those occur, the CVA[from A's perspective] will become negative and deduct from the market-risk-only price of the swap.
• If, in a trial, B's position improves such that B becomes ITM, B's ITM simulated gain in the swap creates future exposure; if A defaults, B experiences a credit loss. A loss to B is a positive in the CVA[from A's perspective] which increases the CVA adjustment from A's perspective.
CVA[from A's perspective] increases to the extent simulations return A's default when B is ITM (gain on value), the adjust reflects A should be willing to pay more in this case, but decreases as simulations return B's default when A is ITM, the adjustment reflects A should be willing to pay less in this case.

From Canabarro's paper:
The fully fledged model of CVA calculation is specified by the Monte Carlo simulation described below:
1. The default events of both counterparties (as well as the evolutions of all market prices) are simulated over the lives of the derivative trades and according to the risk-neutral dynamics (hazard rates, recovery rates, volatilities, correlations, drifts, etc).
2. If and when the first default event (of either A or B) occurs on a simulated market evolution (path), the loss or gain relative to the mid-market valuation of the trades to the non-defaulting counterparty is calculated and discounted to time zero using the appropriate stochastic discount factor (which is simulated along the market path).
3. A's default causes a loss or a gain (relative to mid-market) to B when the trades are closed out. B's default causes a loss or a gain (relative to mid-market) to A. A's losses/gains are negative/positive, respectively. B's losses/gains are positive/negative, respectively. The CVA is the average of all losses and gains over all simulated paths.