# CVA for options

I am trying to do a simple unilateral CVA for call and put options. I found this discretised formula online: $$CVA = \sum_{i=1}^m \frac{EE(t_{i-1})DF(t_{i-1}) + EE(t_i)DF(t_i)}{2} \left( PD(t_i) - PD(t_{i-1}) \right)$$

And I found online that $$EE = \max (S - K,0)$$ for a call and $$EE= \max(K - S,0)$$ for a put. Then, I found that $$PD = N(d_2)$$ for a call and that $$PD = N(-d_2)$$ for a put. Where N is the cumulative distribution function and d2 is from the Black-Sholes framework.

Is this way of calculating the CVA correct ?

Moreover $$DF$$ should be a Discounting Factor and I was thinking of using the risk-free rate used to price the options. Does it make sense? Moreover, I do not see the LGD anywhere, why?

I am looking for a basic approach involving Monte-Carlo that I could code in Python. I have not studied quantitative finance and I have not advanced Mathematical knowledge.

• My 2cents: 1) Everything should be multiplied by LGD indeed, 2) when being (long) call/puts EPE(t)DF(t) = MtM(0) assuming deterministic rates, 3) PD(t) is the probability of default of the counterparty, it has (a priori) nothing to do with the risk-neutral probability of the underlying of the option ending up ITM as you seem to indicate 4) you don't need Monte-Carlo here since everything is closed form (note that the formula can be simplified since the discounted expected positive exposure is constant) – Quantuple Oct 29 '19 at 20:25