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) Commented Oct 29, 2019 at 20:25

Default probability

The CVA is the price adjustment to take into account the default of the counterpart. So, it is obtained by taking the sum of future expected exposures multiplied by the default probabilities of the counterpart on each period (and by the LGD): $$CVA = LGD \times \sum_{i=1}^m \frac{DiscountedEE(t_{i-1}) + DiscountedEE(t_i)}{2} \times PD(t_{i-1},t_i)$$

So, $$PD$$ is not related to the option's underlying, but to the counterpart! You can get this from CDS quotes, or proxy it by the (sector x area x rating) credit curve usually provided by data vendors.

Expected exposure

If the counterpart defaults, you lose the option's value, not it's intrinsic value. So, the exposure is not equal to the intrinsic value of the option, but rather to its price!

The expected exposure is then equal to the expectation taken accross the Monte Carlo paths. For a call for example: $$DiscountedEE(t) = \mathbb{E}\left[D(0, t) \underbrace{\mathbb{E} \left[D(t, T) (S(T) - K)^+ |\mathcal{F}_t \right]}_{\text{Call option's price at } t} \right]$$

You could for example do a Monte-Carlo for the outer expectation, but compute the call price at each date and path of this Monte-Carlo using a closed-form formula.

You could also try to derive a closed-form formula for the expected exposure (it's a compounded option or option on an option) depending on the models you are using.

In this formula (from Basel III reg), EE is the future value of the option, not the payoff of the option. To compute EE at a given time, you need, in the most naive way, simulate the value of the underlying and plug it into an option pricing formula, also plugging smaller time to maturity.

If you live in the Black-Scholes economy with flat rate and vol, then you merely need to generate a sample of the stock values at time t (when you compute expected exposure), plug them into BS with original rate and vol and remaining time to maturity, and then take the average of those values to arrive at EE(t).