Calculating Option CVA from Future Exposure and Counterparty Default Risk
Summary
The document explains why a simple unilateral credit valuation adjustment for an option must combine the option’s future value with the counterparty’s probability of default. The suggested discretized calculation averages discounted expected exposure at adjacent dates, weights it by default probability over that interval, and applies loss given default. Default probabilities should come from counterparty credit information, such as credit default swap quotes or a suitable credit curve, rather than from the option’s underlying price distribution.
Exposure is the option’s market value at the future date, not its intrinsic payoff. A basic Monte Carlo approach simulates underlying prices at each exposure date, reprices the remaining option using a pricing model, and averages those values across paths. In a flat-rate, constant-volatility Black–Scholes setting, the option can be repriced analytically at each simulated state. The discussion is introductory: it does not specify detailed calibration, dependence between exposure and default, or broader model and collateral considerations.
Key ideas
- CVA reflects expected losses from counterparty default over the life of a trade.
- Default probabilities describe the counterparty’s credit risk, not the option’s likelihood of finishing in the money.
- Option exposure is its future market value, rather than its intrinsic payoff at that date.
- Monte Carlo paths can estimate exposure by simulating the underlying and repricing the option at each date.
- The adjustment includes loss given default and discounting, with accuracy depending on the credit and pricing assumptions.
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Full text
# CVA for options
# 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.
## Answer by byouness (score 5)
https://quant.stackexchange.com/a/64131
To expand on Quantuple's comments:
#### 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.
## Answer by achirikhin (score 1)
https://quant.stackexchange.com/a/79311
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).Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.