Estimating CVA for a Purchased European Call Option
Summary
The document discusses counterparty credit valuation adjustment for a European call bought from a counterparty. The key intuition is that the buyer has paid the premium and remains exposed to losing the value of the option if the seller defaults before expiry. Under the stated simplifications of constant loss given default and independence between default probabilities and option value, CVA is expressed as an integral of discounted expected option value weighted by incremental default probability.
It also presents a time-discretized approximation that sums expected discounted option values over intervals, weighted by the counterparty’s default probability in each interval. The response does not fully validate the question’s proposed conditional formula; its integral starts at time zero and depends on the independence assumption. In more general settings, dependence between exposure and default, recovery variation, and the precise filtration and discounting conventions require additional treatment.
Key ideas
- A purchased option can create positive counterparty exposure because the seller owes the future payoff.
- CVA represents expected loss from counterparty default, discounted to valuation time.
- With constant LGD and independence assumptions, expected discounted option value is weighted by default probability over time.
- A discrete-time approximation sums exposure contributions across intervals using interval default probabilities.
- Exposure-default dependence and other modeling choices can change the calculation.
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Full text
# CVA formula for a call option
# CVA formula for a call option
I have a very quick question. Suppose that I buy a European call option from party S with expiry $T$. I want to determine the general formula for the CVA of the option, at time $t$. If I let $T_1\leq T$ be the default time of $S$ and I denote by $\xi=S$ the default event of $S$ (I am following here the notation of Quantitative Risk Management by McNeil, Frey, Embrechts, Chapter 17), and I denote by $c(t,T)$ the risk-neutral, default-free (Black-Scholes) price of the option at time $t$, is the following formula correct: $$ CVA(t) = LGD\cdot\mathbb E^Q[\mathbb 1_{\{T_1\leq T\}}\cdot\mathbb 1_{\{\xi=S\}}D(t,T_1)c(T_1,T) |\mathcal F_t] ? $$
My point here is that the evaluation of the default-free expected cash flow of the option is just the price of the option at time $t$, and since this cannot be negative, the term $c(T_1,T)$ is always positive, so there's no need to take its positive part.
So is my formula above correct?
## Answer by byouness (score 3)
https://quant.stackexchange.com/a/32063
No need to overcomplexify things. The CVA gives you simply the amount that you expect to lose if and when your counterparty defaults, discounted to today.
In your case, you are buying an option. So, you already paid a premium but still expect to receive the payoff at expiry. In a sense, your counterparty owes you the payoff and you can lose this amount in case of a default.
Assuming the LGD is constant, and that the counteparty default probabilities and the call value are independant, the CVA can be expressed as follows: $$ \mathrm{CVA} = LGD \int_0^T \mathbb{E} [ D(0,t) c(t,T) ] dPD(0,t)$$ Which can be discretized as follows ($t_0 = 0,\dots \ , t_n = T$): $$ \mathrm{CVA} \approx LGD \sum_{i=0}^{n-1} \mathbb{E} [ D(0,t_i) c(t_i,T) ] PD(t_i,t_{i+1}) \\ $$ So it is equal to the sum of the expected call value at each time-step (until its expiry), weighted by the default probability of the counterparty during this time-step.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.