i'm struggling with the idea of the time default random variable in the Unilateral CVA.

While the CVA in discrete model is only the sum of the discounted exposure of your financial position (the amount of what you must receive from the other part) multiplied by the probability of deafault. This is an easy expected value to calculate..

I'm not understanding why changes the approach in time continuos CVA calculation. In particular i'm not understanding what is a time of default.

Can you help whit a simple example? Thank You

The approach is the same in discrete and continuous time.

You have $$\text{CVA} = E\left[e^{-\int_0^{\tau} r_u du} \text{EAD}_{\tau} (1-R)\mathbf{1}_{\tau \leq T}\right]$$ where

• $\tau$ = stochastic time of default ($+\infty$ if the counterparty never defaults)
• $T$ = deal maturity
• $\text{EAD}_t$ = exposure at default = exposure (stochastic) to the counterparty if the counterparty defaults on time $t$
• $R$ = recovery rate

This formula simply states that CVA is the present value of a flow that represents the Loss Given Default upon default.

If you now assume that default time and discounted EAD are independent (no "wrong way risk") then $$\text{CVA} = \int_0^T E\left[e^{-\int_0^{t} r_u du} \text{EAD}_{t} (1-R)\right] \phi(t) dt$$ where $\phi()$ is the density of the distribution of $\tau$.

If you then discretize time with a discrete time line $t_i$, $t_0=0$, $t_N=T$, the integral is approximated as $$\text{CVA} = \sum_{i=1}^{N} E\left[e^{-\int_0^{t_i} r_u du} \text{EAD}_{t_i} (1-R)\right] P(t_{i-1} < \tau \leq t_i)$$ which is the discrete model CVA formula you are familiar with.