Explanation

Probability Integral Transform (PIT) derivation:

For each day, the bank:

1. Collects the actual portfolio Profit/Loss realization
2. Calculates the cumulative probability of observing a P/L value below the actual realization using the risk model

This is mathematically expressed as: $PIT_t = F_t(L_t)$ where:

• $F_t$ is the cumulative distribution function of the risk model
• $L_t$ is the actual loss realization

Why Option A is correct:

• PIT represents the cumulative probability up to the actual realization
• It's the probability of observing a value below the actual P/L
• If the model is correct, PIT values should be uniformly distributed U(0,1)

Why Option B is incorrect:

• PIT is not the probability of observing the exact P/L value
• It's the cumulative probability up to and including the actual value
• For continuous distributions, the probability of any single value is zero

This PIT-based approach allows regulators to test both unconditional coverage and independence properties of the VaR model.