The simulation of distributions of financial assets is an important issue for financial institutions. If risk measures are evaluated for a simulated distribution instead of the model-implied distribution, the errors in the risk measurements need to be analyzed. For distribution-invariant risk measures which are continuous on compacts, we employ the theory of large deviations to study the probability of large errors. If the approximate risk measurements are based on the empirical distribution of independent samples, then the rate function equals the minimal relative entropy under a risk measure constraint. We solve this minimization problem explicitly for shortfall risk and average value at risk.

Let (S k )k≥0 be a Markov chain with state space E and (ξx )x∊E be a family of ℝp -valued random vectors assumed independent of the Markov chain. The ξx could be assumed independent and identically distributed or could be Gaussian with reasonable correlations. We study the large deviations of the sum

Let {Y n | n=1,2,…} be a stochastic process and M a positive real number. Define the time of ruin by T = inf{n | Y n > M} (T = +∞ if Y n ≤ M for n=1,2,…). We are interested in the ruin probabilities for large M. Define the family of measures {P M | M > 0} by P M (B) = P(T/M ∊ B) for B ∊ ℬ (ℬ = Borel sets of ℝ). We prove that for a wide class of processes {Y n }, the family {P M } satisfies a large deviations principle. The rate function will correspond to the approximation P(T/M ≈ x) ≈ P(Y ⌈xM⌉/M ≈ 1) for x > 0. We apply the result to a simulation problem.