Estimating Monte Carlo VaR from the Worst Simulated P&Ls
Summary
The discussion explains how to estimate portfolio Value at Risk from simulated scenario P&Ls. Reprice the portfolio under each scenario using the same valuation approach used for marking the book, then identify the loss at the chosen tail percentile. In the example, the worst one percent of 50,000 simulated P&Ls is retained, and the cutoff loss in that tail is the VaR estimate.
The reply describes a memory-efficient selection approach: maintain only the worst tail observations, replacing its current cutoff when a more severe loss arrives. The same retained tail can support an expected shortfall estimate. Full repricing is preferred for accuracy; faster approximations may be used when computation is costly, but the answer specifically cautions against delta-gamma estimates. The result depends on the simulated scenarios, tail level, and valuation quality, so the method does not itself establish that the scenario distribution is realistic.
Key ideas
- VaR is read from the loss threshold at a selected tail percentile of scenario P&Ls.
- Each scenario requires repricing the portfolio to obtain its P&L.
- A running set of the worst tail observations can be used to find the VaR cutoff without storing every outcome.
- The retained worst tail can also be used to estimate expected shortfall.
- Full repricing is preferred, while shortcuts trade accuracy for speed.
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# Computing VaR in a Monte Carlo simulation (question from Joshi's book) # Computing VaR in a Monte Carlo simulation (question from Joshi's book) I am studying Joshi's book on C++ for derivatives pricing. I am at chapter 5 on implementing a statistics gatherers class to use in a (simple) MC routine for pricing vanilla options, where it is assumed that the underlying follows GBM. My question concerns one of the exercises where it is asked to compute the Value at risk for a sample. I am confused about what this is actually asking. My basic guess would be this: for each path compute the difference of the final value of the option and the initial value. Then collect all the information together and run some statistics to calculate the VaR based on that data. But this doesn't really seem correct... Any thoughts? thanks in advance! ## Answer by Dimitri Vulis (score 1) https://quant.stackexchange.com/a/67792 Let's suppose for concreteness that you're generating 50,000 possible P&L's, and you're looking for the worst 1%. You only need to remember at one time the P&L's of the worst-case tail, which is 1% $\times$ 50,000 = 500 P&L's. You do need to reprice your book under each of the 50,000 scenarios. Ideally, you should fully reprice using the same calculation that you use to mark to market. However, if this takes too much calculation, then there are shortcuts to get faster but less accurate P&L estimates, but you definitely must not use delta-gamma. For every P&L, if it is better than your current 500th worst, then just discard it and go to the next scenario. If it is worse, then insert the new P&L in your worst-case tail and discard the 501st worst (previous 500th worst). Once you have compared all 50,000 P&Ls, the 500th worst is the VaR. You can also use the entire worst-case tail to calculate the expected shortfall (ES).
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