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Monte Carlo simulations are more powerful because they explore comprehensively market outcomes. It is common to contrast historical simulations to forward looking Monte Carlo simulations. Historical simulations use actually observed sets of parameter values embedding all volatilities and correlations, over the horizon used for collecting the historical data. The historical simulations use samples from historical data. Future price calculations use historical changes of market parameters applied to the current prices. The last step is to revalue the portfolio using all historically observed sets of values of market parameters over the selected horizon to construct the portfolio value distribution. Using daily observations over 2 years, for example, will provide around 500 sets of values of all market parameters. These observed values embed all correlations observed in the market over the period. They serve for marking-to-market the portfolio 500 times. The risk statistics result from the distribution of these 500 values.

The drawback is that the technique is essentially backward looking. In addition, it averages each observation of market parameters independently of how representative it is of current conditions. The technique is slow in including latest data. In some cases, it is conservative, because historical observations capture market shocks of large magnitude. In others, it averages so much the past values that it misses the current market conditions, which might be an outlier compared to historical scenarios. A large number of observations are necessary to search the fat tail of the VaR measure, and past information over a recent period might not provide enough data. On the other hand, moving too far backwards into the past might capture market behaviour that has become irrelevant. For instance, market liquidity could be poor in the past for new products, and improve when new products disseminate in the market.

Historical simulations provide two benefits:

• They serve for back testing purposes, making sure that the portfolio values include those that would have appeared if the past repeats itself on the current portfolio structure.

• They capture in the simplest possible way all volatilities and correlations linking market parameters to each other.

Full Monte Carlo Simulations

Under full Monte Carlo simulations, the process starts by modelling the stochastic processes of market parameters, doing the best to ensure that they capture recent worst-case situations. It is similar to historical simulations except that we look forward using simulated values rather than historical values. Modelling the inputs is a crucial issue. Inputs include volatilities and correlations whose measures raise issues with respect to their changes over time.

The next stage is the generation of random values of market parameters complying with this input structure. The last stage is the portfolio revaluation for each set of values generated. Since we capture all forward looking information, as well as past information embedded in modelling inputs, we have the best of both worlds. The main drawback of the full-blown simulation is that it is calculation intensive. All instruments should be valued for all sets of market parameter values. The number of runs has to be high enough to provide a sufficient accuracy for the portfolio value distribution. Each set of values implies revaluing each individual instrument within the portfolio.

By contrast, the correlation (Delta VaR) methodology uses a unique variance-co-variance matrix for all portfolios and the calculation requires only a product of matrices to obtain the portfolio volatility.