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Background documentationStructurized Monte Carlo for CFaR

 

Monte Carlo simulation is a simulation method used to describe potential changes to the values of risk factors using random walks. Random walks are generated using a stochastic process. The random numbers required for this process are generated using a random number generator, and various methods can be used to transform these random numbers into the standard normal distribution.

Monte Carlo simulation is based on the assumption that changes to risk factors are distributed normally with an expected value of zero and positive variance. The parameters required for the stochastic process can be calculated using historical values (structurized Monte Carlo).

Monte Carlo simulation comprises the following steps:

  1. Generation of a time series of independent random numbers, distributed using standard normal distribution, for each risk factor:

    X1, X2, X3, ..., Xn

    To generate random numbers, you can choose one of the following methods:

  2. With the random numbers determined in this way and distributed using standard normal distribution, a random walk is determined for a risk factor rf(t1), rf(t1), rf(t2), rf(t3), and so on. Determination of the random walk depends on the element type selected in the statistics type:

    • Relative

      rf(ti) = rf(ti-1)*(1+Xi)

    • Logarithmed

      rf(ti) = rf(ti-1)*exp(Xi)

    where the start value rf(t0) is always the actual value of the risk factor as per the market data table on the evaluation date.

  3. CFaR calculation is then performed on the basis of the random walks.

Integration

The Cash Flow at Risk values are represented on the basis of the portfolio hierarchy, the risk hierarchy, and the maturity band.

It is possible to calculate Cash Flow at Risk using the following methods:

  • With the full valuation approach, the profit/loss on the hierarchy levels of the risk hierarchy is calculated for the position on each portfolio hierarchy level by recalculating the cash flow afresh for each random walk.

    This approach delivers precise results. However, in cases involving a large number of random walks, calculations can require very long runtimes.

  • If the cash flow has a linear dependency on the risk factor, you can use approximations that also lead to precise results but require shorter calculation runtimes:

    • With the delta approach, the reactability of the cash flow to risk factors is calculated for each portfolio on a risk hierarchy level independently of the historical market prices (delta positions). For this, the function CF(rf) is approximated using a linear function.

    • The delta/gamma approach operates in the same way as the delta approach but, in addition, includes the non-linear terms of the second order (gamma positions) at the risk factor level. In this way, the function CF(rf) is approximated by a quadratic function.

    The delta approach and the delta/gamma approach are also referred to as risk factor mapping.

    Note Note

    When calculating Cash Flow at Risk using the variance/covariance approach, you can use only the delta approach and the delta/gamma approach.

    End of the note.