Stop-losses, maximum drawdown-at-risk and replicating financial time series with the stationary bootstrap

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Notes

†Closed-form formulas for computing the MDaR levels are available for certain stochastic process such as the Geometric Brownian Motion. However, in the case where one does not know the true form of the data generation process (which is typical for financial time series) or if the closed-form formula is too difficult to determine, then one will have to resort to a numerical approach.

‡Note that from a theoretical standpoint, it might be more accurate to resample the underlying financial time series (e.g. exchange rates) and then apply the trading rules rather than to bootstrap the model or portfolio returns directly. However, this procedure is computationally intensive and is complicated by issues of cross-correlation between the underlying assets.

†Note that this is almost like pricing a down-and-out barrier option without a strike price. Also, the drift plays an important role here and we are not discounting the future payoff.

†Not to mention that each of these patterns will also need to be verified statistically before it will be considered as part of the price model.

†Note, however, that if the original data is not strictly stationary, the samples generated by the stationary bootstrap will also not be strictly stationary. Mathematically, if a time series (where Z is an integer set) exhibits the property then no bootstrap algorithm can resample a series such that the strict stationarity condition is satisfied: that is, the joint distribution only depends on the time difference, k, and not on time t, where the asterisk ‘*’ indicates a resampled observation. Indeed, it is difficult to imagine any bootstrap algorithm that is able to generate a stationary series from non-stationary data.

‡The corrected version of this algorithm is given by Patton et al. (2009).

†These indices are available on Bloomberg.

‡It is worth noting that the MDaR _α cost curves for both the CitiFX Beta Trend and Carry models are also similar to those illustrated in figures 2 and 3.

§In BLL, the study was based on the DJIA index and the scenario generating models tested included the random walk, AR(1), GARCH-M and E-GARCH. None of these price models was found to be consistent with the trading rule returns obtained from the original data.

§A Matlab code for this algorithm is publicly available for download on Andrew Patton's website: http://econ.duke.edu/~ap172/code.html

¶We consulted Prof. Politis regarding the use of his algorithm in this manner and he told us that although it will not yield an ‘optimal’ result, the computed block lengths will still be reasonable estimates.