Historical Simulation: Theoretical Background 
Historical Simulation: Theoretical Background
The purpose of historical simulation is to determine what profits or losses would be incurred if a market price development from the past were to occur today. In the process, the correlations between the risk factors are implicitly taken into account. The calculation methods available can roughly be divided into the following procedures:
Full valuation:
In the case of full valuation within historical simulation, n comparative NPV calculations are made over the historical period. Hence market data changes (DMD) are determined for all time points in the historical period for a certain holding period.
First the NPV is calculated using current market data. Then the NPV is also calculated for each change to the current market data brought about by the historical market data changes (DMD).This is known as the simulated net present value. These simulated NPVs are compared with the NPV calculated from the current market data. This results in n potential profits/losses.
This calculation is done at each risk factor level in the risk hierarchy, taking into account the respective historical changes. Every time a new node in the risk hierarchy is reached, a new calculation is carried out, which takes into account all the historical changes to the risk factors below that particular node.
The correlations of the individual market prices and the relationships between positions are implicitly taken into account since the NPVs at each time point in the historical period are calculated based on all the market data valid for that time point.
Profits and losses are sorted by amount.
The relative frequency of the profits and losses is calculated. If there is a large enough sample (n), this is an unbiased estimator for predicting the actual probability distribution of profits and losses.
When you enter a confidence level, the system calculates the value at risk from the distribution of profits and losses. This value at risk amount represents a floor based on the specified probability.
With a sample of 200 values and a confidence level of 99%, the third largest loss represents the value at risk.
Delta valuation:
With the delta approach, the NPV is not calculated for every point in time in the historical period. Instead, the elasticity of the price function is estimated for the different price parameters, independent of historical market prices (
delta positions). The NPV differences result from weighting this reactivity with the price differences from the historical market data. As with full valuation, this results in n potential profits/losses, whose relative frequency distribution can be displayed.This approach is based on the assumption that the NPV function is linear. Due to this assumption, the number of calculations is reduced because the implicit correlation between the risk factors is no longer taken into account.
Delta-Gamma approach:
Here, in contrast to the delta valuation, non-linear terms of the second order (gamma positions) are additionally taken into account at the risk factor level.
The process used for both delta valuation and the delta-gamma approach is known as risk factor mapping.