Historical Simulation

 Historical simulation is the method of prediction in terms of which the forecast probability density depends directly on the past empirical distribution. In general the length of the past used for forecasting can be a parameter - which is adjusted for best performance in the in-sample period. The Bank of International Settlements in Basle (BIS), Switzerland, has however recommended [1] the use of at least 250 day memory. In view of this we present this model with a 250 day memory since the comparative performance of the BIS recommendation is likely to be of general topical interest.

Since no optimization is involved, the 10000 univariate and 1000 multivariate prediction realization pairs are constructed using the last 1250 data points $t\in[-248,1001]$ in the 10 log differenced price change series k.

To construct p_t^(k,P) we first define  

$$X_t^{_{(k,P)}} = \{x_{t}^{_{(k,P)}},x_{t-1}^{_{(k,P)}},\ldots ,x_{t-249}^{_{(k,P)}}\}$$ (5)

as the set of 250 points to be used in constructing p_t^(k,P) at time t. Then we introduce ranking notation using parenthetical subscripts (to be distinguished from x_t^(k,P)) such that  

$$x_{(i,t)}^{_{(k,P)}} \le x_{(i+1,t)}^{_{(k,P)}},\quad i\in[1,250], \quad x_{(i,t)}^{_{(k,P)}}\in X_t^{_{(k,P)}}.$$ (6)

Thus x_(1,t)^(k,P) is the minimum and x_{(250,t)^(k,P)} is the maximum element in X_t^(k,P).

We then construct p_t^(k,P) to be the probability density function such that:  

$$\int_{-\infty}^{x_{(i,t)}^{_{(k,P)}}}p_t^{_{(k,P)}}(x)d\!x = {i-{1\over2}\over 250} \quad\forall i\in[1,250].$$ (7)

The above density function is assumed to remain constant between x_(i,t)^(k,P) and x_{(i+1,t)^(k,P)} so that:

$$\int_{x_{(i,t)}^{_{(k,P)}}}^{x_{(i+1,t)}^{_{(k,P)}}}p_t^{_{(k,P)}}(x)d\!x = {1\over250}.$$ (8)

If x_(1,t)^(k,P) < x_t+1^(k,P) < x_{(250,t)^(k,P)} then the probability density and fractile value associated with x_t+1^(k,P) is well defined. If however x_t+1^(k,P) < x_(1,t)^(k,P) or x_t+1^(k,P) > x_{(250,t)^(k,P)} the probability density function is not defined and the fractile value of such an event cannot be determined. To resolve this - we extend this model by asserting that p_t^(k,P) has Gaussian tails and hence the form implied by a normal distribution with mean $\bar x = {\bar x}_t^{_{(k,P)}}$estimated from set X_t^(k,P) and where $\sigma = \sigma_t^{l_{(k,P)}}$ for the left side of the distribution which satisfies

$$\int_{-\infty}^{x_{(1,t)}^{_{(k,P)}}} {1\over\sqrt{2\pi}\ \s... ...r x\over\sigma_t^{l_{(k,P)}}}\right]^2\right)d\!x = {1\over500}$$ (9)

and with $\sigma = \sigma_t^{r_{(k,P)}}$ for the right side of the distribution which satisfies

$$\int_{x_{(250,t)}^{_{(k,P)}}}^{\infty} {1\over\sqrt{2\pi}\ \... ...r x\over\sigma_t^{r_{(k,P)}}}\right]^2\right)d\!x = {1\over500}$$ (10)

This completely defines the construction of the forecast distribution p_t^(k,P) and through this the out-of-sample prediction-realization pairs that form the starting point for performance assessment of this model.