Validation Results of a Parametric Model for European Equity Index Futures

Introduction

This article presents the validation results of a parametric trading model, which has shown remarkable effectiveness, particularly in the context of European equity index futures.

To ensure the model's reliability and performance, a rigorous out-of-sample validation procedure was meticulously devised. This validation process involved multiple rounds of calibration and thorough testing on unseen data, aiming to establish the model's tradability and evaluate its real-world performance.

The validation results provide compelling evidence of the model's viability as a practical trading tool, as it successfully passed all validation tests. Furthermore, the validation procedure highlighted a significant diversification effect within the portfolio, underscoring the advantages of incorporating multiple instruments into the trading strategy.

Parametric models:

A parametric model is a model that depends on a data set and on a set of parameters.

These parameters have predefined values that determine the model’s combination space, which is represented by all possible n_plet combinations [p1, p2….pn].

Each of these unique n_plets generates a specific trading signal, and every signal corresponds to a particular value of the fitness function, which we select to calibrate the model.

For better understanding, let's define some basic notations and rules of calculation: t_i is the i_th timestamp and it is referred to as i.

The conditions for entering trades, denoted as condentry long and condentry short, depend on a set of parameters: p_1, p_2, ..., p_n, as well as the available data up to time i.

If there is a signal generated by condentry long being true at time i, then a position will be open at time i+1. The position will remain the same until there is a signal from condentry short

The return corresponding to the signal generated at time i, will be known at time i+2.

The resulting profit or loss, pl, is obtained by multiplying the return by a certain notional.

Finally, the equity at time i, is the cumulative sum of all profit or loss values up to time i:

For instance, a very basic seasonal model in which every day I go long at a specific hour and go short at another specific hour, is described by 2 parameters: the entry hour and the exit hour, and the parameter space is formed by all the pairs (T entry long , T entry short).

If we assume the instruments trades 23 hours every day, there are 23*22 possible pairs or 2_plets.

Validating a model always implies generating a Matrix that encompasses all the time series of returns associated with the parametric space. The number of columns in this Matrix is equivalent to the total possible parameter combinations (e.g., 23*22 in the previous simple example). Meanwhile, the number of rows is determined by the number of timestamps in the return time series.

For instance, suppose we have ten years of price observations, and each year has 250 trading days. In that case, the number of columns in the Matrix would be (250*10)-1 (excluding one because the calculation transitions from price to return).

Validation Methodology:

The out-of-sample period spans from the minimum year to the maximum year, and it is divided into intervals of one-year length, covering the entire range from year min to year max.

For each out-of-sample year, denoted as k, a series of N calibrations are conducted. Each calibration is performed within an in-sample period that concludes at year k-1. The length of the in-sample calibration period varies, starting from a minimum duration and incrementing by x months at each iteration, until it reaches a maximum length.

The calibration process entails finding the n_plet that optimizes a specific target function within the parameter space. As a result, for each run, a set of associated parameters is generated, as presented in the table below:

In the subsequent charts, the equity lines representing the solutions for each run and year are displayed. Specifically, the first chart illustrates the equity lines for each solution during the in-sample period, assuming the target function is the Sharpe ratio. The second chart presents the equity line of each solution during the out-of-sample period.

Out of sample results:

The validation technique discussed earlier adopts an out-of-sample approach to evaluate the model's performance expectations. To facilitate this evaluation, specific definitions are necessary. For each instrument and every out-of-sample year, the validation methodology generates multiple equity lines. To calculate the expected return at a given timestamp, an average is taken over these equity lines.

Similarly, to determine the expected return and standard deviation, an averaging process is conducted across all the equity lines and out-of-sample years. The notation T(k) represents the number of timestamps in year k.

Where T(k) indicates the number of timestamps at year k.

From which we can define the expected sharpe ratio as:

Another significant metric for assessing the model's quality is the average trade, which quantifies the average number of basis points per trade. This metric is computed by averaging across all solutions and out-of-sample years. If N trades(I, k) denotes the number of trades corresponding to solution I in year k, then the average trade can be calculated accordingly.

?In the chart below the expected out of sample equity line consistent with the definitions above are shown:

Aggregated Out of Sampple results

The correlation matrix displayed below illustrates the relationship between the returns of different instruments. It reveals a range of medium to low correlations among the instruments, indicating a favorable diversification effect.

?The observed diversification effect implies that the instruments exhibit relatively independent performance. Even without incorporating any risk weighting or complex optimization techniques, a simple portfolio constructed by averaging the returns of each instrument already demonstrates a notable enhancement in the Sharpe ratio.

Conclusions

The validation results of the parametric model presented in this article provide compelling evidence of its viability as a practical trading tool, particularly in the context of European equity index futures. Through a rigorous out-of-sample validation procedure, the model demonstrated remarkable effectiveness, passing all validation tests. The validation process highlighted a significant diversification effect within the portfolio, showcasing the advantages of incorporating multiple instruments into the trading strategy.