Hierarchical robust aggregation of sales forecasts at aggregated levels in e-commerce, based on exponential smoothing and Holt's linear trend method

We revisit the interest of classical statistical techniques for sales forecasting like exponential smoothing and extensions thereof (as Holt's linear trend method). We do so by considering ensemble forecasts, given by several instances of these classical techniques tuned with different (sets of) parameters, and by forming convex combinations of the elements of ensemble forecasts over time, in a robust and sequential manner. The machine-learning theory behind this is called "robust online aggregation", or "prediction with expert advice", or "prediction of individual sequences" (see Cesa-Bianchi and Lugosi, 2006). We apply this methodology to a hierarchical data set of sales provided by the e-commerce company Cdiscount and output forecasts at the levels of subsubfamilies, subfamilies and families of items sold, for various forecasting horizons (up to 6-week-ahead). The performance achieved is better than what would be obtained by optimally tuning the classical techniques on a train set and using their forecasts on the test set. The performance is also good from an intrinsic point of view (in terms of mean absolute percentage of error). While getting these better forecasts of sales at the levels of subsubfamilies, subfamilies and families is interesting per se, we also suggest to use them as additional features when forecasting demand at the item level

Model-free bounds for multi-asset options using option-implied information and their exact computation

We consider derivatives written on multiple underlyings in a one-period financial market, and we are interested in the computation of model-free upper and lower bounds for their arbitrage-free prices. We work in a completely realistic setting, in that we only assume the knowledge of traded prices for other single- and multi-asset derivatives, and even allow for the presence of bid-ask spread in these prices. We provide a fundamental theorem of asset pricing for this market model, as well as a superhedging duality result, that allows to transform the abstract maximization problem over probability measures into a more tractable minimization problem over vectors, subject to certain constraints. Then, we recast this problem into a linear semi-infinite optimization problem, and provide two algorithms for its solution. These algorithms provide upper and lower bounds for the prices that are $\varepsilon$-optimal, as well as a characterization of the optimal pricing measures. Moreover, these algorithms are efficient and allow the computation of bounds in high-dimensional scenarios (e.g. when $d=60$). Numerical experiments using synthetic data showcase the efficiency of these algorithms, while they also allow to understand the reduction of model-risk by including additional information, in the form of known derivative prices

Hierarchical robust aggregation of sales forecasts at aggregated levels in e-commerce, based on exponential smoothing and Holt's linear trend method

We revisit the interest of classical statistical techniques for sales forecasting like exponential smoothing and extensions thereof (as Holt's linear trend method). We do so by considering ensemble forecasts, given by several instances of these classical techniques tuned with different (sets of) parameters, and by forming convex combinations of the elements of ensemble forecasts over time, in a robust and sequential manner. The machine-learning theory behind this is called "robust online aggregation", or "prediction with expert advice", or "prediction of individual sequences" (see Cesa-Bianchi and Lugosi, 2006). We apply this methodology to a hierarchical data set of sales provided by the e-commerce company Cdiscount and output forecasts at the levels of subsubfamilies, subfamilies and families of items sold, for various forecasting horizons (up to 6-week-ahead). The performance achieved is better than what would be obtained by optimally tuning the classical techniques on a train set and using their forecasts on the test set. The performance is also good from an intrinsic point of view (in terms of mean absolute percentage of error). While getting these better forecasts of sales at the levels of subsubfamilies, subfamilies and families is interesting per se, we also suggest to use them as additional features when forecasting demand at the item level