Physical returns in a pricing world: Towards forward-looking market risk measures

Most risk measures are calculated using the physical measure, $ \mathbb{P} $. This measure is commonly estimated using historical data, often under the implicit assumption that realized returns are homogeneous in time. Since risk measures are used to assess future risks, they should consider sources of forward-looking information as well. One natural source of this information is option data since it reflects the market's perception of future returns under the pricing measure $ \mathbb{Q} $. Several methodologies incorporating option data have been proposed in the literature. In this paper, we build on the established tilting framework that links $ \mathbb{P} $ and $ \mathbb{Q} $, and propose a refined implementation that preserves the structure of the pricing kernel.One of the key contributions is the introduction of the Adjusted Tilted Sato Bilateral Gamma (ATSBG) process, a new model that fits the S&P 500 option surface competitively relative to standard benchmarks. Within the adapted tilting framework, we use this process to infer an estimate of the physical distribution from option data and evaluate its performance across multiple return horizons. A second key contribution is that our results suggest that the physical measure can be reasonably well extracted from option surfaces, and that our model performs comparably to, and in some respects better than, existing models based on historical data.Additionally, we provide a theoretical reformulation of the tilting framework within the setting of mixture models, offering structural insights into the relation between $ \mathbb{P} $ and $ \mathbb{Q} $.