Intra-Horizon Expected Shortfall and Risk Structure in Models with Jumps

Abstract: The present article deals with intra-horizon risk in models with jumps. Our general understanding of intra-horizon risk is along the lines of the approach taken in Boudoukh, Richardson, Stanton and Whitelaw (2004), Rossello (2008), Bhattacharyya, Misra and Kodase (2009), Bakshi and Panayotov (2010), and Leippold and Vasiljevi\'c (2019). In particular, we believe that quantifying market risk by strictly relying on point-in-time measures cannot be deemed a satisfactory approach in general. Instead, we argue that complementing this approach by studying measures of risk that capture the magnitude of losses potentially incurred at any time of a trading horizon is necessary when dealing with (m)any financial position(s). To address this issue, we propose an intra-horizon analogue of the expected shortfall for general profit and loss processes and discuss its key properties. Our intra-horizon expected shortfall is well-defined for (m)any popular class(es) of L\'evy processes encountered when modeling market dynamics and constitutes a coherent measure of risk, as introduced in Cheridito, Delbaen and Kupper (2004). On the computational side, we provide a simple method to derive the intra-horizon risk inherent to popular L\'evy dynamics. Our general technique relies on results for maturity-randomized first-passage probabilities and allows for a derivation of diffusion and single jump risk contributions. These theoretical results are complemented with an empirical analysis, where popular L\'evy dynamics are calibrated to S&P 500 index data and an analysis of the resulting intra-horizon risk is presented.