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 [BRSW04], [Ro08], [BMK09], [BP10], and [LV19]. 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 Lévy processes encountered when modeling market dynamics and constitutes a coherent measure of risk, as introduced in [CDK04]. On the computational side, we provide a simple method to derive the intra-horizon risk inherent to popular Lévy 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 Lévy dynamics are calibrated to S&P 500 index data and an analysis of the resulting intra-horizon risk is presented.

JDOI Variance Reduction Method and the Pricing of American-Style Options

“…, n}, (A.33) are not directly required to hold under general mixed-exponential jump-diffusion dynamics, however these will be naturally satisfied in practice (cf. [LV20], [FMV20]).…”

“…Indeed, it is well-known that mixed-exponential distributions are dense -in the sense of weak convergence -in the class of all distributions, and that they, therefore, offer the possibility to approximate any jump distribution (cf. [BH86], [CK11], [LV20], [FMV20], [FM20]). We provide an extensive discussion of the JDOI method for both American standard and barrier options in this model and extensively test its least-squares Monte Carlo (LSMC) version, i.e.…”

A refined measure of conditional maximum drawdown

“…From the perspective of portfolio management, it is the lack of convexity the main disadvantage for optimization problems. As a remedy for the lack of coherence, one might use instead the static AV@R as suggested by Farkas et al (2021). The drawdown of X up to time t ∈ [0, T] is D X t ∶= M X t − X t , i.e.…”

“…The recent market risk framework of the Basel Accords admits the AV@R as an adequate risk measure because of its ability in providing sufficiently conservative risk estimates. As byproduct, Farkas et al (2021) studied intra-horizon risk based on running minimum and AV@R to better account for all extremes in a trading horizon since V@R does not account for the whole magnitude of potential losses within it. Thus the literate on quantitative risk assessment devotes increasing interest in developing indices defined in term of the whole path of returns rather than point-in-time at the end of horizon.…”