Discrete approximation of risk-based prices under volatility uncertainty

Paper: arXiv 2411.00713

Authors: Jonas Blessing, Michael Kupper, Alessandro Sgarabottolo

Abstract

We discuss the asymptotic behaviour of risk-based indifference prices of European contingent claims in discrete-time financial markets under volatility uncertainty as the number of intermediate trading periods tends to infinity. The asymptotic risk-based prices form a strongly continuous convex monotone semigroup which is uniquely determined by its infinitesimal generator and therefore only depends on the covariance of the random factors but not on the particular choice of the model. We further compare the risk-based prices with the worst-case prices given by the $G$-expectation and investigate their asymptotic behaviour as the risk aversion of the agent tends to infinity. The theoretical results are illustrated with several examples and numerical simulations showing, in particular, that the risk-based prices lead to a significant reduction of the bid-ask spread compared to the worst-case prices.

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 3.0/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper presents a rigorous theoretical framework using non-linear semigroups, Chernoff approximations, and sublinear expectations to analyze risk-based indifference pricing under volatility uncertainty. While it provides strong mathematical derivations and functional analysis proofs, the empirical section relies primarily on toy numerical simulations rather than empirical data or backtesting.

Research Flowchart

  flowchart TD
    A[Research Goal: Analyze asymptotic risk-based prices under volatility uncertainty] --> B[Input: Discrete-time market model with uncertain covariance]
    B --> C[Compute discrete indifference prices using risk-averse utility]
    C --> D[Take limit as trading periods tend to infinity N → ∞]
    D --> E[Outcome 1: Prices converge to a continuous convex monotone semigroup]
    D --> F[Outcome 2: As risk aversion → ∞, prices converge to G-expectation worst-case]
    E --> G[Outcome 3: Numerical simulations show significantly reduced bid-ask spreads]
    F --> G