Mathematical Finance最新文献

Distortion risk measures (DRM) are risk measures that are law invariant and comonotonic additive. The present paper is an extensive inquiry into this class of risk measures in light of new ideas such as qualitative robustness, prudence and no reward for concentration, and tail relevance. Results include several characterizations of prudent DRMs, a novel representation of coherent DRMs as well as an axiomatization of the Expected Shortfall alternative to the one recently provided by Wang and Zitikis. By linking the two axiomatizations, the paper provides a new perspective on the idea of no reward for concentration. The paper also contains results of independent interest such as the lower semicontinuity with respect to convergence in distribution of the Haezendonck–Goovaerts risk measures, the extension of non-necessarily convex risk measures as well as the structure of the core of a general submodular distortion.

We investigate the stability of the Epstein–Zin problem with respect to small distortions in the dynamics of the traded securities. We work in incomplete market model settings, where our parametrization of perturbations allows for joint distortions in returns and volatility of the risky assets and the interest rate. Considering empirically the most relevant specifications of risk aversion and elasticity of intertemporal substitution, we provide a condition that guarantees the convexity of the domain of the underlying problem and results in the existence and uniqueness of a solution to it. Then, we prove the convergence of the optimal consumption streams, the associated wealth processes, the indirect utility processes, and the value functions in the limit when the model perturbations vanish.

When prices of assets traded in a financial market are determined by nonlinear pricing rules, different parities between call and put options have been considered. We show that, under monotonicity, parities between call and put options and discount certificates characterize ambiguity-sensitive (Choquet and/or Šipoš) pricing rules, that is, pricing rules that can be represented via discounted expectations with respect to non-additive probability measures. We analyze how nonadditivity relates to arbitrage opportunities and we give necessary and sufficient conditions for Choquet and Šipoš pricing rules to be arbitrage free. Finally, we identify violations of the Call-Put Parity with the presence of bid–ask spreads.

We study an extension of the Heston stochastic volatility model that incorporates rough volatility and jump clustering phenomena. In our model, named the rough Hawkes Heston stochastic volatility model, the spot variance is a rough Hawkes-type process proportional to the intensity process of the jump component appearing in the dynamics of the spot variance itself and the log returns. The model belongs to the class of affine Volterra models. In particular, the Fourier-Laplace transform of the log returns and the square of the volatility index can be computed explicitly in terms of solutions of deterministic Riccati-Volterra equations, which can be efficiently approximated using a multi-factor approximation technique. We calibrate a parsimonious specification of our model characterized by a power kernel and an exponential law for the jumps. We show that our parsimonious setup is able to simultaneously capture, with a high precision, the behavior of the implied volatility smile for both S&P 500 and VIX options. In particular, we observe that in our setting the usual shift in the implied volatility of VIX options is explained by a very low value of the power in the kernel. Our findings demonstrate the relevance, under an affine framework, of rough volatility and self-exciting jumps in order to capture the joint evolution of the S&P 500 and VIX.