Quantitative Finance

The paper proposes a new approach to model risk measurement based on the Wasserstein distance between two probability measures. It formulates the theoretical motivation resulting from the interpretation of fictitious adversary of robust risk management. The proposed approach accounts for equivalent and non-equivalent probability measures and incorporates the economic reality of the fictitious adversary. It provides practically feasible results that overcome the restriction of considering only models implying probability measures equivalent to the reference model. The Wasserstein approach suits for various types of model risk problems, ranging from the single-asset hedging risk problem to the multi-asset allocation problem. The robust capital market line, accounting for the correlation risk, is not achievable with other non-parametric approaches.

The issue of model risk in default modeling has been known since inception of the Academic literature in the field. However, a rigorous treatment requires a description of all the possible models, and a measure of the distance between a single model and the alternatives, consistent with the applications. This is the purpose of the current paper. We first analytically describe all possible joint models for default, in the class of finite sequences of exchangeable Bernoulli random variables. We then measure how the model risk of choosing or calibrating one of them affects the portfolio loss from default, using two popular and economically sensible metrics, Value-at-Risk (VaR) and Expected Shortfall (ES).

In this paper we extend discrete time semi-static trading strategies by also allowing for dynamic trading in a finite amount of options, and we study the consequences for the model-independent super-replication prices of exotic derivatives. These include duality results as well as a precise characterization of pricing rules for the dynamically tradable options triggering an improvement of the price bounds for exotic derivatives in comparison with the conventional price bounds obtained through the martingale optimal transport approach.

In this paper, we provide a model-independent extension of the paradigm of dynamic hedging of derivative claims. We relate model-independent replication strategies to local martingales having a closed form which we can characterise via solutions of coupled PDEs. We provide a general framework and then apply it to a market with no traded claims, a market with an underlying asset and a convex claim and a market with an underlying asset and a set of co-maturing call options. The results encompass known examples of model-independent identities and provide a methodology for deriving new identities.

We introduce a model for the evolution of emissions and the price of emissions allowances in a carbon market such as the European Union Emissions Trading System (EU ETS). The model accounts for multiple trading periods, or phases, with multiple times at which compliance can occur. At the end of each trading period, the participating firms must surrender allowances for their emissions made during that period, and additional allowances can be used for compliance in the following periods. We show that the multi-period allowance pricing problem is well-posed for various mechanisms (such as banking, borrowing and withdrawal of allowances) linking the trading periods. The results are based on the analysis of a forward-backward stochastic differential equation with coupled forward and backward components, a discontinuous terminal condition and a forward component that is degenerate. We also introduce an infinite period model, for a carbon market with a sequence of compliance times and with no end date. We show that, under appropriate conditions, the value function for the multi-period pricing problem converges, as the number of periods increases, to a value function for this infinite period model, and that such functions are unique.

We show that the moment explosion time in the rough Heston model [El Euch, Rosenbaum 2016, arXiv:1609.02108] is finite if and only if it is finite for the classical Heston model. Upper and lower bounds for the explosion time are established, as well as an algorithm to compute the explosion time (under some restrictions). We show that the critical moments are finite for all maturities. For negative correlation, we apply our algorithm for the moment explosion time to compute the lower critical moment.

In this paper, we study general monetary risk measures (without any convexity or weak convexity). A monetary (respectively, positively homogeneous) risk measure can be characterized as the lower envelope of a family of convex (respectively, coherent) risk measures. The proof does not depend on but easily leads to the classical representation theorems for convex and coherent risk measures. When the law-invariance and the SSD (second-order stochastic dominance)-consistency are involved, it is not the convexity (respectively, coherence) but the comonotonic convexity (respectively, comonotonic coherence) of risk measures that can be used for such kind of lower envelope characterizations in a unified form. The representation of a law-invariant risk measure in terms of VaR is provided.

Bernard et al. (2015) studied an insurance contract design problem under rank-dependent utility (RDU) theory. Their results, however, suffer from a moral hazard problem, namely, providing incentives for the insured to falsely report the actual loss. Xu et al. (2019) investigated the same problem, but took the {incentive compatibility} constraint into consideration to avoid that moral hazard. Mathematically speaking, the model reduces to a quantile optimisation problem with a compatibility constraint. They solved the problem by imposing assumptions on the loss and the probability weighting function. This paper solves the problem completely by a new quantile optimisation approach under general setting. The optimal solution is expressed by the solution of an obstacle problem for a semilinear second-order elliptic operator with mixed boundary conditions. Surprisingly, it is shown that every reasonable moral-hazard-free contract is optimal for infinitely many RDU maximisers with different utility functions and probability weighting functions.

This paper studies optimal consumption, investment, and healthcare spending under Epstein-Zin preferences. Given consumption and healthcare spending plans, Epstein-Zin utilities are defined over an agent's random lifetime, partially controllable by the agent as healthcare reduces mortality growth. To the best of our knowledge, this is the first time Epstein-Zin utilities are formulated on a controllable random horizon, via an infinite-horizon backward stochastic differential equation with superlinear growth. A new comparison result is established for the uniqueness of associated utility value processes. In a Black-Scholes market, the stochastic control problem is solved through the related Hamilton-Jacobi-Bellman (HJB) equation. The verification argument features a delicate containment of the growth of the controlled morality process, which is unique to our framework, relying on a combination of probabilistic arguments and analysis of the HJB equation. In contrast to prior work under time-separable utilities, Epstein-Zin preferences largely facilitate calibration. In four countries we examined, the model-generated mortality closely approximates actual mortality data; moreover, the calibrated efficacy of healthcare is in broad agreement with empirical studies on healthcare across countries.

This paper addresses the risk-minimization problem, with and without mortality securitization, à la Föllmer-Sondermann for a large class of equity-linked mortality contracts when no model for the death time is specified. This framework includes the situation where the correlation between the market model and the time of death is arbitrary general, and hence leads to the case of a market model where there are two levels of information. The public information which is generated by the financial assets, and a larger flow of information that contains additional knowledge about a death time of an insured. By enlarging the filtration, the death uncertainty and its entailed risk are fully considered without any mathematical restriction. Our key tool lies in our optional martingale representation that states that any martingale in the large filtration stopped at the death time can be decomposed into precise orthogonal local martingales. This allows us to derive the dynamics of the value processes of the mortality/longevity securities used for the securitization, and to decompose any mortality/longevity liability into the sum of orthogonal risks by means of a risk basis. The first main contribution of this paper resides in quantifying, as explicit as possible, the effect of mortality uncertainty on the risk-minimizing strategy by determining the optimal strategy in the enlarged filtration in terms of strategies in the smaller filtration. Our second main contribution consists of finding risk-minimizing strategies with insurance securitization by investing in stocks and one (or more) mortality/longevity derivatives such as longevity bonds. This generalizes the existing literature on risk-minimization using mortality securitization in many directions.