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Register a new userModel-free Superhedging Duality
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Marco Maggis Doctor
Publication date
2015
fields
Financial
and research's language is
English
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In a model free discrete time financial market, we prove the superhedging duality theorem, where trading is allowed with dynamic and semi-static strategies. We also show that the initial cost of the cheapest portfolio that dominates a contingent claim on every possible path $omega in Omega$, might be strictly greater than the upper bound of the no-arbitrage prices. We therefore characterize the subset of trajectories on which this duality gap disappears and prove that it is an analytic set.
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In a discrete-time financial market, a generalized duality is established for model-free superhedging, given marginal distributions of the underlying asset. Contrary to prior studies, we do not require contingent claims to be upper semicontinuous, allowing for upper semi-analytic ones. The generalized duality stipulates an extended version of risk-neutral pricing. To compute the model-free superhedging price, one needs to find the supremum of expected values of a contingent claim, evaluated not directly under martingale (risk-neutral) measures, but along sequences of measures that converge, in an appropriate sense, to martingale ones. To derive the main result, we first establish a portfolio-constrained duality for upper semi-analytic contingent claims, relying on Choquets capacitability theorem. As we gradually fade out the portfolio constraint, the generalized duality emerges through delicate probabilistic estimations.
We prove the superhedging duality for a discrete-time financial market with proportional transaction costs under model uncertainty. Frictions are modeled through solvency cones as in the original model of [Kabanov, Y., Hedging and liquidation under transaction costs in currency markets. Fin. Stoch., 3(2):237-248, 1999] adapted to the quasi-sure setup of [Bouchard, B. and Nutz, M., Arbitrage and duality in nondominated discrete-time models. Ann. Appl. Probab., 25(2):823-859, 2015]. Our approach allows to remove the restrictive assumption of No Arbitrage of the Second Kind considered in [Bouchard, B., Deng, S. and Tan, X., Super-replication with proportional transaction cost under model uncertainty, Math. Fin., 29(3):837-860, 2019] and to show the duality under the more natural condition of No Strict Arbitrage. In addition, we extend the results to models with portfolio constraints.
We study the Fundamental Theorem of Asset Pricing for a general financial market under Knightian Uncertainty. We adopt a functional analytic approach which require neither specific assumptions on the class of priors $mathcal{P}$ nor on the structure of the state space. Several aspects of modeling under Knightian Uncertainty are considered and analyzed. We show the need for a suitable adaptation of the notion of No Free Lunch with Vanishing Risk and discuss its relation to the choice of an appropriate filtration. In an abstract setup, we show that absence of arbitrage is equivalent to the existence of emph{approximate} martingale measures sharing the same polar set of $mathcal{P}$. We then specialize the results to a discrete-time framework in order to obtain true martingale measures.
We propose a hybrid method for generating arbitrage-free implied volatility (IV) surfaces consistent with historical data by combining model-free Variational Autoencoders (VAEs) with continuous time stochastic differential equation (SDE) driven models. We focus on two classes of SDE models: regime switching models and Levy additive processes. By projecting historical surfaces onto the space of SDE model parameters, we obtain a distribution on the parameter subspace faithful to the data on which we then train a VAE. Arbitrage-free IV surfaces are then generated by sampling from the posterior distribution on the latent space, decoding to obtain SDE model parameters, and finally mapping those parameters to IV surfaces.
This paper investigates Pareto optimal (PO, for short) insurance contracts in a behavioral finance framework, in which the insured evaluates contracts by the rank-dependent utility (RDU) theory and the insurer by the expected value premium principle. The incentive compatibility constraint is taken into account, so the contracts are free of moral hazard. The problem is initially formulated as a non-concave maximization problem involving Choquet expectation, then turned into a quantile optimization problem and tackled by calculus of variations method. The optimal contracts are expressed by a double-obstacle ordinary differential equation for a semi-linear second-order elliptic operator with nonlocal boundary conditions. We provide a simple numerical scheme as well as a numerical example to calculate the optimal contracts. Let $theta$ and $m_0$ denote the relative safety loading and the mass of the potential loss at 0. We find that every moral-hazard-free contract is optimal for infinitely many RDU insureds if $0<theta<frac{m_0}{1-m_0}$; by contrast, some contracts such as the full coverage contract are never optimal for any RDU insured if $theta>frac{m_0}{1-m_0}$. We also derive all the PO contracts when either the compensations or the retentions loss monotonicity.
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