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We study the leading term in the small-time asymptotics of at-the-money call option prices when the stock price process follows a general martingale. This is equivalent to studying the rst centered absolute moment of. We show that if has a continuous part, the leading term is of order √ in time and depends only on the initial value of the volatility.(More)
We propose a method to construct the stochastic integral simultaneously under a non-dominated family of probability measures. Path-by-path, and without referring to a probability measure, we construct a sequence of Lebesgue-Stieltjes integrals whose medial limit coincides with the usual stochastic integral under essentially any probability measure such that(More)
J o u r n a l o f P r o b a b i l i t y Electron. Abstract We study martingale inequalities from an analytic point of view and show that a general martingale inequality can be reduced to a pair of deterministic inequalities in a small number of variables. More precisely, the optimal bound in the martingale inequality is determined by a fixed point of a(More)
We consider a nondominated model of a discrete-time financial market where stocks are traded dynamically and options are available for static hedging. In a general measure-theoretic setting, we show that absence of arbitrage in a quasi-sure sense is equivalent to the existence of a suitable family of martingale measures. In the arbitrage-free case, we show(More)
We provide a general construction of time-consistent sublinear expectations on the space of continuous paths. It yields the existence of the conditional-expectation of a Borel-measurable (rather than quasi-continuous) random variable, a generalization of the random-expectation, and an optional sampling theorem that holds without exceptional set. Our results(More)
We consider the economic problem of optimal consumption and investment with power utility. We study the optimal strategy as the relative risk aversion tends to innity or to one. The convergence of the optimal consumption is obtained for general semimartingale models while the convergence of the optimal trading strategy is obtained for continuous models. The(More)
We consider dynamic sublinear expectations (i.e., time-consistent coherent risk measures) whose scenario sets consist of singular measures corresponding to a general form of volatility uncertainty. We derive a càdlàg nonlinear martingale which is also the value process of a superhedging problem. The superhedging strategy is obtained from a representation(More)