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In this paper we introduce a completely continuous and time-variate model of the evolution of market limit orders based on the existence, uniqueness, and regularity of the solutions to a type of stochastic partial differential equations…
We develop two alternate approaches to arbitrage-free, market-complete, option pricing. The first approach requires no riskless asset. We develop the general framework for this approach and illustrate it with two specific examples. The…
The following optimization problem is considered. For a linear vector Ito equation. it is required to find an optimal deterministic control vector which minimizes a quadratic the functional. A necessary and sufficient condition for the…
We consider a general path-dependent version of the hedging problem with price impact of Bouchard et al. (2019), in which a dual formulation for the super-hedging price is obtained by means of PDE arguments, in a Markovian setting and under…
The recent crash demonstrated (once again) that the description of the financial market by present financial mathematics cannot be considered as totally satisfactory. We remind that nowadays financial mathematics is heavily based on the use…
We analyze multiline pricing and capital allocation in equilibrium no-arbitrage markets. Existing theories often assume a perfect complete market, but when pricing is linear, there is no diversification benefit from risk pooling and…
We propose a method for extending a given asset pricing formula to account for two additional sources of risk: the risk associated with future changes in market--calibrated parameters and the remaining risk associated with idiosyncratic…
We study risk-sharing economies where heterogenous agents trade subject to quadratic transaction costs. The corresponding equilibrium asset prices and trading strategies are characterised by a system of nonlinear, fully-coupled…
With model uncertainty characterized by a convex, possibly non-dominated set of probability measures, the agent minimizes the cost of hedging a path dependent contingent claim with given expected success ratio, in a discrete-time,…
We provide a model-free pricing-hedging duality in continuous time. For a frictionless market consisting of $d$ risky assets with continuous price trajectories, we show that the purely analytic problem of finding the minimal superhedging…
We generalize classical results on the existence of optimal portfolios in discrete time frictionless market models to models with capital gains taxes. We consider the realistic but mathematically challenging rule that losses do not trigger…
Empirical evidence suggests that even the most competitive markets are not strictly efficient. Price histories can be used to predict near future returns with a probability better than random chance. Many markets can be considered as {\it…
We show, by studying in detail the market prices of options on liquid markets, that the market has empirically corrected the simple, but inadequate Black-Scholes formula to account for two important statistical features of asset…
This paper studies an equity market of stochastic dimension, where the number of assets fluctuates over time. In such a market, we develop the fundamental theorem of asset pricing, which provides the equivalence of the following statements:…
We consider a financial market in which two securities are traded: a stock and an index. Their prices are assumed to satisfy the Black-Scholes model. Besides assuming that the index is a tradable security, we also assume that it is…
We study specific nonlinear transformations of the Black-Scholes implied volatility to show remarkable properties of the volatility surface. Model-free bounds on the implied volatility skew are given. Pricing formulas for the European…
We consider optimal investment problems for a diffusion market model with non-observable random drifts that evolve as an Ito's process. Admissible strategies do not use direct observations of the market parameters, but rather use historical…
We consider the problem of optimal hedging in an incomplete market with an established pricing kernel. In such a market, prices are uniquely determined, but perfect hedges are usually not available. We work in the rather general setting of…
We contrast Arbitrage Pricing Theory (APT), the theoretical basis for the development of financial instruments, with a dynamical picture of an interacting market, in a simple setting. The proliferation of financial instruments apparently…
This paper does not suppose a priori that the evolution of the price of a financial asset is a semimartingale. Since possible strategies of investors are self-financing, previous prices are forced to be finite quadratic variation processes.…