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Register a new userPointwise Arbitrage Pricing Theory in Discrete Time
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Marco Maggis Doctor
Publication date
2016
fields
Financial
and research's language is
English
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We develop a robust framework for pricing and hedging of derivative securities in discrete-time financial markets. We consider markets with both dynamically and statically traded assets and make minimal measurability assumptions. We obtain an abstract (pointwise) Fundamental Theorem of Asset Pricing and Pricing--Hedging Duality. Our results are general and in particular include so-called model independent results of Acciao et al. (2016), Burzoni et al. (2016) as well as seminal results of Dalang et al. (1990) in a classical probabilistic approach. Our analysis is scenario--based: a model specification is equivalent to a choice of scenarios to be considered. The choice can vary between all scenarios and the set of scenarios charged by a given probability measure. In this way, our framework interpolates between a model with universally acceptable broad assumptions and a model based on a specific probabilistic view of future asset dynamics.
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In a model independent discrete time financial market, we discuss the richness of the family of martingale measures in relation to different notions of Arbitrage, generated by a class $mathcal{S}$ of significant sets, which we call Arbitrage de la classe $mathcal{S}$. The choice of $mathcal{S}$ reflects into the intrinsic properties of the class of polar sets of martingale measures. In particular: for S=${Omega}$ absence of Model Independent Arbitrage is equivalent to the existence of a martingale measure; for $mathcal{S}$ being the open sets, absence of Open Arbitrage is equivalent to the existence of full support martingale measures. These results are obtained by adopting a technical filtration enlargement and by constructing a universal aggregator of all arbitrage opportunities. We further introduce the notion of market feasibility and provide its characterization via arbitrage conditions. We conclude providing a dual representation of Open Arbitrage in terms of weakly open sets of probability measures, which highlights the robust nature of this concept.
We design three continuous--time models in finite horizon of a commodity price, whose dynamics can be affected by the actions of a representative risk--neutral producer and a representative risk--neutral trader. Depending on the model, the producer can control the drift and/or the volatility of the price whereas the trader can at most affect the volatility. The producer can affect the volatility in two ways: either by randomizing her production rate or, as the trader, using other means such as spreading false information. Moreover, the producer contracts at time zero a fixed position in a European convex derivative with the trader. The trader can be price-taker, as in the first two models, or she can also affect the volatility of the commodity price, as in the third model. We solve all three models semi--explicitly and give closed--form expressions of the derivative price over a small time horizon, preventing arbitrage opportunities to arise. We find that when the trader is price-taker, the producer can always compensate the loss in expected production profit generated by an increase of volatility by a gain in the derivative position by driving the price at maturity to a suitable level. Finally, in case the trader is active, the model takes the form of a nonzero-sum linear-quadratic stochastic differential game and we find that when the production rate is already at its optimal stationary level, there is an amount of derivative position that makes both players better off when entering the game.
We examine Kreps (2019) conjecture that optimal expected utility in the classic Black--Scholes--Merton (BSM) economy is the limit of optimal expected utility for a sequence of discrete-time economies that approach the BSM economy in a natural sense: The $n$th discrete-time economy is generated by a scaled $n$-step random walk, based on an unscaled random variable $zeta$ with mean zero, variance one, and bounded support. We confirm Kreps conjecture if the consumers utility function $U$ has asymptotic elasticity strictly less than one, and we provide a counterexample to the conjecture for a utility function $U$ with asymptotic elasticity equal to 1, for $zeta$ such that $E[zeta^3] > 0.$
Consider a financial market with nonnegative semimartingales which does not need to have a num{e}raire. We are interested in the absence of arbitrage in the sense that no self-financing portfolio gives rise to arbitrage opportunities, where we are allowed to add a savings account to the market. We will prove that in this sense the market is free of arbitrage if and only if there exists an equivalent local martingale deflator which is a multiplicative special semimartingale. In this case, the additional savings account relates to the finite variation part of the multiplicative decomposition of the deflator.
In a multi-dimensional diffusion framework, the price of a financial derivative can be expressed as an iterated conditional expectation, where the inner conditional expectation conditions on the future of an auxiliary process that enters into the dynamics for the spot. Inspired by results from non-linear filtering theory, we show that this inner conditional expectation solves a backward SPDE (a so-called `conditional Feynman-Kac formula), thereby establishing a connection between SPDE and derivative pricing theory. The benefits of this representation are potentially significant and of both theoretical and practical interest. In particular, this representation leads to an alternative class of so-called mixed Monte-Carlo / PDE numerical methods.
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