No Free Lunch With Vanishing Risk Research Articles

Can You Hear the Shape of a Market? Geometric Arbitrage and Spectral Theory

Utilizing gauge symmetries, the Geometric Arbitrage Theory reformulates any asset model, allowing for arbitrage by means of a stochastic principal fibre bundle with a connection whose curvature measures the “instantaneous arbitrage capability”. The cash flow bundle is the associated vector bundle. The zero eigenspace of its connection Laplacian parameterizes all risk-neutral measures equivalent to the statistical one. A market satisfies the No-Free-Lunch-with-Vanishing-Risk (NFLVR) condition if and only if 0 is in the discrete spectrum of the Laplacian. The Jarrow–Protter–Shimbo theory of asset bubbles and their classification and decomposition extend to markets not satisfying the NFLVR. Euler’s characteristic of the asset nominal space and non-vanishing of the homology group of the cash flow bundle are both topological obstructions to NFLVR.

The Black–Scholes equation in the presence of arbitrage

We apply Geometric Arbitrage Theory to obtain results in Mathematical Finance, which do not need stochastic differential geometry in their formulation. First, for generic market dynamics given by a subclass of multidimensional Itô processes we specify and prove the equivalence between No-Free-Lunch-with-Vanishing-Risk (NFLVR) and expected utility maximization. As a by-product, we provide a geometric characterization of the No-Unbounded-Profit-with-Bounded-Risk (NUPBR) condition given by the zero curvature (ZC) condition for this subclass of Itô processes. Finally, we extend the Black–Scholes partial differential equation to markets allowing arbitrage.

Informational Efficiency with Trading Constraints: A Characterization

Given a market with a price process S populated by heterogeneous traders with differential information, beliefs, and trading constraints, let the smallest information set containing all of the traders’ information be denoted F. This market is defined to be informationally efficient (Fama [2]) with respect to an information set G ⊆ F if the price process S arises as an outcome of a competitive market equilibrium. We provide a characterization of such an informationally efficient market. Roughly stated, a market is informationally efficient with respect to G if and only if there exists an equivalent probability measure that makes the price process normalized by the value of a money market account a martingale with respect to G if and only if the market satisfies no free lunch with vanishing risk (NFLVR) and no dominance (ND) with respect G. This characterization subsumes the economies and characterizations contained in the more restrictive market settings of Jarrow and Larsson [7, 8].

Some no-arbitrage rules under short-sales constraints, and applications to converging asset prices

Under short-sales prohibitions, no free lunch with vanishing risk (NFLVRS) is known to be equivalent to the existence of an equivalent supermartingale measure for the price process (Pulido in Ann. Appl. Probab. 24:54–75, 2014). We give a necessary condition for the drift of a price process to satisfy NFLVRS. For two given price processes, we introduce the concept of fundamental supermartingale measure, and when a certain condition necessary for the construction of this fundamental supermartingale measure is not fulfilled, we provide the corresponding arbitrage portfolios. The motivation of our study lies in understanding the particular case of converging prices, i.e., two prices that coincide at a bounded random time.

Some No-Arbitrage Rules under Short-Sales Constraints and Applications to Converging Asset Prices

Under short sales prohibitions, no free lunch with vanishing risk (NFLVRS) is known to be equivalent to the existence of an equivalent supermartingale measure for the price processes (Pulido, 2014).We give a necessary condition for the drift of a price process to satsify (NFLVRS). For two given price processes, we introduce the concept of fundamental supermartingale measure, and when a certain condition necessary to the construction of this fundamental supermartingale measure is not fulfilled, we provide the corresponding arbitrage portfolios. The motivation of our study lies in understanding the particular case of converging prices, i.e., two prices that coincide at a bounded random time.

Change of measure up to a random time: Details

This paper extends results of Mortimer and Williams (1991) about changes of probability measure up to a random time under the assumptions that all martingales are continuous and that the random time avoids stopping times. We consider locally absolutely continuous measure changes up to a random time, changes of probability measure up to and after an honest time, and changes of probability measure up to a pseudo-stopping time. Moreover, we apply our results to construct a change of probability measure that is equivalent to the enlargement formula and to build, for a certain class of pseudo-stopping times, a class of measure changes that preserve the pseudo-stopping time property. Furthermore, we study for a price process modeled by a continuous semimartingale the stability of the No Free Lunch with Vanishing Risk (NFLVR) property up to a random time, that avoids stopping times, in the progressively enlarged filtration and provide sufficient conditions for this stability in terms of the Azéma supermartingale.

WEAK AND STRONG NO-ARBITRAGE CONDITIONS FOR CONTINUOUS FINANCIAL MARKETS

We propose a unified analysis of a whole spectrum of no-arbitrage conditions for financial market models based on continuous semimartingales. In particular, we focus on no-arbitrage conditions weaker than the classical notions of No Arbitrage opportunity (NA) and No Free Lunch with Vanishing Risk (NFLVR). We provide a complete characterization of the considered no-arbitrage conditions, linking their validity to the characteristics of the discounted asset price process and to the existence and the properties of (weak) martingale deflators, and review classical as well as recent results.

Locally Risk-Free Asset Implied by a Finite Set of Assets

This document is the translation of the thesis "Titolo localmente non rishioso implicito per un insieme finito di titoli" which I presented at the university of Padua, Italy, in November 1999, for the award of the degree in Pure and Applied Mathematics (Laurea in Matematica Pura ed Applicata). In this work, I provide an extension of the Fundamental Theorem of Asset Pricing (FTAP), which establishes the equivalence between the No Free Lunch with Vanishing Risk (NFLVR) condition and the Existence of an equivalent Separating Measure (ESM). I extend the theorem to the case where some of the assets are subject to short-selling constraints, by showing that the equivalence between NFLVR and ESM continues to hold, and that the assets subject to short-selling constraints turn out to be supermartingales under the equivalent separating measures. A key result allowing this extension, and which is also of independent interest, is a particular extension of Memin's theorem, which establishes the closure of the set of stochastic integrals with respect to a given semimartingale, within the Frechet space of real semimartingales starting from zero. I extend Memin's theorem to the case where the integrands are subject to certain geometric constraints. Finally, I use this extension of the FTAP to show that the NFLVR condition, together with the existence of cash (which is seen as a particular asset subject to short-selling constraints), implies that for every separating measure there exist a unique locally risk-free asset (or "implied bank account") which can be added to the initial set of assets in such a way that the final set also satisfies the NFLVR condition (and in this sense the locally risk-free asset is "implied" by the initial set).

Negative call prices

We show that the existence of an equivalent local martingale measure for asset prices does not prevent negative prices for European calls written on positive stock prices. In particular, we illustrate that many standard no-arbitrage arguments implicitly rely on conditions stronger than the No Free Lunch With Vanishing Risk (NFLVR) assumption. The discrepancy between replicating prices and market prices for a contingent claim may be observed in a model satisfying NFLVR since certain trading strategies of buying one portfolio and selling another one are often excluded by standard admissibility constraints.

Pricing and Modeling Credit Derivatives

The market involving credit derivatives has become increasingly popular and extremely liquid in the most recent years. The pricing of such instruments offers a myriad of new challenges to the research community as the dimension of credit risk should be explicitly taken into account by a quantitative model. In this paper, we describe a doubly stochastic model with the purpose of pricing and hedging derivatives on securities sub ject to default risk. The default event is modeled by the first jump of a counting process Nt , doubly stochastic with respect to the Brownian filtration which drives the uncertainty of the level of the underlying state process conditional on no-default event. By assuming a condition slightly stronger than no arbitrage, i.e., that there is no free lunch with vanishing risk (NFLVR) from Delbaen and Scharchermayer (1994), we provide all the possible equivalent martingale measures under this setting. In order to illustrate the method, two simple examples are presented: the pricing of defaultable stocks, and a framework to price multi-name credit derivatives such as basket defaults.

SOME REMARKS ON ARBITRAGE AND PREFERENCES IN SECURITIES MARKET MODELS

We introduce the notion of a market-free-lunch that depends on the preferences of all agents participating in the market. In semimartingale models of securities markets, we characterize no arbitrage (NA) and no-free-lunch-with-vanishing-risk (NFLVR) in terms of the market-free-lunch and show that the difference between NA and NFLVR consists in the selection of the class of monotone, respectively monotone and continuous, utility functions that determines the absence of the market-free-lunch. We also provide a direct proof of the equivalence between the absence of a market-free-lunch, with respect to monotone concave preferences, and the existence of an equivalent (local/sigma) martingale measure.