Final Report Summary - MATHFI BEYOND NFLVR (Mathematical Finance beyond NFLVR: weak no-arbitrage-type conditions, information and credit risk)
The present research project consists in an advanced investigation of several foundational issues of mathematical finance. More specifically, the overall goal is to study non-classical no-arbitrage-type conditions (in a sense which will be made precise in the following) and their relations with the information available to market participants. The recent turmoil experienced by financial markets has dramatically shown the urgency of a critical rethinking of classical models of quantitative finance. In particular, the risk-neutral paradigm, which is one of the fundamental pillars of quantitative finance, may no longer be an indisputable truth or, at least, should be properly re-addressed. Furthermore, the notion of arbitrage strategy is intrinsically linked to the information available to market participants (mathematically represented by some filtration on a probability space) and, hence, it is natural to study the relationships between different information sets and no-arbitrage-type conditions. By its own nature, the present research project takes an interdisciplinary perspective, being at the intersection between stochastic analysis, finance and economic theory.
One of the fundamental results of mathematical finance (fundamental theorem of asset pricing) shows that the existence of a risk-neutral probability measure is equivalent to the No Free Lunch with Vanishing Risk (NFLVR) condition (see Delbaen & Schachermayer (2006) for a detailed account). Now, the previous discussion suggests that, despite its mathematical elegance, the NFLVR condition might not be the most natural and robust no-arbitrage-type restriction to impose on a stochastic model for a financial market. In the present research project, we will adopt this point of view and investigate no-arbitrage-type conditions which are less restrictive than NFLVR.
Besides studying the mathematical properties of no-arbitrage-type conditions going beyond NFLVR, a major goal of the present project consists in examining the relation between no-arbitrage and different information sets, mathematically represented by different filtrations on the same probability space. We investigate the effects of a change of the reference filtration (in particular, an enlargement of the information set available to market participants) on the validity of weak no-arbitrage-type conditions. This has significant implications for the study of financial markets where agents have access to different information sets (e.g. insider trading phenomena) and is of interest in view of credit risk modeling, where multiple filtrations naturally arise.
In the first part of the present project, we have focused our attention on the characterization of no-arbitrage-type conditions weaker than the classical NFLVR condition. In the case of continuous semimartingale financial models, we established a hierarchy of different no-arbitrage-type conditions weaker than NFLVR (in particular, the No Increasing Profit condition, the No Strong Arbitrage condition and the No Arbitrage of the First Kind condition, together with their equivalent formulations; see the publication "Weak and strong no-arbitrage conditions for continuous financial markets"). In particular, we have studied which no-arbitrage-type conditions only ban pathological forms of arbitrage (and, therefore, only correspond to “sanity checks” of a model) and which conditions imply instead a viable financial market, in the sense that portfolio optimization, valuation and hedging problems can be meaningfully solved. These results show that the No Increasing Profit and No Strong Arbitrage conditions are in general too weak for the purposes of financial modeling. On the contrary, relying also on the recent literature, the No Arbitrage of the First Kind condition, while weaker than NFLVR, is sufficient for the solution of most of the key problems of mathematical finance. Moreover, we have provided a complete characterization of a whole spectrum of weak no-arbitrage-type conditions in terms of the semimartingale characteristics of the discounted price process, while this is in general not possible in the case of the NFLVR condition. The study of the no-arbitrage literature also led to a critical analysis of the paper entitled “Arbitrage, approximate arbitrage and the fundamental theorem of asset pricing”, by B. Wong & C.C. Heyde (2010, Stochastics, 82, 189-200), showing that the proof proposed by the authors can only work in the special and well-known case of a complete financial market. This analysis is also completed by an explicit counterexample (see the publication "A note on arbitrage, approximate arbitrage and the fundamental theorem of asset pricing").
Motivated by the study of insider trading phenomena, we then shifted our attention to the stability of different notions of no-arbitrage with respect to an enlargement of the reference filtration. In particular, we have focused on the case where the financial market for the uninformed agents is well-behaved, in the sense that it does not contain arbitrage possibilities (according to one of the notions of arbitrage discussed above), while the private information of the insider trader can be used to generate arbitrage profits. In a first study, in the context of diffusion models, we have focused on the case where the additional information is represented by the observation of a random change point in the dynamics of asset prices. In that context, we have characterized the validity of no-arbitrage (according to the different conditions studied above) and of market completeness with respect to several different information layers (see the publication "Information, no-arbitrage and completeness for asset price models with a random change point"). Then, in a second study, in the context of general semimartingale financial models, we have studied two main cases for the additional information of the insider trader: the case where the insider trader can observe through time the occurrence of an event which is not observable by regular market participants (and is allowed to trade only until the occurrence of that event, which can for instance represent a default time) and the case where the insider trader knows from the beginning some information not available in the public market. In both cases, the main message is that, on the one hand, the insider trader will typically be able to realise arbitrage opportunities (thus violating the NFLVR condition), but weaker no-arbitrage-type conditions will continue to hold. On the other hand, there are very specific situations where the insider trading behaviour can also lead to violations of weak no-arbitrage-type conditions, thus leading to a failure of market equilibrium. In that context, focusing on the No Arbitrage of the First Kind condition (which is weaker than NFLVR), we have provided conditions in order to preserve the validity of such a condition in the financial market where the insider trader operates (see the preprint "Arbitrage of the first kind and filtration enlargements in semimartingale financial models"). The same analysis has been also performed assuming that the additional information of the insider trader is represented by an initial information, not shared with the rest of the market. Related results have been obtained concerning the stability of weak and strong no-arbitrage conditions with respect to absolutely continuous but not equivalent changes of measure (see the publication "No-arbitrage conditions and absolutely continuous changes of measure").
During this project, motivated by the mathematical tools needed for the analysis of the problems mentioned above, the researcher has acquired experience in the general theory of stochastic processes as well as in specific topics of martingale theory which are especially relevant in arbitrage theory. Moreover, the study of insider trading phenomena, has allowed the researcher to become familiar with the theory of enlargement of filtrations.
Besides the research goals originally foreseen in the research proposal, we also have developed a new and general framework for interest rate modeling in the presence of multiple interest rate curves, as observed in reality. This study has been motivated by the recent changes in fixed income markets starting from the last financial crisis. In that context, we have analyzed the no-arbitrage implications and provided a general modeling construction which allows to generate arbitrage-free dynamics for a whole family of liquidly traded interest rate financial products (see the preprint "A general HJM framework for multiple yield curve modeling").
One of the fundamental results of mathematical finance (fundamental theorem of asset pricing) shows that the existence of a risk-neutral probability measure is equivalent to the No Free Lunch with Vanishing Risk (NFLVR) condition (see Delbaen & Schachermayer (2006) for a detailed account). Now, the previous discussion suggests that, despite its mathematical elegance, the NFLVR condition might not be the most natural and robust no-arbitrage-type restriction to impose on a stochastic model for a financial market. In the present research project, we will adopt this point of view and investigate no-arbitrage-type conditions which are less restrictive than NFLVR.
Besides studying the mathematical properties of no-arbitrage-type conditions going beyond NFLVR, a major goal of the present project consists in examining the relation between no-arbitrage and different information sets, mathematically represented by different filtrations on the same probability space. We investigate the effects of a change of the reference filtration (in particular, an enlargement of the information set available to market participants) on the validity of weak no-arbitrage-type conditions. This has significant implications for the study of financial markets where agents have access to different information sets (e.g. insider trading phenomena) and is of interest in view of credit risk modeling, where multiple filtrations naturally arise.
In the first part of the present project, we have focused our attention on the characterization of no-arbitrage-type conditions weaker than the classical NFLVR condition. In the case of continuous semimartingale financial models, we established a hierarchy of different no-arbitrage-type conditions weaker than NFLVR (in particular, the No Increasing Profit condition, the No Strong Arbitrage condition and the No Arbitrage of the First Kind condition, together with their equivalent formulations; see the publication "Weak and strong no-arbitrage conditions for continuous financial markets"). In particular, we have studied which no-arbitrage-type conditions only ban pathological forms of arbitrage (and, therefore, only correspond to “sanity checks” of a model) and which conditions imply instead a viable financial market, in the sense that portfolio optimization, valuation and hedging problems can be meaningfully solved. These results show that the No Increasing Profit and No Strong Arbitrage conditions are in general too weak for the purposes of financial modeling. On the contrary, relying also on the recent literature, the No Arbitrage of the First Kind condition, while weaker than NFLVR, is sufficient for the solution of most of the key problems of mathematical finance. Moreover, we have provided a complete characterization of a whole spectrum of weak no-arbitrage-type conditions in terms of the semimartingale characteristics of the discounted price process, while this is in general not possible in the case of the NFLVR condition. The study of the no-arbitrage literature also led to a critical analysis of the paper entitled “Arbitrage, approximate arbitrage and the fundamental theorem of asset pricing”, by B. Wong & C.C. Heyde (2010, Stochastics, 82, 189-200), showing that the proof proposed by the authors can only work in the special and well-known case of a complete financial market. This analysis is also completed by an explicit counterexample (see the publication "A note on arbitrage, approximate arbitrage and the fundamental theorem of asset pricing").
Motivated by the study of insider trading phenomena, we then shifted our attention to the stability of different notions of no-arbitrage with respect to an enlargement of the reference filtration. In particular, we have focused on the case where the financial market for the uninformed agents is well-behaved, in the sense that it does not contain arbitrage possibilities (according to one of the notions of arbitrage discussed above), while the private information of the insider trader can be used to generate arbitrage profits. In a first study, in the context of diffusion models, we have focused on the case where the additional information is represented by the observation of a random change point in the dynamics of asset prices. In that context, we have characterized the validity of no-arbitrage (according to the different conditions studied above) and of market completeness with respect to several different information layers (see the publication "Information, no-arbitrage and completeness for asset price models with a random change point"). Then, in a second study, in the context of general semimartingale financial models, we have studied two main cases for the additional information of the insider trader: the case where the insider trader can observe through time the occurrence of an event which is not observable by regular market participants (and is allowed to trade only until the occurrence of that event, which can for instance represent a default time) and the case where the insider trader knows from the beginning some information not available in the public market. In both cases, the main message is that, on the one hand, the insider trader will typically be able to realise arbitrage opportunities (thus violating the NFLVR condition), but weaker no-arbitrage-type conditions will continue to hold. On the other hand, there are very specific situations where the insider trading behaviour can also lead to violations of weak no-arbitrage-type conditions, thus leading to a failure of market equilibrium. In that context, focusing on the No Arbitrage of the First Kind condition (which is weaker than NFLVR), we have provided conditions in order to preserve the validity of such a condition in the financial market where the insider trader operates (see the preprint "Arbitrage of the first kind and filtration enlargements in semimartingale financial models"). The same analysis has been also performed assuming that the additional information of the insider trader is represented by an initial information, not shared with the rest of the market. Related results have been obtained concerning the stability of weak and strong no-arbitrage conditions with respect to absolutely continuous but not equivalent changes of measure (see the publication "No-arbitrage conditions and absolutely continuous changes of measure").
During this project, motivated by the mathematical tools needed for the analysis of the problems mentioned above, the researcher has acquired experience in the general theory of stochastic processes as well as in specific topics of martingale theory which are especially relevant in arbitrage theory. Moreover, the study of insider trading phenomena, has allowed the researcher to become familiar with the theory of enlargement of filtrations.
Besides the research goals originally foreseen in the research proposal, we also have developed a new and general framework for interest rate modeling in the presence of multiple interest rate curves, as observed in reality. This study has been motivated by the recent changes in fixed income markets starting from the last financial crisis. In that context, we have analyzed the no-arbitrage implications and provided a general modeling construction which allows to generate arbitrage-free dynamics for a whole family of liquidly traded interest rate financial products (see the preprint "A general HJM framework for multiple yield curve modeling").