Abstracts
Wednesday June 4, 11:30-12:00 | session 4.8 | Trading (Strategies) | room 1+2
We propose the concept of a multi-person game contingent claim, which extends the two-person game option introduced by Kifer (2000). A multi-person game contingent claim involves several holders of tranches, as well as an issuer. Each party is allowed to make a wide array of decisions at any time, not restricted to simply `exercising the option'. Since each party is able to observe the actions of counterparties, we consider the class of \emph{adaptive trading strategies}, which react to the observable actions. Arbitrage is redefined as the opportunity to make a guaranteed profit by holding tranches along with an adaptive trading strategy until a random time when the claim is settled.
We first focus on arbitrage pricing of a particular combined tranche (i.e., a fixed collection of tranches). As a main tool, we extend to the present framework the classic notion of the Snell envelope. It is shown that no arbitrage is possible from holding a fixed combined tranche if and only if its market price lies within explicitly specified bounds, which can be given an interpretation in terms of super-hedging strategies for holders and/or issuer.
In the second step, the interaction between different combined tranches of a given contract is examined in detail. In particular, it is demonstrated that the price functional for tranches must be additive in order to avoid a second type of arbitrage, dubbed the \emph{reselling arbitrage}. This additivity also eliminates any price disagreement between the parties. We specify necessary and sufficient conditions that exclude both types of arbitrage, and we prove that if an optimal equilibrium exist for the associated multi-player stochastic game under the martingale measure for the underlying market model, then the individual tranches of a contract have unique arbitrage prices given by the value of the game.
Finally, we show that if the payoffs satisfy a sub-zero-sum condition, then every combined tranche has a unique arbitrage price and the pricing functional is additive. As an important example, we discuss the affine game contingent claims, that is, financial derivatives associated with the discrete-time affine stopping game studied in Guo and Rutkowski (2013) (for the continuous-time case, see Nie and Rutkowski (2013)).
Wednesday June 4, 12:00-12:30 | session 4.8 | Trading (Strategies) | room 1+2
We study a financial market in which some assets, with prices adapted w.r.t. a reference filtration ${\mathbb F}$ are traded. In this presentation, we shall restrict our attention to the case where $\ff$ is generated by a Brownian motion. One then assumes that an agent has some extra information, and may use strategies adapted to a larger filtration ${\mathbb G}$. This extra information is modeled by the knowledge of some random time $\tau$, when this time occurs. We restrict our study to progressive enlargement setting, and we pay a particular attention to honest times. Our goal is to detect if the knowledge of $\tau$ allows for some arbitrage (classical arbitrages and arbitrages of the first kind), i.e., if using ${\mathbb G}$-adapted strategies, one can make profit.
The results presented here are based on two joint papers with Aksamit, Choulli, Deng, in which the authors study No Unbounded Profit with Bounded Risk (NUPBR) in a general filtration ${\mathbb F}$ and the case of classical arbitrages in the case of honest times, density framework and immersion setting. We shall also study the information drift and the growth optimal portfolio resulting from that model (forthcoming work with T. Schmidt).
Wednesday June 4, 12:30-13:00 | session 4.8 | Trading (Strategies) | room 1+2
We construct and study market models admitting optimal arbitrage. We say that a model admits optimal arbitrage if it is possible, in a zero-interest rate setting, starting with an initial wealth of 1 and using only positive portfolios, to superreplicate a constant $c>1$. The optimal arbitrage strategy is the strategy for which this constant has the highest possible value. Our definition of optimal arbitrage is similar to the one given in [2], where optimal relative arbitrage with respect to the market portfolio is studied. In this work we present a systematic method to construct market models where the optimal arbitrage strategy exists and is known explicitly. The arbitrage model P is constructed from a reference arbitrage-free model Q by a non-equivalent change of measure. This procedure is not new and goes back to the construction of the Bessel process by [1]. However, we extend it in two directions. Firstly, from the theoretical point of view, we provide a characterization of the superhedging price of a claim under P in terms of the superhedging price of a related claim under Q. This allows us to characterize the optimal arbitrage profit under P in terms of the superhedging price under Q. Secondly, from the economic point of view, we provide an economic intuition for the new arbitrage model as a model implementing the view of the economic agent concerning the impossibility of certain market events. In other words, if an economic agent considers that a certain event (such as a sovereign default) is impossible, but it is actually priced in the market, our method can be used to construct a new model incorporating this arbitrage opportunity, and to compute the associated optimal arbitrage strategy. We then combine these two ideas to develop several new classes of models with optimal arbitrage, allowing for a clear economic interpretation. We also discuss the issue of robustness of these arbitrages to small transaction costs/small observation errors and show that some of our examples are not fragile in the sense of [3].
[1] Delbaen, F. and Schachermayer, W. (1995). Arbitrage possibilities in Bessel processes and their relations to local martingales. Probab. Theory Related Fields, 102(3):357-366
[2] Fernholz, D. and Karatzas, I. (2010). On optimal arbitrage. Ann. Appl. Probab., 20(4):1179-1204.
[3] Guasoni, P. and Rasonyi, M. (2012). Fragility of arbitrage and bubbles in diffusion models. Technical report.