Mathematical Finance最新文献

In this paper, we investigate the robust models for $�\Lambda$-quantiles with partial information regarding the loss distribution, where $�\Lambda$-quantiles extend the classical quantiles by replacing the fixed probability level with a probability/loss function $�\Lambda$. We find that, under some assumptions, the robust $�\Lambda$-quantiles equal the $�\Lambda$-quantiles of the extremal distributions. This finding allows us to obtain the robust $�\Lambda$-quantiles by applying the results of robust quantiles in the literature. Our results are applied to uncertainty sets characterized by the following three different constraints, respectively: moment constraints, probability distance constraints via the Wasserstein metric, and marginal constraints in risk aggregation. We obtain some explicit expressions for robust $�\Lambda$-quantiles by deriving the extremal distributions for each uncertainty set. These results are applied to optimal portfolio selection under model uncertainty.

FX fixings are an indispensable and widely used reference rate in a market that trades continuously without an official close. Yet, a dealer's handling of fix transactions is a much debated topic. Especially when exposure to the fix is large relative to available market liquidity and hedging may extend to the pre-fix window, an inherent conflict of interest can arise between dealer and client. In this paper we use a model with permanent and transient market impact to characterize a dealer's optimal strategy to hedge fixing exposure. We show that smaller fix exposures are fully hedged over the calculation window, but that larger fix transactions are optimally hedged over a longer horizon that includes the pre-fix window. A client's all-in transaction costs can be lowered by pre-fix hedging provided that transient impact decays sufficiently quickly and dominates permanent impact.

We consider a class of optimal liquidation problems where the agent's transactions create transient price impact driven by a Volterra-type propagator along with temporary price impact. We formulate these problems as maximization of a revenue-risk functionals, where the agent also exploits available information on a progressively measurable price predicting signal. By using an infinite dimensional stochastic control approach, we characterize the value function in terms of a solution to a free-boundary $�L^2$-valued backward stochastic differential equation and an operator-valued Riccati equation. We then derive analytic solutions to these equations, which yields an explicit expression for the optimal trading strategy. We show that our formulas can be implemented in a straightforward and efficient way for a large class of price impact kernels with possible singularities such as the power-law kernel.

We determine the optimal affine contract for a client who delegates their order execution to a dealer. Existence and uniqueness are established for general linear price impact dynamics of the dealer's trades. Explicit solutions are available for the model of Obizhaeva and Wang, for example, and a simple gradient descent algorithm is applicable in general. The optimal contract allows the client to almost achieve the first-best performance without any agency conflicts for many reasonable parameter values. Common trading arrangements arise as limiting cases. In particular, optimal contracts for many reasonable model parameters resemble the “fixing contract” common in FX markets, in that they only incorporate market prices briefly before the conclusion of the trade. Price manipulation by the dealer is avoided by only putting a sufficiently small weight on these prices, and complementing this part of the contract with a sufficiently large fixed fee.

This paper develops a new dual approach to compute the hedging portfolio of a Bermudan option and its initial value. It gives a “purely dual” algorithm following the spirit of Rogers in the sense that it only relies on the dual pricing formula. The key is to rewrite the dual formula as an excess reward representation and to combine it with a strict convexification technique. The hedging strategy is then obtained by using a Monte-Carlo method, solving backward a sequence of least square problems. We show convergence results for our algorithm and test it on many different Bermudan options. Beyond giving directly the hedging portfolio, the strength of the algorithm is to assess both the relevance of including financial instruments in the hedging portfolio and the effect of the rebalancing frequency.

This paper is concerned with the problem of capital provision in a large particle system modeled by stochastic differential equations involving hitting times, which arises from considerations of systemic risk in a financial network. Motivated by Tang and Tsai, we focus on the number or proportion of surviving entities that never default to measure the systemic robustness. First we show that the mean-field particle system and its limit McKean–Vlasov equation are both well-posed by virtue of the notion of minimal solutions. We then establish a connection between the proportion of surviving entities in the large particle system and the probability of default in the McKean–Vlasov equation as the size of the interacting particle system $�N$ tends to infinity. Finally, we study the asymptotic efficiency of capital provision for different drift $�\beta$, which is linked to the economy regime: The expected number of surviving entities has a uniform upper bound if ; it is of order $�\sqrt{�N�}$ if $��\beta �=�0�$; and it is of order $�N$ if $��\beta �>�0�$, where the effect of capital provision is negligible.