Path-dependent SDEs in Hilbert spaces

We study path-dependent SDEs in Hilbert spaces. By using methods based on contractions in Banach spaces, we prove existence and uniqueness of mild solutions, continuity of mild solutions with respect to perturbations of all the data of the system, G\^ateaux differentiability of generic order n of mild solutions with respect to the starting point, continuity of the G\^ateaux derivatives with respect to all the data. The analysis is performed for generic spaces of paths that do not necessarily coincide with the space of continuous functions

The end of the waterfall: Default resources of central counterparties

Central counterparties (CCPs) have become pillars of the new global financial architecture following the financial crisis of 2008. The key role of CCPs in mitigating counterparty risk and contagion has in turn cast them as systemically important financial institutions whose eventual failure may lead to potentially serious consequences for financial stability, and prompted discussions on CCP risk management standards and safeguards for recovery and resolutions of CCPs in case of failure. We contribute to the debate on CCP default resources by focusing on the incentives generated by the CCP loss allocation rules for the CCP and its members and discussing how the design of loss allocation rules may be used to align these incentives in favor of outcomes which benefit financial stability. After reviewing the ingredients of the CCP loss waterfall and various proposals for loss recovery provisions for CCPs, we examine the risk management incentives created by different ingredients in the loss waterfall and discuss possible approaches for validating the design of the waterfall. We emphasize the importance of CCP stress tests and argue that such stress tests need to account for the interconnectedness of CCPs through common members and cross-margin agreements. A key proposal is that capital charges on assets held against CCP Default Funds should depend on the quality of the risk management of the CCP, as assessed through independent stress tests

Credit default swaps and financial stability

Credit default swaps (CDSs), initially intended as instruments for hedging and managing credit risk, have been pinpointed during the recent crisis as being detrimental to financial stability. We argue that the impact of credit default swap markets on financial stability crucially depends on clearing mechanisms and capital and liquidity requirements for large protection sellers. In particular, the culprits are not so much speculative or “naked” credit default swaps but inadequate risk management and supervision of protection sellers. When protection sellers are inadequately capitalised, OTC (over-the-counter) CDS markets may act as channels for contagion and systemic risk. On the other hand, a CDS market where all major dealers participate in a central clearing facility with adequate reserves can actually contribute to mitigating systemic risk. In the latter case, a key element is the risk management of the central counterparties, for which we outline some recommendations.

Credit default swaps and systemic risk

We present a network model for investigating the impact on systemic risk of central clearing of over the counter (OTC) credit default swaps (CDS). We model contingent cash flows resulting from CDS and other OTC derivatives by a multi-layered network with a core-periphery structure, which is flexible enough to reproduce the gross and net exposures as well as the heterogeneity of market shares of participating institutions. We analyze illiquidity cascades resulting from liquidity shocks and show that the contagion of illiquidity takes place along a sub-network constituted by links identified as ’critical receivables’. A key role is played by the long intermediation chains inherent to the structure of the OTC network, which may turn into chains of critical receivables. We calibrate our model to data representing net and gross OTC exposures of large dealer banks and use this model to investigate the impact of central clearing on network stability. We find that, when interest rate swaps are cleared, central clearing of credit default swaps through a well-capitalized CCP can reduce the probability and the magnitude of a systemic illiquidity spiral by reducing the length of the chains of critical receivables within the financial network. These benefits are reduced, however, if some large intermediaries are not included as clearing members

Cubature on Wiener space in infinite dimension

We prove a stochastic Taylor expansion for SPDEs and apply this result to obtain cubature methods, i. e. high order weak approximation schemes for SPDEs, in the spirit of T. Lyons and N. Victoir. We can prove a high-order weak convergence for well-defined classes of test functions if the process starts at sufficiently regular initial values. We can also derive analogous results in the presence of L\'evy processes of finite type, here the results seem to be new even in finite dimension. Several numerical examples are added.Comment: revised version, accepted for publication in Proceedings Roy. Soc.

Ergodic transition in a simple model of the continuous double auction

We study a phenomenological model for the continuous double auction, whose aggregate order process is equivalent to two independent M/M/1 queues. The continuous double auction defines a continuous-time random walk for trade prices. The conditions for ergodicity of the auction are derived and, as a consequence, three possible regimes in the behavior of prices and logarithmic returns are observed. In the ergodic regime, prices are unstable and one can observe a heteroskedastic behavior in the logarithmic returns. On the contrary, non-ergodicity triggers stability of prices, even if two different regimes can be seen

A model-free approach to continuous-time finance

We present a non-probabilistic, pathwise approach to continuous-time finance based on causal functional calculus. We introduce a definition of self-financing, free from any integration concept and show that the value of a self-financing portfolio is a pathwise integral (every self-financing strategy is a gradient) and that generic domain of functional calculus is inherently arbitrage-free. We then consider the problem of hedging a path-dependent payoff across a generic set of scenarios. We apply the transition principle of Isaacs in differential games and obtain a verification theorem for the optimal solution, which is characterised by a fully non-linear path-dependent equation. For the Asian option, we obtain explicit solution

On the support of solutions of stochastic differential equations with path-dependent coefficients

Given a stochastic differential equation with path-dependent coefficients driven by a multidimensional Wiener process, we show that the support of the law of the solution is given by the image of the Cameron-Martin space under the flow of the solutions of a system of path-dependent (ordinary) differential equations. Our result extends the Stroock-Varadhan support theorem for diffusion processes to the case of SDEs with path-dependent coefficients. The proof is based on the Functional Ito calculus and interpolation estimates for stochastic processes in Holder norm

A Reply to Professors Cain and Charles

This Reply follows the responses of Professor Bruce Cain and Professor Guy-Uriel Charles to Professor Lessig’s essay "What an Originalist Would Understand 'Corruption' to Mean," 102 Calif. L. Rev. 1 (2014)