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

Modeling interest rate dynamics: an infinite-dimensional approach

We present a family of models for the term structure of interest rates which describe the interest rate curve as a stochastic process in a Hilbert space. We start by decomposing the deformations of the term structure into the variations of the short rate, the long rate and the fluctuations of the curve around its average shape. This fluctuation is then described as a solution of a stochastic evolution equation in an infinite dimensional space. In the case where deformations are local in maturity, this equation reduces to a stochastic PDE, of which we give the simplest example. We discuss the properties of the solutions and show that they capture in a parsimonious manner the essential features of yield curve dynamics: imperfect correlation between maturities, mean reversion of interest rates and the structure of principal components of term structure deformations. Finally, we discuss calibration issues and show that the model parameters have a natural interpretation in terms of empirically observed quantities.Comment: Keywords: interest rates, stochastic PDE, term structure models, stochastic processes in Hilbert space. Other related works may be retrieved on http://www.eleves.ens.fr:8080/home/cont/papers.htm

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.

Convergent multiplicative processes repelled from zero: power laws and truncated power laws

Random multiplicative processes $w_t =\lambda_1 \lambda_2 ... \lambda_t$ (with 0 ) lead, in the presence of a boundary constraint, to a distribution $P(w_t)$ in the form of a power law $w_t^{-(1+\mu)}$. We provide a simple and physically intuitive derivation of this result based on a random walk analogy and show the following: 1) the result applies to the asymptotic ($t \to \infty$) distribution of $w_t$ and should be distinguished from the central limit theorem which is a statement on the asymptotic distribution of the reduced variable ${1 \over \sqrt{t}}(log w_t -)$; 2) the necessary and sufficient conditions for $P(w_t)$ to be a power law are that < 0 (corresponding to a drift $w_t \to 0$) and that $w_t$ not be allowed to become too small. We discuss several models, previously unrelated, showing the common underlying mechanism for the generation of power laws by multiplicative processes: the variable $\log w_t$ undergoes a random walk biased to the left but is bounded by a repulsive ''force''. We give an approximate treatment, which becomes exact for narrow or log-normal distributions of $\lambda$, in terms of the Fokker-Planck equation. 3) For all these models, the exponent $\mu$ is shown exactly to be the solution of $\langle \lambda^{\mu} \rangle = 1$ and is therefore non-universal and depends on the distribution of $\lambda$.Comment: 19 pages, Latex, 4 figures available on request from [email protected]

Price dynamics in a Markovian limit order market

We propose and study a simple stochastic model for the dynamics of a limit order book, in which arrivals of market order, limit orders and order cancellations are described in terms of a Markovian queueing system. Through its analytical tractability, the model allows to obtain analytical expressions for various quantities of interest such as the distribution of the duration between price changes, the distribution and autocorrelation of price changes, and the probability of an upward move in the price, {\it conditional} on the state of the order book. We study the diffusion limit of the price process and express the volatility of price changes in terms of parameters describing the arrival rates of buy and sell orders and cancelations. These analytical results provide some insight into the relation between order flow and price dynamics in order-driven markets.Comment: 18 pages, 5 figure