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Register a new userOn the Basel Liquidity Formula for Elliptical Distributions
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A justification of the Basel liquidity formula for risk capital in the trading book is given under the assumption that market risk-factor changes form a Gaussian white noise process over 10-day time steps and changes to P&L are linear in the risk-factor changes. A generalization of the formula is derived under the more general assumption that risk-factor changes are multivariate elliptical. It is shown that the Basel formula tends to be conservative when the elliptical distributions are from the heavier-tailed generalized hyperbolic family. As a by-product of the analysis a Fourier approach to calculating expected shortfall for general symmetric loss distributions is developed.
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Expanding on techniques of concentration of measure, we develop a quantitative framework for modeling liquidity risk using convex risk measures. The fundamental objects of study are curves of the form $(rho(lambda X))_{lambda ge 0}$, where $rho$ is a convex risk measure and $X$ a random variable, and we call such a curve a emph{liquidity risk profile}. The shape of a liquidity risk profile is intimately linked with the tail behavior of the underlying $X$ for some notable classes of risk measures, namely shortfall risk measures. We exploit this link to systematically bound liquidity risk profiles from above by other real functions $gamma$, deriving tractable necessary and sufficient conditions for emph{concentration inequalities} of the form $rho(lambda X) le gamma(lambda)$, for all $lambda ge 0$. These concentration inequalities admit useful dual representations related to transport inequalities, and this leads to efficient uniform bounds for liquidity risk profiles for large classes of $X$. On the other hand, some modest new mathematical results emerge from this analysis, including a new characterization of some classical transport-entropy inequalities. Lastly, the analysis is deepened by means of a surprising connection between time consistency properties of law invariant risk measures and the tensorization of concentration inequalities.
We propose a unified structural credit risk model incorporating both insolvency and illiquidity risks, in order to investigate how a firms default probability depends on the liquidity risk associated with its financing structure. We assume the firm finances its risky assets by mainly issuing short- and long-term debt. Short-term debt can have either a discrete or a more realistic staggered tenor structure. At rollover dates of short-term debt, creditors face a dynamic coordination problem. We show that a unique threshold strategy (i.e., a debt run barrier) exists for short-term creditors to decide when to withdraw their funding, and this strategy is closely related to the solution of a non-standard optimal stopping time problem with control constraints. We decompose the total credit risk into an insolvency component and an illiquidity component based on such an endogenous debt run barrier together with an exogenous insolvency barrier.
We propose a Monte Carlo simulation method to generate stress tests by VaR scenarios under Solvency II for dependent risks on the basis of observed data. This is of particular interest for the construction of Internal Models and requirements on evaluation processes formulated in the Commission Delegated Regulation. The approach is based on former work on partition-ofunity copulas, however with a direct scenario estimation of the joint density by product beta distributions after a suitable transformation of the original data.
Latency (i.e., time delay) in electronic markets affects the efficacy of liquidity taking strategies. During the time liquidity takers process information and send marketable limit orders (MLOs) to the exchange, the limit order book (LOB) might undergo updates, so there is no guarantee that MLOs are filled. We develop a latency-optimal trading strategy that improves the marksmanship of liquidity takers. The interaction between the LOB and MLOs is modelled as a marked point process. Each MLO specifies a price limit so the order can receive worse prices and quantities than those the liquidity taker targets if the updates in the LOB are against the interest of the trader. In our model, the liquidity taker balances the tradeoff between missing trades and the costs of walking the book. We employ techniques of variational analysis to obtain the optimal price limit of each MLO the agent sends. The price limit of a MLO is characterized as the solution to a new class of forward-backward stochastic differential equations (FBSDEs) driven by random measures. We prove the existence and uniqueness of the solution to the FBSDE and numerically solve it to illustrate the performance of the latency-optimal strategies.
We study the pricing and hedging of European spread options on correlated assets when, in contrast to the standard framework and consistent with imperfect liquidity markets, the trading in the stock market has a direct impact on stocks prices. We consider a partial-impact and a full-impact model in which the price impact is caused by every trading strategy in the market. The generalized Black-Scholes pricing partial differential equations (PDEs) are obtained and analysed. We perform a numerical analysis to exhibit the illiquidity effect on the replication strategy of the European spread option. Compared to the Black-Scholes model or a partial impact model, the trader in the full impact model buys more stock to replicate the option, and this leads to a higher option price.
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