Eva Lütkebohmert

OPTIMALITY OF PAYOFFS IN LÉVY MODELS

In this paper, we determine the lowest cost strategy for a given payoff in Levy markets where the pricing is based on the Esscher martingale measure. In particular, we consider Levy models where prices are driven by a normal inverse Gaussian (NIG)- or a variance Gamma (VG)-process. Explicit solutions for cost-efficient strategies are derived for a variety of vanilla options, spreads, and forwards. Applications to real financial market data show that the cost savings associated with these strategies can be quite substantial. The empirical findings are supplemented by a result that relates the magnitude of these savings to the strength of the market trend. Moreover, we consider the problem of hedging efficient claims, derive explicit formulas for the deltas of efficient calls and puts and apply the results to German stock market data. Using the time-varying payoff profile of efficient options, we further develop alternative delta hedging strategies for vanilla calls and puts. We find that the latter can provide a more accurate way of replicating the final payoff compared to their classical counterparts.

Treatment of Double Default Effects within the Granularity Adjustment for Basel II

Within the Internal Ratings-Based (IRB) approach of Basel II it is assumed that idiosyncratic risk has been fully diversi?ed away. The impact of undiversi?ed idiosyncratic risk on portfolio Value-at-Risk can be quanti?ed via a granularity adjustment (GA). We provide an analytic formula for the GA in an extended single- factor CreditRisk+ setting incorporating double default e?ects. It accounts for guarantees and their e?ect of reducing credit risk in the portfolio. Our general GA very well suits for application under Pillar 2 of Basel II as the data inputs are drawn from quantities already required for the calculation of IRB capital charges.

A Continuous Time Structural Model for Insolvency, Recovery, and Rollover Risks

We propose a unified structural credit risk model incorporating insolvency, recovery and rollover risks. The firm finances itself mainly by issuing short- and long-term debt. Short-term debt can have either a discrete or a more realistic staggered tenor structure. We show that a unique threshold strategy (i.e., a bank 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 bank run barrier together with an exogenous insolvency barrier.

Rollover risk and credit risk under time-varying margin

For a firm financed by a mixture of collateralized (short-term) debt and uncollateralized (long-term) debt, we show that fluctuations in margin requirements, reflecting funding liquidity shocks, lead to increasing the firm’s default risk and credit spreads. The severity with which a firm is hit by increasing margin requirements highly depends on both its financing structure and debt maturity structure. Our results imply that an additional premium should be added when evaluating debt in order to account for rollover risks, especially for short-matured bonds. In terms of policy implications, our results strongly indicate that regulators should intervene fast to curtail margins in crisis periods and maintain a reasonably low margin level in order to effectively prevent creditors’ run on debt.

Collateralized Borrowing and Default Risk

We study how margin requirements in the collateralized borrowing affect banks’ risk exposure. In a model where a firm’s asset value and margin requirement follow correlated geometric Brownian motions, we derive analytic expressions for firm’s default probability and debt value. Our results show that variations in margin requirements, reflecting funding liquidity shocks in the short-term collateralized lending market, can lead to a significant increase in firms’ default risks, in particular for those firms heavily relying on short-term collateralized borrowing. Moreover, our results imply that reducing margin in liquidity crises can be very effective to restore market lending confidence.