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Interconnections among financial institutions create potential channels for contagion and amplification of shocks to the financial system. We estimate the extent to which interconnections increase expected losses, with minimal information about network topology, under a wide range of shock distributions. Expected losses from network effects are small without substantial heterogeneity in bank sizes and a high degree of reliance on interbank funding. They are also small unless shocks are magnified by some mechanism beyond simple spillover effects; these include bankruptcy costs, fire sales, and mark-to-market revaluations of assets. We illustrate the results with data on the European banking system.

Monte Carlo simulation is widely used to measure the credit risk in portfolios of loans, corporate bonds, and other instruments subject to possible default. The accurate measurement of credit risk is often a rare-event simulation problem because default probabilities are low for highly rated obligors and because risk management is particularly concerned with rare but significant losses resulting from a large number of defaults. This makes importance sampling (IS) potentially attractive. But the application of IS is complicated by the mechanisms used to model dependence between obligors; and capturing this dependence is essential to a portfolio view of credit risk. This paper provides an IS procedure for the widely used normal copula model of portfolio credit risk. The procedure has two parts: one applies IS conditional on a set of common factors affecting multiple obligors, the other applies IS to the factors themselves. The relative importance of the two parts of the procedure is determined by the strength of the dependence between obligors. We provide both theoretical and numerical support for the method.

Simulation has proved to be a valuable tool for estimating security prices for which simple closed form solutions do not exist. In this paper we present two direct methods, a pathwise method and a likelihood ratio method, for estimating derivatives of security prices using simulation. With the direct methods, the information from a single simulation can be used to estimate multiple derivatives along with a security's price. The main advantage of the direct methods over re-simulation is increased computational speed. Another advantage is that the direct methods give unbiased estimates of derivatives, whereas the estimates obtained by resimulation are biased. Computational results are given for both direct methods and comparisons are made to the standard method of re-simulation to estimate derivatives. The methods are illustrated for a path independent model (European options), a path dependent model (Asian options), and a model with multiple state variables (options with stochastic volatility).

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The payoff of a barrier option depends on whether or not a specified asset price, index, or rate reaches a specified level during the life of the option. Most models for pricing barrier options assume continuous monitoring of the barrier; under this assumption, the option can often be priced in closed form. Many (if not most) real contracts with barrier provisions specify discrete monitoring instants; there are essentially no formulas for pricing these options, and even numerical pricing is difficult. We show, however, that discrete barrier options can be priced with remarkable accuracy using continuous barrier formulas by applying a simple continuity correction to the barrier. The correction shifts the barrier away from the underlying by a factor of exp(βσ √ t), where β ≈ 0.5826, σ is the underlying volatility, and t is the time between monitoring instants. The correction is justified both theoretically and experimentally.

The recent financial crisis has prompted much new research on the interconnectedness of the modern financial system and the extent to which it contributes to systemic fragility. Network connections diversify firms' risk exposures, but they also create channels through which shocks can spread by contagion. We review the extensive literature on this issue, with the focus on how network structure interacts with other key variables such as leverage, size, common exposures, and short-term funding. We discuss various metrics that have been proposed for evaluating the susceptibility of the system to contagion and suggest directions for future research. (JEL D85, E44, G01, G21, G22, G23, G28)