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Working Paper
Computationally convenient distributional assumptions for common value auctions
Although the mathematical foundations of common value auctions have been well understood since Milgrom & Weber (1982), equilibrium bidding strategies are computationally complex. Very few calculated examples can be found in the literature, and only for highly specialized cases. This paper introduces two sets of distributional assumptions that are flexible enough for theoretical and empirical applications and yet permit straightforward calculation of equilibrium bidding strategies.
Working Paper
Expectations of functions of stochastic time with application to credit risk modeling
We develop two novel approaches to solving for the Laplace transform of a time-changed stochastic process. We discard the standard assumption that the background process (Xt) is Levy. Maintaining the assumption that the business clock (Tt) and the background process are independent, we develop two different series solutions for the Laplace transform of the time-changed process X-tildet=X(Tt). In fact, our methods apply not only to Laplace transforms, but more generically to expectations of smooth functions of random time. We apply the methods to introduce stochastic time change to the ...
Working Paper
A generalization of generalized beta distributions
This paper introduces the ``compound confluent hypergeometric'' (CCH) distribution. The CCH unifies and generalizes three recently introduced generalizations of the beta distribution: the Gauss hypergeometric (GH) distribution of Armero and Bayarri (1994), the generalized beta (GB) distribution of McDonald and Xu (1995), and the confluent hypergeometric (CH) distribution of Gordy (forthcoming). Unlike the beta, GB and GH, the CCH allows for conditioning on explanatory variables in a natural and convenient way. The CCH family is conjugate for gamma distributed signals, and so may also prove ...
Working Paper
Bayesian Estimation of Time-Changed Default Intensity Models
We estimate a reduced-form model of credit risk that incorporates stochastic volatility in default intensity via stochastic time-change. Our Bayesian MCMC estimation method overcomes nonlinearity in the measurement equation and state-dependent volatility in the state equation. We implement on firm-level time-series of CDS spreads, and find strong in-sample evidence of stochastic volatility in this market. Relative to the widely-used CIR model for the default intensity, we find that stochastic time-change offers modest benefit in fitting the cross-section of CDS spreads at each point in time, ...
Working Paper
Granularity adjustment for mark-to-market credit risk models
The impact of undiversified idiosyncratic risk on value-at-risk and expected shortfall can be approximated analytically via a methodology known as granularity adjustment (GA). In principle, the GA methodology can be applied to any risk-factor model of portfolio risk. Thus far, however, analytical results have been derived only for simple models of actuarial loss, i.e., credit loss due to default. We demonstrate that the GA is entirely tractable for single-factor versions of a large class of models that includes all the commonly used mark-to-market approaches. Our approach covers both finite ...
Working Paper
Switching costs and adverse selection in the market for credit cards: new evidence
To explain persistence of credit card interest rates at relatively high levels, Calem and Mester (AER, 1995) argued that informational barriers create switching costs for high-balance customers. As evidence, using data from the 1989 Survey of Consumer Finances, they showed that these households were more likely to be rejected when applying for new credit. In this paper, they revisit the question using the 1998 and 2001 SCF. Further, they use new information on card interest rates to test for pricing effects consistent with information-based switching costs. The authors find that informational ...
Working Paper
On the distribution of a discrete sample path of a square-root diffusion
We derive the multivariate moment generating function (mgf) for the stationary distribution of a discrete sample path of n observations of a square-root diffusion (CIR) process, X(t). The form of the mgf establishes that the stationary joint distribution of (X(t(1)),...,X(t(n))) for any fixed vector of observation times (t(1),...,t(n)) is a Krishnamoorthy-Parthasarathy multivariate gamma distribution. As a corollary, we obtain the mgf for the increment X(t+dt)-X(t), and show that the increment is equivalent in distribution to a scaled difference of two independent draws from a gamma ...
Working Paper
Constant proportion debt obligations: a post-mortem analysis of rating models
In its complexity and its vulnerability to market volatility, the CPDO might be viewed as the poster child for the excesses of financial engineering in the credit market. This paper examines the CPDO as a case study in model risk in the rating of complex structured products. We demonstrate that the models used by S&P and Moody's would have assigned very low probability to the spread levels realized in the investment grade corporate credit default swap market in late 2007, even though these spread levels were comparable to those of 2002. The spread levels realized in the first quarter of 2008 ...
Working Paper
Nested simulation in portfolio risk measurement
Risk measurement for derivative portfolios almost invariably calls for nested simulation. In the outer step one draws realizations of all risk factors up to the horizon, and in the inner step one re-prices each instrument in the portfolio at the horizon conditional on the drawn risk factors. Practitioners may perceive the computational burden of such nested schemes to be unacceptable, and adopt a variety of second-best pricing techniques to avoid the inner simulation. In this paper, we question whether such short cuts are necessary. We show that a relatively small number of trials in the ...
Working Paper
Spectral Backtests of Forecast Distributions with Application to Risk Management
We study a class of backtests for forecast distributions in which the test statistic is a spectral transformation that weights exceedance events by a function of the modeled probability level. The choice of the kernel function makes explicit the user's priorities for model performance. The class of spectral backtests includes tests of unconditional coverage and tests of conditional coverage. We show how the class embeds a wide variety of backtests in the existing literature, and propose novel variants as well. In an empirical application, we backtest forecast distributions for the overnight ...