Abstract
The impact of a stress scenario of default events on the loss distribution of a credit portfolio can be assessed by determining the loss distribution conditional on these events. While it is conceptually easy to estimate loss distributions conditional on default events by means of Monte Carlo simulation, it becomes impractical for two or more simultaneous defaults as then the conditioning event is extremely rare. We provide an analytical approach to the calculation of the conditional loss distribution for the CreditRisk portfolio model with independent random loss given default distributions. The analytical solution for this case can be used to check the accuracy of an approximation to the conditional loss distribution whereby the unconditional model is run with stressed input probabilities of default (PDs). It turns out that this approximation is unbiased. Numerical examples, however, suggest that the approximation may be seriously inaccurate but that the inaccuracy leads to overestimation of tail losses and hence the approach errs on the conservative side.
Highlights
The successes of the VIX [1] and the SKEW [2] indices of the Chicago Board of Options Exchange show that investors are interested in the information on the distribution of future returns carried by option prices
When the horizon goes to zero, it follows that f converges to g, because both of them approach the state price density. This condition, equivalent to requiring that the rate of return exists in each state when the time horizon approaches zero, ensures that option prices provide a valid approximation of Value at Risk (VaR) and Conditional Value at Risk (CVaR), computed at short horizons under the real world measure
The derivative of a put option price with respect to its strike price allows for the immediate computation of VaR and CVaR under the pricing measure
Summary
The successes of the VIX [1] and the SKEW [2] indices of the Chicago Board of Options Exchange show that investors are interested in the information on the distribution of future returns carried by option prices. Expected volatility and skewness under the objective measure may differ from the two indices, the ease of information available in the option market motivates the success of VIX and SKEW. Option markets provide forward-looking information about the distribution of future asset return that cannot be gleaned using just historical prices. The impossibility of making reliable statistical comparisons about CVaR estimates from portfolio returns motivates its direct computation from option prices. Gneiting [3] argues that the lack of elicitability of CVaR is a likely explanation for the VaR receiving more attention in the empirical literature. We discuss applications and the implementation of our proposed estimators
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