Calibrating probability distributions
used for risk measurement purposes to market-implied data: 2. Multi-instrument
calibration – Section Introduction
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2.1 Risk models in practice need to cater for
multiple instruments. The most common framework involves assuming that the
underlying (log) return distribution is multivariate normal. Traditionally, the
corresponding covariance matrix is derived from historical observations
although usually a parsimonious factor structure is imposed to limit the number
of terms in the covariance matrix that need to be estimated from past history,
see e.g. Kemp
(2005) and Kemp
(2009).
2.2 Calibrating a multivariate normal prior
distribution to market-implied (or other) data is typically different to the
univariate case because we will usually have fewer calibration points than we
have degrees of freedom in relation to the number of terms in the now
multi-dimensional covariance matrix. However, some of the principles noted in
the univariate case still carry through to the multivariate case. In
particular, if we are calibrating a multivariate normal prior distribution
merely to market implied volatilities and covariances for a given fixed period
(i.e. merely to second moments) then the resulting calibrated distribution will
still be multivariate normal distribution, just with a different covariance
matrix.
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