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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