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Calibrating probability distributions used for risk measurement purposes to market-implied data: 2. Multi-instrument calibration – Section Analysis

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2.3          A multivariate normal distribution  with mean  (a vector of random variables) and covariance matrix  has a probability density function,  as follows, where  is the number of entries in the vector  and  the number of entries in ):

2.4          We note that:

(a)    Any -dimensional multivariate normal distribution has a probability density function expressible as  where  is some suitable constant and  is a positive definite symmetric quadratic form (with possibly non-zero drift) in  different variables, and vice versa.

(b)   Applying analytical weighted Monte Carlo (using relative entropy) to the sort of calibration problem referred to above will therefore return (unless the calibration problem is ill-posed) a calibrated probability distribution which also has multivariate normal form. This is because the problem can be restated using Lagrange multipliers to one that involves minimising  defined as follows, where the  refer to whatever calibrations there are on the means and  to those on covariance terms (in general there will be fewer than  of the ):

The solution to this minimisation problem is given by the following:

subject to  (i.e. that  is a probability distribution) and other constraints derived directly from calibration requirements, e.g. that  etc.

Thus if  is expressible as  as above, then  will be too, just for a different .

(c)    Applying the principle of no arbitrage we may therefore expect  to have zero mean (more precisely for each element of  to be the same, which without loss of generality we may take as zero if we are focusing on relative returns) and therefore to have the form:

where the  are symmetric zero-drift quadratic forms (, say) corresponding to each of the implied volatilities/implied correlations to which we wish to calibrate.

(d)   The calibrated distribution will therefore be multivariate normal with zero mean and probability distribution as follows, for suitably chosen  that reproduce for the calibrations the relevant market implied variances or covariances (where  is some constant the value of which ensures that ):

(e)   Thus the calibrated probability distribution will be characterised by a covariance matrix   as follows: