Chapter 2: FAQs 45
In CrashMetrics the risk in this portfolio is measured as
the worst case over some range of equity moves:
worst-case loss= min
−δS−≤δS≤δS+F(δS).This is the number that would be quoted as the possible
downside during a dramatic move in the market.
This downside can be reduced by adding derivatives
to the portfolio in an optimal fashion. This is called
Platinum Hedging. For example, if you want to use
some out-of-the-money puts to make this worst case not
so bad then you could optimize by choosingλso that
theworstcaseof
F(δS)+λF∗(δS)−|λ|Crepresents an acceptable level of downside risk. Here
F∗(·) is the ‘formula’ for the change in value of the
hedging contract,Cis the ‘cost’ associated with each
hedging contract andλis the quantity of the contract
which is to be determined. In practice there would be
many such hedging contracts, not necessarily just an
out-of-the-money put, so you would sum over all of
them and then optimize.
CrashMetrics deals with any number of underlyings
by exploiting the high degree of correlation between
equities during extreme markets. We can relate the
return on theith stock to the return on a representative
index,x, during a crash by
δSi
Si=κix,whereκiis a constantcrash coefficient. For example,
if the kappa for stock XYZ is 1.2 it means that when