Читать книгу Perturbation Methods in Credit Derivatives - Colin Turfus - Страница 16

Market risk management are in the throes of a comprehensive re‐working of their risk framework to address the new Basel III regulations. Counterparty risk calculations are causing something of a headache. Previously interest rate and credit curves were evolved by identifying principal components, allowing each of these components to evolve along a Monte Carlo path, then reconstructing the implied shape of the curves at each exposure time of interest. The pricing engine can then be used taking the evolved curves along with the evolved values of other required spot variables (equity, FX, inflation, etc.) as input to obtain conditional prices for each relevant portfolio for each path at each exposure time. The distribution of positive exposure values can then be considered and expected positive exposures (EPE) and or potential future exposures (PFE) calculated by considering the tail of this distribution. Unfortunately, auditors are unhappy that a somewhat ad hoc method used for evolving the principal components hitherto is inadequately justified and would like to see something which is more industry-standard implemented in preference.

An option being considered is to try and evolve the curves in their entirety rather than just principal components. Standard models such as HJM‐based or Black–Karasinski present themselves as candidates. But to be useful, there has to be a convenient mechanism for constructing the entire forward curve at each exposure time of interest, for use as input to the pricing model. The Black–Karasinski lognormal model is preferred to the simpler normal Hull–White alternative on the basis that, as the curve evolves upward or downward, lognormal volatilities rise or fall in proportion, which is intuitively sensible: if a Hull–White model is used instead, consideration would have to be given also to how the volatilities evolved as the associated curve moved up or down. Further, for the curve evolution to work in a credit curve context it must ensure positive values of all forward rates, which Black–Karasinski does, HJM‐based models can, but Hull–White does not.

Encouragement is taken from the availability of the highly accurate analytic conditional bond formulae for the Black–Karasinski model set out in Chapter 5. But it is felt that the simple one‐factor model does not do justice to the range of possible evolutions of the shape, not just the level, of interest rate and credit curves. Specifically, there should be fluctuations which impact mainly at the short end of the curve which decay relatively quickly (rapid mean reversion), whereas fluctuations affecting the long end are likely to be longer‐lived (slow mean reversion). So the multi‐factor Black–Karasinski model framework derived in §6.4 and expounded in greater detail in §15.5 is of interest. Work is initiated to implement the forward rate formula (15.49) to allow forward interest rate and credit curves to be generated from simple evolved Brownian variables, mutually correlated as necessary.