April 25th, 2003, 4:44 pm
It depends on what you mean by "several" and how many risk numbers you want to look at.With one option, you're willing to look at two numbers. With n options you could look at 2*n deltas and gammas, plus n*(n-1)/2 cross deltas or n*(n-1) cross-deltas and cross-gammas.If you want to reduce the number of numbers, you can start with market factors rather than option prices. For example, if all the options are on one underlying, the spot price of the underlying is one market factor that all options have a delta exposure to. You could use either a single vega and get back to two total numbers, or a vega surface with values at different strikes and expiries. For the latter you could use a few or many parameters. If you have more than one underlying, you can treat each one separately, or model each as a function of market factors (for example, each stock return could be Beta*Index_Return+Idiosyncratic_Risk, with all the idiosyncratic risk independent).If you want to collapse everything to a single number, such as VaR, you need to estimate volatilties and correlations for all your market factors (this assumes you're willing to use a multivariate Normal assumption, if not you may need more parameters). This is a good reason to keep the number of market factors small and easy to measure.