January 30th, 2014, 2:48 pm
Since no answer, I'll give it a shot ...In general, the presumption here is that the option value can be expressedin terms of the GBM value with 'implied parameters'. That's demonstrablyfalse in general, but accepting it for the purpose of your question, let[$]V_{gbm}(K_1,K_2,\sigma_1,\sigma_2,\rho)[$] be the GBM value for some payoff thatdepends on asset1 reaching [$]K_1[$] and asset2 reaching [$]K_2[$].Then, you could attempt to fit market values to the functional form[$]V_{mkt}(K_1,K_2) = V_{gbm} \left( K_1,K_2,\sigma_1(K_1),\sigma_2(K_2),\rho(K_1,K_2) \right)[$]In other words, there are implied vols and an implied correlation.So, [$]\frac{\partial}{\partial K_1}[$] applied to the lhs can also be applied to the rhs with the chainrule and similarly for [$]K_2[$]. That means, in addition to the 'univariate smile correction term' youhad in your post, there are `implied correlation correction terms' with [$]\displaystyle{\frac{\partial V_{gbm}}{\partial \rho} \frac{\partial \rho}{\partial K_i}}, \quad (i=1,2)[$]. Corrections to the two deltas would be similar via scaling relations.
Last edited by
Alan on January 29th, 2014, 11:00 pm, edited 1 time in total.