Serving the Quantitative Finance Community

 
User avatar
HOOK
Topic Author
Posts: 0
Joined: October 10th, 2008, 5:15 pm

Smile correction for dual/double digital

January 29th, 2014, 3:30 pm

Dear all,Consider a binary option that pays $1 if at maturity asset A is above a level Xa and asset B is above a level Xb.To incorporate smile effects, in the univarite case we need to add -vega*d(vol)/d(strile). Does anybody know the smile correction term for this derivative on 2 assets?Cheers
 
User avatar
Alan
Posts: 2958
Joined: December 19th, 2001, 4:01 am
Location: California
Contact:

Smile correction for dual/double digital

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.
 
User avatar
HOOK
Topic Author
Posts: 0
Joined: October 10th, 2008, 5:15 pm

Smile correction for dual/double digital

February 7th, 2014, 6:31 pm

NICE!I was woking on the derivatives of the bivariate lognormal with respect to each strike, but there was one missing term, I knew that.It was easily verifiable when rho=1 and rho=0 (situations that we know the result) that one term was missing.I did not consider the correlation might also change! That might be "missing link".Thanks Alan,Have a nice weekend,Hook