Correlation Calibration for a call on spread CMS
Posted: June 15th, 2005, 11:48 am
Hi, I must quote at T(0) a call on a spread CMS, MAX ( CMS10Y(T(i)) - CMS2Y(T(i)) -K, 0) where K is the strike and the fixing date is T(i), payment at T(i+1).I use two methods :1// A closed form approximation , generalised Black Scholes, and my inputs are CMS10y(0,T(i)), CMS2y(0,T(i)), the instantanneous correlation between CMS10y(t,T(i)), CMS2y(t,T(i)), supposed constant " corr_inst", two swaption volatilities vol(T(i),10Y) and vol(T(i),2y).I used convexity to have CMS10y(0,T(i)), CMS2y(0,T(i)) (Hagan formula or Pessler ).To be marked to market , i calibrate my input "corr_inst" from brokers prices.It seems like there are no skew of correlation . When i asked to brokers the same product for others fixings, and others payments, i approximatively get the same correlation : It' a trouble for me.2// A LMM, along a path, i generate the CMS with the forward libor, and i price the call on a spread CMS with Montecarlo.How can I calibrate my instantaneous correlation of forward libor to obtain the same prices???? my instantaneous correlation in my LMM is given by Rebonato approx : cor(i,j) = longcorr+ (1-longcorr)*exp(-beta[T(i)-T(j)])I need to adjust "longcorr" and "beta" to retrieve the market price.As i don't have any closed form, how can i calibrated my "longcorr" and my "beta" Without Recusively run a too long Monte-carlo for price and optimized the "beta" and "Longcorr"??Thank's in advance for yours answers,Regards,Kipi.