June 7th, 2004, 2:17 pm
QuoteOriginally posted by: Noniusnot sure I understand. local sigmas are the infinitesimal specifications. global are the cumulative specifications. for vol, it should be clear for caps (and to some extent swaptions). for alpha, which, I assume you mean mean reversion rate, it isn't obvious at all. Can you provide more clarity to your question?(I've been away for a week, but thank you for the answer.)It's not important with different reversion rates, since the reversion rate normally is set to the same for the entire volatility surface.So what I really wonder is wow to calculate local (piecewise constant) Hull-White volatilities. Maybe this question should have been in the Student Forum, because I think my it's probably very "basic".For Black-Scholes, the "total variance" is vola^2*T, i.e. if you have a (global) volatility vol1 for expiration T1 and a (global) volatility vol2 for T2 (where T2 > T1) the local (constant) volatility V12 between T1 and T2 can be calculated simply by: V1^2 * T1 + V12^2 * (T2-T1) = V2^2 * T2And what I wonder is how to calculate the local (constant) sigma between T1 and T2 in a Hull-White universe, provided that sigma1 and sigma2 and alpa (reversion rate) are known.This is probably "standard knowledge", but I haven't been able to find it anywhere. In "Options, Futures & Other Derivatives" (ed 5) it's said, though, that the instantaneous standard deviation of the T-maturity instantaneous forward rate is: sigma * exp(-alpha * T)And integrating this squared from 0 to T (which I assume is what you'd to get the variance of the short rate state variable at time T) you get sigma^2 * [1 - exp(-2*alpha*T)]/[2*alpha]and if this is correct, the local (constant) volatility sigma12 between T1 and T2 is calculated by using the relationship: sigma1^2 * [1 - exp(-2*alpha*T1)]/[2*alpha] + sigma12^2 * [1 - exp(-2*alpha*(T2-T1))]/[2*alpha] = sigma2^2 * [1 - exp(-2*alpha*T2)]/[2*alpha]However, I'm not sure this is correct, since I haven't seen it written down explicitly somewhere, and I'm a kind of a newbie to Hull-White.(The Hull-White solution I'm looking at at the moment uses a completely different approach, where the sigma is constant from 0 to T, but is calibrated in a least-square sense to the different (implied) volatilities, so I'm trying to figure out if a solution with piecewise constant local volatilities wolud be faster and/or more reliable.The only reason I involved alpha in my original question is that i figured there might be a solutuion/relationship where not only local sigma but also local alpha is piecewise constant, but never mind about that...)Regards,/Samuel, Stockholm