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stampeding
Topic Author
Posts: 164
Joined: July 14th, 2002, 3:00 am

### A simple Hull-White calibration question

(This might be a typical "newbie/FAQ/whatever" question...)Is there a way to calculate "local" sigmas and alphas from "global" sigmas and alphas for Hull-White (lattice methods) that is as simple as calculating the local volatilities from global volatilities in the Black-Scholes universe?For a Black-Scholes process, if we know global volatilities v_Global[j] for T[j], i: 1 -> n , then the (piecewise constant) local volatilities are trivially calculated, because sincev_Global[j]^2*T[j] = v_Local[j]^2*(T[j]-T[j-1]) + v_Global[j-1]^2*T[j-1]we can thus very easily calculate the (piecewise constant) local volatilities asv_Local[1] = v_Global[1]v_Local[j] = sqrt( (v_Global[j]^2*T[j] - v_Global[j-1]^2*T[j-1]) / (T[j]-T[j-1]) )Now, is there an equally simple way for Hull-White, to calculate local sigmas and alphas?I.e. if we know sigma_Global[j] and alpha_Global[j] for i: 1 -> n , is there some simple way to (e.g. iterativerly, as for the B&S volas above) calculate sigma_Local[j] and alpha_Local[j] for i: 1 -> n ?Regards,/Samuel, Stockholm

Nonius
Posts: 5115
Joined: January 22nd, 2003, 6:48 am

### A simple Hull-White calibration question

QuoteOriginally posted by: stampeding(This might be a typical "newbie/FAQ/whatever" question...)Is there a way to calculate "local" sigmas and alphas from "global" sigmas and alphas for Hull-White (lattice methods) that is as simple as calculating the local volatilities from global volatilities in the Black-Scholes universe?For a Black-Scholes process, if we know global volatilities v_Global[j] for T[j], i: 1 -> n , then the (piecewise constant) local volatilities are trivially calculated, because sincev_Global[j]^2*T[j] = v_Local[j]^2*(T[j]-T[j-1]) + v_Global[j-1]^2*T[j-1]we can thus very easily calculate the (piecewise constant) local volatilities asv_Local[1] = v_Global[1]v_Local[j] = sqrt( (v_Global[j]^2*T[j] - v_Global[j-1]^2*T[j-1]) / (T[j]-T[j-1]) )Now, is there an equally simple way for Hull-White, to calculate local sigmas and alphas?I.e. if we know sigma_Global[j] and alpha_Global[j] for i: 1 -> n , is there some simple way to (e.g. iterativerly, as for the B&S volas above) calculate sigma_Local[j] and alpha_Local[j] for i: 1 -> n ?Regards,/Samuel, Stockholmnot 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?

stampeding
Topic Author
Posts: 164
Joined: July 14th, 2002, 3:00 am

### A simple Hull-White calibration question

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

kv
Posts: 11
Joined: May 19th, 2004, 1:28 pm

### A simple Hull-White calibration question

I'm not sure to have understood everything you wrote as you didn't know Latex but maybe this could help you :http://www.wilmott.com/messageview.cfm? ... adid=17499