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kelang
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Posts: 46
Joined: November 14th, 2011, 4:53 pm

question about shifted-BGM...

May 22nd, 2014, 6:06 pm

Hi all,May I ask a question on the calibration of a shifted BGM model? To keep it simple, the model is assumed to be the following (which is equivalent to Dr. Piterbarg's when ignoring the stoc-vol component)dF(t)=(...)dt+vol*(F+shift)dW, where F is the forward rate, and the volatility is decomposed as vol(t)=h(t)*g(T-t)The dW is a multidimenstional independent Brownian motion [dW1, dW2, ..., dWn], and vol is thus [vol1, vol2, ..., voln]So the question is, how "bad" if we allow some component of [h(t)*h(t)] (e.g. h2(t)*h2(t)) to be negative while I still make sure the forward rate variance vol*vol to be positive through the time even if it becomes slightly negative in some circumstances. Or in practice, is it really normal to let the model parameter h(t)*h(t) be negative for some of its components?The reason I am asking is, when I tried to calibrate a 3F factor shifted-BGM to 3 columns of the swaption matrix (USD 2Y, 10Y, and 30Y) under the condition of shift~1/dt (i.e. reducing to a Gaussian HJM model), some components of h(t)*h(t) turn out to be negative...Thank you
Last edited by kelang on May 22nd, 2014, 10:00 pm, edited 1 time in total.
 
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BenjG
Posts: 28
Joined: November 25th, 2013, 3:17 pm

question about shifted-BGM...

May 24th, 2014, 8:25 am

Why do you need the volatility to be a function of time ? If you consider a shifted BGM, then the forward follows a shifted LN diffusion where vol is a constant (possibly a term structure).I don't see why you need this functional vol(t)=h(t)*g(T-t) ?Ben
 
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BenjG
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Joined: November 25th, 2013, 3:17 pm

question about shifted-BGM...

May 24th, 2014, 3:30 pm

Forget my stupid question... You are just using an homogeneous form of volatility: vol(t,T) = f(T-t) like an integrated Hull and White volatility I suppose (to get a non constant forward volatility).About your problem, having a negative volatility is not incorrect. However, a negative variance doesn't seem right to me !
Last edited by BenjG on May 23rd, 2014, 10:00 pm, edited 1 time in total.
 
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kelang
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Posts: 46
Joined: November 14th, 2011, 4:53 pm

question about shifted-BGM...

May 24th, 2014, 3:35 pm

QuoteOriginally posted by: BenjGWhy do you need the volatility to be a function of time ? If you consider a shifted BGM, then the forward follows a shifted LN diffusion where vol is a constant (possibly a term structure).I don't see why you need this functional vol(t)=h(t)*g(T-t) ?Benthanks, apparently, the first term h(t) is the common time-dependent vol component (typically piecewise constant), and the second term is the time-homogenous vol component (typically a parametric function, e.g. abcd or other forms).
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