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slslsl
Topic Author
Posts: 19
Joined: July 12th, 2009, 5:42 pm

### Choice of LMM skew function

In Piterbarg's 2003 paper: http://www.javaquant.net/papers/piterba ... hastic.pdf, equation (4.2) gives the volatility of each LIBOR at time t as being proportional to a weighted function of its present and its starting value L(0). I've seen this form used in a few other places also.Does anyone know why this form is used? It seems strange to me to have the Libor's behavior dependent on L(0) even at times long in the future. Why would you ever not just replace L(0) with a generic (possibly LIBOR-dependent) constant c, to end up with a typical shifted-LMM type of model?

pcaspers
Posts: 695
Joined: June 6th, 2005, 9:49 am

### Choice of LMM skew function

You want to multiply $\sigma(t)$ with a mixture of a constant and $L(t)$ to produce something between a flat smile and a skew. Since the scale does not matter, $\alpha L(0) + (1-\alpha) L(t)$ is a clever choice because this factor is always in the order of magnitude of the Libor rate itself and $\sigma(t)$ stays therefore always at the same level too, which is good for calibration, since $\alpha$ and $\sigma$ are not interacting, but are "orthogonal". You can also take $L(t)+d$ for example, but then $\sigma$ varies between a normal volatility if $d\approx 0$, i.e. something like $0.0050$ for example and a lognormal volaility if $d\approx 1$, so something like $0.30$. That's not nice for calibration obviously.