mean reversion question (joey or bchadwik)

joey since my project is only to come up with an estimate of the mean reversion parameter, im not totally sure about the whole big picture. but this is what i know: it turns out the swaptions and caps historical information are not volatilities but rates of volatilities. the final goal is to come up with stochastic value at risk numbers for different products. one part of this is to take these historical rates and come up with 10,000 different scenarios of projections of these rates into the future. this is done using the BK model. the model is solved using trees. from these scenarios, we get PV’s,a nd then we somehow get the distributions of different products and then from that the VAR parameters are estimated. to come up with the intial scenarios, we need a measure of mean reversion. so i figured that since the BK model is used to come up with the scenarios, that I could estimate the mean reversion parameter by regressing the BK equation. or an AR(1) model which should give similar results. i cant think of any other way to estimate what the mean reversion parameter should be. any advice is greatly appreciated.

Well, I think it is not an easy problem but I would personally bag the mean reversion/short rate model altogether and just assume either forward LIBOR rates are log-normal or forward swap rates are log-normal (turns out that both can’t be true and it also turns out that there is no known way to get a short rate model compatible with either of those). You should also know that vols from the swaptions markets and vols from the caps markets are not directly comparable and certainly not mixable because they are based off different assumptions. If you are going to use both, you need to transform one into the other. But you have a different problem, I guess. So you are trying to calibrate a B-K model from caps and swaption data to come up with VaR numbers. The problem with the whole approach is that your model is going to price the inputs differently than the market does and then when you reprice them in your simulation you have (real price, current simulated price, time T simulated price) which would cause me concern. Anyway, I guess that you need to come up with some way of inverting swaption/cap prices to B-K parameters which sounds like a drag and requires some careful thought. In particular, since you can’t fit all prices you need to come up with some metric for “good fit” and I don’t think that AR least squares sense works here as it is arbitrary and doesn’t correspond to any real world risk sense that I can think of.

BTW - in any VaR situation I can think of correlation among interest rates (whatever interest rates) is more important than short rate mean reversion. Using a single factor model like B-K creates a problem with that.

thanks joey! sorry i forgot to mention the correlations are included as well in the BK model to get the scenarios. so its not a single factor model. i have swap rates, rates of swaptions, rates of caps all for different terms. and all the correlations between these factors are taken into consideration.